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"hardy-course-of-pure-mathematics-1921/eq-1050dbc1f4", "hardy-course-of-pure-mathematics-1921/eq-0d08286ed7", "hardy-course-of-pure-mathematics-1921/eq-e51d8f7636", "hardy-course-of-pure-mathematics-1921/eq-db2303e548", "hardy-course-of-pure-mathematics-1921/eq-113e30761d", "hardy-course-of-pure-mathematics-1921/eq-37ca9f0534", "hardy-course-of-pure-mathematics-1921/eq-84976b1937", "hardy-course-of-pure-mathematics-1921/eq-a6f558ff3d", "hardy-course-of-pure-mathematics-1921/eq-271eb4c1ae", "hardy-course-of-pure-mathematics-1921/eq-2f45892ce9", "hardy-course-of-pure-mathematics-1921/eq-82071283d8", "hardy-course-of-pure-mathematics-1921/eq-5474628d33", "hardy-course-of-pure-mathematics-1921/eq-df5488c548", "hardy-course-of-pure-mathematics-1921/eq-eba7197e71", "hardy-course-of-pure-mathematics-1921/eq-acd8c8e842", "hardy-course-of-pure-mathematics-1921/eq-6211e961ac", "hardy-course-of-pure-mathematics-1921/eq-2470c718f3", "hardy-course-of-pure-mathematics-1921/eq-ac6c1fd12c", "hardy-course-of-pure-mathematics-1921/eq-baf19c7993", "hardy-course-of-pure-mathematics-1921/eq-7c307f868c" ], "exercise_sets": [ "hardy-course-of-pure-mathematics-1921/ex-x", "hardy-course-of-pure-mathematics-1921/ex-xi", "hardy-course-of-pure-mathematics-1921/ex-xii", "hardy-course-of-pure-mathematics-1921/ex-xiii", "hardy-course-of-pure-mathematics-1921/ex-misc-ii", "hardy-course-of-pure-mathematics-1921/ex-xiv", "hardy-course-of-pure-mathematics-1921/ex-xvi", "hardy-course-of-pure-mathematics-1921/ex-xvii", "hardy-course-of-pure-mathematics-1921/ex-xviii", "hardy-course-of-pure-mathematics-1921/ex-xv", "hardy-course-of-pure-mathematics-1921/ex-xix" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ch-iii", "number": "III", "title": "COMPLEX NUMBERS", "name": "Hardy 1921, ch. III: COMPLEX NUMBERS", "pages": [ "84", "98" ], "concepts": [ "concept/amplitude-of-a-complex-number", "concept/argand-diagram", "concept/bilinear-transformation", "concept/cartesian-coordinates", "concept/centre-of-gravity", "concept/closure", "concept/coaxal-circles", "concept/collinear-points", "concept/complex-number", "concept/complex-variable", "concept/conjugate-complex-numbers", "concept/cross-ratio", "concept/cube-root-of-unity", "concept/displacement", "concept/equal-roots", "concept/equilateral-triangle", "concept/fixed-point-of-a-transformation", "concept/harmonic-points", "concept/imaginary-root", "concept/imaginary-straight-line", "concept/imaginary-unit", "concept/limiting-point", "concept/linear-transformation", "concept/magnification", "concept/modulus-of-a-complex-number", "concept/parallelogram", "concept/polar-coordinates", "concept/polynomial", "concept/principal-value-of-a-root", "concept/principal-value-of-amplitude", "concept/quadratic-equation", "concept/rational-function", "concept/rational-number", "concept/real-number", "concept/right-angle", "concept/root-of-a-complex-number", "concept/root-of-an-equation", "concept/root-of-unity", "concept/rotation", "concept/similar-triangles", "concept/transformation", "concept/translation", "concept/zero-displacement", "law/associative-law", "law/commutative-law", "law/distributive-law", "method/addition-of-displacements", "method/equating-real-and-imaginary-parts", "method/euclidean-construction", "method/mathematical-induction", "method/multiplication-by-i", "method/multiplication-of-complex-numbers", "method/multiplication-of-displacements", "method/multiplication-of-displacements-by-numbers", "method/rationalizing-a-complex-denominator", "method/subtraction-of-displacements", "person/euclid", "person/ptolemy", "quantity/amplitude-of-a-complex-number", "quantity/imaginary-part-of-a-complex-number", "quantity/modulus-of-a-complex-number", "quantity/real-part-of-a-complex-number", "theorem/complex-roots-of-a-real-equation-occur-in-conjugate-pairs", "theorem/conjugation-of-real-rational-functions", "theorem/de-moivre-s-theorem", "theorem/de-moivre-s-theorem-for-rational-exponents", "theorem/factor-theorem", "theorem/fundamental-theorem-of-algebra", "theorem/modulus-and-amplitude-of-a-product", "theorem/reduction-of-a-rational-function-to-x-yi", "theorem/relations-between-roots-and-coefficients", "theorem/remainder-theorem", "theorem/triangle-inequality", "theorem/triangle-inequality-for-complex-numbers", "theorem/zero-product-property" ], "excerpts": [ "hardy-course-of-pure-mathematics-1921/x-8ee021cb9a", "hardy-course-of-pure-mathematics-1921/x-27beec6920", "hardy-course-of-pure-mathematics-1921/x-09cc278e74", "hardy-course-of-pure-mathematics-1921/x-2e0e42575a", "hardy-course-of-pure-mathematics-1921/x-4f075be07b", "hardy-course-of-pure-mathematics-1921/x-38700e4624", "hardy-course-of-pure-mathematics-1921/x-0011ff8d8b", "hardy-course-of-pure-mathematics-1921/x-b971722392", "hardy-course-of-pure-mathematics-1921/x-a1eceb2a15", "hardy-course-of-pure-mathematics-1921/x-1623dda3d7", "hardy-course-of-pure-mathematics-1921/x-0a87af19ed", "hardy-course-of-pure-mathematics-1921/x-67e21d5585", "hardy-course-of-pure-mathematics-1921/x-186e0e32dd", "hardy-course-of-pure-mathematics-1921/x-b58dfc300a", "hardy-course-of-pure-mathematics-1921/x-da1a8aed37", "hardy-course-of-pure-mathematics-1921/x-d0410175ef", "hardy-course-of-pure-mathematics-1921/x-bce88ff2da", "hardy-course-of-pure-mathematics-1921/x-8919c874d7", "hardy-course-of-pure-mathematics-1921/x-82afc18dbf", "hardy-course-of-pure-mathematics-1921/x-7e6d37dd96", "hardy-course-of-pure-mathematics-1921/x-81cadd824e", "hardy-course-of-pure-mathematics-1921/x-67454901fe", "hardy-course-of-pure-mathematics-1921/x-86af3bc017", "hardy-course-of-pure-mathematics-1921/x-f500345c11", "hardy-course-of-pure-mathematics-1921/x-39d2ad48e1", "hardy-course-of-pure-mathematics-1921/x-788c198ca3", "hardy-course-of-pure-mathematics-1921/x-76c3235201", "hardy-course-of-pure-mathematics-1921/x-8d4ba67256", "hardy-course-of-pure-mathematics-1921/x-91795b88de", "hardy-course-of-pure-mathematics-1921/x-fc022a425f", "hardy-course-of-pure-mathematics-1921/x-3735f30bab", "hardy-course-of-pure-mathematics-1921/x-d9601b2804", "hardy-course-of-pure-mathematics-1921/x-8fc7c4d150", "hardy-course-of-pure-mathematics-1921/x-7f28a22a45", "hardy-course-of-pure-mathematics-1921/x-ecf6eadca9", "hardy-course-of-pure-mathematics-1921/x-a71919951f", "hardy-course-of-pure-mathematics-1921/x-814012f45b", "hardy-course-of-pure-mathematics-1921/x-154843bdaa", "hardy-course-of-pure-mathematics-1921/x-6443f5add0", "hardy-course-of-pure-mathematics-1921/x-7024a2841f", "hardy-course-of-pure-mathematics-1921/x-f0bdc3bfa6", "hardy-course-of-pure-mathematics-1921/x-6a76ef8834", "hardy-course-of-pure-mathematics-1921/x-8e4c7d4888", "hardy-course-of-pure-mathematics-1921/x-4a525de0d4", "hardy-course-of-pure-mathematics-1921/x-066a3269c6" ], "equations": [ "hardy-course-of-pure-mathematics-1921/eq-a0ac0f607d", "hardy-course-of-pure-mathematics-1921/eq-1ee947d43d", "hardy-course-of-pure-mathematics-1921/eq-1ccade3fb5", "hardy-course-of-pure-mathematics-1921/eq-1decb5173d", "hardy-course-of-pure-mathematics-1921/eq-c766573983", "hardy-course-of-pure-mathematics-1921/eq-1c6ad6e2cc", "hardy-course-of-pure-mathematics-1921/eq-9326a473e9", "hardy-course-of-pure-mathematics-1921/eq-89f9c4faec", "hardy-course-of-pure-mathematics-1921/eq-9e3e7eda9a", "hardy-course-of-pure-mathematics-1921/eq-c4c76bdaf0", "hardy-course-of-pure-mathematics-1921/eq-8e53532f9f", "hardy-course-of-pure-mathematics-1921/eq-e963136739", "hardy-course-of-pure-mathematics-1921/eq-7f09de9701", "hardy-course-of-pure-mathematics-1921/eq-f4e9f4227b", "hardy-course-of-pure-mathematics-1921/eq-86a9738d5f", "hardy-course-of-pure-mathematics-1921/eq-b0f858a963", "hardy-course-of-pure-mathematics-1921/eq-3771af24ef", "hardy-course-of-pure-mathematics-1921/eq-036215f1b7", "hardy-course-of-pure-mathematics-1921/eq-fef6643a35", "hardy-course-of-pure-mathematics-1921/eq-8dc5d1d858", "hardy-course-of-pure-mathematics-1921/eq-94fb865b8e", "hardy-course-of-pure-mathematics-1921/eq-8a69df811b", "hardy-course-of-pure-mathematics-1921/eq-52cb06e032", "hardy-course-of-pure-mathematics-1921/eq-eed30b77f0", "hardy-course-of-pure-mathematics-1921/eq-06fe54cab8", "hardy-course-of-pure-mathematics-1921/eq-c0d8ad3630", "hardy-course-of-pure-mathematics-1921/eq-cca858b0d3", "hardy-course-of-pure-mathematics-1921/eq-84ee7a0c92", "hardy-course-of-pure-mathematics-1921/eq-189b44fa98", "hardy-course-of-pure-mathematics-1921/eq-17269f34a7", "hardy-course-of-pure-mathematics-1921/eq-8028194b49", "hardy-course-of-pure-mathematics-1921/eq-60c6b3a6d5", "hardy-course-of-pure-mathematics-1921/eq-506edf4e9f", "hardy-course-of-pure-mathematics-1921/eq-a4ee600775", "hardy-course-of-pure-mathematics-1921/eq-c5d0f21779", "hardy-course-of-pure-mathematics-1921/eq-00f2388873", "hardy-course-of-pure-mathematics-1921/eq-134704b2b1", "hardy-course-of-pure-mathematics-1921/eq-6d44966a68", "hardy-course-of-pure-mathematics-1921/eq-43f5747fb0", "hardy-course-of-pure-mathematics-1921/eq-4f5efeb54f", "hardy-course-of-pure-mathematics-1921/eq-1d6ca04d59", "hardy-course-of-pure-mathematics-1921/eq-0ed0d6b5c2", "hardy-course-of-pure-mathematics-1921/eq-61945d8237", "hardy-course-of-pure-mathematics-1921/eq-658d69bb6f", "hardy-course-of-pure-mathematics-1921/eq-0b636aa994", "hardy-course-of-pure-mathematics-1921/eq-ae30aef37d", "hardy-course-of-pure-mathematics-1921/eq-cf5461d31f", "hardy-course-of-pure-mathematics-1921/eq-e0da529494", "hardy-course-of-pure-mathematics-1921/eq-9a48920231", "hardy-course-of-pure-mathematics-1921/eq-9c4f225594", "hardy-course-of-pure-mathematics-1921/eq-d42a93ef3b", "hardy-course-of-pure-mathematics-1921/eq-cbec8d0654", "hardy-course-of-pure-mathematics-1921/eq-0466cea82e", "hardy-course-of-pure-mathematics-1921/eq-cd50db4a27", "hardy-course-of-pure-mathematics-1921/eq-d9cd998e61", "hardy-course-of-pure-mathematics-1921/eq-cb17542466", "hardy-course-of-pure-mathematics-1921/eq-2212fcdb6d", "hardy-course-of-pure-mathematics-1921/eq-531183b9c6", "hardy-course-of-pure-mathematics-1921/eq-5a17fadeba", "hardy-course-of-pure-mathematics-1921/eq-4cfeef3415", "hardy-course-of-pure-mathematics-1921/eq-e92f1e1ec8", "hardy-course-of-pure-mathematics-1921/eq-6d2590a78e", "hardy-course-of-pure-mathematics-1921/eq-76516ed5b4", "hardy-course-of-pure-mathematics-1921/eq-681396e93f", "hardy-course-of-pure-mathematics-1921/eq-9913521d71", "hardy-course-of-pure-mathematics-1921/eq-bef13d8959", "hardy-course-of-pure-mathematics-1921/eq-56ead0336f", "hardy-course-of-pure-mathematics-1921/eq-ddc33ee789", "hardy-course-of-pure-mathematics-1921/eq-36b14306f2", "hardy-course-of-pure-mathematics-1921/eq-70a1d5bef8", "hardy-course-of-pure-mathematics-1921/eq-c9b7338322", "hardy-course-of-pure-mathematics-1921/eq-4b3a7c5b4e", "hardy-course-of-pure-mathematics-1921/eq-fcf350bf3a", "hardy-course-of-pure-mathematics-1921/eq-978ca4b7f2", "hardy-course-of-pure-mathematics-1921/eq-97314614df", "hardy-course-of-pure-mathematics-1921/eq-6e4dfa7c71", "hardy-course-of-pure-mathematics-1921/eq-ed4965a3df", "hardy-course-of-pure-mathematics-1921/eq-05b4f5629a", "hardy-course-of-pure-mathematics-1921/eq-d5f877f9e7", "hardy-course-of-pure-mathematics-1921/eq-d8a98886f5", "hardy-course-of-pure-mathematics-1921/eq-163d4d3bd7", "hardy-course-of-pure-mathematics-1921/eq-5de0b39fe3", "hardy-course-of-pure-mathematics-1921/eq-c8bfeb23c2", "hardy-course-of-pure-mathematics-1921/eq-0db5fcd250" ], "exercise_sets": [ "hardy-course-of-pure-mathematics-1921/ex-xxii", "hardy-course-of-pure-mathematics-1921/ex-misc-iii", "hardy-course-of-pure-mathematics-1921/ex-xx", "hardy-course-of-pure-mathematics-1921/ex-xxi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ch-iv", "number": "IV", "title": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "name": "Hardy 1921, ch. IV: LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "pages": [ "154", "162" ], "concepts": [ "concept/absolute-value", "concept/arithmetical-mean", "concept/bounded-function", "concept/complex-number", "concept/conjugate-complex-numbers", "concept/convergent-series", "concept/decimal", "concept/dedekind-section", "concept/divergent-series", "concept/domain-of-definition", "concept/e", "concept/euler-s-number", "concept/expansion", "concept/factorial", "concept/finite-oscillation", "concept/finite-set", "concept/function", "concept/function-of-a-positive-integer-variable", "concept/functional-interpolation", "concept/gamma-function", "concept/geometrical-progression", "concept/harmonic-series", "concept/infinite-oscillation", "concept/infinite-sequence", "concept/infinite-set", "concept/infinity", "concept/integer", "concept/integer-part", "concept/irrational-number", "concept/least-upper-bound", "concept/limit", "concept/limit-inferior", "concept/limit-superior", "concept/modulus-of-a-complex-number", "concept/oscillation", "concept/polar-form-of-a-complex-number", "concept/prime-number", "concept/proper-algebraic-fraction", "concept/recurring-decimal", "concept/representation-of-a-function-by-a-limit", "concept/series-of-complex-terms", "concept/set", "concept/steadily-increasing-function", "concept/sufficiently-large-values", "concept/sum-to-infinity", "concept/tends-to-infinity", "concept/threshold-index", "method/indirect-method", "method/square-root-iteration", "person/euclid", "theorem/binomial-theorem", "theorem/comparison-test", "theorem/convergence-of-a-series-of-positive-terms", "theorem/criterion-for-convergence-of-a-complex-sequence", "theorem/dedekind-s-theorem", "theorem/general-principle-of-convergence", "theorem/grouping-of-terms-in-a-convergent-series", "theorem/infinitude-of-primes", "theorem/limit-of-a-constant-multiple", "theorem/limit-of-a-power", "theorem/limit-of-a-power-of-a-complex-number", "theorem/limit-of-a-product", "theorem/limit-of-a-quotient", "theorem/limit-of-a-rational-function", "theorem/limit-of-a-rational-function-of-n", "theorem/limit-of-a-rational-function-of-sequences", "theorem/limit-of-a-reciprocal", "theorem/limit-of-a-shifted-sequence", "theorem/limit-of-a-sum", "theorem/limit-of-n-nth-root-of-x-1", "theorem/limit-of-the-arithmetic-mean-of-a-sequence", "theorem/limit-of-the-nth-root-of-a-positive-number", "theorem/limit-of-x-n", "theorem/limit-of-z-n", "theorem/monotone-convergence-theorem", "theorem/necessary-condition-for-convergence", "theorem/power-difference-inequality", "theorem/ratio-test-for-sequences", "theorem/steadily-increasing-function-limit-theorem", "theorem/sum-of-an-infinite-geometric-series", "theorem/terms-of-a-convergent-series-tend-to-zero", "theorem/weierstrass-s-theorem" ], "excerpts": [ "hardy-course-of-pure-mathematics-1921/x-27df2b0342", "hardy-course-of-pure-mathematics-1921/x-ccc31e2d74", "hardy-course-of-pure-mathematics-1921/x-f13b200ee5", "hardy-course-of-pure-mathematics-1921/x-ed3216b934", "hardy-course-of-pure-mathematics-1921/x-bd0b0bc969", "hardy-course-of-pure-mathematics-1921/x-df6a9893e1", "hardy-course-of-pure-mathematics-1921/x-dfed52a158", "hardy-course-of-pure-mathematics-1921/x-4a9cbc7dbd", "hardy-course-of-pure-mathematics-1921/x-765646ea2d", "hardy-course-of-pure-mathematics-1921/x-aa7b6c644a", "hardy-course-of-pure-mathematics-1921/x-4a1d889a7e", "hardy-course-of-pure-mathematics-1921/x-43efba2dd7", "hardy-course-of-pure-mathematics-1921/x-f89a92df3b", "hardy-course-of-pure-mathematics-1921/x-bc81274a88", "hardy-course-of-pure-mathematics-1921/x-b15fb71f23", "hardy-course-of-pure-mathematics-1921/x-f81287ae67", "hardy-course-of-pure-mathematics-1921/x-7195191e9b", "hardy-course-of-pure-mathematics-1921/x-ed2f42a845", "hardy-course-of-pure-mathematics-1921/x-7c99adc882", "hardy-course-of-pure-mathematics-1921/x-695f599de5", "hardy-course-of-pure-mathematics-1921/x-51855c17eb", "hardy-course-of-pure-mathematics-1921/x-e970eb4063", "hardy-course-of-pure-mathematics-1921/x-9b6b1ed7fd", "hardy-course-of-pure-mathematics-1921/x-c70f879e42", "hardy-course-of-pure-mathematics-1921/x-042da5543a", "hardy-course-of-pure-mathematics-1921/x-3779019c30", "hardy-course-of-pure-mathematics-1921/x-5c1faddaeb", "hardy-course-of-pure-mathematics-1921/x-a4ee3819e0", "hardy-course-of-pure-mathematics-1921/x-5485ef19ef", "hardy-course-of-pure-mathematics-1921/x-3c0919c195", "hardy-course-of-pure-mathematics-1921/x-1b4fe40821", "hardy-course-of-pure-mathematics-1921/x-785e65a464", "hardy-course-of-pure-mathematics-1921/x-ecb7f4b11e", "hardy-course-of-pure-mathematics-1921/x-a19516a84e", "hardy-course-of-pure-mathematics-1921/x-f565b2eda4", "hardy-course-of-pure-mathematics-1921/x-72b04a0133", "hardy-course-of-pure-mathematics-1921/x-2e9c1d8b31", "hardy-course-of-pure-mathematics-1921/x-64f2da1adc", "hardy-course-of-pure-mathematics-1921/x-0a202f649d", "hardy-course-of-pure-mathematics-1921/x-c68f957677", "hardy-course-of-pure-mathematics-1921/x-fdec29db62", "hardy-course-of-pure-mathematics-1921/x-baba544a79", "hardy-course-of-pure-mathematics-1921/x-7609bf0dc0", "hardy-course-of-pure-mathematics-1921/x-564ebcd161", "hardy-course-of-pure-mathematics-1921/x-865bd1d385", "hardy-course-of-pure-mathematics-1921/x-d9a95da2e7", "hardy-course-of-pure-mathematics-1921/x-3b638d657f", "hardy-course-of-pure-mathematics-1921/x-16f35f7a55", "hardy-course-of-pure-mathematics-1921/x-97a8f08db0", "hardy-course-of-pure-mathematics-1921/x-dad1ae8d16", "hardy-course-of-pure-mathematics-1921/x-b479e331d1", "hardy-course-of-pure-mathematics-1921/x-43ebf85a49", "hardy-course-of-pure-mathematics-1921/x-4a8e547287", "hardy-course-of-pure-mathematics-1921/x-9d3c843bbf", "hardy-course-of-pure-mathematics-1921/x-d8a63f7c14" ], "equations": [ "hardy-course-of-pure-mathematics-1921/eq-6d4b5b8a02", "hardy-course-of-pure-mathematics-1921/eq-72131843bc", "hardy-course-of-pure-mathematics-1921/eq-47a7dd4bea", "hardy-course-of-pure-mathematics-1921/eq-0edd12453e", "hardy-course-of-pure-mathematics-1921/eq-18d8020916", "hardy-course-of-pure-mathematics-1921/eq-2bdd5b47a5", "hardy-course-of-pure-mathematics-1921/eq-73ee399921", "hardy-course-of-pure-mathematics-1921/eq-55586002c3", "hardy-course-of-pure-mathematics-1921/eq-7bc3ec4605", "hardy-course-of-pure-mathematics-1921/eq-c8c35752ba", "hardy-course-of-pure-mathematics-1921/eq-df25b25d7d", "hardy-course-of-pure-mathematics-1921/eq-cb6034f95d", "hardy-course-of-pure-mathematics-1921/eq-fbecf63fcf", "hardy-course-of-pure-mathematics-1921/eq-e64e78d1e0", "hardy-course-of-pure-mathematics-1921/eq-1d674baf2e", "hardy-course-of-pure-mathematics-1921/eq-2859d478d3", "hardy-course-of-pure-mathematics-1921/eq-8356ab3815", "hardy-course-of-pure-mathematics-1921/eq-d775accb12", "hardy-course-of-pure-mathematics-1921/eq-973f1edbbe", "hardy-course-of-pure-mathematics-1921/eq-127b8f88bd", "hardy-course-of-pure-mathematics-1921/eq-b3174e71c5", "hardy-course-of-pure-mathematics-1921/eq-ff0d7d4f3f", "hardy-course-of-pure-mathematics-1921/eq-3c8eada3cb", "hardy-course-of-pure-mathematics-1921/eq-e1a4954646", "hardy-course-of-pure-mathematics-1921/eq-e87e9add02", "hardy-course-of-pure-mathematics-1921/eq-e3a9531bea", "hardy-course-of-pure-mathematics-1921/eq-5fb6e131f2", "hardy-course-of-pure-mathematics-1921/eq-a927e81010", "hardy-course-of-pure-mathematics-1921/eq-f1fd6f67c1", "hardy-course-of-pure-mathematics-1921/eq-07c4b568f1", "hardy-course-of-pure-mathematics-1921/eq-a7eb5fcecd", "hardy-course-of-pure-mathematics-1921/eq-9b023ec2f4", "hardy-course-of-pure-mathematics-1921/eq-18a198b539", "hardy-course-of-pure-mathematics-1921/eq-32b3ecf1c6", "hardy-course-of-pure-mathematics-1921/eq-8720e04a30", "hardy-course-of-pure-mathematics-1921/eq-b274e3d7c2", "hardy-course-of-pure-mathematics-1921/eq-4d3331cb1b", "hardy-course-of-pure-mathematics-1921/eq-5ee96e8e96", "hardy-course-of-pure-mathematics-1921/eq-c1f934c5de", "hardy-course-of-pure-mathematics-1921/eq-d51186254d", "hardy-course-of-pure-mathematics-1921/eq-78033223a3", "hardy-course-of-pure-mathematics-1921/eq-2ceb827003", "hardy-course-of-pure-mathematics-1921/eq-ae75aafe52", "hardy-course-of-pure-mathematics-1921/eq-b1ffb49130", "hardy-course-of-pure-mathematics-1921/eq-dc8abe6dd2", "hardy-course-of-pure-mathematics-1921/eq-f8b5d14936", "hardy-course-of-pure-mathematics-1921/eq-1f005cb506", "hardy-course-of-pure-mathematics-1921/eq-7b248dcb7e", "hardy-course-of-pure-mathematics-1921/eq-cbaf5acf74", "hardy-course-of-pure-mathematics-1921/eq-ba6e8675c9", "hardy-course-of-pure-mathematics-1921/eq-ec8a22f1f1", "hardy-course-of-pure-mathematics-1921/eq-677cebf27f", "hardy-course-of-pure-mathematics-1921/eq-6f69fdb020", "hardy-course-of-pure-mathematics-1921/eq-d61b44e8ef", "hardy-course-of-pure-mathematics-1921/eq-a9482dbff2", "hardy-course-of-pure-mathematics-1921/eq-960dd58fe7", "hardy-course-of-pure-mathematics-1921/eq-87b3725757", "hardy-course-of-pure-mathematics-1921/eq-3fccc8b42b", "hardy-course-of-pure-mathematics-1921/eq-caede7193f", "hardy-course-of-pure-mathematics-1921/eq-711dd0b189", "hardy-course-of-pure-mathematics-1921/eq-9a4a59c1b7", "hardy-course-of-pure-mathematics-1921/eq-965ea2bc61", "hardy-course-of-pure-mathematics-1921/eq-52c51d96ab", "hardy-course-of-pure-mathematics-1921/eq-bc8d51a5ec", "hardy-course-of-pure-mathematics-1921/eq-a57de6f0b8", "hardy-course-of-pure-mathematics-1921/eq-0c4e905fb8", "hardy-course-of-pure-mathematics-1921/eq-3187caff78", "hardy-course-of-pure-mathematics-1921/eq-dc448812b0", "hardy-course-of-pure-mathematics-1921/eq-75d736a5a6", "hardy-course-of-pure-mathematics-1921/eq-9047e0caec", "hardy-course-of-pure-mathematics-1921/eq-6a87c80af3", "hardy-course-of-pure-mathematics-1921/eq-f01d262e0d", "hardy-course-of-pure-mathematics-1921/eq-7ea4a8d07b", "hardy-course-of-pure-mathematics-1921/eq-b7dea93b92", "hardy-course-of-pure-mathematics-1921/eq-50b8e2840a", "hardy-course-of-pure-mathematics-1921/eq-2aa3eb113b", "hardy-course-of-pure-mathematics-1921/eq-670b9c0fcf", "hardy-course-of-pure-mathematics-1921/eq-8c7c184fea", "hardy-course-of-pure-mathematics-1921/eq-f751accd9b", "hardy-course-of-pure-mathematics-1921/eq-5d9e596179", "hardy-course-of-pure-mathematics-1921/eq-de0f42f7aa", "hardy-course-of-pure-mathematics-1921/eq-f22be5bbc3", "hardy-course-of-pure-mathematics-1921/eq-34b58259b8", "hardy-course-of-pure-mathematics-1921/eq-06338a424d", "hardy-course-of-pure-mathematics-1921/eq-d64b7d87c1", "hardy-course-of-pure-mathematics-1921/eq-753468e100", "hardy-course-of-pure-mathematics-1921/eq-05b4f5629a", "hardy-course-of-pure-mathematics-1921/eq-86ab960aca", "hardy-course-of-pure-mathematics-1921/eq-9c3a2cb72c", "hardy-course-of-pure-mathematics-1921/eq-27342cdaf9", "hardy-course-of-pure-mathematics-1921/eq-cfa79400ab", "hardy-course-of-pure-mathematics-1921/eq-9e00f0ef42" ], "exercise_sets": [ "hardy-course-of-pure-mathematics-1921/ex-xxix", "hardy-course-of-pure-mathematics-1921/ex-xxx", "hardy-course-of-pure-mathematics-1921/ex-xxxi", "hardy-course-of-pure-mathematics-1921/ex-xxxii", "hardy-course-of-pure-mathematics-1921/ex-xxiii", "hardy-course-of-pure-mathematics-1921/ex-xxiv", "hardy-course-of-pure-mathematics-1921/ex-xxv", "hardy-course-of-pure-mathematics-1921/ex-xxvi", "hardy-course-of-pure-mathematics-1921/ex-xxvii", "hardy-course-of-pure-mathematics-1921/ex-xxviii", "hardy-course-of-pure-mathematics-1921/ex-xxxiii", "hardy-course-of-pure-mathematics-1921/ex-misc-iv" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ch-v", "number": "V", "title": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "name": "Hardy 1921, ch. V: LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "pages": [ "162", "174" ], "concepts": [ "concept/bounded-function", "concept/continuous-function", "concept/continuous-function-of-two-variables", "concept/continuous-real-variable", "concept/converse-of-a-theorem", "concept/cosine", "concept/dedekind-section", "concept/definite-integral", "concept/discontinuous-function", "concept/domain-of-definition", "concept/function", "concept/function-of-a-positive-integer-variable", "concept/graph-of-a-function", "concept/implicit-function", "concept/indeterminate-form", "concept/infinity", "concept/integer-part-function", "concept/interval", "concept/inverse-circular-function", "concept/inverse-function", "concept/least-upper-bound", "concept/limit", "concept/limit-inferior", "concept/limit-superior", "concept/neighbourhood", "concept/one-sided-limit", "concept/one-valued-function", "concept/order-of-greatness", "concept/order-of-smallness", "concept/oscillation", "concept/oscillatory-discontinuity", "concept/polynomial", "concept/rational-function", "concept/separate-continuity", "concept/set-of-intervals", "concept/simple-discontinuity", "concept/sine", "concept/steadily-increasing-function", "concept/tends-to-infinity", "concept/value-of-a-function", "method/repeated-bisection", "theorem/attainment-of-bounds-by-continuous-functions", "theorem/boundedness-of-a-continuous-function", "theorem/boundedness-of-continuous-functions", "theorem/existence-of-a-continuous-inverse-function", "theorem/extreme-value-theorem", "theorem/finite-subdivision-with-small-oscillation", "theorem/heine-borel-theorem", "theorem/implicit-function-theorem", "theorem/intermediate-value-theorem", "theorem/inverse-function-theorem", "theorem/limit-of-a-product", "theorem/limit-of-a-quotient", "theorem/limit-of-sin-x-over-x", "theorem/principle-of-convergence", "theorem/sign-persistence-of-a-continuous-function", "theorem/sign-persistence-of-continuous-functions", "theorem/small-oscillation-subdivision-theorem", "theorem/uniform-continuity" ], "excerpts": [ "hardy-course-of-pure-mathematics-1921/x-12a0a7d696", "hardy-course-of-pure-mathematics-1921/x-4de50457c1", "hardy-course-of-pure-mathematics-1921/x-be43c7358a", "hardy-course-of-pure-mathematics-1921/x-8cdff9d343", "hardy-course-of-pure-mathematics-1921/x-7878560696", "hardy-course-of-pure-mathematics-1921/x-c67245227e", "hardy-course-of-pure-mathematics-1921/x-7132fce577", "hardy-course-of-pure-mathematics-1921/x-8b2f4d3fc4", "hardy-course-of-pure-mathematics-1921/x-a0fa14271c", "hardy-course-of-pure-mathematics-1921/x-b41a634909", "hardy-course-of-pure-mathematics-1921/x-d8efddb727", "hardy-course-of-pure-mathematics-1921/x-00ed3dc415", "hardy-course-of-pure-mathematics-1921/x-b3f984015f", "hardy-course-of-pure-mathematics-1921/x-3d0413ab71", "hardy-course-of-pure-mathematics-1921/x-1f759f67ed", "hardy-course-of-pure-mathematics-1921/x-e1d4284e75", "hardy-course-of-pure-mathematics-1921/x-83c434d7b3", "hardy-course-of-pure-mathematics-1921/x-1cfae5f090", "hardy-course-of-pure-mathematics-1921/x-b9b5b5b17d", "hardy-course-of-pure-mathematics-1921/x-c38906b44c", "hardy-course-of-pure-mathematics-1921/x-6cefa55de2", "hardy-course-of-pure-mathematics-1921/x-a528198bc2", "hardy-course-of-pure-mathematics-1921/x-478eeefecb", "hardy-course-of-pure-mathematics-1921/x-8f00730d44", "hardy-course-of-pure-mathematics-1921/x-c90089bdf6", "hardy-course-of-pure-mathematics-1921/x-b976622ded", "hardy-course-of-pure-mathematics-1921/x-d5547be8eb", "hardy-course-of-pure-mathematics-1921/x-794d6bec76", "hardy-course-of-pure-mathematics-1921/x-66703a4c2f", "hardy-course-of-pure-mathematics-1921/x-c9099c9e7a", "hardy-course-of-pure-mathematics-1921/x-ec616d9890", "hardy-course-of-pure-mathematics-1921/x-6d7afc912f", "hardy-course-of-pure-mathematics-1921/x-7d226c53a1" ], "equations": [ "hardy-course-of-pure-mathematics-1921/eq-fd818fa0fc", "hardy-course-of-pure-mathematics-1921/eq-62e4d754e9", "hardy-course-of-pure-mathematics-1921/eq-6fe7f44971", "hardy-course-of-pure-mathematics-1921/eq-72a05b191f", "hardy-course-of-pure-mathematics-1921/eq-f3d240eed3", "hardy-course-of-pure-mathematics-1921/eq-41491ea29b", "hardy-course-of-pure-mathematics-1921/eq-e16a95e23c", "hardy-course-of-pure-mathematics-1921/eq-4b6d4ffc85", "hardy-course-of-pure-mathematics-1921/eq-f8126b6910", "hardy-course-of-pure-mathematics-1921/eq-c93ee7d619", "hardy-course-of-pure-mathematics-1921/eq-4fe17211a8", "hardy-course-of-pure-mathematics-1921/eq-173e0b8f6d", "hardy-course-of-pure-mathematics-1921/eq-d22cfaa9c6", "hardy-course-of-pure-mathematics-1921/eq-c4b2c5b181", "hardy-course-of-pure-mathematics-1921/eq-d21fe0ccc9", "hardy-course-of-pure-mathematics-1921/eq-4b6a526511", "hardy-course-of-pure-mathematics-1921/eq-32726bbc64", "hardy-course-of-pure-mathematics-1921/eq-11c25a8710", "hardy-course-of-pure-mathematics-1921/eq-f356980e32", "hardy-course-of-pure-mathematics-1921/eq-bdae8fdd0c", "hardy-course-of-pure-mathematics-1921/eq-32955812a8", "hardy-course-of-pure-mathematics-1921/eq-27965bce4f", "hardy-course-of-pure-mathematics-1921/eq-dee76a8f42", "hardy-course-of-pure-mathematics-1921/eq-5f1c9d1de8", "hardy-course-of-pure-mathematics-1921/eq-13dd732abe", "hardy-course-of-pure-mathematics-1921/eq-f381ef55bd", "hardy-course-of-pure-mathematics-1921/eq-15df879609", "hardy-course-of-pure-mathematics-1921/eq-c9750fc98e", "hardy-course-of-pure-mathematics-1921/eq-5953441da9", "hardy-course-of-pure-mathematics-1921/eq-e1444cb45d", "hardy-course-of-pure-mathematics-1921/eq-0e226d43ac", "hardy-course-of-pure-mathematics-1921/eq-ab462dc14d", "hardy-course-of-pure-mathematics-1921/eq-5e46353463", "hardy-course-of-pure-mathematics-1921/eq-4a93597be2", "hardy-course-of-pure-mathematics-1921/eq-43684d7c82", "hardy-course-of-pure-mathematics-1921/eq-5f131319e9", "hardy-course-of-pure-mathematics-1921/eq-765c57ba10", "hardy-course-of-pure-mathematics-1921/eq-a56e34edc4", "hardy-course-of-pure-mathematics-1921/eq-d299d6057e", "hardy-course-of-pure-mathematics-1921/eq-900c16a3a7", "hardy-course-of-pure-mathematics-1921/eq-1597cea1ff", "hardy-course-of-pure-mathematics-1921/eq-988f8eec7c", "hardy-course-of-pure-mathematics-1921/eq-c8d50f3c9b", "hardy-course-of-pure-mathematics-1921/eq-77f7a27b29", "hardy-course-of-pure-mathematics-1921/eq-69cac481b2", "hardy-course-of-pure-mathematics-1921/eq-92a5d9f916", "hardy-course-of-pure-mathematics-1921/eq-9884327bce", "hardy-course-of-pure-mathematics-1921/eq-d6d192677f", "hardy-course-of-pure-mathematics-1921/eq-985f1d7a62", "hardy-course-of-pure-mathematics-1921/eq-043032cdb1", "hardy-course-of-pure-mathematics-1921/eq-e690d352de", "hardy-course-of-pure-mathematics-1921/eq-1e6c249c34", "hardy-course-of-pure-mathematics-1921/eq-c6b16feeeb", "hardy-course-of-pure-mathematics-1921/eq-29c828c6c8", "hardy-course-of-pure-mathematics-1921/eq-fc5a85d683", "hardy-course-of-pure-mathematics-1921/eq-ad8a7ad3d0", "hardy-course-of-pure-mathematics-1921/eq-3a4d6a61fd", "hardy-course-of-pure-mathematics-1921/eq-cb8106688e", "hardy-course-of-pure-mathematics-1921/eq-437fdd70ef" ], "exercise_sets": [ "hardy-course-of-pure-mathematics-1921/ex-xxxvii", "hardy-course-of-pure-mathematics-1921/ex-xxxviii", "hardy-course-of-pure-mathematics-1921/ex-misc-v", "hardy-course-of-pure-mathematics-1921/ex-xxxiv", "hardy-course-of-pure-mathematics-1921/ex-xxxv", "hardy-course-of-pure-mathematics-1921/ex-xxxvi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ch-vi", "number": "VI", "title": "DERIVATIVES AND INTEGRALS", "name": "Hardy 1921, ch. VI: DERIVATIVES AND INTEGRALS", "pages": [ "226", "240" ], "concepts": [ "concept/algebraic-function", "concept/arbitrary-constant-of-integration", "concept/area", "concept/binomial-coefficient", "concept/binomial-form-of-a-polynomial", "concept/cardioid", "concept/chord", "concept/circle", "concept/complex-number", "concept/conic", "concept/conic-section", "concept/conjugate-complex-numbers", "concept/continuity", "concept/continuous-function", "concept/cosecant", "concept/cosine", "concept/cotangent", "concept/denominator", "concept/derivative", "concept/determinant", "concept/differential-equation", "concept/discontinuous-derivative", "concept/discontinuous-function", "concept/discriminant", "concept/divisibility", "concept/ellipse", "concept/equal-roots", "concept/factor", "concept/function", "concept/graph-of-a-function", "concept/higher-order-derivative", "concept/highest-common-factor", "concept/hyperbola", "concept/hypotenuse", "concept/increment", "concept/integral", "concept/integration-by-substitution", "concept/inverse-circular-function", "concept/inverse-function", "concept/limit", "concept/limiting-case", "concept/locus", "concept/logarithm", "concept/lowest-terms", "concept/mathematical-notation", "concept/maximum", "concept/minimum", "concept/multiple-root", "concept/normal", "concept/one-valued-function", "concept/parabola", "concept/parameter", "concept/perimeter", "concept/pi", "concept/polar-coordinates", "concept/polynomial", "concept/power", "concept/product", "concept/rational-function", "concept/real-root", "concept/right-triangle", "concept/root", "concept/root-of-an-equation", "concept/secant", "concept/sign-of-the-derivative", "concept/sine", "concept/standard-forms-of-integration", "concept/tangent", "concept/tangent-function", "concept/transcendental-function", "concept/variable", "method/differentiation", "method/formulae-of-reduction", "method/implicit-differentiation", "method/integration", "method/integration-by-parts", "method/integration-by-rationalisation", "method/integration-of-algebraic-functions", "method/integration-of-an-inverse-function", "method/integration-of-polynomials-in-cosines-and-sines-of-multiples-of-x", "method/integration-of-rational-functions", "method/mathematical-induction", "method/partial-fractions", "method/reduction-formula", "method/second-derivative-test", "method/substitution", "method/tangent-half-angle-substitution", "quantity/area", "quantity/eccentricity", "quantity/length-of-a-curve", "quantity/velocity", "theorem/chain-rule", "theorem/constant-multiple-rule-for-derivatives", "theorem/derivative-of-an-inverse-function", "theorem/derivative-of-inverse-circular-function", "theorem/derivative-of-trigonometric-functions", "theorem/existence-of-an-integral-of-a-continuous-function", "theorem/fundamental-theorem-of-calculus", "theorem/general-leibniz-rule", "theorem/generalised-mean-value-theorem", "theorem/increasing-function-criterion", "theorem/inverse-function-rule", "theorem/mean-value-theorem", "theorem/necessary-condition-for-a-maximum-or-minimum", "theorem/power-rule", "theorem/power-rule-for-derivatives", "theorem/product-rule", "theorem/product-rule-for-derivatives", "theorem/quotient-rule", "theorem/quotient-rule-for-derivatives", "theorem/reciprocal-rule-for-derivatives", "theorem/rolle-s-theorem", "theorem/sign-of-the-derivative", "theorem/sum-rule-for-derivatives", "theorem/zero-derivative-implies-constant" ], "excerpts": [ "hardy-course-of-pure-mathematics-1921/x-f78022741b", "hardy-course-of-pure-mathematics-1921/x-171c741acf", "hardy-course-of-pure-mathematics-1921/x-35e5ec529c", "hardy-course-of-pure-mathematics-1921/x-9caaa1162e", "hardy-course-of-pure-mathematics-1921/x-46761d54c6", "hardy-course-of-pure-mathematics-1921/x-4a92ceff5d", "hardy-course-of-pure-mathematics-1921/x-466a35aef5", "hardy-course-of-pure-mathematics-1921/x-1553873da7", "hardy-course-of-pure-mathematics-1921/x-a3c2603998", 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"hardy-course-of-pure-mathematics-1921/ex-xlii", "hardy-course-of-pure-mathematics-1921/ex-xliii", "hardy-course-of-pure-mathematics-1921/ex-xliv", "hardy-course-of-pure-mathematics-1921/ex-xlvii", "hardy-course-of-pure-mathematics-1921/ex-xlviii", "hardy-course-of-pure-mathematics-1921/ex-xlix", "hardy-course-of-pure-mathematics-1921/ex-l", "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "hardy-course-of-pure-mathematics-1921/ex-li", "hardy-course-of-pure-mathematics-1921/ex-xlv", "hardy-course-of-pure-mathematics-1921/ex-xlvi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ch-vii", "number": "VII", "title": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "name": "Hardy 1921, ch. VII: ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "pages": [ "299", "308" ], "concepts": [ "concept/approximation", "concept/area", "concept/binomial-series", "concept/centre-of-curvature", "concept/circle-of-curvature", "concept/circular-measure", "concept/circumscribed-circle", "concept/contact-of-the-nth-order", "concept/continuous-function", "concept/cosine", "concept/curvature", "concept/definite-integral", "concept/dependent", "concept/derivative", "concept/determinant", "concept/differential", "concept/ellipse", "concept/exact-differential", "concept/function", "concept/function-of-several-variables", "concept/function-of-two-variables", "concept/functional-relation", "concept/higher-order-derivative", "concept/homogeneous-function", "concept/implicit-function", "concept/increment", "concept/indefinite-integral", "concept/indeterminate-form", "concept/integral-of-a-complex-function", "concept/integrand", "concept/intersection-of-curves", "concept/inverse-circular-function", "concept/jacobian", "concept/least-upper-bound", "concept/limit", "concept/limits-of-integration", "concept/lower-sum", "concept/maclaurin-s-series", "concept/maximum", "concept/minimum", "concept/order-of-smallness", "concept/partial-derivative", "concept/pi", "concept/point-of-inflexion", "concept/polar-coordinates", "concept/polynomial", "concept/principal-part-of-an-increment", "concept/properties-of-the-definite-integral", "concept/radius", "concept/rectangle", "concept/root-of-an-equation", "concept/sine", "concept/tangent", "concept/touching-of-curves", "concept/triangle", "concept/upper-sum", "concept/variable", "method/differentiation", "method/evaluating-limits-by-derivatives", "method/evaluation-of-a-definite-integral-as-the-limit-of-a-sum", "method/higher-derivative-test-for-maxima-and-minima", "method/implicit-differentiation", "method/integration-by-parts", "method/newton-s-method", "method/simpson-s-rule", "method/substitution", 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"hardy-course-of-pure-mathematics-1921/eq-c302768a00", "hardy-course-of-pure-mathematics-1921/eq-edbb136345", "hardy-course-of-pure-mathematics-1921/eq-4f588383f4", "hardy-course-of-pure-mathematics-1921/eq-3c935922eb", "hardy-course-of-pure-mathematics-1921/eq-7e3e0f2929", "hardy-course-of-pure-mathematics-1921/eq-eb080003f2", "hardy-course-of-pure-mathematics-1921/eq-4d8f438300", "hardy-course-of-pure-mathematics-1921/eq-705207139e", "hardy-course-of-pure-mathematics-1921/eq-98f4de2e4e", "hardy-course-of-pure-mathematics-1921/eq-f9b130688c", "hardy-course-of-pure-mathematics-1921/eq-36195831b8", "hardy-course-of-pure-mathematics-1921/eq-b1608e240a", "hardy-course-of-pure-mathematics-1921/eq-64c18584e8", "hardy-course-of-pure-mathematics-1921/eq-41ce9a8bf5", "hardy-course-of-pure-mathematics-1921/eq-15e7a1924c", "hardy-course-of-pure-mathematics-1921/eq-f2767503d1", "hardy-course-of-pure-mathematics-1921/eq-2bb53da838", "hardy-course-of-pure-mathematics-1921/eq-7e21b04086", "hardy-course-of-pure-mathematics-1921/eq-e318568491", "hardy-course-of-pure-mathematics-1921/eq-62b2eed216", "hardy-course-of-pure-mathematics-1921/eq-6cde32c20d", "hardy-course-of-pure-mathematics-1921/eq-fb6b9c4cac", "hardy-course-of-pure-mathematics-1921/eq-ab1600dd79", "hardy-course-of-pure-mathematics-1921/eq-a901ed35fb", "hardy-course-of-pure-mathematics-1921/eq-a3b02b7342", "hardy-course-of-pure-mathematics-1921/eq-8de745a33e", "hardy-course-of-pure-mathematics-1921/eq-a518e2622d", "hardy-course-of-pure-mathematics-1921/eq-bc98991963", "hardy-course-of-pure-mathematics-1921/eq-86fbfc1e0b", "hardy-course-of-pure-mathematics-1921/eq-ca4c2a3e11", "hardy-course-of-pure-mathematics-1921/eq-cc561c4ead", "hardy-course-of-pure-mathematics-1921/eq-1eec198e49", "hardy-course-of-pure-mathematics-1921/eq-b81f10274f", "hardy-course-of-pure-mathematics-1921/eq-1d93993c19", "hardy-course-of-pure-mathematics-1921/eq-03e1506c52", "hardy-course-of-pure-mathematics-1921/eq-fcbea156a4", "hardy-course-of-pure-mathematics-1921/eq-8befb36c3f", "hardy-course-of-pure-mathematics-1921/eq-15d99a5f83", "hardy-course-of-pure-mathematics-1921/eq-b1def553e1", "hardy-course-of-pure-mathematics-1921/eq-697e376b03", "hardy-course-of-pure-mathematics-1921/eq-52f8a41000", "hardy-course-of-pure-mathematics-1921/eq-35e3ffbda3", "hardy-course-of-pure-mathematics-1921/eq-494df6e75d", "hardy-course-of-pure-mathematics-1921/eq-bfb7a4880b", "hardy-course-of-pure-mathematics-1921/eq-1399ff3efb", "hardy-course-of-pure-mathematics-1921/eq-63af1337f7", "hardy-course-of-pure-mathematics-1921/eq-342b9d1453", "hardy-course-of-pure-mathematics-1921/eq-7cd0649a2f", "hardy-course-of-pure-mathematics-1921/eq-455b947f75", "hardy-course-of-pure-mathematics-1921/eq-ec8d98bc9c", "hardy-course-of-pure-mathematics-1921/eq-991d059070", "hardy-course-of-pure-mathematics-1921/eq-26b7f21b71", "hardy-course-of-pure-mathematics-1921/eq-9e9f1cbff1", "hardy-course-of-pure-mathematics-1921/eq-9a8d7025df", "hardy-course-of-pure-mathematics-1921/eq-d4d082d272", "hardy-course-of-pure-mathematics-1921/eq-0243a2624c", "hardy-course-of-pure-mathematics-1921/eq-729e6f7e9a", "hardy-course-of-pure-mathematics-1921/eq-e969d9ad20", "hardy-course-of-pure-mathematics-1921/eq-7c5fd29dbb", "hardy-course-of-pure-mathematics-1921/eq-b6814eb243", "hardy-course-of-pure-mathematics-1921/eq-ee4e71bd85", "hardy-course-of-pure-mathematics-1921/eq-d0dbb97785", "hardy-course-of-pure-mathematics-1921/eq-e8695e73d5", "hardy-course-of-pure-mathematics-1921/eq-0a53fa3f89", "hardy-course-of-pure-mathematics-1921/eq-a8d1fa8567", "hardy-course-of-pure-mathematics-1921/eq-bec150828e", "hardy-course-of-pure-mathematics-1921/eq-0858e2a43c", "hardy-course-of-pure-mathematics-1921/eq-e62004f273", "hardy-course-of-pure-mathematics-1921/eq-23c0182ab2", "hardy-course-of-pure-mathematics-1921/eq-3636549117", "hardy-course-of-pure-mathematics-1921/eq-fa326fef07", "hardy-course-of-pure-mathematics-1921/eq-62f12eea15", "hardy-course-of-pure-mathematics-1921/eq-0f2c31532a", "hardy-course-of-pure-mathematics-1921/eq-70b6709b57", "hardy-course-of-pure-mathematics-1921/eq-4dd0f77bfa", "hardy-course-of-pure-mathematics-1921/eq-c75ad4f187", "hardy-course-of-pure-mathematics-1921/eq-65dbf7f428", "hardy-course-of-pure-mathematics-1921/eq-58f087dd73", "hardy-course-of-pure-mathematics-1921/eq-3fae3c1919", "hardy-course-of-pure-mathematics-1921/eq-22117834f2", "hardy-course-of-pure-mathematics-1921/eq-c8a9afde1e", "hardy-course-of-pure-mathematics-1921/eq-00a79b78fd", "hardy-course-of-pure-mathematics-1921/eq-2541e4e97b", "hardy-course-of-pure-mathematics-1921/eq-60dd7b991c", "hardy-course-of-pure-mathematics-1921/eq-4c279a68d5", "hardy-course-of-pure-mathematics-1921/eq-dda9f21a94", "hardy-course-of-pure-mathematics-1921/eq-832f7b3485", "hardy-course-of-pure-mathematics-1921/eq-a5accd5bef", "hardy-course-of-pure-mathematics-1921/eq-715b9fd00e", "hardy-course-of-pure-mathematics-1921/eq-abab22f7ee", "hardy-course-of-pure-mathematics-1921/eq-1248d92da4", "hardy-course-of-pure-mathematics-1921/eq-15311498cb", "hardy-course-of-pure-mathematics-1921/eq-5cae4be440", "hardy-course-of-pure-mathematics-1921/eq-e6d249137d", "hardy-course-of-pure-mathematics-1921/eq-bb8f38cfcf", "hardy-course-of-pure-mathematics-1921/eq-42db0aa22a", "hardy-course-of-pure-mathematics-1921/eq-4de17d44dc", "hardy-course-of-pure-mathematics-1921/eq-3b87e537b5", "hardy-course-of-pure-mathematics-1921/eq-9b886f3a07", "hardy-course-of-pure-mathematics-1921/eq-101a259fe1", "hardy-course-of-pure-mathematics-1921/eq-a4b3499791", "hardy-course-of-pure-mathematics-1921/eq-56a9fb91cb", "hardy-course-of-pure-mathematics-1921/eq-c554128653", "hardy-course-of-pure-mathematics-1921/eq-8f0c9b4120", "hardy-course-of-pure-mathematics-1921/eq-04df36e3b1", "hardy-course-of-pure-mathematics-1921/eq-658918f35b", "hardy-course-of-pure-mathematics-1921/eq-8d28dffc10", "hardy-course-of-pure-mathematics-1921/eq-7be7687336", "hardy-course-of-pure-mathematics-1921/eq-686b64c6d4", "hardy-course-of-pure-mathematics-1921/eq-b285f386eb", "hardy-course-of-pure-mathematics-1921/eq-974dc8b535", "hardy-course-of-pure-mathematics-1921/eq-bfa55a5ab6" ], "exercise_sets": [ "hardy-course-of-pure-mathematics-1921/ex-lxiii", "hardy-course-of-pure-mathematics-1921/ex-lxiv", "hardy-course-of-pure-mathematics-1921/ex-lxv", "hardy-course-of-pure-mathematics-1921/ex-lxvi", "hardy-course-of-pure-mathematics-1921/ex-misc-vii", "hardy-course-of-pure-mathematics-1921/ex-lx", "hardy-course-of-pure-mathematics-1921/ex-lxi", "hardy-course-of-pure-mathematics-1921/ex-lxii", "hardy-course-of-pure-mathematics-1921/ex-lv", "hardy-course-of-pure-mathematics-1921/ex-lvi", "hardy-course-of-pure-mathematics-1921/ex-lvii", "hardy-course-of-pure-mathematics-1921/ex-lviii", "hardy-course-of-pure-mathematics-1921/ex-lix" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ch-viii", "number": "VIII", "title": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "name": "Hardy 1921, ch. VIII: THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "pages": [ "346", "357" ], "concepts": [ "concept/absolute-convergence", "concept/alternating-series", "concept/binomial-series", "concept/circle-of-convergence", "concept/comparison", "concept/complex-number", "concept/complex-variable", "concept/conditionally-convergent-series", "concept/convergent-series", "concept/converse-of-a-theorem", "concept/cube-root-of-unity", "concept/definite-integral", "concept/divergent-series", "concept/geometrical-progression", "concept/harmonic-series", "concept/infinite-integral", "concept/infinite-integral-of-the-second-kind", "concept/infinity", "concept/linear-difference-equation", "concept/logarithmic-series", "concept/oscillation", "concept/p-series", "concept/power-series", "concept/probability", "concept/product-of-series", "concept/radius-of-convergence", "concept/rational-function", "concept/rearrangement-of-a-series", "concept/recurring-series", "concept/repeated-limit", "concept/scale-of-relation", "concept/series-of-complex-terms", "concept/series-of-positive-and-negative-terms", "concept/series-of-positive-terms", "concept/sufficiently-large-values", "concept/sum-to-infinity", "concept/transformation", "concept/uniqueness", "method/partial-fractions", "method/substitution", "person/alfred-pringsheim", "person/augustin-louis-cauchy", "person/colin-maclaurin", "person/jean-le-rond-d-alembert", "person/niels-henrik-abel", "person/peter-gustav-lejeune-dirichlet", "theorem/abel-s-test", "theorem/abel-s-theorem", "theorem/absolute-convergence-of-a-power-series-inside-a-point-of-convergence", "theorem/binomial-theorem", "theorem/binomial-theorem-for-a-negative-integral-exponent", "theorem/cauchy-s-condensation-test", "theorem/cauchy-s-root-test", "theorem/cauchy-schwarz-inequality", "theorem/comparison-theorem", "theorem/d-alembert-s-ratio-test", "theorem/dirichlet-s-rearrangement-theorem", "theorem/dirichlet-s-test", "theorem/general-principle-of-convergence", "theorem/integral-test", "theorem/trichotomy-of-convergence-of-a-power-series" ], "excerpts": [ "hardy-course-of-pure-mathematics-1921/x-a73f81936c", "hardy-course-of-pure-mathematics-1921/x-7f3db1c9ae", "hardy-course-of-pure-mathematics-1921/x-9f027d6270", "hardy-course-of-pure-mathematics-1921/x-a37d1aa8c8", "hardy-course-of-pure-mathematics-1921/x-6694cdb76a", "hardy-course-of-pure-mathematics-1921/x-d2f08dc4eb", "hardy-course-of-pure-mathematics-1921/x-4b20c68d95", "hardy-course-of-pure-mathematics-1921/x-733fa055db", "hardy-course-of-pure-mathematics-1921/x-b7e96953b3", "hardy-course-of-pure-mathematics-1921/x-37c0e31abc", "hardy-course-of-pure-mathematics-1921/x-773ec13ca6", "hardy-course-of-pure-mathematics-1921/x-1f16498c12", "hardy-course-of-pure-mathematics-1921/x-b577dfc56f", "hardy-course-of-pure-mathematics-1921/x-617aa8f056", "hardy-course-of-pure-mathematics-1921/x-6854c5fbfa", "hardy-course-of-pure-mathematics-1921/x-3ac7798950", "hardy-course-of-pure-mathematics-1921/x-b6ddd3b2da", "hardy-course-of-pure-mathematics-1921/x-251ae42b73", "hardy-course-of-pure-mathematics-1921/x-f136d8c28e", "hardy-course-of-pure-mathematics-1921/x-5497f88e1a", "hardy-course-of-pure-mathematics-1921/x-99581912c1", "hardy-course-of-pure-mathematics-1921/x-ccea0ed1a2", "hardy-course-of-pure-mathematics-1921/x-703020f67a", "hardy-course-of-pure-mathematics-1921/x-a983cd4f5d", "hardy-course-of-pure-mathematics-1921/x-eeffed6e32", "hardy-course-of-pure-mathematics-1921/x-b0b56cb7d2", "hardy-course-of-pure-mathematics-1921/x-790e555930", "hardy-course-of-pure-mathematics-1921/x-476da3035d", "hardy-course-of-pure-mathematics-1921/x-fe469c998e", "hardy-course-of-pure-mathematics-1921/x-33b493c1ae", "hardy-course-of-pure-mathematics-1921/x-c0312ffff2", "hardy-course-of-pure-mathematics-1921/x-e3298f2207", "hardy-course-of-pure-mathematics-1921/x-5a2b92905c", "hardy-course-of-pure-mathematics-1921/x-4890533523", "hardy-course-of-pure-mathematics-1921/x-8d9a321f9d", "hardy-course-of-pure-mathematics-1921/x-fff2e494a7", "hardy-course-of-pure-mathematics-1921/x-eeacb5a5f0", "hardy-course-of-pure-mathematics-1921/x-7c262851b6", "hardy-course-of-pure-mathematics-1921/x-6df0ca4ae2", "hardy-course-of-pure-mathematics-1921/x-c1a22ef130", "hardy-course-of-pure-mathematics-1921/x-cf1ed062e5" ], "equations": [ "hardy-course-of-pure-mathematics-1921/eq-38834fd8b4", "hardy-course-of-pure-mathematics-1921/eq-902f750389", "hardy-course-of-pure-mathematics-1921/eq-e4d99a205a", "hardy-course-of-pure-mathematics-1921/eq-3e788ab1b7", "hardy-course-of-pure-mathematics-1921/eq-e1b471951c", "hardy-course-of-pure-mathematics-1921/eq-f8d4257289", "hardy-course-of-pure-mathematics-1921/eq-8827d24aca", "hardy-course-of-pure-mathematics-1921/eq-6b719c60a1", "hardy-course-of-pure-mathematics-1921/eq-7ef7861f04", "hardy-course-of-pure-mathematics-1921/eq-da7feb4b8f", "hardy-course-of-pure-mathematics-1921/eq-6ec870b328", "hardy-course-of-pure-mathematics-1921/eq-ca4138217a", "hardy-course-of-pure-mathematics-1921/eq-b7b07b154f", "hardy-course-of-pure-mathematics-1921/eq-9362cab691", "hardy-course-of-pure-mathematics-1921/eq-03cac8156c", "hardy-course-of-pure-mathematics-1921/eq-6ab53686eb", "hardy-course-of-pure-mathematics-1921/eq-29aa7fb076", "hardy-course-of-pure-mathematics-1921/eq-e24481e948", "hardy-course-of-pure-mathematics-1921/eq-c17ffdc2e6", "hardy-course-of-pure-mathematics-1921/eq-5b960b7ae6", "hardy-course-of-pure-mathematics-1921/eq-8b11aaf72a", "hardy-course-of-pure-mathematics-1921/eq-b0736d8105", "hardy-course-of-pure-mathematics-1921/eq-244c1cdb94", "hardy-course-of-pure-mathematics-1921/eq-9d9f45da8e", "hardy-course-of-pure-mathematics-1921/eq-ae97425246", "hardy-course-of-pure-mathematics-1921/eq-567e7850a0", "hardy-course-of-pure-mathematics-1921/eq-03dd00067e", "hardy-course-of-pure-mathematics-1921/eq-fe79c967e6", "hardy-course-of-pure-mathematics-1921/eq-c1f6796b61", "hardy-course-of-pure-mathematics-1921/eq-8a2990b4d2", "hardy-course-of-pure-mathematics-1921/eq-2d1e0b9082", "hardy-course-of-pure-mathematics-1921/eq-f0fbadf928", "hardy-course-of-pure-mathematics-1921/eq-3735dd186f", "hardy-course-of-pure-mathematics-1921/eq-cc20ee237d", "hardy-course-of-pure-mathematics-1921/eq-460e8debee", "hardy-course-of-pure-mathematics-1921/eq-c0d5acee59", "hardy-course-of-pure-mathematics-1921/eq-b52333897a", "hardy-course-of-pure-mathematics-1921/eq-60dc235b0d", "hardy-course-of-pure-mathematics-1921/eq-9cde6cbf00", "hardy-course-of-pure-mathematics-1921/eq-55c47f213c", "hardy-course-of-pure-mathematics-1921/eq-fef21e88d9", "hardy-course-of-pure-mathematics-1921/eq-357ca4d096", "hardy-course-of-pure-mathematics-1921/eq-54a817f896", "hardy-course-of-pure-mathematics-1921/eq-a4d67b61ac", "hardy-course-of-pure-mathematics-1921/eq-98ddd3ffc5", "hardy-course-of-pure-mathematics-1921/eq-a6868aa802", "hardy-course-of-pure-mathematics-1921/eq-3a1c3fab01", "hardy-course-of-pure-mathematics-1921/eq-76c13f95dc", "hardy-course-of-pure-mathematics-1921/eq-9f90ceb50b", "hardy-course-of-pure-mathematics-1921/eq-bdcc88f13c", "hardy-course-of-pure-mathematics-1921/eq-99f13dd4e4", "hardy-course-of-pure-mathematics-1921/eq-99c0850d64", "hardy-course-of-pure-mathematics-1921/eq-02be3723c4", "hardy-course-of-pure-mathematics-1921/eq-5b57b6a962", "hardy-course-of-pure-mathematics-1921/eq-0ab8704e5e", "hardy-course-of-pure-mathematics-1921/eq-4f3e49adf5", "hardy-course-of-pure-mathematics-1921/eq-ceb2c24679", "hardy-course-of-pure-mathematics-1921/eq-cfbb063153", "hardy-course-of-pure-mathematics-1921/eq-d7dec9985a", "hardy-course-of-pure-mathematics-1921/eq-425f5305f1", "hardy-course-of-pure-mathematics-1921/eq-8da8affd87", "hardy-course-of-pure-mathematics-1921/eq-0ea90775f0", "hardy-course-of-pure-mathematics-1921/eq-dd4ebb813b", "hardy-course-of-pure-mathematics-1921/eq-82b8dce2d0", "hardy-course-of-pure-mathematics-1921/eq-35b8ac461a", "hardy-course-of-pure-mathematics-1921/eq-b92057290c", "hardy-course-of-pure-mathematics-1921/eq-523398b605", "hardy-course-of-pure-mathematics-1921/eq-a6443df9f5", "hardy-course-of-pure-mathematics-1921/eq-1c0930c267", "hardy-course-of-pure-mathematics-1921/eq-25f7810ef4", "hardy-course-of-pure-mathematics-1921/eq-583d0c481d", "hardy-course-of-pure-mathematics-1921/eq-a56ee4ff4f", "hardy-course-of-pure-mathematics-1921/eq-a2093ee5ce", "hardy-course-of-pure-mathematics-1921/eq-d6d17a64c8", "hardy-course-of-pure-mathematics-1921/eq-c06eef33f4", "hardy-course-of-pure-mathematics-1921/eq-3f2fc7bc64", "hardy-course-of-pure-mathematics-1921/eq-ae58b7fbd0", "hardy-course-of-pure-mathematics-1921/eq-d9f7f836c4", "hardy-course-of-pure-mathematics-1921/eq-2ab1b94b1a", "hardy-course-of-pure-mathematics-1921/eq-6fc81f793d", "hardy-course-of-pure-mathematics-1921/eq-233bd69fb7", "hardy-course-of-pure-mathematics-1921/eq-12b1ed631e", "hardy-course-of-pure-mathematics-1921/eq-aa5bff62cf", "hardy-course-of-pure-mathematics-1921/eq-5b25e7fbcd", "hardy-course-of-pure-mathematics-1921/eq-c6213349b7", "hardy-course-of-pure-mathematics-1921/eq-b4aa3963e4", "hardy-course-of-pure-mathematics-1921/eq-2cb05e003f", "hardy-course-of-pure-mathematics-1921/eq-5948eabcd0", "hardy-course-of-pure-mathematics-1921/eq-21408658da", "hardy-course-of-pure-mathematics-1921/eq-0dda8fc2a1", "hardy-course-of-pure-mathematics-1921/eq-24514bc0e3", "hardy-course-of-pure-mathematics-1921/eq-20a8bed8b7", "hardy-course-of-pure-mathematics-1921/eq-4de0c56ec2", "hardy-course-of-pure-mathematics-1921/eq-3a4f152e7a", "hardy-course-of-pure-mathematics-1921/eq-57332ed8e5", "hardy-course-of-pure-mathematics-1921/eq-85d8dfda6c", "hardy-course-of-pure-mathematics-1921/eq-722c219ea4", "hardy-course-of-pure-mathematics-1921/eq-510ee9f380", "hardy-course-of-pure-mathematics-1921/eq-8e3f01e948", "hardy-course-of-pure-mathematics-1921/eq-7edee13785", "hardy-course-of-pure-mathematics-1921/eq-d6bd18ee9e", "hardy-course-of-pure-mathematics-1921/eq-820aca6aa2", "hardy-course-of-pure-mathematics-1921/eq-108ffc8534", "hardy-course-of-pure-mathematics-1921/eq-a723c7ae56", "hardy-course-of-pure-mathematics-1921/eq-3e3028563d", "hardy-course-of-pure-mathematics-1921/eq-b882dc77ab", "hardy-course-of-pure-mathematics-1921/eq-ae71d4c304", "hardy-course-of-pure-mathematics-1921/eq-ffb45f6b9d", "hardy-course-of-pure-mathematics-1921/eq-84fcf7c305", "hardy-course-of-pure-mathematics-1921/eq-5d4af092cf", "hardy-course-of-pure-mathematics-1921/eq-c23c1d994f", "hardy-course-of-pure-mathematics-1921/eq-505d1c71d3", "hardy-course-of-pure-mathematics-1921/eq-483582be78", "hardy-course-of-pure-mathematics-1921/eq-8c8213b98a", "hardy-course-of-pure-mathematics-1921/eq-b105edb288", "hardy-course-of-pure-mathematics-1921/eq-a30ed911b3", "hardy-course-of-pure-mathematics-1921/eq-b78588f364", "hardy-course-of-pure-mathematics-1921/eq-585cf34373", "hardy-course-of-pure-mathematics-1921/eq-df75c33db6", "hardy-course-of-pure-mathematics-1921/eq-8368cf46ea", "hardy-course-of-pure-mathematics-1921/eq-c261ad2262", "hardy-course-of-pure-mathematics-1921/eq-3838bad0f2", "hardy-course-of-pure-mathematics-1921/eq-37051b676b", "hardy-course-of-pure-mathematics-1921/eq-fde36f8348", "hardy-course-of-pure-mathematics-1921/eq-a0913a27fe", "hardy-course-of-pure-mathematics-1921/eq-be6cfd16f8", "hardy-course-of-pure-mathematics-1921/eq-526c9f4a28", "hardy-course-of-pure-mathematics-1921/eq-63dbe8d1d2", "hardy-course-of-pure-mathematics-1921/eq-ae8b9f040e", "hardy-course-of-pure-mathematics-1921/eq-5554e7def1", "hardy-course-of-pure-mathematics-1921/eq-bd2cdd6e4f", "hardy-course-of-pure-mathematics-1921/eq-47e45f8b14", "hardy-course-of-pure-mathematics-1921/eq-cf5734d245", "hardy-course-of-pure-mathematics-1921/eq-38ff9778a9", "hardy-course-of-pure-mathematics-1921/eq-975d6e2226" ], "exercise_sets": [ "hardy-course-of-pure-mathematics-1921/ex-lxxvii", "hardy-course-of-pure-mathematics-1921/ex-lxxviii", "hardy-course-of-pure-mathematics-1921/ex-lxxix", "hardy-course-of-pure-mathematics-1921/ex-lxxx", "hardy-course-of-pure-mathematics-1921/ex-lxxxi", "hardy-course-of-pure-mathematics-1921/ex-lxvii", "hardy-course-of-pure-mathematics-1921/ex-lxviii", "hardy-course-of-pure-mathematics-1921/ex-lxix", "hardy-course-of-pure-mathematics-1921/ex-lxx", "hardy-course-of-pure-mathematics-1921/ex-lxxi", "hardy-course-of-pure-mathematics-1921/ex-lxxii", "hardy-course-of-pure-mathematics-1921/ex-lxxiii", "hardy-course-of-pure-mathematics-1921/ex-lxxiv", "hardy-course-of-pure-mathematics-1921/ex-lxxv", "hardy-course-of-pure-mathematics-1921/ex-lxxvi", "hardy-course-of-pure-mathematics-1921/ex-misc-viii" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ch-ix", "number": "IX", "title": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "name": "Hardy 1921, ch. IX: THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "pages": [ "382", "395" ], "concepts": [ "concept/base-of-a-logarithm-system", "concept/binomial-series", "concept/cauchy-s-form-of-the-remainder", "concept/common-logarithm", "concept/continuous-function", "concept/convergent-series", "concept/cosine", "concept/divergent-series", "concept/e", "concept/euler-s-number", "concept/exponential-function", "concept/exponential-theorem", "concept/functional-equation", "concept/general-power", "concept/geometrical-progression", "concept/hyperbolic-function", "concept/integral", "concept/inverse-circular-function", "concept/inverse-function", "concept/inverse-hyperbolic-function", "concept/inverse-tangent", "concept/lagrange-s-form-of-the-remainder", "concept/laws-of-indices", "concept/logarithm", "concept/logarithmic-series", "concept/odd-function", "concept/order-of-greatness", "concept/point-of-inflexion", "concept/power-series", "concept/quadratic-surd", "concept/scale-of-infinity", "concept/series-of-positive-terms", "concept/sine", "method/approximation-of-surds-by-the-binomial-series", "method/integration-by-parts", "method/taylor-s-theorem", "person/augustin-louis-cauchy", "person/john-napier", "person/joseph-louis-lagrange", "quantity/euler-s-constant", "theorem/addition-formula-for-the-exponential-function", "theorem/binomial-series", "theorem/binomial-theorem", "theorem/change-of-base-of-logarithms", "theorem/derivative-of-the-exponential-function", "theorem/exponential-as-a-limit", "theorem/exponential-grows-faster-than-any-power", "theorem/exponential-series", "theorem/integral-test", "theorem/inverse-tangent-series", "theorem/irrationality-of-e", "theorem/limit-representation-of-the-exponential-function", "theorem/logarithm-grows-more-slowly-than-any-positive-power", "theorem/logarithmic-series", "theorem/logarithmic-test-of-convergence", "theorem/taylor-s-theorem" ], "excerpts": [ "hardy-course-of-pure-mathematics-1921/x-531ace9542", "hardy-course-of-pure-mathematics-1921/x-a0edd4c0b1", "hardy-course-of-pure-mathematics-1921/x-717bc8769d", "hardy-course-of-pure-mathematics-1921/x-4e24d5b7f0", "hardy-course-of-pure-mathematics-1921/x-251dda7189", "hardy-course-of-pure-mathematics-1921/x-3262d9cde1", "hardy-course-of-pure-mathematics-1921/x-77464815f3", "hardy-course-of-pure-mathematics-1921/x-5f1f1622da", "hardy-course-of-pure-mathematics-1921/x-d4bee5e031", "hardy-course-of-pure-mathematics-1921/x-1a8d2b1e8c", "hardy-course-of-pure-mathematics-1921/x-ef8cf3960a", "hardy-course-of-pure-mathematics-1921/x-f04fce79c2", "hardy-course-of-pure-mathematics-1921/x-d8cfa01763", "hardy-course-of-pure-mathematics-1921/x-7b382ad15e", "hardy-course-of-pure-mathematics-1921/x-42fdf3d995", "hardy-course-of-pure-mathematics-1921/x-ef2fad94d7", "hardy-course-of-pure-mathematics-1921/x-6454195043", "hardy-course-of-pure-mathematics-1921/x-6580ef02c8", "hardy-course-of-pure-mathematics-1921/x-ced4038a6b", "hardy-course-of-pure-mathematics-1921/x-f2c3026afd", "hardy-course-of-pure-mathematics-1921/x-95ca970ed4", "hardy-course-of-pure-mathematics-1921/x-17a97e9a2c", "hardy-course-of-pure-mathematics-1921/x-5d28bfd0c6", "hardy-course-of-pure-mathematics-1921/x-3c104fb01f", "hardy-course-of-pure-mathematics-1921/x-7978249b91", "hardy-course-of-pure-mathematics-1921/x-7a2d6df5ac", "hardy-course-of-pure-mathematics-1921/x-14b53d7fa6", "hardy-course-of-pure-mathematics-1921/x-9a9aa74939", "hardy-course-of-pure-mathematics-1921/x-149528ce78", "hardy-course-of-pure-mathematics-1921/x-706456fdde", "hardy-course-of-pure-mathematics-1921/x-4567fddcc1", "hardy-course-of-pure-mathematics-1921/x-5857bcebc6" ], "equations": [ "hardy-course-of-pure-mathematics-1921/eq-67071498ac", "hardy-course-of-pure-mathematics-1921/eq-8c323ef3cc", "hardy-course-of-pure-mathematics-1921/eq-c51a295a07", "hardy-course-of-pure-mathematics-1921/eq-4654bde320", "hardy-course-of-pure-mathematics-1921/eq-05db6ca01f", "hardy-course-of-pure-mathematics-1921/eq-9112fa9327", "hardy-course-of-pure-mathematics-1921/eq-b91c95345f", "hardy-course-of-pure-mathematics-1921/eq-c67d6cb998", "hardy-course-of-pure-mathematics-1921/eq-e10f0b5935", "hardy-course-of-pure-mathematics-1921/eq-ad054a0478", "hardy-course-of-pure-mathematics-1921/eq-0e119c5541", "hardy-course-of-pure-mathematics-1921/eq-31586c938c", "hardy-course-of-pure-mathematics-1921/eq-b0f6beedc7", "hardy-course-of-pure-mathematics-1921/eq-c3147220ec", "hardy-course-of-pure-mathematics-1921/eq-388281d235", "hardy-course-of-pure-mathematics-1921/eq-5151f6c424", "hardy-course-of-pure-mathematics-1921/eq-b2e7066e77", "hardy-course-of-pure-mathematics-1921/eq-9fbd544332", "hardy-course-of-pure-mathematics-1921/eq-3857592cf1", "hardy-course-of-pure-mathematics-1921/eq-67dd247b1c", "hardy-course-of-pure-mathematics-1921/eq-abe5024ef3", "hardy-course-of-pure-mathematics-1921/eq-0e9b0c3fe7", "hardy-course-of-pure-mathematics-1921/eq-85ccd5216b", "hardy-course-of-pure-mathematics-1921/eq-56690ae681", "hardy-course-of-pure-mathematics-1921/eq-c45d1d0bf4", "hardy-course-of-pure-mathematics-1921/eq-e90e79dd44", "hardy-course-of-pure-mathematics-1921/eq-00692a1075", "hardy-course-of-pure-mathematics-1921/eq-7099a1643a", "hardy-course-of-pure-mathematics-1921/eq-fb3be39d7c", "hardy-course-of-pure-mathematics-1921/eq-43c6bd7957", "hardy-course-of-pure-mathematics-1921/eq-0704c2859d", "hardy-course-of-pure-mathematics-1921/eq-9efb2cc510", "hardy-course-of-pure-mathematics-1921/eq-d8fd5b9f8b", "hardy-course-of-pure-mathematics-1921/eq-c40a30ec41", "hardy-course-of-pure-mathematics-1921/eq-07ce19e33a", "hardy-course-of-pure-mathematics-1921/eq-aa3e300157", "hardy-course-of-pure-mathematics-1921/eq-9dea1624e7", "hardy-course-of-pure-mathematics-1921/eq-906a72d767", "hardy-course-of-pure-mathematics-1921/eq-a1914f46a6", "hardy-course-of-pure-mathematics-1921/eq-73521a181a", "hardy-course-of-pure-mathematics-1921/eq-cee1bdde4d", "hardy-course-of-pure-mathematics-1921/eq-43c4ec854a", "hardy-course-of-pure-mathematics-1921/eq-708f0fa87e", "hardy-course-of-pure-mathematics-1921/eq-fe15121998", "hardy-course-of-pure-mathematics-1921/eq-e99d881ca5", "hardy-course-of-pure-mathematics-1921/eq-71fbae91db", "hardy-course-of-pure-mathematics-1921/eq-3afad81ab4", "hardy-course-of-pure-mathematics-1921/eq-1967672872", "hardy-course-of-pure-mathematics-1921/eq-1a7c4ebb69", "hardy-course-of-pure-mathematics-1921/eq-c25c2f3685", "hardy-course-of-pure-mathematics-1921/eq-d20dfde0cc", "hardy-course-of-pure-mathematics-1921/eq-ce934c2ef1", "hardy-course-of-pure-mathematics-1921/eq-b05cdda98b", "hardy-course-of-pure-mathematics-1921/eq-721efac463", "hardy-course-of-pure-mathematics-1921/eq-bcd0b0feed", "hardy-course-of-pure-mathematics-1921/eq-43831cc0cb", "hardy-course-of-pure-mathematics-1921/eq-fddcdc792f", "hardy-course-of-pure-mathematics-1921/eq-5d580c848f", "hardy-course-of-pure-mathematics-1921/eq-2f3d2f7113", "hardy-course-of-pure-mathematics-1921/eq-22379b1650", "hardy-course-of-pure-mathematics-1921/eq-dc3593a4cc", "hardy-course-of-pure-mathematics-1921/eq-95310d5613", "hardy-course-of-pure-mathematics-1921/eq-0269f67e56", "hardy-course-of-pure-mathematics-1921/eq-ccff19d66e", "hardy-course-of-pure-mathematics-1921/eq-94a9f9ead4", "hardy-course-of-pure-mathematics-1921/eq-e363a7251b", "hardy-course-of-pure-mathematics-1921/eq-63e9c23aa4", "hardy-course-of-pure-mathematics-1921/eq-25bdc4c354", "hardy-course-of-pure-mathematics-1921/eq-248d05baef", "hardy-course-of-pure-mathematics-1921/eq-9a497d48d9", "hardy-course-of-pure-mathematics-1921/eq-9929bdda9c", "hardy-course-of-pure-mathematics-1921/eq-2a1fccc1b3", "hardy-course-of-pure-mathematics-1921/eq-fab4a5d4e1", "hardy-course-of-pure-mathematics-1921/eq-bae6802363" ], "exercise_sets": [ "hardy-course-of-pure-mathematics-1921/ex-lxxxii", "hardy-course-of-pure-mathematics-1921/ex-lxxxvii", "hardy-course-of-pure-mathematics-1921/ex-lxxxviii", "hardy-course-of-pure-mathematics-1921/ex-lxxxix", "hardy-course-of-pure-mathematics-1921/ex-xc", "hardy-course-of-pure-mathematics-1921/ex-xci", "hardy-course-of-pure-mathematics-1921/ex-xcii", "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "hardy-course-of-pure-mathematics-1921/ex-lxxxiii", "hardy-course-of-pure-mathematics-1921/ex-lxxxiv", "hardy-course-of-pure-mathematics-1921/ex-lxxxv", "hardy-course-of-pure-mathematics-1921/ex-lxxxvi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ch-x", "number": "X", "title": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "name": "Hardy 1921, ch. X: THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "pages": [ "421", "433" ], "concepts": [ "concept/absolute-convergence", "concept/addition-formulae", "concept/algebraic-function", "concept/amplitude-of-a-complex-number", "concept/argand-diagram", "concept/argument-of-a-function", "concept/base-of-a-logarithm-system", "concept/binomial-series", "concept/circular-function", "concept/complete-equation", "concept/complex-number", "concept/complex-variable", "concept/conditionally-convergent-series", "concept/corollary", "concept/cosecant", "concept/cosine", "concept/cotangent", "concept/curvilinear-integral", "concept/definite-integral", "concept/discontinuous-function", "concept/equiangular-spiral", "concept/even-function", "concept/exponential-function", "concept/function", "concept/function-of-a-complex-variable", "concept/functional-equation", "concept/general-power", "concept/hyperbolic-function", "concept/inverse-circular-function", "concept/inverse-function", "concept/level-curve", "concept/logarithm", "concept/logarithm-to-any-base", "concept/logarithmic-series", "concept/many-valued-function", "concept/mercator-s-projection", "concept/modulus-of-a-complex-number", "concept/odd-function", "concept/one-valued-function", "concept/origin", "concept/path-of-integration", "concept/periodic-function", "concept/polar-coordinates", "concept/polynomial", "concept/power-series", "concept/principal-value-of-a-logarithm", "concept/principal-value-of-a-power", "concept/rational-function", "concept/secant", "concept/sine", "concept/solution", "concept/stereographic-projection", "concept/tangent-function", "concept/transformation", "concept/trigonometrical-identity", "concept/trigonometry", "method/differentiation", "method/equating-real-and-imaginary-parts", "theorem/binomial-theorem", "theorem/connection-between-the-logarithm-and-the-inverse-circular-functions", "theorem/de-moivre-s-theorem", "theorem/derivative-of-a-complex-power", "theorem/derivative-of-the-complex-exponential", "theorem/euler-s-formula", "theorem/exponential-function-is-one-valued", "theorem/exponential-limit", "theorem/exponential-series", "theorem/logarithmic-series", "theorem/power-series-for-the-exponential-function", "theorem/power-series-for-the-sine-and-cosine", "theorem/taylor-s-theorem-with-integral-remainder" ], "excerpts": [ "hardy-course-of-pure-mathematics-1921/x-489e9d5000", "hardy-course-of-pure-mathematics-1921/x-578c66120e", "hardy-course-of-pure-mathematics-1921/x-897d29f1de", "hardy-course-of-pure-mathematics-1921/x-9c84b8c1e7", "hardy-course-of-pure-mathematics-1921/x-4ac2970531", "hardy-course-of-pure-mathematics-1921/x-737e38d1c4", "hardy-course-of-pure-mathematics-1921/x-78e5a2beba", "hardy-course-of-pure-mathematics-1921/x-2dcbcf23ea", "hardy-course-of-pure-mathematics-1921/x-8954d0489f", "hardy-course-of-pure-mathematics-1921/x-89a956fdd3", "hardy-course-of-pure-mathematics-1921/x-ea7e29707b", "hardy-course-of-pure-mathematics-1921/x-2b2bbd30db", "hardy-course-of-pure-mathematics-1921/x-8c1e20e2f5", "hardy-course-of-pure-mathematics-1921/x-6c705a6dbf", "hardy-course-of-pure-mathematics-1921/x-e0bb7c160b", "hardy-course-of-pure-mathematics-1921/x-3c07ad2ec8", "hardy-course-of-pure-mathematics-1921/x-69505ed6e5", "hardy-course-of-pure-mathematics-1921/x-1d43932253", "hardy-course-of-pure-mathematics-1921/x-c2e0e106f5", "hardy-course-of-pure-mathematics-1921/x-c0d76eff2f", "hardy-course-of-pure-mathematics-1921/x-eac8dc28f5", "hardy-course-of-pure-mathematics-1921/x-d7e839ceb5", "hardy-course-of-pure-mathematics-1921/x-deb8f5be41", "hardy-course-of-pure-mathematics-1921/x-7d942f780e", "hardy-course-of-pure-mathematics-1921/x-cecab47661", "hardy-course-of-pure-mathematics-1921/x-668b828cc5", "hardy-course-of-pure-mathematics-1921/x-05f8ce615e", "hardy-course-of-pure-mathematics-1921/x-7b75252960", "hardy-course-of-pure-mathematics-1921/x-3b380435a6", "hardy-course-of-pure-mathematics-1921/x-29370dd380", "hardy-course-of-pure-mathematics-1921/x-4ddb5faaff", "hardy-course-of-pure-mathematics-1921/x-6dbc417b07", "hardy-course-of-pure-mathematics-1921/x-808132e548", "hardy-course-of-pure-mathematics-1921/x-ee4a968cfc" ], "equations": [ "hardy-course-of-pure-mathematics-1921/eq-7412c75fbf", "hardy-course-of-pure-mathematics-1921/eq-a14adef8cc", "hardy-course-of-pure-mathematics-1921/eq-93d92edb58", "hardy-course-of-pure-mathematics-1921/eq-8c323ef3cc", "hardy-course-of-pure-mathematics-1921/eq-8324d8969c", "hardy-course-of-pure-mathematics-1921/eq-1b9860d31e", "hardy-course-of-pure-mathematics-1921/eq-d8c22cdfb0", "hardy-course-of-pure-mathematics-1921/eq-8bd3c1fa57", "hardy-course-of-pure-mathematics-1921/eq-3b3bf72f35", "hardy-course-of-pure-mathematics-1921/eq-164f5639da", "hardy-course-of-pure-mathematics-1921/eq-96cede1da4", "hardy-course-of-pure-mathematics-1921/eq-0db684eb56", "hardy-course-of-pure-mathematics-1921/eq-8e827e52a7", "hardy-course-of-pure-mathematics-1921/eq-c9403a896c", "hardy-course-of-pure-mathematics-1921/eq-808f2a413b", "hardy-course-of-pure-mathematics-1921/eq-4a15684938", "hardy-course-of-pure-mathematics-1921/eq-e9b0e2db1d", "hardy-course-of-pure-mathematics-1921/eq-4e4c372aed", "hardy-course-of-pure-mathematics-1921/eq-b725935cc4", "hardy-course-of-pure-mathematics-1921/eq-ee592e04bf", "hardy-course-of-pure-mathematics-1921/eq-d1b0d3cc98", "hardy-course-of-pure-mathematics-1921/eq-d853ad2904", "hardy-course-of-pure-mathematics-1921/eq-66e4f1cbc8", "hardy-course-of-pure-mathematics-1921/eq-8a98e6d22d", "hardy-course-of-pure-mathematics-1921/eq-6d645de494", "hardy-course-of-pure-mathematics-1921/eq-d29a40bc99", "hardy-course-of-pure-mathematics-1921/eq-912bd9770e", "hardy-course-of-pure-mathematics-1921/eq-fb756c9603", "hardy-course-of-pure-mathematics-1921/eq-1c6f8cf66f", "hardy-course-of-pure-mathematics-1921/eq-c3b7035d32", "hardy-course-of-pure-mathematics-1921/eq-c478b4d4c3", "hardy-course-of-pure-mathematics-1921/eq-2a9420c7ae", "hardy-course-of-pure-mathematics-1921/eq-993e5f0a44", "hardy-course-of-pure-mathematics-1921/eq-b6fa0df6d8", "hardy-course-of-pure-mathematics-1921/eq-9021b427be", "hardy-course-of-pure-mathematics-1921/eq-b22a4f7b65", "hardy-course-of-pure-mathematics-1921/eq-f1650ef6ad", "hardy-course-of-pure-mathematics-1921/eq-4265396288", "hardy-course-of-pure-mathematics-1921/eq-2d612aa7bd", "hardy-course-of-pure-mathematics-1921/eq-eba14f6f25", "hardy-course-of-pure-mathematics-1921/eq-8f5d09ead1", "hardy-course-of-pure-mathematics-1921/eq-399708c107", "hardy-course-of-pure-mathematics-1921/eq-de6211a0a0", "hardy-course-of-pure-mathematics-1921/eq-2dbdc7923d", "hardy-course-of-pure-mathematics-1921/eq-57fbf9c8a9", "hardy-course-of-pure-mathematics-1921/eq-dd4c6df1c5", "hardy-course-of-pure-mathematics-1921/eq-c91df48771", "hardy-course-of-pure-mathematics-1921/eq-f782912cfa", "hardy-course-of-pure-mathematics-1921/eq-c7b8d250ed", "hardy-course-of-pure-mathematics-1921/eq-0c0df7759e", "hardy-course-of-pure-mathematics-1921/eq-3b352f7d86", "hardy-course-of-pure-mathematics-1921/eq-5c2d2a2d7d", "hardy-course-of-pure-mathematics-1921/eq-a12ab9760f", "hardy-course-of-pure-mathematics-1921/eq-a94084b540", "hardy-course-of-pure-mathematics-1921/eq-93d72b1554", "hardy-course-of-pure-mathematics-1921/eq-06ee5ba57a", "hardy-course-of-pure-mathematics-1921/eq-2da0fde5dd", "hardy-course-of-pure-mathematics-1921/eq-e058ba38ee", "hardy-course-of-pure-mathematics-1921/eq-4bc326b487", "hardy-course-of-pure-mathematics-1921/eq-82757cef79", "hardy-course-of-pure-mathematics-1921/eq-426d320adc", "hardy-course-of-pure-mathematics-1921/eq-34403013fa", "hardy-course-of-pure-mathematics-1921/eq-9ffde08d95", "hardy-course-of-pure-mathematics-1921/eq-4da4219942", "hardy-course-of-pure-mathematics-1921/eq-75f34e8be4", "hardy-course-of-pure-mathematics-1921/eq-7e3caa33a7", "hardy-course-of-pure-mathematics-1921/eq-becab479c4", "hardy-course-of-pure-mathematics-1921/eq-4b6ee51cca", "hardy-course-of-pure-mathematics-1921/eq-31beedaa72", "hardy-course-of-pure-mathematics-1921/eq-c585cceb31", "hardy-course-of-pure-mathematics-1921/eq-b2270fa7eb", "hardy-course-of-pure-mathematics-1921/eq-e9359426a0", "hardy-course-of-pure-mathematics-1921/eq-2625fde525", "hardy-course-of-pure-mathematics-1921/eq-56c058468e", "hardy-course-of-pure-mathematics-1921/eq-0918f60e32", "hardy-course-of-pure-mathematics-1921/eq-d2ced67269", "hardy-course-of-pure-mathematics-1921/eq-c5aaff6190", "hardy-course-of-pure-mathematics-1921/eq-7a0455f910", "hardy-course-of-pure-mathematics-1921/eq-8e25ded8b4", "hardy-course-of-pure-mathematics-1921/eq-d4b87310a5", "hardy-course-of-pure-mathematics-1921/eq-73b6dda4b1", "hardy-course-of-pure-mathematics-1921/eq-412cfdb012", "hardy-course-of-pure-mathematics-1921/eq-8954d0489f" ], "exercise_sets": [ "hardy-course-of-pure-mathematics-1921/ex-xcv", "hardy-course-of-pure-mathematics-1921/ex-xcvi", "hardy-course-of-pure-mathematics-1921/ex-xcvii", "hardy-course-of-pure-mathematics-1921/ex-xcviii", "hardy-course-of-pure-mathematics-1921/ex-misc-x", "hardy-course-of-pure-mathematics-1921/ex-xciii", "hardy-course-of-pure-mathematics-1921/ex-xciv" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ch-appendix-i", "number": "Appendix I", "title": "The Proof that every Equation has a Root", "name": "Hardy 1921, ch. Appendix I: The Proof that every Equation has a Root", "pages": [ "433", "439" ], "concepts": [ "concept/absolute-value", "concept/amplitude-of-a-complex-number", "concept/closed-contour", "concept/complex-number", "concept/continuous-function", "concept/ellipse", "concept/focus", "concept/increment", "concept/infinite-sequence", "concept/least-upper-bound", "concept/limit", "concept/logarithm", "concept/modulus-of-a-complex-number", "concept/multiple-root", "concept/one-valued-function", "concept/origin", "concept/polynomial", "concept/positive-direction", "concept/root-of-an-equation", "method/nested-squares-argument", "theorem/argument-principle", "theorem/fundamental-theorem-of-algebra" ], "excerpts": [ "hardy-course-of-pure-mathematics-1921/x-c8e47abfa6", "hardy-course-of-pure-mathematics-1921/x-9334469a23", "hardy-course-of-pure-mathematics-1921/x-788e292a5c", "hardy-course-of-pure-mathematics-1921/x-5282f70798", "hardy-course-of-pure-mathematics-1921/x-63c7d8f2ab", "hardy-course-of-pure-mathematics-1921/x-0e7fdeaf7b", "hardy-course-of-pure-mathematics-1921/x-2d11f38873", "hardy-course-of-pure-mathematics-1921/x-b717e344b8", "hardy-course-of-pure-mathematics-1921/x-f97d852043", "hardy-course-of-pure-mathematics-1921/x-8bcd76fb5c", "hardy-course-of-pure-mathematics-1921/x-532ef63f0f" ], "equations": [ "hardy-course-of-pure-mathematics-1921/eq-ec86ce3556", "hardy-course-of-pure-mathematics-1921/eq-ac0e59712b", "hardy-course-of-pure-mathematics-1921/eq-361981b4fc", "hardy-course-of-pure-mathematics-1921/eq-42539b078a", "hardy-course-of-pure-mathematics-1921/eq-2e9a31a790", "hardy-course-of-pure-mathematics-1921/eq-94cd082b71", "hardy-course-of-pure-mathematics-1921/eq-c0bd5e2e79", "hardy-course-of-pure-mathematics-1921/eq-8696acd0cb", "hardy-course-of-pure-mathematics-1921/eq-7a49549e21", "hardy-course-of-pure-mathematics-1921/eq-2fe6ce0428", "hardy-course-of-pure-mathematics-1921/eq-5d7a79bef0", "hardy-course-of-pure-mathematics-1921/eq-ec63881b2a", "hardy-course-of-pure-mathematics-1921/eq-d269f8b6ff", "hardy-course-of-pure-mathematics-1921/eq-6354c2b223", "hardy-course-of-pure-mathematics-1921/eq-e8ec2a6e1c", "hardy-course-of-pure-mathematics-1921/eq-310b9aa22d", "hardy-course-of-pure-mathematics-1921/eq-e4579af174", "hardy-course-of-pure-mathematics-1921/eq-a78410e544", "hardy-course-of-pure-mathematics-1921/eq-59c507e61e", "hardy-course-of-pure-mathematics-1921/eq-5f1a29f57f" ], "exercise_sets": [ "hardy-course-of-pure-mathematics-1921/ex-app-i" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ch-appendix-ii", "number": "Appendix II", "title": "A Note on Double Limit Problems", "name": "Hardy 1921, ch. Appendix II: A Note on Double Limit Problems", "pages": [ "439", "443" ], "concepts": [ "concept/commutativity-of-operations", "concept/continuous-function", "concept/definite-integral", "concept/double-limit-problem", "concept/exponential-function", "concept/finite-set", "concept/infinite-sequence", "concept/limit", "concept/limit-operation", "concept/sum", "concept/term", "method/differentiation", "method/integration", "method/multiplication" ], "excerpts": [ "hardy-course-of-pure-mathematics-1921/x-d784c60bec", "hardy-course-of-pure-mathematics-1921/x-759b71e8ce", "hardy-course-of-pure-mathematics-1921/x-dbadc785e2", "hardy-course-of-pure-mathematics-1921/x-ca7d559d41", "hardy-course-of-pure-mathematics-1921/x-30bc3e54a1", "hardy-course-of-pure-mathematics-1921/x-903e114164", "hardy-course-of-pure-mathematics-1921/x-69da6b1285", "hardy-course-of-pure-mathematics-1921/x-e1eced374f" ], "equations": [ "hardy-course-of-pure-mathematics-1921/eq-e3813bf344", "hardy-course-of-pure-mathematics-1921/eq-9552626931", "hardy-course-of-pure-mathematics-1921/eq-a840f7a8bb", "hardy-course-of-pure-mathematics-1921/eq-4f42c1d0ed", "hardy-course-of-pure-mathematics-1921/eq-e363a7251b", "hardy-course-of-pure-mathematics-1921/eq-7cb5f84a9a", "hardy-course-of-pure-mathematics-1921/eq-bb81891aa4", "hardy-course-of-pure-mathematics-1921/eq-975d2fe3d7", "hardy-course-of-pure-mathematics-1921/eq-e1867ae145", "hardy-course-of-pure-mathematics-1921/eq-673073db4a", "hardy-course-of-pure-mathematics-1921/eq-8f955076f9", "hardy-course-of-pure-mathematics-1921/eq-7ac4f47ec0", "hardy-course-of-pure-mathematics-1921/eq-ed12986fe8", "hardy-course-of-pure-mathematics-1921/eq-5612e88b46", "hardy-course-of-pure-mathematics-1921/eq-b99bf33261", "hardy-course-of-pure-mathematics-1921/eq-5ea86a1826" ], "exercise_sets": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ch-appendix-iii", "number": "Appendix III", "title": "The circular functions", "name": "Hardy 1921, ch. Appendix III: The circular functions", "pages": [ "443", "445" ], "concepts": [ "concept/addition-formulae", "concept/circular-function", "concept/continuous-function", "concept/cosine", "concept/domain-of-definition", "concept/function", "concept/infinite-sequence", "concept/interval", "concept/inverse-circular-function", "concept/inverse-function", "concept/pi", "concept/power-series", "concept/sine", "concept/tangent-function", "method/differentiation", "method/substitution", "method/taylor-s-theorem", "theorem/taylor-s-theorem" ], "excerpts": [ "hardy-course-of-pure-mathematics-1921/x-e6cb28bb4c", "hardy-course-of-pure-mathematics-1921/x-01a7ce3cf5", "hardy-course-of-pure-mathematics-1921/x-23d32c17a8", "hardy-course-of-pure-mathematics-1921/x-92b80bed30", "hardy-course-of-pure-mathematics-1921/x-d3cf3c2a39", "hardy-course-of-pure-mathematics-1921/x-fd0ebe15d6", "hardy-course-of-pure-mathematics-1921/x-7d4fe57c8e" ], "equations": [ "hardy-course-of-pure-mathematics-1921/eq-628a00c9db", "hardy-course-of-pure-mathematics-1921/eq-43716da505", "hardy-course-of-pure-mathematics-1921/eq-b02e3eacc3", "hardy-course-of-pure-mathematics-1921/eq-40ff3920ff", "hardy-course-of-pure-mathematics-1921/eq-ccb9ff2d51", "hardy-course-of-pure-mathematics-1921/eq-8d95712e4a", "hardy-course-of-pure-mathematics-1921/eq-d50680d562", "hardy-course-of-pure-mathematics-1921/eq-6979a154b1", "hardy-course-of-pure-mathematics-1921/eq-f9d9ba64c3", "hardy-course-of-pure-mathematics-1921/eq-cbbc63a009", "hardy-course-of-pure-mathematics-1921/eq-2f20c83664" ], "exercise_sets": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ch-appendix-iv", "number": "Appendix IV", "title": "The infinite in analysis and geometry", "name": "Hardy 1921, ch. Appendix IV: The infinite in analysis and geometry", "pages": [ "445", "445" ], "concepts": [ "concept/actual-infinity", "concept/common-cartesian-geometry", "concept/coordinate-geometry", "concept/correlation-between-homogeneous-and-common-geometry", "concept/corresponding-point", "concept/homogeneous-geometry", "concept/incomplete-symbol", "concept/infinity", "concept/limit", "concept/limiting-infinity", "concept/line", "concept/line-at-infinity", "concept/linear-relation", "concept/origin", "concept/point", "concept/real-homogeneous-cartesian-geometry", "concept/special-element", "concept/tends-to-infinity", "concept/triad" ], "excerpts": [ "hardy-course-of-pure-mathematics-1921/x-a5e9d2789d", "hardy-course-of-pure-mathematics-1921/x-58671b2085", "hardy-course-of-pure-mathematics-1921/x-c4f80120ea", "hardy-course-of-pure-mathematics-1921/x-2124d88208", "hardy-course-of-pure-mathematics-1921/x-499174b4ed", "hardy-course-of-pure-mathematics-1921/x-1256770113", "hardy-course-of-pure-mathematics-1921/x-6ec8e6afa1", "hardy-course-of-pure-mathematics-1921/x-f237829a1c", "hardy-course-of-pure-mathematics-1921/x-7b6baaf42b", "hardy-course-of-pure-mathematics-1921/x-31127faa34", "hardy-course-of-pure-mathematics-1921/x-70b2aee08d" ], "equations": [ "hardy-course-of-pure-mathematics-1921/eq-3e84a1d0ec", "hardy-course-of-pure-mathematics-1921/eq-47e4ba5c86", "hardy-course-of-pure-mathematics-1921/eq-ea427caa6a" ], "exercise_sets": [] } ], "excerpts": [ { "id": "hardy-course-of-pure-mathematics-1921/x-7a9f74c9b4", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "2", "location": "REAL VARIABLES", "latex": "The use of geometrical illustrations in this way does not, of course, imply that analysis has any sort of dependence upon geometry: they are illustrations and nothing more, and are employed merely for the sake of clearness of exposition.", "markdown": "The use of geometrical illustrations in this way does not, of course, imply that analysis has any sort of dependence upon geometry: they are illustrations and nothing more, and are employed merely for the sake of clearness of exposition.", "why": "Tells a learner that diagrams here are aids to understanding, not the logical foundation of the argument.", "use": [ "lesson", "website" ], "concepts": [ "concept/geometry", "concept/rational-point" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-1ba6c28b22", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "3", "location": "REAL VARIABLES", "latex": "The assertion `there are an infinity of positive integers' means `given any positive integer~$n$, however large, we can find more than~$n$ positive integers'. This is plainly true whatever $n$~may be, \\eg\\ for $n = 100,000$ or $100,000,000$.", "markdown": "The assertion ‘there are an infinity of positive integers’ means ‘given any positive integer $n$, however large, we can find more than $n$ positive integers’. This is plainly true whatever $n$ may be, *e.g.* for $n = 100,000$ or $100,000,000$.", "why": "Gives a concrete, non-mysterious meaning to 'infinity' as 'more than any n you name'.", "use": [ "lesson", "website" ], "concepts": [ "concept/infinity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-932601a847", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "4", "location": "REAL VARIABLES", "latex": "From these considerations the reader might be tempted to infer that an adequate view of the nature of the line could be obtained by imagining it to be formed simply by the rational points which lie on it.", "markdown": "From these considerations the reader might be tempted to infer that an adequate view of the nature of the line could be obtained by imagining it to be formed simply by the rational points which lie on it.", "why": "Names the tempting but mistaken idea that the rationals alone fill the line, setting up the need for new numbers.", "use": [ "lesson" ], "concepts": [ "concept/irrational-number", "concept/rational-point" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-3066622796", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "6", "location": "REAL VARIABLES", "latex": "Thus $m = p^{2}$, $n = q^{2}$, as was to be proved. In particular it follows, by taking $n = 1$, that an integer cannot be the square of a rational number, unless that rational number is itself integral.", "markdown": "Thus $m = p^{2}$, $n = q^{2}$, as was to be proved. In particular it follows, by taking $n = 1$, that an integer cannot be the square of a rational number, unless that rational number is itself integral.", "why": "Shows the payoff of the lowest-terms argument: an integer's square root is either an integer or not rational.", "use": [ "lesson" ], "concepts": [ "concept/perfect-square", "concept/rational-number", "theorem/irrationality-of-square-root-of-2" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-6170dfabb8", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "14", "location": "REAL VARIABLES", "latex": "A section of the rational numbers, in which both classes exist and the lower class has no greatest member, is called a \\Emph{real number}, or simply a \\Emph{number}.", "markdown": "A section of the rational numbers, in which both classes exist and the lower class has no greatest member, is called a **number**, or simply a ****.", "why": "States the book's definition of a real number as a section of the rationals.", "use": [ "lesson", "website" ], "concepts": [ "concept/dedekind-section", "concept/real-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-ecaf3dff35", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "14", "location": "REAL VARIABLES", "latex": "What is essential in mathematics is that its symbols should be capable of \\emph{some} interpretation; generally they are capable of \\emph{many}, and then, so far as mathematics is concerned, it does not matter which we adopt.", "markdown": "What is essential in mathematics is that its symbols should be capable of *some* interpretation; generally they are capable of *many*, and then, so far as mathematics is concerned, it does not matter which we adopt.", "why": "Reassures a learner that the particular construction of real numbers is a choice, not the one true meaning.", "use": [ "lesson", "website" ], "concepts": [ "concept/mathematics", "concept/real-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-2e2a047e53", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "15", "location": "REAL VARIABLES", "latex": "Moreover, for a beginner, the chief difficulty in the elements of analysis is that of learning to attach precise senses to phrases containing the word `infinity'; and experience seems to show that he is likely to be confused by any addition to their number.", "markdown": "Moreover, for a beginner, the chief difficulty in the elements of analysis is that of learning to attach precise senses to phrases containing the word ‘infinity’; and experience seems to show that he is likely to be confused by any addition to their number.", "why": "Warns about a common source of confusion and explains why infinity is not added as a number.", "use": [ "lesson" ], "concepts": [ "concept/infinity", "concept/real-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-7d985d805e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "29", "location": "REAL VARIABLES", "latex": "A system of real numbers, or of the points on a straight line corresponding to them, defined in any way whatever, is called an \\Emph{aggregate} or \\Emph{set} of numbers or points.", "markdown": "A system of real numbers, or of the points on a straight line corresponding to them, defined in any way whatever, is called an **** or **** of numbers or points.", "why": "Gives the plain definition of a set of numbers or points, which the rest of the chapter builds on.", "use": [ "lesson", "website" ], "concepts": [ "concept/set" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-7e69424422", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "30", "location": "REAL VARIABLES", "latex": "Let us suppose that (ii)~is true. Then any interval $\\DPmod{(\\xi - \\delta, \\xi + \\delta)}{[\\xi - \\delta, \\xi + \\delta]}$, however small its length, contains at least one point~$\\xi_{1}$ which belongs to~$S$ and does not coincide with~$\\xi$; and this whether $\\xi$~itself be a member of~$S$ or not. In this case we shall say that $\\xi$~is a \\Emph{point of accumulation} of~$S$.", "markdown": "Let us suppose that (ii) is true. Then any interval $\\DPmod{(\\xi - \\delta, \\xi + \\delta)}{[\\xi - \\delta, \\xi + \\delta]}$, however small its length, contains at least one point $\\xi_{1}$ which belongs to $S$ and does not coincide with $\\xi$; and this whether $\\xi$ itself be a member of $S$ or not. In this case we shall say that $\\xi$ is a **of accumulation** of $S$.", "why": "Defines a point of accumulation through the idea of an interval that is crowded however small it is made.", "use": [ "lesson", "website" ], "concepts": [ "concept/point-of-accumulation", "concept/set" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-de0f061426", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "29", "location": "REAL VARIABLES", "latex": "Suppose, for example, that $S$~consists of the points corresponding to all the positive integers. If $\\xi$~is itself a positive integer, we can take $\\delta$ to be any number less than~$1$, and (i)~will be true; or, if $\\xi$~is halfway between two positive integers, we can take $\\delta$ to be any number less than~$\\frac{1}{2}$. On the other hand, if $S$~consists of all the rational points, then, whatever the value of~$\\xi$, (ii)~is true; for any interval whatever contains an infinity of rational points.", "markdown": "Suppose, for example, that $S$ consists of the points corresponding to all the positive integers. If $\\xi$ is itself a positive integer, we can take $\\delta$ to be any number less than $1$, and (i) will be true; or, if $\\xi$ is halfway between two positive integers, we can take $\\delta$ to be any number less than $\\frac{1}{2}$. On the other hand, if $S$ consists of all the rational points, then, whatever the value of $\\xi$, (ii) is true; for any interval whatever contains an infinity of rational points.", "why": "Contrasts a set with no accumulation points (the integers) with one where every point is one (the rationals).", "use": [ "lesson" ], "concepts": [ "concept/point-of-accumulation", "concept/set" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-c5206f415a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "31", "location": "REAL VARIABLES", "latex": "This point may of course coincide with $\\alpha$~or~$\\beta$, as for instance when $\\alpha = 0$, $\\beta = 1$, and $S$~consists of the points $1$, $\\frac{1}{2}$, $\\frac{1}{3}, \\dots$. In this case $0$~is the sole point of accumulation.", "markdown": "This point may of course coincide with $\\alpha$ or $\\beta$, as for instance when $\\alpha = 0$, $\\beta = 1$, and $S$ consists of the points $1$, $\\frac{1}{2}$, $\\frac{1}{3}, \\dots$. In this case $0$ is the sole point of accumulation.", "why": "A concrete case showing that the accumulation point promised by Weierstrass's theorem can sit at the edge of the interval.", "use": [ "lesson" ], "concepts": [ "concept/point-of-accumulation", "theorem/weierstrass-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-e4948bad0e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "30", "location": "REAL VARIABLES", "latex": "The general theory of sets of points is of the utmost interest and importance in the higher branches of analysis; but it is for the most part too difficult to be included in a book such as this.", "markdown": "The general theory of sets of points is of the utmost interest and importance in the higher branches of analysis; but it is for the most part too difficult to be included in a book such as this.", "why": "Shows how the author limits the scope of the course and signals that this one theorem is chosen because it is needed later.", "use": [ "history", "website" ], "concepts": [ "concept/set", "theorem/weierstrass-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-cbd8bf0652", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "32", "location": "REAL VARIABLES", "latex": "It is clear that we may repeat this argument until we have replaced each of $a_{1}$, $a_{2}$, \\dots,~$a_{n}$ by~$G$; at most $n$~repetitions will be necessary. As the final value of the arithmetic mean is~$G$, the initial value cannot have been less.", "markdown": "It is clear that we may repeat this argument until we have replaced each of $a_{1}$, $a_{2}$, …, $a_{n}$ by $G$; at most $n$ repetitions will be necessary. As the final value of the arithmetic mean is $G$, the initial value cannot have been less.", "why": "Completes a short proof that the arithmetic mean is at least the geometric mean, by changing the numbers without changing G.", "use": [ "lesson" ], "concepts": [ "concept/arithmetical-mean", "concept/geometrical-mean", "theorem/inequality-of-arithmetic-and-geometric-means" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-ac3b21e0ca", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "32", "location": "REAL VARIABLES", "latex": "When $p = q = 1$ in~\\Eq{(1)}, or $p = 2$ in~\\Eq{(4)}, the inequalities are merely different forms of the inequality $a_{1}^{2} + a_{2}^{2} \\geq 2a_{1} a_{2}$, which expresses the fact that the arithmetic mean of two positive numbers is not less than their geometric mean.", "markdown": "When $p = q = 1$ in (1), or $p = 2$ in (4), the inequalities are merely different forms of the inequality $a_{1}^{2} + a_{2}^{2} \\geq 2a_{1} a_{2}$, which expresses the fact that the arithmetic mean of two positive numbers is not less than their geometric mean.", "why": "Links a family of inequalities back to the familiar two-number comparison of arithmetic and geometric means.", "use": [ "lesson" ], "concepts": [ "concept/arithmetical-mean", "concept/geometrical-mean", "theorem/arithmetic-geometric-mean-inequality", "theorem/inequality-of-arithmetic-and-geometric-means" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-0cf667c01f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "36", "location": "REAL VARIABLES", "latex": "Such irrational numbers are called \\emph{algebraical} numbers: all other irrational numbers, such as~$\\pi$ (\\SecNo[§]{15}), are called \\emph{transcendental} numbers.", "markdown": "Such irrational numbers are called *algebraical* numbers: all other irrational numbers, such as $\\pi$ ([§]15), are called *transcendental* numbers.", "why": "Splits the irrational numbers into algebraical and transcendental, and names pi as an example of the second kind.", "use": [ "lesson", "website" ], "concepts": [ "concept/algebraical-number", "concept/transcendental-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-08ee59a4bf", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "14", "location": "REAL VARIABLES", "latex": "Mr~Bertrand Russell has said that `mathematics is the science in which we do not know what we are talking about, and do not care whether what we say about it is true', a remark which is expressed in the form of a paradox but which in reality embodies a number of important truths.", "markdown": "Mr Bertrand Russell has said that ‘mathematics is the science in which we do not know what we are talking about, and do not care whether what we say about it is true’, a remark which is expressed in the form of a paradox but which in reality embodies a number of important truths.", "why": "A witty historical remark that Hardy uses to explain why the form of a definition matters less than its properties.", "use": [ "history", "website" ], "concepts": [ "concept/mathematics", "concept/real-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-8bf6c132c0", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "1", "location": "REAL VARIABLES", "latex": "A fraction $r = p/q$, where $p$~and~$q$ are positive or negative integers, is called a \\emph{rational number}.", "markdown": "A fraction $r = p/q$, where $p$ and $q$ are positive or negative integers, is called a *rational number*.", "why": "It states the definition a learner needs before any of the chapter's arguments make sense.", "use": [ "lesson", "website" ], "concepts": [ "concept/rational-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-7629812bb1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "5", "location": "REAL VARIABLES", "latex": "Now it is very easy to see that the idea of a straight line as composed of a series of points, each corresponding to a rational number, cannot possibly satisfy all these requirements.", "markdown": "Now it is very easy to see that the idea of a straight line as composed of a series of points, each corresponding to a rational number, cannot possibly satisfy all these requirements.", "why": "It shows why a line cannot be built from rational points alone, which motivates the whole chapter.", "use": [ "lesson" ], "concepts": [ "concept/number-line", "concept/rational-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-273bb2c3a3", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "6", "location": "REAL VARIABLES", "latex": "But it is easy to see that \\emph{there is no rational number such that its square is~$2$}.", "markdown": "But it is easy to see that *there is no rational number such that its square is $2$*.", "why": "It is the central claim behind irrational numbers, stated plainly enough for a learner to test.", "use": [ "lesson" ], "concepts": [ "concept/irrational-number", "concept/rational-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-d15649689d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "7", "location": "REAL VARIABLES", "latex": "We can therefore divide the rational numbers into two classes, one containing the numbers whose squares are less than~$2$, and the other those whose squares are greater than~$2$.", "markdown": "We can therefore divide the rational numbers into two classes, one containing the numbers whose squares are less than $2$, and the other those whose squares are greater than $2$.", "why": "It gives the idea of splitting the rationals into two classes, which is the key step toward defining a new number.", "use": [ "lesson" ], "concepts": [ "concept/dedekind-section", "concept/irrational-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-e347f6c98c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "4", "location": "REAL VARIABLES", "latex": "given any rational number~$r$, and any positive integer~$n$, we can find another rational number lying on either side of~$r$ and differing from~$r$ by less than~$1/n$.", "markdown": "given any rational number $r$, and any positive integer $n$, we can find another rational number lying on either side of $r$ and differing from $r$ by less than $1/n$.", "why": "It makes concrete how densely the rationals fill the line, which is why they seem enough at first.", "use": [ "lesson" ], "concepts": [ "concept/rational-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-15ec1ec725", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "12", "location": "REAL VARIABLES", "latex": "It should be observed that we do not obtain a section at all by taking $P$ to be `$x^{2} < 1$' and $Q$~to be `$x^{2} > 1$'; for the special number~$1$ escapes classification (cf.\\ \\Ex{iii}.~5).", "markdown": "It should be observed that we do not obtain a section at all by taking $P$ to be ‘$x^{2} < 1$’ and $Q$ to be ‘$x^{2} > 1$’; for the special number $1$ escapes classification (cf. % [examples:iii]Ex. iii%. 5).", "why": "It warns the learner that a naive split of the rationals can miss a number, a common mistake.", "use": [ "lesson" ], "concepts": [ "concept/dedekind-section" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-2e6a3c748c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "29", "location": "REAL VARIABLES", "latex": "then there is a number~$\\alpha$, which has the property that all the numbers less than it belong to~$L$ and all the numbers greater than it to~$R$.", "markdown": "then there is a number $\\alpha$, which has the property that all the numbers less than it belong to $L$ and all the numbers greater than it to $R$.", "why": "It states the completeness of the real numbers in one sentence: any cut of the reals is made at a single number.", "use": [ "lesson", "website" ], "concepts": [ "concept/dedekind-section", "theorem/dedekind-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-66c4fa7011", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "19", "location": "REAL VARIABLES", "latex": "A number of the form~$±\\sqrt{a}$, where $a$~is a positive rational number which is not the square of another rational number, is called a \\emph{pure quadratic surd}.", "markdown": "A number of the form $±\\sqrt{a}$, where $a$ is a positive rational number which is not the square of another rational number, is called a *pure quadratic surd*.", "why": "It gives the precise condition on a that makes √a a genuinely new irrational number, which a learner can check on examples.", "use": [ "lesson" ], "concepts": [ "concept/pure-quadratic-surd" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-2a66b15c85", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "21", "location": "REAL VARIABLES", "latex": "Two pure quadratic surds are said to be \\emph{similar} if they can be expressed as rational multiples of the same surd, and otherwise to be \\emph{dissimilar}.", "markdown": "Two pure quadratic surds are said to be *similar* if they can be expressed as rational multiples of the same surd, and otherwise to be *dissimilar*.", "why": "It gives a clear test for when two surds can be combined and compared, which is the basis for simplifying surd sums.", "use": [ "lesson" ], "concepts": [ "concept/pure-quadratic-surd", "concept/similar-surds" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-87db63d560", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "27", "location": "REAL VARIABLES", "latex": "It is important to observe that a pair of properties which suffice to define a section of the rational numbers may not suffice to define one of the real numbers. This is so, for example, with the pair `$x < \\sqrt{2}$' and `$x > \\sqrt{2}$' or (if we confine ourselves to positive numbers) with `$x^{2} < 2$' and `$x^{2} > 2$'.", "markdown": "It is important to observe that a pair of properties which suffice to define a section of the rational numbers may not suffice to define one of the real numbers. This is so, for example, with the pair ‘$x < \\sqrt{2}$’ and ‘$x > \\sqrt{2}$’ or (if we confine ourselves to positive numbers) with ‘$x^{2} < 2$’ and ‘$x^{2} > 2$’.", "why": "It shows why the two-class description must cover every real number, since √2 otherwise escapes classification.", "use": [ "lesson", "website" ], "concepts": [ "concept/dedekind-section", "concept/real-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-37159fb21f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "26", "location": "REAL VARIABLES", "latex": "In the theory of decimals, for instance, we may denote by~$x$ any figure in the expression of any number as a decimal. Then $x$~is a variable, but a variable which has only ten different values, viz.\\ $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$,~$9$.", "markdown": "In the theory of decimals, for instance, we may denote by $x$ any figure in the expression of any number as a decimal. Then $x$ is a variable, but a variable which has only ten different values, viz. $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$.", "why": "A variable with ten values, the digits, makes the idea of a field of variation concrete before any algebra is needed.", "use": [ "lesson" ], "concepts": [ "concept/variable" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-a069a3546c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "27", "location": "REAL VARIABLES", "latex": "He will find interesting examples in ordinary life: policeman~$x$, the driver of cab~$x$, the year~$x$, the $x$th~day of the week.", "markdown": "He will find interesting examples in ordinary life: policeman $x$, the driver of cab $x$, the year $x$, the $x$th day of the week.", "why": "Ordinary-life variables with non-numeric values show learners that a variable is any symbol ranging over a set of values.", "use": [ "website", "history" ], "concepts": [ "concept/variable" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-685e60e2ee", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "30", "location": "REAL VARIABLES", "latex": "In this case we shall say that $\\xi$~is a \\Emph{point of accumulation} of~$S$.", "markdown": "In this case we shall say that $\\xi$ is a **of accumulation** of $S$.", "why": "States the definition of a point of accumulation in one line, after the reasoning that motivates it.", "use": [ "lesson" ], "concepts": [ "concept/point-of-accumulation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-ef2b31d0f1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "29", "location": "REAL VARIABLES", "latex": "On the other hand, if $S$~consists of all the rational points, then, whatever the value of~$\\xi$, (ii)~is true; for any interval whatever contains an infinity of rational points.", "markdown": "On the other hand, if $S$ consists of all the rational points, then, whatever the value of $\\xi$, (ii) is true; for any interval whatever contains an infinity of rational points.", "why": "Shows that the rational numbers are dense enough that every point of the line is an accumulation point, which surprises many learners.", "use": [ "lesson", "website" ], "concepts": [ "concept/point-of-accumulation", "concept/rational-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-017b6bb384", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "30", "location": "REAL VARIABLES", "latex": "If a set~$S$ contains infinitely many points, and is entirely situated in an interval $\\DPmod{(\\alpha, \\beta)}{[\\alpha, \\beta]}$, then at least one point of the interval is a point of accumulation of~$S$.", "markdown": "If a set $S$ contains infinitely many points, and is entirely situated in an interval $\\DPmod{(\\alpha, \\beta)}{[\\alpha, \\beta]}$, then at least one point of the interval is a point of accumulation of $S$.", "why": "Gives the statement of Weierstrass's theorem, the result the rest of the section builds toward.", "use": [ "lesson" ], "concepts": [ "concept/interval", "concept/point-of-accumulation", "theorem/weierstrass-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-ef8f497277", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "35", "location": "REAL VARIABLES", "latex": "As a corollary, if $a + b\\sqrt[3]{2} + c\\sqrt[3]{4} = d + e\\sqrt[3]{2} + f\\sqrt[3]{4}$, then $a = d$, $b = e$, $c = f$.", "markdown": "As a corollary, if $a + b\\sqrt[3]{2} + c\\sqrt[3]{4} = d + e\\sqrt[3]{2} + f\\sqrt[3]{4}$, then $a = d$, $b = e$, $c = f$.", "why": "States the uniqueness of rational coefficients in a cube-root expression, a result a learner can check and use.", "use": [ "lesson" ], "concepts": [ "concept/rational-number", "concept/surd" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-b8e8d4ab94", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "15", "location": "REAL VARIABLES", "latex": "We have to show that these ideas can be applied to the new numbers, and that, when this extension of them is made, all the ordinary laws of algebra retain their validity, so that we can operate with real numbers in general in exactly the same way as with the rational numbers of \\SecNo[§]{1}.", "markdown": "We have to show that these ideas can be applied to the new numbers, and that, when this extension of them is made, all the ordinary laws of algebra retain their validity, so that we can operate with real numbers in general in exactly the same way as with the rational numbers of [§]1.", "why": "States plainly what must be checked when the idea of number is widened, and why it matters that the old algebra still works.", "use": [ "lesson", "website" ], "concepts": [ "concept/real-number", "method/addition", "method/multiplication" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-df67d51697", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "16", "location": "REAL VARIABLES", "latex": "Of the two numbers $\\alpha$~and~$-\\alpha$ one is always positive (unless $\\alpha = 0$). The one which is positive we denote by~$|\\alpha|$ and call the \\emph{modulus} of~$\\alpha$.", "markdown": "Of the two numbers $\\alpha$ and $-\\alpha$ one is always positive (unless $\\alpha = 0$). The one which is positive we denote by $|\\alpha|$ and call the *modulus* of $\\alpha$.", "why": "Gives a clean definition of the modulus, including the exceptional case of zero.", "use": [ "lesson" ], "concepts": [ "concept/absolute-value", "concept/real-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-ed81f635ae", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "26", "location": "REAL VARIABLES", "latex": "The~$x$ which occurs in propositions such as these is called \\emph{the continuous real variable}: and the individual numbers are called the \\emph{values} of the variable.", "markdown": "The $x$ which occurs in propositions such as these is called *the continuous real variable*: and the individual numbers are called the *values* of the variable.", "why": "Names the continuous real variable and separates the variable from its individual values.", "use": [ "lesson", "website" ], "concepts": [ "concept/continuous-real-variable", "concept/variable" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-256e55cdf7", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "26", "location": "REAL VARIABLES", "latex": "The reader should think of other examples of variables with different fields of variation. He will find interesting examples in ordinary life: policeman~$x$, the driver of cab~$x$, the year~$x$, the $x$th~day of the week. The values of these variables are naturally not numbers.", "markdown": "The reader should think of other examples of variables with different fields of variation. He will find interesting examples in ordinary life: policeman $x$, the driver of cab $x$, the year $x$, the $x$th day of the week. The values of these variables are naturally not numbers.", "why": "Shows with everyday cases that a variable need not take numerical values and can have any field of variation.", "use": [ "lesson", "website" ], "concepts": [ "concept/field-of-a-variable", "concept/variable" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-3eb76b5594", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "24", "location": "REAL VARIABLES", "latex": "We add a few further examples to show how very special these particular classes of numbers are, and how, to put it roughly, they comprise only a minute fraction of the infinite variety of numbers which constitute the continuum.", "markdown": "We add a few further examples to show how very special these particular classes of numbers are, and how, to put it roughly, they comprise only a minute fraction of the infinite variety of numbers which constitute the continuum.", "why": "Warns the learner that rationals and quadratic surds are only a small part of the real numbers.", "use": [ "lesson", "website" ], "concepts": [ "concept/arithmetical-continuum", "concept/irrational-number", "concept/quadratic-surd" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-b44afed958", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "25", "location": "REAL VARIABLES", "latex": "It can in fact be proved (though the proof is difficult) that it is \\emph{generally} impossible to find such an expression for the root of an equation of higher degree than~$4$.", "markdown": "It can in fact be proved (though the proof is difficult) that it is *generally* impossible to find such an expression for the root of an equation of higher degree than $4$.", "why": "Honestly flags a result stated without proof: general equations of degree above 4 have roots not expressible in surds.", "use": [ "lesson", "history" ], "concepts": [ "concept/irrational-number", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-c80c693751", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "26", "location": "REAL VARIABLES", "latex": "And this number~$\\pi$ is no isolated or exceptional case. Any number of other examples can be constructed. In fact it is only special classes of irrational numbers which are roots of equations of this kind, just as it is only a still smaller class which can be expressed by means of surds.", "markdown": "And this number $\\pi$ is no isolated or exceptional case. Any number of other examples can be constructed. In fact it is only special classes of irrational numbers which are roots of equations of this kind, just as it is only a still smaller class which can be expressed by means of surds.", "why": "Shows that numbers beyond algebraic equations, like \\pi, are the rule among irrationals rather than a rarity.", "use": [ "lesson", "website" ], "concepts": [ "concept/irrational-number", "concept/pi", "concept/transcendental-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-7188788249", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "28", "location": "REAL VARIABLES", "latex": "This conclusion is of very great importance; for it shows that the consideration of sections of all the real numbers does not lead to any further generalisation of our idea of number.", "markdown": "This conclusion is of very great importance; for it shows that the consideration of sections of all the real numbers does not lead to any further generalisation of our idea of number.", "why": "Explains why the process of extending number by sections stops with the real numbers.", "use": [ "lesson", "history" ], "concepts": [ "concept/completeness-of-the-continuum", "concept/dedekind-section", "theorem/dedekind-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-94e9b92345", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "39", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "Let $y = 0$ whatever be the value of~$x$. Then $y$~is a function of~$x$, for we can give~$x$ any value, and the corresponding value of~$y$ (viz.~$0$) is known.", "markdown": "Let $y = 0$ whatever be the value of $x$. Then $y$ is a function of $x$, for we can give $x$ any value, and the corresponding value of $y$ (viz. $0$) is known.", "why": "It shows a learner that a constant function meets the definition without needing a formula to vary, which corrects the idea that functions must change.", "use": [ "lesson" ], "concepts": [ "concept/constant", "concept/function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-edc6c1a4fa", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "40", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "`The largest prime factor of~$\\frac{11}{3}$ or of~$\\sqrt{2}$ or of~$\\pi$' means nothing, and so our defining relation fails to define for such values of~$x$ as these.", "markdown": "‘The largest prime factor of $\\frac{11}{3}$ or of $\\sqrt{2}$ or of $\\pi$’ means nothing, and so our defining relation fails to define for such values of $x$ as these.", "why": "It shows that a rule can be a perfectly good definition and still have a domain of definition that excludes many values, a point learners often miss.", "use": [ "lesson" ], "concepts": [ "concept/domain-of-definition", "concept/prime-factor" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-ade63c1449", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "47", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "Thus the function~$x/x$ is equal to~$1$ if $x\\neq 0$ and is undefined when $x = 0$.", "markdown": "Thus the function $x/x$ is equal to $1$ if $x\\neq 0$ and is undefined when $x = 0$.", "why": "It warns that cancelling a common factor can silently change a function at the excluded value, a common error in simplifying rational expressions.", "use": [ "lesson" ], "concepts": [ "concept/domain-of-definition", "concept/rational-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-b2ddb5f5f7", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "47", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "A rational function is the quotient of one polynomial by another", "markdown": "A rational function is the quotient of one polynomial by another", "why": "It gives the standard definition of a rational function, which the chapter then builds on in its examples.", "use": [ "lesson", "website" ], "concepts": [ "concept/polynomial", "concept/rational-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-04b516938c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "42", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "We call the aggregate of all these points the \\Emph{graph} of the function~$y$.", "markdown": "We call the aggregate of all these points the **** of the function $y$.", "why": "It defines the graph as the collection of plotted points, which is the bridge between an algebraic rule and the picture a learner draws.", "use": [ "lesson" ], "concepts": [ "concept/graph-of-a-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-f84fa4fb0f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "50", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "It is however often more convenient to regard~$\\sqrt{x}$ as standing for the two-valued function whose two values are the positive and negative square roots of~$x$.", "markdown": "It is however often more convenient to regard $\\sqrt{x}$ as standing for the two-valued function whose two values are the positive and negative square roots of $x$.", "why": "It makes plain that the notation sqrt(x) can mean different things, and that the choice must be stated before one can say how many values a function has.", "use": [ "lesson" ], "concepts": [ "concept/explicit-function", "concept/function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-fe28f5c9b2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "39", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "It is known that a much better approximation to the true relation can then be found by means of what is known as `van~der Waals' law', expressed by the equation", "markdown": "It is known that a much better approximation to the true relation can then be found by means of what is known as ‘van der Waals’ law’, expressed by the equation", "why": "It is a historical remark showing how a simple gas law is refined by a second formula at high compression, which gives a sense of how models are improved.", "use": [ "history" ], "concepts": [ "concept/van-der-waals-equation", "law/boyle-s-law" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-289229b1d2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "38", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "In these circumstances $y$~is said to be a \\emph{function} of~$x$. This notion of functional dependence of one variable upon another is perhaps the most important in the whole range of higher mathematics.", "markdown": "In these circumstances $y$ is said to be a *function* of $x$. This notion of functional dependence of one variable upon another is perhaps the most important in the whole range of higher mathematics.", "why": "States the central definition and tells the learner why it deserves care.", "use": [ "lesson", "website" ], "concepts": [ "concept/function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-b57ea25c89", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "39", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "All that is essential is that there should be some relation between $x$~and~$y$ such that to some values of~$x$ at any rate correspond values of~$y$.", "markdown": "All that is essential is that there should be some relation between $x$ and $y$ such that to some values of $x$ at any rate correspond values of $y$.", "why": "Frees the learner from the false belief that a function must have a formula and be defined everywhere.", "use": [ "lesson" ], "concepts": [ "concept/domain-of-definition", "concept/function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-bcc8264b79", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "39", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "Boyle's law, however, only gives a reasonable approximation to the facts provided the gas is not compressed too much. When $v$~is decreased and $p$~increased beyond a certain point, the relation between them is no longer expressed with tolerable exactness by the equation~\\Eq{(i)}.", "markdown": "Boyle’s law, however, only gives a reasonable approximation to the facts provided the gas is not compressed too much. When $v$ is decreased and $p$ increased beyond a certain point, the relation between them is no longer expressed with tolerable exactness by the equation (i).", "why": "Shows that a physical law is an approximation with limits, and that a function may be defined by different formulas on different ranges.", "use": [ "lesson", "website" ], "concepts": [ "concept/function", "law/boyle-s-law", "law/van-der-waals-law" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-bbc7621d31", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "41", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "Let $y$~be defined as \\emph{the height in inches of policeman~$Cx$, in the Metropolitan Police, at {\\upshape5.30}~\\DPchg{p.m.}{\\textsc{p.m.}}\\ on {\\upshape8}~Aug.~{\\upshape1907}}. Then $y$~is defined for a certain number of integral values of~$x$, viz.\\ $1$, $2$, \\dots,~$N$, where $N$~is the total number of policemen in division~$C$ at that particular moment of time.", "markdown": "Let $y$ be defined as *the height in inches of policeman $Cx$, in the Metropolitan Police, at 5.30 p.m.p.m. on 8 Aug. 1907*. Then $y$ is defined for a certain number of integral values of $x$, viz. $1$, $2$, …, $N$, where $N$ is the total number of policemen in division $C$ at that particular moment of time.", "why": "A vivid, charming example of a function with no formula that is defined only for a finite set of whole numbers.", "use": [ "website", "history", "lesson" ], "concepts": [ "concept/domain-of-definition", "concept/function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-fcd5f47abf", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "44", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "The reader has no doubt some notion as to what is meant by a \\emph{continuous} curve, a curve without breaks or jumps; such a curve, in fact, as is roughly represented in \\Fig{8}.", "markdown": "The reader has no doubt some notion as to what is meant by a *continuous* curve, a curve without breaks or jumps; such a curve, in fact, as is roughly represented in [fig:8]Fig. 8.", "why": "Introduces the intuitive idea of continuity and, with the text that follows, warns that plotted points alone cannot prove it.", "use": [ "lesson" ], "concepts": [ "concept/continuous-function", "concept/graph-of-a-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-b8bbe12f85", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "47", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "Consider for example the function~$x/x$, which is a rational function. On removing the common factor~$x$ we obtain $1/1 = 1$. But the original function is not \\emph{always} equal to~$1$: it is equal to~$1$ only so long as $x\\neq 0$. If $x = 0$ it takes the form~$0/0$, which is meaningless.", "markdown": "Consider for example the function $x/x$, which is a rational function. On removing the common factor $x$ we obtain $1/1 = 1$. But the original function is not *always* equal to $1$: it is equal to $1$ only so long as $x\\neq 0$. If $x = 0$ it takes the form $0/0$, which is meaningless.", "why": "Warns of a common mistake: cancelling a common factor can change a function at exceptional values.", "use": [ "lesson" ], "concepts": [ "concept/domain-of-definition", "concept/rational-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-b968012cc2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "48", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "It is in no way presupposed in the definition of a rational function that the constants which occur as coefficients should be rational \\emph{numbers}. The word rational has reference solely to the way in which the variable~$x$ appears in the function.", "markdown": "It is in no way presupposed in the definition of a rational function that the constants which occur as coefficients should be rational *numbers*. The word rational has reference solely to the way in which the variable $x$ appears in the function.", "why": "Clears up a likely confusion between a rational function and rational numbers.", "use": [ "lesson" ], "concepts": [ "concept/rational-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-74107c82f2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "61", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "We can in this case at once obtain a direct relation between $x$~and~$y$ by squaring and adding: we find that $x^{2} + y^{2} = a^{2}$, $t$~being now eliminated.", "markdown": "We can in this case at once obtain a direct relation between $x$ and $y$ by squaring and adding: we find that $x^{2} + y^{2} = a^{2}$, $t$ being now eliminated.", "why": "It shows how eliminating the auxiliary variable turns a parametric curve back into a familiar equation.", "use": [ "lesson" ], "concepts": [ "concept/parametric-representation-of-a-curve" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-b595546ace", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "67", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "Starting from a unit length we can construct any \\emph{rational} length.", "markdown": "Starting from a unit length we can construct any *rational* length.", "why": "Shows that rational lengths are the starting point of every ruler construction, a useful anchor for the idea of constructible numbers.", "use": [ "lesson" ], "concepts": [ "concept/rational-number", "method/euclidean-construction" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-0c03675fe2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "66", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "Show that if $x$~is a rational function of~$y$, and $y$~is a rational function of~$x$, then $Axy + Bx + Cy + D = 0$.", "markdown": "Show that if $x$ is a rational function of $y$, and $y$ is a rational function of $x$, then $Axy + Bx + Cy + D = 0$.", "why": "A short proof exercise that shows how two functional relations force a symmetric linear form.", "use": [ "lesson" ], "concepts": [ "concept/function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-dd0b922e67", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "65", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "If $f(x) = f(-x)$ for all values of~$x$, $f(x)$~is called an \\emph{even} function.\nIf $f(x) = -f(-x)$, it is called an \\emph{odd} function.", "markdown": "If $f(x) = f(-x)$ for all values of $x$, $f(x)$ is called an *even* function. If $f(x) = -f(-x)$, it is called an *odd* function.", "why": "Gives the two symmetric definitions side by side, ready for the exercise showing every function splits into even and odd parts.", "use": [ "lesson", "website" ], "concepts": [ "concept/even-function", "concept/function", "concept/odd-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-564251fa04", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "66", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "All these constructions\nwere what may be called Euclidean constructions; they depended\non the ruler and compasses only.", "markdown": "All these constructions were what may be called Euclidean constructions; they depended on the ruler and compasses only.", "why": "Names the ruler-and-compasses rules that frame everything the exercise then proves about constructible lengths.", "use": [ "lesson", "website" ], "concepts": [ "method/euclidean-construction" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-61f7c4919f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "67", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "This expression contains a fourth root, but this is of\ncourse the square root of a square root.", "markdown": "This expression contains a fourth root, but this is of course the square root of a square root.", "why": "Shows a learner that a daunting nested radical is only repeated square roots, hence constructible.", "use": [ "lesson" ], "concepts": [ "concept/surd", "method/euclidean-construction" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-b4241a419d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "67", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "Conversely, \\emph{only} irrationals of this kind can be constructed by Euclidean\nmethods. Starting from a unit length we can construct any \\emph{rational} length.", "markdown": "Conversely, *only* irrationals of this kind can be constructed by Euclidean methods. Starting from a unit length we can construct any *rational* length.", "why": "States the converse half of the result and the starting point of the argument: what is constructible is limited, and rational lengths are the base.", "use": [ "lesson" ], "concepts": [ "concept/surd", "method/euclidean-construction" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-f506c64d37", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "67", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "Hence \\emph{Euclidean methods will construct any surd expression involving square\nroots only, and no others}.", "markdown": "Hence *Euclidean methods will construct any surd expression involving square roots only, and no others*.", "why": "The exercise's central conclusion, a clean statement of exactly what ruler and compasses can do.", "use": [ "lesson", "website" ], "concepts": [ "concept/surd", "method/euclidean-construction" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-4133500dab", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "67", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "One of the famous problems of antiquity was that of the duplication of\nthe cube, that is to say of the construction by Euclidean methods of a\nlength measured by~$\\sqrt[3]{2}$. It can be shown that $\\sqrt[3]{2}$~cannot be expressed by\nmeans of any finite combination of rational numbers and square roots, and so\nthat the problem is an impossible one.", "markdown": "One of the famous problems of antiquity was that of the duplication of the cube, that is to say of the construction by Euclidean methods of a length measured by $\\sqrt[3]{2}$. It can be shown that $\\sqrt[3]{2}$ cannot be expressed by means of any finite combination of rational numbers and square roots, and so that the problem is an impossible one.", "why": "Connects an ancient problem to the constructibility result, showing how a proof of impossibility works.", "use": [ "lesson", "website", "history" ], "concepts": [ "concept/duplication-of-the-cube", "method/euclidean-construction" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-1c631332c1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "68", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "If $R$~is the earth's\nradius, the error in supposing $AM$ to be its circumference is less than $11$~yards.", "markdown": "If $R$ is the earth’s radius, the error in supposing $AM$ to be its circumference is less than $11$ yards.", "why": "Makes the accuracy of an approximate construction vivid by scaling it to the size of the earth.", "use": [ "website", "history" ], "concepts": [ "concept/squaring-the-circle" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-b561aa3565", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "63", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "It will at once suggest a contoured Ordnance Survey map: and in fact this is the principle on which such maps are constructed.", "markdown": "It will at once suggest a contoured Ordnance Survey map: and in fact this is the principle on which such maps are constructed.", "why": "It links the abstract graph of a function of two variables to the familiar contour map.", "use": [ "lesson", "website" ], "concepts": [ "concept/contour-line", "concept/function-of-two-variables", "concept/surface" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-5376ee36cb", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "51", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "\\emph{a function $y = f(x)$ will be said to be an algebraical function of~$x$ if it is the root of an equation such as~\\Eq{(1)}, \\ie~the root of an equation of the $m$\\textsuperscript{th}~degree in~$y$, whose coefficients are rational functions of~$x$}.", "markdown": "*a function $y = f(x)$ will be said to be an algebraical function of $x$ if it is the root of an equation such as (1), *i.e.* the root of an equation of the $m$th degree in $y$, whose coefficients are rational functions of $x$*.", "why": "It gives the learner the single definition the whole section turns on: an algebraic function is any root of a polynomial equation whose coefficients are rational in x.", "use": [ "lesson", "website" ], "concepts": [ "concept/algebraic-function", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-0c64f122e2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "54", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "A function is said to be \\emph{periodic}, with period~$a$, if $f(x) = f(x + a)$ for all values of~$x$ for which $f(x)$~is defined.", "markdown": "A function is said to be *periodic*, with period $a$, if $f(x) = f(x + a)$ for all values of $x$ for which $f(x)$ is defined.", "why": "It states the definition of a period precisely, so a learner can test any function against it.", "use": [ "lesson" ], "concepts": [ "concept/periodic-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-81ae2b2397", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "62", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "The locus is the surface formed by drawing lines parallel to~$OZ$ through all points of this curve. Such a surface is called a \\emph{cylinder}.", "markdown": "The locus is the surface formed by drawing lines parallel to $OZ$ through all points of this curve. Such a surface is called a *cylinder*.", "why": "It shows how a cylinder is built from a plane curve, which makes the three-dimensional figure easy to picture.", "use": [ "lesson", "website" ], "concepts": [ "concept/curve", "concept/cylinder", "concept/cylindrical-surface" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-7941a2e75f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "64", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "It is entirely built up of straight lines; but the surface is curved everywhere, and is in general shape not unlike certain forms of table-napkin rings (\\Fig{18c}).", "markdown": "It is entirely built up of straight lines; but the surface is curved everywhere, and is in general shape not unlike certain forms of table-napkin rings ([fig:18c]Fig. 18c).", "why": "It shows that a surface made only of straight lines can still be curved, a surprise that rewards close attention.", "use": [ "history", "website" ], "concepts": [ "concept/ruled-surface" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-090ac09035", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "56", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "The reader may possibly regard this as an unreasonable function. \\emph{Why}, he may ask, if $y$~is equal to~$x$ for all values of~$x$ save integral values, should it not be equal to~$x$ for integral values too?", "markdown": "The reader may possibly regard this as an unreasonable function. *Why*, he may ask, if $y$ is equal to $x$ for all values of $x$ save integral values, should it not be equal to $x$ for integral values too?", "why": "It addresses the learner's natural doubt about a strange function and makes the point that any rule of correspondence counts as a function.", "use": [ "lesson", "history" ], "concepts": [ "concept/function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-270a15857d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "58", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "The abscissae of its intersections with the axis of~$x$ are the roots of the equation.", "markdown": "The abscissae of its intersections with the axis of $x$ are the roots of the equation.", "why": "It states the key step of solving a quadratic graphically, which a learner can carry over to any equation.", "use": [ "lesson" ], "concepts": [ "concept/abscissa", "concept/quadratic-equation", "concept/root-of-an-equation", "method/solving-an-equation-graphically" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-0a3d21532d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "51", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "This function is defined for all values of~$x$ for which $R_{1}^{2} \\geq 4R_{2}$.", "markdown": "This function is defined for all values of $x$ for which $R_{1}^{2} \\geq 4R_{2}$.", "why": "It shows that a function given implicitly can have a domain restriction, so a learner must check where the root is real.", "use": [ "lesson" ], "concepts": [ "concept/algebraic-function", "concept/domain-of-definition", "concept/quadratic-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-e381016ac0", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "51", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "We are therefore led to give the following definition: \\emph{a function $y = f(x)$ will be said to be an algebraical function of~$x$ if it is the root of an equation such as~\\Eq{(1)}, \\ie~the root of an equation of the $m$\\textsuperscript{th}~degree in~$y$, whose coefficients are rational functions of~$x$}.", "markdown": "We are therefore led to give the following definition: *a function $y = f(x)$ will be said to be an algebraical function of $x$ if it is the root of an equation such as (1), *i.e.* the root of an equation of the $m$th degree in $y$, whose coefficients are rational functions of $x$*.", "why": "It gives the book's precise definition of an algebraical function, which learners can compare with the explicit formulas they already know.", "use": [ "lesson", "website" ], "concepts": [ "concept/algebraic-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-57a19e9bf8", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "51", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "For it is known that in general such an equation as~\\Eq{(1)} cannot be solved explicitly for~$y$ in terms of~$x$, when $m$~is greater than~$4$, though such a solution is always possible if $m = 1$, $2$,~$3$, or~$4$ and in special cases for higher values of~$m$.", "markdown": "For it is known that in general such an equation as (1) cannot be solved explicitly for $y$ in terms of $x$, when $m$ is greater than $4$, though such a solution is always possible if $m = 1$, $2$, $3$, or $4$ and in special cases for higher values of $m$.", "why": "It shows why implicit algebraical functions are a real extension of the explicit ones, and where explicit solution breaks down.", "use": [ "lesson" ], "concepts": [ "concept/algebraic-function", "concept/explicit-function", "concept/implicit-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-0432dabffd", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "52", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "All functions of~$x$ which are not rational or even algebraical are called \\emph{transcendental} functions. This class of functions, being defined in so purely negative a manner, naturally includes an infinite variety of whole kinds of functions of varying degrees of simplicity and importance.", "markdown": "All functions of $x$ which are not rational or even algebraical are called *transcendental* functions. This class of functions, being defined in so purely negative a manner, naturally includes an infinite variety of whole kinds of functions of varying degrees of simplicity and importance.", "why": "It names the class and honestly notes that a purely negative definition gathers very different kinds of function together.", "use": [ "lesson", "website" ], "concepts": [ "concept/transcendental-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-0689cee364", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "54", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "It is easy to see that no periodic function can be a rational function, unless it is a constant.", "markdown": "It is easy to see that no periodic function can be a rational function, unless it is a constant.", "why": "It states the key fact behind the proof that cos x and sin x are not rational.", "use": [ "lesson" ], "concepts": [ "concept/periodic-function", "concept/rational-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-f2750f2e8a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "It oscillates up and down, the rapidity of the oscillations becoming greater and greater as $x$~approaches~$0$. For $x = 0$ the function is undefined.", "markdown": "It oscillates up and down, the rapidity of the oscillations becoming greater and greater as $x$ approaches $0$. For $x = 0$ the function is undefined.", "why": "It describes the graph of sin(1/x) vividly and shows a function that is perfectly well behaved yet undefined at a point.", "use": [ "lesson", "website" ], "concepts": [ "concept/circular-function", "concept/graph-of-a-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-770f5e7012", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "56", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "The function~$y$ does in point of fact answer to the definition of a function: there is a relation between $x$~and~$y$ such that when $x$~is known $y$~is known. We are perfectly at liberty to take this relation to be what we please, however arbitrary and apparently futile.", "markdown": "The function $y$ does in point of fact answer to the definition of a function: there is a relation between $x$ and $y$ such that when $x$ is known $y$ is known. We are perfectly at liberty to take this relation to be what we please, however arbitrary and apparently futile.", "why": "It tells learners that a function need not follow a single formula and answers the natural objection to a function that is odd at integers.", "use": [ "lesson", "website" ], "concepts": [ "concept/graph-of-a-function", "concept/integer-part-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-3639abad46", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "61", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "A particle which moves along a straight line has only \\emph{one degree of freedom}. Its direction of motion is fixed; its position can be completely fixed by one measurement of position, \\eg\\ by its distance from a fixed point on the line.", "markdown": "A particle which moves along a straight line has only *one degree of freedom*. Its direction of motion is fixed; its position can be completely fixed by one measurement of position, *e.g.* by its distance from a fixed point on the line.", "why": "It gives an intuitive physical meaning to the number of coordinates needed to fix a point on a line, in a plane, or in space.", "use": [ "lesson", "website" ], "concepts": [ "concept/curve", "concept/degree-of-freedom", "concept/surface-of-a-polyhedron" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-8ee021cb9a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "85", "location": "COMPLEX NUMBERS", "latex": "\\Fig{24} is usually known as Argand's diagram.", "markdown": "[fig:24]Fig. 24 is usually known as Argand’s diagram.", "why": "Tells the learner where the plane picture of complex numbers gets its name, which helps when the diagram is cited in other books.", "use": [ "history", "website" ], "concepts": [ "concept/argand-diagram" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-27beec6920", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "85", "location": "COMPLEX NUMBERS", "latex": "When $y = 0$ we say that \\emph{$z$~is real}, when $x = 0$ that \\emph{$z$~is purely imaginary}.", "markdown": "When $y = 0$ we say that *$z$ is real*, when $x = 0$ that *$z$ is purely imaginary*.", "why": "Gives the two special cases of a complex number in one plain sentence, so the learner can see when the imaginary part vanishes.", "use": [ "lesson" ], "concepts": [ "concept/complex-number", "quantity/imaginary-part-of-a-complex-number", "quantity/real-part-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-09cc278e74", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "85", "location": "COMPLEX NUMBERS", "latex": "Two numbers $x + yi$, $x - yi$ which differ only in the signs of their imaginary parts, we call \\emph{conjugate}.", "markdown": "Two numbers $x + yi$, $x - yi$ which differ only in the signs of their imaginary parts, we call *conjugate*.", "why": "Defines conjugate complex numbers exactly, which the learner needs before rationalizing denominators or reading the theorem on roots.", "use": [ "lesson" ], "concepts": [ "concept/conjugate-complex-numbers" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-2e0e42575a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "100", "location": "COMPLEX NUMBERS", "latex": "There are other special values of~$n$ for which such constructions are possible, the most interesting being~$n = 17$.", "markdown": "There are other special values of $n$ for which such constructions are possible, the most interesting being $n = 17$.", "why": "It links an algebraic question about roots of unity to the classical geometric question of constructing regular polygons with ruler and compasses, and sets the learner a famous case to explore.", "use": [ "history", "website" ], "concepts": [ "concept/root-of-unity", "method/euclidean-construction" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-4f075be07b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "71", "location": "COMPLEX NUMBERS", "latex": "Common sense at once suggests that we should define the sum of two displacements as the displacement which is the result of the successive application of the two given displacements.", "markdown": "Common sense at once suggests that we should define the sum of two displacements as the displacement which is the result of the successive application of the two given displacements.", "why": "It shows the reasoning behind the definition of addition before any formula appears, so the learner sees why the rule is chosen.", "use": [ "lesson" ], "concepts": [ "concept/displacement", "method/addition-of-displacements" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-38700e4624", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "72", "location": "COMPLEX NUMBERS", "latex": "In other words, \\emph{addition of displacements obeys the commutative law} expressed in ordinary algebra by the equation $a + b = b + a$.", "markdown": "In other words, *addition of displacements obeys the commutative law* expressed in ordinary algebra by the equation $a + b = b + a$.", "why": "It links the new law of addition to the familiar commutative law of ordinary algebra in a single sentence.", "use": [ "lesson" ], "concepts": [ "law/commutative-law", "method/addition-of-displacements" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-0011ff8d8b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "77", "location": "COMPLEX NUMBERS", "latex": "The required definition is therefore \\[ [x, y] [x', y'] = [xx' - yy', xy' + yx']. \\Tag{(6)} \\]", "markdown": "The required definition is therefore [x, y] [x’, y’] = [xx’ - yy’, xy’ + yx’]. (6)", "why": "It gives the rule for multiplying displacements, which agrees with ordinary multiplication when one displacement lies along the axis.", "use": [ "lesson" ], "concepts": [ "concept/polar-coordinates", "method/multiplication-of-displacements" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-b971722392", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "83", "location": "COMPLEX NUMBERS", "latex": "We conclude that a quadratic equation with real coefficients has exactly two roots.", "markdown": "We conclude that a quadratic equation with real coefficients has exactly two roots.", "why": "It states the result that a quadratic has exactly two roots in every case, which the reader can check against the earlier discussion of real and complex roots.", "use": [ "lesson" ], "concepts": [ "concept/quadratic-equation", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-a1eceb2a15", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "84", "location": "COMPLEX NUMBERS", "latex": "The application of any of the ordinary algebraical operations to complex numbers will yield only complex numbers.", "markdown": "The application of any of the ordinary algebraical operations to complex numbers will yield only complex numbers.", "why": "It explains why no further kinds of number are needed for solving polynomial equations with complex numbers.", "use": [ "lesson" ], "concepts": [ "concept/closure", "concept/complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-1623dda3d7", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "80", "location": "COMPLEX NUMBERS", "latex": "One most important property of real numbers is that known as \\emph{the factor theorem}, which asserts that \\emph{the product of two numbers cannot be zero unless one of the two is itself zero}.", "markdown": "One most important property of real numbers is that known as *the factor theorem*, which asserts that *the product of two numbers cannot be zero unless one of the two is itself zero*.", "why": "It states the zero-product property that the chapter then extends to complex numbers.", "use": [ "lesson" ], "concepts": [ "theorem/factor-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-0a87af19ed", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "84", "location": "COMPLEX NUMBERS", "latex": "All such theorems as these are true whether $a$,~$b$,~\\dots\\Add{,} $\\alpha$,~$\\beta$,~\\dots\\ are real or complex.", "markdown": "All such theorems as these are true whether $a$, $b$, … $\\alpha$, $\\beta$, … are real or complex.", "why": "It tells learners that algebra proved from the addition and multiplication rules still holds when the numbers are complex, so Vieta-type relations carry over.", "use": [ "lesson" ], "concepts": [ "concept/complex-number", "theorem/relations-between-roots-and-coefficients" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-67e21d5585", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "85", "location": "COMPLEX NUMBERS", "latex": "we call~$z$ the \\emph{complex variable}.", "markdown": "we call $z$ the *complex variable*.", "why": "Gives the learner the name for z and the reason it is called a variable: it stands for any complex number.", "use": [ "lesson" ], "concepts": [ "concept/complex-number", "concept/complex-variable" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-186e0e32dd", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "85", "location": "COMPLEX NUMBERS", "latex": "It must be observed that $\\theta$ or $\\am z$ is a many-valued function of $x$~and~$y$, having an infinity of values, which are angles differing by multiples of~$2\\pi$.", "markdown": "It must be observed that $\\theta$ or $\\am z$ is a many-valued function of $x$ and $y$, having an infinity of values, which are angles differing by multiples of $2\\pi$.", "why": "Warns that the amplitude is not one fixed angle but a family of angles differing by multiples of 2pi.", "use": [ "lesson" ], "concepts": [ "concept/principal-value-of-amplitude", "quantity/amplitude-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-b58dfc300a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "87", "location": "COMPLEX NUMBERS", "latex": "Hence \\emph{De Moivre's Theorem holds for all integral values of~$n$, positive or negative}.", "markdown": "Hence *De Moivre’s Theorem holds for all integral values of $n$, positive or negative*.", "why": "States the extension of De Moivre's theorem from positive to all integer powers, with the reciprocal argument that justifies it.", "use": [ "lesson", "website" ], "concepts": [ "theorem/de-moivre-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-da1a8aed37", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "96", "location": "COMPLEX NUMBERS", "latex": "The four points are said to be \\emph{harmonic} or \\emph{harmonically related} if any one of these is equal to~$-1$.", "markdown": "The four points are said to be *harmonic* or *harmonically related* if any one of these is equal to $-1$.", "why": "Defines harmonic points by a single cross-ratio condition, linking the cross ratio to a named special case.", "use": [ "lesson" ], "concepts": [ "concept/cross-ratio", "concept/harmonic-points" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-d0410175ef", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "86", "location": "COMPLEX NUMBERS", "latex": "in this notation, suggested by Profs.\\ Harkness and Morley, De~Moivre's theorem is expressed by the equation $(\\Cis\\theta)^{n} = \\Cis n\\theta$.", "markdown": "in this notation, suggested by Profs. Harkness and Morley, De Moivre’s theorem is expressed by the equation $(\\Cis\\theta)^{n} = \\Cis n\\theta$.", "why": "A historical remark showing where the compact cis notation came from and how it makes De Moivre's theorem look simple.", "use": [ "history" ], "concepts": [ "theorem/de-moivre-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-bce88ff2da", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "98", "location": "COMPLEX NUMBERS", "latex": "These definitions do not prejudge the question as to whether there are or are not more than one (or any) roots of the equation.", "markdown": "These definitions do not prejudge the question as to whether there are or are not more than one (or any) roots of the equation.", "why": "It warns learners that the definition alone does not say how many roots exist, a point the chapter then settles.", "use": [ "lesson" ], "concepts": [ "concept/root-of-a-complex-number", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-8919c874d7", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "99", "location": "COMPLEX NUMBERS", "latex": "That these $n$~roots are in reality all distinct is easily seen by plotting them on Argand's diagram.", "markdown": "That these $n$ roots are in reality all distinct is easily seen by plotting them on Argand’s diagram.", "why": "It gives a picture check that the n roots are different points, which the learner can see on the diagram.", "use": [ "lesson", "website" ], "concepts": [ "concept/argand-diagram", "concept/root-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-82afc18dbf", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "102", "location": "COMPLEX NUMBERS", "latex": "This is the analytical equivalent of the geometrical theorem that, if $M$~is the middle point of~$PQ$, then $OP^{2} + OQ^{2} = 2OM^{2} + 2MP^{2}$.", "markdown": "This is the analytical equivalent of the geometrical theorem that, if $M$ is the middle point of $PQ$, then $OP^{2} + OQ^{2} = 2OM^{2} + 2MP^{2}$.", "why": "It shows a complex identity as the algebraic form of a familiar geometric theorem about a midpoint.", "use": [ "lesson", "history" ], "concepts": [ "concept/complex-number", "quantity/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-7e6d37dd96", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "102", "location": "COMPLEX NUMBERS", "latex": "[The amplitudes have not necessarily their principal values.]", "markdown": "[The amplitudes have not necessarily their principal values.]", "why": "It warns learners not to assume principal values when solving for roots with a given modulus condition.", "use": [ "lesson" ], "concepts": [ "concept/principal-value-of-amplitude", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-81cadd824e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "98", "location": "COMPLEX NUMBERS", "latex": "Thus we define~$\\sqrt[n]{a}$ or~$a^{1/n}$, where $n$~is a positive integer, as a number~$z$ which satisfies the equation $z^{n} = a$; and $a^{m/n}$, where $m$~is an integer, as~$(a^{1/n})^{m}$. These definitions do not prejudge the question as to whether there are or are not more than one (or any) roots of the equation.", "markdown": "Thus we define $\\sqrt[n]{a}$ or $a^{1/n}$, where $n$ is a positive integer, as a number $z$ which satisfies the equation $z^{n} = a$; and $a^{m/n}$, where $m$ is an integer, as $(a^{1/n})^{m}$. These definitions do not prejudge the question as to whether there are or are not more than one (or any) roots of the equation.", "why": "Shows how the elementary definition of a root is carried over to complex numbers, leaving open how many roots there are.", "use": [ "lesson", "website" ], "concepts": [ "concept/root-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-67454901fe", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "99", "location": "COMPLEX NUMBERS", "latex": "The particular root \\[ \\sqrt[n]{\\rho}\\{\\cos(\\phi/n) + i\\sin(\\phi/n)\\} \\] is called the \\emph{principal value} of~$\\sqrt[n]{a}$.", "markdown": "The particular root [n](/n) + i(/n) is called the *principal value* of $\\sqrt[n]{a}$.", "why": "Names the one root singled out among the n, so that the symbol \\sqrt[n]{a} can have a definite meaning.", "use": [ "lesson" ], "concepts": [ "concept/principal-value-of-a-root" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-86af3bc017", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "99", "location": "COMPLEX NUMBERS", "latex": "These numbers are called the $n$th~roots of unity; the principal value is unity itself.", "markdown": "These numbers are called the $n$th roots of unity; the principal value is unity itself.", "why": "Introduces roots of unity as the special case of nth roots with a = 1.", "use": [ "lesson", "website" ], "concepts": [ "concept/principal-value-of-a-root", "concept/root-of-unity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-f500345c11", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "100", "location": "COMPLEX NUMBERS", "latex": "Euclid gives constructions for $n = 3$, $4$, $5$, $6$, $8$, $10$, $12$, and~$15$. It is evident that the construction is possible for any value of~$n$ which can be found from these by multiplication by any power of~$2$. There are other special values of~$n$ for which such constructions are possible, the most interesting being~$n = 17$.", "markdown": "Euclid gives constructions for $n = 3$, $4$, $5$, $6$, $8$, $10$, $12$, and $15$. It is evident that the construction is possible for any value of $n$ which can be found from these by multiplication by any power of $2$. There are other special values of $n$ for which such constructions are possible, the most interesting being $n = 17$.", "why": "Gives a historical note on which regular polygons Euclid could construct and hints at the surprise of the 17-gon.", "use": [ "history", "website" ], "concepts": [ "concept/root-of-unity", "method/euclidean-construction", "person/euclid" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-39d2ad48e1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "70", "location": "COMPLEX NUMBERS", "latex": "If cricket were a mathematical science, it would be very important to distinguish between the \\emph{motion} of the batsman between the wickets, the \\emph{run} which he scores, and the \\emph{mark} which is put down in the score-book.", "markdown": "If cricket were a mathematical science, it would be very important to distinguish between the *motion* of the batsman between the wickets, the *run* which he scores, and the *mark* which is put down in the score-book.", "why": "A cricket analogy that shows why things closely linked, like a length and the number measuring it, should still be kept apart.", "use": [ "lesson", "website" ], "concepts": [ "concept/displacement", "concept/real-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-788c198ca3", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "69", "location": "COMPLEX NUMBERS", "latex": "To specify a displacement completely three things are needed, its \\emph{magnitude}, its \\emph{sense} forwards or backwards along the line, and what may be called its \\emph{point of application}, \\ie\\ the original position~$P$ of the particle.", "markdown": "To specify a displacement completely three things are needed, its *magnitude*, its *sense* forwards or backwards along the line, and what may be called its *point of application*, *i.e.* the original position $P$ of the particle.", "why": "Lists exactly what information a displacement carries and which part can be ignored.", "use": [ "lesson" ], "concepts": [ "concept/displacement" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-76c3235201", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "75", "location": "COMPLEX NUMBERS", "latex": "In the first place our definition would be futile. We should only be introducing a new method of expressing something which we can perfectly well express without it.", "markdown": "In the first place our definition would be futile. We should only be introducing a new method of expressing something which we can perfectly well express without it.", "why": "Shows how a definition is judged: by whether it adds something useful, here when rejecting 'product = sum'.", "use": [ "lesson" ], "concepts": [ "method/multiplication-of-displacements" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-8d4ba67256", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "78", "location": "COMPLEX NUMBERS", "latex": "For the present the reader must regard $x + yi$ as \\emph{simply another way of writing $[x, y]$}. The expression $x + yi$ is called a \\emph{complex number}.", "markdown": "For the present the reader must regard $x + yi$ as *simply another way of writing $[x, y]$*. The expression $x + yi$ is called a *complex number*.", "why": "Introduces complex numbers plainly as notation for pairs of reals, before any mystery about i.", "use": [ "lesson", "website" ], "concepts": [ "concept/complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-91795b88de", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "80", "location": "COMPLEX NUMBERS", "latex": "The reader will now easily satisfy himself that the upshot of the rules for addition and multiplication of complex numbers is this, that \\emph{we operate with complex numbers in exactly the same way as with real numbers, treating the symbol~$i$ as itself a number, but replacing the product $ii = i^{2}$ by~$-1$ whenever it occurs}.", "markdown": "The reader will now easily satisfy himself that the upshot of the rules for addition and multiplication of complex numbers is this, that *we operate with complex numbers in exactly the same way as with real numbers, treating the symbol $i$ as itself a number, but replacing the product $ii = i^{2}$ by $-1$ whenever it occurs*.", "why": "Gives the working rule for computing with complex numbers.", "use": [ "lesson" ], "concepts": [ "concept/complex-number", "concept/imaginary-unit", "method/multiplication", "method/multiplication-of-complex-numbers" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-fc022a425f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "81", "location": "COMPLEX NUMBERS", "latex": "In other words, \\emph{multiplication of a complex number by~$i$ turns the corresponding displacement through a right angle}.", "markdown": "In other words, *multiplication of a complex number by $i$ turns the corresponding displacement through a right angle*.", "why": "Gives a geometric picture of i as a quarter turn.", "use": [ "lesson", "website" ], "concepts": [ "concept/imaginary-unit", "concept/right-angle", "method/multiplication-by-i" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-3735f30bab", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "82", "location": "COMPLEX NUMBERS", "latex": "It cannot, however, be too strongly impressed upon the reader that an `imaginary number' is no more `imaginary', in any ordinary sense of the word, than a `real' number; and that it is not a number at all, in the sense in which the `real' numbers are numbers, but, as should be clear from the preceding discussion, \\emph{a pair of numbers $(x, y)$}, united symbolically, for purposes of technical convenience, in the form $x + yi$. Such a pair of numbers is no less `real' than any ordinary number such as~$\\frac{1}{2}$, or than the paper on which this is printed, or than the Solar System.", "markdown": "It cannot, however, be too strongly impressed upon the reader that an ‘imaginary number’ is no more ‘imaginary’, in any ordinary sense of the word, than a ‘real’ number; and that it is not a number at all, in the sense in which the ‘real’ numbers are numbers, but, as should be clear from the preceding discussion, *a pair of numbers $(x, y)$*, united symbolically, for purposes of technical convenience, in the form $x + yi$. Such a pair of numbers is no less ‘real’ than any ordinary number such as $\\frac{1}{2}$, or than the paper on which this is printed, or than the Solar System.", "why": "Answers the worry that 'imaginary' numbers are less real than others, and is a historical remark on the name.", "use": [ "website", "history", "lesson" ], "concepts": [ "concept/complex-number", "concept/imaginary-root" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-d9601b2804", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "83", "location": "COMPLEX NUMBERS", "latex": "We can only attach a meaning to~$3 - 7$ if we admit \\emph{negative} numbers, or to~$\\frac{3}{7}$ if we admit \\emph{rational fractions}.", "markdown": "We can only attach a meaning to $3 - 7$ if we admit *negative* numbers, or to $\\frac{3}{7}$ if we admit *rational fractions*.", "why": "Shows that each new kind of number arises because an operation would otherwise be impossible, which prepares for complex numbers.", "use": [ "lesson", "history" ], "concepts": [ "concept/closure", "concept/complex-number", "concept/rational-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-8fc7c4d150", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "86", "location": "COMPLEX NUMBERS", "latex": "It should be observed that it is not always true that the principal value of~$\\am(zz')$ is the sum of the principal values of $\\am z$ and~$\\am z'$. For example, if $z = z' = -1 + i$, then the principal values of the amplitudes of $z$~and~$z'$ are each~$\\frac{3}{4}\\pi$. But $zz' = -2i$, and the principal value of~$\\am(zz')$ is~$-\\frac{1}{2}\\pi$ and not~$\\frac{3}{2}\\pi$.", "markdown": "It should be observed that it is not always true that the principal value of $\\am(zz')$ is the sum of the principal values of $\\am z$ and $\\am z'$. For example, if $z = z' = -1 + i$, then the principal values of the amplitudes of $z$ and $z'$ are each $\\frac{3}{4}\\pi$. But $zz' = -2i$, and the principal value of $\\am(zz')$ is $-\\frac{1}{2}\\pi$ and not $\\frac{3}{2}\\pi$.", "why": "A warning, with a concrete counterexample, that principal values of amplitudes do not simply add.", "use": [ "lesson", "website" ], "concepts": [ "concept/amplitude-of-a-complex-number", "concept/principal-value-of-amplitude", "quantity/amplitude-of-a-complex-number", "theorem/modulus-and-amplitude-of-a-product" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-7f28a22a45", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "85", "location": "COMPLEX NUMBERS", "latex": "A line originally lying along~$OX$ will, if turned through any of these angles, come to lie along~$OP$.", "markdown": "A line originally lying along $OX$ will, if turned through any of these angles, come to lie along $OP$.", "why": "Gives a physical picture of why the amplitude has many values, one for each way of turning the line to OP.", "use": [ "lesson" ], "concepts": [ "concept/amplitude-of-a-complex-number", "concept/rotation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-ecf6eadca9", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "85", "location": "COMPLEX NUMBERS", "latex": "It will be observed that the sum~$2x$ of two conjugate numbers and their product $x^{2} + y^{2}$ are both real, that they have the same modulus $\\sqrtp{x^{2} + y^{2}}$ and that their product is equal to the square of the modulus of either.", "markdown": "It will be observed that the sum $2x$ of two conjugate numbers and their product $x^{2} + y^{2}$ are both real, that they have the same modulus $\\sqrtp{x^{2} + y^{2}}$ and that their product is equal to the square of the modulus of either.", "why": "Collects the key properties of conjugates that later explain why they appear as pairs of roots.", "use": [ "lesson" ], "concepts": [ "concept/conjugate-complex-numbers", "concept/modulus-of-a-complex-number", "quantity/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-a71919951f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "86", "location": "COMPLEX NUMBERS", "latex": "Thus the modulus of the reciprocal of~$z$ is the reciprocal of the modulus of~$z$, and the amplitude of the reciprocal is the negative of the amplitude of~$z$.", "markdown": "Thus the modulus of the reciprocal of $z$ is the reciprocal of the modulus of $z$, and the amplitude of the reciprocal is the negative of the amplitude of $z$.", "why": "States what taking a reciprocal does to modulus and amplitude, the step that extends De Moivre's theorem to negative integers.", "use": [ "lesson" ], "concepts": [ "concept/amplitude-of-a-complex-number", "concept/modulus-of-a-complex-number", "theorem/de-moivre-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-814012f45b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "87", "location": "COMPLEX NUMBERS", "latex": "The length~$OU$ is the modulus of the sum of the complex numbers, whereas the sum of their moduli is the total length of the broken line $OPQR\\dots U$, which is not less than~$OU$.", "markdown": "The length $OU$ is the modulus of the sum of the complex numbers, whereas the sum of their moduli is the total length of the broken line $OPQR\\dots U$, which is not less than $OU$.", "why": "Proves the triangle inequality geometrically: a straight line is shortest.", "use": [ "lesson", "website" ], "concepts": [ "concept/modulus-of-a-complex-number", "theorem/triangle-inequality-for-complex-numbers" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-154843bdaa", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "89", "location": "COMPLEX NUMBERS", "latex": "This theorem is sometimes stated as follows: \\emph{in an equation with real coefficients complex roots occur in conjugate pairs}. It should be compared with the result of \\Exs{viii}.~7, which may be stated as follows: \\emph{in an equation with rational coefficients irrational roots occur in conjugate pairs}.", "markdown": "This theorem is sometimes stated as follows: *in an equation with real coefficients complex roots occur in conjugate pairs*. It should be compared with the result of viii. 7, which may be stated as follows: *in an equation with rational coefficients irrational roots occur in conjugate pairs*.", "why": "Shows that the same pairing pattern appears for complex roots and for irrational roots.", "use": [ "lesson", "history" ], "concepts": [ "concept/conjugate-complex-numbers", "theorem/complex-roots-of-a-real-equation-occur-in-conjugate-pairs", "theorem/imaginary-roots-occur-in-pairs" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-6443f5add0", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "95", "location": "COMPLEX NUMBERS", "latex": "Thus \\emph{the general linear transformation is equivalent to the combination of a translation, a magnification, and a rotation}.", "markdown": "Thus *the general linear transformation is equivalent to the combination of a translation, a magnification, and a rotation*.", "why": "Breaks z = aZ + b into three familiar geometric motions.", "use": [ "lesson", "website" ], "concepts": [ "concept/linear-transformation", "concept/magnification", "concept/rotation", "concept/translation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-7024a2841f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "95", "location": "COMPLEX NUMBERS", "latex": "The general bilinear transformation is the most general type of transformation for which one and only one value of~$z$ corresponds to each value of~$Z$, and conversely.", "markdown": "The general bilinear transformation is the most general type of transformation for which one and only one value of $z$ corresponds to each value of $Z$, and conversely.", "why": "Says why the bilinear transformation matters: it is exactly the one-to-one case.", "use": [ "lesson" ], "concepts": [ "concept/bilinear-transformation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-f0bdc3bfa6", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "99", "location": "COMPLEX NUMBERS", "latex": "The only possible value of~$r$ is~$\\sqrt[n]{\\rho}$, the ordinary arithmetical $n$th~root of~$\\rho$; and in order that the last two equations should be satisfied it is necessary and sufficient that $n\\theta = \\phi + 2k\\pi$, where $k$~is an integer, or \\[ \\theta = (\\phi + 2k\\pi)/n. \\]", "markdown": "The only possible value of $r$ is $\\sqrt[n]{\\rho}$, the ordinary arithmetical $n$th root of $\\rho$; and in order that the last two equations should be satisfied it is necessary and sufficient that $n\\theta = \\phi + 2k\\pi$, where $k$ is an integer, or = (+ 2k)/n.", "why": "Shows the key step in solving z^n = a: the modulus is fixed and the angle is determined up to multiples of 2\\pi/n.", "use": [ "lesson" ], "concepts": [ "concept/root-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-6a76ef8834", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "101", "location": "COMPLEX NUMBERS", "latex": "Raising each of these expressions to the power~$p$ (where $p$~is any integer positive or negative), we obtain the theorem that one of the values of $(\\cos\\theta + i\\sin\\theta)^{p/q}$ is $\\cos(p\\theta/q) + i\\sin(p\\theta/q)$, or that \\emph{if $\\alpha$~is any rational number then one of the values of $(\\cos\\theta + i\\sin\\theta)^{\\alpha}$ is} \\[ \\cos\\alpha\\theta + i\\sin\\alpha\\theta. \\] This is a generalised form of De~Moivre's Theorem (\\SecNo[§]{45}).", "markdown": "Raising each of these expressions to the power $p$ (where $p$ is any integer positive or negative), we obtain the theorem that one of the values of $(\\cos\\theta + i\\sin\\theta)^{p/q}$ is $\\cos(p\\theta/q) + i\\sin(p\\theta/q)$, or that *if $\\alpha$ is any rational number then one of the values of $(\\cos\\theta + i\\sin\\theta)^{\\alpha}$ is* + i. This is a generalised form of De Moivre’s Theorem ([§]45).", "why": "Extends De Moivre's theorem from integer to rational powers, with the careful wording \"one of the values\".", "use": [ "lesson" ], "concepts": [ "concept/root-of-a-complex-number", "theorem/de-moivre-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-8e4c7d4888", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "100", "location": "COMPLEX NUMBERS", "latex": "The problem of finding the accurate value of~$\\omega_{n}$ in a numerical form involving square roots only, as in the formula $\\omega_{3} = \\frac{1}{2}(-1 + i\\sqrt{3})$, is the algebraical equivalent of the geometrical problem of inscribing a regular polygon of $n$~sides in a circle of unit radius by Euclidean methods, \\ie\\ by ruler and compasses.", "markdown": "The problem of finding the accurate value of $\\omega_{n}$ in a numerical form involving square roots only, as in the formula $\\omega_{3} = \\frac{1}{2}(-1 + i\\sqrt{3})$, is the algebraical equivalent of the geometrical problem of inscribing a regular polygon of $n$ sides in a circle of unit radius by Euclidean methods, *i.e.* by ruler and compasses.", "why": "Connects an algebra question about roots of unity to the geometry of ruler-and-compass polygons.", "use": [ "lesson", "website" ], "concepts": [ "concept/root-of-unity", "method/euclidean-construction" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-4a525de0d4", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "102", "location": "COMPLEX NUMBERS", "latex": "Prove that $|a + b|^{2} + |a - b|^{2} = 2\\{|a|^{2} + |b|^{2}\\}$. [This is the analytical equivalent of the geometrical theorem that, if $M$~is the middle point of~$PQ$, then $OP^{2} + OQ^{2} = 2OM^{2} + 2MP^{2}$.]", "markdown": "Prove that $|a + b|^{2} + |a - b|^{2} = 2\\{|a|^{2} + |b|^{2}\\}$. [This is the analytical equivalent of the geometrical theorem that, if $M$ is the middle point of $PQ$, then $OP^{2} + OQ^{2} = 2OM^{2} + 2MP^{2}$.]", "why": "Shows a complex-number identity as the algebraic twin of a classical geometric theorem.", "use": [ "lesson" ], "concepts": [ "concept/complex-number", "quantity/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-066a3269c6", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "103", "location": "COMPLEX NUMBERS", "latex": "The aggregate of pairs of real or complex values of $x$~and~$y$ which satisfy the equation is called an \\emph{imaginary straight line}; the pairs of values are called \\emph{imaginary points}, and are said \\emph{to lie on the line}.", "markdown": "The aggregate of pairs of real or complex values of $x$ and $y$ which satisfy the equation is called an *imaginary straight line*; the pairs of values are called *imaginary points*, and are said *to lie on the line*.", "why": "Shows how lines and points are extended to complex coordinates.", "use": [ "lesson" ], "concepts": [ "concept/imaginary-straight-line" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-27df2b0342", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "131", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "The function $\\phi(n)$ will be said to increase steadily with~$n$ if $\\phi(n + 1) \\geq \\phi(n)$ for all values of~$n$.", "markdown": "The function $\\phi(n)$ will be said to increase steadily with $n$ if $\\phi(n + 1) \\geq \\phi(n)$ for all values of $n$.", "why": "It gives the exact definition of steady increase, with the equality case allowed, so learners see that a constant stretch still counts.", "use": [ "lesson" ], "concepts": [ "concept/steadily-increasing-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-ccc31e2d74", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "143", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "The commonest example\nof an infinite geometric series is given by an ordinary recurring decimal.", "markdown": "The commonest example of an infinite geometric series is given by an ordinary recurring decimal.", "why": "It links a school-familiar object, the recurring decimal, to the infinite geometric series, which makes the abstract series concrete.", "use": [ "lesson", "website" ], "concepts": [ "concept/geometrical-progression", "concept/recurring-decimal" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-f13b200ee5", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "144", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "Thus every proper fraction can be expressed as a recurring decimal, and conversely.", "markdown": "Thus every proper fraction can be expressed as a recurring decimal, and conversely.", "why": "It closes the argument that fractions and recurring decimals are the same thing, a result a learner can check on examples.", "use": [ "lesson" ], "concepts": [ "concept/proper-algebraic-fraction", "concept/recurring-decimal" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-ed3216b934", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "149", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "This number~$M$ we call the \\emph{upper bound} of~$S$, and we may enunciate the following theorem.", "markdown": "This number $M$ we call the *upper bound* of $S$, and we may enunciate the following theorem.", "why": "It introduces the upper bound by name just before its existence theorem, so the reader knows what the theorem is about.", "use": [ "lesson" ], "concepts": [ "concept/least-upper-bound" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-bd0b0bc969", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "147", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "the function which is equal to $1/(1 - x)$ if $-1 < x < 1$ and\nis undefined for all other values of~$x$.", "markdown": "the function which is equal to $1/(1 - x)$ if $-1 < x < 1$ and is undefined for all other values of $x$.", "why": "It shows that a limit can define a function with a restricted domain, the idea behind a geometric series seen as a function of x.", "use": [ "lesson", "history" ], "concepts": [ "concept/domain-of-definition", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-df6a9893e1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "154", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "The reader will find no difficulty in proving such theorems as the following, which are obvious extensions of theorems already proved for real functions and series.", "markdown": "The reader will find no difficulty in proving such theorems as the following, which are obvious extensions of theorems already proved for real functions and series.", "why": "It tells a learner that the complex-number limit theorems are not new in spirit: they carry over the real-number results already learned.", "use": [ "history", "lesson" ], "concepts": [ "concept/complex-number", "concept/convergent-series", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-dfed52a158", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "157", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "If $\\phi(n)$~steadily increases, and $\\psi(n)$~steadily decreases, as $n$~tends to~$\\infty$, and if $\\psi(n) > \\phi(n)$ for all values of~$n$, then both $\\phi(n)$~and~$\\psi(n)$ tend to limits, and $\\lim\\phi(n) \\leq \\lim\\psi(n)$.", "markdown": "If $\\phi(n)$ steadily increases, and $\\psi(n)$ steadily decreases, as $n$ tends to $\\infty$, and if $\\psi(n) > \\phi(n)$ for all values of $n$, then both $\\phi(n)$ and $\\psi(n)$ tend to limits, and $\\lim\\phi(n) \\leq \\lim\\psi(n)$.", "why": "It gives a clear exercise statement of the squeezing principle for monotone sequences, which a learner can test against a concrete example.", "use": [ "lesson", "website" ], "concepts": [ "concept/function-of-a-positive-integer-variable", "concept/infinite-sequence", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-4a9cbc7dbd", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "111", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\emph{Large} is in fact a word which, standing by itself, has no more absolute meaning in mathematics than in the language of common life.", "markdown": "*Large* is in fact a word which, standing by itself, has no more absolute meaning in mathematics than in the language of common life.", "why": "It shows that 'large' in a limit statement always means large enough for the purpose at hand, which prepares the learner for the definitions that follow.", "use": [ "lesson", "website" ], "concepts": [ "concept/sufficiently-large-values" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-765646ea2d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "118", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "On the other hand we cannot as a rule alter an \\emph{infinite} number of the values of~$\\phi(n)$ without affecting fundamentally its behaviour as $n$~tends to~$\\infty$.", "markdown": "On the other hand we cannot as a rule alter an *infinite* number of the values of $\\phi(n)$ without affecting fundamentally its behaviour as $n$ tends to $\\infty$.", "why": "It explains that changing finitely many values of a sequence leaves its limit unchanged, while changing infinitely many can alter it.", "use": [ "lesson" ], "concepts": [ "concept/infinite-set", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-aa7b6c644a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "107", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "Were the problem however merely that of finding \\emph{some} function of~$x$ to fulfil the condition stated, it would of course present no difficulty whatever.", "markdown": "Were the problem however merely that of finding *some* function of $x$ to fulfil the condition stated, it would of course present no difficulty whatever.", "why": "It makes the learner see that interpolation is only interesting when a simple formula is wanted, not when any function will do.", "use": [ "lesson", "history" ], "concepts": [ "concept/functional-interpolation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-4a1d889a7e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "120", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "A function can only tend to~$+\\infty$ or to~$-\\infty$ if, after a certain value of~$n$, it maintains a constant sign.", "markdown": "A function can only tend to $+\\infty$ or to $-\\infty$ if, after a certain value of $n$, it maintains a constant sign.", "why": "It corrects the idea that a function which is always large must tend to infinity, by giving a clear necessary condition.", "use": [ "lesson" ], "concepts": [ "concept/limit", "concept/tends-to-infinity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-43efba2dd7", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "108", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "For mathematicians have succeeded in discovering a function (the Gamma-function) which possesses the desired property and many other interesting and important properties besides.", "markdown": "For mathematicians have succeeded in discovering a function (the Gamma-function) which possesses the desired property and many other interesting and important properties besides.", "why": "It is a historical note showing how the attempt to interpolate n! led to a major new function.", "use": [ "history", "website" ], "concepts": [ "concept/factorial", "concept/functional-interpolation", "concept/gamma-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-f89a92df3b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "120", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "Take for instance the case of $k > 0$. Let $\\Delta$ be any assigned number, however large.", "markdown": "Take for instance the case of $k > 0$. Let $\\Delta$ be any assigned number, however large.", "why": "It shows a learner how to turn the phrase tends to infinity into a formal threshold argument by choosing n_0 for any given bound.", "use": [ "lesson" ], "concepts": [ "concept/limit", "concept/tends-to-infinity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-bc81274a88", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "120", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "This number is evidently not divisible by any of $2$,~$3$, $5$,~\\dots~$N$, since the remainder when it is divided by any of these numbers is~$1$.", "markdown": "This number is evidently not divisible by any of $2$, $3$, $5$, … $N$, since the remainder when it is divided by any of these numbers is $1$.", "why": "It shows the key step of Euclid's argument, where a product plus one leaves remainder 1 on division by each listed prime.", "use": [ "lesson" ], "concepts": [ "concept/divisibility", "concept/prime-number", "concept/proof", "concept/remainder" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-b15fb71f23", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "120", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "There are however, as was first shown by Euclid, infinitely many primes.", "markdown": "There are however, as was first shown by Euclid, infinitely many primes.", "why": "It gives the historical origin of the fact that the list of primes never ends.", "use": [ "history" ], "concepts": [ "concept/prime-number", "person/euclid", "theorem/infinitude-of-primes" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-f81287ae67", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "123", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "Such an assertion is palpably absurd when made of a \\emph{fixed} number such as~$\\cos\\frac{1}{2}\\theta\\pi$, which is not zero.", "markdown": "Such an assertion is palpably absurd when made of a *fixed* number such as $\\cos\\frac{1}{2}\\theta\\pi$, which is not zero.", "why": "It corrects a common error by showing that a product tends to zero only if the varying factor tends to zero, not a fixed nonzero factor.", "use": [ "lesson" ], "concepts": [ "concept/cosine", "concept/limit", "concept/sufficiently-large-values" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-7195191e9b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "121", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "When $\\phi(n)$ does not tend to a limit, nor to~$+\\infty$, nor to~$-\\infty$, as $n$~tends to~$\\infty$, we say that $\\phi(n)$ \\Emph{oscillates} as $n$~tends to~$\\infty$.", "markdown": "When $\\phi(n)$ does not tend to a limit, nor to $+\\infty$, nor to $-\\infty$, as $n$ tends to $\\infty$, we say that $\\phi(n)$ **** as $n$ tends to $\\infty$.", "why": "Gives the plain definition of oscillation as the case that is none of the three tidier behaviours.", "use": [ "lesson", "website" ], "concepts": [ "concept/limit", "concept/oscillation", "concept/tends-to-infinity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-ed2f42a845", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "121", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "Oscillation is defined in a purely negative manner: a function oscillates when it does not do certain other things.", "markdown": "Oscillation is defined in a purely negative manner: a function oscillates when it does not do certain other things.", "why": "Warns learners that 'oscillates' means 'fails to settle', not 'swings back and forth regularly'.", "use": [ "lesson", "website" ], "concepts": [ "concept/oscillation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-7c99adc882", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "132", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "We divide the real numbers~$\\xi$ into two classes $L$~and~$R$, putting $\\xi$~in $L$~or~$R$ according as $\\phi(n) \\geq \\xi$ for some value of~$n$ (and so of course for all greater values), or $\\phi(n) < \\xi$ for all values of~$n$.", "markdown": "We divide the real numbers $\\xi$ into two classes $L$ and $R$, putting $\\xi$ in $L$ or $R$ according as $\\phi(n) \\geq \\xi$ for some value of $n$ (and so of course for all greater values), or $\\phi(n) < \\xi$ for all values of $n$.", "why": "It shows how a Dedekind section is built from a sequence, which is the heart of the proof of the monotone theorem.", "use": [ "lesson", "history" ], "concepts": [ "concept/dedekind-section", "theorem/dedekind-s-theorem", "theorem/monotone-convergence-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-695f599de5", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "143", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "It is perhaps hardly necessary to point out that the theorem is not true if the condition that every~$u_{n}$ is positive is not fulfilled.", "markdown": "It is perhaps hardly necessary to point out that the theorem is not true if the condition that every $u_{n}$ is positive is not fulfilled.", "why": "It warns that the comparison and bounded-sum results need the positivity hypothesis.", "use": [ "lesson" ], "concepts": [ "concept/convergent-series", "theorem/comparison-test" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-51855c17eb", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "111", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "It is a truism that in common life a number which is large in one connection is small in another; $6$~goals is a large score in a football match, but $6$~runs is not a large score in a cricket match; and $400$~runs is a large score, but £$400$~is not a large income: and so of course in mathematics \\emph{large} generally means \\emph{large enough}, and what is large enough for one purpose may not be large enough for another.", "markdown": "It is a truism that in common life a number which is large in one connection is small in another; $6$ goals is a large score in a football match, but $6$ runs is not a large score in a cricket match; and $400$ runs is a large score, but £$400$ is not a large income: and so of course in mathematics *large* generally means *large enough*, and what is large enough for one purpose may not be large enough for another.", "why": "Uses football, cricket and income to show that 'large' in mathematics always means 'large enough for the purpose'.", "use": [ "lesson", "website" ], "concepts": [ "concept/sufficiently-large-values" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-e970eb4063", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "112", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "The reader cannot too strongly impress upon himself that when we say that $n$~`tends to~$\\infty$' we mean simply that $n$~is supposed to assume a series of values which increase continually and without limit.", "markdown": "The reader cannot too strongly impress upon himself that when we say that $n$ ‘tends to $\\infty$’ we mean simply that $n$ is supposed to assume a series of values which increase continually and without limit.", "why": "Warns learners that 'tends to infinity' describes unending growth, not arrival at a number.", "use": [ "lesson", "website" ], "concepts": [ "concept/infinity", "concept/tends-to-infinity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-9b6b1ed7fd", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "114", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "The reader should imagine himself confronted by an opponent who questions the truth of the statement. He would name a series of numbers growing smaller and smaller. He might begin with~$.001$. The reader would reply that $1/n < .001$ as soon as $n > 1000$. The opponent would be bound to admit this, but would try again with some smaller number, such as~$.000\\MS000\\MS1$.", "markdown": "The reader should imagine himself confronted by an opponent who questions the truth of the statement. He would name a series of numbers growing smaller and smaller. He might begin with $.001$. The reader would reply that $1/n < .001$ as soon as $n > 1000$. The opponent would be bound to admit this, but would try again with some smaller number, such as $.000\\MS000\\MS1$.", "why": "Frames the definition of a limit as a challenge-and-response game with an opponent, giving it a concrete meaning.", "use": [ "lesson", "website" ], "concepts": [ "concept/limit", "concept/threshold-index" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-c70f879e42", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "119", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "The reader cannot impress these facts too strongly on his mind. \\Emph{A limit is not a value of the function}: it is something quite distinct from these values, though it is defined by its relations to them and may possibly be equal to some of them.", "markdown": "The reader cannot impress these facts too strongly on his mind. **limit is not a value of the function**: it is something quite distinct from these values, though it is defined by its relations to them and may possibly be equal to some of them.", "why": "Warns against the common mistake of assuming a function must reach its limit.", "use": [ "lesson", "website" ], "concepts": [ "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-042da5543a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "108", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "This is a case in which attempts to solve the problem of interpolation have led to important advances in mathematics. For mathematicians have succeeded in discovering a function (the Gamma-function) which possesses the desired property and many other interesting and important properties besides.", "markdown": "This is a case in which attempts to solve the problem of interpolation have led to important advances in mathematics. For mathematicians have succeeded in discovering a function (the Gamma-function) which possesses the desired property and many other interesting and important properties besides.", "why": "Shows how asking what n! could mean between the integers led to a new and important function.", "use": [ "history", "website" ], "concepts": [ "concept/functional-interpolation", "concept/gamma-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-3779019c30", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "116", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "The function $\\phi(n)$ is said to tend to the limit~$l$ as $n$~tends to~$\\infty$, if, however small be the positive number~$\\DELTA$, $\\phi(n)$~differs from~$l$ by less than~$\\DELTA$ for sufficiently large values of~$n$; that is to say if, however small be the positive number~$\\DELTA$, we can determine a number~$n_{0}(\\DELTA)$ corresponding to~$\\DELTA$, such that $\\phi(n)$~differs from~$l$ by less than~$\\DELTA$ for all values of~$n$ greater than or equal to~$n_{0}(\\DELTA)$.", "markdown": "The function $\\phi(n)$ is said to tend to the limit $l$ as $n$ tends to $\\infty$, if, however small be the positive number $\\DELTA$, $\\phi(n)$ differs from $l$ by less than $\\DELTA$ for sufficiently large values of $n$; that is to say if, however small be the positive number $\\DELTA$, we can determine a number $n_{0}(\\DELTA)$ corresponding to $\\DELTA$, such that $\\phi(n)$ differs from $l$ by less than $\\DELTA$ for all values of $n$ greater than or equal to $n_{0}(\\DELTA)$.", "why": "Gives the book's formal definition of a limit of a function of n.", "use": [ "lesson", "website" ], "concepts": [ "concept/limit", "concept/sufficiently-large-values", "concept/threshold-index" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-5c1faddaeb", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "109", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "If $n$~is any positive integer, such as $1000$, $1,000,000$ or any number we like to think of, then there are more than $n$ positive integers. Thus, if the number we think of is $1,000,000$, there are obviously at least $1,000,001$ positive integers.", "markdown": "If $n$ is any positive integer, such as $1000$, $1,000,000$ or any number we like to think of, then there are more than $n$ positive integers. Thus, if the number we think of is $1,000,000$, there are obviously at least $1,000,001$ positive integers.", "why": "Makes the idea of an infinite class concrete with a simple 'name any number' argument.", "use": [ "lesson" ], "concepts": [ "concept/infinite-set" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-a4ee3819e0", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "118", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "We may obviously alter the values of~$\\phi(n)$ for any finite number of values of~$n$, in any way we please, without in the least affecting the behaviour of~$\\phi(n)$ as $n$~tends to~$\\infty$.", "markdown": "We may obviously alter the values of $\\phi(n)$ for any finite number of values of $n$, in any way we please, without in the least affecting the behaviour of $\\phi(n)$ as $n$ tends to $\\infty$.", "why": "Tells learners that limits depend only on the long-run behaviour, not on any finite set of early values.", "use": [ "lesson" ], "concepts": [ "concept/limit", "concept/sufficiently-large-values" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-5485ef19ef", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "122", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "But now consider $\\phi(n) = (-1)^{n}n$, the values of which are $-1$, $2$, $-3$, $4$, $-5$,~\\dots. This function oscillates, for it does not tend to a limit, nor to~$+\\infty$, nor to~$-\\infty$. And in this case we cannot assign any limit beyond which the numerical value of the terms does not rise.", "markdown": "But now consider $\\phi(n) = (-1)^{n}n$, the values of which are $-1$, $2$, $-3$, $4$, $-5$, …. This function oscillates, for it does not tend to a limit, nor to $+\\infty$, nor to $-\\infty$. And in this case we cannot assign any limit beyond which the numerical value of the terms does not rise.", "why": "A concrete example that motivates the split between finite and infinite oscillation.", "use": [ "lesson" ], "concepts": [ "concept/finite-oscillation", "concept/infinite-oscillation", "concept/oscillation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-3c0919c195", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "123", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "It should be observed that in this case $\\phi(2k + 1)$~is always less than~$\\phi(2k)$, so that the function progresses to infinity by a continual series of steps forwards and backwards. It does not however `oscillate' according to our definition of the term.", "markdown": "It should be observed that in this case $\\phi(2k + 1)$ is always less than $\\phi(2k)$, so that the function progresses to infinity by a continual series of steps forwards and backwards. It does not however ‘oscillate’ according to our definition of the term.", "why": "Shows that a function can wobble on the way up and still tend to infinity, so wobbling alone is not oscillation in this sense.", "use": [ "lesson" ], "concepts": [ "concept/oscillation", "concept/tends-to-infinity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-1b4fe40821", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "120", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "Euclid's proof is as follows. If there are only a finite number of primes, let them be $1$,~$2$, $3$, $5$, $7$, $11$,~\\dots~$N$. Consider the number $1 + (1 · 2 · 3 · 5 · 7 · 11 \\dots N)$. This number is evidently not divisible by any of $2$,~$3$, $5$,~\\dots~$N$, since the remainder when it is divided by any of these numbers is~$1$. It is therefore not divisible by any prime save~$1$, and is therefore itself prime, which is contrary to our hypothesis.", "markdown": "Euclid’s proof is as follows. If there are only a finite number of primes, let them be $1$, $2$, $3$, $5$, $7$, $11$, … $N$. Consider the number $1 + (1 · 2 · 3 · 5 · 7 · 11 \\dots N)$. This number is evidently not divisible by any of $2$, $3$, $5$, … $N$, since the remainder when it is divided by any of these numbers is $1$. It is therefore not divisible by any prime save $1$, and is therefore itself prime, which is contrary to our hypothesis.", "why": "A classic proof by contradiction that the primes never run out (note that Hardy here counts 1 among the primes, which modern usage does not).", "use": [ "lesson", "history", "website" ], "concepts": [ "concept/prime-number", "person/euclid", "theorem/infinitude-of-primes" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-785e65a464", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "125", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "That a theorem is `obvious' in this sense does not prove that it is true, since the most confident of the intuitive judgments of common sense are often found to be mistaken; and even if the theorem is true, the fact that it is also `obvious' is no reason for not proving it, if a proof can be found. The object of mathematics is to prove that certain premises imply certain conclusions; and the fact that the conclusions may be as `obvious' as the premises never detracts from the necessity, and often not even from the interest of the proof.", "markdown": "That a theorem is ‘obvious’ in this sense does not prove that it is true, since the most confident of the intuitive judgments of common sense are often found to be mistaken; and even if the theorem is true, the fact that it is also ‘obvious’ is no reason for not proving it, if a proof can be found. The object of mathematics is to prove that certain premises imply certain conclusions; and the fact that the conclusions may be as ‘obvious’ as the premises never detracts from the necessity, and often not even from the interest of the proof.", "why": "Explains why mathematicians prove things that seem obvious.", "use": [ "lesson", "website" ], "concepts": [ "concept/mathematics", "concept/proof" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-ecb7f4b11e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "125", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "The argument which the reader will at once form in his mind is roughly this: `when $n$~is large, $\\phi(n)$~is nearly equal to~$a$ and $\\psi(n)$ to~$b$, and therefore their sum is nearly equal to $a + b$'. It is well to state the argument quite formally, however.", "markdown": "The argument which the reader will at once form in his mind is roughly this: ‘when $n$ is large, $\\phi(n)$ is nearly equal to $a$ and $\\psi(n)$ to $b$, and therefore their sum is nearly equal to $a + b$’. It is well to state the argument quite formally, however.", "why": "Shows how an intuitive idea becomes a rigorous proof of the limit of a sum.", "use": [ "lesson" ], "concepts": [ "concept/limit", "theorem/limit-of-a-sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-a19516a84e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "123", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "The answer is to be found, of course, in the meaning of the phrase `very small' as used in this connection. When we say `$\\phi(n)$~is very small' for large values of~$n$, we mean that we can choose~$n_{0}$ so that $\\phi(n)$~is numerically smaller than \\emph{any} assigned number, if \\DPchg{$n$~is sufficiently large}{$n \\geq n_{0}$}. Such an assertion is palpably absurd when made of a \\emph{fixed} number such as~$\\cos\\frac{1}{2}\\theta\\pi$, which is not zero.", "markdown": "The answer is to be found, of course, in the meaning of the phrase ‘very small’ as used in this connection. When we say ‘$\\phi(n)$ is very small’ for large values of $n$, we mean that we can choose $n_{0}$ so that $\\phi(n)$ is numerically smaller than *any* assigned number, if $n$ is sufficiently large$n \\geq n_{0}$. Such an assertion is palpably absurd when made of a *fixed* number such as $\\cos\\frac{1}{2}\\theta\\pi$, which is not zero.", "why": "Clears up a common mistake: 'very small' means smaller than any assigned number, which a fixed non-zero number can never be.", "use": [ "lesson", "website" ], "concepts": [ "concept/limit", "concept/oscillation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-f565b2eda4", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "132", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "That is to say, while there are in general \\emph{five} alternatives as to the behaviour of a function, there are \\emph{two} only for this special kind of function.", "markdown": "That is to say, while there are in general *five* alternatives as to the behaviour of a function, there are *two* only for this special kind of function.", "why": "Shows in one sentence what the monotone convergence theorem buys: five possible behaviours shrink to two.", "use": [ "lesson", "website" ], "concepts": [ "concept/limit", "concept/oscillation", "concept/steadily-increasing-function", "concept/tends-to-infinity", "theorem/monotone-convergence-theorem", "theorem/steadily-increasing-function-limit-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-72b04a0133", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "133", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "The great importance of these theorems lies in the fact that they give us (what we have so far been without) a means of deciding, in a great many cases, whether a given function of~$n$ does or does not tend to a limit as $n \\to \\infty$, \\emph{without requiring us to be able to guess or otherwise infer beforehand what the limit is}.", "markdown": "The great importance of these theorems lies in the fact that they give us (what we have so far been without) a means of deciding, in a great many cases, whether a given function of $n$ does or does not tend to a limit as $n \\to \\infty$, *without requiring us to be able to guess or otherwise infer beforehand what the limit is*.", "why": "Explains why proving a limit exists without knowing its value is a real advance.", "use": [ "lesson", "website" ], "concepts": [ "concept/limit", "theorem/monotone-convergence-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-2e9c1d8b31", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "132", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "It is to be observed that we do not exclude the case in which $\\phi(n)$ has the \\emph{same} value for several values of~$n$; all we exclude is possible \\emph{decrease}.", "markdown": "It is to be observed that we do not exclude the case in which $\\phi(n)$ has the *same* value for several values of $n$; all we exclude is possible *decrease*.", "why": "Warns that 'steadily increasing' here allows flat stretches.", "use": [ "lesson" ], "concepts": [ "concept/infinite-sequence", "concept/steadily-increasing-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-64f2da1adc", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "133", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "It should be noticed that the limit may be equal to~$K$: if \\eg\\ $\\phi(n) = 3 - (1/n)$, then every value of~$\\phi(n)$ is less than~$3$, but the limit is equal to~$3$.", "markdown": "It should be noticed that the limit may be equal to $K$: if *e.g.* $\\phi(n) = 3 - (1/n)$, then every value of $\\phi(n)$ is less than $3$, but the limit is equal to $3$.", "why": "A short example showing that a strict bound on every term does not give a strict bound on the limit.", "use": [ "lesson" ], "concepts": [ "concept/limit", "concept/steadily-increasing-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-0a202f649d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "131", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "Since $-\\phi(n)$ always increases if $\\phi(n)$ always decreases, it is not necessary to consider the two kinds of functions separately; for theorems proved for one kind can at once be extended to the other.", "markdown": "Since $-\\phi(n)$ always increases if $\\phi(n)$ always decreases, it is not necessary to consider the two kinds of functions separately; for theorems proved for one kind can at once be extended to the other.", "why": "Shows a reduction trick: prove the increasing case and get the decreasing case for free.", "use": [ "lesson" ], "concepts": [ "concept/steadily-increasing-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-c68f957677", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "142", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "The reader may be tempted to think that the converse of the theorem is true and that if $\\lim u_{n} = 0$ then the series~$\\sum u_{n}$ must be convergent. That this is not the case is easily seen from an example.", "markdown": "The reader may be tempted to think that the converse of the theorem is true and that if $\\lim u_{n} = 0$ then the series $\\sum u_{n}$ must be convergent. That this is not the case is easily seen from an example.", "why": "Flags a common mistake: terms tending to zero is not enough for a series to converge.", "use": [ "lesson", "website" ], "concepts": [ "concept/convergent-series", "concept/harmonic-series", "theorem/necessary-condition-for-convergence", "theorem/terms-of-a-convergent-series-tend-to-zero" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-fdec29db62", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "141", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "The reader should be warned that the words `divergent' and `oscillatory' are used differently by different writers.", "markdown": "The reader should be warned that the words ‘divergent’ and ‘oscillatory’ are used differently by different writers.", "why": "Reminds learners that terminology varies between authors, so definitions must be checked.", "use": [ "lesson", "history" ], "concepts": [ "concept/divergent-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-baba544a79", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "140", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "Later on we shall be able to identify this function with the \\emph{Napierian logarithm} of~$x$.", "markdown": "Later on we shall be able to identify this function with the *Napierian logarithm* of $x$.", "why": "Hints that a limit defined here will turn out to be the logarithm.", "use": [ "history", "website" ], "concepts": [ "concept/logarithm", "theorem/limit-of-n-nth-root-of-x-1" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-7609bf0dc0", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "161", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "The reader, if he desires to become expert in dealing with questions about limits, should study the argument above with great care. It is very often necessary, in proving the limit of some given expression to be zero, to split it into two parts which have to be proved to have the limit zero in slightly different ways. When this is the case the proof is never very easy.", "markdown": "The reader, if he desires to become expert in dealing with questions about limits, should study the argument above with great care. It is very often necessary, in proving the limit of some given expression to be zero, to split it into two parts which have to be proved to have the limit zero in slightly different ways. When this is the case the proof is never very easy.", "why": "Gives the learner honest advice that limit proofs often need splitting and take real effort.", "use": [ "lesson", "website" ], "concepts": [ "concept/arithmetical-mean", "concept/limit", "theorem/limit-of-the-arithmetic-mean-of-a-sequence" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-564ebcd161", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "161", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "The point of the proof is this: we have to prove that $(t_{1} + t_{2} + \\dots + t_{n})/n$ is small when $n$~is large, the~$t$'s being small when their suffixes are large. We split up the terms in the bracket into two groups. The terms in the first group are not all small, but their number is small compared with~$n$. The number in the second group is \\emph{not} small compared with~$n$, but the terms are all small, and their number at any rate less than~$n$, so that their sum is small compared with~$n$. Hence each of the parts into which $(t_{1} + t_{2} + \\dots + t_{n})/n$ has been divided is small when $n$~is large.", "markdown": "The point of the proof is this: we have to prove that $(t_{1} + t_{2} + \\dots + t_{n})/n$ is small when $n$ is large, the $t$’s being small when their suffixes are large. We split up the terms in the bracket into two groups. The terms in the first group are not all small, but their number is small compared with $n$. The number in the second group is *not* small compared with $n$, but the terms are all small, and their number at any rate less than $n$, so that their sum is small compared with $n$. Hence each of the parts into which $(t_{1} + t_{2} + \\dots + t_{n})/n$ has been divided is small when $n$ is large.", "why": "Explains in plain words the strategy behind a two-part epsilon argument.", "use": [ "lesson" ], "concepts": [ "concept/arithmetical-mean", "concept/infinite-sequence", "concept/limit", "theorem/limit-of-the-arithmetic-mean-of-a-sequence" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-865bd1d385", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "156", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "If $z^{n} \\to l$ then $z^{n+1} \\to l$, by~\\Eq{(1)} of \\SecNo[§]{86}. But, by~\\Eq{(4)} of \\SecNo[§]{86}, \\[ z^{n+1} = zz^{n} \\to zl, \\] and therefore $l = zl$, which is only possible if (\\ia)~$l = 0$ or (\\ib)~$z = 1$. If $z = 1$ then $\\lim z^{n} = 1$. Apart from this special case the limit, if it exists, can only be zero.", "markdown": "If $z^{n} \\to l$ then $z^{n+1} \\to l$, by (1) of [§]86. But, by (4) of [§]86, z^n+1 = zz^n zl, and therefore $l = zl$, which is only possible if (*a*) $l = 0$ or (*b*) $z = 1$. If $z = 1$ then $\\lim z^{n} = 1$. Apart from this special case the limit, if it exists, can only be zero.", "why": "Shows a slick way to find what a limit must be before proving that it exists.", "use": [ "lesson" ], "concepts": [ "theorem/limit-of-a-constant-multiple", "theorem/limit-of-a-power-of-a-complex-number", "theorem/limit-of-a-shifted-sequence", "theorem/limit-of-z-n" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-d9a95da2e7", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "155", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "If $\\rho(n)$ and~$\\sigma(n)$ both converge to zero then it is plain that $\\sqrtp{\\rho^{2} + \\sigma^{2}}$ does so. The converse follows from the fact that the numerical value of~$\\rho$ or~$\\sigma$ cannot be greater than $\\sqrtp{\\rho^{2} + \\sigma^{2}}$.", "markdown": "If $\\rho(n)$ and $\\sigma(n)$ both converge to zero then it is plain that $\\sqrtp{\\rho^{2} + \\sigma^{2}}$ does so. The converse follows from the fact that the numerical value of $\\rho$ or $\\sigma$ cannot be greater than $\\sqrtp{\\rho^{2} + \\sigma^{2}}$.", "why": "Shows why convergence of a complex function reduces to convergence of its modulus.", "use": [ "lesson" ], "concepts": [ "concept/modulus-of-a-complex-number", "quantity/modulus-of-a-complex-number", "theorem/criterion-for-convergence-of-a-complex-sequence" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-3b638d657f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "156", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "If we change~$\\theta$ into~$\\theta + \\pi$, we see that these results hold also for negative values of~$r$ numerically less than~$1$. Thus they hold when $-1 < r < 1$.", "markdown": "If we change $\\theta$ into $\\theta + \\pi$, we see that these results hold also for negative values of $r$ numerically less than $1$. Thus they hold when $-1 < r < 1$.", "why": "Shows how a simple substitution extends a result from positive r to the whole interval (-1, 1).", "use": [ "lesson" ], "concepts": [ "concept/geometrical-progression", "concept/polar-form-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-16f35f7a55", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "161", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "This example proves that the converse of Ex.~27 is not true: for $s_{n}$~oscillates as $n \\to \\infty$.", "markdown": "This example proves that the converse of Ex. 27 is not true: for $s_{n}$ oscillates as $n \\to \\infty$.", "why": "Warns that a mean can converge even when the sequence itself does not.", "use": [ "lesson", "website" ], "concepts": [ "concept/arithmetical-mean", "concept/oscillation", "theorem/limit-of-the-arithmetic-mean-of-a-sequence" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-97a8f08db0", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "143", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "the series $1 + r + r^{2} + \\dots$ diverges to~$+\\infty$ if $r \\geq 1$, converges to $1/(1 - r)$ if $-1 < r < 1$, oscillates finitely if $r = -1$, and oscillates infinitely if $r < -1$.", "markdown": "the series $1 + r + r^{2} + \\dots$ diverges to $+\\infty$ if $r \\geq 1$, converges to $1/(1 - r)$ if $-1 < r < 1$, oscillates finitely if $r = -1$, and oscillates infinitely if $r < -1$.", "why": "Gives the complete four-way behaviour of the geometric series in one sentence.", "use": [ "lesson", "website" ], "concepts": [ "concept/convergent-series", "concept/divergent-series", "concept/geometrical-progression", "concept/oscillation", "concept/sum-to-infinity", "theorem/sum-of-an-infinite-geometric-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-dad1ae8d16", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "144", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "For example, $.217 = .216\\DPmod{\\dot{9}}{\\Repeat{9}}$. Thus every proper fraction can be expressed as a recurring decimal, and conversely.", "markdown": "For example, $.217 = .216\\DPmod{\\dot{9}}{\\Repeat{9}}$. Thus every proper fraction can be expressed as a recurring decimal, and conversely.", "why": "Shows that a terminating decimal can be rewritten as a recurring one ending in 9's, so every proper fraction is a recurring decimal.", "use": [ "lesson" ], "concepts": [ "concept/recurring-decimal" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-b479e331d1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "145", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "It also follows that every decimal which does not recur represents some \\emph{irrational} number between $0$~and~$1$. Conversely, any such number can be expressed as such a decimal.", "markdown": "It also follows that every decimal which does not recur represents some *irrational* number between $0$ and $1$. Conversely, any such number can be expressed as such a decimal.", "why": "Links irrational numbers to non-recurring decimals in both directions.", "use": [ "lesson", "website" ], "concepts": [ "concept/irrational-number", "concept/recurring-decimal" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-43ebf85a49", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "145", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "Since the number of primes is infinite the decimal does not terminate. Nor can it recur: for if it did we could determine $m$~and~$p$ so that $m$,~$m + p$, $m + 2p$, $m + 3p$,~\\dots\\ are all prime numbers; and this is absurd, since the series includes $m + mp$.", "markdown": "Since the number of primes is infinite the decimal does not terminate. Nor can it recur: for if it did we could determine $m$ and $p$ so that $m$, $m + p$, $m + 2p$, $m + 3p$, … are all prime numbers; and this is absurd, since the series includes $m + mp$.", "why": "A short, vivid proof that a decimal built from the primes is irrational.", "use": [ "lesson", "website" ], "concepts": [ "concept/irrational-number", "concept/prime-number", "concept/recurring-decimal" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-4a8e547287", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "149", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "An infinite aggregate of numbers does not necessarily possess a least member.", "markdown": "An infinite aggregate of numbers does not necessarily possess a least member.", "why": "Warns that an infinite set need not have a least member, which is why the least upper bound needs proof.", "use": [ "lesson" ], "concepts": [ "concept/least-upper-bound", "concept/set" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-9d3c843bbf", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "149", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "This number~$M$ is not exceeded by any member of~$S$, but every number less than~$M$ is exceeded by at least one member of~$S$.", "markdown": "This number $M$ is not exceeded by any member of $S$, but every number less than $M$ is exceeded by at least one member of $S$.", "why": "States precisely what makes M the least upper bound.", "use": [ "lesson" ], "concepts": [ "concept/bounded-function", "concept/least-upper-bound" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-d8a63f7c14", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "153", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "The theoretical importance of the `general principle of convergence' can hardly be overestimated. Like the theorems of \\SecNo[§]{69}, it gives us a means of deciding whether a function~$\\phi(n)$ tends to a limit or not, without requiring us to be able to tell beforehand what the limit, if it exists, must be; and it has not the limitations inevitable in theorems of such a special character as those of \\SecNo[§]{69}.", "markdown": "The theoretical importance of the ‘general principle of convergence’ can hardly be overestimated. Like the theorems of [§]69, it gives us a means of deciding whether a function $\\phi(n)$ tends to a limit or not, without requiring us to be able to tell beforehand what the limit, if it exists, must be; and it has not the limitations inevitable in theorems of such a special character as those of [§]69.", "why": "Explains why a test for convergence that needs no prior knowledge of the limit is valuable.", "use": [ "lesson", "history" ], "concepts": [ "concept/bounded-function", "concept/limit", "theorem/general-principle-of-convergence" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-12a0a7d696", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "183", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "The proof just given is somewhat subtle and indirect, and it may be well, in view of the great importance of the theorem, to indicate alternative lines of proof.", "markdown": "The proof just given is somewhat subtle and indirect, and it may be well, in view of the great importance of the theorem, to indicate alternative lines of proof.", "why": "Shows an author admitting a proof is roundabout and offering other routes.", "use": [ "history", "website" ], "concepts": [ "method/repeated-bisection", "theorem/attainment-of-bounds-by-continuous-functions" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-4de50457c1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "170", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "This function is equal to~$1$ for all values of~$x$ save $x = 0$. It is \\emph{not} equal to~$1$ when $x = 0$: it is in fact not defined at all for $x = 0$.", "markdown": "This function is equal to $1$ for all values of $x$ save $x = 0$. It is *not* equal to $1$ when $x = 0$: it is in fact not defined at all for $x = 0$.", "why": "It separates a function being undefined at a point from its limit existing there, which resolves the apparent paradox of 0/0.", "use": [ "lesson", "website" ], "concepts": [ "concept/domain-of-definition", "concept/indeterminate-form", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-be43c7358a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "172", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "If it were complete, then every function~$\\phi(x)$ which tends to zero with~$x$ would be of either the first or second or some higher order of smallness. This is obviously not the case.", "markdown": "If it were complete, then every function $\\phi(x)$ which tends to zero with $x$ would be of either the first or second or some higher order of smallness. This is obviously not the case.", "why": "It explains that the integer scale of orders of smallness is useful but incomplete, which guards against overgeneralising it.", "use": [ "lesson", "history" ], "concepts": [ "concept/order-of-smallness" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-8cdff9d343", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "170", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "The graph of this function consists of the axis of~$x$, with the point $x = 0$ left out, and one isolated point, viz.\\ the point $(0, 1)$.", "markdown": "The graph of this function consists of the axis of $x$, with the point $x = 0$ left out, and one isolated point, viz. the point $(0, 1)$.", "why": "A picture of a removable exception makes the idea that a limit ignores an isolated point visible at a glance.", "use": [ "website", "lesson" ], "concepts": [ "concept/graph-of-a-function", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-7878560696", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "172", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "We shall say that $\\phi(x)$~is of the $k$th~order of greatness when $x$~is small if $\\phi(x)/x^{-k} = x^{k}\\phi(x)$ tends to a limit different from~$0$ as $x$~tends to~$0$.", "markdown": "We shall say that $\\phi(x)$ is of the $k$th order of greatness when $x$ is small if $\\phi(x)/x^{-k} = x^{k}\\phi(x)$ tends to a limit different from $0$ as $x$ tends to $0$.", "why": "It gives the precise, checkable test for how fast a function blows up near zero, mirroring the test for smallness.", "use": [ "lesson" ], "concepts": [ "concept/limit", "concept/order-of-greatness" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-c67245227e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "162", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "The only difference between the `tending of~$n$ to~$\\infty$' discussed in the last chapter, and this `tending of~$x$ to~$\\infty$', is that $x$~assumes all values as it tends to~$\\infty$, \\ie\\ that the point~$P$ which corresponds to~$x$ coincides in turn with every point of~$\\Lambda$ to the right of its initial position, whereas $n$~tended to~$\\infty$ by a series of jumps. We can express this distinction by saying that $x$~tends \\emph{continuously} to~$\\infty$.", "markdown": "The only difference between the ‘tending of $n$ to $\\infty$’ discussed in the last chapter, and this ‘tending of $x$ to $\\infty$’, is that $x$ assumes all values as it tends to $\\infty$, *i.e.* that the point $P$ which corresponds to $x$ coincides in turn with every point of $\\Lambda$ to the right of its initial position, whereas $n$ tended to $\\infty$ by a series of jumps. We can express this distinction by saying that $x$ tends *continuously* to $\\infty$.", "why": "It shows plainly how a continuous variable differs from an integer one, so the sequence limits already learned carry over.", "use": [ "lesson", "website" ], "concepts": [ "concept/continuous-real-variable", "concept/function-of-a-positive-integer-variable", "concept/tends-to-infinity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-7132fce577", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "164", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "It is equal to zero whenever $x$~is an integer, so that the function~$\\phi(n)$ derived from it is always zero and so tends to the limit zero.", "markdown": "It is equal to zero whenever $x$ is an integer, so that the function $\\phi(n)$ derived from it is always zero and so tends to the limit zero.", "why": "It shows a learner that a function can tend to a limit along one set of points while failing to do so along all points, so the limit's converse fails.", "use": [ "lesson" ], "concepts": [ "concept/function-of-a-positive-integer-variable", "concept/integer-part-function", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-8b2f4d3fc4", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "191", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "The reader should observe carefully that to assert the continuity of~$\\phi(x, y)$ with respect to the two variables $x$~and~$y$ is to assert much more than its continuity with respect to each variable considered separately. It is plain that if $\\phi(x, y)$~is continuous with respect to $x$~and~$y$ then it is certainly continuous with respect to~$x$ (or~$y$) when any fixed value is assigned to~$y$ (or~$x$). But the converse is by no means true.", "markdown": "The reader should observe carefully that to assert the continuity of $\\phi(x, y)$ with respect to the two variables $x$ and $y$ is to assert much more than its continuity with respect to each variable considered separately. It is plain that if $\\phi(x, y)$ is continuous with respect to $x$ and $y$ then it is certainly continuous with respect to $x$ (or $y$) when any fixed value is assigned to $y$ (or $x$). But the converse is by no means true.", "why": "It warns learners that continuity in two variables together is stronger than continuity in each one separately.", "use": [ "lesson", "website" ], "concepts": [ "concept/continuous-function", "concept/continuous-function-of-two-variables", "concept/separate-continuity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-a0fa14271c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "192", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "We shall call such a square a \\emph{neighbourhood} of~$(a, b)$, and say that the condition in question is satisfied \\emph{in the neighbourhood of~$(a, b)$}, or \\emph{near~$(a, b)$}, meaning by this simply that it is possible to find \\emph{some} square throughout which the condition is satisfied.", "markdown": "We shall call such a square a *neighbourhood* of $(a, b)$, and say that the condition in question is satisfied *in the neighbourhood of $(a, b)$*, or *near $(a, b)$*, meaning by this simply that it is possible to find *some* square throughout which the condition is satisfied.", "why": "It gives the precise meaning of the common phrase 'near a point' and stresses that only the existence of some square is claimed.", "use": [ "lesson", "website" ], "concepts": [ "concept/neighbourhood" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-b41a634909", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "190", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "This theorem is of fundamental importance in the theory of definite integrals (\\okrickRef{Ch.}{VII}). It is impossible, without the use of this or some similar theorem, to prove that a function continuous throughout an interval necessarily possesses an integral over that interval.", "markdown": "This theorem is of fundamental importance in the theory of definite integrals (Ch.VII). It is impossible, without the use of this or some similar theorem, to prove that a function continuous throughout an interval necessarily possesses an integral over that interval.", "why": "It tells the learner why this abstract theorem matters: integration of continuous functions rests on it.", "use": [ "lesson", "history" ], "concepts": [ "concept/definite-integral", "theorem/uniform-continuity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-d8efddb727", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "193", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "If we had supposed that $y^{2} = x$ then the conditions of the theorem would not have been satisfied, for $y^{2}$~is not a steadily increasing function of~$y$ in any interval which includes $y = 0$: it decreases when $y$~is negative and increases when $y$~is positive. And in this case the conclusion of the theorem does not hold, for $y^{2} = x$ defines \\emph{two} functions of~$x$, viz.\\ $y = \\sqrt{x}$ and $y = -\\sqrt{x}$, both of which vanish when $x = 0$, and each of which is defined only for positive values of~$x$, so that the equation has sometimes two solutions and sometimes none.", "markdown": "If we had supposed that $y^{2} = x$ then the conditions of the theorem would not have been satisfied, for $y^{2}$ is not a steadily increasing function of $y$ in any interval which includes $y = 0$: it decreases when $y$ is negative and increases when $y$ is positive. And in this case the conclusion of the theorem does not hold, for $y^{2} = x$ defines *two* functions of $x$, viz. $y = \\sqrt{x}$ and $y = -\\sqrt{x}$, both of which vanish when $x = 0$, and each of which is defined only for positive values of $x$, so that the equation has sometimes two solutions and sometimes none.", "why": "It shows by a failing example why the monotonicity hypothesis in the inverse function theorem cannot be dropped.", "use": [ "lesson" ], "concepts": [ "concept/inverse-function", "concept/steadily-increasing-function", "theorem/inverse-function-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-00ed3dc415", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "193", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "There is nothing to show that the~$\\EPSILON_{1}$ of the conclusion is the~$\\EPSILON$ of the hypotheses, and indeed this is generally untrue.", "markdown": "There is nothing to show that the $\\EPSILON_{1}$ of the conclusion is the $\\EPSILON$ of the hypotheses, and indeed this is generally untrue.", "why": "It warns that the neighbourhood where the conclusion holds can be smaller than the one where the hypotheses hold.", "use": [ "lesson" ], "concepts": [ "concept/neighbourhood", "theorem/implicit-function-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-b3f984015f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "188", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "The reader may be tempted to think that this proof is needlessly elaborate, and that the existence of points of the interval, not in any interval of~$I$, follows at once from the fact that the sum of all these intervals is less than~$1$.", "markdown": "The reader may be tempted to think that this proof is needlessly elaborate, and that the existence of points of the interval, not in any interval of $I$, follows at once from the fact that the sum of all these intervals is less than $1$.", "why": "Hardy anticipates the reader's doubt and explains why the careful argument is needed, which gives the proof a human voice.", "use": [ "history", "website" ], "concepts": [ "concept/set-of-intervals", "theorem/heine-borel-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-3d0413ab71", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "194", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "This class has an upper bound~$\\eta$, and plainly $F(\\eta) \\leq \\xi$. If $F(\\eta)$~were less than~$\\xi$, we could find a value of~$y$ such that $y > \\eta$ and $F(y) < \\xi$, and $\\eta$~would not be the upper bound of the class considered. Hence $F(\\eta) = \\xi$.", "markdown": "This class has an upper bound $\\eta$, and plainly $F(\\eta) \\leq \\xi$. If $F(\\eta)$ were less than $\\xi$, we could find a value of $y$ such that $y > \\eta$ and $F(y) < \\xi$, and $\\eta$ would not be the upper bound of the class considered. Hence $F(\\eta) = \\xi$.", "why": "It is a compact proof by contradiction that builds the inverse function from a least upper bound.", "use": [ "lesson" ], "concepts": [ "concept/least-upper-bound", "theorem/inverse-function-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-1f759f67ed", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "189", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "It is indeed obvious that there are an infinity of such intervals corresponding to every~$\\xi$ and every~$\\DELTA$, for if the condition is satisfied for any particular value of~$\\EPSILON$, then it is satisfied \\textit{a~fortiori} for any smaller value.", "markdown": "It is indeed obvious that there are an infinity of such intervals corresponding to every $\\xi$ and every $\\DELTA$, for if the condition is satisfied for any particular value of $\\EPSILON$, then it is satisfied *a fortiori* for any smaller value.", "why": "It shows learners that a working epsilon can always be shrunk, which is why there are infinitely many Delta-intervals.", "use": [ "lesson" ], "concepts": [ "concept/continuous-function", "concept/interval", "concept/oscillation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-e1d4284e75", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "191", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "This statement is apparently simpler; but it contains phrases the precise meaning of which has not yet been explained and can only be explained by the help of inequalities like those which occur in our original statement.", "markdown": "This statement is apparently simpler; but it contains phrases the precise meaning of which has not yet been explained and can only be explained by the help of inequalities like those which occur in our original statement.", "why": "It cautions that a tidy-sounding 'tends to ... in any manner' definition is only precise once the epsilon-delta inequalities are written out.", "use": [ "lesson", "website" ], "concepts": [ "concept/continuous-function-of-two-variables" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-83c434d7b3", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "191", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "Another method of stating the definition is this: \\emph{$\\phi(x, y)$~is\ncontinuous for $x = \\xi$, $y = \\eta$ if $\\phi(x, y) \\to \\phi(\\xi, \\eta)$ when $x \\to \\xi$, $y \\to \\eta$\nin any manner}. This statement is apparently simpler; but it\ncontains phrases the precise meaning of which has not yet been\nexplained and can only be explained by the help of inequalities\nlike those which occur in our original statement.", "markdown": "Another method of stating the definition is this: *$\\phi(x, y)$ is continuous for $x = \\xi$, $y = \\eta$ if $\\phi(x, y) \\to \\phi(\\xi, \\eta)$ when $x \\to \\xi$, $y \\to \\eta$ in any manner*. This statement is apparently simpler; but it contains phrases the precise meaning of which has not yet been explained and can only be explained by the help of inequalities like those which occur in our original statement.", "why": "It shows a learner that the informal 'approach in any manner' version of continuity hides precise content that only the inequality definition makes exact.", "use": [ "lesson" ], "concepts": [ "concept/continuous-function-of-two-variables", "concept/function-of-two-variables" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-1cfae5f090", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "190", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "It is impossible, without the use of\nthis or some similar theorem, to prove that a function continuous\nthroughout an interval necessarily possesses an integral over that\ninterval.", "markdown": "It is impossible, without the use of this or some similar theorem, to prove that a function continuous throughout an interval necessarily possesses an integral over that interval.", "why": "It explains to a learner why a seemingly obvious fact about continuous functions needs a genuine proof, and why it matters for integration.", "use": [ "history", "lesson" ], "concepts": [ "concept/continuous-function", "concept/integral", "theorem/uniform-continuity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-b9b5b5b17d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "195", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "If $\\phi(x) = 1/q$ when $x = p/q$, and $\\phi(x) = 0$ when $x$~is irrational, then\n$\\phi(x)$~is continuous for all irrational and discontinuous for all rational values\nof~$x$.", "markdown": "If $\\phi(x) = 1/q$ when $x = p/q$, and $\\phi(x) = 0$ when $x$ is irrational, then $\\phi(x)$ is continuous for all irrational and discontinuous for all rational values of $x$.", "why": "It gives a striking example of a function that is continuous at every irrational and discontinuous at every rational point.", "use": [ "website", "lesson" ], "concepts": [ "concept/continuous-function", "concept/discontinuous-function", "concept/irrational-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-c38906b44c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "170", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "It is \\emph{not} a statement about the \\emph{value of~$\\phi(x)$ when $x = 0$}. When we make the statement we assert that, when $x$~is \\emph{nearly} equal to zero, $\\phi(x)$~is nearly equal to~$l$. We assert nothing whatever about what happens when $x$~is \\emph{actually} equal to~$0$.", "markdown": "It is *not* a statement about the *value of $\\phi(x)$ when $x = 0$*. When we make the statement we assert that, when $x$ is *nearly* equal to zero, $\\phi(x)$ is nearly equal to $l$. We assert nothing whatever about what happens when $x$ is *actually* equal to $0$.", "why": "It warns against the common mistake of reading a limit at a point as the value at that point.", "use": [ "lesson", "website" ], "concepts": [ "concept/limit", "concept/value-of-a-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-6cefa55de2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "170", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "The equation~\\Eq{(2)} expresses the fact that if we move along the graph towards the axis of~$y$, from either side, then the ordinate of the curve, being always equal to zero, tends to the limit zero. This fact is in no way affected by the position of the isolated point~$(0, 1)$.", "markdown": "The equation (2) expresses the fact that if we move along the graph towards the axis of $y$, from either side, then the ordinate of the curve, being always equal to zero, tends to the limit zero. This fact is in no way affected by the position of the isolated point $(0, 1)$.", "why": "It uses a graph with one stray point to show that a limit ignores the value at the point itself.", "use": [ "lesson" ], "concepts": [ "concept/limit", "concept/value-of-a-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-a528198bc2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "171", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "When we put $x = 0$ in~$\\phi(x)$ we obtain~$0/0$, which is a meaningless expression. The reader may object `divide numerator and denominator by~$x$'. But he must admit that when $x = 0$ this is impossible.", "markdown": "When we put $x = 0$ in $\\phi(x)$ we obtain $0/0$, which is a meaningless expression. The reader may object ‘divide numerator and denominator by $x$’. But he must admit that when $x = 0$ this is impossible.", "why": "It meets the learner's natural objection head-on and explains why x/x is undefined at 0 though its limit is 1.", "use": [ "lesson", "website" ], "concepts": [ "concept/domain-of-definition", "concept/indeterminate-form", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-478eeefecb", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "171", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "Thus $y = x/x$ is a function which differs from $y = 1$ solely in that it is not defined for $x = 0$. None the less \\[ \\lim(x/x) = 1, \\] for $x/x$~is equal to~$1$ so long as $x$~differs from zero, however small the difference may be.", "markdown": "Thus $y = x/x$ is a function which differs from $y = 1$ solely in that it is not defined for $x = 0$. None the less (x/x) = 1, for $x/x$ is equal to $1$ so long as $x$ differs from zero, however small the difference may be.", "why": "It is a clean worked case of a limit that exists where the function does not.", "use": [ "lesson" ], "concepts": [ "concept/domain-of-definition", "concept/indeterminate-form", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-8f00730d44", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "172", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "It must not be imagined that this scale of orders of smallness is in any way complete. If it were complete, then every function~$\\phi(x)$ which tends to zero with~$x$ would be of either the first or second or some higher order of smallness. This is obviously not the case. For example $\\phi(x) = x^{7/5}$ tends to zero more rapidly than~$x$ and less rapidly than~$x^{2}$.", "markdown": "It must not be imagined that this scale of orders of smallness is in any way complete. If it were complete, then every function $\\phi(x)$ which tends to zero with $x$ would be of either the first or second or some higher order of smallness. This is obviously not the case. For example $\\phi(x) = x^{7/5}$ tends to zero more rapidly than $x$ and less rapidly than $x^{2}$.", "why": "It cautions that the integer scale of smallness leaves gaps and gives a concrete function that falls between two steps.", "use": [ "lesson" ], "concepts": [ "concept/order-of-smallness" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-c90089bdf6", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "164", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "In fact, if $x = -y$ and $\\phi(x) = \\phi(-y) = \\psi(y)$, then $y$~tends to~$\\infty$ as $x$~tends to~$-\\infty$, and the question of the behaviour of~$\\phi(x)$ as $x$~tends to~$-\\infty$ is the same as that of the behaviour of~$\\psi(y)$ as $y$~tends to~$\\infty$.", "markdown": "In fact, if $x = -y$ and $\\phi(x) = \\phi(-y) = \\psi(y)$, then $y$ tends to $\\infty$ as $x$ tends to $-\\infty$, and the question of the behaviour of $\\phi(x)$ as $x$ tends to $-\\infty$ is the same as that of the behaviour of $\\psi(y)$ as $y$ tends to $\\infty$.", "why": "It models the substitution trick that reduces one kind of limit to another already understood.", "use": [ "lesson" ], "concepts": [ "concept/limit", "concept/tends-to-infinity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-b976622ded", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "164", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "The function $\\phi(x) = x - [x]$ oscillates between $0$~and~$1$, as is obvious from the form of its graph. It is equal to zero whenever $x$~is an integer, so that the function~$\\phi(n)$ derived from it is always zero and so tends to the limit zero.", "markdown": "The function $\\phi(x) = x - [x]$ oscillates between $0$ and $1$, as is obvious from the form of its graph. It is equal to zero whenever $x$ is an integer, so that the function $\\phi(n)$ derived from it is always zero and so tends to the limit zero.", "why": "It shows that sampling a function at integers can hide its oscillation, so a limit of phi(n) does not give a limit of phi(x).", "use": [ "lesson" ], "concepts": [ "concept/function-of-a-positive-integer-variable", "concept/integer-part-function", "concept/oscillation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-d5547be8eb", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "174", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "It is natural to call a function \\emph{continuous} if its graph is a continuous curve, and otherwise discontinuous.", "markdown": "It is natural to call a function *continuous* if its graph is a continuous curve, and otherwise discontinuous.", "why": "Starts from the picture a learner already has of an unbroken curve before the precise definition is made.", "use": [ "lesson", "website" ], "concepts": [ "concept/continuous-function", "concept/discontinuous-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-794d6bec76", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "175", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "The function~$\\phi(x)$ is said to be continuous for $x = \\xi$ if it tends to a limit as $x$~tends to~$\\xi$ from either side, and each of these limits is equal to~$\\phi(\\xi)$.", "markdown": "The function $\\phi(x)$ is said to be continuous for $x = \\xi$ if it tends to a limit as $x$ tends to $\\xi$ from either side, and each of these limits is equal to $\\phi(\\xi)$.", "why": "Gives Hardy's formal definition of continuity at a point in terms of limits from both sides.", "use": [ "lesson", "website" ], "concepts": [ "concept/continuous-function", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-66703a4c2f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "176", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "Thus our definition asserts that if we draw two such horizontal lines, no matter how close together, we can always cut off a vertical strip of the plane by two vertical lines in such a way that all that part of the curve which is contained in the strip lies between the two horizontal lines.", "markdown": "Thus our definition asserts that if we draw two such horizontal lines, no matter how close together, we can always cut off a vertical strip of the plane by two vertical lines in such a way that all that part of the curve which is contained in the strip lies between the two horizontal lines.", "why": "Turns the epsilon-delta condition into a picture of a strip around the curve.", "use": [ "lesson" ], "concepts": [ "concept/continuous-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-c9099c9e7a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "177", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "An `infinity' is the kind of discontinuity of most common occurrence in ordinary work.", "markdown": "An ‘infinity’ is the kind of discontinuity of most common occurrence in ordinary work.", "why": "Tells learners which kind of break they will meet most often, such as 1/x at zero.", "use": [ "lesson", "history" ], "concepts": [ "concept/discontinuous-function", "concept/infinity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-ec616d9890", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "179", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "In other words \\emph{as $x$~varies from $x_{0}$ to~$x_{1}$, $y$~must assume at least once every value between $y_{0}$~and~$y_{1}$}.", "markdown": "In other words *as $x$ varies from $x_{0}$ to $x_{1}$, $y$ must assume at least once every value between $y_{0}$ and $y_{1}$*.", "why": "States the intermediate value property of continuous functions in plain words.", "use": [ "lesson", "website" ], "concepts": [ "theorem/intermediate-value-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-6d7afc912f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "181", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "Indeed it is not even true that $\\phi(x)$~must be continuous when it assumes each value \\emph{once and once only}.", "markdown": "Indeed it is not even true that $\\phi(x)$ must be continuous when it assumes each value *once and once only*.", "why": "Warns that taking every intermediate value does not make a function continuous.", "use": [ "lesson" ], "concepts": [ "concept/converse-of-a-theorem", "theorem/intermediate-value-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-7d226c53a1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "181", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "The net result of this and the last section is consequently to show that our common-sense notion of what we mean by continuity is substantially accurate, and capable of precise statement in mathematical terms.", "markdown": "The net result of this and the last section is consequently to show that our common-sense notion of what we mean by continuity is substantially accurate, and capable of precise statement in mathematical terms.", "why": "Shows the purpose of rigorous definition: to confirm and sharpen intuition, not to replace it.", "use": [ "website", "history" ], "concepts": [ "concept/continuous-function", "theorem/intermediate-value-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-f78022741b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "259", "location": "DERIVATIVES AND INTEGRALS", "latex": "A formula such as this is called a \\emph{formula of reduction}. It is most useful when $n$~is a positive integer.", "markdown": "A formula such as this is called a *formula of reduction*. It is most useful when $n$ is a positive integer.", "why": "Names the reduction-formula technique and says when it works best.", "use": [ "lesson", "website" ], "concepts": [ "method/formulae-of-reduction" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-171c741acf", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "254", "location": "DERIVATIVES AND INTEGRALS", "latex": "If $ax + by + c = 0$ then $y_{2} = 0$ (suffixes denoting differentiations with respect to~$x$). We may express this by saying that \\emph{the general differential equation of all straight lines is $y_{2} = 0$}.", "markdown": "If $ax + by + c = 0$ then $y_{2} = 0$ (suffixes denoting differentiations with respect to $x$). We may express this by saying that *the general differential equation of all straight lines is $y_{2} = 0$*.", "why": "Introduces the idea of a differential equation that describes a whole family of curves.", "use": [ "lesson", "website" ], "concepts": [ "concept/differential-equation", "method/differentiation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-35e5ec529c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "254", "location": "DERIVATIVES AND INTEGRALS", "latex": "In each case we have only to write down the general equation of the curves in question, and differentiate until we have enough equations to eliminate all the arbitrary constants.", "markdown": "In each case we have only to write down the general equation of the curves in question, and differentiate until we have enough equations to eliminate all the arbitrary constants.", "why": "States in one sentence the method for finding a family's differential equation.", "use": [ "lesson" ], "concepts": [ "concept/differential-equation", "method/differentiation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-9caaa1162e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "254", "location": "DERIVATIVES AND INTEGRALS", "latex": "The constituents of a determinant are functions of~$x$. Show that its differential coefficient is the sum of the determinants formed by differentiating the constituents of one row only, leaving the rest unaltered.", "markdown": "The constituents of a determinant are functions of $x$. Show that its differential coefficient is the sum of the determinants formed by differentiating the constituents of one row only, leaving the rest unaltered.", "why": "Poses how to differentiate a determinant, a rule that later exercises use.", "use": [ "lesson" ], "concepts": [ "concept/derivative", "concept/determinant" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-46761d54c6", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "199", "location": "DERIVATIVES AND INTEGRALS", "latex": "The existence of a derived function~$\\phi'(x)$ for all values of~$x$ in the interval $a \\leq x \\leq b$ implies that $\\phi(x)$~is continuous at every point of this interval.", "markdown": "The existence of a derived function $\\phi'(x)$ for all values of $x$ in the interval $a \\leq x \\leq b$ implies that $\\phi(x)$ is continuous at every point of this interval.", "why": "It tells the learner that a derivative can exist only where the function is continuous, so continuity must be checked first.", "use": [ "lesson" ], "concepts": [ "concept/continuous-function", "concept/derivative" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-4a92ceff5d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "205", "location": "DERIVATIVES AND INTEGRALS", "latex": "the reader must however be careful to remember that $dy/dx$ does not mean `a certain number~$dy$ divided by another number~$dx$': it means `the result of a certain operation~$D_{x}$ or~$d/dx$ applied to $y = \\phi(x)$', the operation being that of forming the quotient $\\{\\phi(x + h) - \\phi(x)\\}/h$ and making $h \\to 0$.", "markdown": "the reader must however be careful to remember that $dy/dx$ does not mean ‘a certain number $dy$ divided by another number $dx$’: it means ‘the result of a certain operation $D_{x}$ or $d/dx$ applied to $y = \\phi(x)$’, the operation being that of forming the quotient $\\{\\phi(x + h) - \\phi(x)\\}/h$ and making $h \\to 0$.", "why": "It warns the learner that dy/dx is a limit of a quotient and not a ratio of two separate numbers.", "use": [ "lesson" ], "concepts": [ "concept/derivative", "method/differentiation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-466a35aef5", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "201", "location": "DERIVATIVES AND INTEGRALS", "latex": "The reader should observe that this method cannot be applied to~$x^{p/q}$, where $p/q$~is a rational fraction, as we have no means of expressing $(x + h)^{p/q}$ as a finite series of powers of~$h$.", "markdown": "The reader should observe that this method cannot be applied to $x^{p/q}$, where $p/q$ is a rational fraction, as we have no means of expressing $(x + h)^{p/q}$ as a finite series of powers of $h$.", "why": "It shows the learner the limits of the binomial-expansion method for differentiating powers.", "use": [ "lesson" ], "concepts": [ "concept/derivative", "theorem/power-rule" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-1553873da7", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "200", "location": "DERIVATIVES AND INTEGRALS", "latex": "The geometry of curves is merely one of many departments of mathematics in which the idea of a derivative finds an application.", "markdown": "The geometry of curves is merely one of many departments of mathematics in which the idea of a derivative finds an application.", "why": "It places the derivative in a wider mathematical setting, showing that its meaning does not depend on geometry.", "use": [ "history" ], "concepts": [ "concept/derivative", "concept/geometry" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-a3c2603998", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "200", "location": "DERIVATIVES AND INTEGRALS", "latex": "The notion of `velocity' is in fact merely a special case of that of the derivative of a function.", "markdown": "The notion of ‘velocity’ is in fact merely a special case of that of the derivative of a function.", "why": "It links the derivative to the physical idea of velocity.", "use": [ "lesson", "website" ], "concepts": [ "concept/derivative", "quantity/velocity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-bb69e23e33", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "217", "location": "DERIVATIVES AND INTEGRALS", "latex": "If $\\phi(a) = 0$ and $\\phi(b) = 0$, then there must be at least one value of~$x$ which lies between $a$ and~$b$ and for which $\\phi'(x) = 0$.", "markdown": "If $\\phi(a) = 0$ and $\\phi(b) = 0$, then there must be at least one value of $x$ which lies between $a$ and $b$ and for which $\\phi'(x) = 0$.", "why": "It states the condition that forces a turning point between two equal values, which a learner can test on a sketch before proving it.", "use": [ "lesson" ], "concepts": [ "concept/derivative", "theorem/rolle-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-77644e4f5f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "214", "location": "DERIVATIVES AND INTEGRALS", "latex": "This function is called the \\emph{second derivative} or \\emph{second differential coefficient} of~$\\phi(x)$.", "markdown": "This function is called the *second derivative* or *second differential coefficient* of $\\phi(x)$.", "why": "It gives the learner the naming of the second derivative and the notation that goes with it, step by step from the first derivative.", "use": [ "lesson" ], "concepts": [ "concept/higher-order-derivative" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-6f424e68d5", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "219", "location": "DERIVATIVES AND INTEGRALS", "latex": "A \\Emph{necessary} condition for a maximum or minimum value of~$\\phi(x)$ at $x = \\xi$ is that $\\phi'(\\xi) = 0$.", "markdown": "A **** condition for a maximum or minimum value of $\\phi(x)$ at $x = \\xi$ is that $\\phi'(\\xi) = 0$.", "why": "It warns that a zero derivative is necessary but not sufficient for a maximum or minimum, a mistake learners often make.", "use": [ "lesson" ], "concepts": [ "concept/derivative", "concept/maximum", "concept/minimum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-758de01772", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "221", "location": "DERIVATIVES AND INTEGRALS", "latex": "There is no derivative for $x = 0$, and no tangent to the graph at~$P$.", "markdown": "There is no derivative for $x = 0$, and no tangent to the graph at $P$.", "why": "It shows that a function can have a maximum where the derivative does not exist, so the derivative test cannot be applied there.", "use": [ "lesson" ], "concepts": [ "concept/derivative", "concept/tangent" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-b6904b26af", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "221", "location": "DERIVATIVES AND INTEGRALS", "latex": "The reader, if he considers what the question means and tries to answer it in the light of common sense, will probably incline to the answer \\emph{No}. It is, however, not difficult to see that this answer is wrong.", "markdown": "The reader, if he considers what the question means and tries to answer it in the light of common sense, will probably incline to the answer *No*. It is, however, not difficult to see that this answer is wrong.", "why": "It invites the learner to test a plausible intuition about continuity of the derivative before the counterexample is given.", "use": [ "website", "history" ], "concepts": [ "concept/continuous-function", "concept/derivative" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-6ebc9f7340", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "211", "location": "DERIVATIVES AND INTEGRALS", "latex": "But there is no practical difficulty in the actual calculation of the derivative of such a function: the method to be adopted will be illustrated sufficiently by an example.", "markdown": "But there is no practical difficulty in the actual calculation of the derivative of such a function: the method to be adopted will be illustrated sufficiently by an example.", "why": "It tells the learner that implicit differentiation is a routine procedure, not a source of difficulty in practice.", "use": [ "lesson" ], "concepts": [ "method/implicit-differentiation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-f750079d88", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "215", "location": "DERIVATIVES AND INTEGRALS", "latex": "It will then follow by the principle of mathematical induction that $a_{n, r} = \\dbinom{n}{r}$ for all values of $n$ and~$r$ in question.", "markdown": "It will then follow by the principle of mathematical induction that $a_{n, r} = \\dbinom{n}{r}$ for all values of $n$ and $r$ in question.", "why": "It shows the induction argument in its conclusion form, letting a learner see how a formula for all n is established from one step.", "use": [ "lesson" ], "concepts": [ "method/mathematical-induction", "theorem/general-leibniz-rule" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-095f217df5", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "217", "location": "DERIVATIVES AND INTEGRALS", "latex": "Of course from a geometrical point of view the result is intuitive, the inequality $\\phi'(x) > 0$ expressing the fact that the tangent to the curve $y = \\phi(x)$ makes a positive acute angle with the axis of~$x$.", "markdown": "Of course from a geometrical point of view the result is intuitive, the inequality $\\phi'(x) > 0$ expressing the fact that the tangent to the curve $y = \\phi(x)$ makes a positive acute angle with the axis of $x$.", "why": "Gives the picture behind Theorem A: a positive derivative means the tangent slopes upward.", "use": [ "lesson", "website" ], "concepts": [ "concept/sign-of-the-derivative", "concept/tangent", "theorem/sign-of-the-derivative" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-8c92041bd2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "228", "location": "DERIVATIVES AND INTEGRALS", "latex": "It is natural to consider the converse question, that of \\emph{determining a function whose derivative is a given function}.", "markdown": "It is natural to consider the converse question, that of *determining a function whose derivative is a given function*.", "why": "It motivates integration as the reverse of differentiation in a single clear sentence.", "use": [ "lesson", "history" ], "concepts": [ "concept/integral", "method/differentiation", "method/integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-df67d80e11", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "231", "location": "DERIVATIVES AND INTEGRALS", "latex": "There is however one case of exception to the first formula, that in which $m = -1$.", "markdown": "There is however one case of exception to the first formula, that in which $m = -1$.", "why": "It alerts the learner that the power rule for integrals fails for m = -1, which leads to the logarithm.", "use": [ "lesson" ], "concepts": [ "concept/logarithm", "concept/standard-forms-of-integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-f6b275e91c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "230", "location": "DERIVATIVES AND INTEGRALS", "latex": "the~$\\int$ and the~$dx$ no more mean anything when taken by themselves than do the~$d$ and~$dx$ of the other operative symbol~$d/dx$.", "markdown": "the $\\int$ and the $dx$ no more mean anything when taken by themselves than do the $d$ and $dx$ of the other operative symbol $d/dx$.", "why": "It explains that the integral sign and dx are operator symbols and carry no meaning alone.", "use": [ "lesson" ], "concepts": [ "concept/mathematical-notation", "method/differentiation", "method/integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-588d2ac54b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "236", "location": "DERIVATIVES AND INTEGRALS", "latex": "If the equation $Q(x) = 0$ cannot be solved algebraically, then the method of partial fractions naturally fails and recourse must be had to other methods.", "markdown": "If the equation $Q(x) = 0$ cannot be solved algebraically, then the method of partial fractions naturally fails and recourse must be had to other methods.", "why": "It gives a clear limit on when partial fractions can integrate a rational function.", "use": [ "lesson" ], "concepts": [ "concept/rational-function", "method/integration", "method/partial-fractions" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-8e0d53b77c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "240", "location": "DERIVATIVES AND INTEGRALS", "latex": "Suppose, for example, that $\\phi(x) = x\\psi(x)$, where $\\psi(x)$~is the second derivative of a known function~$\\chi(x)$.", "markdown": "Suppose, for example, that $\\phi(x) = x\\psi(x)$, where $\\psi(x)$ is the second derivative of a known function $\\chi(x)$.", "why": "It shows how to choose the factor to differentiate and the factor to integrate, using a function whose second derivative is known.", "use": [ "lesson" ], "concepts": [ "concept/derivative", "method/integration-by-parts" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-f779a30245", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "249", "location": "DERIVATIVES AND INTEGRALS", "latex": "It is indeed one which needs and has received the most careful mathematical analysis: later on we shall return to it and explain precisely what is meant by ascribing an `area' to such a region of space as~$ONPP_{0}$.", "markdown": "It is indeed one which needs and has received the most careful mathematical analysis: later on we shall return to it and explain precisely what is meant by ascribing an ‘area’ to such a region of space as $ONPP_{0}$.", "why": "It is an honest remark that the notion of area is assumed here and will be made rigorous later, which warns learners not to treat it as settled.", "use": [ "history" ], "concepts": [ "quantity/area" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-c212a39b29", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "250", "location": "DERIVATIVES AND INTEGRALS", "latex": "Calculate $\\Phi(x)$, the integral of~$\\phi(x)$. This involves an arbitrary constant, which we suppose so chosen that $\\Phi(0) = 0$.", "markdown": "Calculate $\\Phi(x)$, the integral of $\\phi(x)$. This involves an arbitrary constant, which we suppose so chosen that $\\Phi(0) = 0$.", "why": "It gives the rule for finding an area by integrating and fixing the arbitrary constant by a condition.", "use": [ "lesson" ], "concepts": [ "concept/arbitrary-constant-of-integration", "quantity/area" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-f3c6556fad", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "251", "location": "DERIVATIVES AND INTEGRALS", "latex": "It is however easy to see what the \\emph{formula} must be.", "markdown": "It is however easy to see what the *formula* must be.", "why": "It signals that the arc-length formula is first obtained by a heuristic argument, which the learner should later check.", "use": [ "lesson" ], "concepts": [ "quantity/length-of-a-curve" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-f22ae4b4f4", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "DERIVATIVES AND INTEGRALS", "latex": "This integral cannot however be evaluated in terms of such functions as are at present at our disposal.", "markdown": "This integral cannot however be evaluated in terms of such functions as are at present at our disposal.", "why": "It warns that some integrals of transcendental type cannot be done with the methods so far available.", "use": [ "history" ], "concepts": [ "concept/integral", "concept/transcendental-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-c1b58b7453", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "259", "location": "DERIVATIVES AND INTEGRALS", "latex": "Put $x + \\frac{1}{2}p = t$, $q - \\frac{1}{4}p^{2} = \\lambda$: then we obtain", "markdown": "Put $x + \\frac{1}{2}p = t$, $q - \\frac{1}{4}p^{2} = \\lambda$: then we obtain", "why": "It shows the substitution that turns a quadratic-denominator integral into a standard form, a method a learner can copy.", "use": [ "lesson" ], "concepts": [ "method/formulae-of-reduction", "method/integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-54247cc4a6", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "257", "location": "DERIVATIVES AND INTEGRALS", "latex": "This equation has three real roots if $s^{4} > 27\\Delta^{2}$, and one in the contrary case.", "markdown": "This equation has three real roots if $s^{4} > 27\\Delta^{2}$, and one in the contrary case.", "why": "It shows how a discriminant-type condition decides how many real roots an equation has, a useful idea for root-counting.", "use": [ "lesson", "website" ], "concepts": [ "concept/real-root", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-13a749bc1f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "258", "location": "DERIVATIVES AND INTEGRALS", "latex": "If $\\phi(x) \\to a$ as $x \\to \\infty$, then $\\phi'(x)$~cannot tend to any limit other than zero.", "markdown": "If $\\phi(x) \\to a$ as $x \\to \\infty$, then $\\phi'(x)$ cannot tend to any limit other than zero.", "why": "It gives a striking result about limits of derivatives that surprises most learners and rewards careful thought.", "use": [ "website", "history" ], "concepts": [ "concept/derivative", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-3bb9e68d13", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "258", "location": "DERIVATIVES AND INTEGRALS", "latex": "This theorem reduces to the Mean Value Theorem (\\SecNo[§]{125}) when $\\phi(x) = x$ and $\\psi(x) = 1$.", "markdown": "This theorem reduces to the Mean Value Theorem ([§]125) when $\\phi(x) = x$ and $\\psi(x) = 1$.", "why": "Shows how a general result contains a familiar one as a special case.", "use": [ "lesson", "history" ], "concepts": [ "theorem/generalised-mean-value-theorem", "theorem/mean-value-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-84e79e0edc", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "257", "location": "DERIVATIVES AND INTEGRALS", "latex": "In an equilateral triangle (the triangle of minimum perimeter for a given area) $s^{4} = 27\\Delta^{2}$; thus it is impossible that $s^{4} < 27\\Delta^{2}$.", "markdown": "In an equilateral triangle (the triangle of minimum perimeter for a given area) $s^{4} = 27\\Delta^{2}$; thus it is impossible that $s^{4} < 27\\Delta^{2}$.", "why": "Shows a geometrical minimum being used to rule out a case in a root-counting argument.", "use": [ "lesson" ], "concepts": [ "concept/equal-roots", "concept/minimum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-9613d15e91", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "197", "location": "DERIVATIVES AND INTEGRALS", "latex": "The reader will probably remember that in elementary geometry the tangent to a curve at~$P$ is defined to be `the limiting position of the chord~$PQ$, when $Q$~moves up towards coincidence with~$P$'.", "markdown": "The reader will probably remember that in elementary geometry the tangent to a curve at $P$ is defined to be ‘the limiting position of the chord $PQ$, when $Q$ moves up towards coincidence with $P$’.", "why": "It begins from the tangent definition the learner already knows and then asks what that definition requires, which leads to the derivative.", "use": [ "lesson", "website" ], "concepts": [ "concept/derivative", "concept/tangent" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-fb2b2b3042", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "200", "location": "DERIVATIVES AND INTEGRALS", "latex": "But $\\phi(x)$~has no derivative for $x = 0$. For $\\phi'(0)$~would be, by definition, $\\lim\\{\\phi(h) - \\phi(0)\\}/h$ or $\\lim\\sin(1/h)$; and no such limit exists.", "markdown": "But $\\phi(x)$ has no derivative for $x = 0$. For $\\phi'(0)$ would be, by definition, $\\lim\\{\\phi(h) - \\phi(0)\\}/h$ or $\\lim\\sin(1/h)$; and no such limit exists.", "why": "It shows that a continuous function can still lack a derivative, by applying the definition to a concrete case.", "use": [ "lesson" ], "concepts": [ "concept/continuous-function", "concept/derivative", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-6ad252a68c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "200", "location": "DERIVATIVES AND INTEGRALS", "latex": "The notion of a derivative or differential coefficient was suggested to us by geometrical considerations. But there is nothing geometrical in the notion itself.", "markdown": "The notion of a derivative or differential coefficient was suggested to us by geometrical considerations. But there is nothing geometrical in the notion itself.", "why": "It tells the learner that the derivative is an idea about limits that applies beyond curves.", "use": [ "lesson", "website" ], "concepts": [ "concept/derivative" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-0647b131a5", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "205", "location": "DERIVATIVES AND INTEGRALS", "latex": "Of these the last is the most usual and convenient: the reader must however be careful to remember that $dy/dx$ does not mean `a certain number~$dy$ divided by another number~$dx$': it means `the result of a certain operation~$D_{x}$ or~$d/dx$ applied to $y = \\phi(x)$', the operation being that of forming the quotient $\\{\\phi(x + h) - \\phi(x)\\}/h$ and making $h \\to 0$.", "markdown": "Of these the last is the most usual and convenient: the reader must however be careful to remember that $dy/dx$ does not mean ‘a certain number $dy$ divided by another number $dx$’: it means ‘the result of a certain operation $D_{x}$ or $d/dx$ applied to $y = \\phi(x)$’, the operation being that of forming the quotient $\\{\\phi(x + h) - \\phi(x)\\}/h$ and making $h \\to 0$.", "why": "It warns against reading dy/dx as an ordinary fraction.", "use": [ "lesson", "website" ], "concepts": [ "concept/derivative", "concept/mathematical-notation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-1524dc3edb", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "205", "location": "DERIVATIVES AND INTEGRALS", "latex": "These differences may be called the \\emph{increments} of $x$~and~$y$ respectively, and denoted by $\\delta x$~and~$\\delta y$.", "markdown": "These differences may be called the *increments* of $x$ and $y$ respectively, and denoted by $\\delta x$ and $\\delta y$.", "why": "It explains where the dy/dx notation comes from by naming the differences in the numerator and denominator.", "use": [ "lesson", "history" ], "concepts": [ "concept/derivative", "concept/increment" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-8f302b2247", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "210", "location": "DERIVATIVES AND INTEGRALS", "latex": "Incidentally we have proved that \\emph{the derivative of~$x^{m}$ is~$mx^{m-1}$, for all integral values of~$m$ positive or negative}.", "markdown": "Incidentally we have proved that *the derivative of $x^{m}$ is $mx^{m-1}$, for all integral values of $m$ positive or negative*.", "why": "It states that the power rule holds for negative integers as well as positive ones.", "use": [ "lesson", "website" ], "concepts": [ "theorem/power-rule-for-derivatives" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-cae905b9d5", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "210", "location": "DERIVATIVES AND INTEGRALS", "latex": "We have seen already (\\SecNo[§]{117}) that the derivative of this function is~$mx^{m-1}$ when $m$~is an integer positive or negative; and we shall now prove that this result is true for all rational values of~$m$.", "markdown": "We have seen already ([§]117) that the derivative of this function is $mx^{m-1}$ when $m$ is an integer positive or negative; and we shall now prove that this result is true for all rational values of $m$.", "why": "States the power rule and the plan to extend it from integer to rational exponents.", "use": [ "lesson" ], "concepts": [ "concept/derivative", "theorem/power-rule" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-59c42d472e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "211", "location": "DERIVATIVES AND INTEGRALS", "latex": "The differentiation of \\emph{implicit} algebraical functions involves certain theoretical difficulties to which we shall return in \\okrickRef{Ch.}{VII}\\@. But there is no practical difficulty in the actual calculation of the derivative of such a function: the method to be adopted will be illustrated sufficiently by an example.", "markdown": "The differentiation of *implicit* algebraical functions involves certain theoretical difficulties to which we shall return in Ch.VII. But there is no practical difficulty in the actual calculation of the derivative of such a function: the method to be adopted will be illustrated sufficiently by an example.", "why": "Reassures a learner that implicit differentiation is practical now even though its theory comes later.", "use": [ "lesson" ], "concepts": [ "method/implicit-differentiation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-331bef581e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "217", "location": "DERIVATIVES AND INTEGRALS", "latex": "An immediate deduction from Theorem~A is the following important theorem, generally known as Rolle's Theorem. In view of the great importance of this theorem it may be well to repeat that its truth depends on the assumption of the existence of the derivative~$\\phi'(x)$ for all values of~$x$ in question.", "markdown": "An immediate deduction from Theorem A is the following important theorem, generally known as Rolle’s Theorem. In view of the great importance of this theorem it may be well to repeat that its truth depends on the assumption of the existence of the derivative $\\phi'(x)$ for all values of $x$ in question.", "why": "Names Rolle's theorem and warns that it needs the derivative to exist everywhere in the interval.", "use": [ "lesson", "history" ], "concepts": [ "theorem/rolle-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-99a41d97cd", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "220", "location": "DERIVATIVES AND INTEGRALS", "latex": "Thus if $y = x^{3}$ then $\\phi'(x) = 3x^{2}$, which vanishes when $x = 0$. But $x = 0$ does not give either a maximum or a minimum of~$x^{3}$, as is obvious from the form of the graph of~$x^{3}$ (\\Fig{10}, \\PageRef{p.}{45}).", "markdown": "Thus if $y = x^{3}$ then $\\phi'(x) = 3x^{2}$, which vanishes when $x = 0$. But $x = 0$ does not give either a maximum or a minimum of $x^{3}$, as is obvious from the form of the graph of $x^{3}$ ([fig:10]Fig. 10, p.45).", "why": "A standard counterexample showing that a zero derivative alone does not give a maximum or minimum.", "use": [ "lesson" ], "concepts": [ "concept/maximum", "concept/minimum", "theorem/necessary-condition-for-a-maximum-or-minimum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-80010c7df5", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "220", "location": "DERIVATIVES AND INTEGRALS", "latex": "If the sign of~$\\phi'(x)$ changes at $x = \\xi$ from positive to negative, then $x = \\xi$ gives a maximum of~$\\phi(x)$: and if the sign of~$\\phi'(x)$ changes in the opposite sense, then $x = \\xi$ gives a minimum.", "markdown": "If the sign of $\\phi'(x)$ changes at $x = \\xi$ from positive to negative, then $x = \\xi$ gives a maximum of $\\phi(x)$: and if the sign of $\\phi'(x)$ changes in the opposite sense, then $x = \\xi$ gives a minimum.", "why": "Gives a usable test for maxima and minima based on the sign change of the derivative.", "use": [ "lesson" ], "concepts": [ "concept/maximum", "concept/minimum", "theorem/sign-of-the-derivative" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-6d93bb8b96", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "220", "location": "DERIVATIVES AND INTEGRALS", "latex": "Suppose, \\eg, that $\\phi''(\\xi) < 0$. Then, by Theorem~A, $\\phi'(x)$~is negative when $x$~is less than~$\\xi$ but sufficiently near to~$\\xi$, and positive when $x$~is greater than~$\\xi$ but sufficiently near to~$\\xi$. Thus $x = \\xi$ gives a maximum.", "markdown": "Suppose, *e.g.*, that $\\phi''(\\xi) < 0$. Then, by Theorem A, $\\phi'(x)$ is negative when $x$ is less than $\\xi$ but sufficiently near to $\\xi$, and positive when $x$ is greater than $\\xi$ but sufficiently near to $\\xi$. Thus $x = \\xi$ gives a maximum.", "why": "Shows why the second derivative test works by reducing it to the sign of the first derivative.", "use": [ "lesson" ], "concepts": [ "concept/maximum", "method/second-derivative-test" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-3f9bcb9f22", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "221", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\emph{Can} a function~$\\phi(x)$ have a derivative for all values of~$x$ which is not itself continuous? In other words can a curve have a tangent at every point, and yet the direction of the tangent not vary continuously? The reader, if he considers what the question means and tries to answer it in the light of common sense, will probably incline to the answer \\emph{No}. It is, however, not difficult to see that this answer is wrong.", "markdown": "*Can* a function $\\phi(x)$ have a derivative for all values of $x$ which is not itself continuous? In other words can a curve have a tangent at every point, and yet the direction of the tangent not vary continuously? The reader, if he considers what the question means and tries to answer it in the light of common sense, will probably incline to the answer *No*. It is, however, not difficult to see that this answer is wrong.", "why": "Shows that intuition about smooth curves can be wrong and leads into the x^2 sin(1/x) example.", "use": [ "lesson", "website" ], "concepts": [ "concept/derivative", "concept/discontinuous-derivative" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-eac03d2e12", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "240", "location": "DERIVATIVES AND INTEGRALS", "latex": "The theorem of \\emph{integration by parts} is merely another way of stating the rule for the differentiation of a product proved in \\SecNo[§]{113}.", "markdown": "The theorem of *integration by parts* is merely another way of stating the rule for the differentiation of a product proved in [§]113.", "why": "Shows that integration by parts is not a new fact but the product rule read backwards.", "use": [ "lesson", "website" ], "concepts": [ "method/differentiation", "method/integration-by-parts" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-46b739e02e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "247", "location": "DERIVATIVES AND INTEGRALS", "latex": "The integral of any rational function of $\\cos x$~and $\\sin x$ may be calculated by the substitution $\\tan \\frac{1}{2}x = t$.", "markdown": "The integral of any rational function of $\\cos x$ and $\\sin x$ may be calculated by the substitution $\\tan \\frac{1}{2}x = t$.", "why": "Gives learners one reliable substitution that handles a whole class of trigonometric integrals.", "use": [ "lesson" ], "concepts": [ "method/substitution", "method/tangent-half-angle-substitution" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-e4ac174c14", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "248", "location": "DERIVATIVES AND INTEGRALS", "latex": "It is easy to see that if we can find the integral of $y = f(x)$ then we can always find that of $x = \\phi(y)$, where $\\phi$~is the function inverse to~$f$.", "markdown": "It is easy to see that if we can find the integral of $y = f(x)$ then we can always find that of $x = \\phi(y)$, where $\\phi$ is the function inverse to $f$.", "why": "Shows how knowing one integral unlocks the integral of the inverse function.", "use": [ "lesson" ], "concepts": [ "concept/inverse-function", "method/integration-of-an-inverse-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-2c68d30839", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "250", "location": "DERIVATIVES AND INTEGRALS", "latex": "Thus \\emph{the ordinate of the curve is the derivative of the area, and the area is the integral of the ordinate}.", "markdown": "Thus *the ordinate of the curve is the derivative of the area, and the area is the integral of the ordinate*.", "why": "States the link between area and integration in one memorable line.", "use": [ "lesson", "website" ], "concepts": [ "concept/area", "concept/derivative", "concept/integral", "quantity/area", "theorem/fundamental-theorem-of-calculus" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-d2d7d0ce8b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "249", "location": "DERIVATIVES AND INTEGRALS", "latex": "The reader is of course familiar with the idea of an `area', and in particular with that of an area such as~$ONPP_{0}$. This idea we shall at present take for granted.", "markdown": "The reader is of course familiar with the idea of an ‘area’, and in particular with that of an area such as $ONPP_{0}$. This idea we shall at present take for granted.", "why": "Models honesty about what is assumed rather than proved, with a promise to return to it.", "use": [ "lesson", "history" ], "concepts": [ "concept/area" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-1fb1aa1c65", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "251", "location": "DERIVATIVES AND INTEGRALS", "latex": "The notion of the length of a curve, other than a straight line, is in reality a more difficult one even than that of an area.", "markdown": "The notion of the length of a curve, other than a straight line, is in reality a more difficult one even than that of an area.", "why": "Warns that a curve's length is subtler than it looks, and that the formula rests on an assumption.", "use": [ "lesson", "history" ], "concepts": [ "concept/area", "quantity/length-of-a-curve" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-1c2069ca30", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "252", "location": "DERIVATIVES AND INTEGRALS", "latex": "The explanation of this is of course that between $x = \\pi$ and $x = 2\\pi$ the curve lies below the axis of~$x$, and so the corresponding part of the area is counted negative in applying the method.", "markdown": "The explanation of this is of course that between $x = \\pi$ and $x = 2\\pi$ the curve lies below the axis of $x$, and so the corresponding part of the area is counted negative in applying the method.", "why": "Explains why the integral of sin x over a full period is zero though the area is not.", "use": [ "lesson" ], "concepts": [ "concept/area", "method/integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-0987a6a14b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "248", "location": "DERIVATIVES AND INTEGRALS", "latex": "The integrals of the inverse sine and tangent and of the logarithm can easily be calculated by integration by parts.", "markdown": "The integrals of the inverse sine and tangent and of the logarithm can easily be calculated by integration by parts.", "why": "Shows a surprising use of integration by parts on functions that look as if they have nothing to multiply.", "use": [ "lesson" ], "concepts": [ "method/integration", "method/integration-by-parts" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-821128939e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "226", "location": "DERIVATIVES AND INTEGRALS", "latex": "Before we give a strict proof of this theorem, which is perhaps the most important theorem in the Differential Calculus, it will be well to point out its obvious geometrical meaning.", "markdown": "Before we give a strict proof of this theorem, which is perhaps the most important theorem in the Differential Calculus, it will be well to point out its obvious geometrical meaning.", "why": "Tells the learner how much weight Hardy places on the theorem and that a picture comes before the proof.", "use": [ "lesson", "website" ], "concepts": [ "theorem/mean-value-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-a184042fb3", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "227", "location": "DERIVATIVES AND INTEGRALS", "latex": "For $\\phi'(\\xi)$~is the tangent of the angle which the tangent at~$P$ makes with~$OX$, and $\\{\\phi(b) - \\phi(a)\\}/(b - a)$ the tangent of the angle which $AB$ makes with~$OX$.", "markdown": "For $\\phi'(\\xi)$ is the tangent of the angle which the tangent at $P$ makes with $OX$, and $\\{\\phi(b) - \\phi(a)\\}/(b - a)$ the tangent of the angle which $AB$ makes with $OX$.", "why": "Shows the geometric meaning of the theorem as the slope of the tangent equalling the slope of the chord.", "use": [ "lesson" ], "concepts": [ "concept/chord", "concept/tangent", "theorem/mean-value-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-0992a9d814", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "227", "location": "DERIVATIVES AND INTEGRALS", "latex": "It should be observed that it has not been assumed in this proof that $\\phi'(x)$~is continuous.", "markdown": "It should be observed that it has not been assumed in this proof that $\\phi'(x)$ is continuous.", "why": "Warns the learner that the theorem needs only the existence of the derivative, not its continuity.", "use": [ "lesson" ], "concepts": [ "concept/continuous-function", "concept/derivative", "theorem/mean-value-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-6e3c43f393", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "228", "location": "DERIVATIVES AND INTEGRALS", "latex": "In the first place we want to know whether such a function as $\\phi(x)$ \\emph{actually exists}. This question must be carefully distinguished from the question as to whether (supposing that there is such a function) we can find any simple formula to express it.", "markdown": "In the first place we want to know whether such a function as $\\phi(x)$ *actually exists*. This question must be carefully distinguished from the question as to whether (supposing that there is such a function) we can find any simple formula to express it.", "why": "Separates the question of whether a solution exists from the question of whether we can write it down.", "use": [ "lesson", "website" ], "concepts": [ "method/integration", "theorem/existence-of-an-integral-of-a-continuous-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-5ce26a510f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "229", "location": "DERIVATIVES AND INTEGRALS", "latex": "Whether there are continuous functions which \\emph{never} have derivatives, or continuous curves which never have tangents, is a further question which is at present beyond us. Common-sense says \\emph{No}: but, as we have already stated in \\SecNo[§]{111}, this is one of the cases in which higher mathematics has proved common-sense to be mistaken.", "markdown": "Whether there are continuous functions which *never* have derivatives, or continuous curves which never have tangents, is a further question which is at present beyond us. Common-sense says *No*: but, as we have already stated in [§]111, this is one of the cases in which higher mathematics has proved common-sense to be mistaken.", "why": "Gives an honest historical caution that intuition about curves and tangents can be wrong.", "use": [ "history", "website" ], "concepts": [ "concept/derivative", "concept/tangent" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-8666fffac5", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "230", "location": "DERIVATIVES AND INTEGRALS", "latex": "It is hardly necessary to point out that $\\int\\dots dx$ like $d/dx$ must, at present at any rate, be regarded purely as a symbol of operation: the~$\\int$ and the~$dx$ no more mean anything when taken by themselves than do the~$d$ and~$dx$ of the other operative symbol~$d/dx$.", "markdown": "It is hardly necessary to point out that $\\int\\dots dx$ like $d/dx$ must, at present at any rate, be regarded purely as a symbol of operation: the $\\int$ and the $dx$ no more mean anything when taken by themselves than do the $d$ and $dx$ of the other operative symbol $d/dx$.", "why": "Warns learners not to read the integral sign and dx as separate objects with meanings of their own.", "use": [ "lesson" ], "concepts": [ "concept/integral", "method/integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-a52fe76597", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "230", "location": "DERIVATIVES AND INTEGRALS", "latex": "These formulae must be understood as meaning that the function on the right-hand side is \\emph{one} integral of that under the sign of integration. The \\emph{most general} integral is of course obtained by adding to the former a constant~$C$, known as the \\Emph{arbitrary constant} of integration.", "markdown": "These formulae must be understood as meaning that the function on the right-hand side is *one* integral of that under the sign of integration. The *most general* integral is of course obtained by adding to the former a constant $C$, known as the **constant** of integration.", "why": "Explains why every integral formula carries a +C and what the formula really states.", "use": [ "lesson" ], "concepts": [ "concept/arbitrary-constant-of-integration", "concept/integral", "concept/standard-forms-of-integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-f03aca2f54", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "237", "location": "DERIVATIVES AND INTEGRALS", "latex": "Thus the integral of~$R(\\sqrt{x})$, where $R$~denotes a rational function, is reduced by the substitution $x = t^{2}$ to the integral of~$2tR(t^{2})$, \\ie\\ to the integral of a rational function of~$t$. This method of integration is called \\Emph{integration by rationalisation}, and is of extremely wide application.", "markdown": "Thus the integral of $R(\\sqrt{x})$, where $R$ denotes a rational function, is reduced by the substitution $x = t^{2}$ to the integral of $2tR(t^{2})$, *i.e.* to the integral of a rational function of $t$. This method of integration is called **by rationalisation**, and is of extremely wide application.", "why": "A small worked example showing how a substitution turns a square-root integral into a rational one.", "use": [ "lesson" ], "concepts": [ "concept/integration-by-substitution", "concept/rational-function", "method/integration-by-rationalisation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-465b066400", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "266", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "This expansion of~$f(a + h)$ is known as \\Emph{Taylor's Series}.", "markdown": "This expansion of $f(a + h)$ is known as **’s Series**.", "why": "It names the series that the chapter builds from the remainder argument, so a learner can connect the name to the formula that follows.", "use": [ "lesson" ], "concepts": [ "theorem/taylor-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-c83c7d390b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "272", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "A point at which the tangent to a curve has contact of the second order is called a \\Emph{point of inflexion}.", "markdown": "A point at which the tangent to a curve has contact of the second order is called a **of inflexion**.", "why": "It gives the definition of a point of inflexion in terms of contact, so learners see why the second derivative vanishes there.", "use": [ "lesson", "website" ], "concepts": [ "concept/contact-of-the-nth-order", "concept/point-of-inflexion", "concept/tangent" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-6918bbf023", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "268", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "if there is to be a maximum or minimum the first derivative which does not vanish must be an even derivative, and there will be a maximum if it is negative, a minimum if it is positive.", "markdown": "if there is to be a maximum or minimum the first derivative which does not vanish must be an even derivative, and there will be a maximum if it is negative, a minimum if it is positive.", "why": "It states a test for maxima and minima that works even when the second derivative vanishes, which the earlier test could not handle.", "use": [ "lesson" ], "concepts": [ "concept/higher-order-derivative", "concept/maximum", "concept/minimum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-d92f0f582c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "264", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "The formula $f(x + h) = f(x) + hf'(x + \\theta_{1}h)$ is not true if $f(x) = 1/x$ and $x < 0 < x + h$.", "markdown": "The formula $f(x + h) = f(x) + hf'(x + \\theta_{1}h)$ is not true if $f(x) = 1/x$ and $x < 0 < x + h$.", "why": "It shows that the mean value theorem fails when its hypotheses are not met, so learners learn to check conditions before applying it.", "use": [ "lesson", "history" ], "concepts": [ "theorem/mean-value-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-16e8961eb8", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "276", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "The fact is of course that \\emph{$\\dd x/\\dd r$ and $\\dd r/\\dd x$ are not formed upon the same hypothesis as to the variation of~$P$.}", "markdown": "The fact is of course that *$\\dd x/\\dd r$ and $\\dd r/\\dd x$ are not formed upon the same hypothesis as to the variation of $P$.*", "why": "It names the common error of treating reciprocal partial derivatives as reciprocals, and says why they differ.", "use": [ "website", "lesson" ], "concepts": [ "concept/partial-derivative" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-b0d070702c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "276", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "Thus if $u = x + y + z$, $x$,~$y$, and~$z$ being the independent variables, then $\\dd u/\\dd x = 1$. But if we regard $u$ as a function of the variables $x$, $x + y = \\eta$, and $x + y + z = \\zeta$, so that $u = \\zeta$, then $\\dd u/\\dd x = 0$.", "markdown": "Thus if $u = x + y + z$, $x$, $y$, and $z$ being the independent variables, then $\\dd u/\\dd x = 1$. But if we regard $u$ as a function of the variables $x$, $x + y = \\eta$, and $x + y + z = \\zeta$, so that $u = \\zeta$, then $\\dd u/\\dd x = 0$.", "why": "It shows that a partial derivative is only defined once every independent variable is specified, a point learners often miss.", "use": [ "lesson" ], "concepts": [ "concept/function-of-several-variables", "concept/partial-derivative" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-00348b36a2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "280", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "We have up to the present attributed no meaning of any kind to the symbol~$dy$ standing by itself. We now agree to \\emph{define}~$dy$ by the equation", "markdown": "We have up to the present attributed no meaning of any kind to the symbol $dy$ standing by itself. We now agree to *define* $dy$ by the equation", "why": "It gives a plain account of what a differential is: a definition, not a limit, introduced to make equations exact.", "use": [ "lesson" ], "concepts": [ "concept/differential" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-0e3bf586a2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "280", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "The advantages of the `differential' notation are in reality of a purely technical character.", "markdown": "The advantages of the ‘differential’ notation are in reality of a purely technical character.", "why": "It is a candid remark that the differential notation is a convenience, which helps a learner judge what the notation does and does not add.", "use": [ "history" ], "concepts": [ "concept/differential" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-229fd39a33", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "287", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "The number \\[ \\int_{a}^{b} f(x)\\, dx \\] is called a \\Emph{definite integral}; $a$~and~$b$ are called its \\Emph{lower and upper limits}; $f(x)$~is called the \\Emph{subject of integration} or \\Emph{integrand}; and the interval~$\\DPmod{(a, b)}{[a, b]}$ the \\Emph{range of integration}.", "markdown": "The number _a^b f(x)  dx is called a **integral**; $a$ and $b$ are called its **and upper limits**; $f(x)$ is called the **of integration** or ****; and the interval $\\DPmod{(a, b)}{[a, b]}$ the **of integration**.", "why": "It gives the standard names for the parts of a definite integral in one place, which a learner needs before reading any integral.", "use": [ "lesson" ], "concepts": [ "concept/definite-integral", "concept/limits-of-integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-b7a05b230e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "287", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "The distinction between the definite and the indefinite integral is merely one of point of view.", "markdown": "The distinction between the definite and the indefinite integral is merely one of point of view.", "why": "It tells the learner that the definite and indefinite integral are two views of one object, not two different operations.", "use": [ "lesson" ], "concepts": [ "concept/definite-integral", "concept/indefinite-integral" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-009ce096e9", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "287", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "And when we are considering a `definite integral' we are not as a rule concerned with any possible variation of the limits.", "markdown": "And when we are considering a ‘definite integral’ we are not as a rule concerned with any possible variation of the limits.", "why": "It warns that a definite integral with constant limits is a number, not a function, which heads off a common confusion.", "use": [ "lesson" ], "concepts": [ "concept/definite-integral" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-1930f35ad1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "295", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "which is the formula for the transformation of a definite integral by \\Emph{substitution}.", "markdown": "which is the formula for the transformation of a definite integral by ****.", "why": "It names the substitution rule for definite integrals and states it as a change of limits, which a learner can apply directly.", "use": [ "lesson" ], "concepts": [ "concept/definite-integral", "method/substitution" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-4e45df982a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "275", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "The reader must not suppose, however, that these new notations imply any essential novelty of idea: `partial differentiation' with respect to~$x$ is exactly the same process as ordinary differentiation, the only novelty lying in the presence in~$f$ of a second variable~$y$ independent of~$x$.", "markdown": "The reader must not suppose, however, that these new notations imply any essential novelty of idea: ‘partial differentiation’ with respect to $x$ is exactly the same process as ordinary differentiation, the only novelty lying in the presence in $f$ of a second variable $y$ independent of $x$.", "why": "Reassures a learner that partial differentiation is ordinary differentiation with the other variable held still.", "use": [ "lesson", "website" ], "concepts": [ "concept/derivative", "concept/partial-derivative", "method/differentiation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-5594b7ead2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "276", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "But the reader must be careful to impress on his mind that the notion of the partial derivative of a function of several variables is only determinate when \\emph{all} the independent variables are specified.", "markdown": "But the reader must be careful to impress on his mind that the notion of the partial derivative of a function of several variables is only determinate when *all* the independent variables are specified.", "why": "Warns against a common mistake: a partial derivative depends on which other variables are held fixed.", "use": [ "lesson" ], "concepts": [ "concept/function-of-two-variables", "concept/partial-derivative" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-37a658040d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "278", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "Suppose for example that $y = 1 - x$ and $f(x, y) = x + y$. Then $D_{x}f(x, 1 - x) = D_{x}1 = 0$, but $D_{x}f(x, y) = 1$.", "markdown": "Suppose for example that $y = 1 - x$ and $f(x, y) = x + y$. Then $D_{x}f(x, 1 - x) = D_{x}1 = 0$, but $D_{x}f(x, y) = 1$.", "why": "A tiny worked example showing why df/dx and the partial derivative must be told apart.", "use": [ "lesson" ], "concepts": [ "concept/partial-derivative", "theorem/total-differential" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-3f1bec515e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "280", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "The symbol $dy/dx$ thus acquires a double meaning; but there is no inconvenience in this, since \\Eq{(6)}~is true whichever meaning we choose.", "markdown": "The symbol $dy/dx$ thus acquires a double meaning; but there is no inconvenience in this, since (6) is true whichever meaning we choose.", "why": "Candidly explains how dy/dx can be both a derivative and a quotient of differentials.", "use": [ "lesson", "history" ], "concepts": [ "concept/derivative", "concept/differential" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-978278d377", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "280", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "This is sometimes expressed by saying that $dy$~is the \\emph{principal part} of~$\\delta y$ when $\\delta x$~is small, just as we might say that $ax$~is the `principal part' of $ax + bx^{2}$ when $x$~is small.", "markdown": "This is sometimes expressed by saying that $dy$ is the *principal part* of $\\delta y$ when $\\delta x$ is small, just as we might say that $ax$ is the ‘principal part’ of $ax + bx^{2}$ when $x$ is small.", "why": "Gives a concrete analogy for what a differential is relative to a small increment.", "use": [ "lesson", "website" ], "concepts": [ "concept/differential", "concept/principal-part-of-an-increment" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-85e79a45e1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "281", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "Thus \\emph{the formula which expresses~$dz$ in terms of $dx$~and~$dy$ is the same whether the variables $x$~and~$y$ are independent or not}. This remark is of great importance in applications.", "markdown": "Thus *the formula which expresses $dz$ in terms of $dx$ and $dy$ is the same whether the variables $x$ and $y$ are independent or not*. This remark is of great importance in applications.", "why": "States the key payoff of the differential notation: one formula for dz whether or not the variables are independent.", "use": [ "lesson", "website" ], "concepts": [ "concept/differential", "theorem/total-differential" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-df7215ab28", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "284", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "But nothing which we know so far provides us with a direct definition of the area of a figure bounded by curved lines. We shall now show how to give a definition of~$F(x)$ which will enable us to \\emph{prove} its existence.", "markdown": "But nothing which we know so far provides us with a direct definition of the area of a figure bounded by curved lines. We shall now show how to give a definition of $F(x)$ which will enable us to *prove* its existence.", "why": "Honestly names the gap in the earlier treatment of area and motivates the rigorous construction.", "use": [ "lesson", "history" ], "concepts": [ "concept/area" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-89912e2f29", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "286", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "We define the area of~$PpqQ$ as being \\emph{the common limit of $s$~and~$S$, that is to say~$J$}.", "markdown": "We define the area of $PpqQ$ as being *the common limit of $s$ and $S$, that is to say $J$*.", "why": "Gives the precise definition of the area under a curve as the limit of inner and outer rectangle sums.", "use": [ "lesson", "website" ], "concepts": [ "concept/area", "concept/limit", "concept/lower-sum", "concept/upper-sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-2ff921f815", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "267", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "The reader should be careful to guard himself against supposing that the\ncontinuity of all the derivatives of~$f(x)$ is a sufficient condition for the validity\nof Taylor's series. A direct discussion of the behaviour of~$R_{n}$ is always\nessential.", "markdown": "The reader should be careful to guard himself against supposing that the continuity of all the derivatives of $f(x)$ is a sufficient condition for the validity of Taylor’s series. A direct discussion of the behaviour of $R_{n}$ is always essential.", "why": "Warns learners that having every derivative continuous does not by itself guarantee that Taylor's series represents the function; the remainder must be checked.", "use": [ "lesson", "website" ], "concepts": [ "theorem/lagrange-s-form-of-the-remainder", "theorem/taylor-s-series", "theorem/taylor-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-028a4a1f73", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "264", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "In view of the great importance of this theorem we shall give\nat the end of this chapter another proof, not essentially distinct\nfrom that given above, but different in form and depending on\nthe method of integration by parts.", "markdown": "In view of the great importance of this theorem we shall give at the end of this chapter another proof, not essentially distinct from that given above, but different in form and depending on the method of integration by parts.", "why": "Shows the author flagging a result as central and promising a second proof by a different route.", "use": [ "history", "website" ], "concepts": [ "theorem/taylor-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-12adf02b79", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "265", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "Apply this process to the equation $x^{2} = 2$, taking $\\xi = 3/2$ as the first\napproximation. [We find $h = -1/12$, $\\xi + h = 17/12 = 1.417\\dots$, which is quite a\ngood approximation, in spite of the roughness of the first. If now we repeat\nthe process, taking $\\xi = 17/12$, we obtain $\\xi + h = 577/408 = 1.414\\MS215\\dots$, which\nis correct to $5$~places of decimals.\\Add{]}", "markdown": "Apply this process to the equation $x^{2} = 2$, taking $\\xi = 3/2$ as the first approximation. [We find $h = -1/12$, $\\xi + h = 17/12 = 1.417\\dots$, which is quite a good approximation, in spite of the roughness of the first. If now we repeat the process, taking $\\xi = 17/12$, we obtain $\\xi + h = 577/408 = 1.414\\MS215\\dots$, which is correct to $5$ places of decimals.", "why": "A short worked example in which two steps of Newton's method turn a rough guess into the square root of 2 correct to five decimal places.", "use": [ "lesson", "website" ], "concepts": [ "concept/approximation", "concept/root-of-an-equation", "method/newton-s-method" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-cac89f7c66", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "271", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "It is evident that the degree of smallness of~$QR$ may be taken\nas a kind of measure of the \\emph{closeness of the contact} of the curves.", "markdown": "It is evident that the degree of smallness of $QR$ may be taken as a kind of measure of the *closeness of the contact* of the curves.", "why": "Gives learners a way to think about contact: the smaller the gap between curves near a point, the closer the contact.", "use": [ "lesson" ], "concepts": [ "concept/contact-of-the-nth-order", "concept/order-of-smallness" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-8592e9daea", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "268", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "In order that there should be a maximum or a minimum this\nexpression must be of constant sign for all sufficiently small\nvalues of~$h$, positive or negative. This evidently requires that $n$~should\nbe even.", "markdown": "In order that there should be a maximum or a minimum this expression must be of constant sign for all sufficiently small values of $h$, positive or negative. This evidently requires that $n$ should be even.", "why": "Explains why the first non-vanishing derivative must be of even order, using the sign of the Taylor term.", "use": [ "lesson" ], "concepts": [ "concept/maximum", "concept/minimum", "method/higher-derivative-test-for-maxima-and-minima" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-e6f0c66b5a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "270", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "Two curves are said to \\emph{intersect} (or \\emph{cut}) at a point if the point lies on each of them. They are said to \\emph{touch} at the point if they have the same tangent\nat the point.", "markdown": "Two curves are said to *intersect* (or *cut*) at a point if the point lies on each of them. They are said to *touch* at the point if they have the same tangent at the point.", "why": "Separates cutting from touching, which sets up the idea of order of contact.", "use": [ "lesson", "website" ], "concepts": [ "concept/intersection-of-curves", "concept/touching-of-curves" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-8d7d446b69", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "267", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "But a difficulty arises if $-1 < x < 0$, since $1 + \\theta_{n}x < 1$ and $(1 + \\theta_{n}x)^{m-n} > 1$\nif $n > m$; knowing only that $0 < \\theta_{n} < 1$, we cannot be assured that $1 + \\theta_{n}x$~is not\nquite small and $(1 + \\theta _{n}x)^{m-n}$ quite large.", "markdown": "But a difficulty arises if $-1 < x < 0$, since $1 + \\theta_{n}x < 1$ and $(1 + \\theta_{n}x)^{m-n} > 1$ if $n > m$; knowing only that $0 < \\theta_{n} < 1$, we cannot be assured that $1 + \\theta_{n}x$ is not quite small and $(1 + \\theta _{n}x)^{m-n}$ quite large.", "why": "Shows honestly where an attempted proof of the binomial series breaks down, and why a different remainder form is needed.", "use": [ "lesson" ], "concepts": [ "concept/binomial-series", "theorem/lagrange-s-form-of-the-remainder" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-1e65c2c74f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "273", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "The circle which has contact of the second order with the curve at the point\n$(\\xi, \\eta)$ is called the \\Emph{circle of curvature}, and its radius the \\Emph{radius of curvature}.", "markdown": "The circle which has contact of the second order with the curve at the point $(\\xi, \\eta)$ is called the **of curvature**, and its radius the **of curvature**.", "why": "Defines the circle that best fits a curve at a point, linking curvature to contact.", "use": [ "lesson", "website" ], "concepts": [ "concept/circle-of-curvature", "concept/contact-of-the-nth-order", "quantity/radius-of-curvature" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-8a14b08931", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "286", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "As it is not always practicable actually to determine the form of~$F(x)$, it is convenient to have a formula which represents the area~$PpqQ$ and contains no explicit reference to~$F(x)$.", "markdown": "As it is not always practicable actually to determine the form of $F(x)$, it is convenient to have a formula which represents the area $PpqQ$ and contains no explicit reference to $F(x)$.", "why": "Motivates the definite integral notation as a way to name an area without first finding an integral function.", "use": [ "lesson", "website" ], "concepts": [ "concept/definite-integral", "quantity/area" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-6436ccac57", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "287", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "But when we are considering `indefinite integrals' or `integral functions' we are usually thinking of \\emph{a relation between two functions}, in virtue of which one is the derivative of the other.", "markdown": "But when we are considering ‘indefinite integrals’ or ‘integral functions’ we are usually thinking of *a relation between two functions*, in virtue of which one is the derivative of the other.", "why": "Clarifies how the indefinite integral, a relation between functions, differs in usage from the definite integral, a number.", "use": [ "lesson" ], "concepts": [ "concept/definite-integral", "concept/indefinite-integral" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-a00866be0e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "288", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "The whole difficulty lies in the question, \\emph{what is the~$x$ which occurs in $\\cos x$ and $\\sin x$}? To answer this question, we must define the measure of an angle, and we are now in a position to do so.", "markdown": "The whole difficulty lies in the question, *what is the $x$ which occurs in $\\cos x$ and $\\sin x$*? To answer this question, we must define the measure of an angle, and we are now in a position to do so.", "why": "Shows a learner that the x in sin x and cos x needs a definition, not just an assumption.", "use": [ "lesson", "website" ], "concepts": [ "concept/circular-function", "concept/circular-measure" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-39c3a4dd26", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "288", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "It has however, for our present purpose, a fatal defect; for we have not proved that the arc of a curve, even of a circle, possesses a length.", "markdown": "It has however, for our present purpose, a fatal defect; for we have not proved that the arc of a curve, even of a circle, possesses a length.", "why": "A candid statement of why the familiar arc-length definition of angle measure cannot yet be used.", "use": [ "lesson", "history" ], "concepts": [ "concept/circular-measure", "quantity/length-of-a-curve" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-f4956cb1aa", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "288", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "We must therefore found our definition on the notion not of length but of \\emph{area}. We define the measure of the angle~$AOP$ as \\emph{twice the area of the sector~$AOP$ of the unit circle}.", "markdown": "We must therefore found our definition on the notion not of length but of *area*. We define the measure of the angle $AOP$ as *twice the area of the sector $AOP$ of the unit circle*.", "why": "Gives the book's rigorous definition of angle measure through area.", "use": [ "lesson", "website" ], "concepts": [ "concept/circular-measure", "concept/circular-sector", "quantity/area" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-4c006bdc1e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "292", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "This follows from~(7). For we can take $H$ to be the least and $K$~the greatest value of~$f(x)$ in~$\\DPmod{(a, b)}{[a, b]}$. Then the integral is equal to~$\\eta(b - a)$, where $\\eta$~lies between $H$ and~$K$. But, since $f(x)$~is continuous, there must be a value of~$\\xi$ for which $f(\\xi) = \\eta$~(\\SecNo[§]{100}).", "markdown": "This follows from (7). For we can take $H$ to be the least and $K$ the greatest value of $f(x)$ in $\\DPmod{(a, b)}{[a, b]}$. Then the integral is equal to $\\eta(b - a)$, where $\\eta$ lies between $H$ and $K$. But, since $f(x)$ is continuous, there must be a value of $\\xi$ for which $f(\\xi) = \\eta$ ([§]100).", "why": "A short worked proof showing how the mean value theorem for integrals follows from bounds and continuity.", "use": [ "lesson" ], "concepts": [ "concept/continuous-function", "theorem/first-mean-value-theorem-for-integrals" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-f14c861585", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "295", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "That the value of a definite integral may sometimes be found without a knowledge of the integral function is only to be expected, for the fact that we cannot determine the general form of a function~$F(x)$ in no way precludes the possibility that we may be able to determine the difference $F(b) - F(a)$ between two of its particular values.", "markdown": "That the value of a definite integral may sometimes be found without a knowledge of the integral function is only to be expected, for the fact that we cannot determine the general form of a function $F(x)$ in no way precludes the possibility that we may be able to determine the difference $F(b) - F(a)$ between two of its particular values.", "why": "Explains why definite integrals can sometimes be evaluated even when no formula for the integral function exists.", "use": [ "lesson", "website" ], "concepts": [ "concept/definite-integral", "method/integration-by-parts", "method/substitution" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-031e400b96", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "299", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "It will be remembered that the difficulty in using Lagrange's form, in \\Ex{lvi}.~2, arose in connection with negative values of~$x$.", "markdown": "It will be remembered that the difficulty in using Lagrange’s form, in % [examples:lvi]Ex. lvi%. 2, arose in connection with negative values of $x$.", "why": "Connects the earlier difficulty with Lagrange's remainder to the reason Cauchy's form is used for the binomial series.", "use": [ "history", "lesson" ], "concepts": [ "theorem/binomial-theorem", "theorem/cauchy-s-form-of-the-remainder", "theorem/lagrange-s-form-of-the-remainder" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-812159a07d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "299", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "We define the integral of a complex function $f(x) = \\DPtypo{\\psi}{\\phi}(x) + i\\psi(x)$ of the real variable~$x$, between the limits $a$~and~$b$, by the equations", "markdown": "We define the integral of a complex function $f(x) = \\DPtypo{\\psi}{\\phi}(x) + i\\psi(x)$ of the real variable $x$, between the limits $a$ and $b$, by the equations", "why": "Shows how a real-variable integral extends to complex values by splitting into real and imaginary parts.", "use": [ "lesson", "website" ], "concepts": [ "concept/integral-of-a-complex-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-de87f38f5b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "299", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "This inequality may be deduced without difficulty from the definitions of \\SecNo[§§]{156}~and~\\SecNo{157}.", "markdown": "This inequality may be deduced without difficulty from the definitions of [§§]156 and 157.", "why": "Shows that the modulus inequality comes straight from the limit-of-sums definition of the integral.", "use": [ "lesson" ], "concepts": [ "concept/integral-of-a-complex-function", "theorem/modulus-inequality-for-integrals" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-b12adb6c92", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "302", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "In these circumstances $u$~is called a \\emph{homogeneous function of degree~$n$} in the variables $x$,~$y$, $z$,~\\dots.", "markdown": "In these circumstances $u$ is called a *homogeneous function of degree $n$* in the variables $x$, $y$, $z$, ….", "why": "Names homogeneity after the scaling behaviour just described.", "use": [ "lesson" ], "concepts": [ "concept/homogeneous-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-e7d82bd84d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "302", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "This result is known as \\Emph{Euler's Theorem} on homogeneous functions.", "markdown": "This result is known as **’s Theorem** on homogeneous functions.", "why": "Attaches Euler's name to the identity x u_x + y u_y + ... = n u.", "use": [ "history" ], "concepts": [ "theorem/euler-s-theorem-on-homogeneous-functions" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-93927056b7", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "303", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "Two functions $u$~and~$v$ are said to be \\emph{dependent} or \\emph{independent} according as they are or are not connected by such a relation as~\\Eq{(1)}.", "markdown": "Two functions $u$ and $v$ are said to be *dependent* or *independent* according as they are or are not connected by such a relation as (1).", "why": "Gives a precise meaning to functional dependence, which the Jacobian then tests.", "use": [ "lesson", "website" ], "concepts": [ "concept/dependent", "concept/functional-relation", "concept/jacobian" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-84d51e9b6f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "303", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "This condition is therefore \\emph{necessary} for the existence of a relation such as~\\Eq{(1)}.", "markdown": "This condition is therefore *necessary* for the existence of a relation such as (1).", "why": "Models careful reasoning: the vanishing Jacobian is shown to be necessary, and sufficiency is left to a reference.", "use": [ "lesson" ], "concepts": [ "concept/functional-relation", "concept/jacobian" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-246cd666ef", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "307", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "This rule, which gives a very good approximation, is known as \\Emph{Simpson's Rule}.", "markdown": "This rule, which gives a very good approximation, is known as **’s Rule**.", "why": "Introduces Simpson's rule as a practical way to approximate an integral.", "use": [ "lesson", "history" ], "concepts": [ "method/simpson-s-rule" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-94c8253472", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "307", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "It should be observed that if $\\phi(x)$~is any cubic polynomial then $\\phi^{(4)}(x) = 0$, and Simpson's Rule is exact.", "markdown": "It should be observed that if $\\phi(x)$ is any cubic polynomial then $\\phi^{(4)}(x) = 0$, and Simpson’s Rule is exact.", "why": "Explains why the error bound for Simpson's rule makes it exact for cubics.", "use": [ "lesson" ], "concepts": [ "method/simpson-s-rule" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-a73f81936c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "313", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "This theorem asserts that if we have a convergent series of positive terms, $u_{0} + u_{1} + u_{2} + \\dots$ say, and form any other series", "markdown": "This theorem asserts that if we have a convergent series of positive terms, $u_{0} + u_{1} + u_{2} + \\dots$ say, and form any other series", "why": "It states the rearrangement idea in plain words, so a learner sees what is being claimed before the proof.", "use": [ "lesson" ], "concepts": [ "concept/rearrangement-of-a-series", "theorem/dirichlet-s-rearrangement-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-7f3db1c9ae", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "315", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "The tests derived from comparison with it are therefore naturally very crude, and much more delicate tests are often wanted.", "markdown": "The tests derived from comparison with it are therefore naturally very crude, and much more delicate tests are often wanted.", "why": "It tells the learner that the simple tests are a starting point and that finer tests are needed for borderline series.", "use": [ "lesson" ], "concepts": [ "theorem/comparison-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-9f027d6270", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "310", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "This hardly requires proof, for $v_{n}^{1/n} \\geq 1$ involves $v_{n} \\geq 1$.", "markdown": "This hardly requires proof, for $v_{n}^{1/n} \\geq 1$ involves $v_{n} \\geq 1$.", "why": "It shows the reasoning behind a divergence test in one line, which helps a learner check the idea.", "use": [ "lesson" ], "concepts": [ "theorem/cauchy-s-root-test" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-a37d1aa8c8", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "312", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "If $0 < a < b < 1$, then the series $a + b + a^{2} + b^{2} + a^{3} + \\dots$ is convergent.", "markdown": "If $0 < a < b < 1$, then the series $a + b + a^{2} + b^{2} + a^{3} + \\dots$ is convergent.", "why": "It gives a practice case where Cauchy's test works and d'Alembert's test fails, which teaches when to choose each test.", "use": [ "lesson" ], "concepts": [ "concept/convergent-series", "theorem/cauchy-s-root-test", "theorem/d-alembert-s-ratio-test" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-6694cdb76a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "335", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "The explanation is to be found in a closer consideration of the relation between $x$~and~$y$.", "markdown": "The explanation is to be found in a closer consideration of the relation between $x$ and $y$.", "why": "It tells the learner that a substitution in a definite integral can fail silently, so the relation between the variables must be checked before trusting the result.", "use": [ "lesson" ], "concepts": [ "concept/definite-integral", "method/substitution" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-d2f08dc4eb", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "341", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "This example shows that the condition that $\\phi_{n}$~should tend \\emph{steadily} to zero is essential to the truth of the theorem.", "markdown": "This example shows that the condition that $\\phi_{n}$ should tend *steadily* to zero is essential to the truth of the theorem.", "why": "It shows the learner the common mistake of dropping the monotonicity hypothesis in the alternating series test.", "use": [ "lesson" ], "concepts": [ "concept/alternating-series", "concept/convergent-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-4b20c68d95", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "338", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "Dirichlet's Theorem (\\SecNo[§]{169}) shows that the terms of a series of positive terms may be rearranged in any way without affecting its sum.", "markdown": "Dirichlet’s Theorem ([§]169) shows that the terms of a series of positive terms may be rearranged in any way without affecting its sum.", "why": "It states the rearrangement property for positive series that the learner can then contrast with conditional convergence.", "use": [ "lesson" ], "concepts": [ "concept/rearrangement-of-a-series", "concept/series-of-positive-terms", "theorem/dirichlet-s-rearrangement-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-733fa055db", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "337", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "if $\\sum u_{n}$~is absolutely convergent then it is convergent; so are the series formed by its positive and negative terms taken separately; and the sum of the series is equal to the sum of the positive terms plus the sum of the negative terms.", "markdown": "if $\\sum u_{n}$ is absolutely convergent then it is convergent; so are the series formed by its positive and negative terms taken separately; and the sum of the series is equal to the sum of the positive terms plus the sum of the negative terms.", "why": "It collects in one statement what absolute convergence guarantees, which is the summary a learner needs before the examples.", "use": [ "lesson" ], "concepts": [ "concept/absolute-convergence", "concept/convergent-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-b7e96953b3", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "342", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "There is another test, due to Abel, which, though of less frequent application than Dirichlet's, is sometimes useful.", "markdown": "There is another test, due to Abel, which, though of less frequent application than Dirichlet’s, is sometimes useful.", "why": "It credits the test to Abel and places it relative to Dirichlet's test, giving the learner a sense of the historical lineage.", "use": [ "history" ], "concepts": [ "person/niels-henrik-abel", "theorem/abel-s-test", "theorem/dirichlet-s-test" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-37c0e31abc", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "349", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "We can now extend this result to all cases in which $\\sum u_{n}$ and $\\sum v_{n}$ are \\emph{absolutely} convergent; for our proof was merely a simple application of Dirichlet's Theorem, which we have already extended to all absolutely convergent series.", "markdown": "We can now extend this result to all cases in which $\\sum u_{n}$ and $\\sum v_{n}$ are *absolutely* convergent; for our proof was merely a simple application of Dirichlet’s Theorem, which we have already extended to all absolutely convergent series.", "why": "It shows how a result proved for positive series is carried to absolutely convergent series by a single application of an earlier theorem.", "use": [ "lesson", "history" ], "concepts": [ "concept/absolute-convergence", "concept/product-of-series", "theorem/dirichlet-s-rearrangement-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-773ec13ca6", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "354", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "Deduce that \\emph{if $\\sum c_{n}$ is convergent then its sum is~$AB$}.", "markdown": "Deduce that *if $\\sum c_{n}$ is convergent then its sum is $AB$*.", "why": "It states the conclusion of Abel's theorem on multiplying series, which removes the need for absolute convergence.", "use": [ "history", "lesson" ], "concepts": [ "concept/convergent-series", "concept/product-of-series", "theorem/abel-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-1f16498c12", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "347", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "In case~(3) the circle is called the \\Emph{circle of convergence} and its radius the \\Emph{radius of convergence} of the power series.", "markdown": "In case (3) the circle is called the **of convergence** and its radius the **of convergence** of the power series.", "why": "It gives the names for the circle and radius of convergence at the point where they are first introduced.", "use": [ "lesson" ], "concepts": [ "concept/circle-of-convergence", "concept/radius-of-convergence" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-b577dfc56f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "309", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "Moreover, in inferring the convergence or divergence of~$\\sum v_{n}$ by means of one of these tests, it is sufficient to know that the test is satisfied for \\emph{sufficiently large} values of~$n$, \\ie\\ for all values of~$n$ greater than a definite value~$n_{0}$.", "markdown": "Moreover, in inferring the convergence or divergence of $\\sum v_{n}$ by means of one of these tests, it is sufficient to know that the test is satisfied for *sufficiently large* values of $n$, *i.e.* for all values of $n$ greater than a definite value $n_{0}$.", "why": "Shows that a test need only hold from some point on, so a few early odd terms never spoil a conclusion.", "use": [ "lesson" ], "concepts": [ "concept/sufficiently-large-values", "theorem/comparison-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-617aa8f056", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "311", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "None the less d'Alembert's test is very useful in practice, because when $v_{n}$~is a complicated function $v_{n+1}/v_{n}$~is often much less complicated and so easier to work with.", "markdown": "None the less d’Alembert’s test is very useful in practice, because when $v_{n}$ is a complicated function $v_{n+1}/v_{n}$ is often much less complicated and so easier to work with.", "why": "Explains honestly why a theoretically weaker test is still the one reached for in practice.", "use": [ "lesson" ], "concepts": [ "concept/common-ratio", "theorem/d-alembert-s-ratio-test" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-6854c5fbfa", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "313", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "Let $s$~be the sum of the series of~$u'$s. Then the sum of any number of terms, selected from the~$u'$s, is not greater than~$s$. But every~$v$ is a~$u$, and therefore the sum of any number of terms selected from the~$v'$s is not greater than~$s$. Hence $\\sum v_{n}$~is convergent, and its sum~$t$ is not greater than~$s$. But we can show in exactly the same way that $s \\leq t$. Thus $s = t$.", "markdown": "Let $s$ be the sum of the series of $u'$s. Then the sum of any number of terms, selected from the $u'$s, is not greater than $s$. But every $v$ is a $u$, and therefore the sum of any number of terms selected from the $v'$s is not greater than $s$. Hence $\\sum v_{n}$ is convergent, and its sum $t$ is not greater than $s$. But we can show in exactly the same way that $s \\leq t$. Thus $s = t$.", "why": "A short two-sided inequality argument that shows why reordering positive terms cannot change the sum.", "use": [ "lesson" ], "concepts": [ "concept/rearrangement-of-a-series", "theorem/dirichlet-s-rearrangement-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-3ac7798950", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "316", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "This theorem was discovered by Abel but forgotten, and rediscovered by Pringsheim.", "markdown": "This theorem was discovered by Abel but forgotten, and rediscovered by Pringsheim.", "why": "A small piece of history showing that results get lost and found again, and why a theorem can carry two names.", "use": [ "history" ], "concepts": [ "theorem/abel-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-b6ddd3b2da", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "317", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\emph{The converse of Abel's theorem is not true}, \\ie\\ it is not true that, if $u_{n}$~decreases with~$n$ and $\\lim nu_{n} = 0$, then $\\sum u_{n}$~is convergent.", "markdown": "*The converse of Abel’s theorem is not true*, *i.e.* it is not true that, if $u_{n}$ decreases with $n$ and $\\lim nu_{n} = 0$, then $\\sum u_{n}$ is convergent.", "why": "Warns learners that Abel's theorem only rules out convergence and can never confirm it.", "use": [ "lesson", "website" ], "concepts": [ "concept/converse-of-a-theorem", "theorem/abel-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-251ae42b73", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "315", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "The fact is that the geometric series, by comparison with which the tests of \\SecNo[§]{168} were obtained, is not only convergent but \\emph{very rapidly} convergent, far more rapidly than is necessary in order to ensure convergence. The tests derived from comparison with it are therefore naturally very crude, and much more delicate tests are often wanted.", "markdown": "The fact is that the geometric series, by comparison with which the tests of [§]168 were obtained, is not only convergent but *very rapidly* convergent, far more rapidly than is necessary in order to ensure convergence. The tests derived from comparison with it are therefore naturally very crude, and much more delicate tests are often wanted.", "why": "Explains why the ratio and root tests fail on slowly convergent series and motivates finer tests.", "use": [ "lesson", "website" ], "concepts": [ "concept/geometrical-progression", "theorem/cauchy-s-root-test", "theorem/d-alembert-s-ratio-test" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-f136d8c28e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "315", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "But this is far from being the case; if only we go far enough into the sequences we shall find the terms of the first sequence very much the smaller.", "markdown": "But this is far from being the case; if only we go far enough into the sequences we shall find the terms of the first sequence very much the smaller.", "why": "Corrects the intuition that a large power n^-12 shrinks faster than (2/3)^n, which needs far-out terms to see.", "use": [ "lesson", "website" ], "concepts": [ "concept/geometrical-progression", "concept/p-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-5497f88e1a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "320", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "The second of the\ntwo tests mentioned in \\SecNo[§]{172} is as follows: \\begin{Result}if $u_{n} = \\phi(n)$ is a\ndecreasing function of~$n$, then the series $\\sum \\phi(n)$~is convergent or\ndivergent according as $\\sum 2^{n}\\phi(2^{n})$~is convergent or divergent.\n\\end{Result}", "markdown": "The second of the two tests mentioned in [§]172 is as follows: Resultif $u_{n} = \\phi(n)$ is a decreasing function of $n$, then the series $\\sum \\phi(n)$ is convergent or divergent according as $\\sum 2^{n}\\phi(2^{n})$ is convergent or divergent. Result", "why": "States Cauchy's condensation test plainly, so a learner sees how the terms of a series can be grouped in blocks of doubling length.", "use": [ "lesson", "website" ], "concepts": [ "theorem/cauchy-s-condensation-test" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-99581912c1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "321", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "For our present purposes the field of application of this test is\npractically the same as that of the Integral Test. It enables us\nto discuss the series $\\sum n^{-s}$ with equal ease. For $\\sum n^{-s}$ will converge\nor diverge according as $\\sum 2^{n}2^{-ns}$ converges or diverges, \\ie\\ according\nas $s > 1$ or $s \\leq 1$.", "markdown": "For our present purposes the field of application of this test is practically the same as that of the Integral Test. It enables us to discuss the series $\\sum n^{-s}$ with equal ease. For $\\sum n^{-s}$ will converge or diverge according as $\\sum 2^{n}2^{-ns}$ converges or diverges, *i.e.* according as $s > 1$ or $s \\leq 1$.", "why": "Shows the condensation test giving the p-series result a second way, which helps a learner see that two methods can agree.", "use": [ "lesson" ], "concepts": [ "concept/p-series", "theorem/cauchy-s-condensation-test", "theorem/integral-test" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-ccea0ed1a2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "322", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "Of course the reader will not be puzzled by the use of the term\n\\emph{infinite integral} to denote something which has a definite value such as\n$2$ or~$\\frac{1}{2}\\pi$. The distinction between an infinite integral and a finite integral\nis similar to that between an infinite series and a finite series: no one supposes\nthat an infinite series is necessarily divergent.", "markdown": "Of course the reader will not be puzzled by the use of the term *infinite integral* to denote something which has a definite value such as $2$ or $\\frac{1}{2}\\pi$. The distinction between an infinite integral and a finite integral is similar to that between an infinite series and a finite series: no one supposes that an infinite series is necessarily divergent.", "why": "Clears up the common worry that 'infinite' in the name means the value is infinite, by pointing to the same word in infinite series.", "use": [ "lesson", "website" ], "concepts": [ "concept/convergent-series", "concept/infinite-integral" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-703020f67a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "322", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "The integral $\\ds\\int_{a}^{x} \\phi(t)\\, dt$ was defined in \\SecNo[§§]{156}~and~\\SecNo{157} as a \\emph{simple}\nlimit, \\ie\\ the limit of a certain finite sum. The infinite integral is therefore\n\\emph{the limit of a limit}, or what is known as a \\emph{repeated} limit. The notion of the\ninfinite integral is in fact essentially more complex than that of the finite\nintegral, of which it is a development.", "markdown": "The integral $\\ds\\int_{a}^{x} \\phi(t)\\, dt$ was defined in [§§]156 and 157 as a *simple* limit, *i.e.* the limit of a certain finite sum. The infinite integral is therefore *the limit of a limit*, or what is known as a *repeated* limit. The notion of the infinite integral is in fact essentially more complex than that of the finite integral, of which it is a development.", "why": "Explains why infinite integrals are a step harder than ordinary ones: they stack one limit on another.", "use": [ "lesson", "website" ], "concepts": [ "concept/definite-integral", "concept/infinite-integral", "concept/repeated-limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-a983cd4f5d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "324", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "There is one fundamental property of a convergent infinite series in\nregard to which the analogy between infinite series and infinite integrals\nbreaks down. If $\\sum \\phi(n)$~is convergent then $\\phi(n) \\to 0$; but it is \\emph{not} always\ntrue, even when $\\phi(x)$~is always positive, that if $\\ds\\int_{a}^{\\infty} \\phi(x)\\, dx$ is convergent\nthen $\\phi(x) \\to 0$.", "markdown": "There is one fundamental property of a convergent infinite series in regard to which the analogy between infinite series and infinite integrals breaks down. If $\\sum \\phi(n)$ is convergent then $\\phi(n) \\to 0$; but it is *not* always true, even when $\\phi(x)$ is always positive, that if $\\ds\\int_{a}^{\\infty} \\phi(x)\\, dx$ is convergent then $\\phi(x) \\to 0$.", "why": "Warns learners that a rule true for series does not carry over to integrals, a mistake that is easy to make.", "use": [ "lesson", "website" ], "concepts": [ "concept/convergent-series", "concept/infinite-integral" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-eeffed6e32", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "333", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "It often happens that the subject of integration has a discontinuity which\nis due simply to a failure in its definition at a particular point in the range\nof integration, and can be removed by attaching a particular value to it at\nthat point. In this case it is usual to suppose the definition of the subject\nof integration completed in this way.", "markdown": "It often happens that the subject of integration has a discontinuity which is due simply to a failure in its definition at a particular point in the range of integration, and can be removed by attaching a particular value to it at that point. In this case it is usual to suppose the definition of the subject of integration completed in this way.", "why": "Teaches that a gap in a formula's definition at one point is not always a real discontinuity, and that the integrand can be completed there.", "use": [ "lesson" ], "concepts": [ "concept/definite-integral", "concept/infinite-integral-of-the-second-kind" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-b0b56cb7d2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "330", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "An integral in which the subject of integration tends to~$\\infty$\nor to~$-\\infty$ as $x$~tends to some value or values included in the range\nof integration will be called an \\emph{infinite integral of the second kind}:\nthe \\emph{first kind} of infinite integrals being the class discussed in\n\\SecNo[§§]{177}~\\textit{et~seq.}", "markdown": "An integral in which the subject of integration tends to $\\infty$ or to $-\\infty$ as $x$ tends to some value or values included in the range of integration will be called an *infinite integral of the second kind*: the *first kind* of infinite integrals being the class discussed in [§§]177 *et seq.*", "why": "Separates the two kinds of infinite integral: an infinite range, and an integrand that blows up inside a finite range.", "use": [ "lesson", "website" ], "concepts": [ "concept/infinite-integral", "concept/infinite-integral-of-the-second-kind" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-790e555930", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "334", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "Some care has occasionally to be exercised in applying the rule for transformation by substitution. The following example affords a good illustration of this.", "markdown": "Some care has occasionally to be exercised in applying the rule for transformation by substitution. The following example affords a good illustration of this.", "why": "Warns learners that substitution in a definite integral can give a wrong answer if the sign of dx/dy is not checked.", "use": [ "lesson", "website" ], "concepts": [ "method/substitution" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-476da3035d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "337", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "The reader should carefully guard himself against supposing that the statement `an absolutely convergent series is convergent' is a mere tautology. When we say that $\\sum u_{n}$~is `absolutely convergent' we do \\emph{not} assert directly that $\\sum u_{n}$~is convergent: we assert the convergence of \\emph{another} series $\\sum |u_{n}|$, and it is by no means evident \\textit{a~priori} that this precludes oscillation on the part of~$\\sum u_{n}$.", "markdown": "The reader should carefully guard himself against supposing that the statement ‘an absolutely convergent series is convergent’ is a mere tautology. When we say that $\\sum u_{n}$ is ‘absolutely convergent’ we do *not* assert directly that $\\sum u_{n}$ is convergent: we assert the convergence of *another* series $\\sum |u_{n}|$, and it is by no means evident *a priori* that this precludes oscillation on the part of $\\sum u_{n}$.", "why": "Explains why absolute convergence implying convergence is a real theorem and not a restatement.", "use": [ "lesson", "website" ], "concepts": [ "concept/absolute-convergence", "concept/convergent-series", "concept/oscillation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-fe469c998e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "338", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "In the first place we note that, if $\\sum u_{n}$ is conditionally convergent, then the series $\\sum v_{n}$, $\\sum w_{n}$ of \\SecNo[§]{184} must both diverge to~$\\infty$.", "markdown": "In the first place we note that, if $\\sum u_{n}$ is conditionally convergent, then the series $\\sum v_{n}$, $\\sum w_{n}$ of [§]184 must both diverge to $\\infty$.", "why": "Shows that a conditionally convergent series converges only because two divergent series cancel.", "use": [ "lesson" ], "concepts": [ "concept/conditionally-convergent-series", "concept/divergent-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-33b493c1ae", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "339", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "In the first instance \\emph{there are no comparison tests for convergence of conditionally convergent series}.", "markdown": "In the first instance *there are no comparison tests for convergence of conditionally convergent series*.", "why": "Warns learners that the comparison methods they know for positive series do not carry over to conditional convergence.", "use": [ "lesson", "website" ], "concepts": [ "concept/conditionally-convergent-series", "concept/series-of-positive-and-negative-terms" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-c0312ffff2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "340", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "We shall see shortly that the series $1 - \\frac{1}{2} + \\frac{1}{3} - \\frac{1}{4} + \\dots$ is convergent. But the series $\\frac{1}{2} + \\frac{1}{3} + \\frac{1}{4} + \\frac{1}{5} + \\dots$ is divergent, although each of its terms is numerically less than the corresponding term of the former series.", "markdown": "We shall see shortly that the series $1 - \\frac{1}{2} + \\frac{1}{3} - \\frac{1}{4} + \\dots$ is convergent. But the series $\\frac{1}{2} + \\frac{1}{3} + \\frac{1}{4} + \\frac{1}{5} + \\dots$ is divergent, although each of its terms is numerically less than the corresponding term of the former series.", "why": "A concrete counterexample showing comparison fails once terms change sign.", "use": [ "lesson", "website" ], "concepts": [ "concept/alternating-series", "concept/conditionally-convergent-series", "concept/divergent-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-e3298f2207", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "342", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "It can indeed be proved that a conditionally convergent series can always be so rearranged as to converge to any sum whatever, or to diverge to~$\\infty$ or to~$-\\infty$.", "markdown": "It can indeed be proved that a conditionally convergent series can always be so rearranged as to converge to any sum whatever, or to diverge to $\\infty$ or to $-\\infty$.", "why": "States the surprising fact that reordering a conditionally convergent series can change its sum.", "use": [ "lesson", "website", "history" ], "concepts": [ "concept/conditionally-convergent-series", "concept/rearrangement-of-a-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-5a2b92905c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "343", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "This theorem may be stated as follows: \\begin{Result}a convergent series remains convergent if we multiply its terms by any sequence of positive and decreasing factors. \\end{Result}", "markdown": "This theorem may be stated as follows: Resulta convergent series remains convergent if we multiply its terms by any sequence of positive and decreasing factors. Result", "why": "Gives Abel's test in a plain one-sentence form a learner can remember.", "use": [ "lesson" ], "concepts": [ "theorem/abel-s-test" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-4890533523", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "345", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "It is obvious that \\emph{an absolutely convergent series is convergent}, since its real and imaginary parts converge separately.", "markdown": "It is obvious that *an absolutely convergent series is convergent*, since its real and imaginary parts converge separately.", "why": "Shows how the real-series result extends to complex series by treating the real and imaginary parts separately.", "use": [ "lesson" ], "concepts": [ "concept/absolute-convergence", "concept/series-of-complex-terms" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-8d9a321f9d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "346", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "For $\\lim a_{n}z_{1}^{n} = 0$, since $\\sum a_{n}z_{1}^{n}$~is convergent, and therefore we can certainly find a constant~$K$ such that $|a_{n}z_{1}^{n}| < K$ for all values of~$n$.", "markdown": "For $\\lim a_{n}z_{1}^{n} = 0$, since $\\sum a_{n}z_{1}^{n}$ is convergent, and therefore we can certainly find a constant $K$ such that $|a_{n}z_{1}^{n}| < K$ for all values of $n$.", "why": "Shows the key first step of the proof: a convergent series has bounded terms, which sets up comparison with a geometric series.", "use": [ "lesson" ], "concepts": [ "concept/comparison", "theorem/absolute-convergence-of-a-power-series-inside-a-point-of-convergence" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-fff2e494a7", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "346", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "In other words, if the series converges at~$P$ \\emph{then it converges absolutely at all points nearer to the origin than~$P$}.", "markdown": "In other words, if the series converges at $P$ *then it converges absolutely at all points nearer to the origin than $P$*.", "why": "Restates the theorem in a picture a learner can hold: convergence spreads inward toward the origin.", "use": [ "lesson", "website" ], "concepts": [ "concept/absolute-convergence", "concept/convergent-series", "theorem/absolute-convergence-of-a-power-series-inside-a-point-of-convergence" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-eeacb5a5f0", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "347", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "It should be observed that this general result gives absolutely no information about the behaviour of the series \\emph{on} the circle of convergence. The examples which follow show that as a matter of fact there are very diverse possibilities as to this.", "markdown": "It should be observed that this general result gives absolutely no information about the behaviour of the series *on* the circle of convergence. The examples which follow show that as a matter of fact there are very diverse possibilities as to this.", "why": "Warns learners that the circle's boundary needs separate investigation.", "use": [ "lesson", "website" ], "concepts": [ "concept/circle-of-convergence", "concept/power-series", "concept/radius-of-convergence" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-7c262851b6", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "348", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "Thus the logarithmic series converges at all points of its circle of convergence except the point $z = -1$.", "markdown": "Thus the logarithmic series converges at all points of its circle of convergence except the point $z = -1$.", "why": "A concrete example of a series that converges on almost all of its boundary circle.", "use": [ "lesson" ], "concepts": [ "concept/circle-of-convergence", "concept/divergent-series", "concept/logarithmic-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-6df0ca4ae2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "349", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "It shows that \\emph{the same function~$f(z)$ cannot be represented by two different power series}.", "markdown": "It shows that *the same function $f(z)$ cannot be represented by two different power series*.", "why": "States the uniqueness property that makes comparing coefficients legitimate.", "use": [ "lesson", "website" ], "concepts": [ "concept/power-series", "concept/uniqueness" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-c1a22ef130", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "353", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "Such equations may be solved by a method which will be sufficiently explained by an example.", "markdown": "Such equations may be solved by a method which will be sufficiently explained by an example.", "why": "Introduces solving a difference-equation by summing a recurring power series, taught through a worked case.", "use": [ "lesson" ], "concepts": [ "concept/linear-difference-equation", "concept/recurring-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-cf1ed062e5", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "353", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "A player tossing a coin is to score one point for every head he turns up and two for every tail, and is to play on until his score reaches or passes a total~$n$.", "markdown": "A player tossing a coin is to score one point for every head he turns up and two for every tail, and is to play on until his score reaches or passes a total $n$.", "why": "An old-fashioned probability puzzle that turns into a recurrence, giving a learner a concrete reason for difference-equations.", "use": [ "website", "lesson" ], "concepts": [ "concept/linear-difference-equation", "concept/probability" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-531ace9542", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "361", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "The logarithm of~$x$ tends to infinity with~$x$, but more slowly than \\Emph{any} positive power of~$x$, integral or fractional.", "markdown": "The logarithm of $x$ tends to infinity with $x$, but more slowly than **** positive power of $x$, integral or fractional.", "why": "It states the striking fact that the logarithm grows more slowly than any power, which surprises most beginners.", "use": [ "website", "lesson" ], "concepts": [ "concept/logarithm", "concept/order-of-greatness", "concept/tends-to-infinity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-a0edd4c0b1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "363", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "Since $\\log x$~is an increasing function of~$x$, in the stricter sense of \\SecNo[§]{95}, it can only pass once through the value~$1$. Hence our definition does in fact define one definite number.", "markdown": "Since $\\log x$ is an increasing function of $x$, in the stricter sense of [§]95, it can only pass once through the value $1$. Hence our definition does in fact define one definite number.", "why": "It models the habit of checking that a definition picks out exactly one object before using it.", "use": [ "lesson" ], "concepts": [ "concept/euler-s-number", "concept/logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-717bc8769d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "357", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "The process may fairly be compared with that by which the irrational and complex numbers were first introduced, when it was found that certain algebraical equations could not be solved by means of the numbers already recognised.", "markdown": "The process may fairly be compared with that by which the irrational and complex numbers were first introduced, when it was found that certain algebraical equations could not be solved by means of the numbers already recognised.", "why": "It gives the historical reason new functions were introduced, by analogy with the extension of the number system.", "use": [ "history" ], "concepts": [ "concept/function", "concept/logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-4e24d5b7f0", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "378", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "The series on the right-hand side of this equation is known as\nthe \\Emph{exponential series}.", "markdown": "The series on the right-hand side of this equation is known as the **series**.", "why": "It names the power series for e^x so a learner can recognise it wherever it appears.", "use": [ "lesson" ], "concepts": [ "concept/exponential-function", "theorem/exponential-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-251dda7189", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "381", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "Another very important\nexpansion in powers of~$x$ is that for~$\\log(1 + x)$.", "markdown": "Another very important expansion in powers of $x$ is that for $\\log(1 + x)$.", "why": "It signals that the logarithm has its own power series, which is the next expansion a learner should master.", "use": [ "lesson" ], "concepts": [ "concept/logarithm", "concept/logarithmic-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-3262d9cde1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "377", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "The value of~$\\gamma$ is in fact~$.577\\dots$, and $\\gamma$~is usually called \\Emph{Euler's constant}.", "markdown": "The value of $\\gamma$ is in fact $.577\\dots$, and $\\gamma$ is usually called **’s constant**.", "why": "It gives the numerical value of Euler's constant and the name by which it is known, useful for a history or website feature.", "use": [ "history", "website" ], "concepts": [ "quantity/euler-s-constant" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-77464815f3", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "379", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "The power series for~$e^{x}$ is so important that it is worth while to investigate\nit by an alternative method which does not depend upon Taylor's Theorem.", "markdown": "The power series for $e^{x}$ is so important that it is worth while to investigate it by an alternative method which does not depend upon Taylor’s Theorem.", "why": "It tells a learner that the exponential series can be derived by a second route, which shows the result is not an artefact of one method.", "use": [ "lesson" ], "concepts": [ "theorem/exponential-series", "theorem/taylor-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-5f1f1622da", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "382", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "If $x$~lies outside these limits the series\nis not convergent.", "markdown": "If $x$ lies outside these limits the series is not convergent.", "why": "It warns that the logarithmic series only converges for -1 < x <= 1, a common mistake when applying it.", "use": [ "lesson" ], "concepts": [ "concept/logarithmic-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-d4bee5e031", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "386", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "We know that this series is convergent for all values of~$x$, and we may therefore define the function $\\exp x$ by the equation", "markdown": "We know that this series is convergent for all values of $x$, and we may therefore define the function $\\exp x$ by the equation", "why": "It shows the book choosing the power series as the foundation for the exponential function, a route the learner can follow from first principles.", "use": [ "lesson" ], "concepts": [ "concept/exponential-function", "theorem/exponential-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-1a8d2b1e8c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "357", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "These new functions have generally been introduced because it appeared that some problem which was occupying the attention of mathematicians was incapable of solution by means of the functions already known. The process may fairly be compared with that by which the irrational and complex numbers were first introduced, when it was found that certain algebraical equations could not be solved by means of the numbers already recognised.", "markdown": "These new functions have generally been introduced because it appeared that some problem which was occupying the attention of mathematicians was incapable of solution by means of the functions already known. The process may fairly be compared with that by which the irrational and complex numbers were first introduced, when it was found that certain algebraical equations could not be solved by means of the numbers already recognised.", "why": "It explains why mathematics invents new functions and compares this to the earlier extension of the number system.", "use": [ "lesson", "website", "history" ], "concepts": [ "concept/function", "concept/logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-ef8cf3960a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "358", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "We define $\\log x$, the logarithm of~$x$, by the equation \\[ \\log x = \\int_{1}^{x} \\frac{dt}{t}. \\]", "markdown": "We define $\\log x$, the logarithm of $x$, by the equation x = _1^x dtt.", "why": "It gives the integral definition from which every property of the logarithm in the chapter is derived.", "use": [ "lesson", "website" ], "concepts": [ "concept/definite-integral", "concept/logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-f04fce79c2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "361", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "Perhaps the most interesting feature of the function $\\log x$ is its behaviour as $x$~tends to infinity. It shows that the presupposition stated above, which seems so natural, is unfounded. \\emph{The logarithm of~$x$ tends to infinity with~$x$, but more slowly than \\Emph{any} positive power of~$x$, integral or fractional.}", "markdown": "Perhaps the most interesting feature of the function $\\log x$ is its behaviour as $x$ tends to infinity. It shows that the presupposition stated above, which seems so natural, is unfounded. *The logarithm of $x$ tends to infinity with $x$, but more slowly than **** positive power of $x$, integral or fractional.*", "why": "It overturns the natural assumption that powers of x capture every rate of growth and states the surprising fact plainly.", "use": [ "lesson", "website" ], "concepts": [ "concept/order-of-greatness", "concept/scale-of-infinity", "theorem/logarithm-grows-more-slowly-than-any-positive-power" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-d8cfa01763", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "361", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "This fact is sometimes expressed loosely by saying that the `order of infinity of~$\\log x$ is infinitely small'; but the reader will hardly require at this stage to be warned against such modes of expression.", "markdown": "This fact is sometimes expressed loosely by saying that the ‘order of infinity of $\\log x$ is infinitely small’; but the reader will hardly require at this stage to be warned against such modes of expression.", "why": "It warns learners against a common loose way of speaking about infinity.", "use": [ "lesson", "history" ], "concepts": [ "concept/infinity", "theorem/logarithm-grows-more-slowly-than-any-positive-power" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-7b382ad15e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "363", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "We define~$e$ as \\emph{the number whose logarithm is~$1$}. In other words $e$~is defined by the equation \\[ 1 = \\int_{1}^{e} \\frac{dt}{t}. \\]", "markdown": "We define $e$ as *the number whose logarithm is $1$*. In other words $e$ is defined by the equation 1 = _1^e dtt.", "why": "It shows how e arises from the logarithm instead of from a limit or a series.", "use": [ "lesson", "website" ], "concepts": [ "concept/e", "concept/euler-s-number", "concept/logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-42fdf3d995", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "364", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "We now define the \\emph{exponential function}~$e^{y}$ for all real values of~$y$ as the inverse of the logarithmic function. In other words we write \\[ x = e^{y} \\] if $y = \\log x$.", "markdown": "We now define the *exponential function* $e^{y}$ for all real values of $y$ as the inverse of the logarithmic function. In other words we write x = e^y if $y = \\log x$.", "why": "It defines the exponential function for all real y as the inverse of the logarithm.", "use": [ "lesson", "website" ], "concepts": [ "concept/exponential-function", "concept/logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-ef2fad94d7", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "365", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "Thus \\emph{the derivative of the exponential function is equal to the function itself}. More generally, if $x = e^{ay}$ then $dx/dy = ae^{ay}$.", "markdown": "Thus *the derivative of the exponential function is equal to the function itself*. More generally, if $x = e^{ay}$ then $dx/dy = ae^{ay}$.", "why": "It states the best-known property of e^x, which follows from the derivative of the logarithm.", "use": [ "lesson", "website" ], "concepts": [ "concept/exponential-function", "theorem/derivative-of-the-exponential-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-6454195043", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "367", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "We take this as our \\emph{definition} of~$a^{x}$ when $x$~is irrational. Thus $10^{\\sqrt{2}} = e^{\\sqrt{2}\\log 10}$.", "markdown": "We take this as our *definition* of $a^{x}$ when $x$ is irrational. Thus $10^{\\sqrt{2}} = e^{\\sqrt{2}\\log 10}$.", "why": "It shows how a power with an irrational index gets a meaning, using a concrete example.", "use": [ "lesson", "website" ], "concepts": [ "concept/exponential-function", "concept/general-power", "concept/logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-6580ef02c8", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "374", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "We saw however in \\SecNo[§]{200} that with the aid of logarithms we can construct functions which tend to zero, as $n \\to \\infty$, more rapidly than~$1/n$, yet less rapidly than~$n^{-1-\\alpha}$, however small $\\alpha$ may be, provided of course that it is positive.", "markdown": "We saw however in [§]200 that with the aid of logarithms we can construct functions which tend to zero, as $n \\to \\infty$, more rapidly than $1/n$, yet less rapidly than $n^{-1-\\alpha}$, however small $\\alpha$ may be, provided of course that it is positive.", "why": "Explains why comparison with the series of n^-s is not enough and why logarithmic tests are needed.", "use": [ "lesson" ], "concepts": [ "concept/logarithm", "theorem/logarithmic-test-of-convergence" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-ced4038a6b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "376", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "The reader should observe the extreme rapidity with which the higher exponential functions, such as $e^{e^{x}}$ and~$e^{e^{e^{x}}}$, increase with~$x$.", "markdown": "The reader should observe the extreme rapidity with which the higher exponential functions, such as $e^{e^{x}}$ and $e^{e^{e^{x}}}$, increase with $x$.", "why": "Gives a vivid sense of how fast iterated exponentials grow.", "use": [ "lesson", "website" ], "concepts": [ "concept/exponential-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-f2c3026afd", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "376", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "Conversely, the rate of increase of the higher logarithmic functions is extremely slow. Thus to make $\\log\\log\\log\\log x > 1$ we have to suppose $x$~a number with over $8000$~figures.", "markdown": "Conversely, the rate of increase of the higher logarithmic functions is extremely slow. Thus to make $\\log\\log\\log\\log x > 1$ we have to suppose $x$ a number with over $8000$ figures.", "why": "Shows concretely how slowly iterated logarithms grow, as the mirror of the previous remark.", "use": [ "lesson", "website" ], "concepts": [ "concept/logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-95ca970ed4", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "379", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "The reader will observe that the exponential series has the property of reproducing itself when every term is differentiated, and that no other series of powers of~$x$ would possess this property: for some further remarks in this connection see \\okrickRef{Appendix}{II}\\@.", "markdown": "The reader will observe that the exponential series has the property of reproducing itself when every term is differentiated, and that no other series of powers of $x$ would possess this property: for some further remarks in this connection see AppendixII.", "why": "Names the key property that makes the exponential series special.", "use": [ "lesson", "website" ], "concepts": [ "theorem/exponential-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-17a97e9a2c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "374", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "The approximations are of course very rough, but suffice to give us a good idea of the scale of magnitude of the root.", "markdown": "The approximations are of course very rough, but suffice to give us a good idea of the scale of magnitude of the root.", "why": "Models honest use of rough approximation: state its roughness and what it is still good for.", "use": [ "lesson" ], "concepts": [ "concept/approximation", "concept/exponential-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-5d28bfd0c6", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "372", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "Verify that these formulae may be deduced from the corresponding formulae in $\\cos x$ and $\\sin x$, by writing $\\cosh x$ for $\\cos x$ and $i\\sinh x$ for~$\\sin x$.", "markdown": "Verify that these formulae may be deduced from the corresponding formulae in $\\cos x$ and $\\sin x$, by writing $\\cosh x$ for $\\cos x$ and $i\\sinh x$ for $\\sin x$.", "why": "Points out the formal analogy between hyperbolic and circular functions.", "use": [ "lesson" ], "concepts": [ "concept/cosine", "concept/hyperbolic-function", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-3c104fb01f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "381", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "We require to show that the limit of~$R_{m}$, when $m$~tends to~$\\infty$, is zero.", "markdown": "We require to show that the limit of $R_{m}$, when $m$ tends to $\\infty$, is zero.", "why": "Shows that a series expansion is justified only by showing the remainder tends to zero.", "use": [ "lesson" ], "concepts": [ "concept/limit", "concept/logarithmic-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-7978249b91", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "382", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "The only difference is that the proof is a little simpler; for, since $\\arctan x$~is an odd function of~$x$, we need only consider positive values of~$x$.", "markdown": "The only difference is that the proof is a little simpler; for, since $\\arctan x$ is an odd function of $x$, we need only consider positive values of $x$.", "why": "Shows how symmetry (an odd function) halves the work of a proof.", "use": [ "lesson" ], "concepts": [ "concept/inverse-circular-function", "concept/odd-function", "theorem/inverse-tangent-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-7a2d6df5ac", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "382", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "If $x = 1$, we obtain the formula \\[ \\tfrac{1}{4}\\pi = 1 - \\tfrac{1}{3} + \\tfrac{1}{5} - \\dots. \\]", "markdown": "If $x = 1$, we obtain the formula 14= 1 - 13 + 15 - ….", "why": "A surprising series for pi that falls out of a general series at a single value.", "use": [ "lesson", "website" ], "concepts": [ "theorem/inverse-tangent-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-14b53d7fa6", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "383", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "This series may be used to calculate~$\\log 2$, a purpose for which the series $1 - \\frac{1}{2} + \\frac{1}{3} - \\dots$, owing to the slowness of its convergence, is practically useless.", "markdown": "This series may be used to calculate $\\log 2$, a purpose for which the series $1 - \\frac{1}{2} + \\frac{1}{3} - \\dots$, owing to the slowness of its convergence, is practically useless.", "why": "Warns that a correct series can still be a poor tool for computing, and that the choice of series matters.", "use": [ "lesson", "website" ], "concepts": [ "concept/logarithm", "concept/logarithmic-series", "theorem/logarithmic-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-9a9aa74939", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "385", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "Let us consider the error committed in taking~$8\\frac{3}{16}$ (the value given by the first two terms) as an approximate value. After the second term the terms alternate in sign and decrease. Hence the error is one of excess, and is less than~$3^{2}/64^{2}$, which is less than~$.003$.", "markdown": "Let us consider the error committed in taking $8\\frac{3}{16}$ (the value given by the first two terms) as an approximate value. After the second term the terms alternate in sign and decrease. Hence the error is one of excess, and is less than $3^{2}/64^{2}$, which is less than $.003$.", "why": "A worked example showing how an alternating series gives a firm bound on the error of an approximation.", "use": [ "lesson" ], "concepts": [ "concept/approximation", "concept/quadratic-surd", "method/approximation-of-surds-by-the-binomial-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-149528ce78", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "386", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "We shall now give an outline of a method of investigation of the properties of $e^{x}$ and $\\log x$ entirely different in logical order from that followed in the preceding pages.", "markdown": "We shall now give an outline of a method of investigation of the properties of $e^{x}$ and $\\log x$ entirely different in logical order from that followed in the preceding pages.", "why": "Shows that the same theory can be built in a different logical order, starting from the series.", "use": [ "lesson", "website" ], "concepts": [ "concept/exponential-function", "concept/logarithm", "theorem/exponential-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-706456fdde", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "386", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "Incidentally we have proved that $\\exp x$ is a continuous function.", "markdown": "Incidentally we have proved that $\\exp x$ is a continuous function.", "why": "Shows how continuity can come as a by-product of a derivative argument.", "use": [ "lesson" ], "concepts": [ "concept/continuous-function", "concept/exponential-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-4567fddcc1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "392", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "This formula is interesting historically as having been employed by Napier for the numerical calculation of logarithms.", "markdown": "This formula is interesting historically as having been employed by Napier for the numerical calculation of logarithms.", "why": "Connects an approximation formula to the historical computation of the first logarithm tables.", "use": [ "history", "website" ], "concepts": [ "concept/logarithm", "person/john-napier" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-5857bcebc6", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "If $n$~is not divisible by~$10$, and $\\log_{10}n = p/q$, we have $10^{p} = n^{q}$, which is impossible, since $10^{p}$~ends with~$0$ and $n^{q}$~does not.", "markdown": "If $n$ is not divisible by $10$, and $\\log_{10}n = p/q$, we have $10^{p} = n^{q}$, which is impossible, since $10^{p}$ ends with $0$ and $n^{q}$ does not.", "why": "A short, elegant proof by contradiction that common logarithms of most integers are irrational.", "use": [ "lesson", "website" ], "concepts": [ "concept/logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-489e9d5000", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "409", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "The left-hand sides of these equations are defined, by the ordinary geometrical definitions adopted in elementary Trigonometry, only for real values of~$\\zeta$. The right-hand sides have, on the other hand, been defined for all values of~$\\zeta$, real or complex. We are therefore naturally led to adopt the formulae~\\Eq{(1)} as the \\emph{definitions} of $\\cos \\zeta$ and~$\\sin \\zeta$ for all values of~$\\zeta$.", "markdown": "The left-hand sides of these equations are defined, by the ordinary geometrical definitions adopted in elementary Trigonometry, only for real values of $\\zeta$. The right-hand sides have, on the other hand, been defined for all values of $\\zeta$, real or complex. We are therefore naturally led to adopt the formulae (1) as the *definitions* of $\\cos \\zeta$ and $\\sin \\zeta$ for all values of $\\zeta$.", "why": "Shows why a definition valid only for real angles is replaced by an exponential one that covers complex arguments while agreeing with the old one.", "use": [ "lesson", "website" ], "concepts": [ "concept/cosine", "concept/exponential-function", "concept/sine", "concept/trigonometry" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-578c66120e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "410", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "All the ordinary formulae of elementary Trigonometry are algebraical corollaries of the equations~\\Eq{(2)}--\\Eq{(6)}; and so all such relations hold also for the generalised trigonometrical functions defined in this section.", "markdown": "All the ordinary formulae of elementary Trigonometry are algebraical corollaries of the equations (2)--(6); and so all such relations hold also for the generalised trigonometrical functions defined in this section.", "why": "Tells the learner that nothing learned in elementary trigonometry is lost when the functions are generalised.", "use": [ "lesson" ], "concepts": [ "concept/addition-formulae", "concept/circular-function", "concept/corollary", "concept/trigonometrical-identity", "concept/trigonometry" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-897d29f1de", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "414", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "where $-1 \\leq x \\leq 1$: each of these formulae also is `completely' true.", "markdown": "where $-1 \\leq x \\leq 1$: each of these formulae also is ‘completely’ true.", "why": "It warns that the inverse-circular-logarithm formulae hold on all branches, not only the principal one.", "use": [ "lesson" ], "concepts": [ "concept/inverse-circular-function", "concept/principal-value-of-a-logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-9c84b8c1e7", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "420", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "The function is discontinuous for $\\theta = (2k + 1)\\pi$.", "markdown": "The function is discontinuous for $\\theta = (2k + 1)\\pi$.", "why": "It gives a concrete graph-reading example of a function that jumps at odd multiples of pi.", "use": [ "website" ], "concepts": [ "concept/discontinuous-function", "concept/logarithmic-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-4ac2970531", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "409", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "It is evident that $\\cos \\zeta$ and~$\\sec \\zeta$ are even functions of~$\\zeta$, and $\\sin \\zeta$, $\\tan \\zeta$, $\\cot \\zeta$, and~$\\cosec \\zeta$ odd functions.", "markdown": "It is evident that $\\cos \\zeta$ and $\\sec \\zeta$ are even functions of $\\zeta$, and $\\sin \\zeta$, $\\tan \\zeta$, $\\cot \\zeta$, and $\\cosec \\zeta$ odd functions.", "why": "It gives learners a short rule for sorting the trigonometric ratios into even and odd.", "use": [ "lesson" ], "concepts": [ "concept/cosecant", "concept/cosine", "concept/cotangent", "concept/even-function", "concept/odd-function", "concept/secant", "concept/sine", "concept/tangent-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-737e38d1c4", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "414", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "the question is suggested whether, now that we have defined the logarithm of a complex number, this equation will not be found to be actually true.", "markdown": "the question is suggested whether, now that we have defined the logarithm of a complex number, this equation will not be found to be actually true.", "why": "It shows the historical habit of testing a formula that held for real numbers by extending it to complex ones.", "use": [ "history" ], "concepts": [ "concept/complex-number", "concept/logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-78e5a2beba", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "421", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "Let $z$~be any complex number, and $h$~a real number small enough to ensure that $|hz| < 1$.", "markdown": "Let $z$ be any complex number, and $h$ a real number small enough to ensure that $|hz| < 1$.", "why": "It sets up the small-parameter device that gives the logarithm's limit (1) and the exponential limit that follows.", "use": [ "lesson" ], "concepts": [ "concept/complex-number", "concept/limit", "concept/logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-2dcbcf23ea", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "421", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\lim_{h\\to 0} \\frac{\\log(1 + hz)}{h} = z.", "markdown": "_h0 (1 + hz)h = z.", "why": "It states the key limit from which the exponential limit and the derivative of log(1 + tz) are derived.", "use": [ "lesson", "website" ], "concepts": [ "concept/limit", "concept/logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-8954d0489f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "422", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\frac{d}{dt}(1 + tz)^{m} = mz(1 + tz)^{m-1}", "markdown": "ddt(1 + tz)^m = mz(1 + tz)^m-1", "why": "It gives the power rule for a complex exponent, which a learner can compare with the familiar rule for real powers.", "use": [ "lesson" ], "concepts": [ "concept/derivative", "concept/power", "method/differentiation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-89a956fdd3", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "422", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "Here both $(1 + tz)^{m}$ and~$(1 + tz)^{m-1}$ have their principal values\\Add{.}", "markdown": "Here both $(1 + tz)^{m}$ and $(1 + tz)^{m-1}$ have their principal values", "why": "It warns that complex powers only satisfy the rule once each side is taken with its principal value.", "use": [ "lesson" ], "concepts": [ "concept/power", "concept/principal-value-of-a-logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-ea7e29707b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "423", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "for all values of~$m$, real or complex, and all values of~$z$ such that", "markdown": "for all values of $m$, real or complex, and all values of $z$ such that", "why": "It states the range of validity of the binomial theorem for complex exponents, which a learner must keep in mind before applying it.", "use": [ "lesson", "website" ], "concepts": [ "concept/binomial-series", "theorem/binomial-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-2b2bbd30db", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "400", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "We shall call this particular value of~$\\Log \\zeta$ the \\Emph{principal value}. When $\\zeta$~is real and positive, $\\zeta = \\rho$ and $\\phi = 0$, so that the principal value of~$\\Log \\zeta$ is the ordinary logarithm~$\\log \\zeta$.", "markdown": "We shall call this particular value of $\\Log \\zeta$ the **value**. When $\\zeta$ is real and positive, $\\zeta = \\rho$ and $\\phi = 0$, so that the principal value of $\\Log \\zeta$ is the ordinary logarithm $\\log \\zeta$.", "why": "It shows the learner how the familiar real logarithm sits inside the many-valued complex logarithm as its principal value.", "use": [ "lesson" ], "concepts": [ "concept/logarithm", "concept/principal-value-of-a-logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-8c1e20e2f5", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "402", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "The equation $\\Log z^{m} = m\\Log z$, where $m$~is an integer, is not completely true: every value of the right-hand side is a value of the left-hand side, but the converse is not true.", "markdown": "The equation $\\Log z^{m} = m\\Log z$, where $m$ is an integer, is not completely true: every value of the right-hand side is a value of the left-hand side, but the converse is not true.", "why": "It warns the learner that a power law can hold for one direction of a many-valued equation and fail in the other.", "use": [ "lesson" ], "concepts": [ "concept/complete-equation", "concept/logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-6c705a6dbf", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "395", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "The definition, although perfectly legitimate, is futile because it does not really define a new idea at all.", "markdown": "The definition, although perfectly legitimate, is futile because it does not really define a new idea at all.", "why": "It shows the learner why a naive extension of a definition to complex z can be empty, and why the chapter restricts the class of functions.", "use": [ "lesson", "history" ], "concepts": [ "concept/complex-variable", "concept/function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-e0bb7c160b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "406", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "This is a further generalisation of De~Moivre's Theorem", "markdown": "This is a further generalisation of De Moivre’s Theorem", "why": "It places the general power in the historical line of De Moivre's formula, which the learner already knows.", "use": [ "history" ], "concepts": [ "concept/general-power", "theorem/de-moivre-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-3c07ad2ec8", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "395", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "But it will be found, on closer examination, that this definition is not one from which any profit can be derived. For if $z$~is given, so are $x$~and~$y$, and conversely: to assign a value of~$z$ is precisely the same thing as to assign a pair of values of $x$~and~$y$.", "markdown": "But it will be found, on closer examination, that this definition is not one from which any profit can be derived. For if $z$ is given, so are $x$ and $y$, and conversely: to assign a value of $z$ is precisely the same thing as to assign a pair of values of $x$ and $y$.", "why": "Shows honestly why a definition can be legitimate yet useless, and why 'function of z' needs more than 'a relation between z and Z'.", "use": [ "lesson", "website" ], "concepts": [ "concept/complex-variable", "concept/function-of-a-complex-variable" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-69505ed6e5", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "401", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "Since $|\\zeta | = \\rho$, and the different angles~$2k\\pi + \\phi$ are the different values of~$\\am \\zeta$, we conclude that every value of~$\\log |\\zeta| + i\\am \\zeta$ is a value of~$\\Log \\zeta$; and it is clear from the preceding discussion that every value of~$\\Log \\zeta$ must be of this form.", "markdown": "Since $|\\zeta | = \\rho$, and the different angles $2k\\pi + \\phi$ are the different values of $\\am \\zeta$, we conclude that every value of $\\log |\\zeta| + i\\am \\zeta$ is a value of $\\Log \\zeta$; and it is clear from the preceding discussion that every value of $\\Log \\zeta$ must be of this form.", "why": "States the key conclusion that the many values of Log zeta come from the many amplitudes of zeta.", "use": [ "lesson" ], "concepts": [ "concept/amplitude-of-a-complex-number", "concept/logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-1d43932253", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "402", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "An equation such as~\\Eq{(1)}, in which every value of either side is a value of the other, we shall call a \\emph{complete} equation, or an equation which is \\emph{completely true}.", "markdown": "An equation such as (1), in which every value of either side is a value of the other, we shall call a *complete* equation, or an equation which is *completely true*.", "why": "Gives the precise meaning of equality between many-valued expressions, needed throughout the exercises.", "use": [ "lesson", "website" ], "concepts": [ "concept/complete-equation", "concept/logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-c2e0e106f5", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "403", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "It would not be unnatural to suppose that, conversely, to any given value of~$\\zeta$ correspond infinitely many values of~$z$, or in other words that $\\exp \\zeta$~is an infinitely many-valued function of~$\\zeta$. This is however not the case, as is proved by the following theorem.", "markdown": "It would not be unnatural to suppose that, conversely, to any given value of $\\zeta$ correspond infinitely many values of $z$, or in other words that $\\exp \\zeta$ is an infinitely many-valued function of $\\zeta$. This is however not the case, as is proved by the following theorem.", "why": "Sets up a surprising asymmetry: the inverse of a many-valued function is one-valued.", "use": [ "lesson" ], "concepts": [ "concept/exponential-function", "concept/many-valued-function", "theorem/exponential-function-is-one-valued" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-c0d76eff2f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "404", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "It might seem natural, as $\\exp \\zeta = e^{\\zeta}$ when $\\zeta$~is real, to adopt the same notation when $\\zeta$~is complex and to drop the notation $\\exp \\zeta$ altogether. We shall not follow this course because we shall have to give a more general definition of the meaning of the symbol~$e^{\\zeta}$: we shall find then that $e^{\\zeta}$~represents a function with infinitely many values of which $\\exp \\zeta$~is only one.", "markdown": "It might seem natural, as $\\exp \\zeta = e^{\\zeta}$ when $\\zeta$ is real, to adopt the same notation when $\\zeta$ is complex and to drop the notation $\\exp \\zeta$ altogether. We shall not follow this course because we shall have to give a more general definition of the meaning of the symbol $e^{\\zeta}$: we shall find then that $e^{\\zeta}$ represents a function with infinitely many values of which $\\exp \\zeta$ is only one.", "why": "Warns that e^zeta and exp zeta are not the same thing for complex zeta, and explains the book's notation.", "use": [ "lesson", "website" ], "concepts": [ "concept/exponential-function", "concept/general-power" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-eac8dc28f5", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "406", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "We conclude that \\emph{$a^{\\zeta}$~is infinitely many-valued unless $\\zeta$~is real and rational}. On the other hand we have already seen that, when $\\zeta$~is real and rational, $a^{\\zeta}$~has but a finite number of values.", "markdown": "We conclude that *$a^{\\zeta}$ is infinitely many-valued unless $\\zeta$ is real and rational*. On the other hand we have already seen that, when $\\zeta$ is real and rational, $a^{\\zeta}$ has but a finite number of values.", "why": "Summarises exactly when the general power has finitely many values and when it has infinitely many.", "use": [ "lesson" ], "concepts": [ "concept/general-power", "concept/many-valued-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-d7e839ceb5", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "408", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "Explain the fallacy in the following argument: since $e^{2m\\pi i} = e^{2n\\pi i} = 1$, where $m$~and~$n$ are any integers, therefore, raising each side to the power~$i$ we obtain $e^{-2m\\pi} = e^{-2n\\pi}$.", "markdown": "Explain the fallacy in the following argument: since $e^{2m\\pi i} = e^{2n\\pi i} = 1$, where $m$ and $n$ are any integers, therefore, raising each side to the power $i$ we obtain $e^{-2m\\pi} = e^{-2n\\pi}$.", "why": "A good puzzle that makes learners see how careless use of many-valued powers produces false results.", "use": [ "lesson", "website" ], "concepts": [ "concept/complete-equation", "concept/exponential-function", "concept/general-power", "concept/many-valued-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-deb8f5be41", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "410", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "The same process of transformation may be applied to any trigonometrical identity. It is of course this fact which explains the correspondence noted in \\Ex{lxxxvii}.~21 between the formulae for the hyperbolic and those for the ordinary trigonometrical functions.", "markdown": "The same process of transformation may be applied to any trigonometrical identity. It is of course this fact which explains the correspondence noted in % [examples:lxxxvii]Ex. lxxxvii%. 21 between the formulae for the hyperbolic and those for the ordinary trigonometrical functions.", "why": "Explains why hyperbolic formulae mirror the circular ones: both come from the same identities with ζ replaced by iζ.", "use": [ "lesson", "website" ], "concepts": [ "concept/circular-function", "concept/hyperbolic-function", "concept/trigonometrical-identity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-7d942f780e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "413", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "These facts suggest the existence of some functional connection between the logarithmic and the inverse circular functions. That there is such a connection may also be inferred from the facts that we have expressed the circular functions of~$\\zeta$ in terms of~$\\exp i\\zeta$, and that the logarithm is the inverse of the exponential function.", "markdown": "These facts suggest the existence of some functional connection between the logarithmic and the inverse circular functions. That there is such a connection may also be inferred from the facts that we have expressed the circular functions of $\\zeta$ in terms of $\\exp i\\zeta$, and that the logarithm is the inverse of the exponential function.", "why": "Shows the reasoning that leads a mathematician to guess a link between logarithms and inverse trigonometric functions before proving it.", "use": [ "lesson", "history" ], "concepts": [ "concept/exponential-function", "concept/inverse-circular-function", "concept/logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-cecab47661", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "414", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "Moreover we saw in \\SecNo[§]{191} that the series on the right-hand side remains convergent (indeed absolutely convergent) when $z$~is complex. It is naturally suggested that the equation~\\Eq{(1)} also remains true, and we shall now prove that this is the case.", "markdown": "Moreover we saw in [§]191 that the series on the right-hand side remains convergent (indeed absolutely convergent) when $z$ is complex. It is naturally suggested that the equation (1) also remains true, and we shall now prove that this is the case.", "why": "Models the move from a real-variable result to a conjecture about complex values, and the need to prove it rather than assume it.", "use": [ "lesson" ], "concepts": [ "concept/absolute-convergence", "concept/exponential-function", "concept/power-series", "theorem/power-series-for-the-exponential-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-668b828cc5", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "418", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "We shall in fact prove rather more than this, viz.\\ that \\Eq{(1)}~is true for all values of~$z$ such that $|z| \\leq 1$, with the exception of the value~$-1$.", "markdown": "We shall in fact prove rather more than this, viz. that (1) is true for all values of $z$ such that $|z| \\leq 1$, with the exception of the value $-1$.", "why": "States the exact range of validity of the logarithmic series on the complex plane, including the one excluded point.", "use": [ "lesson", "website" ], "concepts": [ "theorem/logarithmic-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-05f8ce615e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "420", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "The sums of the series, for other values of~$\\theta$, are easily found from the consideration that they are periodic functions of~$\\theta$ with the period~$2\\pi$.", "markdown": "The sums of the series, for other values of $\\theta$, are easily found from the consideration that they are periodic functions of $\\theta$ with the period $2\\pi$.", "why": "Illustrates using periodicity to extend a sum known on one interval to all values of the variable.", "use": [ "lesson" ], "concepts": [ "concept/periodic-function", "theorem/logarithmic-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-7b75252960", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "421", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "This is a generalisation of the result proved in \\SecNo[§]{208} for real values of~$z$.", "markdown": "This is a generalisation of the result proved in [§]208 for real values of $z$.", "why": "Shows that the familiar limit of (1 + z/n)^n carries over unchanged from real to complex z.", "use": [ "lesson" ], "concepts": [ "theorem/exponential-limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-3b380435a6", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "423", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "A more complete discussion of the binomial series, taking account of the more difficult case in which $|z| = 1$, will be found on pp.~225~\\textit{et~seq.}\\ of Bromwich's \\textit{Infinite Series}.", "markdown": "A more complete discussion of the binomial series, taking account of the more difficult case in which $|z| = 1$, will be found on pp. 225 *et seq.* of Bromwich’s *Infinite Series*.", "why": "Tells a learner where the theorem stops (|z| < 1) and where to read further.", "use": [ "lesson", "history" ], "concepts": [ "concept/binomial-series", "theorem/binomial-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-29370dd380", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "429", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "If $f(z)$~is a function of the complex variable~$z$, we call the curves for which $|f(z)|$~is constant the \\emph{level curves} of~$f(z)$.", "markdown": "If $f(z)$ is a function of the complex variable $z$, we call the curves for which $|f(z)|$ is constant the *level curves* of $f(z)$.", "why": "Gives a plain definition of level curves, which the exercises then sketch for several functions.", "use": [ "lesson", "website" ], "concepts": [ "concept/level-curve" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-4ddb5faaff", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "428", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "This projection gives a map of the sphere on the tangent plane, generally known as the \\emph{Stereographic Projection}.", "markdown": "This projection gives a map of the sphere on the tangent plane, generally known as the *Stereographic Projection*.", "why": "Names the projection and states what it does in a single sentence.", "use": [ "website", "history" ], "concepts": [ "concept/stereographic-projection" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-6dbc417b07", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "428", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "In this map parallels of latitude and longitude are represented by straight lines parallel to the axes of $X$ and $Y$ respectively.", "markdown": "In this map parallels of latitude and longitude are represented by straight lines parallel to the axes of $X$ and $Y$ respectively.", "why": "Shows the practical payoff of the complex logarithm: curved lines on the sphere become straight lines on the map.", "use": [ "lesson", "website" ], "concepts": [ "concept/logarithm", "concept/mercator-s-projection" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-808132e548", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "426", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "To one value of~$Z$ corresponds one of~$z$, but to one of~$z$ infinitely many of~$Z$.", "markdown": "To one value of $Z$ corresponds one of $z$, but to one of $z$ infinitely many of $Z$.", "why": "Makes the many-valuedness of the complex logarithm concrete as a mapping between planes.", "use": [ "lesson" ], "concepts": [ "concept/logarithm", "concept/transformation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-ee4a968cfc", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "429", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "The reader will probably find but little difficulty in arriving at a general idea of the forms of the level curves of any given rational function; but to enter into details would carry us into the general theory of functions of a complex variable.", "markdown": "The reader will probably find but little difficulty in arriving at a general idea of the forms of the level curves of any given rational function; but to enter into details would carry us into the general theory of functions of a complex variable.", "why": "An honest note from the author on how far this course goes and where a deeper theory begins.", "use": [ "website", "history" ], "concepts": [ "concept/level-curve" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-c8e47abfa6", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "433", "location": "The Proof that every Equation has a Root", "latex": "We can represent the values of $z$ and~$Z$ by points in two planes, which we may call the $z$-plane and the $Z$-plane respectively. It is evident that if $z$~describes a closed path~$\\gamma$ in the $z$-plane, then $Z$~describes a corresponding closed path~$\\Gamma$ in the $Z$-plane.", "markdown": "We can represent the values of $z$ and $Z$ by points in two planes, which we may call the $z$-plane and the $Z$-plane respectively. It is evident that if $z$ describes a closed path $\\gamma$ in the $z$-plane, then $Z$ describes a corresponding closed path $\\Gamma$ in the $Z$-plane.", "why": "It sets up the geometric picture the whole proof rests on: a polynomial maps a path in one plane to a path in another.", "use": [ "lesson" ], "concepts": [ "concept/closed-contour", "concept/complex-number", "concept/polynomial" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-9334469a23", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "433", "location": "The Proof that every Equation has a Root", "latex": "Thus $\\am Z$~denotes a one-valued and continuous function of $X$~and~$Y$, the real and imaginary parts of~$Z$.", "markdown": "Thus $\\am Z$ denotes a one-valued and continuous function of $X$ and $Y$, the real and imaginary parts of $Z$.", "why": "It explains why the chosen amplitude can be treated as an honest function of the point, which the argument needs.", "use": [ "lesson" ], "concepts": [ "concept/amplitude-of-a-complex-number", "concept/continuous-function", "concept/one-valued-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-788e292a5c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "436", "location": "The Proof that every Equation has a Root", "latex": "when $z$~describes any contour~$\\gamma$ in the positive sense the increment of~$\\am Z$ is~$2k\\pi$, where $k$~is the number of roots of $Z = 0$ inside~$\\gamma$, multiple roots being counted multiply.", "markdown": "when $z$ describes any contour $\\gamma$ in the positive sense the increment of $\\am Z$ is $2k\\pi$, where $k$ is the number of roots of $Z = 0$ inside $\\gamma$, multiple roots being counted multiply.", "why": "This is the central statement a learner should carry away: winding of the amplitude counts the roots enclosed.", "use": [ "lesson", "website" ], "concepts": [ "concept/multiple-root", "concept/root-of-an-equation", "theorem/argument-principle" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-5282f70798", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "438", "location": "The Proof that every Equation has a Root", "latex": "Show that the roots of $f'(z) = 0$ are the foci of the ellipse which touches the sides of the triangle $(z_{1}, z_{2}, z_{3})$ at their middle points.", "markdown": "Show that the roots of $f'(z) = 0$ are the foci of the ellipse which touches the sides of the triangle $(z_{1}, z_{2}, z_{3})$ at their middle points.", "why": "A striking exercise that links the roots of a derivative to a geometric figure, inviting a learner to explore beyond the proof.", "use": [ "website" ], "concepts": [ "concept/ellipse", "concept/focus", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-63c7d8f2ab", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "434", "location": "The Proof that every Equation has a Root", "latex": "Thus if its path is like~(\\ib) in \\Fig{B}, winding once round the origin in the positive direction, then its amplitude will have increased by~$2\\pi$.", "markdown": "Thus if its path is like (*b*) in [fig:B]Fig. B, winding once round the origin in the positive direction, then its amplitude will have increased by $2\\pi$.", "why": "Shows a learner concretely how winding once round the origin changes the amplitude by 2\\pi.", "use": [ "lesson", "website" ], "concepts": [ "concept/amplitude-of-a-complex-number", "concept/increment", "concept/origin", "concept/positive-direction" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-0e7fdeaf7b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "434", "location": "The Proof that every Equation has a Root", "latex": "Thus $PQ$~will have been described twice, once from $P$ to~$Q$ and once from $Q$ to~$P$. As $z$~moves from $P$ to~$Q$, $\\am Z$~varies continuously, since $Z$~does not pass through the origin; and if the increment of~$\\am Z$ is in this case~$\\theta$, then its increment when $z$~moves from $Q$ to~$P$ is~$-\\theta$; so that, when we add up the increments of~$\\am Z$ due to the description of the various parts of the smaller contours, all cancel one another, save the increments due to the description of parts of $\\gamma$~itself.", "markdown": "Thus $PQ$ will have been described twice, once from $P$ to $Q$ and once from $Q$ to $P$. As $z$ moves from $P$ to $Q$, $\\am Z$ varies continuously, since $Z$ does not pass through the origin; and if the increment of $\\am Z$ is in this case $\\theta$, then its increment when $z$ moves from $Q$ to $P$ is $-\\theta$; so that, when we add up the increments of $\\am Z$ due to the description of the various parts of the smaller contours, all cancel one another, save the increments due to the description of parts of $\\gamma$ itself.", "why": "Explains why interior boundaries cancel, so the increment around the whole contour is the sum over the small ones.", "use": [ "lesson" ], "concepts": [ "concept/closed-contour", "concept/increment", "method/nested-squares-argument" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-2d11f38873", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "435", "location": "The Proof that every Equation has a Root", "latex": "Hence, if $\\am Z$~is changed when $z$~describes~$\\gamma$, there must be \\emph{at least one} of the smaller contours, say~$\\gamma_{1}$, such that $\\am Z$~is changed when $z$~describes~$\\gamma_{1}$.", "markdown": "Hence, if $\\am Z$ is changed when $z$ describes $\\gamma$, there must be *at least one* of the smaller contours, say $\\gamma_{1}$, such that $\\am Z$ is changed when $z$ describes $\\gamma_{1}$.", "why": "States the pivotal step of the subdivision argument, which can be repeated indefinitely.", "use": [ "lesson" ], "concepts": [ "concept/closed-contour", "method/nested-squares-argument" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-b717e344b8", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "435", "location": "The Proof that every Equation has a Root", "latex": "But the latter contour evidently lies inside the circle whose centre is~$a$ and whose radius is~$\\frac{1}{2}\\rho$, and this circle does not include the origin. Hence the amplitude of~$Z$ is unchanged.", "markdown": "But the latter contour evidently lies inside the circle whose centre is $a$ and whose radius is $\\frac{1}{2}\\rho$, and this circle does not include the origin. Hence the amplitude of $Z$ is unchanged.", "why": "Delivers the contradiction: near a point where P is not zero, the amplitude cannot change.", "use": [ "lesson" ], "concepts": [ "concept/amplitude-of-a-complex-number", "concept/continuous-function", "theorem/fundamental-theorem-of-algebra" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-f97d852043", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "436", "location": "The Proof that every Equation has a Root", "latex": "We can then show, by an argument similar to that used above, that $\\am(1 + \\rho)$~is unchanged as $z$~describes $\\gamma$~in the positive sense, while $\\am z^{n}$ on the other hand is increased by~$2n\\pi$. Hence $\\am Z$~is increased by~$2n\\pi$, and the proof that $Z = 0$ has a root is completed.", "markdown": "We can then show, by an argument similar to that used above, that $\\am(1 + \\rho)$ is unchanged as $z$ describes $\\gamma$ in the positive sense, while $\\am z^{n}$ on the other hand is increased by $2n\\pi$. Hence $\\am Z$ is increased by $2n\\pi$, and the proof that $Z = 0$ has a root is completed.", "why": "Shows how a large circle forces the amplitude to change, completing the proof.", "use": [ "lesson" ], "concepts": [ "concept/amplitude-of-a-complex-number", "concept/polynomial", "theorem/fundamental-theorem-of-algebra" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-8bcd76fb5c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "436", "location": "The Proof that every Equation has a Root", "latex": "This assumption is obviously legitimate, for to suppose the contrary, at any stage of the argument, is to admit the truth of the theorem.", "markdown": "This assumption is obviously legitimate, for to suppose the contrary, at any stage of the argument, is to admit the truth of the theorem.", "why": "Models how a proof can dispose of an awkward assumption by noting that its failure already gives the result.", "use": [ "lesson", "website" ], "concepts": [ "concept/origin", "theorem/fundamental-theorem-of-algebra" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-532ef63f0f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "436", "location": "The Proof that every Equation has a Root", "latex": "There is another proof, proceeding on different lines, which is often given. It depends, however, on an extension to functions of two or more variables of the results of \\SecNo[§§]{102}~\\textit{et~seq.}", "markdown": "There is another proof, proceeding on different lines, which is often given. It depends, however, on an extension to functions of two or more variables of the results of [§§]102 *et seq.*", "why": "Introduces the second, minimum-modulus proof and notes what it depends on.", "use": [ "history", "lesson" ], "concepts": [ "concept/least-upper-bound", "theorem/fundamental-theorem-of-algebra" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-d784c60bec", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "439", "location": "A Note on Double Limit Problems", "latex": "Another way of expressing this fact is to say that the operations of summation from $0$ to~$\\infty$, and of integration from $0$ to~$x$, are \\emph{commutative} when applied to the function $(-1)^{n}t^{n}$, \\ie\\ that it does not matter in what order they are performed on the function.", "markdown": "Another way of expressing this fact is to say that the operations of summation from $0$ to $\\infty$, and of integration from $0$ to $x$, are *commutative* when applied to the function $(-1)^{n}t^{n}$, *i.e.* that it does not matter in what order they are performed on the function.", "why": "It states in plain words what commutativity of two operations means, using a series and an integral the learner already knows.", "use": [ "lesson", "website" ], "concepts": [ "concept/commutativity-of-operations", "concept/definite-integral", "concept/infinite-sequence", "method/integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-759b71e8ce", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "441", "location": "A Note on Double Limit Problems", "latex": "We can always, by the exercise of a little ingenuity, find~$z$ so that $LL'z$ and~$L'Lz$ shall differ from one another.", "markdown": "We can always, by the exercise of a little ingenuity, find $z$ so that $LL'z$ and $L'Lz$ shall differ from one another.", "why": "It warns the learner that a formula which seems to commute may still fail for some specially chosen input, so commutativity cannot be assumed.", "use": [ "lesson" ], "concepts": [ "concept/commutativity-of-operations", "concept/limit-operation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-dbadc785e2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "442", "location": "A Note on Double Limit Problems", "latex": "Of course, in an exact science like pure mathematics, we cannot be satisfied with an answer of this kind; and in the higher branches of mathematics the detailed investigation of these questions is an absolute necessity.", "markdown": "Of course, in an exact science like pure mathematics, we cannot be satisfied with an answer of this kind; and in the higher branches of mathematics the detailed investigation of these questions is an absolute necessity.", "why": "Shows why rigour matters even when a heuristic usually works.", "use": [ "lesson", "website" ], "concepts": [ "concept/commutativity-of-operations", "concept/double-limit-problem", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-ca7d559d41", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "440", "location": "A Note on Double Limit Problems", "latex": "The operations of proceeding to the limit zero with each of two variables $x$~and~$y$ may or may not be commutative when applied to a function~$f(x, y)$.", "markdown": "The operations of proceeding to the limit zero with each of two variables $x$ and $y$ may or may not be commutative when applied to a function $f(x, y)$.", "why": "Introduces the idea that the order of two limits can matter, before the examples show both outcomes.", "use": [ "lesson", "website" ], "concepts": [ "concept/commutativity-of-operations", "concept/double-limit-problem", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-30bc3e54a1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "440", "location": "A Note on Double Limit Problems", "latex": "It is natural to suppose so: but that is all that we have a right to say at present.", "markdown": "It is natural to suppose so: but that is all that we have a right to say at present.", "why": "Models the honest distinction between a plausible guess and something proved.", "use": [ "lesson", "website" ], "concepts": [ "concept/commutativity-of-operations", "concept/limit-operation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-903e114164", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "441", "location": "A Note on Double Limit Problems", "latex": "The preceding examples suggest that there are three possibilities with respect to the commutation of two given operations, viz.:\\ (1)~the operations may \\emph{always} be commutative; (2)~they may \\emph{never} be commutative, \\emph{except in very special circumstances}; (3)~they may be commutative \\emph{in most of the ordinary cases which occur practically}.", "markdown": "The preceding examples suggest that there are three possibilities with respect to the commutation of two given operations, viz.: (1) the operations may *always* be commutative; (2) they may *never* be commutative, *except in very special circumstances*; (3) they may be commutative *in most of the ordinary cases which occur practically*.", "why": "Gives learners a simple three-way classification for whether two operations can be swapped.", "use": [ "lesson" ], "concepts": [ "concept/commutativity-of-operations" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-69da6b1285", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "442", "location": "A Note on Double Limit Problems", "latex": "In practice, a result obtained by assuming that two limit-operations are commutative is \\emph{probably} true: it at any rate affords a valuable \\emph{suggestion} as to the answer to the problem under consideration.", "markdown": "In practice, a result obtained by assuming that two limit-operations are commutative is *probably* true: it at any rate affords a valuable *suggestion* as to the answer to the problem under consideration.", "why": "Teaches that swapping limit operations is a useful heuristic for finding answers, not a proof.", "use": [ "lesson", "website" ], "concepts": [ "concept/commutativity-of-operations", "concept/double-limit-problem", "concept/limit-operation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-e1eced374f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "442", "location": "A Note on Double Limit Problems", "latex": "Detailed investigations of a large number of important double limit problems will be found in Bromwich's \\textit{Infinite Series}.", "markdown": "Detailed investigations of a large number of important double limit problems will be found in Bromwich’s *Infinite Series*.", "why": "Points a curious learner to the book's recommended source for going deeper.", "use": [ "history" ], "concepts": [ "concept/double-limit-problem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-e6cb28bb4c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "443", "location": "The circular functions", "latex": "The cosine and sine are continuous for all values of~$y$, the tangent except at the points where its definition fails.", "markdown": "The cosine and sine are continuous for all values of $y$, the tangent except at the points where its definition fails.", "why": "It warns that the tangent, unlike cosine and sine, can fail to be defined at certain points.", "use": [ "lesson" ], "concepts": [ "concept/continuous-function", "concept/cosine", "concept/sine", "concept/tangent-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-01a7ce3cf5", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "444", "location": "The circular functions", "latex": "The formulae for differentiation of the circular functions may now be deduced in the ordinary way, and the power series derived from Taylor's Theorem.", "markdown": "The formulae for differentiation of the circular functions may now be deduced in the ordinary way, and the power series derived from Taylor’s Theorem.", "why": "It links the derivatives of the circular functions to Taylor's theorem and the power series that follow from it.", "use": [ "lesson" ], "concepts": [ "concept/circular-function", "method/differentiation", "method/taylor-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-23d32c17a8", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "443", "location": "The circular functions", "latex": "The equation~\\Eq{(1)} defines a unique value of~$y$ corresponding to every real\nvalue of~$x$. As $y$~is continuous and strictly increasing, there is an inverse\nfunction $x = x(y)$, also continuous and steadily increasing.", "markdown": "The equation (1) defines a unique value of $y$ corresponding to every real value of $x$. As $y$ is continuous and strictly increasing, there is an inverse function $x = x(y)$, also continuous and steadily increasing.", "why": "Shows how a function's inverse is obtained from nothing more than monotonicity and continuity, which is the move the whole construction rests on.", "use": [ "lesson", "website" ], "concepts": [ "concept/continuous-function", "concept/inverse-circular-function", "concept/inverse-function", "concept/tangent-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-92b80bed30", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "443", "location": "The circular functions", "latex": "We have thus defined $\\cos y$ and~$\\sin y$ for all values of~$y$, and $\\tan y$~for all\nvalues of~$y$ other than odd multiples of~$\\frac{1}{2}\\pi$. The cosine and sine are continuous\nfor all values of~$y$, the tangent except at the points where its definition fails.", "markdown": "We have thus defined $\\cos y$ and $\\sin y$ for all values of $y$, and $\\tan y$ for all values of $y$ other than odd multiples of $\\frac{1}{2}\\pi$. The cosine and sine are continuous for all values of $y$, the tangent except at the points where its definition fails.", "why": "States plainly where each circular function is defined and continuous, a good check on a learner's picture of the graphs.", "use": [ "lesson" ], "concepts": [ "concept/continuous-function", "concept/cosine", "concept/domain-of-definition", "concept/sine", "concept/tangent-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-d3cf3c2a39", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "443", "location": "The circular functions", "latex": "The further development of the theory depends merely on the addition\nformulae.", "markdown": "The further development of the theory depends merely on the addition formulae.", "why": "Tells the learner that the addition formulae are the hinge of the whole theory.", "use": [ "lesson" ], "concepts": [ "concept/addition-formulae", "concept/circular-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-fd0ebe15d6", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "444", "location": "The circular functions", "latex": "To determine the sign put $y_{2} = 0$. The equation reduces to $\\cos y_{1} = ±\\cos y_{1}$,\nwhich shows that the positive sign must be chosen for at least one value of~$y_{2}$,\nviz.\\ $y_{2} = 0$. It follows from considerations of continuity that the positive sign\nmust be chosen in all cases.", "markdown": "To determine the sign put $y_{2} = 0$. The equation reduces to $\\cos y_{1} = ±\\cos y_{1}$, which shows that the positive sign must be chosen for at least one value of $y_{2}$, viz. $y_{2} = 0$. It follows from considerations of continuity that the positive sign must be chosen in all cases.", "why": "Models a small argument in which a special case and continuity together fix an ambiguous sign.", "use": [ "lesson" ], "concepts": [ "concept/addition-formulae", "concept/continuous-function", "concept/cosine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-7d4fe57c8e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "444", "location": "The circular functions", "latex": "An alternative theory of the circular functions is based on the theory of\ninfinite series.", "markdown": "An alternative theory of the circular functions is based on the theory of infinite series.", "why": "Shows that the same functions can be built from a different starting point, here power series, and points to a further reference.", "use": [ "history", "website" ], "concepts": [ "concept/circular-function", "concept/infinite-sequence", "concept/power-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-a5e9d2789d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "445", "location": "The infinite in analysis and geometry", "latex": "The point of importance is this. The infinite of analysis is a `limiting' and not an `actual' infinite.", "markdown": "The point of importance is this. The infinite of analysis is a ‘limiting’ and not an ‘actual’ infinite.", "why": "It states the chapter's central distinction: the infinity of analysis is a limit, not a completed object.", "use": [ "lesson" ], "concepts": [ "concept/infinity", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-58671b2085", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "445", "location": "The infinite in analysis and geometry", "latex": "But \\emph{the infinite of geometry is an actual and not a limiting infinite}. The `line at infinity' is a line in precisely the same sense in which other lines are lines.", "markdown": "But *the infinite of geometry is an actual and not a limiting infinite*. The ‘line at infinity’ is a line in precisely the same sense in which other lines are lines.", "why": "It tells learners that the geometric infinite is a genuine object, so the line at infinity needs no limit to be a line.", "use": [ "website", "lesson" ], "concepts": [ "concept/infinity", "concept/line-at-infinity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-c4f80120ea", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "445", "location": "The infinite in analysis and geometry", "latex": "This correlation is historically important, for it is from it that the vocabulary of the subject has been derived, and it is often useful for purposes of illustration. It is however no more than an illustration, and no rational account of the geometrical infinite can be based upon it.", "markdown": "This correlation is historically important, for it is from it that the vocabulary of the subject has been derived, and it is often useful for purposes of illustration. It is however no more than an illustration, and no rational account of the geometrical infinite can be based upon it.", "why": "It explains why the familiar vocabulary of infinite points came from a correlation, while warning that the correlation is only illustration.", "use": [ "history", "lesson" ], "concepts": [ "concept/coordinate-geometry", "concept/homogeneous-geometry", "concept/infinity", "concept/line-at-infinity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-2124d88208", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "445", "location": "The infinite in analysis and geometry", "latex": "The confusion about these matters so prevalent among students arises from the fact that, in the commonly used text books of analytical geometry, the illustration is taken for the reality.", "markdown": "The confusion about these matters so prevalent among students arises from the fact that, in the commonly used text books of analytical geometry, the illustration is taken for the reality.", "why": "It names a common student mistake, mistaking the correlation for the geometry it illustrates, so a learner can watch for it.", "use": [ "lesson", "website" ], "concepts": [ "concept/coordinate-geometry", "concept/infinity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-499174b4ed", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "445", "location": "The infinite in analysis and geometry", "latex": "The object of this brief note is to point out that these concepts are in no way dependent upon the analytical doctrine of limits.", "markdown": "The object of this brief note is to point out that these concepts are in no way dependent upon the analytical doctrine of limits.", "why": "States the note's single aim: the infinite of geometry does not rest on limits.", "use": [ "lesson", "website" ], "concepts": [ "concept/coordinate-geometry", "concept/infinity", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-1256770113", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "445", "location": "The infinite in analysis and geometry", "latex": "In what may be called `common Cartesian geometry', a \\emph{point} is \\emph{a pair of real numbers $(x, y)$}. A \\emph{line} is the class of points which satisfy a linear relation $ax + by + c=0$, in which $a$~and~$b$ are not both zero.", "markdown": "In what may be called ‘common Cartesian geometry’, a *point* is *a pair of real numbers $(x, y)$*. A *line* is the class of points which satisfy a linear relation $ax + by + c=0$, in which $a$ and $b$ are not both zero.", "why": "Gives the plain definitions of point and line that a learner already half-knows, as a baseline for the homogeneous system.", "use": [ "lesson" ], "concepts": [ "concept/common-cartesian-geometry", "concept/line", "concept/linear-relation", "concept/point" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-6ec8e6afa1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "445", "location": "The infinite in analysis and geometry", "latex": "In a system of real homogeneous geometry a point is \\emph{a class of triads of real numbers $(x, y, z)$}, not all zero, triads being classed together when their constituents are proportional.", "markdown": "In a system of real homogeneous geometry a point is *a class of triads of real numbers $(x, y, z)$*, not all zero, triads being classed together when their constituents are proportional.", "why": "Shows how a point can be defined as a class of proportional triples rather than a single pair.", "use": [ "lesson" ], "concepts": [ "concept/homogeneous-geometry", "concept/point", "concept/triad" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-f237829a1c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "445", "location": "The infinite in analysis and geometry", "latex": "Thus, in what may be called `real homogeneous Cartesian geometry', those points are special for which $z = 0$, and there is one special line, viz.\\ the line $z = 0$. This special line is called `the line at infinity'.", "markdown": "Thus, in what may be called ‘real homogeneous Cartesian geometry’, those points are special for which $z = 0$, and there is one special line, viz. the line $z = 0$. This special line is called ‘the line at infinity’.", "why": "Shows exactly where the line at infinity comes from: it is the ordinary line z = 0 that has been singled out.", "use": [ "lesson", "website" ], "concepts": [ "concept/homogeneous-geometry", "concept/line-at-infinity", "concept/point", "concept/real-homogeneous-cartesian-geometry", "concept/special-element" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-7b6baaf42b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "445", "location": "The infinite in analysis and geometry", "latex": "The infinite of analysis is a `limiting' and not an `actual' infinite. The symbol~`$\\infty$' has, throughout this book, been regarded as an `incomplete symbol', a symbol to which no independent meaning has been attached, though one has been attached to certain phrases containing it. But \\emph{the infinite of geometry is an actual and not a limiting infinite}. The `line at infinity' is a line in precisely the same sense in which other lines are lines.", "markdown": "The infinite of analysis is a ‘limiting’ and not an ‘actual’ infinite. The symbol ‘$\\infty$’ has, throughout this book, been regarded as an ‘incomplete symbol’, a symbol to which no independent meaning has been attached, though one has been attached to certain phrases containing it. But *the infinite of geometry is an actual and not a limiting infinite*. The ‘line at infinity’ is a line in precisely the same sense in which other lines are lines.", "why": "Separates two senses of 'infinite' that learners tend to blur: a limiting process in analysis and an actual object in geometry.", "use": [ "lesson", "website" ], "concepts": [ "concept/actual-infinity", "concept/incomplete-symbol", "concept/infinity", "concept/limiting-infinity", "concept/line-at-infinity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-31127faa34", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "445", "location": "The infinite in analysis and geometry", "latex": "When $(x, y, z)$ varies on the first line, in such a manner as to tend in the limit to the special point for which $z = 0$, the corresponding point on the second line varies so that its distance from the origin tends to infinity.", "markdown": "When $(x, y, z)$ varies on the first line, in such a manner as to tend in the limit to the special point for which $z = 0$, the corresponding point on the second line varies so that its distance from the origin tends to infinity.", "why": "Describes the correlation that explains where the geometric vocabulary of 'infinity' comes from.", "use": [ "lesson", "history" ], "concepts": [ "concept/correlation-between-homogeneous-and-common-geometry", "concept/corresponding-point", "concept/origin", "concept/tends-to-infinity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/x-70b2aee08d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "445", "location": "The infinite in analysis and geometry", "latex": "It is however no more than an illustration, and no rational account of the geometrical infinite can be based upon it. The confusion about these matters so prevalent among students arises from the fact that, in the commonly used text books of analytical geometry, the illustration is taken for the reality.", "markdown": "It is however no more than an illustration, and no rational account of the geometrical infinite can be based upon it. The confusion about these matters so prevalent among students arises from the fact that, in the commonly used text books of analytical geometry, the illustration is taken for the reality.", "why": "Warns against a common student error: treating a helpful picture as the definition of the geometric infinite.", "use": [ "lesson", "website", "history" ], "concepts": [ "concept/actual-infinity", "concept/correlation-between-homogeneous-and-common-geometry", "concept/line-at-infinity" ] } ], "equations": [ { "id": "hardy-course-of-pure-mathematics-1921/eq-cd02b1069d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "1", "location": "REAL VARIABLES", "latex": "(-p)/(-q) = p/q", "name": null, "statement": "A fraction with both numerator and denominator negated equals the original fraction.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "p", "meaning": "integer numerator" }, { "unit": null, "symbol": "q", "meaning": "integer denominator" } ], "sympy": "Eq((-p)/(-q), p/q)", "physics": false, "states": [], "concepts": [ "concept/algebraic-fraction", "concept/negative-number", "concept/rational-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-89f759d1c2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "2", "location": "REAL VARIABLES", "latex": "A_{0}A_{r}/A_{0}A_{1} = r", "name": null, "statement": "The point A_r for a positive rational r is chosen so that the ratio of the segment A_0A_r to A_0A_1 equals r.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "A_{0}", "meaning": "origin point on the line" }, { "unit": null, "symbol": "A_{r}", "meaning": "point representing the rational number r" }, { "unit": null, "symbol": "A_{1}", "meaning": "point representing the number 1" }, { "unit": null, "symbol": "r", "meaning": "positive rational number" } ], "sympy": "Eq(A0Ar/A0A1, r)", "physics": false, "states": [], "concepts": [ "concept/common-ratio", "concept/dimension", "concept/line-segment", "concept/origin", "concept/point", "concept/rational-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-2de524e8cd", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "2", "location": "REAL VARIABLES", "latex": "AB = -BA", "name": null, "statement": "Length is signed, so reversing the direction of a segment changes the sign of its length.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "AB", "meaning": "signed length of the segment from A to B" }, { "unit": null, "symbol": "BA", "meaning": "signed length of the segment from B to A" } ], "sympy": "Eq(AB, -BA)", "physics": false, "states": [], "concepts": [ "concept/dimension", "concept/negative-number", "concept/zodiacal-sign" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-1f52444f80", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "2", "location": "REAL VARIABLES", "latex": "A_{0}A_{-s} = -A_{-s}A_{0}", "name": null, "statement": "The point representing the negative rational -s is placed so that its signed distance from the origin is minus the distance of A_s.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "A_{0}", "meaning": "origin point" }, { "unit": null, "symbol": "A_{-s}", "meaning": "point representing the negative rational number -s" }, { "unit": null, "symbol": "s", "meaning": "positive rational number" } ], "sympy": "Eq(A0Am, -AmA0)", "physics": false, "states": [], "concepts": [ "concept/dimension", "concept/negative-number", "concept/origin", "concept/point", "concept/rational-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-062e127eca", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "3", "location": "REAL VARIABLES", "latex": "A_{0}A_{r} = r · A_{0}A_{1}", "name": null, "statement": "The signed length from the origin to the point for r equals r times the unit length A_0A_1.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "A_{0}A_{r}", "meaning": "signed length from origin to the point for r" }, { "unit": null, "symbol": "r", "meaning": "rational number" }, { "unit": "unit length", "symbol": "A_{0}A_{1}", "meaning": "unit segment (unit length)" } ], "sympy": "Eq(A0Ar, r*A0A1)", "physics": false, "states": [], "concepts": [ "concept/dimension", "concept/origin", "concept/point", "concept/rational-number", "unit/unit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e56422ab1d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "1", "location": "REAL VARIABLES", "latex": "r = p/q", "name": null, "statement": "A rational number r is defined as the fraction p/q, with p and q integers and q positive.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "r", "meaning": "rational number" }, { "unit": null, "symbol": "p", "meaning": "integer numerator" }, { "unit": null, "symbol": "q", "meaning": "integer denominator (taken positive)" } ], "sympy": "Eq(r, p/q)", "physics": false, "states": [], "concepts": [ "concept/algebraic-fraction", "concept/integer", "concept/rational-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-9b5e363b14", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "1", "location": "REAL VARIABLES", "latex": "p/(-q) = (-p)/q", "name": null, "statement": "Moving a minus sign from the denominator to the numerator leaves a fraction unchanged.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "p", "meaning": "integer numerator" }, { "unit": null, "symbol": "q", "meaning": "integer denominator" } ], "sympy": "Eq(p/(-q), (-p)/q)", "physics": false, "states": [], "concepts": [ "concept/algebraic-fraction", "concept/negative-number", "concept/rational-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-aca0a6219f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "3", "location": "REAL VARIABLES", "latex": "A_{0}A_{r} = r", "name": null, "statement": "With the unit segment A_0A_1 taken as 1, the length from the origin to the point for r is the number r.", "kind": "definition", "symbols": [ { "unit": "unit length", "symbol": "A_{0}A_{r}", "meaning": "length from origin to the point for r" }, { "unit": null, "symbol": "r", "meaning": "rational number" } ], "sympy": "Eq(A0Ar, r)", "physics": false, "states": [], "concepts": [ "concept/dimension", "concept/rational-number", "concept/rational-point", "unit/unit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-f06381569d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "3", "location": "REAL VARIABLES", "latex": "k · BC > 1", "name": null, "statement": "For any segment BC there is a positive integer k with k times BC greater than the unit length (the Axiom of Archimedes, assumed).", "kind": "rule", "symbols": [ { "unit": null, "symbol": "k", "meaning": "positive integer" }, { "unit": "unit length", "symbol": "BC", "meaning": "length of the segment BC" } ], "sympy": "Gt(k*BC, 1)", "physics": false, "states": [], "concepts": [ "concept/axiom", "concept/dimension", "concept/inequality", "concept/integer", "theorem/archimedean-property" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-27148deeca", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "7", "location": "REAL VARIABLES", "latex": "x^{2} = 1", "name": null, "statement": "The equation whose two rational roots are 1 and -1.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "unknown number" } ], "sympy": "Eq(x**2, 1)", "physics": false, "states": [], "concepts": [ "concept/equation", "concept/rational-number", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ae785d6659", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "5", "location": "REAL VARIABLES", "latex": "x^{2} = 2", "name": null, "statement": "The equation with no rational root, whose solution is the irrational number sqrt(2).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "unknown number" } ], "sympy": "Eq(x**2, 2)", "physics": false, "states": [], "concepts": [ "concept/equation", "concept/irrational-number", "concept/real-number", "concept/root-of-an-equation", "concept/square", "concept/surd" ], "pages": [ "5", "19" ], "chapters": [ "hardy-course-of-pure-mathematics-1921/ch-i" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a4c4b75016", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "5", "location": "REAL VARIABLES", "latex": "a + b = b + a", "name": null, "statement": "Addition of lengths (numbers) is commutative.", "kind": "law", "symbols": [ { "unit": null, "symbol": "a", "meaning": "length or number" }, { "unit": null, "symbol": "b", "meaning": "length or number" } ], "sympy": "Eq(a + b, b + a)", "physics": false, "states": [], "concepts": [ "concept/algebra", "concept/axiom", "concept/sum", "method/addition" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ffef7ae279", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "5", "location": "REAL VARIABLES", "latex": "a + (b + c) = (a + b) + c", "name": null, "statement": "Addition of lengths (numbers) is associative.", "kind": "law", "symbols": [ { "unit": null, "symbol": "a", "meaning": "length or number" }, { "unit": null, "symbol": "b", "meaning": "length or number" }, { "unit": null, "symbol": "c", "meaning": "length or number" } ], "sympy": "Eq(a + (b + c), (a + b) + c)", "physics": false, "states": [], "concepts": [ "concept/algebra", "concept/axiom", "concept/sum", "method/addition" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-05e1c6e871", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "5", "location": "REAL VARIABLES", "latex": "ab = ba", "name": null, "statement": "Multiplication of lengths (numbers) is commutative.", "kind": "law", "symbols": [ { "unit": null, "symbol": "a", "meaning": "length or number" }, { "unit": null, "symbol": "b", "meaning": "length or number" } ], "sympy": "Eq(a*b, b*a)", "physics": false, "states": [], "concepts": [ "concept/algebra", "concept/axiom", "concept/product", "method/multiplication" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-7b1b828802", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "5", "location": "REAL VARIABLES", "latex": "a(bc) = (ab)c", "name": null, "statement": "Multiplication of lengths (numbers) is associative.", "kind": "law", "symbols": [ { "unit": null, "symbol": "a", "meaning": "length or number" }, { "unit": null, "symbol": "b", "meaning": "length or number" }, { "unit": null, "symbol": "c", "meaning": "length or number" } ], "sympy": "Eq(a*(b*c), (a*b)*c)", "physics": false, "states": [], "concepts": [ "concept/algebra", "concept/axiom", "concept/product", "method/multiplication" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b437ed4649", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "5", "location": "REAL VARIABLES", "latex": "a(b + c) = ab + ac", "name": null, "statement": "Multiplication distributes over addition for lengths (numbers).", "kind": "law", "symbols": [ { "unit": null, "symbol": "a", "meaning": "length or number" }, { "unit": null, "symbol": "b", "meaning": "length or number" }, { "unit": null, "symbol": "c", "meaning": "length or number" } ], "sympy": "Eq(a*(b + c), a*b + a*c)", "physics": false, "states": [], "concepts": [ "concept/algebra", "concept/axiom", "concept/product", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-7a49b0b08d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "5", "location": "REAL VARIABLES", "latex": "A_{0}P = x", "name": null, "statement": "The required point P on the line is at distance x from the origin, where x is the length whose square is 2.", "kind": "definition", "symbols": [ { "unit": "unit length", "symbol": "A_{0}P", "meaning": "length from origin to point P" }, { "unit": "unit length", "symbol": "x", "meaning": "length (real number) required by the geometry" } ], "sympy": "Eq(A0P, x)", "physics": false, "states": [], "concepts": [ "concept/dimension", "concept/origin", "concept/point", "concept/real-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-21ae12af75", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "6", "location": "REAL VARIABLES", "latex": "x = \\sqrt{2}", "name": null, "statement": "The number x whose square is 2 is denoted by the symbol sqrt(2); it is not rational.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x", "meaning": "real number whose square is 2" } ], "sympy": "Eq(x, sqrt(2))", "physics": false, "states": [], "concepts": [ "concept/irrational-number", "concept/real-number", "concept/root" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-7dc3e8e2f7", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "10", "location": "REAL VARIABLES", "latex": "x^{q} = n", "name": null, "statement": "The q-th root of n is the number x whose q-th power is n, written n^{1/q} or the radical form.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x", "meaning": "q-th root of n" }, { "unit": null, "symbol": "n", "meaning": "integer or number" }, { "unit": null, "symbol": "q", "meaning": "positive integer index of the root" } ], "sympy": "Eq(x**q, n)", "physics": false, "states": [], "concepts": [ "concept/power", "concept/root", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-27b0b5fc44", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "10", "location": "REAL VARIABLES", "latex": "n^{p/q} = (n^{1/q})^{p}", "name": null, "statement": "A fractional power is defined as the p-th power of the q-th root of n.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "n", "meaning": "positive number" }, { "unit": null, "symbol": "p", "meaning": "integer exponent numerator" }, { "unit": null, "symbol": "q", "meaning": "positive integer root index" } ], "sympy": "Eq(n**(p/q), (n**(1/q))**p)", "physics": false, "states": [], "concepts": [ "concept/exponent", "concept/power", "concept/rational-number", "concept/root" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-0b6271637c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "10", "location": "REAL VARIABLES", "latex": "n^{p/q} n^{-p/q} = 1", "name": null, "statement": "A number raised to a rational power times the same number raised to the negative of that power equals 1.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "n", "meaning": "positive number" }, { "unit": null, "symbol": "p", "meaning": "integer exponent numerator" }, { "unit": null, "symbol": "q", "meaning": "positive integer root index" } ], "sympy": "Eq(n**(p/q)*n**(-p/q), 1)", "physics": false, "states": [], "concepts": [ "concept/exponent", "concept/power", "concept/rational-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-19ccb4fe51", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "10", "location": "REAL VARIABLES", "latex": "n^{r} × n^{s} = n^{r+s}", "name": "laws of indices (product)", "statement": "Multiplying powers of the same base adds their exponents, extended to rational exponents.", "kind": "law", "symbols": [ { "unit": null, "symbol": "n", "meaning": "positive number base" }, { "unit": null, "symbol": "r", "meaning": "rational exponent" }, { "unit": null, "symbol": "s", "meaning": "rational exponent" } ], "sympy": "Eq(n**r*n**s, n**(r+s))", "physics": false, "states": [ "law/laws-of-indices-product" ], "concepts": [ "concept/exponent", "concept/laws-of-indices", "concept/power", "concept/rational-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-5b63f004b6", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "10", "location": "REAL VARIABLES", "latex": "(n^{r})^{s} = n^{rs}", "name": "laws of indices (power of a power)", "statement": "Raising a power to a further power multiplies the exponents, extended to rational exponents.", "kind": "law", "symbols": [ { "unit": null, "symbol": "n", "meaning": "positive number base" }, { "unit": null, "symbol": "r", "meaning": "rational exponent" }, { "unit": null, "symbol": "s", "meaning": "rational exponent" } ], "sympy": "Eq((n**r)**s, n**(r*s))", "physics": false, "states": [ "law/laws-of-indices-power-of-a-power" ], "concepts": [ "concept/exponent", "concept/laws-of-indices", "concept/power", "concept/rational-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c9267123a9", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "6", "location": "REAL VARIABLES", "latex": "x^{n} + p_{1}x^{n-1} + p_{2}x^{n-2} + \\dots + p_{n} = 0", "name": "Gauss's theorem on rational roots (general form)", "statement": "An algebraic equation with integral coefficients cannot have a rational but non-integral root.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "unknown number (root)" }, { "unit": null, "symbol": "n", "meaning": "degree of the equation" }, { "unit": null, "symbol": "p_{1}, ..., p_{n}", "meaning": "integral coefficients of the equation" } ], "sympy": null, "physics": false, "states": [ "theorem/gauss-s-theorem-on-rational-roots-general-form" ], "concepts": [ "concept/coefficient", "concept/equation", "concept/integer", "concept/rational-number", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-7cbae75279", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "8", "location": "REAL VARIABLES", "latex": "y^{2} - x^{2} = (y - x)(y + x)", "name": null, "statement": "The difference of two squares factorises as the product of the difference and the sum of the numbers.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "real number (lower approximation)" }, { "unit": null, "symbol": "y", "meaning": "real number (upper approximation)" } ], "sympy": "Eq(y**2 - x**2, (y - x)*(y + x))", "physics": false, "states": [], "concepts": [ "concept/algebra", "concept/difference", "concept/factor", "concept/square" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-53161f6310", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "10", "location": "REAL VARIABLES", "latex": "x^{2} = N", "name": null, "statement": "For an integer N that is not a perfect square, the same argument shows that the root x of this equation is irrational.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "unknown number" }, { "unit": null, "symbol": "N", "meaning": "integer that is not a perfect square" } ], "sympy": "Eq(x**2, N)", "physics": false, "states": [], "concepts": [ "concept/equation", "concept/integer", "concept/irrational-number", "concept/perfect-square" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-1af659d15b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "16", "location": "REAL VARIABLES", "latex": "-(-\\alpha) = \\alpha", "name": null, "statement": "The negative of the negative of a real number is the number itself.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "\\alpha", "meaning": "a real number" } ], "sympy": "Eq(-(-alpha), alpha)", "physics": false, "states": [], "concepts": [ "concept/negative-number", "concept/real-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-1bc81f11ea", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "17", "location": "REAL VARIABLES", "latex": "\\gamma = \\alpha + \\beta", "name": null, "statement": "The sum of two real numbers is defined as the real number given by the section of all sums of their lower and upper classes.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "\\alpha", "meaning": "a real number" }, { "unit": null, "symbol": "\\beta", "meaning": "a real number" }, { "unit": null, "symbol": "\\gamma", "meaning": "the sum of alpha and beta" } ], "sympy": "Eq(gamma, alpha + beta)", "physics": false, "states": [], "concepts": [ "concept/real-number", "concept/sum", "method/addition" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d50828eadc", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "17", "location": "REAL VARIABLES", "latex": "\\alpha - \\beta = \\alpha + (-\\beta)", "name": null, "statement": "Subtraction of real numbers is defined as adding the negative of the second number.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "\\alpha", "meaning": "a real number" }, { "unit": null, "symbol": "\\beta", "meaning": "a real number" } ], "sympy": "Eq(alpha - beta, alpha + (-beta))", "physics": false, "states": [], "concepts": [ "concept/negative-number", "concept/real-number", "method/subtraction" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-763bd56411", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "18", "location": "REAL VARIABLES", "latex": "(-\\alpha)\\beta = -\\alpha\\beta", "name": null, "statement": "For positive alpha and beta, a negative times a positive gives the negative of the product.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "\\alpha", "meaning": "a positive real number" }, { "unit": null, "symbol": "\\beta", "meaning": "a positive real number" } ], "sympy": "Eq((-alpha)*beta, -alpha*beta)", "physics": false, "states": [], "concepts": [ "concept/negative-number", "concept/product", "method/multiplication" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-6a9fe49173", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "18", "location": "REAL VARIABLES", "latex": "\\alpha(-\\beta) = -\\alpha\\beta", "name": null, "statement": "For positive alpha and beta, a positive times a negative gives the negative of the product.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "\\alpha", "meaning": "a positive real number" }, { "unit": null, "symbol": "\\beta", "meaning": "a positive real number" } ], "sympy": "Eq(alpha*(-beta), -alpha*beta)", "physics": false, "states": [], "concepts": [ "concept/negative-number", "concept/product", "method/multiplication" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-0c2a6c134f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "18", "location": "REAL VARIABLES", "latex": "(-\\alpha)(-\\beta) = \\alpha\\beta", "name": null, "statement": "For positive alpha and beta, the product of two negatives is the product of the positive numbers.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "\\alpha", "meaning": "a positive real number" }, { "unit": null, "symbol": "\\beta", "meaning": "a positive real number" } ], "sympy": "Eq((-alpha)*(-beta), alpha*beta)", "physics": false, "states": [], "concepts": [ "concept/negative-number", "concept/product", "method/multiplication" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-fef2cacbe7", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "18", "location": "REAL VARIABLES", "latex": "1/(-\\alpha) = -(1/\\alpha)", "name": null, "statement": "The reciprocal of a negative number is the negative of the reciprocal of its positive counterpart.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "\\alpha", "meaning": "a positive real number" } ], "sympy": "Eq(1/(-alpha), -(1/alpha))", "physics": false, "states": [], "concepts": [ "concept/negative-number", "concept/reciprocal", "method/division" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-7d46eeeced", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "18", "location": "REAL VARIABLES", "latex": "\\alpha/\\beta = \\alpha × (1/\\beta)", "name": null, "statement": "Division of real numbers is defined as multiplication by the reciprocal of the divisor.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "\\alpha", "meaning": "a real number" }, { "unit": null, "symbol": "\\beta", "meaning": "a non-zero real number" } ], "sympy": "Eq(alpha/beta, alpha*(1/beta))", "physics": false, "states": [], "concepts": [ "concept/reciprocal", "method/division", "method/multiplication" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-498c97fafd", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "19", "location": "REAL VARIABLES", "latex": "(\\sqrt{2})^{2} = \\sqrt{2}\\sqrt{2} = 2.", "name": null, "statement": "The square of the number sqrt(2), defined as a section of the rational numbers, equals 2.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\sqrt{2}", "meaning": "the real number defined by the section x^2 < 2, x^2 > 2 of the rational numbers" } ], "sympy": "Eq(sqrt(2)**2, 2)", "physics": false, "states": [], "concepts": [ "concept/real-number", "concept/square", "concept/surd" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-9b363987a0", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "20", "location": "REAL VARIABLES", "latex": "x^{2} - 2ax + a^{2} - b = 0", "name": null, "statement": "The quadratic equation whose roots are a + sqrt(b) and a - sqrt(b), for rational a and b.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the unknown (variable)" }, { "unit": null, "symbol": "a", "meaning": "a rational number" }, { "unit": null, "symbol": "b", "meaning": "a positive rational number that is not a perfect square" } ], "sympy": "Eq(x**2 - 2*a*x + a**2 - b, 0)", "physics": false, "states": [], "concepts": [ "concept/mixed-quadratic-surd", "concept/quadratic-equation", "concept/quadratic-surd", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-82071283d8", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "20", "location": "REAL VARIABLES", "latex": "ax^{2} + 2bx + c = 0", "name": null, "statement": "The general quadratic equation with rational coefficients, whose real roots are {-b ± sqrt(b^2 - ac)}/a when b^2 - ac > 0.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "a", "meaning": "coefficient of x^2 (rational, taken as integer)" }, { "unit": null, "symbol": "b", "meaning": "half the coefficient of x (rational)" }, { "unit": null, "symbol": "c", "meaning": "constant term (rational)" }, { "unit": null, "symbol": "x", "meaning": "the unknown (variable)" } ], "sympy": null, "physics": false, "states": [ "theorem/quadratic-formula-setting" ], "concepts": [ "concept/coefficient", "concept/quadratic-equation", "concept/real-root", "concept/root-of-an-equation", "method/euclidean-construction" ], "pages": [ "20", "66" ], "chapters": [ "hardy-course-of-pure-mathematics-1921/ch-i", "hardy-course-of-pure-mathematics-1921/ch-ii" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-04e14423a8", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "21", "location": "REAL VARIABLES", "latex": "\\sqrt{8} = 2\\sqrt{2}", "name": null, "statement": "The surd sqrt(8) is a rational multiple of sqrt(2), so the two surds are similar.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\sqrt{8}", "meaning": "the square root of 8" } ], "sympy": "Eq(sqrt(8), 2*sqrt(2))", "physics": false, "states": [], "concepts": [ "concept/rational-number", "concept/similar-surds", "concept/surd" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ffc9f70bba", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "22", "location": "REAL VARIABLES", "latex": "A + \\sqrt{B} = C + \\sqrt{D}", "name": null, "statement": "If A, B, C, D are rational and this holds, then either A = C and B = D, or B and D are both squares of rational numbers.", "kind": "result", "symbols": [ { "unit": null, "symbol": "A", "meaning": "rational number" }, { "unit": null, "symbol": "B", "meaning": "rational number" }, { "unit": null, "symbol": "C", "meaning": "rational number" }, { "unit": null, "symbol": "D", "meaning": "rational number" } ], "sympy": "Eq(A + sqrt(B), C + sqrt(D))", "physics": false, "states": [], "concepts": [ "concept/rational-number", "concept/surd", "theorem/work-done-in-charging-a-conductor" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-df46acf064", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "22", "location": "REAL VARIABLES", "latex": "A - \\sqrt{B} = C - \\sqrt{D}", "name": null, "statement": "Corollary: if A + sqrt(B) = C + sqrt(D), then A - sqrt(B) = C - sqrt(D), unless sqrt(B) and sqrt(D) are both rational.", "kind": "result", "symbols": [ { "unit": null, "symbol": "A", "meaning": "rational number" }, { "unit": null, "symbol": "B", "meaning": "rational number" }, { "unit": null, "symbol": "C", "meaning": "rational number" }, { "unit": null, "symbol": "D", "meaning": "rational number" } ], "sympy": "Eq(A - sqrt(B), C - sqrt(D))", "physics": false, "states": [], "concepts": [ "concept/corollary", "concept/rational-number", "concept/surd" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-7431d6e597", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "24", "location": "REAL VARIABLES", "latex": "z = \\sqrtp[3]{4 + \\sqrt{15}} + \\sqrtp[3]{4 - \\sqrt{15}}", "name": null, "statement": "Defines z as the sum of the real cube roots of 4 + sqrt(15) and 4 - sqrt(15); z is shown to satisfy z^3 = 3z + 8.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "z", "meaning": "the real number defined as the sum of two cube roots" } ], "sympy": "Eq(z, cbrt(4 + sqrt(15)) + cbrt(4 - sqrt(15)))", "physics": false, "states": [], "concepts": [ "concept/cubic-equation", "concept/real-number", "concept/surd" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c928f7a75d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "24", "location": "REAL VARIABLES", "latex": "z^{3} = 3z + 8", "name": null, "statement": "The unique real number z satisfying this cubic equation exists, is positive, and is not rational.", "kind": "result", "symbols": [ { "unit": null, "symbol": "z", "meaning": "the real number root of the cubic equation (the continuous real variable)" } ], "sympy": "Eq(z**3, 3*z + 8)", "physics": false, "states": [], "concepts": [ "concept/cubic-equation", "concept/irrational-number", "concept/real-number", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-2c5f1a9e62", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "25", "location": "REAL VARIABLES", "latex": "x^{5} = x + 16\\DPtypo{.}{,}", "name": null, "statement": "This quintic has a unique positive real root, which cannot in general be expressed by surds.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the continuous real variable (unknown real number)" } ], "sympy": "Eq(x**5, x + 16)", "physics": false, "states": [], "concepts": [ "concept/irrational-number", "concept/real-number", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a30bd5fbde", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "25", "location": "REAL VARIABLES", "latex": "\\pi^{5} = \\pi + n", "name": null, "statement": "pi is not the root of any algebraic equation with integer coefficients, such as this one, where n is an integer.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\pi", "meaning": "the length of the circumference of a circle of unit diameter" }, { "unit": null, "symbol": "n", "meaning": "an integer" } ], "sympy": "Eq(pi**5, pi + n)", "physics": false, "states": [], "concepts": [ "concept/integer", "concept/irrational-number", "concept/root-of-an-equation", "quantity/pi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-2acdfb1442", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "29", "location": "REAL VARIABLES", "latex": "\\beta \\leq x \\leq \\gamma", "name": null, "statement": "The closed interval of real numbers x lying from beta to gamma inclusive.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x", "meaning": "a real number in the interval" }, { "unit": null, "symbol": "\\beta", "meaning": "the lower end of the interval" }, { "unit": null, "symbol": "\\gamma", "meaning": "the upper end of the interval" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/continuous-real-variable", "concept/interval" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-6e73c613f3", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "32", "location": "REAL VARIABLES", "latex": "n \\tsum{a^{p+q}} \\geq \\tsum a^{p} \\tsum a^{q}", "name": null, "statement": "Summing the two-number inequality over all pairs of n positive numbers gives n times the sum of their (p+q)th powers is at least the product of the sums of their pth and qth powers (Miscellaneous Example 8, equation (5)).", "kind": "inequality", "symbols": [ { "unit": null, "symbol": "a", "meaning": "positive number (including zero) from the set a_1, ..., a_n" }, { "unit": null, "symbol": "n", "meaning": "number of numbers in the set" }, { "unit": null, "symbol": "p", "meaning": "positive integer" }, { "unit": null, "symbol": "q", "meaning": "positive integer" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/inequality", "concept/power", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-21b86b0b3d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "32", "location": "REAL VARIABLES", "latex": "a_{1}^{p+q} + a_{2}^{p+q} \\geq a_{1}^{p} a_{2}^{q} + a_{1}^{q} a_{2}^{p}", "name": null, "statement": "For positive numbers a_1, a_2 and positive integers p, q, the sum of the (p+q)th powers is at least the sum of the mixed products a_1^p a_2^q + a_1^q a_2^p (derived in the text of Miscellaneous Example 7, equation (1)).", "kind": "inequality", "symbols": [ { "unit": null, "symbol": "a_1", "meaning": "positive number (including zero)" }, { "unit": null, "symbol": "a_2", "meaning": "positive number (including zero)" }, { "unit": null, "symbol": "p", "meaning": "positive integer" }, { "unit": null, "symbol": "q", "meaning": "positive integer" } ], "sympy": "Ge(a1**(p+q) + a2**(p+q), a1**p*a2**q + a1**q*a2**p)", "physics": false, "states": [], "concepts": [ "concept/inequality", "concept/positive-number", "concept/power" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-9cd67eb446", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "32", "location": "REAL VARIABLES", "latex": "\\frac{a_{1}^{p+q} + a_{2}^{p+q}}{2} \\geq \\left(\\frac{a_{1}^{p} + a_{2}^{p}}{2}\\right) \\left(\\frac{a_{1}^{q} + a_{2}^{q}}{2}\\right)", "name": null, "statement": "The mean of the (p+q)th powers of two positive numbers is at least the product of the means of their pth and qth powers (Miscellaneous Example 7, equation (2)).", "kind": "inequality", "symbols": [ { "unit": null, "symbol": "a_1", "meaning": "positive number (including zero)" }, { "unit": null, "symbol": "a_2", "meaning": "positive number (including zero)" }, { "unit": null, "symbol": "p", "meaning": "positive integer" }, { "unit": null, "symbol": "q", "meaning": "positive integer" } ], "sympy": "Ge((a1**(p+q) + a2**(p+q))/2, ((a1**p + a2**p)/2)*((a1**q + a2**q)/2))", "physics": false, "states": [], "concepts": [ "concept/arithmetical-mean", "concept/inequality", "concept/power" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-31a6e6bc73", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "32", "location": "REAL VARIABLES", "latex": "\\frac{a_{1}^{p} + a_{2}^{p}}{2} \\geq \\left(\\frac{a_{1} + a_{2}}{2}\\right)^{p}", "name": null, "statement": "The arithmetic mean of the pth powers of two positive numbers is at least the pth power of their arithmetic mean (Miscellaneous Example 7, equation (4)).", "kind": "inequality", "symbols": [ { "unit": null, "symbol": "a_1", "meaning": "positive number (including zero)" }, { "unit": null, "symbol": "a_2", "meaning": "positive number (including zero)" }, { "unit": null, "symbol": "p", "meaning": "positive integer" } ], "sympy": "Ge((a1**p + a2**p)/2, ((a1 + a2)/2)**p)", "physics": false, "states": [], "concepts": [ "concept/arithmetical-mean", "concept/inequality", "concept/power" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-463b0de2d3", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "32", "location": "REAL VARIABLES", "latex": "\\left(\\tsum a^{p}\\right)/n \\geq \\left\\{\\left(\\tsum a\\right)/n\\right\\}^{p}", "name": null, "statement": "For n positive numbers, the arithmetic mean of their pth powers is at least the pth power of their arithmetic mean (Miscellaneous Example 8, equation (7)).", "kind": "inequality", "symbols": [ { "unit": null, "symbol": "a", "meaning": "positive number (including zero) from the set a_1, ..., a_n" }, { "unit": null, "symbol": "n", "meaning": "number of numbers in the set" }, { "unit": null, "symbol": "p", "meaning": "positive integer" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/arithmetical-mean", "concept/inequality", "concept/power", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-3dfc7516a9", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "32", "location": "REAL VARIABLES", "latex": "a_{r}' + a_{s}' - a_{r} - a_{s} = (a_{r} - G)(a_{s} - G)/G", "name": null, "statement": "Replacing a_r and a_s by G and a_r a_s/G changes their sum by the product (a_r - G)(a_s - G)/G, used in the proof that the arithmetic mean is not less than the geometric mean (Miscellaneous Example 9).", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a_r", "meaning": "the greatest of the numbers a_1, ..., a_n" }, { "unit": null, "symbol": "a_s", "meaning": "the least of the numbers a_1, ..., a_n" }, { "unit": null, "symbol": "a_r'", "meaning": "new value G replacing a_r" }, { "unit": null, "symbol": "a_s'", "meaning": "new value a_r a_s / G replacing a_s" }, { "unit": null, "symbol": "G", "meaning": "geometric mean of a_r and a_s" } ], "sympy": "Eq(ar1 + as1 - ar - as, (ar - G)*(as - G)/G)", "physics": false, "states": [], "concepts": [ "concept/arithmetical-mean", "concept/geometrical-mean", "concept/identity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-22583f6776", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "33", "location": "REAL VARIABLES", "latex": "\\left(\\tsum a_{r} b_{r}\\right)^{2} = \\tsum a_{r}^{2} \\tsum a_{s}^{2} - \\tsum (a_{r} b_{s} - a_{s} b_{r})^{2}", "name": null, "statement": "The square of the sum of products a_r b_r equals the product of the sums of squares minus a sum of squared cross-differences; this identity is used to prove Schwarz's inequality (Miscellaneous Example 10).", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a_r", "meaning": "number of the first set (positive or negative)" }, { "unit": null, "symbol": "b_r", "meaning": "number of the second set (positive or negative)" }, { "unit": null, "symbol": "r", "meaning": "index running over 1, 2, ..., n" }, { "unit": null, "symbol": "s", "meaning": "index running over 1, 2, ..., n" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/identity", "concept/root", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-880a76a6f3", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "33", "location": "REAL VARIABLES", "latex": "\\left(\\tsum a_{r} b_{r}\\right)^{2} \\leq \\tsum a_{r}^{2} \\tsum b_{r}^{2}", "name": "Schwarz's inequality", "statement": "The square of the sum of products of two sets of n numbers is at most the product of the sums of their squares; the inequality is usually known as Schwarz's, though due originally to Cauchy (Miscellaneous Example 10).", "kind": "result", "symbols": [ { "unit": null, "symbol": "a_r", "meaning": "number of the first set (positive or negative)" }, { "unit": null, "symbol": "b_r", "meaning": "number of the second set (positive or negative)" }, { "unit": null, "symbol": "r", "meaning": "index running over 1, 2, ..., n" } ], "sympy": null, "physics": false, "states": [ "theorem/cauchy-schwarz-inequality" ], "concepts": [ "concept/inequality", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-6501297543", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "36", "location": "REAL VARIABLES", "latex": "a_{0}x^{n} + a_{1}x^{n-1} + \\dots + a_{n} = 0", "name": null, "statement": "An irrational number that is a root of a polynomial equation with integer coefficients is called an algebraical number; all other irrational numbers are transcendental (Miscellaneous Example 32, the defining equation).", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a_0, a_1, ..., a_n", "meaning": "integers (coefficients)" }, { "unit": null, "symbol": "x", "meaning": "the irrational number being classed as algebraical" }, { "unit": null, "symbol": "n", "meaning": "degree of the equation" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/algebraical-number", "concept/coefficient", "concept/equation", "concept/integer", "concept/irrational-number", "concept/transcendental-number", "unit/degree-of-angle" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-784473b83c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "42", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "Ax + By + C = 0", "name": "general equation of the first degree", "statement": "A straight line in the (x, y) plane is the locus of all points whose coordinates satisfy this linear equation, with A, B, C fixed numbers.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "A", "meaning": "fixed number (coefficient of x)" }, { "unit": null, "symbol": "B", "meaning": "fixed number (coefficient of y)" }, { "unit": null, "symbol": "C", "meaning": "fixed number (constant term)" }, { "unit": null, "symbol": "x", "meaning": "abscissa, distance measured along OX" }, { "unit": null, "symbol": "y", "meaning": "ordinate, distance measured along OY" } ], "sympy": "Eq(A*x + B*y + C, 0)", "physics": false, "states": [ "concept/linear-equation" ], "concepts": [ "concept/cartesian-coordinates", "concept/coefficient", "concept/line", "concept/locus", "method/euclidean-construction" ], "pages": [ "42", "67" ], "chapters": [ "hardy-course-of-pure-mathematics-1921/ch-ii" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-76d3aa6695", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "42", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "(x - \\alpha)^{2} + (y - \\beta)^{2} = \\rho^{2}", "name": null, "statement": "A circle with centre (alpha, beta) and radius rho is the locus of points satisfying this equation.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "alpha", "meaning": "x-coordinate of the centre of the circle" }, { "unit": null, "symbol": "beta", "meaning": "y-coordinate of the centre of the circle" }, { "unit": null, "symbol": "rho", "meaning": "radius of the circle" } ], "sympy": "Eq((x - alpha)**2 + (y - beta)**2, rho**2)", "physics": false, "states": [], "concepts": [ "concept/cartesian-coordinates", "concept/circle", "concept/locus" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d0e17f1921", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "43", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "x^{2} + y^{2} + 2Gx + 2Fy + C = 0", "name": null, "statement": "A circle, provided G^2 + F^2 - C > 0, is represented by this equation in general form.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "G", "meaning": "fixed number (coefficient of 2x)" }, { "unit": null, "symbol": "F", "meaning": "fixed number (coefficient of 2y)" }, { "unit": null, "symbol": "C", "meaning": "fixed number (constant term)" } ], "sympy": "Eq(x**2 + y**2 + 2*G*x + 2*F*y + C, 0)", "physics": false, "states": [], "concepts": [ "concept/cartesian-coordinates", "concept/circle", "concept/inequality", "concept/locus" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-de413e8886", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "43", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "Ax^{2} + 2Hxy + By^{2} + 2Gx + 2Fy + C = 0", "name": "general equation of the second degree", "statement": "This general second-degree equation represents, under certain inequalities on the coefficients, a conic section (ellipse, parabola or hyperbola).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "A", "meaning": "fixed number (coefficient of x^2)" }, { "unit": null, "symbol": "H", "meaning": "fixed number (coefficient of xy)" }, { "unit": null, "symbol": "B", "meaning": "fixed number (coefficient of y^2)" }, { "unit": null, "symbol": "G", "meaning": "fixed number (coefficient of 2x)" }, { "unit": null, "symbol": "F", "meaning": "fixed number (coefficient of 2y)" }, { "unit": null, "symbol": "C", "meaning": "fixed number (constant term)" } ], "sympy": "Eq(A*x**2 + 2*H*x*y + B*y**2 + 2*G*x + 2*F*y + C, 0)", "physics": false, "states": [ "concept/conic-section" ], "concepts": [ "concept/ellipse", "concept/hyperbola", "concept/locus", "concept/parabola" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-17269f34a7", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "43", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "x = r\\cos\\theta", "name": null, "statement": "The Cartesian abscissa of a point equals its polar distance r times the cosine of the polar angle theta.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "abscissa, OM" }, { "unit": null, "symbol": "r", "meaning": "polar coordinate OP, essentially positive" }, { "unit": "radian", "symbol": "theta", "meaning": "polar angle MOP, between 0 and 2pi measured positively" } ], "sympy": "Eq(x, r*cos(theta))", "physics": false, "states": [], "concepts": [ "concept/cartesian-coordinates", "concept/complex-number", "concept/cosine", "concept/polar-coordinates", "quantity/amplitude-of-a-complex-number", "quantity/modulus-of-a-complex-number", "quantity/real-part-of-a-complex-number" ], "pages": [ "43", "84" ], "chapters": [ "hardy-course-of-pure-mathematics-1921/ch-ii", "hardy-course-of-pure-mathematics-1921/ch-iii" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-8028194b49", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "43", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "y = r\\sin\\theta", "name": null, "statement": "The Cartesian ordinate of a point equals its polar distance r times the sine of the polar angle theta.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "y", "meaning": "ordinate, MP" }, { "unit": null, "symbol": "r", "meaning": "polar coordinate OP, essentially positive" }, { "unit": "radian", "symbol": "theta", "meaning": "polar angle MOP, between 0 and 2pi measured positively" } ], "sympy": "Eq(y, r*sin(theta))", "physics": false, "states": [], "concepts": [ "concept/cartesian-coordinates", "concept/complex-number", "concept/polar-coordinates", "concept/sine", "quantity/amplitude-of-a-complex-number", "quantity/imaginary-part-of-a-complex-number", "quantity/modulus-of-a-complex-number" ], "pages": [ "43", "84" ], "chapters": [ "hardy-course-of-pure-mathematics-1921/ch-ii", "hardy-course-of-pure-mathematics-1921/ch-iii" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-60c6b3a6d5", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "43", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "r = \\sqrtp{x^{2} + y^{2}}", "name": null, "statement": "The polar distance r of a point is the square root of the sum of the squares of its Cartesian coordinates.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "r", "meaning": "polar coordinate OP, essentially positive" }, { "unit": null, "symbol": "x", "meaning": "abscissa" }, { "unit": null, "symbol": "y", "meaning": "ordinate" } ], "sympy": "Eq(r, sqrt(x**2 + y**2))", "physics": false, "states": [], "concepts": [ "concept/cartesian-coordinates", "concept/polar-coordinates", "concept/root", "person/pythagoras", "quantity/imaginary-part-of-a-complex-number", "quantity/modulus-of-a-complex-number", "quantity/real-part-of-a-complex-number" ], "pages": [ "43", "84" ], "chapters": [ "hardy-course-of-pure-mathematics-1921/ch-ii", "hardy-course-of-pure-mathematics-1921/ch-iii" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-7fcb54123a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "43", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "r\\cos(\\theta - \\alpha) = p", "name": null, "statement": "The polar equation of a straight line, with p and alpha constants.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "p", "meaning": "constant (perpendicular distance of the line from the pole)" }, { "unit": null, "symbol": "alpha", "meaning": "constant (polar angle of the perpendicular)" } ], "sympy": "Eq(r*cos(theta - alpha), p)", "physics": false, "states": [], "concepts": [ "concept/constant", "concept/cosine", "concept/line", "concept/polar-coordinates" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-8fa23504a1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "43", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "r = 2a\\cos\\theta", "name": null, "statement": "This polar equation represents a circle passing through the origin.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "a", "meaning": "constant (radius-like parameter)" } ], "sympy": "Eq(r, 2*a*cos(theta))", "physics": false, "states": [], "concepts": [ "concept/circle", "concept/cosine", "concept/origin", "concept/polar-coordinates" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a76400b775", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "43", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "r^{2} + c^{2} - 2rc\\cos(\\theta - \\alpha) = A^{2}", "name": null, "statement": "The general polar equation of a circle, with A, c and alpha constants.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "A", "meaning": "constant (radius of the circle)" }, { "unit": null, "symbol": "c", "meaning": "constant (distance of the centre from the pole)" }, { "unit": null, "symbol": "alpha", "meaning": "constant (polar angle of the centre)" } ], "sympy": "Eq(r**2 + c**2 - 2*r*c*cos(theta - alpha), A**2)", "physics": false, "states": [], "concepts": [ "concept/circle", "concept/constant", "concept/cosine", "concept/polar-coordinates" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-02f4b9b50c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "43", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "l/r = 1 - e\\cos\\theta", "name": null, "statement": "With r positive, this polar equation (l > 0, e > 1) represents only one branch of a hyperbola; the other branch is given by -l/r = 1 - e cos theta.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "l", "meaning": "positive constant (semi-latus rectum)" }, { "unit": null, "symbol": "e", "meaning": "constant greater than 1 (eccentricity)" } ], "sympy": "Eq(l/r, 1 - e*cos(theta))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/curve", "concept/hyperbola", "concept/polar-coordinates" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-817c6600eb", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "43", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "-l/r = 1 - e\\cos\\theta", "name": null, "statement": "With negative values of r allowed, this polar equation gives the other branch of the hyperbola, the two branches together forming the whole hyperbola.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "l", "meaning": "positive constant (semi-latus rectum)" }, { "unit": null, "symbol": "e", "meaning": "constant greater than 1 (eccentricity)" } ], "sympy": "Eq(-l/r, 1 - e*cos(theta))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/curve", "concept/hyperbola", "concept/polar-coordinates" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-abda9c0403", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "44", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "a_{0}x^{m} + a_{1}x^{m-1} + \\dots + a_{m}", "name": null, "statement": "The general form of a polynomial in x, with constant coefficients a_0 to a_m.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a_0, ..., a_m", "meaning": "constants (coefficients of the polynomial)" }, { "unit": null, "symbol": "m", "meaning": "degree of the polynomial" }, { "unit": null, "symbol": "x", "meaning": "independent variable" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/constant", "concept/polynomial", "concept/variable", "unit/degree-of-angle" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-54adebf06c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "47", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "R(x) = \\frac{P(x)}{Q(x)}", "name": null, "statement": "A rational function is the quotient of one polynomial by another.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "R(x)", "meaning": "rational function of x" }, { "unit": null, "symbol": "P(x)", "meaning": "polynomial in x (numerator)" }, { "unit": null, "symbol": "Q(x)", "meaning": "polynomial in x (denominator)" } ], "sympy": "Eq(R, P/Q)", "physics": false, "states": [], "concepts": [ "concept/denominator", "concept/function-notation", "concept/numerator", "concept/polynomial", "concept/quotient", "concept/rational-function" ], "pages": [ "47", "209" ], "chapters": [ "hardy-course-of-pure-mathematics-1921/ch-ii", "hardy-course-of-pure-mathematics-1921/ch-vi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-2230f48d01", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "41", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "y = f(x)", "name": null, "statement": "y is a function of x: the dependent variable y is determined by the independent variable x through the rule f (other letters such as F, phi, psi may be used).", "kind": "definition", "symbols": [ { "unit": null, "symbol": "y", "meaning": "dependent variable" }, { "unit": null, "symbol": "x", "meaning": "independent variable" }, { "unit": null, "symbol": "f", "meaning": "rule (function) relating y to x" } ], "sympy": "Eq(y, f(x))", "physics": false, "states": [], "concepts": [ "concept/function", "concept/function-notation", "concept/variable" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-2d3137ab63", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "46", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "y - \\{(ac - b^{2})/a\\} = a\\{x + (b/a)\\}^{2}", "name": null, "statement": "Completing the square shows that the graph of ax^2 + 2bx + c is a parabola, with new axes through the point x = -b/a, y = (ac - b^2)/a.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "constant (coefficient of x^2)" }, { "unit": null, "symbol": "b", "meaning": "constant (half the coefficient of x)" }, { "unit": null, "symbol": "c", "meaning": "constant (constant term)" }, { "unit": null, "symbol": "x", "meaning": "abscissa" }, { "unit": null, "symbol": "y", "meaning": "ordinate" } ], "sympy": "Eq(y - (a*c - b**2)/a, a*(x + b/a)**2)", "physics": false, "states": [], "concepts": [ "concept/cartesian-coordinates", "concept/parabola", "concept/quadratic-equation", "method/completing-the-square" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b450dfe11a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "45", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "(-x)^{m} = x^{m}", "name": null, "statement": "For even m the function x^m is symmetrical about the axis OY, since (-x)^m equals x^m.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "m", "meaning": "even integer (power)" }, { "unit": null, "symbol": "x", "meaning": "real variable" } ], "sympy": "Eq((-x)**m, x**m)", "physics": false, "states": [], "concepts": [ "concept/even-function", "concept/polynomial", "concept/power", "concept/symmetry" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-be2f5bde5f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "45", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "(-x)^{m} = -x^{m}", "name": null, "statement": "For odd m, (-x)^m equals -x^m, so y = x^m is negative when x is negative.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "m", "meaning": "odd integer (power)" }, { "unit": null, "symbol": "x", "meaning": "real variable" } ], "sympy": "Eq((-x)**m, -x**m)", "physics": false, "states": [], "concepts": [ "concept/odd-function", "concept/polynomial", "concept/power" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-2477242342", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "39", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "W = Ap_{0}", "name": null, "statement": "The weight W of the piston is balanced by the gas pressure p_0 acting over the piston's cross-sectional area A in equilibrium.", "kind": "law", "symbols": [ { "unit": "weight unit (not specified)", "symbol": "W", "meaning": "weight of the piston" }, { "unit": null, "symbol": "A", "meaning": "area of the cross section of the piston" }, { "unit": "pressure per unit area (not specified)", "symbol": "p_0", "meaning": "pressure of the gas per unit area on the piston in equilibrium" } ], "sympy": "Eq(W, A*p0)", "physics": true, "states": [], "concepts": [ "concept/area", "concept/cross-section", "concept/mechanical-equilibrium", "concept/pressure", "concept/weight" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-4537d415bf", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "39", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "pv = a", "name": "Boyle's law", "statement": "Boyle's experimental law, to approximation: at constant temperature the product of pressure p and volume v is very nearly constant, equal to a number a fixed by experiment; it holds only for moderate compression.", "kind": "law", "symbols": [ { "unit": null, "symbol": "p", "meaning": "pressure exerted by the gas per unit area of the piston" }, { "unit": null, "symbol": "v", "meaning": "volume of the gas" }, { "unit": null, "symbol": "a", "meaning": "number determined approximately by experiment" } ], "sympy": "Eq(p*v, a)", "physics": true, "states": [ "law/boyle-s-law" ], "concepts": [ "concept/approximation", "concept/constant", "concept/pressure", "quantity/volume" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d2b6447cd4", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "40", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "\\left(p + \\frac{\\alpha}{v^{2}}\\right)(v - \\beta) = \\gamma", "name": "van der Waals' law", "statement": "A better approximation to the relation between pressure and volume of a gas under strong compression, with alpha, beta, gamma constants determined by experiment.", "kind": "law", "symbols": [ { "unit": null, "symbol": "p", "meaning": "pressure exerted by the gas per unit area of the piston" }, { "unit": null, "symbol": "v", "meaning": "volume of the gas" }, { "unit": null, "symbol": "alpha", "meaning": "constant determined approximately by experiment" }, { "unit": null, "symbol": "beta", "meaning": "constant determined approximately by experiment" }, { "unit": null, "symbol": "gamma", "meaning": "constant determined approximately by experiment" } ], "sympy": "Eq((p + alpha/v**2)*(v - beta), gamma)", "physics": true, "states": [ "concept/van-der-waals-equation" ], "concepts": [ "concept/approximation", "concept/constant", "concept/pressure", "quantity/volume" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-f82c697c2a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "40", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "h = \\tfrac{1}{2}g(2n\\tau - t)^{2}", "name": null, "statement": "The depth of the bouncing ball below its original position at time t, for (2n-1)tau <= t <= (2n+1)tau, from the elementary formulae of dynamics for an elastic ball dropped from height (1/2) g tau^2.", "kind": "formula", "symbols": [ { "unit": "length (not specified)", "symbol": "h", "meaning": "depth of the ball below its original position at time t" }, { "unit": null, "symbol": "g", "meaning": "acceleration due to gravity" }, { "unit": "time (not specified)", "symbol": "tau", "meaning": "time for the first fall from the initial height" }, { "unit": null, "symbol": "t", "meaning": "time" }, { "unit": null, "symbol": "n", "meaning": "integer index of the bounce interval" } ], "sympy": "Eq(h, Rational(1, 2)*g*(2*n*tau - t)**2)", "physics": true, "states": [], "concepts": [ "concept/dimension", "concept/dynamics", "concept/function", "concept/rebound", "quantity/time" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-3c39bc4ad8", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "50", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "y = \\frac{\\sqrtp{1 + x} - \\sqrtp[3]{1 - x}} {\\sqrtp{1 + x} + \\sqrtp[3]{1 - x}}", "name": null, "statement": "The stated ratio of a root expression in x to another root expression defines y as an example of an algebraic function.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y", "meaning": "the function of x defined by the ratio" }, { "unit": null, "symbol": "x", "meaning": "independent variable" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/algebraic-expression", "concept/algebraic-function", "concept/function", "concept/variable" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-6612408ed2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "50", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "\\left(\\frac{1 + y}{1 - y}\\right)^{6} = \\frac{(1 + x)^{3}}{(1 - x)^{2}}", "name": null, "statement": "The example function y satisfies an algebraic equation with coefficients rational in x.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y", "meaning": "the example algebraic function of x" }, { "unit": null, "symbol": "x", "meaning": "independent variable" } ], "sympy": "Eq(((1 + y)/(1 - y))**6, (1 + x)**3/(1 - x)**2)", "physics": false, "states": [], "concepts": [ "concept/algebraic-function", "concept/equation", "concept/rational-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-1ee0d81ec5", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "50", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "y = \\sqrt{x} + \\sqrtp{x + \\sqrt{x}}", "name": null, "statement": "The example function y is the sum of a square root of x and a root of x plus its square root.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "y", "meaning": "the example algebraic function of x" }, { "unit": null, "symbol": "x", "meaning": "independent variable" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/algebraic-expression", "concept/algebraic-function", "concept/root" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-f2fd082c65", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "50", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "y^{4} - (4y^{2} + 4y + 1)x = 0", "name": null, "statement": "The example function y of the preceding formula is a root of a quartic equation whose coefficients are rational in x.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y", "meaning": "the example algebraic function of x" }, { "unit": null, "symbol": "x", "meaning": "independent variable" } ], "sympy": "Eq(y**4 - (4*y**2 + 4*y + 1)*x, 0)", "physics": false, "states": [], "concepts": [ "concept/algebraic-function", "concept/equation", "concept/rational-function", "unit/degree-of-angle" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e37735b278", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "50", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "y^{m} + R_{1}y^{m-1} + \\dots + R_{m} = 0", "name": null, "statement": "The general form of an equation of degree m in y whose coefficients are rational functions of x, whose root defines an algebraical function.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "y", "meaning": "the dependent variable, the algebraical function of x" }, { "unit": null, "symbol": "m", "meaning": "the degree of the equation in y" }, { "unit": null, "symbol": "R_{1}, ..., R_{m}", "meaning": "rational functions of x (the coefficients)" }, { "unit": null, "symbol": "x", "meaning": "independent variable" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/algebraic-function", "concept/coefficient", "concept/equation", "concept/rational-function", "concept/variable", "unit/degree-of-angle" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-74b6524873", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "51", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "y = \\tfrac{1}{2}\\{-R_{1} ± \\sqrtp{R_{1}^{2} - 4R_{2}}\\}", "name": "quadratic formula", "statement": "For a quadratic equation in y with coefficients R_1 and R_2, y is half the negative of R_1 plus or minus the square root of R_1 squared minus 4R_2.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "y", "meaning": "the unknown root of the quadratic in y" }, { "unit": null, "symbol": "R_{1}", "meaning": "coefficient of y (rational function of x)" }, { "unit": null, "symbol": "R_{2}", "meaning": "constant term (rational function of x)" } ], "sympy": null, "physics": false, "states": [ "method/quadratic-formula" ], "concepts": [ "concept/algebraic-function", "concept/coefficient", "concept/quadratic-equation", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-44f1beed63", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "58", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "f(x) = \\phi(x)", "name": null, "statement": "Many equations can be written in this form, and their roots are the abscissae where the graphs of f and phi meet.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "f", "meaning": "a function whose graph is easy to draw" }, { "unit": null, "symbol": "phi", "meaning": "a second function whose graph is easy to draw" }, { "unit": null, "symbol": "x", "meaning": "the unknown, the abscissa of an intersection point" } ], "sympy": "Eq(f(x), phi(x))", "physics": false, "states": [], "concepts": [ "concept/abscissa", "concept/function", "concept/graph-of-a-function", "concept/root-of-an-equation", "method/graphical-solution-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-9ce4847511", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "60", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "f(x, y) = 0", "name": null, "statement": "The standard form in which a plane curve is given: a relation between x and y expressed by equating a function of two variables to zero.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "f(x, y)", "meaning": "a function of the two variables x and y" }, { "unit": null, "symbol": "x", "meaning": "abscissa, first coordinate" }, { "unit": null, "symbol": "y", "meaning": "ordinate, second coordinate" } ], "sympy": "Eq(f(x, y), 0)", "physics": false, "states": [], "concepts": [ "concept/cartesian-coordinates", "concept/curve", "concept/equation", "concept/function", "concept/implicit-function", "concept/locus" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-50cad42c65", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "60", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "x = f(t)", "name": null, "statement": "A curve is described by giving x and y each as a function of an auxiliary variable t (the x-equation of the pair).", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x", "meaning": "abscissa of a point on the curve" }, { "unit": null, "symbol": "t", "meaning": "auxiliary variable with no geometrical significance" }, { "unit": null, "symbol": "f", "meaning": "function giving x in terms of t" } ], "sympy": "Eq(x, f(t))", "physics": false, "states": [], "concepts": [ "concept/curve", "concept/function", "concept/locus", "concept/variable" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-45d8b989b6", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "61", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "x = a\\cos t", "name": null, "statement": "With y = a sin t, as t varies from 0 to 2 pi the point (x, y) describes a circle with centre at the origin and radius a.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "abscissa of the point" }, { "unit": null, "symbol": "a", "meaning": "radius of the circle" }, { "unit": null, "symbol": "t", "meaning": "auxiliary variable (angle)" } ], "sympy": "Eq(x, a*cos(t))", "physics": false, "states": [], "concepts": [ "concept/cartesian-coordinates", "concept/centre-of-a-circle", "concept/circle", "concept/curve", "concept/radius" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d896ff8fc8", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "61", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "y = a\\sin t", "name": null, "statement": "The second parametric equation of the circle of radius a centred at the origin, paired with x = a cos t.", "kind": "result", "symbols": [ { 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z**2 + 2*F*x + 2*G*y + 2*H*z + C, 0)", "physics": false, "states": [], "concepts": [ "concept/cartesian-coordinates", "concept/equation", "concept/locus", "concept/sphere" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-5f3ca93a44", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "59", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "F^{2} + G^{2} + H^{2} - C > 0", "name": null, "statement": "The condition on the constants that makes the general quadratic equation in three variables represent a sphere.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "F, G, H, C", "meaning": "constants of the sphere equation" } ], "sympy": "Gt(F**2 + G**2 + H**2 - C, 0)", "physics": false, "states": [], "concepts": [ "concept/inequality", "concept/radius", "concept/sphere" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-4b86569ba4", "chapter": 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1921", "page": "60", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "x^{2} + y^{2} + 2Gx + 2Fy+ C = 0", "name": null, "statement": "The general equation of a circle in the plane, from which y can be solved explicitly as a function of x but is better kept in this implicit form.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x, y", "meaning": "cartesian coordinates of a point" }, { "unit": null, "symbol": "G, F, C", "meaning": "constants of the circle equation" } ], "sympy": "Eq(x**2 + y**2 + 2*G*x + 2*F*y + C, 0)", "physics": false, "states": [], "concepts": [ "concept/cartesian-coordinates", "concept/circle", "concept/curve", "concept/equation", "concept/implicit-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d4ca52196d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "60", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "y = -F + \\sqrtp{F^{2} - x^{2} - 2Gx - C}", "name": null, "statement": "Solving the circle equation for y gives y as an explicit function of x.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y", "meaning": "ordinate, explicit function of x" }, { "unit": null, "symbol": "x", "meaning": "abscissa" }, { "unit": null, "symbol": "F, G, C", "meaning": "constants of the circle equation" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/algebraic-function", "concept/circle", "concept/explicit-function", "concept/function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c7f06c7575", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "60", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "x^{5} + y^{5} - ay = 0", "name": null, "statement": "An example of a curve whose equation cannot be solved explicitly for y as an algebraical 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"concept/cartesian-coordinates", "concept/circle", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-0e632d93da", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "65", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "x^{4} + nx^{3} + px^{2} + qx + r = 0", "name": null, "statement": "The quartic equation whose roots are the abscissae of the intersections of a parabola and a circle.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable (root)" }, { "unit": null, "symbol": "n", "meaning": "coefficient" }, { "unit": null, "symbol": "p", "meaning": "coefficient" }, { "unit": null, "symbol": "q", "meaning": "coefficient" }, { "unit": null, "symbol": "r", "meaning": "coefficient" } ], "sympy": "Eq(x**4 + n*x**3 + p*x**2 + q*x + r, 0)", "physics": false, "states": [], "concepts": [ "concept/equation", "concept/polynomial", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-6533b30b96", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "65", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "x^{2} = y - \\frac{1}{2}nx", "name": null, "statement": "The parabola used with the circle in Item 9 to find the roots of the quartic.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "abscissa" }, { "unit": null, "symbol": "y", "meaning": "ordinate" }, { "unit": null, "symbol": "n", "meaning": "coefficient" } ], "sympy": "Eq(x**2, y - n*x/2)", "physics": false, "states": [], "concepts": [ "concept/cartesian-coordinates", "concept/parabola" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-dc184fa7c3", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", 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"hardy-course-of-pure-mathematics-1921/eq-378b9fd1a7", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "65", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "x^{m} + ax^{2} + bx + c = 0", "name": null, "statement": "The equation whose roots are found graphically as intersections of y = x^m with y = -ax^2 - bx - c.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable (root)" }, { "unit": null, "symbol": "m", "meaning": "exponent" }, { "unit": null, "symbol": "a", "meaning": "coefficient" }, { "unit": null, "symbol": "b", "meaning": "coefficient" }, { "unit": null, "symbol": "c", "meaning": "coefficient" } ], "sympy": "Eq(x**m + a*x**2 + b*x + c, 0)", "physics": false, "states": [], "concepts": [ "concept/equation", "concept/root-of-an-equation", "method/graphical-solution-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-cabd31f31f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "65", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "y = x^{m}", "name": null, "statement": "One of the two curves used for the graphical solution of x^m + ax^2 + bx + c = 0.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "y", "meaning": "ordinate" }, { "unit": null, "symbol": "x", "meaning": "abscissa" }, { "unit": null, "symbol": "m", "meaning": "exponent" } ], "sympy": "Eq(y, x**m)", "physics": false, "states": [], "concepts": [ "concept/cartesian-coordinates", "concept/graph-of-a-function", "concept/power" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-9eeedd6c6e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "65", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "y = -ax^{2} - bx - c", "name": null, "statement": "The second curve used for the graphical solution of x^m + ax^2 + bx + c = 0.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "y", "meaning": "ordinate" }, { "unit": null, "symbol": "x", "meaning": "abscissa" }, { "unit": null, "symbol": "a", "meaning": "coefficient" }, { "unit": null, "symbol": "b", "meaning": "coefficient" }, { "unit": null, "symbol": "c", "meaning": "coefficient" } ], "sympy": "Eq(y, -a*x**2 - b*x - c)", "physics": false, "states": [], "concepts": [ "concept/cartesian-coordinates", "concept/graph-of-a-function", "concept/polynomial" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-3260c9af14", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "66", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "2x = (2n + 1)\\pi(1 - \\cos x)", "name": null, "statement": "The equation, with n a positive integer, that is shown to have 2n + 3 roots.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable (root)" }, { "unit": null, "symbol": "n", "meaning": "positive integer" } ], "sympy": "Eq(2*x, (2*n + 1)*pi*(1 - cos(x)))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/equation", "concept/integer", "concept/root-of-an-equation", "quantity/pi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-071bea3fcb", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "66", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "\\frac{2}{3}x\\sin x = 1", "name": null, "statement": "The equation stated to have four roots between -pi and pi.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable (root)" } ], "sympy": "Eq(Rational(2,3)*x*sin(x), 1)", "physics": false, "states": [], "concepts": [ "concept/equation", "concept/root-of-an-equation", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a58d711d19", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "66", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "\\cot x + x - \\frac{3}{2}\\pi = 0", "name": null, "statement": "Equation (1) of Item 14, whose number and values of roots are to be discussed.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable (root)" } ], "sympy": "Eq(cot(x) + x - Rational(3,2)*pi, 0)", "physics": false, "states": [], "concepts": [ "concept/cotangent", "concept/equation", "concept/root-of-an-equation", "quantity/pi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-4c7228ac31", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "66", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "x^{2} + \\sin^{2} x = 1", "name": null, "statement": "Equation (2) of Item 14, whose roots are to be discussed.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable (root)" } ], "sympy": "Eq(x**2 + sin(x)**2, 1)", "physics": false, "states": [], "concepts": [ "concept/equation", "concept/root-of-an-equation", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-582b07c65b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "66", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "\\tan x = 2x/(1 + x^{2})", "name": null, "statement": "Equation (3) of Item 14, whose roots are to be discussed.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable (root)" } ], "sympy": "Eq(tan(x), 2*x/(1 + x**2))", "physics": false, "states": [], "concepts": [ "concept/equation", "concept/root-of-an-equation", "concept/tangent-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-82f488c8fd", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "66", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "\\sin x - x + \\frac{1}{6}x^{3} = 0", "name": null, "statement": "Equation (4) of Item 14, whose roots are to be discussed.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable (root)" } ], "sympy": "Eq(sin(x) - x + Rational(1,6)*x**3, 0)", "physics": false, "states": [], "concepts": [ "concept/equation", "concept/root-of-an-equation", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-1050dbc1f4", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "66", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "(1 - \\cos x)\\tan\\alpha - x + \\sin x = 0", "name": null, "statement": "Equation (5) of Item 14, whose roots are to be discussed, with alpha as a parameter.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable (root)" }, { "unit": null, "symbol": "alpha", "meaning": "parameter angle" } ], "sympy": "Eq((1 - cos(x))*tan(alpha) - x + sin(x), 0)", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/plane-angle", "concept/root-of-an-equation", "concept/sine", "concept/tangent-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-0d08286ed7", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "66", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "\\alpha\\frac{(x - b)(x - c)}{(a - b)(a - c)} + \\beta \\frac{(x - c)(x - a)}{(b - c)(b - a)} + \\gamma\\frac{(x - a)(x - b)}{(c - a)(c - b)}", "name": null, "statement": "The second-degree polynomial taking the values alpha, beta, gamma at x = a, b, c (Lagrange-type interpolation formula).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable" }, { "unit": null, "symbol": "a", "meaning": "point of interpolation" }, { "unit": null, "symbol": "b", "meaning": "point of interpolation" }, { "unit": null, "symbol": "c", "meaning": "point of interpolation" }, { "unit": null, "symbol": "alpha", "meaning": "value of the polynomial at x = a" }, { "unit": null, "symbol": "beta", "meaning": "value of the polynomial at x = b" }, { "unit": null, "symbol": "gamma", "meaning": "value of the polynomial at x = c" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/function", "concept/polynomial", "concept/value-of-a-function", "unit/degree-of-angle" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e51d8f7636", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "66", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "Axy + Bx + Cy + D = 0", "name": null, "statement": "If x is a rational function of y and y a rational function of x, then x and y satisfy this bilinear relation.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable" }, { "unit": null, "symbol": "y", "meaning": "variable" }, { "unit": null, "symbol": "A", "meaning": "coefficient" }, { "unit": null, "symbol": "B", "meaning": "coefficient" }, { "unit": null, "symbol": "C", "meaning": "coefficient" }, { "unit": null, "symbol": "D", "meaning": "coefficient" } ], "sympy": "Eq(A*x*y + B*x + C*y + D, 0)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/equation", "concept/rational-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-db2303e548", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "66", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "\\cos\\tfrac{1}{2}\\pi x = 1 - \\frac{x^{2}}{x + (x - 1)\\bigsqrtp{\\dfrac{2 - x}{3}}}", "name": null, "statement": "A formula stated to be approximately true for all x between 0 and 1 (checked numerically at seven points).", "kind": "approximation", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable between 0 and 1" } ], "sympy": "Eq(cos(pi*x/2), 1 - x**2/(x + (x - 1)*sqrt((2 - x)/3)))", "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/cosine", "concept/function", "quantity/pi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-113e30761d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "66", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "z = [x] + [y]", "name": null, "statement": "A function whose graph is to be described; the integer part of x and y are added.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "z", "meaning": "value of the function" }, { "unit": null, "symbol": "x", "meaning": "variable" }, { "unit": null, "symbol": "y", "meaning": "variable" } ], "sympy": "Eq(z, floor(x) + floor(y))", "physics": false, "states": [], "concepts": [ "concept/function", "concept/graph-of-a-function", "concept/integer-part-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-37ca9f0534", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "66", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "z = x + y - [x] - [y]", "name": null, "statement": "A second function whose graph is to be described; it equals x + y minus the integer parts of x and y.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "z", "meaning": "value of the function" }, { "unit": null, "symbol": "x", "meaning": "variable" }, { "unit": null, "symbol": "y", "meaning": "variable" } ], "sympy": "Eq(z, x + y - floor(x) - floor(y))", "physics": false, "states": [], "concepts": [ "concept/function", "concept/graph-of-a-function", "concept/integer-part-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-84976b1937", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "66", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "z = \\sin x + \\sin y", "name": null, "statement": "One of four functions whose graph form is asked for in Item 21.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "z", "meaning": "value of the function" }, { "unit": null, "symbol": "x", "meaning": "variable" }, { "unit": null, "symbol": "y", "meaning": "variable" } ], "sympy": "Eq(z, sin(x) + sin(y))", "physics": false, "states": [], "concepts": [ "concept/function", "concept/graph-of-a-function", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a6f558ff3d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "66", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "z = \\sin x\\sin y", "name": null, "statement": "One of four functions whose graph form is asked for in Item 21.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "z", "meaning": "value of the function" }, { "unit": null, "symbol": "x", "meaning": "variable" }, { "unit": null, "symbol": "y", "meaning": "variable" } ], "sympy": "Eq(z, sin(x)*sin(y))", "physics": false, "states": [], "concepts": [ "concept/function", "concept/graph-of-a-function", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-271eb4c1ae", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "66", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "z = \\sin xy", "name": null, "statement": "One of four functions whose graph form is asked for in Item 21.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "z", "meaning": "value of the function" }, { "unit": null, "symbol": "x", "meaning": "variable" }, { "unit": null, "symbol": "y", "meaning": "variable" } ], "sympy": "Eq(z, sin(x*y))", "physics": false, "states": [], "concepts": [ "concept/function", "concept/graph-of-a-function", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-2f45892ce9", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "66", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "z = \\sin(x^{2} + y^{2})", "name": null, "statement": "One of four functions whose graph form is asked for in Item 21.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "z", "meaning": "value of the function" }, { "unit": null, "symbol": "x", "meaning": "variable" }, { "unit": null, "symbol": "y", "meaning": "variable" } ], "sympy": "Eq(z, sin(x**2 + y**2))", "physics": false, "states": [], "concepts": [ "concept/function", "concept/graph-of-a-function", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-5474628d33", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "67", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "(x - \\alpha)^{2} + (y - \\beta)^{2} = \\rho ^{2}", "name": null, "statement": "A circle of centre (alpha, beta) and radius rho, constructible when alpha, beta, rho are rational.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "cartesian coordinate" }, { "unit": null, "symbol": "y", "meaning": "cartesian coordinate" }, { "unit": null, "symbol": "alpha", "meaning": "x-coordinate of centre" }, { "unit": null, "symbol": "beta", "meaning": "y-coordinate of centre" }, { "unit": null, "symbol": "rho", "meaning": "radius" } ], "sympy": "Eq((x - alpha)**2 + (y - beta)**2, rho**2)", "physics": false, "states": [], "concepts": [ "concept/cartesian-coordinates", "concept/centre-of-a-circle", "concept/circle", "concept/radius" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-df5488c548", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "67", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "x^{2} + y^{2} + 2gx + 2fy + c = 0", "name": null, "statement": "The general circle equation; its coefficients g, f, c are rational when the centre and radius are rational.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "cartesian coordinate" }, { "unit": null, "symbol": "y", "meaning": "cartesian coordinate" }, { "unit": null, "symbol": "g", "meaning": "coefficient" }, { "unit": null, "symbol": "f", "meaning": "coefficient" }, { "unit": null, "symbol": "c", "meaning": "constant term" } ], "sympy": "Eq(x**2 + y**2 + 2*g*x + 2*f*y + c, 0)", "physics": false, "states": [], "concepts": [ "concept/cartesian-coordinates", "concept/circle", "concept/coefficient" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-eba7197e71", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "67", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "x^{2} - 34x + 190 = 0", "name": null, "statement": "The quadratic whose roots are the two mixed surds (17 + 3 sqrt(11)) and (17 - 3 sqrt(11)) of the example.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable (root)" } ], "sympy": "Eq(x**2 - 34*x + 190, 0)", "physics": false, "states": [], "concepts": [ "concept/quadratic-equation", "concept/quadratic-surd", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-acd8c8e842", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "68", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "AM/R = \\tfrac{13}{25}\\sqrt{146}", "name": null, "statement": "The length AM, divided by the circle radius R, equals 13/25 times the square root of 146, which approximates pi/4-type circumference comparison.", "kind": "result", "symbols": [ { "unit": "unit length", "symbol": "AM", "meaning": "length of the constructed segment AM" }, { "unit": "unit length", "symbol": "R", "meaning": "radius of the circle" } ], "sympy": "Eq(AM/R, Rational(13,25)*sqrt(146))", "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/quadratic-surd", "concept/radius", "quantity/circumference", "quantity/pi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-6211e961ac", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "68", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "y^{2} = 4x", "name": null, "statement": "The parabola with vertex O and focus S used in the construction of cube root of 2.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "y", "meaning": "ordinate" }, { "unit": null, "symbol": "x", "meaning": "abscissa" } ], "sympy": "Eq(y**2, 4*x)", "physics": false, "states": [], "concepts": [ "concept/cartesian-coordinates", "concept/focus", "concept/parabola", "concept/vertex-of-a-conic-section" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-2470c718f3", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "68", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "x^{2} = 2y", "name": null, "statement": "The second parabola, meeting the first at P, used in the construction of cube root of 2.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "abscissa" }, { "unit": null, "symbol": "y", "meaning": "ordinate" } ], "sympy": "Eq(x**2, 2*y)", "physics": false, "states": [], "concepts": [ "concept/cartesian-coordinates", "concept/parabola" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ac6c1fd12c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "68", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "SQ = \\sqrt[3]{2}", "name": null, "statement": "The length SQ equals the cube root of 2, so this construction gives the duplication of the cube.", "kind": "result", "symbols": [ { "unit": "unit length", "symbol": "SQ", "meaning": "length from focus S to point Q" } ], "sympy": "Eq(SQ, 2**Rational(1,3))", "physics": false, "states": [], "concepts": [ "concept/duplication-of-the-cube", "concept/root", "method/euclidean-construction" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-baf19c7993", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "68", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "(x^{2} + y^{2})x - y^{2} = 0", "name": "Cissoid of Diocles", "statement": "The locus of M, with O as origin and OA as x-axis, is the Cissoid of Diocles.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "cartesian coordinate" }, { "unit": null, "symbol": "y", "meaning": "cartesian coordinate" } ], "sympy": "Eq((x**2 + y**2)*x - y**2, 0)", "physics": false, "states": [ "concept/cissoid-of-diocles" ], "concepts": [ "concept/cartesian-coordinates", "concept/curve", "concept/locus" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-7c307f868c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "68", "location": "FUNCTIONS OF REAL VARIABLES", "latex": "AQ = \\sqrt[3]{2}", "name": null, "statement": "The length AQ equals the cube root of 2, obtained from the Cissoid of Diocles construction.", "kind": "result", "symbols": [ { "unit": "unit length", "symbol": "AQ", "meaning": "length from A to Q" } ], "sympy": "Eq(AQ, 2**Rational(1,3))", "physics": false, "states": [], "concepts": [ "concept/cissoid-of-diocles", "concept/duplication-of-the-cube", "concept/root" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a0ac0f607d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "78", "location": "COMPLEX NUMBERS", "latex": "x + yi = x' + y'i", "name": null, "statement": "Two complex numbers are equal (equivalent) exactly when their real parts and their coefficients of i agree.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x", "meaning": "real part of the first complex number" }, { "unit": null, "symbol": "y", "meaning": "coefficient of i in the first complex number" }, { "unit": null, "symbol": "x'", "meaning": "real part of the second complex number" }, { "unit": null, "symbol": "y'", "meaning": "coefficient of i in the second complex number" } ], "sympy": "Eq(x + y*I, xp + yp*I)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/equality", "concept/equivalent-solids" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-1ee947d43d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "78", "location": "COMPLEX NUMBERS", "latex": "(x + yi) + (x' + y'i) = (x + x') + (y + y')i", "name": null, "statement": "The sum of two complex numbers is the complex number whose real and imaginary coefficients are the sums of the corresponding parts.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x", "meaning": "real part of the first complex number" }, { "unit": null, "symbol": "y", "meaning": "coefficient of i in the first complex number" }, { "unit": null, "symbol": "x'", "meaning": "real part of the second complex number" }, { "unit": null, "symbol": "y'", "meaning": "coefficient of i in the second complex number" } ], "sympy": "Eq((x + y*I) + (xp + yp*I), (x + xp) + (y + yp)*I)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/sum", "method/addition" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-1ccade3fb5", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "78", "location": "COMPLEX NUMBERS", "latex": "(x + yi) (x' + y'i) = xx' - yy' + (xy' + yx')i", "name": null, "statement": "The product of two complex numbers is the complex number with real part xx' - yy' and imaginary coefficient xy' + yx'.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x", "meaning": "real part of the first complex number" }, { "unit": null, "symbol": "y", "meaning": "coefficient of i in the first complex number" }, { "unit": null, "symbol": "x'", "meaning": "real part of the second complex number" }, { "unit": null, "symbol": "y'", "meaning": "coefficient of i in the second complex number" } ], "sympy": "Eq((x + y*I)*(xp + yp*I), x*xp - y*yp + (x*yp + y*xp)*I)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/product", "method/multiplication" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-1decb5173d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "79", "location": "COMPLEX NUMBERS", "latex": "(x + yi) (x' + y'i) = (x' + y'i) (x + yi)", "name": null, "statement": "Multiplication of complex numbers obeys the commutative law.", "kind": "law", "symbols": [ { "unit": null, "symbol": "x", "meaning": "real part of the first complex number" }, { "unit": null, "symbol": "y", "meaning": "coefficient of i in the first complex number" }, { "unit": null, "symbol": "x'", "meaning": "real part of the second complex number" }, { "unit": null, "symbol": "y'", "meaning": "coefficient of i in the second complex number" } ], "sympy": "Eq((x + y*I)*(xp + yp*I), (xp + yp*I)*(x + y*I))", "physics": false, "states": [], "concepts": [ "concept/commutativity-of-operations", "concept/complex-number", "law/commutative-law", "method/multiplication" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c766573983", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "80", "location": "COMPLEX NUMBERS", "latex": "i^{2} = ii = (0 + 1i) (0 + 1i) = (0 · 0 - 1 · 1) + (0 · 1 + 1 · 0)i = -1", "name": null, "statement": "The imaginary unit i, multiplied by itself, gives -1, so i and -i both satisfy x^2 = -1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "i", "meaning": "the complex number 1i, corresponding to a unit displacement along OY" } ], "sympy": "Eq(I**2, -1)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/imaginary-root", "concept/root-of-an-equation", "method/multiplication" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-1c6ad6e2cc", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "81", "location": "COMPLEX NUMBERS", "latex": "(x + yi)i = -y + xi", "name": null, "statement": "Multiplying a complex number by i turns its displacement through a positive right angle.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "real part of the complex number" }, { "unit": null, "symbol": "y", "meaning": "coefficient of i in the complex number" } ], "sympy": "Eq((x + y*I)*I, -y + x*I)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "method/multiplication", "quantity/right-angle" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-9326a473e9", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "80", "location": "COMPLEX NUMBERS", "latex": "(x + yi)(x - yi) = x^{2} + y^{2}", "name": null, "statement": "The product of a complex number and its conjugate is the real number x^2 + y^2.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "real part of the complex number" }, { "unit": null, "symbol": "y", "meaning": "coefficient of i in the complex number" } ], "sympy": "Eq((x + y*I)*(x - y*I), x**2 + y**2)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/conjugate-complex-numbers", "concept/real-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-89f9c4faec", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "81", "location": "COMPLEX NUMBERS", "latex": "az^{2} + 2bz + c = 0", "name": null, "statement": "The general quadratic equation with real coefficients a, b, c, whose roots may be real or complex.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a", "meaning": "real coefficient of z^2" }, { "unit": null, "symbol": "b", "meaning": "real half-coefficient of z" }, { "unit": null, "symbol": "c", "meaning": "real constant term" }, { "unit": null, "symbol": "z", "meaning": "complex variable x + yi" } ], "sympy": "Eq(a*z**2 + 2*b*z + c, 0)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/equation", "concept/quadratic-equation", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-9e3e7eda9a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "82", "location": "COMPLEX NUMBERS", "latex": "\\{z + (b/a)\\}^{2} = -(ac - b^{2})/a^{2}", "name": null, "statement": "The quadratic az^2 + 2bz + c = 0 written in completed-square form, so that its roots are z = (-b ± i√(ac - b^2))/a when b^2 < ac.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "real coefficient of z^2" }, { "unit": null, "symbol": "b", "meaning": "real half-coefficient of z" }, { "unit": null, "symbol": "c", "meaning": "real constant term" }, { "unit": null, "symbol": "z", "meaning": "complex variable" } ], "sympy": "Eq((z + b/a)**2, -(a*c - b**2)/a**2)", "physics": false, "states": [], "concepts": [ "concept/imaginary-root", "concept/quadratic-equation", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c4c76bdaf0", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "84", "location": "COMPLEX NUMBERS", "latex": "\\alpha + \\beta = -(2b/a)", "name": null, "statement": "The sum of the two roots of az^2 + 2bz + c = 0 equals -2b/a, and this holds for complex as well as real roots.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\alpha", "meaning": "a root of az^2 + 2bz + c = 0" }, { "unit": null, "symbol": "\\beta", "meaning": "the other root of az^2 + 2bz + c = 0" }, { "unit": null, "symbol": "a", "meaning": "real coefficient of z^2" }, { "unit": null, "symbol": "b", "meaning": "real half-coefficient of z" } ], "sympy": "Eq(alpha + beta, -(2*b/a))", "physics": false, "states": [], "concepts": [ "concept/quadratic-equation", "concept/root-of-an-equation", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-8e53532f9f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "84", "location": "COMPLEX NUMBERS", "latex": "\\alpha\\beta = (c/a)", "name": null, "statement": "The product of the two roots of az^2 + 2bz + c = 0 equals c/a.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\alpha", "meaning": "a root of az^2 + 2bz + c = 0" }, { "unit": null, "symbol": "\\beta", "meaning": "the other root" }, { "unit": null, "symbol": "a", "meaning": "real coefficient of z^2" }, { "unit": null, "symbol": "c", "meaning": "real constant term" } ], "sympy": "Eq(alpha*beta, c/a)", "physics": false, "states": [], "concepts": [ "concept/product", "concept/quadratic-equation", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e963136739", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "84", "location": "COMPLEX NUMBERS", "latex": "\\alpha + \\beta + \\gamma = -(3b/a)", "name": null, "statement": "The sum of the three roots of az^3 + 3bz^2 + 3cz + d = 0 equals -3b/a.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\alpha", "meaning": "a root of the cubic" }, { "unit": null, "symbol": "\\beta", "meaning": "a root of the cubic" }, { "unit": null, "symbol": "\\gamma", "meaning": "a root of the cubic" }, { "unit": null, "symbol": "a", "meaning": "coefficient of z^3" }, { "unit": null, "symbol": "b", "meaning": "coefficient in 3bz^2" } ], "sympy": "Eq(alpha + beta + gamma, -(3*b/a))", "physics": false, "states": [], "concepts": [ "concept/root-of-an-equation", "concept/sum", "theorem/work-done-in-charging-a-conductor" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-7f09de9701", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "84", "location": "COMPLEX NUMBERS", "latex": "\\beta\\gamma + \\gamma\\alpha + \\alpha\\beta = (3c/a)", "name": null, "statement": "The sum of the pairwise products of the three roots of the cubic az^3 + 3bz^2 + 3cz + d = 0 equals 3c/a.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\alpha", "meaning": "a root of the cubic" }, { "unit": null, "symbol": "\\beta", "meaning": "a root of the cubic" }, { "unit": null, "symbol": "\\gamma", "meaning": "a root of the cubic" }, { "unit": null, "symbol": "a", "meaning": "coefficient of z^3" }, { "unit": null, "symbol": "c", "meaning": "coefficient in 3cz" } ], "sympy": "Eq(beta*gamma + gamma*alpha + alpha*beta, 3*c/a)", "physics": false, "states": [], "concepts": [ "concept/product", "concept/root-of-an-equation", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-f4e9f4227b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "84", "location": "COMPLEX NUMBERS", "latex": "\\alpha\\beta\\gamma = -(d/a)", "name": null, "statement": "The product of the three roots of the cubic az^3 + 3bz^2 + 3cz + d = 0 equals -d/a.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\alpha", "meaning": "a root of the cubic" }, { "unit": null, "symbol": "\\beta", "meaning": "a root of the cubic" }, { "unit": null, "symbol": "\\gamma", "meaning": "a root of the cubic" }, { "unit": null, "symbol": "a", "meaning": "coefficient of z^3" }, { "unit": null, "symbol": "d", "meaning": "constant term" } ], "sympy": "Eq(alpha*beta*gamma, -(d/a))", "physics": false, "states": [], "concepts": [ "concept/polynomial", "concept/product", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-86a9738d5f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "83", "location": "COMPLEX NUMBERS", "latex": "f(z) = A(z - a_{1}) (z - a_{2}) \\dots (z - a_{n})", "name": null, "statement": "A polynomial of degree n with roots a_1,...,a_n factors as A times the product of (z - a_k), where A is its leading coefficient.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f(z)", "meaning": "polynomial in z of degree n" }, { "unit": null, "symbol": "A", "meaning": "constant, the coefficient of z^n in f(z)" }, { "unit": null, "symbol": "a_{k}", "meaning": "the n roots of f(z) = 0" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/factor", "concept/polynomial", "concept/root-of-an-equation", "theorem/factor-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b0f858a963", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "79", "location": "COMPLEX NUMBERS", "latex": "x' \\xi - y' \\eta = x", "name": null, "statement": "The quotient (x + yi)/(x' + y'i) is the complex number xi + eta i satisfying this real part condition of the product equation.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "\\xi", "meaning": "real part of the quotient" }, { "unit": null, "symbol": "\\eta", "meaning": "coefficient of i in the quotient" }, { "unit": null, "symbol": "x'", "meaning": "real part of the divisor" }, { "unit": null, "symbol": "y'", "meaning": "coefficient of i in the divisor" }, { "unit": null, "symbol": "x", "meaning": "real part of the dividend" } ], "sympy": "Eq(xp*xi - yp*eta, x)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/quotient", "method/division" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-3771af24ef", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "79", "location": "COMPLEX NUMBERS", "latex": "x' \\eta + y' \\xi = y", "name": null, "statement": "The quotient (x + yi)/(x' + y'i) is the complex number xi + eta i satisfying this imaginary part condition of the product equation.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "\\xi", "meaning": "real part of the quotient" }, { "unit": null, "symbol": "\\eta", "meaning": "coefficient of i in the quotient" }, { "unit": null, "symbol": "x'", "meaning": "real part of the divisor" }, { "unit": null, "symbol": "y'", "meaning": "coefficient of i in the divisor" }, { "unit": null, "symbol": "y", "meaning": "coefficient of i in the dividend" } ], "sympy": "Eq(xp*eta + yp*xi, y)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/quotient", "method/division" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-036215f1b7", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "79", "location": "COMPLEX NUMBERS", "latex": "\\xi = \\frac{xx' + yy'}{x'^{2} + y'^{2}}", "name": null, "statement": "The real part of the quotient of two complex numbers, obtained by solving the division equations; it fails when x' + y'i = 0.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\xi", "meaning": "real part of the quotient" }, { "unit": null, "symbol": "x", "meaning": "real part of the dividend" }, { "unit": null, "symbol": "y", "meaning": "coefficient of i in the dividend" }, { "unit": null, "symbol": "x'", "meaning": "real part of the divisor" }, { "unit": null, "symbol": "y'", "meaning": "coefficient of i in the divisor" } ], "sympy": "Eq(xi, (x*xp + y*yp)/(xp**2 + yp**2))", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/quotient", "method/division" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-fef6643a35", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "79", "location": "COMPLEX NUMBERS", "latex": "\\eta = \\frac{yx' - xy'}{x'^{2} + y'^{2}}", "name": null, "statement": "The imaginary coefficient of the quotient of two complex numbers, obtained by solving the division equations.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\eta", "meaning": "coefficient of i in the quotient" }, { "unit": null, "symbol": "x", "meaning": "real part of the dividend" }, { "unit": null, "symbol": "y", "meaning": "coefficient of i in the dividend" }, { "unit": null, "symbol": "x'", "meaning": "real part of the divisor" }, { "unit": null, "symbol": "y'", "meaning": "coefficient of i in the divisor" } ], "sympy": "Eq(eta, (y*xp - x*yp)/(xp**2 + yp**2))", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/quotient", "method/division" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-8dc5d1d858", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "77", "location": "COMPLEX NUMBERS", "latex": "x = \\rho\\cos\\theta", "name": null, "statement": "The real part of a complex number written in polar form with modulus rho and angle theta.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x", "meaning": "real part of the complex number" }, { "unit": null, "symbol": "\\rho", "meaning": "modulus, the length of the displacement" }, { "unit": "radian", "symbol": "\\theta", "meaning": "polar angle of the displacement" } ], "sympy": "Eq(x, rho*cos(theta))", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/polar-coordinates" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-94fb865b8e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "77", "location": "COMPLEX NUMBERS", "latex": "y = \\rho\\sin\\theta", "name": null, "statement": "The imaginary coefficient of a complex number written in polar form with modulus rho and angle theta.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "y", "meaning": "coefficient of i in the complex number" }, { "unit": null, "symbol": "\\rho", "meaning": "modulus, the length of the displacement" }, { "unit": "radian", "symbol": "\\theta", "meaning": "polar angle of the displacement" } ], "sympy": "Eq(y, rho*sin(theta))", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/polar-coordinates", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-8a69df811b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "73", "location": "COMPLEX NUMBERS", "latex": "[x, y] = [x, 0] + [0, y]", "name": null, "statement": "A displacement [x, y] is the sum of its components [x, 0] along OX and [0, y] along OY.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "coordinate of the displacement along OX" }, { "unit": null, "symbol": "y", "meaning": "coordinate of the displacement along OY" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/displacement", "concept/independent-constituent", "method/addition-of-displacements" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-52cb06e032", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "72", "location": "COMPLEX NUMBERS", "latex": "[x, y] + [x', y'] = [x + x', y + y']", "name": null, "statement": "The sum of two displacements has coordinates equal to the sums of their coordinates.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x", "meaning": "coordinate of the first displacement along OX" }, { "unit": null, "symbol": "y", "meaning": "coordinate of the first displacement along OY" }, { "unit": null, "symbol": "x'", "meaning": "coordinate of the second displacement along OX" }, { "unit": null, "symbol": "y'", "meaning": "coordinate of the second displacement along OY" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/displacement", "law/commutative-law", "method/addition-of-displacements" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-eed30b77f0", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "71", "location": "COMPLEX NUMBERS", "latex": "\\alpha[x, y] = [\\alpha x, \\alpha y]", "name": null, "statement": "Multiplying a displacement by a real number multiplies each of its coordinates by that number.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "\\alpha", "meaning": "any real number, positive or negative" }, { "unit": null, "symbol": "x", "meaning": "coordinate of the displacement along OX" }, { "unit": null, "symbol": "y", "meaning": "coordinate of the displacement along OY" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/displacement", "concept/real-number", "method/multiplication-of-displacements-by-numbers" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-06fe54cab8", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "73", "location": "COMPLEX NUMBERS", "latex": "[x, y] - [x', y'] = [x, y] + (-[x', y'])", "name": null, "statement": "Subtraction of displacements is defined as adding the reversed displacement.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x", "meaning": "coordinate of the first displacement along OX" }, { "unit": null, "symbol": "y", "meaning": "coordinate of the first displacement along OY" }, { "unit": null, "symbol": "x'", "meaning": "coordinate of the second displacement along OX" }, { "unit": null, "symbol": "y'", "meaning": "coordinate of the second displacement along OY" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/displacement", "method/addition-of-displacements", "method/subtraction" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c0d8ad3630", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "73", "location": "COMPLEX NUMBERS", "latex": "[0, 0] = 0", "name": null, "statement": "The zero displacement, which leaves the particle where it was, is written as the number 0.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "0", "meaning": "the zero displacement" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/zero-displacement" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-cca858b0d3", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "77", "location": "COMPLEX NUMBERS", "latex": "[x, y] [x', y'] = [xx' - yy', xy' + yx']", "name": null, "statement": "The product of two displacements is the displacement obtained by similar-triangle construction, with coordinates xx' - yy' and xy' + yx'.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x", "meaning": "coordinate of the first displacement along OX" }, { "unit": null, "symbol": "y", "meaning": "coordinate of the first displacement along OY" }, { "unit": null, "symbol": "x'", "meaning": "coordinate of the second displacement along OX" }, { "unit": null, "symbol": "y'", "meaning": "coordinate of the second displacement along OY" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/similar-triangles", "method/multiplication-of-displacements" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-84ee7a0c92", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "76", "location": "COMPLEX NUMBERS", "latex": "[x, 0] [x', y'] = [xx', xy']", "name": null, "statement": "Multiplying a displacement along OX by a displacement agrees with ordinary multiplication by the real number x.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "x", "meaning": "coordinate of the displacement along OX, a real number" }, { "unit": null, "symbol": "x'", "meaning": "coordinate of the second displacement along OX" }, { "unit": null, "symbol": "y'", "meaning": "coordinate of the second displacement along OY" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/displacement", "concept/real-number", "method/multiplication-of-displacements" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-189b44fa98", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "86", "location": "COMPLEX NUMBERS", "latex": "1/z = (\\cos\\theta - i\\sin\\theta)/r", "name": null, "statement": "The reciprocal of z has modulus 1/r and amplitude minus theta, where z = r(cos theta + i sin theta).", "kind": "result", "symbols": [ { "unit": null, "symbol": "z", "meaning": "complex number r(cos theta + i sin theta)" }, { "unit": null, "symbol": "r", "meaning": "modulus of z" }, { "unit": null, "symbol": "\\theta", "meaning": "amplitude of z" } ], "sympy": "Eq(1/z, (cos(theta) - I*sin(theta))/r)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/reciprocal", "method/division", "quantity/amplitude-of-a-complex-number", "quantity/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-506edf4e9f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "85", "location": "COMPLEX NUMBERS", "latex": "x = \\Real(z)", "name": null, "statement": "The real part x of the complex number z is its real component.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x", "meaning": "real part of z" }, { "unit": null, "symbol": "z", "meaning": "complex variable x + yi" } ], "sympy": "Eq(x, re(z))", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/complex-variable", "quantity/real-part-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a4ee600775", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "85", "location": "COMPLEX NUMBERS", "latex": "y = \\Imag(z)", "name": null, "statement": "The imaginary part y of the complex number z is its imaginary component.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "y", "meaning": "imaginary part of z" }, { "unit": null, "symbol": "z", "meaning": "complex variable x + yi" } ], "sympy": "Eq(y, im(z))", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/complex-variable", "quantity/imaginary-part-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c5d0f21779", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "85", "location": "COMPLEX NUMBERS", "latex": "r = |z|", "name": null, "statement": "The modulus r of z is the absolute value of z.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "r", "meaning": "modulus of z" }, { "unit": null, "symbol": "z", "meaning": "complex variable x + yi" } ], "sympy": "Eq(r, Abs(z))", "physics": false, "states": [], "concepts": [ "concept/complex-variable", "quantity/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-00f2388873", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "85", "location": "COMPLEX NUMBERS", "latex": "\\theta = \\am z", "name": null, "statement": "The angle theta is the amplitude of z.", "kind": "definition", "symbols": [ { "unit": "angle", "symbol": "\\theta", "meaning": "amplitude of z" }, { "unit": null, "symbol": "z", "meaning": "complex variable x + yi" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/complex-variable", "concept/plane-angle", "quantity/amplitude-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-134704b2b1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "86", "location": "COMPLEX NUMBERS", "latex": "(\\cos\\theta + i\\sin\\theta)^{n} = \\cos n\\theta + i\\sin n\\theta", "name": "De Moivre's Theorem", "statement": "The n-th power of cos theta plus i sin theta equals cos n theta plus i sin n theta, for any positive integer n.", "kind": "law", "symbols": [ { "unit": null, "symbol": "n", "meaning": "positive integer (later extended to all integers)" }, { "unit": null, "symbol": "\\theta", "meaning": "angle (amplitude)" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": "Eq((cos(theta) + I*sin(theta))**n, cos(n*theta) + I*sin(n*theta))", "physics": false, "states": [ "theorem/de-moivre-s-theorem" ], "concepts": [ "concept/complex-number", "concept/cosine", "concept/integer", "concept/sine", "quantity/amplitude-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-6d44966a68", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "86", "location": "COMPLEX NUMBERS", "latex": "r(\\cos\\theta + i\\sin\\theta) × \\rho(\\cos\\phi + i\\sin\\phi) = r\\rho\\{\\cos(\\theta + \\phi) + i\\sin(\\theta + \\phi)\\}", "name": null, "statement": "The product of two complex numbers has modulus equal to the product of the moduli and amplitude equal to the sum of the amplitudes.", "kind": "result", "symbols": [ { "unit": null, "symbol": "r, \\rho", "meaning": "moduli of the two complex numbers" }, { "unit": null, "symbol": "\\theta, \\phi", "meaning": "amplitudes of the two complex numbers" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": "Eq(r*(cos(theta) + I*sin(theta))*rho*(cos(phi) + I*sin(phi)), r*rho*(cos(theta + phi) + I*sin(theta + phi)))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/sine", "method/addition", "method/multiplication", "quantity/amplitude-of-a-complex-number", "quantity/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-43f5747fb0", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "96", "location": "COMPLEX NUMBERS", "latex": "\\frac{(z_{1} - z_{3}) (z_{2} - z_{4})}{(z_{1} - z_{4}) (z_{2} - z_{3})} = -1", "name": null, "statement": "The four points z1, z2, z3, z4 are harmonic when this cross ratio equals -1.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "z_1, z_2, z_3, z_4", "meaning": "complex numbers denoting four points" } ], "sympy": "Eq((z1 - z3)*(z2 - z4)/((z1 - z4)*(z2 - z3)), -1)", "physics": false, "states": [], "concepts": [ "concept/complex-variable", "concept/cross-ratio", "concept/harmonic-points" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-4f5efeb54f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "88", "location": "COMPLEX NUMBERS", "latex": "a_{0}z^{n} + a_{1}z^{n-1} + \\dots + a_{n} = 0", "name": null, "statement": "A polynomial equation of degree n in the complex variable z with coefficients a_0, a_1, ..., a_n.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "z", "meaning": "unknown complex variable" }, { "unit": null, "symbol": "a_0, ..., a_n", "meaning": "coefficients of the equation" }, { "unit": null, "symbol": "n", "meaning": "degree of the equation" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/equation", "concept/polynomial", "concept/real-number", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-1d6ca04d59", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "91", "location": "COMPLEX NUMBERS", "latex": "z^{2} + 2(b + Bi)z + (c + Ci) = 0", "name": null, "statement": "The standard form of the quadratic equation with complex coefficients, obtained after dividing by a + iA.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "z", "meaning": "unknown complex variable" }, { "unit": null, "symbol": "b, B", "meaning": "real and imaginary parts of the middle coefficient" }, { "unit": null, "symbol": "c, C", "meaning": "real and imaginary parts of the constant term" } ], "sympy": "Eq(z**2 + 2*(b + B*I)*z + (c + C*I), 0)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/complex-number", "concept/quadratic-equation", "concept/standard-form" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-0ed0d6b5c2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "91", "location": "COMPLEX NUMBERS", "latex": "x^{2} - y^{2} + 2(bx - By) + c = 0", "name": null, "statement": "The real part of the quadratic equation with complex coefficients, after substituting z = x + yi.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x, y", "meaning": "real and imaginary parts of z" }, { "unit": null, "symbol": "b, B, c", "meaning": "coefficients of the standard quadratic" } ], "sympy": "Eq(x**2 - y**2 + 2*(b*x - B*y) + c, 0)", "physics": false, "states": [], "concepts": [ "concept/quadratic-equation", "concept/simultaneous-equations", "method/equating-real-and-imaginary-parts", "quantity/real-part-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-61945d8237", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "91", "location": "COMPLEX NUMBERS", "latex": "2xy + 2(by + Bx) + C = 0", "name": null, "statement": "The imaginary part of the quadratic equation with complex coefficients, after substituting z = x + yi.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x, y", "meaning": "real and imaginary parts of z" }, { "unit": null, "symbol": "b, B, C", "meaning": "coefficients of the standard quadratic" } ], "sympy": "Eq(2*x*y + 2*(b*y + B*x) + C, 0)", "physics": false, "states": [], "concepts": [ "concept/quadratic-equation", "concept/simultaneous-equations", "method/equating-real-and-imaginary-parts", "quantity/imaginary-part-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-658d69bb6f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "91", "location": "COMPLEX NUMBERS", "latex": "\\xi^{2} - \\eta^{2} = h", "name": null, "statement": "The shifted real part of the quadratic equation, with xi = x + b and eta = y + B.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\xi, \\eta", "meaning": "shifted real and imaginary parts (x + b, y + B)" }, { "unit": null, "symbol": "h", "meaning": "b^2 - B^2 - c" } ], "sympy": "Eq(xi**2 - eta**2, h)", "physics": false, "states": [], "concepts": [ "concept/quadratic-equation", "concept/simultaneous-equations" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-0b636aa994", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "91", "location": "COMPLEX NUMBERS", "latex": "2\\xi\\eta = k", "name": null, "statement": "The shifted imaginary part of the quadratic equation.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\xi, \\eta", "meaning": "shifted real and imaginary parts" }, { "unit": null, "symbol": "k", "meaning": "2bB - C" } ], "sympy": "Eq(2*xi*eta, k)", "physics": false, "states": [], "concepts": [ "concept/quadratic-equation", "concept/simultaneous-equations" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ae30aef37d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "92", "location": "COMPLEX NUMBERS", "latex": "\\xi^{2} + \\eta^{2} = \\sqrtp{h^{2} + k^{2}}", "name": null, "statement": "Squaring and adding the two shifted equations gives the sum of the squares equal to the square root of h squared plus k squared.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\xi, \\eta", "meaning": "shifted real and imaginary parts" }, { "unit": null, "symbol": "h, k", "meaning": "constants defined from the coefficients" } ], "sympy": "Eq(xi**2 + eta**2, sqrt(h**2 + k**2))", "physics": false, "states": [], "concepts": [ "concept/quadratic-equation", "concept/root" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-cf5461d31f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "92", "location": "COMPLEX NUMBERS", "latex": "c + Ci = (b + Bi)^{2}", "name": null, "statement": "The two roots of the quadratic are equal exactly when the constant term is the square of the middle coefficient, i.e. the left side is a perfect square.", "kind": "result", "symbols": [ { "unit": null, "symbol": "b, B, c, C", "meaning": "coefficients of the standard quadratic" } ], "sympy": "Eq(c + C*I, (b + B*I)**2)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/equal-roots", "concept/perfect-square", "concept/quadratic-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e0da529494", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "92", "location": "COMPLEX NUMBERS", "latex": "C^{2} - 4bBC + 4cB^{2} = 0", "name": null, "statement": "Condition on the coefficients for the quadratic to have a real root.", "kind": "result", "symbols": [ { "unit": null, "symbol": "b, B, c, C", "meaning": "coefficients of the standard quadratic" } ], "sympy": "Eq(C**2 - 4*b*B*C + 4*c*B**2, 0)", "physics": false, "states": [], "concepts": [ "concept/quadratic-equation", "concept/real-root", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-9a48920231", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "92", "location": "COMPLEX NUMBERS", "latex": "C^{2} - 4bBC - 4b^{2}c = 0", "name": null, "statement": "Condition on the coefficients for the quadratic to have a purely imaginary root.", "kind": "result", "symbols": [ { "unit": null, "symbol": "b, B, c, C", "meaning": "coefficients of the standard quadratic" } ], "sympy": "Eq(C**2 - 4*b*B*C - 4*b**2*c, 0)", "physics": false, "states": [], "concepts": [ "concept/imaginary-root", "concept/quadratic-equation", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-9c4f225594", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "92", "location": "COMPLEX NUMBERS", "latex": "z^{3} + 3Hz + G = 0", "name": null, "statement": "The cubic equation with complex coefficients in the reduced form studied in Example 15.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "z", "meaning": "unknown complex variable" }, { "unit": null, "symbol": "H, G", "meaning": "complex coefficients" } ], "sympy": "Eq(z**3 + 3*H*z + G, 0)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/complex-number", "concept/cubic-equation", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d42a93ef3b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "92", "location": "COMPLEX NUMBERS", "latex": "\\sigma^{3} + 27\\lambda\\mu^{2}\\sigma - 27\\mu^{3}\\rho = 0", "name": null, "statement": "Condition for the cubic z^3 + 3Hz + G = 0 to have a real root when mu is not zero.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\lambda, \\mu", "meaning": "real and imaginary parts of H" }, { "unit": null, "symbol": "\\rho, \\sigma", "meaning": "real and imaginary parts of G" } ], "sympy": "Eq(sigma**3 + 27*lambda_*mu**2*sigma - 27*mu**3*rho, 0)", "physics": false, "states": [], "concepts": [ "concept/cubic-equation", "concept/real-root", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-cbec8d0654", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "92", "location": "COMPLEX NUMBERS", "latex": "\\rho^{3} - 27\\lambda\\mu^{2}\\rho - 27\\mu^{3}\\sigma = 0", "name": null, "statement": "Condition for the cubic z^3 + 3Hz + G = 0 to have a purely imaginary root when mu is not zero.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\lambda, \\mu", "meaning": "real and imaginary parts of H" }, { "unit": null, "symbol": "\\rho, \\sigma", "meaning": "real and imaginary parts of G" } ], "sympy": "Eq(rho**3 - 27*lambda_*mu**2*rho - 27*mu**3*sigma, 0)", "physics": false, "states": [], "concepts": [ "concept/cubic-equation", "concept/imaginary-root", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-0466cea82e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "92", "location": "COMPLEX NUMBERS", "latex": "y^{2} - 3x^{2} = 3H", "name": null, "statement": "Relation between the real part x and imaginary part y of a complex-pair root of the cubic and H.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x, y", "meaning": "real and imaginary parts of a complex root x + yi" }, { "unit": null, "symbol": "H", "meaning": "complex coefficient of the cubic" } ], "sympy": "Eq(y**2 - 3*x**2, 3*H)", "physics": false, "states": [], "concepts": [ "concept/conjugate-complex-numbers", "concept/cubic-equation", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-cd50db4a27", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "92", "location": "COMPLEX NUMBERS", "latex": "2x(x^{2} + y^{2}) = G", "name": null, "statement": "Relation between the real part x and imaginary part y of a complex-pair root of the cubic and G.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x, y", "meaning": "real and imaginary parts of a complex root x + yi" }, { "unit": null, "symbol": "G", "meaning": "complex constant term of the cubic" } ], "sympy": "Eq(2*x*(x**2 + y**2), G)", "physics": false, "states": [], "concepts": [ "concept/conjugate-complex-numbers", "concept/cubic-equation", "concept/root-of-an-equation", "quantity/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d9cd998e61", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "91", "location": "COMPLEX NUMBERS", "latex": "\\alpha z + \\beta = 0", "name": null, "statement": "The general linear equation with complex coefficients, which has one solution unless alpha is zero.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "\\alpha, \\beta", "meaning": "complex coefficients" }, { "unit": null, "symbol": "z", "meaning": "unknown complex variable" } ], "sympy": "Eq(alpha*z + beta, 0)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/linear-equation", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-cb17542466", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "91", "location": "COMPLEX NUMBERS", "latex": "z = -(\\beta/\\alpha)", "name": null, "statement": "The unique solution of the general linear equation with complex coefficients, when alpha is not zero.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\alpha, \\beta", "meaning": "complex coefficients" }, { "unit": null, "symbol": "z", "meaning": "unknown complex variable" } ], "sympy": "Eq(z, -(beta/alpha))", "physics": false, "states": [], "concepts": [ "concept/linear-equation", "concept/root-of-an-equation", "concept/solution" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-2212fcdb6d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "91", "location": "COMPLEX NUMBERS", "latex": "aB - bA = 0", "name": null, "statement": "Consistency condition for the two real equations ax + b = 0 and Ax + B = 0 to have a common real root.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a, A", "meaning": "real and imaginary parts of alpha" }, { "unit": null, "symbol": "b, B", "meaning": "real and imaginary parts of beta" } ], "sympy": "Eq(a*B - b*A, 0)", "physics": false, "states": [], "concepts": [ "concept/linear-equation", "concept/real-root", "theorem/consistency-of-a-linear-system" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-531183b9c6", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "93", "location": "COMPLEX NUMBERS", "latex": "8\\alpha^{3} + 6\\alpha H - G = 0", "name": null, "statement": "The real part alpha of the complex roots of z^3 + 3Hz + G = 0 is a root of this real-coefficient cubic.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\alpha", "meaning": "real part of the complex roots of the original cubic" }, { "unit": null, "symbol": "H, G", "meaning": "complex coefficients of the original cubic" } ], "sympy": "Eq(8*alpha**3 + 6*alpha*H - G, 0)", "physics": false, "states": [], "concepts": [ "concept/cubic-equation", "concept/root-of-an-equation", "quantity/real-part-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-5a17fadeba", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "93", "location": "COMPLEX NUMBERS", "latex": "\\left|\\frac{z - b}{z - a}\\right| = \\lambda", "name": null, "statement": "The locus of P is a circle when the ratio of distances PA to PB is a constant lambda.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a, b, z", "meaning": "arguments of the points A, B, P" }, { "unit": null, "symbol": "\\lambda", "meaning": "constant ratio of distances PA/PB" } ], "sympy": "Eq(Abs((z - b)/(z - a)), lambda_)", "physics": false, "states": [], "concepts": [ "concept/circle", "concept/common-ratio", "concept/locus", "quantity/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-4cfeef3415", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "94", "location": "COMPLEX NUMBERS", "latex": "z = Z + a", "name": null, "statement": "The translation: z is Z shifted by the complex number a, so figures are moved without change of size or orientation.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "z", "meaning": "complex variable in the xoy plane" }, { "unit": null, "symbol": "Z", "meaning": "complex variable in the XOY plane" }, { "unit": null, "symbol": "a", "meaning": "complex shift a = alpha + beta i" } ], "sympy": "Eq(z, Z + a)", "physics": false, "states": [], "concepts": [ "concept/complex-variable", "concept/transformation", "concept/translation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e92f1e1ec8", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "94", "location": "COMPLEX NUMBERS", "latex": "z = \\rho Z", "name": null, "statement": "The magnification: z is Z multiplied by the positive real rho, scaling figures by rho.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "\\rho", "meaning": "real scale factor" }, { "unit": null, "symbol": "z, Z", "meaning": "complex variables in the two planes" } ], "sympy": "Eq(z, rho*Z)", "physics": false, "states": [], "concepts": [ "concept/magnification", "concept/scale-of-a-map", "concept/transformation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-6d2590a78e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "94", "location": "COMPLEX NUMBERS", "latex": "z = (\\cos\\phi + i \\sin\\phi)Z", "name": null, "statement": "The rotation: z is Z multiplied by a unit complex number, turning the figure through angle phi about the origin.", "kind": "definition", "symbols": [ { "unit": "angle", "symbol": "\\phi", "meaning": "angle of rotation" }, { "unit": null, "symbol": "z, Z", "meaning": "complex variables in the two planes" } ], "sympy": "Eq(z, (cos(phi) + I*sin(phi))*Z)", "physics": false, "states": [], "concepts": [ "concept/rotation", "concept/transformation", "quantity/amplitude-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-76516ed5b4", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "94", "location": "COMPLEX NUMBERS", "latex": "z = aZ + b", "name": null, "statement": "The general linear transformation, equivalent to a translation, a magnification and a rotation combined.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a, b", "meaning": "complex constants" }, { "unit": null, "symbol": "z, Z", "meaning": "complex variables in the two planes" } ], "sympy": "Eq(z, a*Z + b)", "physics": false, "states": [], "concepts": [ "concept/linear-transformation", "concept/magnification", "concept/rotation", "concept/transformation", "concept/translation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-681396e93f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "95", "location": "COMPLEX NUMBERS", "latex": "z = 1/Z", "name": null, "statement": "The inversion transformation: modulus becomes 1/R and amplitude becomes minus Theta, followed by reflection in the axis ox.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "z, Z", "meaning": "complex variables in the two planes" } ], "sympy": "Eq(z, 1/Z)", "physics": false, "states": [], "concepts": [ "concept/reciprocal", "concept/transformation", "quantity/amplitude-of-a-complex-number", "quantity/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-9913521d71", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "95", "location": "COMPLEX NUMBERS", "latex": "z = \\frac{aZ + b}{cZ + d}", "name": "general bilinear transformation", "statement": "The general bilinear transformation, the most general one-to-one transformation between z and Z in the plane.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a, b, c, d", "meaning": "complex constants" }, { "unit": null, "symbol": "z, Z", "meaning": "complex variables in the two planes" } ], "sympy": "Eq(z, (a*Z + b)/(c*Z + d))", "physics": false, "states": [ "concept/bilinear-transformation" ], "concepts": [ "concept/linear-transformation", "concept/transformation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-bef13d8959", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "95", "location": "COMPLEX NUMBERS", "latex": "Z = \\frac{dz - b}{cz - a}", "name": null, "statement": "The inverse of the general bilinear transformation, solving for Z in terms of z.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a, b, c, d", "meaning": "complex constants of the bilinear transformation" }, { "unit": null, "symbol": "z, Z", "meaning": "complex variables in the two planes" } ], "sympy": "Eq(Z, (d*z - b)/(c*z - a))", "physics": false, "states": [], "concepts": [ "concept/bilinear-transformation", "concept/transformation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-56ead0336f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "96", "location": "COMPLEX NUMBERS", "latex": "ac' + a'c - 2bb' = 0", "name": null, "statement": "Condition under which OA1 and OA2 are equally inclined to A3A4 with OA1·OA2 = OA3^2 = OA4^2.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a, b, c", "meaning": "coefficients of the quadratic az^2 + 2bz + c = 0 with roots A1, A2" }, { "unit": null, "symbol": "a', b', c'", "meaning": "coefficients of the quadratic a'z^2 + 2b'z + c' = 0 with roots A3, A4" } ], "sympy": "Eq(a*c_p + a_p*c - 2*b*b_p, 0)", "physics": false, "states": [], "concepts": [ "concept/circle", "concept/cross-ratio", "concept/harmonic-points", "concept/quadratic-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ddc33ee789", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "90", "location": "COMPLEX NUMBERS", "latex": "\\cot\\omega = \\cot A + \\cot B + \\cot C", "name": null, "statement": "For the point P inside a triangle with equal angles omega at the vertices, cot omega equals the sum of the cotangents of the triangle's angles.", "kind": "result", "symbols": [ { "unit": "angle", "symbol": "\\omega", "meaning": "common angle at the vertices defined by the point P" }, { "unit": "angle", "symbol": "A, B, C", "meaning": "angles of the triangle" } ], "sympy": "Eq(cot(omega), cot(A) + cot(B) + cot(C))", "physics": false, "states": [], "concepts": [ "concept/cotangent", "concept/plane-angle", "concept/triangle" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-36b14306f2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "90", "location": "COMPLEX NUMBERS", "latex": "z = \\frac{1}{3}(\\alpha + \\beta + \\gamma)", "name": null, "statement": "The centre of gravity of a triangle with complex vertices alpha, beta, gamma is one third of their sum.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\alpha, \\beta, \\gamma", "meaning": "complex numbers giving the three vertices of a triangle" }, { "unit": null, "symbol": "z", "meaning": "complex number of the centre of gravity" } ], "sympy": "Eq(z, Rational(1,3)*(alpha + beta + gamma))", "physics": false, "states": [], "concepts": [ "concept/centre-of-gravity", "concept/complex-number", "concept/triangle" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-70a1d5bef8", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "98", "location": "COMPLEX NUMBERS", "latex": "z = c + \\rho\\left(\\frac{1 + ti}{1 - ti}\\right)", "name": null, "statement": "As the real parameter t varies, z describes the circle with centre c and radius rho.", "kind": "result", "symbols": [ { "unit": null, "symbol": "c", "meaning": "centre of the circle" }, { "unit": null, "symbol": "\\rho", "meaning": "positive radius of the circle" }, { "unit": null, "symbol": "t", "meaning": "real parameter" } ], "sympy": "Eq(z, c + rho*(1 + t*I)/(1 - t*I))", "physics": false, "states": [], "concepts": [ "concept/centre-of-a-circle", "concept/circle", "concept/complex-function-of-a-real-variable", "concept/radius" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c9b7338322", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "98", "location": "COMPLEX NUMBERS", "latex": "z = a + 2bt + ct^{2}", "name": null, "statement": "As the real parameter t varies, z describes a parabola in general, and a straight line if b/c is real.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a, b, c", "meaning": "complex constants" }, { "unit": null, "symbol": "t", "meaning": "real parameter" } ], "sympy": "Eq(z, a + 2*b*t + c*t**2)", "physics": false, "states": [], "concepts": [ "concept/complex-function-of-a-real-variable", "concept/line", "concept/parabola" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-4b3a7c5b4e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "90", "location": "COMPLEX NUMBERS", "latex": "\\am\\left(\\frac{a - b}{c - d}\\right) = ±\\tfrac{1}{2} \\pi", "name": null, "statement": "The lines joining z=a to z=b and z=c to z=d are perpendicular when this amplitude is plus or minus pi/2.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a, b, c, d", "meaning": "complex numbers giving four points" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/line", "concept/perpendicular", "quantity/amplitude-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-fcf350bf3a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "97", "location": "COMPLEX NUMBERS", "latex": "\\tan\\phi_{m+n} &= \\tan\\phi_{m} \\sec\\phi_{n} &&+ \\sec\\phi_{m} \\tan\\phi_{n}", "name": null, "statement": "The analogue of De Moivre's theorem for the tangent: tan of phi(m+n) equals tan phi_m sec phi_n plus sec phi_m tan phi_n.", "kind": "identity", "symbols": [ { "unit": "angle", "symbol": "\\phi_m, \\phi_n", "meaning": "positive acute angles in the series" }, { "unit": null, "symbol": "m, n", "meaning": "integers indexing the series" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/acute-angle", "concept/secant", "concept/tangent-function", "method/mathematical-induction", "theorem/de-moivre-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-978ca4b7f2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "97", "location": "COMPLEX NUMBERS", "latex": "\\sec\\phi_{m+n} &= \\sec\\phi_{m} \\sec\\phi_{n} &&+ \\tan\\phi_{m} \\tan\\phi_{n}", "name": null, "statement": "The analogue of De Moivre's theorem for the secant: sec of phi(m+n) equals sec phi_m sec phi_n plus tan phi_m tan phi_n.", "kind": "identity", "symbols": [ { "unit": "angle", "symbol": "\\phi_m, \\phi_n", "meaning": "positive acute angles in the series" }, { "unit": null, "symbol": "m, n", "meaning": "integers indexing the series" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/acute-angle", "concept/secant", "concept/tangent-function", "method/mathematical-induction", "theorem/de-moivre-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-97314614df", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "97", "location": "COMPLEX NUMBERS", "latex": "\\tan\\phi_{m} + \\sec\\phi_{m} = (\\tan\\phi_{1} + \\sec\\phi_{1})^{m}", "name": null, "statement": "The sum of tan and sec of phi_m equals the m-th power of tan phi_1 plus sec phi_1.", "kind": "result", "symbols": [ { "unit": "angle", "symbol": "\\phi_m, \\phi_1", "meaning": "positive acute angles in the series" }, { "unit": null, "symbol": "m", "meaning": "positive integer index" } ], "sympy": "Eq(tan(phi_m) + sec(phi_m), (tan(phi_1) + sec(phi_1))**m)", "physics": false, "states": [], "concepts": [ "concept/acute-angle", "concept/secant", "concept/tangent-function", "method/mathematical-induction", "theorem/de-moivre-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-6e4dfa7c71", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "98", "location": "COMPLEX NUMBERS", "latex": "z^{n} = a", "name": null, "statement": "A number z is an n-th root of a when its n-th power equals a; the book uses this to define the symbol a^{1/n} for complex a.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "z", "meaning": "a number defined as an n-th root of a" }, { "unit": null, "symbol": "n", "meaning": "positive integer" }, { "unit": null, "symbol": "a", "meaning": "complex number" } ], "sympy": "Eq(z**n, a)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/power", "concept/root-of-a-complex-number", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ed4965a3df", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "98", "location": "COMPLEX NUMBERS", "latex": "a = \\rho(\\cos\\phi + i\\sin\\phi)", "name": null, "statement": "A non-zero complex number a is written in modulus-amplitude form, with positive modulus rho and angle phi.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "a", "meaning": "complex number" }, { "unit": null, "symbol": "\\rho", "meaning": "positive modulus of a" }, { "unit": null, "symbol": "\\phi", "meaning": "angle with -pi < phi <= pi (amplitude of a)" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": "Eq(a, rho*(cos(phi) + I*sin(phi)))", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/principal-value-of-amplitude", "quantity/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-05b4f5629a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "99", "location": "COMPLEX NUMBERS", "latex": "z = r(\\cos\\theta + i\\sin\\theta)", "name": null, "statement": "The unknown complex number z is written in modulus-amplitude form with modulus r and angle theta.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "z", "meaning": "unknown complex number" }, { "unit": null, "symbol": "r", "meaning": "modulus of z" }, { "unit": null, "symbol": "\\theta", "meaning": "angle (amplitude of z)" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": "Eq(z, r*(cos(theta) + I*sin(theta)))", "physics": false, "states": [], "concepts": [ "concept/argand-diagram", "concept/complex-number", "concept/cosine", "concept/plane-angle", "concept/sine", "quantity/modulus-of-a-complex-number" ], "pages": [ "99", "156" ], "chapters": [ "hardy-course-of-pure-mathematics-1921/ch-iii", "hardy-course-of-pure-mathematics-1921/ch-iv" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d5f877f9e7", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "99", "location": "COMPLEX NUMBERS", "latex": "r^{n} = \\rho", "name": null, "statement": "The modulus of an n-th root z of a must satisfy r^n equal to the modulus rho of a.", "kind": "result", "symbols": [ { "unit": null, "symbol": "r", "meaning": "modulus of z" }, { "unit": null, "symbol": "n", "meaning": "positive integer" }, { "unit": null, "symbol": "\\rho", "meaning": "positive modulus of a" } ], "sympy": "Eq(r**n, rho)", "physics": false, "states": [], "concepts": [ "concept/root-of-a-complex-number", "quantity/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d8a98886f5", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "99", "location": "COMPLEX NUMBERS", "latex": "n\\theta = \\phi + 2k\\pi", "name": null, "statement": "The angle of an n-th root of a satisfies n theta equal to phi plus a whole number of turns 2k pi, where k is an integer.", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "positive integer" }, { "unit": null, "symbol": "\\theta", "meaning": "angle of the root z" }, { "unit": null, "symbol": "\\phi", "meaning": "angle of a, with -pi < phi <= pi" }, { "unit": null, "symbol": "k", "meaning": "integer" } ], "sympy": "Eq(n*theta, phi + 2*k*pi)", "physics": false, "states": [], "concepts": [ "concept/integer", "concept/root-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-163d4d3bd7", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "99", "location": "COMPLEX NUMBERS", "latex": "r = \\sqrt[n]{\\rho}", "name": null, "statement": "The only possible modulus of an n-th root of a is the ordinary positive n-th root of rho.", "kind": "result", "symbols": [ { "unit": null, "symbol": "r", "meaning": "modulus of the root z" }, { "unit": null, "symbol": "n", "meaning": "positive integer" }, { "unit": null, "symbol": "\\rho", "meaning": "positive modulus of a" } ], "sympy": "Eq(r, rho**(1/n))", "physics": false, "states": [], "concepts": [ "concept/real-number", "concept/root-of-a-complex-number", "quantity/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-5de0b39fe3", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "99", "location": "COMPLEX NUMBERS", "latex": "\\theta = (\\phi + 2q\\pi)/n", "name": null, "statement": "The angles of the n distinct roots of a are (phi + 2q pi)/n for q = 0, 1, ..., n-1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\theta", "meaning": "angle of the root z" }, { "unit": null, "symbol": "\\phi", "meaning": "angle of a, with -pi < phi <= pi" }, { "unit": null, "symbol": "n", "meaning": "positive integer" }, { "unit": null, "symbol": "q", "meaning": "integer with 0 <= q < n" } ], "sympy": "Eq(theta, (phi + 2*q*pi)/n)", "physics": false, "states": [], "concepts": [ "concept/integer", "concept/root-of-a-complex-number", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c8bfeb23c2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "99", "location": "COMPLEX NUMBERS", "latex": "z^{n} = a = \\rho(\\cos\\phi + i\\sin\\phi)", "name": null, "statement": "The equation z^n = a, with a in modulus-amplitude form, has exactly n roots, all of which are given by the formulas for r and theta.", "kind": "result", "symbols": [ { "unit": null, "symbol": "z", "meaning": "unknown complex number" }, { "unit": null, "symbol": "n", "meaning": "positive integer" }, { "unit": null, "symbol": "a", "meaning": "complex number" }, { "unit": null, "symbol": "\\rho", "meaning": "positive modulus of a" }, { "unit": null, "symbol": "\\phi", "meaning": "angle of a, with -pi < phi <= pi" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": "Eq(z**n, rho*(cos(phi) + I*sin(phi)))", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/root-of-a-complex-number", "concept/root-of-an-equation", "quantity/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-0db5fcd250", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "101", "location": "COMPLEX NUMBERS", "latex": "\\cos\\alpha\\theta + i\\sin\\alpha\\theta", "name": "generalised form of De Moivre's theorem", "statement": "For rational alpha, one of the values of (cos theta + i sin theta)^alpha is cos(alpha theta) + i sin(alpha theta).", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\alpha", "meaning": "any rational number" }, { "unit": null, "symbol": "\\theta", "meaning": "angle" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": "Eq((cos(theta) + I*sin(theta))**alpha, cos(alpha*theta) + I*sin(alpha*theta))", "physics": false, "states": [ "theorem/generalised-form-of-de-moivre-s-theorem" ], "concepts": [ "concept/complex-number", "concept/cosine", "concept/power", "concept/rational-number", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-6d4b5b8a02", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "107", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "y = \\phi(n)", "name": null, "statement": "The function of the positive integer variable n is written as y = phi(n), with y regarded as a function of n defined for all values of n.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "y", "meaning": "the value of the function of n" }, { "unit": null, "symbol": "n", "meaning": "positive integral variable" }, { "unit": null, "symbol": "\\phi", "meaning": "function of n" } ], "sympy": "Eq(y, phi(n))", "physics": false, "states": [], "concepts": [ "concept/function-notation", "concept/function-of-a-positive-integer-variable" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-72131843bc", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "107", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "y = \\phi(x) + \\sin x\\pi", "name": null, "statement": "A function of x that takes the value phi(n) at each positive integer x = n, because sin(n pi) = 0.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "y", "meaning": "value of the interpolating function of x" }, { "unit": null, "symbol": "x", "meaning": "real variable" }, { "unit": null, "symbol": "\\phi", "meaning": "function of x" } ], "sympy": "Eq(y, phi(x) + sin(x*pi))", "physics": false, "states": [], "concepts": [ "concept/function", "concept/functional-interpolation", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-47a7dd4bea", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "107", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\sin n\\pi = 0", "name": null, "statement": "The sine of n times pi vanishes for every integer n.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "n", "meaning": "integer" } ], "sympy": "Eq(sin(n*pi), 0)", "physics": false, "states": [], "concepts": [ "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-0edd12453e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "108", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "(-1)^{n} = \\cos n\\pi", "name": null, "statement": "For integer n, (-1) to the power n equals cos(n pi), giving a form of (-1)^n defined for all real x.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "n", "meaning": "integer" } ], "sympy": "Eq((-1)**n, cos(n*pi))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/functional-interpolation", "concept/power" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-18d8020916", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "108", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "y = 1 · 2 \\dots n = n!", "name": null, "statement": "The product of the first n positive integers is written n!, which has no obvious formula in x that reduces to it at x = n.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "y", "meaning": "the product 1 · 2 … n" }, { "unit": null, "symbol": "n", "meaning": "positive integer" } ], "sympy": "Eq(y, factorial(n))", "physics": false, "states": [], "concepts": [ "concept/functional-interpolation", "concept/gamma-function", "concept/integer", "concept/product" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-2bdd5b47a5", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "114", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\lim_{n\\to\\infty} \\frac{1}{n} = 0", "name": null, "statement": "The limit of 1/n as n tends to infinity is zero.", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "positive integral variable tending to infinity" } ], "sympy": "Eq(Limit(1/n, n, oo), 0)", "physics": false, "states": [], "concepts": [ "concept/limit", "concept/tends-to-infinity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-73ee399921", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "115", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\lim_{n\\to\\infty} \\left(1 - \\frac{1}{n}\\right) = 1", "name": null, "statement": "The limit of 1 - 1/n as n tends to infinity is one.", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "positive integral variable tending to infinity" } ], "sympy": "Eq(Limit(1 - 1/n, n, oo), 1)", "physics": false, "states": [], "concepts": [ "concept/limit", "concept/tends-to-infinity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-55586002c3", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "114", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "1 - \\phi(n) = 1/n", "name": null, "statement": "For phi(n) = 1 - 1/n, the difference 1 - phi(n) equals 1/n, which is why statement (ib) is true.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "n", "meaning": "positive integral variable" }, { "unit": null, "symbol": "\\phi", "meaning": "function of n, here 1 - 1/n" } ], "sympy": "Eq(1 - phi(n), 1/n)", "physics": false, "states": [], "concepts": [ "concept/function-of-a-positive-integer-variable", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-7bc3ec4605", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "115", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "n^2 \\to \\infty", "name": null, "statement": "n squared tends to infinity as n tends to infinity, meaning n squared is large for large n.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "n", "meaning": "positive integral variable" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/infinity", "concept/tends-to-infinity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c8c35752ba", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "115", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "-n^{2} \\to -\\infty", "name": null, "statement": "Minus n squared tends to negative infinity as n tends to infinity.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "n", "meaning": "positive integral variable" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/infinity", "concept/tends-to-infinity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-df25b25d7d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "116", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\lim_{n \\to \\infty} \\phi(n) = l", "name": null, "statement": "phi(n) tends to the limit l as n tends to infinity: for any positive Delta, phi(n) differs from l by less than Delta for all n at or beyond some n_0(Delta).", "kind": "definition", "symbols": [ { "unit": null, "symbol": "\\phi", "meaning": "function of n" }, { "unit": null, "symbol": "l", "meaning": "the limit" }, { "unit": null, "symbol": "n", "meaning": "positive integral variable" } ], "sympy": "Eq(Limit(phi(n), n, oo), l)", "physics": false, "states": [], "concepts": [ "concept/function-of-a-positive-integer-variable", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-cb6034f95d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "116", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "|\\phi(n) - l| < \\DELTA", "name": null, "statement": "The distance of phi(n) from its limit l is less than Delta for all n greater than or equal to n_0(Delta); this is the test in Definition I.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "\\phi", "meaning": "function of n" }, { "unit": null, "symbol": "l", "meaning": "the limit" }, { "unit": null, "symbol": "\\DELTA", "meaning": "any given positive number, however small" } ], "sympy": "Lt(Abs(phi(n) - l), Delta)", "physics": false, "states": [], "concepts": [ "concept/absolute-value", "concept/congruence", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-fbecf63fcf", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "117", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\phi(n) > \\Delta", "name": null, "statement": "phi(n) exceeds any given number Delta for all n greater than or equal to n_0(Delta); this is the test in Definition II for tending to plus infinity.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "\\phi", "meaning": "function of n" }, { "unit": null, "symbol": "\\Delta", "meaning": "any given number, however large" } ], "sympy": "Gt(phi(n), Delta)", "physics": false, "states": [], "concepts": [ "concept/inequality", "concept/infinity", "concept/tends-to-infinity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e64e78d1e0", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "118", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\phi(n) \\to +\\infty", "name": null, "statement": "phi(n) tends to positive infinity as n tends to infinity.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "\\phi", "meaning": "function of n" }, { "unit": null, "symbol": "n", "meaning": "positive integral variable" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/infinity", "concept/limit", "concept/tends-to-infinity" ], "pages": [ "118", "120" ], "chapters": [ "hardy-course-of-pure-mathematics-1921/ch-iv" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-1d674baf2e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "116", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "n_{0} = 1 + [1/\\DELTA]", "name": null, "statement": "For phi(n) = 1/n, the choice n_0 = 1 + [1/Delta] (with [x] the greatest integer not greater than x) satisfies the condition 1/n < Delta for n at or above n_0.", "kind": "result", "symbols": [ { "unit": null, "symbol": "n_{0}", "meaning": "the index beyond which the limit condition holds" }, { "unit": null, "symbol": "\\DELTA", "meaning": "given positive number" }, { "unit": null, "symbol": "[x]", "meaning": "greatest integer not greater than x (integer part)" } ], "sympy": "Eq(n_0, 1 + floor(1/Delta))", "physics": false, "states": [], "concepts": [ "concept/integer-part", "concept/integer-part-function", "concept/limit", "concept/sufficiently-large-values" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-2859d478d3", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "113", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "1/n < \\DELTA", "name": null, "statement": "For phi(n) = 1/n the inequality 1/n < Delta holds for all n greater than 1/Delta, so the sufficiently large values of n need only exceed 1/Delta.", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "positive integral variable" }, { "unit": null, "symbol": "\\DELTA", "meaning": "given positive number" } ], "sympy": "Lt(1/n, Delta)", "physics": false, "states": [], "concepts": [ "concept/inequality", "concept/limit", "concept/sufficiently-large-values" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-8356ab3815", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "111", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "1000\\{1 + (-1)^{n}\\}/n < 1", "name": null, "statement": "The inequality 1000(1 + (-1)^n)/n < 1 holds for all n greater than 2000, the exceptions being the even values 2 to 2000.", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "positive integral variable" } ], "sympy": "Lt(1000*(1 + (-1)**n)/n, 1)", "physics": false, "states": [], "concepts": [ "concept/even-function", "concept/inequality", "concept/sufficiently-large-values" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d775accb12", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "120", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\phi(n) = n^{k}", "name": null, "statement": "The function phi(n) is the power n^k, where k is a positive or negative integer or rational fraction.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "function of the positive integer variable n" }, { "unit": null, "symbol": "n", "meaning": "positive integer variable" }, { "unit": null, "symbol": "k", "meaning": "a positive or negative integer or rational fraction (the exponent)" } ], "sympy": "Eq(phi(n), n**k)", "physics": false, "states": [], "concepts": [ "concept/function-of-a-positive-integer-variable", "concept/infinity", "concept/integer", "concept/limit", "concept/rational-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-973f1edbbe", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "120", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\lim n^{k} = 0", "name": null, "statement": "If k is negative, n^k tends to the limit 0 as n tends to infinity.", "kind": "result", "symbols": [ { "unit": null, "symbol": "k", "meaning": "negative exponent" }, { "unit": null, "symbol": "n", "meaning": "positive integer variable" } ], "sympy": "Eq(Limit(n**k, n, oo), 0)", "physics": false, "states": [], "concepts": [ "concept/limit", "concept/tends-to-infinity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-127b8f88bd", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "120", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\lim n^{k} = 1", "name": null, "statement": "If k = 0, then n^k = 1 for every n, so the limit is 1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "k", "meaning": "exponent equal to zero" }, { "unit": null, "symbol": "n", "meaning": "positive integer variable" } ], "sympy": "Eq(Limit(n**k, n, oo), 1)", "physics": false, "states": [], "concepts": [ "concept/constant", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b3174e71c5", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "120", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\phi(n) = p_{n}", "name": null, "statement": "phi(n) is the n-th prime number p_n.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "function of the positive integer variable n" }, { "unit": null, "symbol": "p_{n}", "meaning": "the n-th prime number" }, { "unit": null, "symbol": "n", "meaning": "positive integer index" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/function-of-a-positive-integer-variable", "concept/prime-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ff0d7d4f3f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "120", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\phi(n) > n", "name": null, "statement": "For the prime-number function, phi(n) is greater than n for all n except n = 1, 2, 3.", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "the n-th prime number p_n" }, { "unit": null, "symbol": "n", "meaning": "positive integer variable" } ], "sympy": "Gt(phi(n), n)", "physics": false, "states": [], "concepts": [ "concept/inequality", "concept/prime-number", "concept/tends-to-infinity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-3c8eada3cb", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "121", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\phi(n) = [\\alpha n]", "name": null, "statement": "phi(n) is the integer part of alpha times n, where alpha is any positive number.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "[x]", "meaning": "integer part (greatest integer not exceeding x)" }, { "unit": null, "symbol": "alpha", "meaning": "any positive number" }, { "unit": null, "symbol": "n", "meaning": "positive integer variable" } ], "sympy": "Eq(phi(n), floor(alpha*n))", "physics": false, "states": [], "concepts": [ "concept/function-of-a-positive-integer-variable", "concept/integer-part" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e1a4954646", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "121", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\phi(n) = 0\\quad (0 \\leq n < 1 / \\alpha)", "name": null, "statement": "For alpha positive, the integer-part function phi(n) = [alpha n] is 0 while n lies between 0 and 1/alpha.", "kind": "result", "symbols": [ { "unit": null, "symbol": "alpha", "meaning": "any positive number" }, { "unit": null, "symbol": "n", "meaning": "positive integer variable" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/integer-part" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e87e9add02", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "121", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\phi(n) = 1/\\{n - (-1)^{n}\\}", "name": null, "statement": "One example function in the list of behaviours as n tends to infinity; it tends to 0.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "function of the 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"states": [], "concepts": [ "concept/absolute-value", "concept/inequality", "concept/limit", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-5fb6e131f2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "122", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\phi(n) = (-1)^{n}", "name": null, "statement": "The simplest oscillatory function: equal to +1 for even n and to -1 for odd n.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "function of the positive integer variable n" }, { "unit": null, "symbol": "n", "meaning": "positive integer variable" } ], "sympy": "Eq(phi(n), (-1)**n)", "physics": false, "states": [], "concepts": [ "concept/function-of-a-positive-integer-variable", "concept/oscillation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a927e81010", "chapter": 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"page": "122", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\phi(n) = (-1)^{n}n", "name": null, "statement": "An oscillating function with no bound on its numerical value, so it oscillates infinitely.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "function of the positive integer variable n" }, { "unit": null, "symbol": "n", "meaning": "positive integer variable" } ], "sympy": "Eq(phi(n), (-1)**n*n)", "physics": false, "states": [], "concepts": [ "concept/infinity", "concept/oscillation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-07c4b568f1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "123", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\lim \\sin n\\theta\\pi = l", "name": null, "statement": "Supposition tested in the argument: sin(n theta pi) tends to a limit l, which the argument shows is impossible for irrational theta.", "kind": "result", "symbols": [ { "unit": null, "symbol": "l", "meaning": "a supposed limit of sin(n theta pi)" }, { "unit": null, "symbol": "theta", "meaning": "a real number" }, { "unit": null, "symbol": "n", "meaning": "positive integer variable" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/limit", "concept/oscillation", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a7eb5fcecd", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "123", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\cos(n + \\tfrac{1}{2})\\theta\\pi = \\cos n\\theta\\pi \\cos\\tfrac{1}{2}\\theta\\pi - \\sin n\\theta\\pi \\sin\\tfrac{1}{2}\\theta\\pi", "name": null, "statement": "The cosine of a sum of angles expands as cos a cos b minus sin a sin b, with a = n theta pi and b = theta pi / 2.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "n", "meaning": "positive integer variable" }, { "unit": null, "symbol": "theta", "meaning": "a real number" }, { "unit": null, "symbol": "pi", "meaning": "the ratio of circumference to diameter of a circle" } ], "sympy": "Eq(cos((n + Rational(1,2))*theta*pi), cos(n*theta*pi)*cos(Rational(1,2)*theta*pi) - sin(n*theta*pi)*sin(Rational(1,2)*theta*pi))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-9b023ec2f4", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "123", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\cos(n - \\tfrac{1}{2})\\theta\\pi = \\cos n\\theta\\pi \\cos\\tfrac{1}{2}\\theta\\pi + \\sin n\\theta\\pi \\sin\\tfrac{1}{2}\\theta\\pi", "name": null, "statement": "The cosine of a difference of angles expands as cos a cos b plus sin a sin b, with a = n theta pi and b = theta pi / 2.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "n", "meaning": "positive integer variable" }, { "unit": null, "symbol": "theta", "meaning": "a real number" }, { "unit": null, "symbol": "pi", "meaning": "the ratio of circumference to diameter of a circle" } ], "sympy": "Eq(cos((n - Rational(1,2))*theta*pi), cos(n*theta*pi)*cos(Rational(1,2)*theta*pi) + sin(n*theta*pi)*sin(Rational(1,2)*theta*pi))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-18a198b539", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "121", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\sin(np\\pi/q) = (-1)^{ap}\\sin(bp\\pi/q)", "name": null, "statement": "With n = aq + b, the sine of n p pi / q equals (-1)^(ap) times the sine of b p pi / q.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "n", "meaning": "positive integer variable" }, { "unit": null, "symbol": "p", "meaning": "positive integer numerator of theta = p/q" }, { "unit": null, "symbol": "q", "meaning": "positive integer denominator of theta = p/q" }, { "unit": null, "symbol": "a", "meaning": "the quotient of n divided by q" }, { "unit": null, "symbol": "b", "meaning": "the remainder of n divided by q" } ], "sympy": "Eq(sin(n*p*pi/q), (-1)**(a*p)*sin(b*p*pi/q))", "physics": false, "states": [], "concepts": [ "concept/integer", "concept/quotient", "concept/remainder", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-32b3ecf1c6", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "126", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "|\\phi(n) + \\psi(n) - a - b| < \\DELTA", "name": null, "statement": "The sum phi(n) + psi(n) lies within any assigned positive number Delta of a + b once n is large enough.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "function of the positive integer variable n" }, { "unit": null, "symbol": "psi", "meaning": "a second function of the positive integer variable n" }, { "unit": null, "symbol": "a", "meaning": "the limit of phi(n)" }, { "unit": null, "symbol": "b", "meaning": "the limit of psi(n)" }, { "unit": null, "symbol": "Delta", "meaning": "any assigned positive number" } ], "sympy": "Lt(Abs(phi(n) + psi(n) - a - b), Delta)", "physics": false, "states": [], "concepts": [ "concept/absolute-value", "concept/limit", "theorem/limit-of-a-sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-8720e04a30", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "126", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\lim\\{\\phi(n) + \\psi(n)\\} = a + b", "name": "Theorem I", "statement": "If phi(n) and psi(n) tend to limits a and b, then phi(n) + psi(n) tends to a + b.", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "function of the positive integer variable n" }, { "unit": null, "symbol": "psi", "meaning": "a second function of the positive integer variable n" }, { "unit": null, "symbol": "a", "meaning": "limit of phi(n)" }, { "unit": null, "symbol": "b", "meaning": "limit of psi(n)" } ], "sympy": "Eq(Limit(phi(n) + psi(n), n, oo), a + b)", "physics": false, "states": [ "theorem/intersecting-planes-meet-in-a-straight-line" ], "concepts": [ "concept/limit", "theorem/limit-of-a-sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b274e3d7c2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "126", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "|\\phi(n) + \\psi(n) - a - b| \\leq |\\phi(n) - a| + |\\psi(n) - b|", "name": null, "statement": "The modulus of a sum is at most the sum of the moduli, which bounds the error of phi + psi by the errors of phi and psi.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "function of the positive integer variable n" }, { "unit": null, "symbol": "psi", "meaning": "a second function of the positive integer variable n" }, { "unit": null, "symbol": "a", "meaning": "limit of phi(n)" }, { "unit": null, "symbol": "b", "meaning": "limit of psi(n)" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/absolute-value", "concept/inequality", "theorem/limit-of-a-sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-4d3331cb1b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "128", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\phi(n)\\psi(n) = ab + a\\psi_{1}(n) + b\\phi_{1}(n) + \\phi_{1}(n)\\psi_{1}(n)", "name": null, "statement": "Writing phi = a + phi1 and psi = b + psi1, the product phi psi expands into ab plus three error terms.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "limit of phi(n)" }, { "unit": null, "symbol": "b", "meaning": "limit of psi(n)" }, { "unit": null, "symbol": "phi1", "meaning": "phi(n) minus its limit a, which tends to 0" }, { "unit": null, "symbol": "psi1", "meaning": "psi(n) minus its limit b, which tends to 0" } ], "sympy": "Eq(phi*psi, a*b + a*psi1 + b*phi1 + phi1*psi1)", "physics": false, "states": [], "concepts": [ "concept/limit", "concept/product", "theorem/limit-of-a-product" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-5ee96e8e96", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "128", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\lim\\phi(n)\\psi(n) = ab", "name": "Theorem II", "statement": "If phi(n) tends to a and psi(n) tends to b, then the product phi(n) psi(n) tends to ab.", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "function of the positive integer variable n" }, { "unit": null, "symbol": "psi", "meaning": "a second function of the positive integer variable n" }, { "unit": null, "symbol": "a", "meaning": "limit of phi(n)" }, { "unit": null, "symbol": "b", "meaning": "limit of psi(n)" } ], "sympy": "Eq(Limit(phi(n)*psi(n), n, oo), a*b)", "physics": false, "states": [ "theorem/mutually-bounded-increasing-sequences-have-the-same-limit" ], "concepts": [ "concept/limit", "theorem/limit-of-a-product" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c1f934c5de", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "129", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\lim k\\phi(n) = ka", "name": null, "statement": "If phi(n) tends to a limit a, then k times phi(n) tends to k a, for a constant k.", "kind": "result", "symbols": [ { "unit": null, "symbol": "k", "meaning": "a constant multiplier" }, { "unit": null, "symbol": "a", "meaning": "limit of phi(n)" } ], "sympy": "Eq(Limit(k*phi(n), n, oo), k*a)", "physics": false, "states": [], "concepts": [ "concept/constant", "concept/limit", "theorem/limit-of-a-constant-multiple" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d51186254d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "130", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\left|\\frac{1}{\\phi(n)} - \\frac{1}{a}\\right| = \\frac{|\\phi_{1}(n)|}{|a| |a + \\phi_{1}(n)|}", "name": null, "statement": "The difference between 1/phi(n) and 1/a equals |phi1(n)| divided by |a| times |phi(n)|, which shows it becomes small when phi1 does.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "function of the positive integer variable n" }, { "unit": null, "symbol": "a", "meaning": "nonzero limit of phi(n)" }, { "unit": null, "symbol": "phi1", "meaning": "phi(n) minus a, which tends to 0" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/absolute-value", "concept/quotient", "concept/reciprocal" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-78033223a3", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "129", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\lim\\frac{1}{\\phi(n)} = \\frac{1}{a}", "name": "Theorem III", "statement": "If phi(n) tends to a nonzero limit a, then 1/phi(n) tends to 1/a.", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "function of the positive integer variable n" }, { "unit": null, "symbol": "a", "meaning": "nonzero limit of phi(n)" } ], "sympy": "Eq(Limit(1/phi(n), n, oo), 1/a)", "physics": false, "states": [ "theorem/mutually-bounded-decreasing-sequences-have-the-same-limit" ], "concepts": [ "concept/limit", "concept/quotient", "theorem/limit-of-a-quotient" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-2ceb827003", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "130", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\lim\\frac{\\phi(n)}{\\psi(n)} = \\frac{a}{b}", "name": "Theorem IV", "statement": "If phi(n) tends to a and psi(n) tends to a nonzero b, then phi(n)/psi(n) tends to a/b.", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "function of the positive integer variable n" }, { "unit": null, "symbol": "psi", "meaning": "a second function of the positive integer variable n" }, { "unit": null, "symbol": "a", "meaning": "limit of phi(n)" }, { "unit": null, "symbol": "b", "meaning": "nonzero limit of psi(n)" } ], "sympy": "Eq(Limit(phi(n)/psi(n), n, oo), a/b)", "physics": false, "states": [ "theorem/through-a-point-there-is-one-plane-perpendicular-to-a-line" ], "concepts": [ "concept/limit", "concept/quotient", "theorem/limit-of-a-quotient" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ae75aafe52", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "130", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\lim R\\{\\phi(n), \\psi(n), \\chi(n), \\dots\\} = R(a, b, c, \\dots)", "name": "Theorem V", "statement": "If each of phi, psi, chi tends to a limit and the denominator of the rational function R does not vanish at those limits, then R tends to R evaluated at the limits.", "kind": "result", "symbols": [ { "unit": null, "symbol": "R", "meaning": "rational function of phi(n), psi(n), chi(n), ..." }, { "unit": null, "symbol": "a, b, c", "meaning": "limits of phi(n), psi(n), chi(n)" } ], "sympy": null, "physics": false, "states": [ "theorem/through-a-point-there-is-one-line-perpendicular-to-a-plane" ], "concepts": [ "concept/limit", "concept/rational-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b1ffb49130", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "130", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "S(n) = \\frac{a_{0}n^{p} + a_{1}n^{p-1} + \\dots + a_{p}} {b_{0}n^{q} + b_{1}n^{q-1} + \\dots + b_{q}}", "name": null, "statement": "The most general rational function of n, with leading coefficients a0 and b0 not zero, whose behaviour as n tends to infinity is studied.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "S(n)", "meaning": "rational function of n" }, { "unit": null, "symbol": "n", "meaning": "positive integer variable" }, { "unit": null, "symbol": "a_{0}, ..., a_{p}", "meaning": "coefficients of the numerator polynomial" }, { "unit": null, "symbol": "b_{0}, ..., b_{q}", "meaning": "coefficients of the denominator polynomial" }, { "unit": null, "symbol": "p", "meaning": "degree of the numerator" }, { "unit": null, "symbol": "q", "meaning": "degree of the denominator" } ], "sympy": "Eq(S(n), (a0*n**p + a1*n**(p-1) + a_p)/(b0*n**q + b1*n**(q-1) + b_q))", "physics": false, "states": [], "concepts": [ "concept/limit", "concept/rational-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-dc8abe6dd2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "131", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\lim S(n) = 0\\quad (p < q)", "name": null, "statement": "If the numerator's degree p is less than the denominator's degree q, S(n) tends to 0.", "kind": "result", "symbols": [ { "unit": null, "symbol": "S(n)", "meaning": "rational function of n" }, { "unit": null, "symbol": "p", "meaning": "degree of the numerator" }, { "unit": null, "symbol": "q", "meaning": "degree of the denominator" } ], "sympy": "Eq(Limit(S(n), n, oo), 0)", "physics": false, "states": [], "concepts": [ "concept/limit", "concept/rational-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-f8b5d14936", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "131", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\lim S(n) = a_{0}/b_{0}\\quad (p = q)", "name": null, "statement": "If the numerator and denominator have equal degree p = q, S(n) tends to the ratio a0/b0 of leading coefficients.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a_{0}", "meaning": "leading coefficient of the numerator" }, { "unit": null, "symbol": "b_{0}", "meaning": "leading coefficient of the denominator" }, { "unit": null, "symbol": "p", "meaning": "degree of the numerator" }, { "unit": null, "symbol": "q", "meaning": "degree of the denominator" } ], "sympy": "Eq(Limit(S(n), n, oo), a0/b0)", "physics": false, "states": [], "concepts": [ "concept/limit", "concept/quotient", "concept/rational-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-1f005cb506", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "143", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "u_{n} = r^{n-1}", "name": null, "statement": "The general term of the geometrical series is r raised to the power n-1.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "u_{n}", "meaning": "general term of the series" }, { "unit": null, "symbol": "r", "meaning": "common ratio of the geometrical series" }, { "unit": null, "symbol": "n", "meaning": "index of the term (positive integer)" } ], "sympy": "Eq(u_n, r**(n-1))", "physics": false, "states": [], "concepts": [ "concept/geometrical-progression", "concept/infinite-sequence" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-7b248dcb7e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "143", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "s_{n} = 1 + r + r^{2} + \\dots + r^{n-1} = (1 - r^{n})/(1 - r)", "name": null, "statement": "The sum of the first n terms of the geometrical series is (1 - r^n)/(1 - r), for r not equal to 1.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "s_{n}", "meaning": "sum of the first n terms of the geometrical series" }, { "unit": null, "symbol": "r", "meaning": "common ratio" }, { "unit": null, "symbol": "n", "meaning": "number of terms" } ], "sympy": "Eq(s_n, (1 - r**n)/(1 - r))", "physics": false, "states": [], "concepts": [ "concept/geometrical-progression", "concept/limit", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-cbaf5acf74", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "143", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "s_{n} = 1 + 1 + \\dots + 1 = n", "name": null, "statement": "In the special case r = 1, the sum of the first n terms equals n.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "s_{n}", "meaning": "sum of the first n terms" }, { "unit": null, "symbol": "n", "meaning": "number of terms" } ], "sympy": "Eq(s_n, n)", "physics": false, "states": [], "concepts": [ "concept/geometrical-progression", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ba6e8675c9", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "143", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "s_{n} \\to +\\infty", "name": null, "statement": "When r = 1 the partial sums s_n increase without limit.", "kind": "result", "symbols": [ { "unit": null, "symbol": "s_{n}", "meaning": "sum of the first n terms" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/infinity", "concept/limit", "concept/tends-to-infinity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ec8a22f1f1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "143", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "s_{n} \\geq n", "name": null, "statement": "If r is at least 1, the partial sum s_n is at least n, so s_n tends to plus infinity.", "kind": "result", "symbols": [ { "unit": null, "symbol": "s_{n}", "meaning": "sum of the first n terms" }, { "unit": null, "symbol": "n", "meaning": "number of terms" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/divergent-series", "concept/geometrical-progression", "concept/tends-to-infinity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-677cebf27f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "143", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "1/(1 - r)", "name": null, "statement": "The series 1 + r + r^2 + ... is convergent with this sum if and only if -1 < r < 1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "r", "meaning": "common ratio" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/convergent-series", "concept/geometrical-progression", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-6f69fdb020", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "147", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "f(x) = \\lim_{n \\to \\infty} n(\\sqrt[n]{x} - 1)", "name": null, "statement": "The function f(x) is defined as the limit as n tends to infinity of n times (the n-th root of x minus 1); this is the function encountered in Section 75.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "f(x)", "meaning": "function of x defined as a limit" }, { "unit": null, "symbol": "x", "meaning": "independent real variable" }, { "unit": null, "symbol": "n", "meaning": "positive integer variable tending to infinity" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/function", "concept/function-of-a-positive-integer-variable", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d61b44e8ef", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "147", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "u_{1}(x) + u_{2}(x) + \\dots = \\lim_{n \\to \\infty}\\{u_{1}(x) + u_{2}(x) + \\dots + u_{n}(x)\\}", "name": null, "statement": "The sum of an infinite series of functions of x is defined as the limit of its partial sums as n tends to infinity.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "u_{n}(x)", "meaning": "n-th term of the series, a function of x" }, { "unit": null, "symbol": "n", "meaning": "number of terms in the partial sum" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/convergent-series", "concept/infinite-sequence", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a9482dbff2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "147", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\lim_{n \\to \\infty} \\phi_{n}(x)", "name": null, "statement": "The limit as n tends to infinity of phi_n(x) is, for each x, a function of x; this is the function represented by the limit.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "phi_{n}(x)", "meaning": "function of n and x" }, { "unit": null, "symbol": "x", "meaning": "continuous real variable" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/continuous-real-variable", "concept/function-of-a-positive-integer-variable", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-960dd58fe7", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "149", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "s \\leq K", "name": null, "statement": "A set S of real numbers is bounded above if some number K satisfies s at most K for every member s of S.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "s", "meaning": "member of the aggregate S" }, { "unit": null, "symbol": "K", "meaning": "a number bounding S above" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/bounded-function", "concept/least-upper-bound", "concept/set" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-87b3725757", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "149", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "s \\geq k", "name": null, "statement": "A set S of real numbers is bounded below if some number k satisfies s at least k for every member s of S.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "s", "meaning": "member of the aggregate S" }, { "unit": null, "symbol": "k", "meaning": "a number bounding S below" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/bounded-function", "concept/least-upper-bound", "concept/set" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-3fccc8b42b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "149", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "k \\leq m \\leq M \\leq K", "name": null, "statement": "When a set is bounded, its lower bound m and upper bound M lie between any lower bound k and any upper bound K.", "kind": "result", "symbols": [ { "unit": null, "symbol": "k", "meaning": "a lower bound of the set" }, { "unit": null, "symbol": "m", "meaning": "lower bound of the set" }, { "unit": null, "symbol": "M", "meaning": "upper bound of the set" }, { "unit": null, "symbol": "K", "meaning": "an upper bound of the set" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/bounded-function", "concept/least-upper-bound" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-caede7193f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "150", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\phi(n) \\leq M", "name": null, "statement": "The upper bound M of a bounded-above function phi(n) is not exceeded by any of its values.", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi(n)", "meaning": "bounded function of the positive integer variable n" }, { "unit": null, "symbol": "M", "meaning": "upper bound of phi(n)" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/bounded-function", "concept/function-of-a-positive-integer-variable", "concept/least-upper-bound" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-711dd0b189", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "150", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\phi(n) > M - \\DELTA", "name": null, "statement": "For any positive number DELTA, some value of phi(n) exceeds M minus DELTA.", "kind": "result", "symbols": [ { "unit": null, "symbol": "DELTA", "meaning": "any positive number" }, { "unit": null, "symbol": "M", "meaning": "upper bound of phi(n)" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/bounded-function", "concept/least-upper-bound" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-9a4a59c1b7", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "150", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\phi(n) \\geq m", "name": null, "statement": "The lower bound m of a bounded-below function phi(n) is not exceeded from below by any of its values.", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi(n)", "meaning": "bounded function of the positive integer variable n" }, { "unit": null, "symbol": "m", "meaning": "lower bound of phi(n)" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/bounded-function", "concept/function-of-a-positive-integer-variable", "concept/least-upper-bound" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-965ea2bc61", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "150", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\phi(n) < m + \\DELTA", "name": null, "statement": "For any positive number DELTA, some value of phi(n) is less than m plus DELTA.", "kind": "result", "symbols": [ { "unit": null, "symbol": "DELTA", "meaning": "any positive number" }, { "unit": null, "symbol": "m", "meaning": "lower bound of phi(n)" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/bounded-function", "concept/least-upper-bound" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-52c51d96ab", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "150", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\phi(n) \\leq k", "name": null, "statement": "The book states the bounded-below condition with this inequality, but the lower bound is defined by values at least k; this appears to be a misprint for phi(n) >= k (flagged, not corrected).", "kind": "definition", "symbols": [ { "unit": null, "symbol": "phi(n)", "meaning": "function of the positive integer variable n" }, { "unit": null, "symbol": "k", "meaning": "a number bounding phi(n) below" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/bounded-function", "concept/least-upper-bound" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-bc8d51a5ec", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "150", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "m \\leq \\lambda \\leq \\Lambda \\leq M", "name": null, "statement": "The lower and upper limits of indetermination lie between the lower and upper bounds of the function.", "kind": "result", "symbols": [ { "unit": null, "symbol": "m", "meaning": "lower bound of phi(n)" }, { "unit": null, "symbol": "lambda", "meaning": "lower limit of indetermination of phi(n)" }, { "unit": null, "symbol": "Lambda", "meaning": "upper limit of indetermination of phi(n)" }, { "unit": null, "symbol": "M", "meaning": "upper bound of phi(n)" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/bounded-function", "concept/limit-inferior", "concept/limit-superior" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a57de6f0b8", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "150", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\Lambda = \\limsup \\phi(n)", "name": null, "statement": "The upper limit of indetermination of phi(n) as n tends to infinity is defined as the limit superior.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "Lambda", "meaning": "upper limit of indetermination of phi(n)" }, { "unit": null, "symbol": "phi(n)", "meaning": "bounded function of n" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/limit", "concept/limit-superior" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-0c4e905fb8", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "150", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\lambda = \\liminf \\phi(n)", "name": null, "statement": "The lower limit of indetermination of phi(n) as n tends to infinity is defined as the limit inferior.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "lambda", "meaning": "lower limit of indetermination of phi(n)" }, { "unit": null, "symbol": "phi(n)", "meaning": "bounded function of n" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/limit", "concept/limit-inferior" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-3187caff78", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "150", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\phi(n) < \\Lambda + \\DELTA", "name": null, "statement": "For any positive DELTA, phi(n) is less than Lambda plus DELTA for all sufficiently large n.", "kind": "result", "symbols": [ { "unit": null, "symbol": "DELTA", "meaning": "any positive number" }, { "unit": null, "symbol": "Lambda", "meaning": "upper limit of indetermination" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/limit-superior", "concept/sufficiently-large-values" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-dc448812b0", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "150", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\phi(n) > \\Lambda - \\DELTA", "name": null, "statement": "For any positive DELTA, phi(n) exceeds Lambda minus DELTA for infinitely many values of n.", "kind": "result", "symbols": [ { "unit": null, "symbol": "DELTA", "meaning": "any positive number" }, { "unit": null, "symbol": "Lambda", "meaning": "upper limit of indetermination" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/infinity", "concept/limit-superior" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-75d736a5a6", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "151", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "l - \\DELTA < \\phi(n) < l + \\DELTA", "name": null, "statement": "phi(n) tends to the limit l exactly when, for every positive DELTA, phi(n) lies between l minus DELTA and l plus DELTA for all sufficiently large n.", "kind": "result", "symbols": [ { "unit": null, "symbol": "l", "meaning": "the limit of phi(n)" }, { "unit": null, "symbol": "DELTA", "meaning": "any positive number" }, { "unit": null, "symbol": "phi(n)", "meaning": "bounded function of n" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/limit", "concept/sufficiently-large-values" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-9047e0caec", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "151", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\Lambda - \\lambda \\leq 2\\DELTA", "name": null, "statement": "If phi(n) tends to l, the gap between the upper and lower limits of indetermination is at most 2 DELTA for every positive DELTA, so they are equal.", "kind": "result", "symbols": [ { "unit": null, "symbol": "Lambda", "meaning": "upper limit of indetermination" }, { "unit": null, "symbol": "lambda", "meaning": "lower limit of indetermination" }, { "unit": null, "symbol": "DELTA", "meaning": "any positive number" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/limit", "concept/limit-inferior", "concept/limit-superior" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-6a87c80af3", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "152", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "|\\phi(n_{2}) - \\phi(n_{1})| < \\DELTA", "name": "general principle of convergence", "statement": "A bounded function tends to a limit if and only if, for any positive DELTA, phi(n_2) and phi(n_1) differ by less than DELTA whenever n_2 > n_1 >= n_0(DELTA).", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi(n)", "meaning": "bounded function of the positive integer variable n" }, { "unit": null, "symbol": "DELTA", "meaning": "any positive number" }, { "unit": null, "symbol": "n_{0}(DELTA)", "meaning": "a number depending on DELTA beyond which the condition holds" }, { "unit": null, "symbol": "n_{1}, n_{2}", "meaning": "positive integers" } ], "sympy": null, "physics": false, "states": [ "theorem/general-principle-of-convergence" ], "concepts": [ "concept/bounded-function", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-f01d262e0d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "153", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "|u_{n_{1}+1} + u_{n_{1}+2} + \\dots + u_{n_{2}}| < \\DELTA", "name": null, "statement": "The series u_1 + u_2 + ... converges if and only if, for any positive DELTA, the sums of consecutive blocks of terms beyond some n_0 are all less than DELTA in absolute value.", "kind": "result", "symbols": [ { "unit": null, "symbol": "u_{n}", "meaning": "n-th term of the series" }, { "unit": null, "symbol": "DELTA", "meaning": "any positive number" }, { "unit": null, "symbol": "n_{1}, n_{2}", "meaning": "positive integers with n_2 > n_1 >= n_0" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/convergent-series", "concept/infinite-sequence", "theorem/general-principle-of-convergence" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-7ea4a8d07b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "152", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\phi(n_{1}) - \\DELTA < \\phi(n_{2}) < \\phi(n_{1}) + \\DELTA", "name": null, "statement": "Fixing one value n_1 beyond n_0 bounds every later value of phi(n) within DELTA of phi(n_1), so phi(n) is bounded.", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi(n)", "meaning": "function of the positive integer variable n" }, { "unit": null, "symbol": "DELTA", "meaning": "any positive number" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/bounded-function", "concept/unbounded-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b7dea93b92", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "153", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\rho(n) + i\\sigma(n)", "name": null, "statement": "A complex function phi(n) is written as its real part rho(n) plus i times its imaginary part sigma(n), both real functions of n.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "rho(n)", "meaning": "real part of phi(n)" }, { "unit": null, "symbol": "sigma(n)", "meaning": "imaginary part of phi(n), a real function" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/function-of-a-positive-integer-variable" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-50b8e2840a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "153", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\lim\\phi(n) = l", "name": null, "statement": "A complex function phi(n) converges to l = r + is when its real and imaginary parts converge to r and s respectively.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "phi(n)", "meaning": "complex function of n" }, { "unit": null, "symbol": "l", "meaning": "complex limit" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-2aa3eb113b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "153", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "l = r + is", "name": null, "statement": "The complex limit l is the real part r plus i times the imaginary part s.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "l", "meaning": "complex limit" }, { "unit": null, "symbol": "r", "meaning": "real part of the limit" }, { "unit": null, "symbol": "s", "meaning": "imaginary part of the limit (real number)" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": "Eq(l, r + I*s)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-670b9c0fcf", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "153", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "s_{n} = u_{1} + u_{2} + \\dots + u_{n}", "name": null, "statement": "The partial sum s_n is the sum of the first n terms of the series.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "s_{n}", "meaning": "partial sum of the first n terms" }, { "unit": null, "symbol": "u_{n}", "meaning": "n-th term of the series" } ], "sympy": "Eq(s_n, u_1 + u_2 + u_n)", "physics": false, "states": [], "concepts": [ "concept/convergent-series", "concept/infinite-sequence", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-8c7c184fea", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "154", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "(v_{1} + v_{2} + \\dots + v_{n}) + i(w_{1} + w_{2} + \\dots + w_{n})", "name": null, "statement": "The partial sum of a complex series equals the sum of its real parts plus i times the sum of its imaginary parts, so it converges to l exactly when those real and imaginary series converge.", "kind": "result", "symbols": [ { "unit": null, "symbol": "v_{n}", "meaning": "real part of u_n" }, { "unit": null, "symbol": "w_{n}", "meaning": "imaginary part of u_n" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/convergent-series", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-f751accd9b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "154", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\lim\\phi(n + p) = l", "name": null, "statement": "If φ(n) tends to l, then φ(n + p) also tends to l for any fixed p.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\phi(n)", "meaning": "function of a positive integer variable n" }, { "unit": null, "symbol": "n", "meaning": "positive integer variable" }, { "unit": null, "symbol": "p", "meaning": "fixed value (shift)" }, { "unit": null, "symbol": "l", "meaning": "limit of φ(n)" } ], "sympy": "Eq(Limit(phi(n + p), n, oo), l)", "physics": false, "states": [], "concepts": [ "concept/function-of-a-positive-integer-variable", "concept/limit", "theorem/limit-of-a-shifted-sequence" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-5d9e596179", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "154", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\lim\\{\\phi(n) + \\psi(n)\\} = l + m", "name": null, "statement": "The limit of a sum of two sequences is the sum of their limits.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\phi(n)", "meaning": "function of a positive integer variable" }, { "unit": null, "symbol": "\\psi(n)", "meaning": "second function of a positive integer variable" }, { "unit": null, "symbol": "l", "meaning": "limit of φ(n)" }, { "unit": null, "symbol": "m", "meaning": "limit of ψ(n)" } ], "sympy": "Eq(Limit(phi(n) + psi(n), n, oo), l + m)", "physics": false, "states": [], "concepts": [ "concept/function-of-a-positive-integer-variable", "concept/limit", "theorem/limit-of-a-sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-de0f42f7aa", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "154", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\lim k\\phi(n) = kl", "name": null, "statement": "A constant k times a sequence tends to k times its limit.", "kind": "result", "symbols": [ { "unit": null, "symbol": "k", "meaning": "constant factor" }, { "unit": null, "symbol": "\\phi(n)", "meaning": "function of a positive integer variable" }, { "unit": null, "symbol": "l", "meaning": "limit of φ(n)" } ], "sympy": "Eq(Limit(k*phi(n), n, oo), k*l)", "physics": false, "states": [], "concepts": [ "concept/function-of-a-positive-integer-variable", "concept/limit", "theorem/limit-of-a-constant-multiple" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-f22be5bbc3", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "154", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\lim \\phi(n)\\psi(n) = lm", "name": null, "statement": "The limit of a product of two sequences is the product of their limits.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\phi(n)", "meaning": "function of a positive integer variable" }, { "unit": null, "symbol": "\\psi(n)", "meaning": "second function of a positive integer variable" }, { "unit": null, "symbol": "l", "meaning": "limit of φ(n)" }, { "unit": null, "symbol": "m", "meaning": "limit of ψ(n)" } ], "sympy": "Eq(Limit(phi(n)*psi(n), n, oo), l*m)", "physics": false, "states": [], "concepts": [ "concept/function-of-a-positive-integer-variable", "concept/limit", "theorem/limit-of-a-product" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-34b58259b8", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "154", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\lim u_{n} = 0", "name": null, "statement": "If the series u_1 + u_2 + u_3 + … converges, its terms u_n tend to zero.", "kind": "result", "symbols": [ { "unit": null, "symbol": "u_{n}", "meaning": "n-th term of a series" }, { "unit": null, "symbol": "n", "meaning": "positive integer index" } ], "sympy": "Eq(Limit(u_n, n, oo), 0)", "physics": false, "states": [], "concepts": [ "concept/convergent-series", "concept/limit", "concept/term" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-06338a424d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "155", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "|\\phi(n)| = \\sqrtbr{\\{\\rho(n)\\}^{2} + \\{\\sigma(n)\\}^{2}}", "name": null, "statement": "The modulus of a complex number φ(n) = ρ(n) + iσ(n) is the square root of the sum of the squares of its real and imaginary parts.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "\\phi(n)", "meaning": "complex function of a positive integer variable" }, { "unit": null, "symbol": "\\rho(n)", "meaning": "real part of φ(n)" }, { "unit": null, "symbol": 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n, equal to ρ'(n) + iσ'(n)" }, { "unit": null, "symbol": "\\rho", "meaning": "real part of φ(n)" }, { "unit": null, "symbol": "\\sigma", "meaning": "imaginary part of φ(n)" }, { "unit": null, "symbol": "\\rho'", "meaning": "real part of ψ(n)" }, { "unit": null, "symbol": "\\sigma'", "meaning": "imaginary part of ψ(n)" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": "Eq(phi*psi, rho*rho_p - sigma*sigma_p + I*(rho*sigma_p + rho_p*sigma))", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/product", "theorem/limit-of-a-product" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-753468e100", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "155", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\phi(n) = \\rho(n) + i\\sigma(n)", "name": null, "statement": "A complex function φ(n) is written as its real part ρ(n) plus i times its imaginary part σ(n).", "kind": "definition", "symbols": [ { "unit": null, "symbol": "\\phi(n)", "meaning": "complex function of a positive integer variable" }, { "unit": null, "symbol": "\\rho(n)", "meaning": "real part" }, { "unit": null, "symbol": "\\sigma(n)", "meaning": "imaginary part" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": "Eq(phi(n), rho(n) + I*sigma(n))", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/function-of-a-positive-integer-variable", "concept/real-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-86ab960aca", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "156", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "z^{n} = r^{n} (\\cos n\\theta + i\\sin n\\theta)", "name": null, "statement": "The n-th power of a complex number in polar form has modulus r^n and angle nθ.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "z", "meaning": "complex number" }, { "unit": null, "symbol": "n", "meaning": "positive integer exponent" }, { "unit": null, "symbol": "r", "meaning": "positive modulus of z" }, { "unit": null, "symbol": "\\theta", "meaning": "angle of z" } ], "sympy": "Eq(z**n, r**n*(cos(n*theta) + I*sin(n*theta)))", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/cosine", "concept/power", "concept/sine", "quantity/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-9c3a2cb72c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "156", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "|z^{n}| = r^{n}", "name": null, "statement": "The modulus of z^n equals r^n, where r is the modulus of z.", "kind": "result", "symbols": [ { "unit": null, "symbol": "z", "meaning": "complex number" }, { "unit": null, "symbol": "n", "meaning": "positive integer exponent" }, { "unit": null, "symbol": "r", "meaning": "modulus of z" } ], "sympy": "Eq(Abs(z**n), r**n)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/power", "quantity/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-27342cdaf9", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "156", "location": "LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE", "latex": "\\lim z^{n} = 0", "name": null, "statement": "For complex z with modulus r < 1, z^n tends to zero as n tends to infinity.", "kind": "result", "symbols": [ { "unit": null, "symbol": "z", "meaning": "complex number with |z| = r" }, { "unit": null, 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"id": "hardy-course-of-pure-mathematics-1921/eq-fd818fa0fc", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "163", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. 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CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\phi(x) \\to -\\infty", "name": null, "statement": "phi(x) tends to -infinity with x, the mirror of the definition of tending to +infinity.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "phi(x)", "meaning": "one-valued function of the continuous variable x" }, { "unit": null, "symbol": "x", "meaning": "continuous real variable" } ], "sympy": "Eq(Limit(phi(x), x, oo), -oo)", "physics": false, "states": [], "concepts": [ "concept/function", "concept/infinity", "concept/tends-to-infinity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-72a05b191f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "163", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. 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CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\lim_{y \\to 0} \\phi(y) = l", "name": null, "statement": "phi(y) tends to the limit l as y tends to 0 from both sides, with y different from zero.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "phi(y)", "meaning": "function of the variable y" }, { "unit": null, "symbol": "l", "meaning": "the limit" } ], "sympy": "Eq(Limit(phi(y), y, 0), l)", "physics": false, "states": [], "concepts": [ "concept/function", "concept/limit", "concept/neighbourhood" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e16a95e23c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "167", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. 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CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\lim_{x \\to a+0} \\phi(x) = l", "name": null, "statement": "phi(x) tends to l as x approaches a from the right, through values greater than a.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "phi(x)", "meaning": "function of the continuous variable x" }, { "unit": null, "symbol": "a", "meaning": "the point approached" }, { "unit": null, "symbol": "l", "meaning": "the limit" } ], "sympy": "Eq(Limit(phi(x), x, a, '+'), l)", "physics": false, "states": [], "concepts": [ "concept/function", "concept/limit", "concept/variable" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-f8126b6910", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "168", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\lim_{x \\to a+0} \\phi(x) = \\phi(a+0)", "name": null, "statement": "The right-hand limit of phi at a is written phi(a+0).", "kind": "definition", "symbols": [ { "unit": null, "symbol": "phi(a+0)", "meaning": "the limit of phi(x) as x tends to a from the right" }, { "unit": null, "symbol": "a", "meaning": "the point approached" } ], "sympy": "Eq(Limit(phi(x), x, a, '+'), phi(a + 0))", "physics": false, "states": [], "concepts": [ "concept/function", "concept/limit", "concept/mathematical-notation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c93ee7d619", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "168", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\phi(a-0) \\leq \\phi(a) \\leq \\phi(a+0)", "name": null, "statement": "For a function that is steadily increasing near a, its left-hand limit at a is no greater than its value at a, which is no greater than its right-hand limit.", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi(a-0)", "meaning": "left-hand limit of phi at a" }, { "unit": null, "symbol": "phi(a)", "meaning": "value of phi at a" }, { "unit": null, "symbol": "phi(a+0)", "meaning": "right-hand limit of phi at a" } ], "sympy": "And(Le(phi(a - 0), phi(a)), Le(phi(a), phi(a + 0)))", "physics": false, "states": [], "concepts": [ "concept/inequality", "concept/limit", "concept/steadily-increasing-function", "concept/value-of-a-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-4fe17211a8", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "170", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\lim\\phi(x) = 0", "name": null, "statement": "If phi(x) is identically zero, its limit as x tends to zero is zero.", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi(x)", "meaning": "the function defined as zero for all x" }, { "unit": null, "symbol": "x", "meaning": "continuous real variable" } ], "sympy": "Eq(Limit(phi(x), x, 0), 0)", "physics": false, "states": [], "concepts": [ "concept/function", "concept/limit", "concept/value-of-a-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-173e0b8f6d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "170", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\lim\\psi(x) = 0", "name": null, "statement": "A function equal to phi(x) except that psi(0) = 1 still has limit zero at x = 0, since the limit ignores the value at 0.", "kind": "result", "symbols": [ { "unit": null, "symbol": "psi(x)", "meaning": "function equal to zero except psi(0) = 1" }, { "unit": null, "symbol": "x", "meaning": "continuous real variable" } ], "sympy": "Eq(Limit(psi(x), x, 0), 0)", "physics": false, "states": [], "concepts": [ "concept/function", "concept/limit", "concept/value-of-a-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d22cfaa9c6", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "170", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\psi(x) = [1 - x^{2}]", "name": null, "statement": "psi(x) is the greatest integer not greater than 1 - x^2; it equals 1 at x = 0 and 0 for 0 < |x| < 1.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "psi(x)", "meaning": "function defined by the greatest-integer bracket" }, { "unit": null, "symbol": "[ ]", "meaning": "the greatest integer not greater than the enclosed number" }, { "unit": null, "symbol": "x", "meaning": "continuous real variable" } ], "sympy": "Eq(psi(x), floor(1 - x**2))", "physics": false, "states": [], "concepts": [ "concept/discontinuous-function", "concept/function", "concept/integer-part-function", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c4b2c5b181", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "171", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\lim(x/x) = 1", "name": null, "statement": "The function x/x has limit 1 as x tends to zero, although it is not defined at x = 0.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "continuous real variable" } ], "sympy": "Eq(Limit(x/x, x, 0), 1)", "physics": false, "states": [], "concepts": [ "concept/domain-of-definition", "concept/indeterminate-form", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d21fe0ccc9", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "171", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\lim\\phi(x) = 2", "name": null, "statement": "For phi(x) = ((x+1)^2 - 1)/x, which equals x + 2 for x not zero, the limit as x tends to zero is 2.", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi(x)", "meaning": "the function ((x+1)^2 - 1)/x, undefined at x = 0" }, { "unit": null, "symbol": "x", "meaning": "continuous real variable" } ], "sympy": "Eq(Limit(phi(x), x, 0), 2)", "physics": false, "states": [], "concepts": [ "concept/domain-of-definition", "concept/function", "concept/indeterminate-form", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-4b6a526511", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "171", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\phi(x) = \\{(x + 1)^{2} - 1\\}/x = x + 2", "name": null, "statement": "For x not zero, phi(x) equals x + 2, so it is defined by the expression except at x = 0.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "phi(x)", "meaning": "the function ((x+1)^2 - 1)/x" }, { "unit": null, "symbol": "x", "meaning": "continuous real variable" } ], "sympy": "Eq(phi(x), ((x + 1)**2 - 1)/x)", "physics": false, "states": [], "concepts": [ "concept/algebraic-expression", "concept/domain-of-definition", "concept/function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-32726bbc64", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "173", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\lim\\dfrac{\\sin x}{x} = 1", "name": null, "statement": "The ratio of sine x to x tends to 1 as x tends to zero.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "angle in radians" } ], "sympy": "Eq(Limit(sin(x)/x, x, 0), 1)", "physics": false, "states": [], "concepts": [ "concept/circular-function", "concept/indeterminate-form", "concept/limit", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-11c25a8710", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "173", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\sin x < x < \\tan x", "name": null, "statement": "For x positive and less than pi/2, sine x is less than x, which is less than tangent x; this is the inequality used to prove the limit of sin x over x.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "angle in radians, 0 < x < pi/2" } ], "sympy": "And(Lt(sin(x), x), Lt(x, tan(x)))", "physics": false, "states": [], "concepts": [ "concept/circular-function", "concept/inequality", "concept/sine", "concept/tangent-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-f356980e32", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "173", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\lim \\dfrac{1 - \\cos x}{x^{2}} = \\frac{1}{2}", "name": null, "statement": "The quantity 1 minus cosine x, divided by x squared, tends to one half as x tends to zero.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "angle in radians" } ], "sympy": "Eq(Limit((1 - cos(x))/x**2, x, 0), Rational(1, 2))", "physics": false, "states": [], "concepts": [ "concept/circular-function", "concept/cosine", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-bdae8fdd0c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "173", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\lim \\dfrac{\\arcsin x}{x} = 1", "name": null, "statement": "The ratio of the inverse sine of x to x tends to 1 as x tends to zero.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "number tending to zero" } ], "sympy": "Eq(Limit(asin(x)/x, x, 0), 1)", "physics": false, "states": [], "concepts": [ "concept/inverse-circular-function", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-32955812a8", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "173", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\lim \\dfrac{\\sin \\alpha x}{x} = \\alpha", "name": null, "statement": "The ratio of sin(alpha x) to x tends to alpha as x tends to zero.", "kind": "result", "symbols": [ { "unit": null, "symbol": "alpha", "meaning": "constant multiplier" }, { "unit": null, "symbol": "x", "meaning": "angle in radians" } ], "sympy": "Eq(Limit(sin(alpha*x)/x, x, 0), alpha)", "physics": false, "states": [], "concepts": [ "concept/circular-function", "concept/constant", "concept/limit", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-27965bce4f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "173", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\lim \\dfrac{\\tan \\alpha x}{x}= \\alpha", "name": null, "statement": "The ratio of tan(alpha x) to x tends to alpha as x tends to zero.", "kind": "result", "symbols": [ { "unit": null, "symbol": "alpha", "meaning": "constant multiplier" }, { "unit": null, "symbol": "x", "meaning": "angle in radians" } ], "sympy": "Eq(Limit(tan(alpha*x)/x, x, 0), alpha)", "physics": false, "states": [], "concepts": [ "concept/circular-function", "concept/limit", "concept/tangent-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-dee76a8f42", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "173", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\lim \\dfrac{\\cosec x - \\cot x}{x} = \\frac{1}{2}", "name": null, "statement": "The difference cosecant x minus cotangent x, divided by x, tends to one half as x tends to zero.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "angle in radians" } ], "sympy": "Eq(Limit((1/sin(x) - cos(x)/sin(x))/x, x, 0), Rational(1, 2))", "physics": false, "states": [], "concepts": [ "concept/circular-function", "concept/cosecant", "concept/cotangent", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-5f1c9d1de8", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "173", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\lim\\limits_{x \\to 1} \\dfrac{1 + \\cos \\pi x}{\\tan^{2}\\pi x} = \\frac{1}{2}", "name": null, "statement": "The ratio of 1 + cos(pi x) to tan^2(pi x) tends to one half as x tends to 1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "continuous real variable tending to 1" } ], "sympy": "Eq(Limit((1 + cos(pi*x))/tan(pi*x)**2, x, 1), Rational(1, 2))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/indeterminate-form", "concept/limit", "concept/tangent-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-13dd732abe", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "171", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\lim\\limits_{x \\to a} (x^{2} - a^{2})/(x - a) = 2a", "name": null, "statement": "The difference quotient of x squared tends to 2a as x tends to a.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "fixed number, the point approached" }, { "unit": null, "symbol": "x", "meaning": "continuous real variable" } ], "sympy": "Eq(Limit((x**2 - a**2)/(x - a), x, a), 2*a)", "physics": false, "states": [], "concepts": [ "concept/difference", "concept/indeterminate-form", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-f381ef55bd", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "171", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\lim\\limits_{x \\to a} (x^{m} - a^{m})/(x - a) = ma^{m-1}", "name": null, "statement": "For any integer m, the difference quotient of x^m tends to m a^(m-1) as x tends to a.", "kind": "result", "symbols": [ { "unit": null, "symbol": "m", "meaning": "integer exponent" }, { "unit": null, "symbol": "a", "meaning": "fixed number, the point approached" } ], "sympy": "Eq(Limit((x**m - a**m)/(x - a), x, a), m*a**(m - 1))", "physics": false, "states": [], "concepts": [ "concept/indeterminate-form", "concept/integer", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-15df879609", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "171", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\lim\\limits_{x \\to 1} (x^{7} - 2x^{5} + 1)/(x^{3} - 3x^{2} + 2) = 1", "name": null, "statement": "The ratio of the two polynomials tends to 1 as x tends to 1, since x - 1 is a factor of both.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "continuous real variable tending to 1" } ], "sympy": "Eq(Limit((x**7 - 2*x**5 + 1)/(x**3 - 3*x**2 + 2), x, 1), 1)", "physics": false, "states": [], "concepts": [ "concept/factor", "concept/indeterminate-form", "concept/limit", "concept/polynomial", "concept/rational-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c9750fc98e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "169", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\lim\\limits_{x \\to a} x^{m} = a^{m}", "name": null, "statement": "For any integer m (except a = 0 with m negative), x^m tends to a^m as x tends to a.", "kind": "result", "symbols": [ { "unit": null, "symbol": "m", "meaning": "integer exponent" }, { "unit": null, "symbol": "a", "meaning": "fixed number, the point approached" } ], "sympy": "Eq(Limit(x**m, x, a), a**m)", "physics": false, "states": [], "concepts": [ "concept/integer", "concept/limit", "concept/power" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-5953441da9", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "169", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\lim\\limits_{x \\to a} R(x) = R(a)", "name": null, "statement": "A rational function R tends to its value at a whenever a is not a root of its denominator.", "kind": "result", "symbols": [ { "unit": null, "symbol": "R(x)", "meaning": "rational function of x" }, { "unit": null, "symbol": "a", "meaning": "point approached, not a root of the denominator" } ], "sympy": "Eq(Limit(R(x), x, a), R(a))", "physics": false, "states": [], "concepts": [ "concept/limit", "concept/rational-function", "concept/value-of-a-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e1444cb45d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "169", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\lim\\limits_{x \\to 0} (a + bx + cx^{2} + \\dots + kx^{m}) = a", "name": null, "statement": "A polynomial in x tends to its constant term as x tends to zero.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a, b, c, k", "meaning": "constant coefficients" }, { "unit": null, "symbol": "m", "meaning": "positive integer, degree of the polynomial" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/limit", "concept/polynomial" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-0e226d43ac", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "169", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\lim\\limits_{x \\to 0} \\left\\{(a + bx + \\dots + kx^{m})/(\\alpha + \\beta x + \\dots + \\kappa x^{\\mu})\\right\\} = a/\\alpha", "name": null, "statement": "The ratio of two polynomials in x tends to a over alpha as x tends to zero, provided alpha is not zero.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a, alpha", "meaning": "constant leading coefficients of numerator and denominator" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/limit", "concept/polynomial", "concept/rational-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ab462dc14d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "168", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\phi(x) + \\psi(x) \\to l + l'", "name": null, "statement": "The sum of two functions tending to limits l and l' tends to l + l'.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "phi(x), psi(x)", "meaning": "functions tending to limits as x tends to a" }, { "unit": null, "symbol": "l, l'", "meaning": "their respective limits" } ], "sympy": "Eq(Limit(phi(x) + psi(x), x, a), l + lp)", "physics": false, "states": [], "concepts": [ "concept/function", "concept/limit", "theorem/limit-of-a-sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-5e46353463", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "168", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\phi(x)\\psi(x) \\to ll'", "name": null, "statement": "The product of two functions tending to limits l and l' tends to ll'.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "phi(x), psi(x)", "meaning": "functions tending to limits as x tends to a" }, { "unit": null, "symbol": "l, l'", "meaning": "their respective limits" } ], "sympy": "Eq(Limit(phi(x)*psi(x), x, a), l*lp)", "physics": false, "states": [], "concepts": [ "concept/function", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-4a93597be2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "168", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\phi(x)/\\psi(x) \\to l/l'", "name": null, "statement": "The quotient of two functions tending to limits l and l' tends to l/l', unless l' is zero.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "phi(x), psi(x)", "meaning": "functions tending to limits as x tends to a" }, { "unit": null, "symbol": "l, l'", "meaning": "their respective limits, l' not zero" } ], "sympy": "Eq(Limit(phi(x)/psi(x), x, a), l/lp)", "physics": false, "states": [], "concepts": [ "concept/function", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-43684d7c82", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "172", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\lim_{x\\to 0} (x^{2}/x) = 0", "name": null, "statement": "x^2 is of smaller order than x as x tends to zero, since x^2/x tends to zero.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "continuous real variable tending to zero" } ], "sympy": "Eq(Limit(x**2/x, x, 0), 0)", "physics": false, "states": [], "concepts": [ "concept/indeterminate-form", "concept/limit", "concept/order-of-smallness" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-5f131319e9", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "172", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\phi(x)/x^{-k} = x^{k}\\phi(x)", "name": null, "statement": "Dividing phi(x) by x^(-k) is the same as multiplying it by x^k; the order of greatness of phi when x is small is decided by this product tending to a nonzero limit.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "k", "meaning": "integer order of greatness" }, { "unit": null, "symbol": "phi(x)", "meaning": "function large when x is small" }, { "unit": null, "symbol": "x", "meaning": "continuous real variable tending to zero" } ], "sympy": "Eq(phi(x)/x**(-k), x**k*phi(x))", "physics": false, "states": [], "concepts": [ "concept/function", "concept/order-of-greatness", "concept/order-of-smallness", "concept/power" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-765c57ba10", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "172", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\lim\\sqrtp{1 + x} = \\lim\\sqrtp{1 - x} = 1", "name": null, "statement": "The square root of 1 + x and the square root of 1 - x both tend to 1 as x tends to zero.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "continuous real variable tending to zero" } ], "sympy": "Eq(Limit(sqrt(1 + x), x, 0), 1)", "physics": false, "states": [], "concepts": [ "concept/limit", "concept/value-of-a-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a56e34edc4", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "172", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\lim\\{\\sqrtp{1 + x} - \\sqrtp{1 - x}\\}/x = 1", "name": null, "statement": "The difference of the two square roots divided by x tends to 1 as x tends to zero.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "continuous real variable tending to zero" } ], "sympy": "Eq(Limit((sqrt(1 + x) - sqrt(1 - x))/x, x, 0), 1)", "physics": false, "states": [], "concepts": [ "concept/difference", "concept/indeterminate-form", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d299d6057e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "173", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\lim\\{\\sqrtp{1 + x + x^{2}} - 1\\}/x = \\frac{1}{2}", "name": null, "statement": "The square root of 1 + x + x^2, less 1, divided by x tends to one half as x tends to zero.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "continuous real variable tending to zero" } ], "sympy": "Eq(Limit((sqrt(1 + x + x**2) - 1)/x, x, 0), Rational(1, 2))", "physics": false, "states": [], "concepts": [ "concept/algebraic-expression", "concept/indeterminate-form", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-900c16a3a7", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "168", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "H < \\phi(x) < K", "name": null, "statement": "phi(x) is bounded in an interval around a when it lies between two fixed constants H and K there.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "H, K", "meaning": "constant lower and upper bounds" }, { "unit": null, "symbol": "phi(x)", "meaning": "function of x" } ], "sympy": "And(Lt(H, phi(x)), Lt(phi(x), K))", "physics": false, "states": [], "concepts": [ "concept/bounded-function", "concept/inequality", "concept/interval" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-1597cea1ff", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "168", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\lambda = \\Lambda = l", "name": null, "statement": "phi(x) tends to l as x tends to a exactly when its lower and upper limits of indetermination both equal l.", "kind": "result", "symbols": [ { "unit": null, "symbol": "lambda", "meaning": "lower limit of indetermination of phi(x) as x tends to a" }, { "unit": null, "symbol": "Lambda", "meaning": "upper limit of indetermination of phi(x) as x tends to a" }, { "unit": null, "symbol": "l", "meaning": "the limit" } ], "sympy": "And(Eq(lam, l), Eq(Lam, l))", "physics": false, "states": [], "concepts": [ "concept/bounded-function", "concept/limit", "concept/limit-inferior", "concept/limit-superior" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-988f8eec7c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "168", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "|\\phi(x_{2}) - \\phi(x_{1})| < \\DELTA", "name": null, "statement": "The principle of convergence: phi(x) tends to a limit as x tends to a exactly when values of phi at two points close to a differ by less than any given Delta.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x_1, x_2", "meaning": "two values of x close to a" }, { "unit": null, "symbol": "\\DELTA", "meaning": "any given positive number" } ], "sympy": "Lt(Abs(phi(x2) - phi(x1)), Delta)", "physics": false, "states": [], "concepts": [ "concept/function", "concept/inequality", "concept/limit", "theorem/principle-of-convergence" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c8d50f3c9b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "165", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\phi(x_{2}) \\geq \\phi(x_{1})", "name": null, "statement": "phi(x) is steadily increasing with x when its value at any larger x_2 is at least its value at any smaller x_1.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x_1, x_2", "meaning": "values of the continuous variable with x_2 > x_1" }, { "unit": null, "symbol": "phi(x)", "meaning": "one-valued function of x" } ], "sympy": "Ge(phi(x2), phi(x1))", "physics": false, "states": [], "concepts": [ "concept/function", "concept/inequality", "concept/steadily-increasing-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-77f7a27b29", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "165", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\phi(x_{2}) > \\phi(x_{1})", "name": null, "statement": "phi(x) is steadily increasing in the stricter sense when, for x_2 > x_1, its value at x_2 is strictly greater than at x_1.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x_1, x_2", "meaning": "values of the continuous variable with x_2 > x_1" }, { "unit": null, "symbol": "phi(x)", "meaning": "one-valued function of x" } ], "sympy": "Gt(phi(x2), phi(x1))", "physics": false, "states": [], "concepts": [ "concept/inequality", "concept/steadily-increasing-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-69cac481b2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "175", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "|\\phi(x) - \\phi(\\xi)| < \\DELTA", "name": null, "statement": "The value of the function at x stays within a given small amount DELTA of its value at xi; this is the fundamental inequality of continuity, which the book shows is equivalent to the limit definition.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "the function phi(x)" }, { "unit": null, "symbol": "x", "meaning": "the continuous real variable" }, { "unit": null, "symbol": "xi", "meaning": "the particular value of x at which continuity is tested" }, { "unit": null, "symbol": "DELTA", "meaning": "an arbitrarily small positive number bounding the difference of function values" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/continuity", "concept/continuous-function", "concept/limit", "concept/variable" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-92a5d9f916", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "181", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\phi(\\xi - 0) = \\phi(\\xi) = \\phi(\\xi + 0)", "name": null, "statement": "A function is continuous at xi exactly when its left-hand limit, its value at xi, and its right-hand limit at xi are all equal.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "the function phi(x)" }, { "unit": null, "symbol": "xi", "meaning": "the point at which continuity is tested" }, { "unit": null, "symbol": "xi - 0", "meaning": "limit of phi as x approaches xi from values less than xi" }, { "unit": null, "symbol": "xi + 0", "meaning": "limit of phi as x approaches xi from values greater than xi" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/continuity", "concept/continuous-function", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-9884327bce", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "178", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\phi(a - 0) = \\phi(a) = \\phi(a + 0)", "name": null, "statement": "For continuity at x = a, the one-sided limits from below and above must both exist and equal the value phi(a).", "kind": "rule", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "the function phi(x)" }, { "unit": null, "symbol": "a", "meaning": "the point of the variable at which continuity is examined" }, { "unit": null, "symbol": "a - 0", "meaning": "limit of phi as x approaches a from values less than a" }, { "unit": null, "symbol": "a + 0", "meaning": "limit of phi as x approaches a from values greater than a" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/continuous-function", "concept/discontinuous-function", "concept/limit", "concept/simple-discontinuity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d6d192677f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "182", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\phi(x) \\leq K", "name": null, "statement": "The function is bounded above when some fixed number K is never exceeded by any of its values in the interval.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "the function phi(x)" }, { "unit": null, "symbol": "K", "meaning": "a number bounding the function from above" }, { "unit": null, "symbol": "x", "meaning": "the continuous real variable" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/bounded-function", "concept/least-upper-bound", "concept/variable" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-985f1d7a62", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "184", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "O(a, b) = M(a, b) - m(a, b)", "name": null, "statement": "The oscillation of a bounded function in an interval is the difference between its upper bound and its lower bound in that interval.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "O(a, b)", "meaning": "oscillation of the function in the interval (a, b)" }, { "unit": null, "symbol": "M(a, b)", "meaning": "upper bound of the function in the interval (a, b)" }, { "unit": null, "symbol": "m(a, b)", "meaning": "lower bound of the function in the interval (a, b)" } ], "sympy": "Eq(O(a, b), M(a, b) - m(a, b))", "physics": false, "states": [], "concepts": [ "concept/bounded-function", "concept/least-upper-bound", "concept/oscillation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-043032cdb1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "184", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "O(a, b) \\leq O(a, c) + O(c, b)", "name": null, "statement": "The oscillation over the whole interval from a to b is at most the sum of the oscillations over the two parts a to c and c to b.", "kind": "result", "symbols": [ { "unit": null, "symbol": "O(a, b)", "meaning": "oscillation of the function in the interval (a, b)" }, { "unit": null, "symbol": "a", "meaning": "left end of the interval" }, { "unit": null, "symbol": "b", "meaning": "right end of the interval" }, { "unit": null, "symbol": "c", "meaning": "a point between a and b dividing the interval" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/interval", "concept/oscillation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e690d352de", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "177", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "R(x) = P(x)/Q(x)", "name": null, "statement": "The standard rational function R(x) is the quotient of two polynomials P(x) and Q(x).", "kind": "definition", "symbols": [ { "unit": null, "symbol": "R(x)", "meaning": "the standard rational function of x" }, { "unit": null, "symbol": "P(x)", "meaning": "polynomial numerator" }, { "unit": null, "symbol": "Q(x)", "meaning": "polynomial denominator" }, { "unit": null, "symbol": "x", "meaning": "the continuous real variable" } ], "sympy": "Eq(R(x), P(x)/Q(x))", "physics": false, "states": [], "concepts": [ "concept/denominator", "concept/polynomial", "concept/quotient", "concept/rational-expression", "concept/rational-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-1e6c249c34", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "177", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\sin(x + h) - \\sin x = 2\\sin \\tfrac{1}{2}h \\cos(x + \\tfrac{1}{2}h)", "name": null, "statement": "The difference between sin of x plus h and sin of x equals twice sin of half h times cos of x plus half h, which shows sin x is continuous.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the continuous real variable (an angle)" }, { "unit": null, "symbol": "h", "meaning": "a small increment of x" } ], "sympy": "Eq(sin(x + h) - sin(x), 2*sin(h/2)*cos(x + h/2))", "physics": false, "states": [], "concepts": [ "concept/continuous-function", "concept/cosine", "concept/difference", "concept/sine", "concept/trigonometrical-identity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c6b16feeeb", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "190", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "|\\phi(x, y) - \\phi(\\xi, \\eta) | < \\DELTA", "name": null, "statement": "The function phi(x, y) is continuous at (xi, eta) when every value within a small enough square around that point differs from phi(xi, eta) by less than any given positive Delta.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "phi(x, y)", "meaning": "function of the two variables x and y" }, { "unit": null, "symbol": "xi, eta", "meaning": "the point (xi, eta) at which continuity is tested" }, { "unit": null, "symbol": "Delta", "meaning": "an arbitrary positive number, however small" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/continuity", "concept/continuous-function-of-two-variables", "concept/function-of-two-variables", "concept/neighbourhood" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-29c828c6c8", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "191", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\phi(x, y) = \\frac{2xy}{x^{2} + y^{2}}", "name": null, "statement": "Example function of two variables, defined as 2xy/(x^2+y^2) away from the axes and 0 when x or y is zero, which is continuous in each variable separately but not jointly at (0, 0).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "phi(x, y)", "meaning": "the example function of two variables" }, { "unit": null, "symbol": "x", "meaning": "first independent variable" }, { "unit": null, "symbol": "y", "meaning": "second independent variable" } ], "sympy": "Eq(phi(x, y), 2*x*y/(x**2 + y**2))", "physics": false, "states": [], "concepts": [ "concept/continuity", "concept/continuous-function-of-two-variables", "concept/discontinuous-function", "concept/function-of-two-variables" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-fc5a85d683", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "191", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "\\lim\\phi(x, y) = \\frac{2a}{1 + a^{2}}", "name": null, "statement": "Along the straight line y = ax, the example function tends to 2a/(1+a^2), which depends on a, so the joint limit at (0, 0) does not exist.", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi(x, y)", "meaning": "the example function of two variables" }, { "unit": null, "symbol": "a", "meaning": "slope of the line y = ax along which x and y tend to zero" } ], "sympy": "Eq(phi(x, y), 2*a/(1 + a**2))", "physics": false, "states": [], "concepts": [ "concept/discontinuous-function", "concept/function-of-two-variables", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ad8a7ad3d0", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "191", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "y^{5} - xy - y - x = 0", "name": null, "statement": "The relation between x and y that defines y as an implicit function of x, which by the theorem has a unique continuous solution vanishing with x.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "independent variable" }, { "unit": null, "symbol": "y", "meaning": "implicit function of x defined by the relation" } ], "sympy": "Eq(y**5 - x*y - y - x, 0)", "physics": false, "states": [], "concepts": [ "concept/equation", "concept/function", "concept/implicit-function", "concept/relation-between-variables" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-3a4d6a61fd", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "193", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "f(x, y) - f(x, y') = (y - y') (y^{4} + y^{3}y' + y^{2}y'^{2} + yy'^{3} + y'^{4} - x - 1)", "name": null, "statement": "For f(x, y) = y^5 - xy - y - x, the difference f(x, y) - f(x, y') factors as (y - y') times a second factor, which is the basis for checking that f is steadily decreasing in y.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "f(x, y)", "meaning": "the left-hand function y^5 - xy - y - x of the implicit equation" }, { "unit": null, "symbol": "x", "meaning": "independent variable" }, { "unit": null, "symbol": "y", "meaning": "variable of the function" }, { "unit": null, "symbol": "y'", "meaning": "a second value of the variable y" } ], "sympy": "Eq(f(x, y) - f(x, yp), (y - yp)*(y**4 + y**3*yp + y**2*yp**2 + y*yp**3 + yp**4 - x - 1))", "physics": false, "states": [], "concepts": [ "concept/function-of-two-variables", "concept/implicit-function", "concept/steadily-increasing-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-cb8106688e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "193", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "y = \\tfrac{1}{2}\\{1 + x - \\sqrtp{1 + 6 x + x^{2}}\\}", "name": null, "statement": "Solution of the quadratic y^2 - xy - y - x = 0 as an explicit function of x, using the positive square root.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "independent variable" }, { "unit": null, "symbol": "y", "meaning": "implicit function of x given by the quadratic" } ], "sympy": "Eq(y, Rational(1,2)*(1 + x - sqrt(1 + 6*x + x**2)))", "physics": false, "states": [], "concepts": [ "concept/function", "concept/implicit-function", "concept/root" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-437fdd70ef", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "192", "location": "LIMITS OF FUNCTIONS OF A CONTINUOUS VARIABLE. CONTINUOUS AND DISCONTINUOUS FUNCTIONS", "latex": "f(a, \\lambda) = 0", "name": null, "statement": "Because f{x, phi(x)} = 0 and f is continuous, the value of f at the limit point (a, lambda) is zero, which forces lambda = b.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f", "meaning": "the continuous function of x and y in the neighbourhood of (a, b)" }, { "unit": null, "symbol": "a", "meaning": "the x-value at which continuity of the implicit function is tested" }, { "unit": null, "symbol": "lambda", "meaning": "limit of indetermination (inferior limit) of phi(x) as x tends to a" } ], "sympy": "Eq(f(a, lambda), 0)", "physics": false, "states": [], "concepts": [ "concept/continuous-function-of-two-variables", "concept/implicit-function", "concept/limit", "concept/limit-inferior" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-f341da88eb", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "198", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\lim_{h \\to 0} \\frac{\\phi(x + h) - \\phi(x)}{h} = \\tan\\psi", "name": null, "statement": "The limit of the difference quotient of phi at x, as h tends to zero, equals the tangent of the angle psi that the tangent at P makes with OX.", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "function whose graph is the curve" }, { "unit": null, "symbol": "x", "meaning": "abscissa of the fixed point P on the curve" }, { "unit": null, "symbol": "h", "meaning": "increment of x (signed; Q has abscissa x + h)" }, { "unit": "radian", "symbol": "psi", "meaning": "angle the tangent PT makes with OX" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/function", "concept/increment", "concept/limit", "concept/tangent", "concept/tangent-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-9d83f30ec6", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "200", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\phi'(x) = \\lim_{h \\to 0} \\frac{\\phi(x + h) - \\phi(x)}{h}", "name": null, "statement": "The derivative of phi at x is defined as the limit of the difference quotient as h tends to zero, without any geometrical picture.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "function of x" }, { "unit": null, "symbol": "x", "meaning": "independent variable" }, { "unit": null, "symbol": "h", "meaning": "increment of x" }, { "unit": null, "symbol": "phi'(x)", "meaning": "derivative (differential coefficient) of phi at x" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/derivative", "concept/function", "concept/increment", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-02dc9f4a51", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "207", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\phi(x) = a_{0}x^{n} + a_{1}x^{n-1} + \\dots + a_{n}", "name": null, "statement": "A polynomial of degree n in x written as a sum of powers of x with constant coefficients.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "polynomial in x" }, { "unit": null, "symbol": "n", "meaning": "degree of the polynomial" }, { "unit": null, "symbol": "a_{0}, a_{1}, ..., a_{n}", "meaning": "coefficients of the polynomial" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/polynomial", "concept/power", "unit/degree-of-angle" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b72250a6a6", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "207", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\phi'(x) = na_{0}x^{n-1} + (n - 1)a_{1}x^{n-2} + \\dots + a_{n-1}", "name": null, "statement": "The derivative of a polynomial of degree n is obtained by multiplying each term's coefficient by its power and lowering that power by one.", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi'(x)", "meaning": "derivative of the polynomial" }, { "unit": null, "symbol": "n", "meaning": "degree of the polynomial" }, { "unit": null, "symbol": "a_{0}, ..., a_{n-1}", "meaning": "coefficients of the polynomial" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/polynomial", "concept/power", "method/differentiation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a6b0135800", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "207", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\phi'(x) = n \\left\\{ a_{0}x^{n-1} + \\binom{n - 1}{1} a_{1}x^{n-2} + \\binom{n - 1}{2} a_{2}x^{n-3} + \\dots + a_{n-1} \\right\\}", "name": null, "statement": "The derivative of a polynomial written in binomial form, with binomial coefficients, is n times a polynomial of degree n-1 of the same binomial type.", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi'(x)", "meaning": "derivative of the polynomial" }, { "unit": null, "symbol": "n", "meaning": "degree of the polynomial" }, { "unit": null, "symbol": "a_{0}, ..., a_{n-1}", "meaning": "coefficients of the polynomial" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/binomial", "concept/coefficient", "concept/derivative", "concept/polynomial" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c009902e90", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "207", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\phi(x) = a_{0}(x - \\alpha_{1})(x - \\alpha_{2}) \\dots (x - \\alpha_{n})", "name": null, "statement": "A polynomial of degree n factorises into n linear factors, with real or complex roots alpha.", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "polynomial in x" }, { "unit": null, "symbol": "a_{0}", "meaning": "leading coefficient" }, { "unit": null, "symbol": "alpha_{1}, ..., alpha_{n}", "meaning": "roots of phi(x) = 0, real or complex" }, { "unit": null, "symbol": "n", "meaning": "degree of the polynomial" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/factor", "concept/polynomial", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-3621f7612e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "207", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\phi'(x) = a_{0}\\tsum (x - \\alpha_{2})(x - \\alpha_{3}) \\dots (x - \\alpha_{n})", "name": null, "statement": "The derivative of a factored polynomial is a_0 times the sum of all products of n-1 of its linear factors.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "phi'(x)", "meaning": "derivative of the polynomial" }, { "unit": null, "symbol": "a_{0}", "meaning": "leading coefficient" }, { "unit": null, "symbol": "alpha_{i}", "meaning": "roots of the polynomial" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/factor", "concept/polynomial", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-97734f487a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "207", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\phi'(x) = a_{0} \\tsum m_{1}(x - \\alpha_{1})^{m_{1}-1} (x - \\alpha_{2})^{m_{2}}\\dots (x - \\alpha_{\\nu})^{m_{\\nu}}", "name": null, "statement": "The derivative of a polynomial with repeated roots, each factor raised to multiplicity m, is a sum over the roots with the multiplicity lowering by one for the differentiated factor.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "phi'(x)", "meaning": "derivative of the polynomial" }, { "unit": null, "symbol": "a_{0}", "meaning": "leading coefficient" }, { "unit": null, "symbol": "alpha_{i}", "meaning": "distinct roots of the polynomial" }, { "unit": null, "symbol": "m_{i}", "meaning": "multiplicity of the root alpha_i" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/multiple-root", "concept/polynomial", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-5def5cb8e6", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "209", "location": "DERIVATIVES AND INTEGRALS", "latex": "R'(x) = \\frac{P'(x)Q(x) - P(x)Q'(x)}{\\{Q(x)\\}^{2}}", "name": null, "statement": "The derivative of a quotient of two polynomials equals (P'Q - PQ') divided by Q squared.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "R'(x)", "meaning": "derivative of the rational function" }, { "unit": null, "symbol": "P'(x), Q'(x)", "meaning": "derivatives of the polynomials P and Q" }, { "unit": null, "symbol": "P(x), Q(x)", "meaning": "numerator and denominator polynomials" } ], "sympy": "Eq(Rp, (Pp*Q - P*Qp)/Q**2)", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/quotient", "concept/rational-function", "concept/rule", "method/differentiation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-7bf632eb97", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "210", "location": "DERIVATIVES AND INTEGRALS", "latex": "-\\frac{pA(x -\\alpha)^{p-1}}{(x - \\alpha)^{2p}} = -\\frac{pA}{(x - \\alpha)^{p+1}}", "name": null, "statement": "The derivative of the partial-fraction term A over (x - alpha) to the power p is -pA over (x - alpha) to the power p+1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "A", "meaning": "constant numerator of the partial fraction term" }, { "unit": null, "symbol": "alpha", "meaning": "root of Q(x) = 0" }, { "unit": null, "symbol": "p", "meaning": "power of the denominator (x - alpha)" } ], "sympy": "Eq(-p*A*(x - alpha)**(p-1)/(x - alpha)**(2*p), -p*A/(x - alpha)**(p+1))", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/power", "concept/rational-function", "method/partial-fractions" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-58ae381ee4", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "203", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\phi'(x) = f'(x) + F'(x)", "name": null, "statement": "The derivative of a sum of two differentiable functions is the sum of their derivatives.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "phi(x)", "meaning": "f(x) + F(x)" }, { "unit": null, "symbol": "f'(x), F'(x)", "meaning": "derivatives of f and F" } ], "sympy": "Eq(phip, fp + Fp)", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/rule", "concept/sum", "method/differentiation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-2068343824", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "203", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\phi'(x) = kf'(x)", "name": null, "statement": "The derivative of a constant multiple of a function is the constant times the derivative of the function.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "k", "meaning": "constant" }, { "unit": null, "symbol": "f'(x)", "meaning": "derivative of f" } ], "sympy": "Eq(phip, k*fp)", "physics": false, "states": [], "concepts": [ "concept/constant", "concept/derivative", "concept/rule", "method/differentiation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ad5e8e6771", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "203", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\phi'(x) = f(x)F'(x) + f'(x)F(x)", "name": null, "statement": "The derivative of a product of two differentiable functions is f times F-prime plus f-prime times F.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "phi(x)", "meaning": "f(x)F(x)" }, { "unit": null, "symbol": "f(x), F(x)", "meaning": "the two functions" }, { "unit": null, "symbol": "f'(x), F'(x)", "meaning": "their derivatives" } ], "sympy": "Eq(phip, f*Fp + fp*F)", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/product", "concept/rule", "method/differentiation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-6c875f4e2a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "204", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\phi'(x) = -\\frac{f'(x)}{\\{f(x)\\}^{2}}", "name": null, "statement": "The derivative of the reciprocal of a function is minus its derivative divided by the square of the function, where f is non-zero.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "phi(x)", "meaning": "1/f(x)" }, { "unit": null, "symbol": "f'(x)", "meaning": "derivative of f" } ], "sympy": "Eq(phip, -fp/f**2)", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/reciprocal", "concept/rule", "method/differentiation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-819ca38553", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "204", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\phi'(x) = \\frac{f'(x)F(x) - f(x)F'(x)}{\\{F(x)\\}^{2}}", "name": null, "statement": "The derivative of a quotient f over F is (f'F - fF') divided by F squared.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "phi(x)", "meaning": "f(x)/F(x)" }, { "unit": null, "symbol": "f'(x), F'(x)", "meaning": "derivatives of f and F" } ], "sympy": "Eq(phip, (fp*F - f*Fp)/F**2)", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/quotient", "concept/rule", "method/differentiation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a4377b8037", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "204", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\phi'(x) = F'\\{f(x)\\} f'(x)", "name": null, "statement": "The derivative of a composite function F of f(x) is F-prime evaluated at f(x), times f-prime(x) (chain rule).", "kind": "rule", "symbols": [ { "unit": null, "symbol": "phi(x)", "meaning": "F{f(x)}" }, { "unit": null, "symbol": "F'{f(x)}", "meaning": "derivative of F evaluated at f(x)" }, { "unit": null, "symbol": "f'(x)", "meaning": "derivative of f" } ], "sympy": "Eq(phip, Fp_of_f*fp)", "physics": false, "states": [], "concepts": [ "concept/composite-function", "concept/derivative", "concept/function", "concept/rule" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b9269f29fc", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "204", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\phi'(x) = \\frac{1}{\\psi'(y)}", "name": null, "statement": "The derivative of the inverse function phi is the reciprocal of the derivative of psi, evaluated at y = phi(x).", "kind": "rule", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "inverse function of psi" }, { "unit": null, "symbol": "psi", "meaning": "continuous strictly monotone function with x = psi(y)" }, { "unit": null, "symbol": "psi'(y)", "meaning": "derivative of psi, not zero" } ], "sympy": "Eq(phip, 1/psip)", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/inverse-function", "concept/reciprocal", "concept/rule" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-809d734789", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "206", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\frac{dy}{dx} = \\frac{dy_{1}}{dx} + \\frac{dy_{2}}{dx}", "name": null, "statement": "In differential notation, the derivative of a sum y = y1 + y2 is the sum of the derivatives.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "y", "meaning": "dependent variable, y = y1 + y2" }, { "unit": null, "symbol": "y_1, y_2", "meaning": "dependent variables that are functions of x" }, { "unit": null, "symbol": "x", "meaning": "independent variable" } ], "sympy": "Eq(dy_dx, dy1_dx + dy2_dx)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/derivative", "concept/mathematical-notation", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a59990528a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "206", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\frac{dy}{dx} = k\\frac{dy_{1}}{dx}", "name": null, "statement": "In differential notation, the derivative of k times y1 is k times the derivative of y1.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "k", "meaning": "constant" }, { "unit": null, "symbol": "y", "meaning": "k y1" }, { "unit": null, "symbol": "y_1", "meaning": "function of x" }, { "unit": null, "symbol": "x", "meaning": "independent variable" } ], "sympy": "Eq(dy_dx, k*dy1_dx)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/constant", "concept/derivative", "concept/mathematical-notation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-2aedb80959", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "206", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\frac{dy}{dx} = y_{1}\\frac{dy_{2}}{dx} + y_{2}\\frac{dy_{1}}{dx}", "name": null, "statement": "In differential notation, the derivative of a product y = y1 y2 is y1 times dy2/dx plus y2 times dy1/dx.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "y", "meaning": "product y1 y2" }, { "unit": null, "symbol": "y_1, y_2", "meaning": "functions of x" }, { "unit": null, "symbol": "x", "meaning": "independent variable" } ], "sympy": "Eq(dy_dx, y1*dy2_dx + y2*dy1_dx)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/derivative", "concept/mathematical-notation", "concept/product" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c27fac1e4a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "206", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\frac{dy}{dx} = -\\frac{1}{y_{1}^{2}}\\, \\frac{dy_{1}}{dx}", "name": null, "statement": "In differential notation, the derivative of 1/y1 is minus dy1/dx divided by y1 squared.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "y", "meaning": "1/y1" }, { "unit": null, "symbol": "y_1", "meaning": "function of x, non-zero" }, { "unit": null, "symbol": "x", "meaning": "independent variable" } ], "sympy": "Eq(dy_dx, -dy1_dx/y1**2)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/derivative", "concept/mathematical-notation", "concept/reciprocal" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c7226dd428", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "206", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\frac{dy}{dx} = \\biggl(y_{2}\\frac{dy_{1}}{dx} - y_{1}\\frac{dy_{2}}{dx}\\biggr) \\bigg/ y_{2}^{2}", "name": null, "statement": "In differential notation, the derivative of the quotient y1/y2 is (y2 dy1/dx - y1 dy2/dx) divided by y2 squared.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "y", "meaning": "quotient y1/y2" }, { "unit": null, "symbol": "y_1, y_2", "meaning": "functions of x" }, { "unit": null, "symbol": "x", "meaning": "independent variable" } ], "sympy": "Eq(dy_dx, (y2*dy1_dx - y1*dy2_dx)/y2**2)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/derivative", "concept/mathematical-notation", "concept/quotient" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-4a30ab2dda", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "206", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\frac{dz}{dx} = \\frac{dz}{dy}\\, \\frac{dy}{dx}", "name": null, "statement": "In differential notation, the chain rule: dz/dx equals dz/dy times dy/dx when z is a function of y and y of x.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "z", "meaning": "function of y" }, { "unit": null, "symbol": "y", "meaning": "function of x" }, { "unit": null, "symbol": "x", "meaning": "independent variable" } ], "sympy": "Eq(dz_dx, dz_dy*dy_dx)", "physics": false, "states": [], "concepts": [ "concept/composite-function", "concept/derivative", "concept/function", "concept/mathematical-notation", "concept/rule" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-259bf41fcb", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "206", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\dfrac{dy}{dx} = 1 \\bigg/ \\biggl(\\dfrac{dx}{dy}\\biggr)", "name": null, "statement": "The derivative of y with respect to x is the reciprocal of the derivative of x with respect to y, for an inverse function.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "y", "meaning": "dependent variable" }, { "unit": null, "symbol": "x", "meaning": "independent variable (x = psi(y))" } ], "sympy": "Eq(dy_dx, 1/dx_dy)", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/inverse-function", "concept/mathematical-notation", "concept/reciprocal", "concept/rule" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-26903dc53f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "201", "location": "DERIVATIVES AND INTEGRALS", "latex": "y - y_{0} = (x - x_{0}) \\phi'(x_{0})", "name": "equation of the tangent", "statement": "The tangent to the curve y = phi(x) at (x0, y0) is the line through that point with slope phi'(x0).", "kind": "result", "symbols": [ { "unit": null, "symbol": "(x_0, y_0)", "meaning": "point of contact on the curve" }, { "unit": null, "symbol": "phi'(x_0)", "meaning": "derivative at x0, the slope of the tangent" } ], "sympy": "Eq(y - y0, (x - x0)*phip0)", "physics": false, "states": [ "theorem/equation-of-the-tangent" ], "concepts": [ "concept/derivative", "concept/line", "concept/point-of-tangency", "concept/tangent" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-3d86c0f0f8", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "201", "location": "DERIVATIVES AND INTEGRALS", "latex": "(y - y_{0}) \\phi'(x_{0}) + x - x_{0} = 0", "name": "equation of the normal", "statement": "The normal at (x0, y0) is the line through that point perpendicular to the tangent.", "kind": "result", "symbols": [ { "unit": null, "symbol": "(x_0, y_0)", "meaning": "point of contact on the curve" }, { "unit": null, "symbol": "phi'(x_0)", "meaning": "derivative at x0, slope of the tangent" } ], "sympy": "Eq((y - y0)*phip0 + x - x0, 0)", "physics": false, "states": [ "theorem/equation-of-the-normal" ], "concepts": [ "concept/derivative", "concept/normal", "concept/perpendicular", "concept/tangent" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c0a72b3ee8", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "211", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\frac{dy}{dx} = \\biggl(\\frac{dy}{dz}\\biggr) \\bigg/ \\biggl(\\frac{dx}{dz}\\biggr) = \\frac{p}{q} z^{p-q} = mx^{m-1}", "name": null, "statement": "For y = x^m with m = p/q, the derivative with respect to x is mx^(m-1), obtained through the substitution z = x^(1/q) and the chain rule.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y", "meaning": "x^m, the function being differentiated" }, { "unit": null, "symbol": "x", "meaning": "the independent variable" }, { "unit": null, "symbol": "m", "meaning": "a rational number, m = p/q" }, { "unit": null, "symbol": "p", "meaning": "an integer" }, { "unit": null, "symbol": "q", "meaning": "a positive integer" }, { "unit": null, "symbol": "z", "meaning": "x^(1/q), so that x = z^q and y = z^p" } ], "sympy": "Eq(Derivative(y, x), m*x**(m - 1))", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/derivative", "concept/power", "concept/rational-number", "theorem/chain-rule" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-39633c3aa2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "211", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\phi'(x) = \\lim_{h \\to 0} \\frac{(x + h)^{m} - x^{m}}{h}", "name": null, "statement": "The derivative of phi(x) is the limit of the difference quotient as h tends to zero.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "function of x" }, { "unit": null, "symbol": "h", "meaning": "increment of x tending to zero" }, { "unit": null, "symbol": "m", "meaning": "rational exponent" } ], "sympy": "Eq(Derivative(phi(x), x), limit(((x + h)**m - x**m)/h, h, 0))", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/function", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-2d3bba3762", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "211", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\lim_{\\xi \\to x} \\frac{\\xi^{m} - x^{m}}{\\xi - x} = mx^{m-1}", "name": null, "statement": "The limit of the difference quotient taken with xi tending to x gives the derivative mx^(m-1) of x^m.", "kind": "result", "symbols": [ { "unit": null, "symbol": "xi", "meaning": "a variable point approaching x" }, { "unit": null, "symbol": "m", "meaning": "rational exponent" } ], "sympy": "Eq(limit((xi**m - x**m)/(xi - x), xi, x), m*x**(m - 1))", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/limit", "concept/power", "concept/variable" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-9f403d3b67", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "211", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\frac{d}{dx} (ax + b)^{m} = ma(ax + b)^{m-1}", "name": null, "statement": "The derivative of (ax + b) raised to the power m is ma(ax + b)^(m-1), valid for all rational m.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "a", "meaning": "constant coefficient of x" }, { "unit": null, "symbol": "b", "meaning": "constant term" }, { "unit": null, "symbol": "m", "meaning": "rational number" }, { "unit": null, "symbol": "x", "meaning": "independent variable" } ], "sympy": "Eq(Derivative((a*x + b)**m, x), m*a*(a*x + b)**(m - 1))", "physics": false, "states": [], "concepts": [ "concept/constant", "concept/derivative", "concept/power", "concept/rational-number", "theorem/chain-rule" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-9337aa93d4", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "211", "location": "DERIVATIVES AND INTEGRALS", "latex": "x^{3} + y^{3} - 3axy = 0", "name": null, "statement": "The implicit relation between x and y used as the example for differentiating an implicit algebraic function.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "independent variable" }, { "unit": null, "symbol": "y", "meaning": "dependent variable defined implicitly by the equation" }, { "unit": null, "symbol": "a", "meaning": "constant" } ], "sympy": "Eq(x**3 + y**3 - 3*a*x*y, 0)", "physics": false, "states": [], "concepts": [ "concept/algebraic-function", "concept/equation", "concept/variable", "method/implicit-differentiation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-1060cda363", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "211", "location": "DERIVATIVES AND INTEGRALS", "latex": "x^{2} + y^{2} \\frac{dy}{dx} - a\\left(y + x \\frac{dy}{dx}\\right) = 0", "name": null, "statement": "The result of differentiating x^3 + y^3 - 3axy = 0 with respect to x, which contains dy/dx.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "independent variable" }, { "unit": null, "symbol": "y", "meaning": "dependent variable" }, { "unit": null, "symbol": "a", "meaning": "constant" } ], "sympy": "Eq(x**2 + y**2*Derivative(y, x) - a*(y + x*Derivative(y, x)), 0)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/derivative", "method/implicit-differentiation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-19dc168d58", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "211", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\frac{dy}{dx} = -\\frac{x^{2} - ay}{y^{2} - ax}", "name": null, "statement": "The derivative of the implicitly defined y with respect to x, found by solving the differentiated equation.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y", "meaning": "dependent variable defined implicitly" }, { "unit": null, "symbol": "x", "meaning": "independent variable" }, { "unit": null, "symbol": "a", "meaning": "constant" } ], "sympy": "Eq(Derivative(y, x), -(x**2 - a*y)/(y**2 - a*x))", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/derivative", "concept/implicit-function", "method/implicit-differentiation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e872fea371", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "212", "location": "DERIVATIVES AND INTEGRALS", "latex": "D_{x} \\sin x = \\cos x", "name": null, "statement": "The derivative of sin x with respect to x is cos x.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "angle variable, in radians" } ], "sympy": "Eq(Derivative(sin(x), x), cos(x))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/derivative", "concept/sine", "method/differentiation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-2353069910", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "212", "location": "DERIVATIVES AND INTEGRALS", "latex": "D_{x} \\cos x = -\\sin x", "name": null, "statement": "The derivative of cos x with respect to x is minus sin x.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "angle variable" } ], "sympy": "Eq(Derivative(cos(x), x), -sin(x))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/derivative", "concept/sine", "method/differentiation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-75b0665818", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "212", "location": "DERIVATIVES AND INTEGRALS", "latex": "D_{x} \\tan x = \\sec^{2} x", "name": null, "statement": "The derivative of tan x is sec^2 x.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "angle variable" } ], "sympy": "Eq(Derivative(tan(x), x), sec(x)**2)", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/secant", "concept/tangent-function", "method/differentiation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-222aa0473b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "212", "location": "DERIVATIVES AND INTEGRALS", "latex": "D_{x} \\cot x = -\\cosec^{2} x", "name": null, "statement": "The derivative of cot x is minus cosec^2 x.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "angle variable" } ], "sympy": "Eq(Derivative(cot(x), x), -csc(x)**2)", "physics": false, "states": [], "concepts": [ "concept/cosecant", "concept/cotangent", "concept/derivative", "method/differentiation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-1ba0d29d78", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "212", "location": "DERIVATIVES AND INTEGRALS", "latex": "D_{x} \\sec x = \\tan x \\sec x", "name": null, "statement": "The derivative of sec x is tan x sec x.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "angle variable" } ], "sympy": "Eq(Derivative(sec(x), x), tan(x)*sec(x))", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/secant", "concept/tangent-function", "method/differentiation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-20bd599b40", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "212", "location": "DERIVATIVES AND INTEGRALS", "latex": "D_{x} \\cosec x = -\\cot x\\cosec x", "name": null, "statement": "The derivative of cosec x is minus cot x cosec x.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "angle variable" } ], "sympy": "Eq(Derivative(csc(x), x), -cot(x)*csc(x))", "physics": false, "states": [], "concepts": [ "concept/cosecant", "concept/cotangent", "concept/derivative", "method/differentiation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-fe9494ba04", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "212", "location": "DERIVATIVES AND INTEGRALS", "latex": "D_{x} \\arcsin x = ±1/\\sqrtp{1 - x^{2}}", "name": null, "statement": "The derivative of the inverse sine is 1 over the square root of 1 - x^2, with the sign fixed by cos(arcsin x).", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "argument of the inverse sine" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/ambiguous-sign", "concept/derivative", "concept/inverse-circular-function", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-886ad49ca3", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "212", "location": "DERIVATIVES AND INTEGRALS", "latex": "D_{x} \\arccos x = \\mp 1/\\sqrtp{1 - x^{2}}", "name": null, "statement": "The derivative of the inverse cosine is minus 1 over the square root of 1 - x^2, with the sign fixed by sin(arccos x).", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "argument of the inverse cosine" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/ambiguous-sign", "concept/cosine", "concept/derivative", "concept/inverse-circular-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-34f067fa70", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "212", "location": "DERIVATIVES AND INTEGRALS", "latex": "D_{x} \\arctan x = 1/(1 + x^{2})", "name": null, "statement": "The derivative of the inverse tangent is 1 over (1 + x^2).", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "argument of the inverse tangent" } ], "sympy": "Eq(Derivative(atan(x), x), 1/(1 + x**2))", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/inverse-circular-function", "concept/tangent-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-87d9d63b3c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "212", "location": "DERIVATIVES AND INTEGRALS", "latex": "D_{x} \\arccot x = -1/(1 + x^{2})", "name": null, "statement": "The derivative of the inverse cotangent is minus 1 over (1 + x^2).", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "argument of the inverse cotangent" } ], "sympy": "Eq(Derivative(acot(x), x), -1/(1 + x**2))", "physics": false, "states": [], "concepts": [ "concept/cotangent", "concept/derivative", "concept/inverse-circular-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ffc07ff567", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "212", "location": "DERIVATIVES AND INTEGRALS", "latex": "D_{x} \\arcsec x = ± 1/\\{x\\sqrtp{x^{2} - 1}\\}", "name": null, "statement": "The derivative of the inverse secant is ±1 over x times the square root of x^2 - 1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "argument of the inverse secant" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/ambiguous-sign", "concept/derivative", "concept/inverse-circular-function", "concept/secant" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c8c56a8e3d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "212", "location": "DERIVATIVES AND INTEGRALS", "latex": "D_{x} \\arccosec x = \\mp 1/\\{x\\sqrtp{x^{2} - 1}\\}", "name": null, "statement": "The derivative of the inverse cosecant is minus or plus 1 over x times the square root of x^2 - 1, with a sign convention.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "argument of the inverse cosecant" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/ambiguous-sign", "concept/cosecant", "concept/derivative", "concept/inverse-circular-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-9e77107220", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "212", "location": "DERIVATIVES AND INTEGRALS", "latex": "D_{x} \\arcsin(x/a) = ±1/\\sqrtp{a^{2} - x^{2}}", "name": null, "statement": "The more general derivative of the inverse sine of x/a, with the sign given by a cos{arcsin(x/a)}.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable" }, { "unit": null, "symbol": "a", "meaning": "nonzero constant, positive or negative" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/ambiguous-sign", "concept/constant", "concept/derivative", "concept/inverse-circular-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-30bb09d5df", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "212", "location": "DERIVATIVES AND INTEGRALS", "latex": "D_{x} \\arctan(x/a) = a/(x^{2} + a^{2})", "name": null, "statement": "The derivative of the inverse tangent of x/a is a over (x^2 + a^2).", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable" }, { "unit": null, "symbol": "a", "meaning": "constant" } ], "sympy": "Eq(Derivative(atan(x/a), x), a/(x**2 + a**2))", "physics": false, "states": [], "concepts": [ "concept/constant", "concept/derivative", "concept/inverse-circular-function", "concept/tangent-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-6dea22caa2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "212", "location": "DERIVATIVES AND INTEGRALS", "latex": "a\\sqrtb{1 - (x^{2}/a^{2})} = ±\\sqrtp{a^{2} - x^{2}}", "name": null, "statement": "The square root of 1 - x^2/a^2 times a equals plus or minus the square root of a^2 - x^2, according as a is positive or negative.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "nonzero constant" }, { "unit": null, "symbol": "x", "meaning": "variable" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/ambiguous-sign", "concept/constant", "concept/root", "concept/square" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-04f82a280a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "217", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\phi'(x) = 0", "name": null, "statement": "Rolle's theorem: if phi vanishes at a and b, then phi' vanishes at some point between a and b.", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "function of x with a derivative throughout the interval" }, { "unit": null, "symbol": "x", "meaning": "point between a and b" } ], "sympy": "Eq(Derivative(phi(x), x), 0)", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/interval", "concept/root", "theorem/rolle-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-edfae2f6c5", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "217", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\phi'(x) > 0", "name": null, "statement": "If the derivative is positive throughout an interval, phi is an increasing function throughout that interval, in the stricter sense.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "function of x" }, { "unit": null, "symbol": "x", "meaning": "point in the interval" } ], "sympy": "Eq(Derivative(phi(x), x) > 0, True)", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/inequality", "concept/interval", "concept/steadily-increasing-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-fdf4695acd", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "219", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\phi'(\\xi) = 0", "name": null, "statement": "A necessary condition for a maximum or minimum of phi at x = xi is that the derivative vanishes at xi.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "function of x" }, { "unit": null, "symbol": "xi", "meaning": "point where the maximum or minimum occurs" } ], "sympy": "Eq(Derivative(phi(xi), xi), 0)", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/maximum", "concept/minimum", "concept/necessary-condition" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-88eb0962de", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "220", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\phi'(x) = 3x^{2}", "name": null, "statement": "The derivative of y = x^3 is 3x^2, which vanishes at x = 0 without giving a maximum or minimum there.", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "the function x^3" }, { "unit": null, "symbol": "x", "meaning": "independent variable" } ], "sympy": "Eq(Derivative(x**3, x), 3*x**2)", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/inflection", "concept/maximum", "concept/minimum", "concept/power" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-25807df4db", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "220", "location": "DERIVATIVES AND INTEGRALS", "latex": "y = 1 - \\sqrtp{x^{2}}", "name": null, "statement": "The example function, with the positive square root, for which Rolle's theorem fails because there is no derivative at x = 0.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "y", "meaning": "value of the function" }, { "unit": null, "symbol": "x", "meaning": "independent variable" } ], "sympy": "Eq(y, 1 - sqrt(x**2))", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/function", "concept/root", "theorem/rolle-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d4652908f9", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "214", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\phi(x) = x^{2}\\sin(1/x)", "name": null, "statement": "Definition of a function that has a derivative everywhere but whose derivative is discontinuous at x = 0.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "function of x, with phi(0) = 0" }, { "unit": null, "symbol": "x", "meaning": "independent variable, nonzero here" } ], "sympy": "Eq(phi(x), x**2*sin(1/x))", "physics": false, "states": [], "concepts": [ "concept/continuous-function", "concept/derivative", "concept/discontinuous-function", "concept/function", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-62671e421f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "214", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\phi'(x) = 2x \\sin(1/x) - \\cos(1/x)", "name": null, "statement": "The derivative of x^2 sin(1/x) for x not equal to zero, which oscillates near zero and so is discontinuous at x = 0.", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "function x^2 sin(1/x)" }, { "unit": null, "symbol": "x", "meaning": "nonzero variable" } ], "sympy": "Eq(Derivative(x**2*sin(1/x), x), 2*x*sin(1/x) - cos(1/x))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/derivative", "concept/discontinuous-function", "concept/oscillation", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-87ea73ff88", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "221", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\phi'(0) = \\lim_{h \\to 0} \\frac{h^{2}\\sin(1/h)}{h} = 0", "name": null, "statement": "The derivative at zero exists and equals zero, obtained as a limit of the difference quotient.", "kind": "result", "symbols": [ { "unit": null, "symbol": "h", "meaning": "increment tending to zero" }, { "unit": null, "symbol": "phi", "meaning": "function x^2 sin(1/x)" } ], "sympy": "Eq(limit(h**2*sin(1/h)/h, h, 0), 0)", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/limit", "concept/sine", "concept/zero" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-79e89e77fc", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "221", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\phi(x) = x^{2}\\sin(1/x) + ax", "name": null, "statement": "The modified example function with a linear term, which has positive derivative at zero but is not steadily increasing on any interval containing zero.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "function of x" }, { "unit": null, "symbol": "a", "meaning": "constant with 0 < a < 1" }, { "unit": null, "symbol": "x", "meaning": "independent variable" } ], "sympy": "Eq(phi(x), x**2*sin(1/x) + a*x)", "physics": false, "states": [], "concepts": [ "concept/constant", "concept/function", "concept/sine", "concept/steadily-increasing-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ff42e19701", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "221", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\phi'(x) = 2x\\sin(1/x) - \\cos(1/x) + a", "name": null, "statement": "The derivative of the modified example for x not equal to zero, which oscillates between a - 1 and a + 1 as x tends to zero.", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "modified example function" }, { "unit": null, "symbol": "a", "meaning": "constant with 0 < a < 1" }, { "unit": null, "symbol": "x", "meaning": "nonzero variable" } ], "sympy": "Eq(Derivative(x**2*sin(1/x) + a*x, x), 2*x*sin(1/x) - cos(1/x) + a)", "physics": false, "states": [], "concepts": [ "concept/constant", "concept/cosine", "concept/derivative", "concept/oscillation", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a293242816", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "232", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\log (1/x) = -\\log x", "name": null, "statement": "The logarithm of the reciprocal of x is minus the logarithm of x.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "positive real variable" } ], "sympy": "Eq(log(1/x), -log(x))", "physics": false, "states": [], "concepts": [ "concept/logarithm", "concept/reciprocal" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-46f193eeeb", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "226", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\phi(b) - \\phi(a) = (b - a)\\phi'(\\xi)", "name": "Mean Value Theorem", "statement": "If φ has a derivative throughout the interval from a to b, then at some value ξ between a and b the derivative equals the average rate of change over the interval.", "kind": "result", "symbols": [ { "unit": null, "symbol": "φ", "meaning": "a function of x with a derivative on the interval" }, { "unit": null, "symbol": "a", "meaning": "lower end of the interval" }, { "unit": null, "symbol": "b", "meaning": "upper end of the interval" }, { "unit": null, "symbol": "ξ", "meaning": "a value of x lying between a and b" } ], "sympy": "Eq(phi(b) - phi(a), (b - a)*diff(phi(x), x).subs(x, xi))", "physics": false, "states": [ "theorem/mean-value-theorem" ], "concepts": [ "concept/derivative", "concept/function", "concept/interval" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-38f962ce53", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "227", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\phi(b) = \\phi(a) + (b - a) \\phi'\\{a + \\theta(b - a)\\}", "name": null, "statement": "The mean value theorem restated: φ(b) equals φ(a) plus (b − a) times the derivative at some point a + θ(b − a), with θ between 0 and 1.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "θ", "meaning": "a number lying between 0 and 1" }, { "unit": null, "symbol": "φ'", "meaning": "derivative of φ" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/function", "theorem/mean-value-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-1dcca4b25f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "227", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\phi(a + h) = \\phi(a) + h\\phi'(a + \\theta h)", "name": null, "statement": "The mean value theorem in increment form: φ(a+h) equals φ(a) plus h times the derivative at a point between a and a+h.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "h", "meaning": "increment b − a" }, { "unit": null, "symbol": "θ", "meaning": "a number lying between 0 and 1" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/increment", "theorem/mean-value-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-6b46c3ac6c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "230", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\phi(x) = \\int \\psi(x)\\, dx", "name": null, "statement": "φ is an integral (integral function) of ψ, meaning φ'(x) = ψ(x); this is the notation for integration.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "φ", "meaning": "integral function of ψ" }, { "unit": null, "symbol": "ψ", "meaning": "given function whose integral is sought" } ], "sympy": "Eq(phi(x), Integral(psi(x), x))", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/function", "concept/integral", "method/integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-f8e4650609", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "230", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int x^{m}\\, dx = \\frac{x^{m+1}}{m + 1}", "name": null, "statement": "The integral of x to the power m is x to the power m+1 divided by m+1, for m not equal to −1.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "m", "meaning": "a constant exponent, not equal to −1" } ], "sympy": "Eq(Integral(x**m, x), x**(m+1)/(m+1))", "physics": false, "states": [], "concepts": [ "concept/arbitrary-constant-of-integration", "concept/integral", "concept/standard-forms-of-integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-4394e2ee53", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "230", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int \\cos x\\, dx = \\sin x", "name": null, "statement": "The integral of cos x is sin x (one integral; the arbitrary constant C may be added).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "independent variable" } ], "sympy": "Eq(Integral(cos(x), x), sin(x))", "physics": false, "states": [], "concepts": [ "concept/arbitrary-constant-of-integration", "concept/cosine", "concept/integral", "concept/standard-forms-of-integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-f0ef56cd30", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "230", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int \\sin x\\, dx = -\\cos x", "name": null, "statement": "The integral of sin x is minus cos x (one integral; the arbitrary constant C may be added).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "independent variable" } ], "sympy": "Eq(Integral(sin(x), x), -cos(x))", "physics": false, "states": [], "concepts": [ "concept/arbitrary-constant-of-integration", "concept/cosine", "concept/integral", "concept/sine", "concept/standard-forms-of-integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-f0d3f2539b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "231", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int \\frac{dx}{x} = \\log x", "name": null, "statement": "For positive x, the integral of 1/x is the logarithmic function log x, which is defined by this equation.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x", "meaning": "a positive real variable" }, { "unit": null, "symbol": "log", "meaning": "the logarithmic function, defined here as this integral" } ], "sympy": "Eq(Integral(1/x, x), log(x))", "physics": false, "states": [], "concepts": [ "concept/integral", "concept/logarithm", "concept/real-number", "concept/standard-forms-of-integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e03bdeef09", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "231", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int \\frac{dx}{x} = \\log(-x)", "name": null, "statement": "For negative x, the integral of 1/x is log(−x), since the derivative of log(−x) is 1/x.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "a negative real variable" } ], "sympy": "Eq(Integral(1/x, x), log(-x))", "physics": false, "states": [], "concepts": [ "concept/ambiguous-sign", "concept/logarithm", "concept/real-number", "concept/standard-forms-of-integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b5d004d098", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "231", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int \\frac{dx}{x} = \\log(±x) = \\log|x|", "name": null, "statement": "Combining the positive and negative cases, the integral of 1/x is log|x| for all real x other than zero, where the ambiguous sign is chosen to make ±x positive.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "any real number other than zero" }, { "unit": null, "symbol": "±", "meaning": "ambiguous sign chosen so that ±x is positive" } ], "sympy": "Eq(Integral(1/x, x), log(Abs(x)))", "physics": false, "states": [], "concepts": [ "concept/absolute-value", "concept/ambiguous-sign", "concept/logarithm", "concept/standard-forms-of-integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-408281831a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "232", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int \\frac{dx}{x} = \\tfrac{1}{2}\\log x^{2}", "name": null, "statement": "Equivalent single form for the integral of 1/x, since log x² equals 2 log|x|.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "any real number other than zero" } ], "sympy": "Eq(Integral(1/x, x), log(x**2)/2)", "physics": false, "states": [], "concepts": [ "concept/ambiguous-sign", "concept/logarithm", "concept/standard-forms-of-integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d4eff72c43", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "232", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int \\frac{dx}{1 + x^{2}} = \\arctan x", "name": null, "statement": "The integral of 1/(1 + x²) is the inverse tangent of x.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "real variable" } ], "sympy": "Eq(Integral(1/(1 + x**2), x), atan(x))", "physics": false, "states": [], "concepts": [ "concept/inverse-circular-function", "concept/inverse-function", "concept/standard-forms-of-integration", "concept/tangent-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-0d5a338121", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "232", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int \\frac{x}{\\sqrtp{1 - x^{2}}} = ±\\arcsin x", "name": null, "statement": "The integral of x over the square root of 1 − x² is ± arcsin x, the sign being fixed by the rule given in section 119.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "real variable" }, { "unit": null, "symbol": "±", "meaning": "ambiguous sign fixed by the rule of section 119" } ], "sympy": "Eq(Integral(x/sqrt(1 - x**2), x), asin(x))", "physics": false, "states": [], "concepts": [ "concept/ambiguous-sign", "concept/inverse-circular-function", "concept/standard-forms-of-integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-401be09736", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "232", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\log 1 = 0", "name": null, "statement": "The logarithm of 1 is zero.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "log", "meaning": "logarithmic function" } ], "sympy": "Eq(log(1), 0)", "physics": false, "states": [], "concepts": [ "concept/logarithm", "theorem/logarithm-of-1-is-zero" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b91c95345f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "232", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\log xy = \\log x + \\log y", "name": null, "statement": "The logarithm of a product is the sum of the logarithms of the factors.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "positive real variable" }, { "unit": null, "symbol": "y", "meaning": "positive real variable" } ], "sympy": "Eq(log(x*y), log(x) + log(y))", "physics": false, "states": [], "concepts": [ "concept/functional-equation", "concept/logarithm", "concept/product" ], "pages": [ "232", "363" ], "chapters": [ "hardy-course-of-pure-mathematics-1921/ch-vi", "hardy-course-of-pure-mathematics-1921/ch-ix" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-3d50b49f33", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "232", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int \\{f(x) + F(x)\\}\\, dx = \\int f(x) dx + \\int F(x)\\, dx", "name": null, "statement": "The integral of a sum is the sum of the integrals, up to the arbitrary constants.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "f", "meaning": "a function of x" }, { "unit": null, "symbol": "F", "meaning": "a second function of x" } ], "sympy": "Eq(Integral(f(x) + F(x), x), Integral(f(x), x) + Integral(F(x), x))", "physics": false, "states": [], "concepts": [ "concept/arbitrary-constant-of-integration", "concept/function", "concept/integral", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-6c4bff4a7e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "232", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int kf(x)\\, dx = k\\int f(x)\\, dx", "name": null, "statement": "A constant factor can be taken outside the integral sign.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "k", "meaning": "a constant" }, { "unit": null, "symbol": "f", "meaning": "a function of x" } ], "sympy": "Eq(Integral(k*f(x), x), k*Integral(f(x), x))", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/constant", "concept/integral" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-7a175027ca", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "232", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int (a_{0}x^{n} + a_{1}x^{n-1} + \\dots + a_{n})\\, dx = \\frac{a_{0}x^{n+1}}{n + 1} + \\frac{a_{1}x^{n}}{n} + \\dots + a_{n}x", "name": null, "statement": "The integral of a polynomial is obtained term by term, each power x^k becoming x^(k+1)/(k+1).", "kind": "result", "symbols": [ { "unit": null, "symbol": "a_{0}, ..., a_{n}", "meaning": "coefficients of the polynomial" }, { "unit": null, "symbol": "n", "meaning": "degree of the polynomial" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/degree", "concept/integral", "concept/polynomial" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a9a6b159e7", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "233", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int \\frac{A}{(x - \\alpha)^{p}}\\, dx = -\\frac{A}{p - 1}\\, \\frac{1}{(x - \\alpha)^{p-1}}", "name": null, "statement": "The integral of A over (x − α) to the power p, for p not equal to 1, is minus A/(p−1) times 1/(x−α) to the power p−1; this holds whether α is real or complex.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "A", "meaning": "constant numerator" }, { "unit": null, "symbol": "α", "meaning": "a constant, real or complex, the root of the denominator" }, { "unit": null, "symbol": "p", "meaning": "a positive integer, not equal to 1" } ], "sympy": "Eq(Integral(A/(x - alpha)**p, x), -A/(p - 1)/(x - alpha)**(p - 1))", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/integral", "concept/multiple-root", "concept/standard-forms-of-integration", "method/partial-fractions" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b7d4f19a67", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "233", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int F'\\{f(x)\\}\\, f'(x)\\, dx = F\\{f(x)\\}", "name": null, "statement": "The integral of the derivative of F at f(x), multiplied by the derivative of f, is F of f(x): the chain rule read backwards.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "F", "meaning": "a function whose derivative is F'" }, { "unit": null, "symbol": "f", "meaning": "an inner function of x" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/function", "concept/integral", "concept/integration-by-substitution" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-cbe550f7c7", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "233", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int \\psi(ax + b)\\, dx = \\frac{1}{a}\\phi(ax + b)", "name": null, "statement": "If φ is an integral of ψ, then the integral of ψ(ax+b) is φ(ax+b) divided by a.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "a", "meaning": "a real constant, not zero" }, { "unit": null, "symbol": "b", "meaning": "a real constant" }, { "unit": null, "symbol": "φ", "meaning": "an integral of ψ" } ], "sympy": "Eq(Integral(psi(a*x + b), x), phi(a*x + b)/a)", "physics": false, "states": [], "concepts": [ "concept/function", "concept/integral", "concept/linear-substitution" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-8f1ce82386", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "233", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int \\frac{dx}{ax + b} = \\frac{1}{a} \\log|ax + b|", "name": null, "statement": "The integral of 1/(ax+b) is log of the absolute value of ax+b, divided by a.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "a", "meaning": "a real constant, not zero" }, { "unit": null, "symbol": "b", "meaning": "a real constant" } ], "sympy": "Eq(Integral(1/(a*x + b), x), log(Abs(a*x + b))/a)", "physics": false, "states": [], "concepts": [ "concept/absolute-value", "concept/logarithm", "concept/standard-forms-of-integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-03c855c260", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "233", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int \\frac{dx}{x - \\alpha} = \\log|x - \\alpha|", "name": null, "statement": "For real α, the integral of 1/(x−α) is the logarithm of the absolute value of x−α.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "α", "meaning": "a real constant" } ], "sympy": "Eq(Integral(1/(x - alpha), x), log(Abs(x - alpha)))", "physics": false, "states": [], "concepts": [ "concept/logarithm", "concept/standard-forms-of-integration", "method/partial-fractions" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-6df063ce58", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "234", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\lambda = A/2a", "name": null, "statement": "The constant λ in the partial fraction form equals A divided by 2a.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "λ", "meaning": "real part coefficient in the combined complex partial fraction" }, { "unit": null, "symbol": "A", "meaning": "numerator coefficient of Ax + B" }, { "unit": null, "symbol": "a", "meaning": "leading coefficient of the quadratic" } ], "sympy": "Eq(lambda_, A/(2*a))", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/complex-number", "method/partial-fractions" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-2b7fc05ecf", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "234", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\mu = -D/(2a\\sqrt{\\Delta})", "name": null, "statement": "The constant μ in the combined partial fraction equals minus D over 2a times the square root of Δ.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "μ", "meaning": "imaginary-part coefficient in the combined partial fraction" }, { "unit": null, "symbol": "D", "meaning": "aB − bA" }, { "unit": null, "symbol": "Δ", "meaning": "ac − b²" } ], "sympy": "Eq(mu, -D/(2*a*sqrt(Delta)))", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/discriminant", "method/partial-fractions" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-0973eeccc2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "234", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\gamma = -b/a", "name": null, "statement": "The real part γ of the complex root equals minus b over a.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "γ", "meaning": "real part of the complex root α = γ + δi" }, { "unit": null, "symbol": "b", "meaning": "middle coefficient of the quadratic ax² + 2bx + c" }, { "unit": null, "symbol": "a", "meaning": "leading coefficient of the quadratic" } ], "sympy": "Eq(gamma, -b/a)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/complex-number", "concept/root" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-334da62431", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "234", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\delta = \\sqrt{\\Delta}/a", "name": null, "statement": "The imaginary part δ of the complex root equals the square root of Δ over a.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "δ", "meaning": "imaginary part of the complex root α = γ + δi" }, { "unit": null, "symbol": "Δ", "meaning": "ac − b²" }, { "unit": null, "symbol": "a", "meaning": "leading coefficient of the quadratic" } ], "sympy": "Eq(delta, sqrt(Delta)/a)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/discriminant", "concept/root" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-da195dd392", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "234", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\Delta = ac - b^{2}", "name": null, "statement": "Δ is defined as ac minus b squared for the quadratic ax² + 2bx + c.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "Δ", "meaning": "ac − b², the discriminant-type quantity of the quadratic" }, { "unit": null, "symbol": "a, b, c", "meaning": "coefficients of the quadratic ax² + 2bx + c" } ], "sympy": "Eq(Delta, a*c - b**2)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/discriminant", "concept/quadratic-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-bee92aa2cf", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "234", "location": "DERIVATIVES AND INTEGRALS", "latex": "D = aB - bA", "name": null, "statement": "D is defined as aB minus bA, a combination of the coefficients of the numerator Ax + B and the quadratic.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "D", "meaning": "aB − bA" }, { "unit": null, "symbol": "A, B", "meaning": "coefficients of the numerator Ax + B" } ], "sympy": "Eq(D, a*B - b*A)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "method/partial-fractions" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-29b95145cf", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "234", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int \\frac{f'(x)}{f(x)}\\, dx = \\log |f(x)|", "name": null, "statement": "The integral of a logarithmic derivative f'(x)/f(x) is the logarithm of the absolute value of f(x).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "f", "meaning": "a real function of x, not zero" } ], "sympy": "Eq(Integral(Derivative(f(x), x)/f(x), x), log(Abs(f(x))))", "physics": false, "states": [], "concepts": [ "concept/absolute-value", "concept/derivative", "concept/integral", "concept/logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-0d0350eb0d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "234", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int \\frac{2(x - \\lambda)}{(x - \\lambda)^{2} + \\mu^{2}}\\, dx = \\log\\{(x - \\lambda)^{2} + \\mu^{2}\\}", "name": null, "statement": "The integral of 2(x−λ)/((x−λ)²+μ²) is the logarithm of (x−λ)²+μ².", "kind": "formula", "symbols": [ { "unit": null, "symbol": "λ", "meaning": "real constant (real part of the complex root)" }, { "unit": null, "symbol": "μ", "meaning": "real constant (imaginary part coefficient)" } ], "sympy": "Eq(Integral(2*(x - lam)/((x - lam)**2 + mu**2), x), log((x - lam)**2 + mu**2))", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/integral", "concept/logarithm", "method/partial-fractions" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a829ff1619", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "234", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int \\frac{-2\\delta\\mu}{(x - \\lambda)^{2} + \\mu^{2}}\\, dx = -2\\delta \\arctan \\left(\\frac{x - \\lambda}{\\mu}\\right)", "name": null, "statement": "The integral of −2δμ/((x−λ)²+μ²) is −2δ times the inverse tangent of (x−λ)/μ.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "δ", "meaning": "imaginary part of the complex root" }, { "unit": null, "symbol": "μ", "meaning": "real constant" }, { "unit": null, "symbol": "λ", "meaning": "real constant" } ], "sympy": "Eq(Integral(-2*delta*mu/((x - lam)**2 + mu**2), x), -2*delta*atan((x - lam)/mu))", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/integral", "concept/inverse-circular-function", "concept/inverse-tangent", "method/partial-fractions" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b07cecfdaa", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "237", "location": "DERIVATIVES AND INTEGRALS", "latex": "ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0", "name": null, "statement": "The general equation of the second degree in x and y, whose graph is a conic section.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a, b, c, f, g, h", "meaning": "constant coefficients of the conic" }, { "unit": null, "symbol": "x, y", "meaning": "coordinates of a point on the conic" } ], "sympy": "Eq(a*x**2 + 2*h*x*y + b*y**2 + 2*g*x + 2*f*y + c, 0)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/conic-section", "concept/graph-of-a-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-31ff66cc6e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "238", "location": "DERIVATIVES AND INTEGRALS", "latex": "aX^{2} + 2hXY + bY^{2} + 2GX + 2FY = 0", "name": null, "statement": "The conic written in the shifted variables X = x − ξ and Y = y − η.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "X", "meaning": "x − ξ" }, { "unit": null, "symbol": "Y", "meaning": "y − η" }, { "unit": null, "symbol": "F, G", "meaning": "coefficients defined in terms of the point (ξ, η)" } ], "sympy": "Eq(a*X**2 + 2*h*X*Y + b*Y**2 + 2*G*X + 2*F*Y, 0)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/conic-section" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-4ea7627168", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "238", "location": "DERIVATIVES AND INTEGRALS", "latex": "F = h\\xi + b\\eta + f", "name": null, "statement": "The coefficient F is defined as hξ + bη + f for a point (ξ, η) on the conic.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "F", "meaning": "coefficient in the shifted conic" }, { "unit": null, "symbol": "(ξ, η)", "meaning": "a point on the conic" } ], "sympy": "Eq(F, h*xi + b*eta + f)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/conic-section" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-4b33d3173e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "238", "location": "DERIVATIVES AND INTEGRALS", "latex": "G = a\\xi + h\\eta + g", "name": null, "statement": "The coefficient G is defined as aξ + hη + g for a point (ξ, η) on the conic.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "G", "meaning": "coefficient in the shifted conic" }, { "unit": null, "symbol": "(ξ, η)", "meaning": "a point on the conic" } ], "sympy": "Eq(G, a*xi + h*eta + g)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/conic-section" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-46aa0121c2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "238", "location": "DERIVATIVES AND INTEGRALS", "latex": "x - \\xi = -\\frac{2 (G + Ft)}{a + 2ht + bt^{2}}", "name": null, "statement": "Rational parametrisation of x in terms of the parameter t = Y/X along the conic.", "kind": "result", "symbols": [ { "unit": null, "symbol": "t", "meaning": "parameter Y/X" }, { "unit": null, "symbol": "ξ", "meaning": "x-coordinate of a fixed point on the conic" } ], "sympy": "Eq(x - xi, -2*(G + F*t)/(a + 2*h*t + b*t**2))", "physics": false, "states": [], "concepts": [ "concept/conic-section", "concept/parameter", "method/integration-by-rationalisation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-69d4504380", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "238", "location": "DERIVATIVES AND INTEGRALS", "latex": "y - \\eta = -\\frac{2t(G + Ft)}{a + 2ht + bt^{2}}", "name": null, "statement": "Rational parametrisation of y in terms of the parameter t along the conic.", "kind": "result", "symbols": [ { "unit": null, "symbol": "t", "meaning": "parameter Y/X" }, { "unit": null, "symbol": "η", "meaning": "y-coordinate of a fixed point on the conic" } ], "sympy": "Eq(y - eta, -2*t*(G + F*t)/(a + 2*h*t + b*t**2))", "physics": false, "states": [], "concepts": [ "concept/conic-section", "concept/parameter", "method/integration-by-rationalisation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-5a08bfbf62", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "238", "location": "DERIVATIVES AND INTEGRALS", "latex": "hx + by + f = -\\tfrac{1}{2}(a + 2ht + bt^{2}) \\frac{dx}{dt}", "name": null, "statement": "The linear expression hx + by + f equals minus half of the quadratic in t times dx/dt.", "kind": "result", "symbols": [ { "unit": null, "symbol": "t", "meaning": "parameter along the conic" } ], "sympy": "Eq(h*x + b*y + f, -Rational(1, 2)*(a + 2*h*t + b*t**2)*Derivative(x, t))", "physics": false, "states": [], "concepts": [ "concept/conic-section", "concept/derivative", "concept/differential" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-7d4b582b40", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "238", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int \\frac{dx}{hx + by + f}= -2\\int \\frac{dt}{a + 2ht + bt^{2}}", "name": null, "statement": "The integral over the conic of dx/(hx+by+f) equals minus two times the integral of dt over the quadratic in t.", "kind": "result", "symbols": [ { "unit": null, "symbol": "t", "meaning": "parameter along the conic" } ], "sympy": "Eq(Integral(1/(h*x + b*y + f), x), -2*Integral(1/(a + 2*h*t + b*t**2), t))", "physics": false, "states": [], "concepts": [ "concept/conic-section", "concept/integral", "method/integration-by-rationalisation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-29b0476ca8", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "238", "location": "DERIVATIVES AND INTEGRALS", "latex": "y^{2} = ax^{2} + 2bx + c", "name": null, "statement": "The curve y² = ax² + 2bx + c, the graph of y as a function of x, is treated as a conic in the integrals of section 135.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a, b, c", "meaning": "constant coefficients, with a > 0 in section 135" } ], "sympy": "Eq(y**2, a*x**2 + 2*b*x + c)", "physics": false, "states": [], "concepts": [ "concept/algebraic-function", "concept/conic-section", "concept/curve", "concept/equation" ], "pages": [ "238", "242" ], "chapters": [ "hardy-course-of-pure-mathematics-1921/ch-vi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-cbfb755ae8", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "238", "location": "DERIVATIVES AND INTEGRALS", "latex": "2\\frac{dx}{dt} = \\frac{(t^{2} + c)\\sqrt{a} + 2bt}{(t\\sqrt{a} + b)^{2}}", "name": null, "statement": "Derivative of x with respect to t after the substitution y + x√a = t.", "kind": "result", "symbols": [ { "unit": null, "symbol": "t", "meaning": "new variable y + x√a" } ], "sympy": "Eq(2*Derivative(x, t), ((t**2 + c)*sqrt(a) + 2*b*t)/(t*sqrt(a) + b)**2)", "physics": false, "states": [], "concepts": [ "concept/derivative", "method/integration-by-rationalisation", "method/substitution" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d3c80cf3ec", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "238", "location": "DERIVATIVES AND INTEGRALS", "latex": "2y = \\frac{(t^{2} + c)\\sqrt{a} + 2bt}{t\\sqrt{a} + b}", "name": null, "statement": "y expressed as a rational function of t after the substitution y + x√a = t.", "kind": "result", "symbols": [ { "unit": null, "symbol": "t", "meaning": "new variable y + x√a" } ], "sympy": "Eq(2*y, ((t**2 + c)*sqrt(a) + 2*b*t)/(t*sqrt(a) + b))", "physics": false, "states": [], "concepts": [ "method/integration-by-rationalisation", "method/substitution" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a66c184613", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "238", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int \\frac{dx}{y} = \\int \\frac{dt}{t\\sqrt{a} + b} = \\frac{1}{\\sqrt{a}} \\log \\left|x\\sqrt{a} + y + \\frac{b}{\\sqrt{a}}\\right|", "name": null, "statement": "The integral of dx/y, for y² = ax² + 2bx + c with a > 0, equals (1/√a) log of the absolute value of x√a + y + b/√a.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y", "meaning": "square root of ax² + 2bx + c" }, { "unit": null, "symbol": "a, b, c", "meaning": "constant coefficients, a > 0" } ], "sympy": "Eq(Integral(1/y, x), log(Abs(x*sqrt(a) + y + b/sqrt(a)))/sqrt(a))", "physics": false, "states": [], "concepts": [ "concept/absolute-value", "concept/integral", "concept/logarithm", "method/integration-by-rationalisation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-6c8e85acee", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "239", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int \\frac{dx}{\\sqrtp{x^{2} + a^{2}}} = \\log \\{x + \\sqrtp{x^{2} + a^{2}}\\}", "name": null, "statement": "The integral of 1/√(x²+a²) is the logarithm of x plus √(x²+a²).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "a", "meaning": "a real constant" } ], "sympy": "Eq(Integral(1/sqrt(x**2 + a**2), x), log(x + sqrt(x**2 + a**2)))", "physics": false, "states": [], "concepts": [ "concept/algebraic-function", "concept/logarithm", "concept/standard-forms-of-integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-0da67353ab", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "239", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int \\frac{dx}{\\sqrtp{x^{2} - a^{2}}} = \\log |x + \\sqrtp{x^{2} - a^{2}}|", "name": null, "statement": "The integral of 1/√(x²−a²) is the logarithm of the absolute value of x + √(x²−a²).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "a", "meaning": "a real constant" } ], "sympy": "Eq(Integral(1/sqrt(x**2 - a**2), x), log(Abs(x + sqrt(x**2 - a**2))))", "physics": false, "states": [], "concepts": [ "concept/absolute-value", "concept/algebraic-function", "concept/logarithm", "concept/standard-forms-of-integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-6d67995f75", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "239", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int \\frac{dx}{\\sqrtp{a^{2} - x^{2}}} = \\arcsin(x/a)", "name": null, "statement": "The integral of 1/√(a²−x²) is the inverse sine of x/a.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "a", "meaning": "a positive real constant" } ], "sympy": "Eq(Integral(1/sqrt(a**2 - x**2), x), asin(x/a))", "physics": false, "states": [], "concepts": [ "concept/algebraic-function", "concept/inverse-circular-function", "concept/sine", "concept/standard-forms-of-integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-6baf3ff312", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "239", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\lambda x + \\mu = (\\lambda/a) (ax + b) + \\mu - (\\lambda b/a)", "name": null, "statement": "Rewrites the linear numerator λx + μ in terms of ax + b, to reduce the integral in section 136.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "λ, μ", "meaning": "constants in the numerator" }, { "unit": null, "symbol": "a, b", "meaning": "constants of the quadratic ax² + 2bx + c" } ], "sympy": "Eq(lam*x + mu, (lam/a)*(a*x + b) + mu - lam*b/a)", "physics": false, "states": [], "concepts": [ "concept/algebraic-identity", "concept/linear-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-980b70a11b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "239", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int \\frac{ax + b}{\\sqrtp{ax^{2} + 2bx + c}}\\, dx = \\sqrtp{ax^{2} + 2bx + c}", "name": null, "statement": "The integral of (ax+b)/√(ax²+2bx+c) is √(ax²+2bx+c).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "a, b, c", "meaning": "constant coefficients of the quadratic" } ], "sympy": "Eq(Integral((a*x + b)/sqrt(a*x**2 + 2*b*x + c), x), sqrt(a*x**2 + 2*b*x + c))", "physics": false, "states": [], "concepts": [ "concept/algebraic-function", "concept/coefficient", "concept/integral" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-f786c92a8c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "239", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int \\frac{(\\lambda x + \\mu)\\, dx}{\\sqrtp{ax^{2} + 2bx + c}} = \\frac{\\lambda}{a} \\sqrtp{ax^{2} + 2bx + c} + \\left(\\mu - \\frac{\\lambda b}{a}\\right) \\int \\frac{dx}{\\sqrtp{ax^{2} + 2bx + c}}", "name": null, "statement": "The integral of (λx+μ)/√(ax²+2bx+c) reduces to a square-root term plus a constant multiple of the integral of 1/√(ax²+2bx+c).", "kind": "result", "symbols": [ { "unit": null, "symbol": "λ, μ", "meaning": "constants in the numerator" }, { "unit": null, "symbol": "a, b, c", "meaning": "constant coefficients of the quadratic" } ], "sympy": "Eq(Integral((lam*x + mu)/sqrt(a*x**2 + 2*b*x + c), x), lam/a*sqrt(a*x**2 + 2*b*x + c) + (mu - lam*b/a)*Integral(1/sqrt(a*x**2 + 2*b*x + c), x))", "physics": false, "states": [], "concepts": [ "concept/algebraic-function", "concept/integral", "method/integration-by-rationalisation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-16fa454b03", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "239", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\kappa = (ac - b^{2})/a", "name": null, "statement": "κ is defined as (ac − b²)/a, the constant that appears after the substitution x√a + b/√a = t.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "κ", "meaning": "(ac − b²)/a" }, { "unit": null, "symbol": "a, b, c", "meaning": "constant coefficients of the quadratic" } ], "sympy": "Eq(kappa, (a*c - b**2)/a)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "method/substitution" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-00d4e3f617", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "240", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int(\\lambda x + \\mu) \\sqrtp{ax^{2} + 2bx + c}\\, dx \\\\ = \\left(\\frac{\\lambda}{3a}\\right) (ax^{2} + 2bx + c)^{3/2} + \\left(\\Add{\\mu} - \\frac{\\lambda b}{a}\\right) \\int \\sqrtp{ax^{2} + 2bx + c}\\, dx", "name": null, "statement": "The integral of (λx+μ)√(ax²+2bx+c) equals a (3/2)-power term plus a constant multiple of the integral of √(ax²+2bx+c).", "kind": "result", "symbols": [ { "unit": null, "symbol": "λ, μ", "meaning": "constants in the numerator" }, { "unit": null, "symbol": "a, b, c", "meaning": "constant coefficients of the quadratic" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/algebraic-function", "concept/integral", "concept/power" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-fadb59cc27", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "242", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int R(x, \\sqrt{X})\\, dx", "name": null, "statement": "The most general integral of a real rational function R of x and the square root of X, where X = y^2 = ax^2 + 2bx + c; this is equation (1).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "R", "meaning": "a real rational function of x and sqrt(X)" }, { "unit": null, "symbol": "X", "meaning": "the quadratic ax^2 + 2bx + c, equal to y^2" }, { "unit": null, "symbol": "x", "meaning": "the variable of integration" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/algebraic-function", "concept/rational-function", "method/integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-575153a90c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "240", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int f'(x)F(x)\\, dx = f(x)F(x) - \\int f(x)F'(x)\\, dx", "name": "integration by parts", "statement": "The integral of f'(x) times F(x) equals f(x)F(x) minus the integral of f(x) times F'(x).", "kind": "rule", "symbols": [ { "unit": null, "symbol": "f", "meaning": "a function of x whose derivative appears in the integrand" }, { "unit": null, "symbol": "F", "meaning": "a function of x" }, { "unit": null, "symbol": "x", "meaning": "the variable of integration" } ], "sympy": "Eq(Integral(Derivative(f(x), x)*F(x), x), f(x)*F(x) - Integral(f(x)*Derivative(F(x), x), x))", "physics": false, "states": [ "method/integration-by-parts" ], "concepts": [ "concept/derivative", "concept/product", "method/integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-2e7cfb1d4d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "240", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int\\phi(x)\\, dx = \\int x\\chi''(x)\\, dx = x\\chi'(x) - \\int \\chi'(x)\\, dx = x\\chi'(x) - \\chi(x)", "name": null, "statement": "Worked case of integration by parts: when phi(x) = x times the second derivative of chi, the integral of phi is x chi'(x) minus chi(x).", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "the function to be integrated, phi(x) = x psi(x)" }, { "unit": null, "symbol": "chi", "meaning": "a known function whose second derivative is psi(x)" }, { "unit": null, "symbol": "x", "meaning": "the variable" } ], "sympy": "Eq(Integral(phi(x), x), x*Derivative(chi(x), x) - chi(x))", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/function", "method/integration", "method/integration-by-parts" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-6b20dc6dbe", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "240", "location": "DERIVATIVES AND INTEGRALS", "latex": "F(x) = \\sqrtp{ax^{2} + 2bx + c} = y", "name": null, "statement": "In the worked illustration F(x) is the square root of the quadratic in x, and this is written as y.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "F", "meaning": "the square-root function in the integration-by-parts illustration" }, { "unit": null, "symbol": "y", "meaning": "the square root of ax^2 + 2bx + c" }, { "unit": null, "symbol": "a", "meaning": "coefficient of x^2 in the quadratic" }, { "unit": null, "symbol": "b", "meaning": "half the coefficient of x in the quadratic" }, { "unit": null, "symbol": "c", "meaning": "constant term of the quadratic" } ], "sympy": "Eq(F(x), sqrt(a*x**2 + 2*b*x + c))", "physics": false, "states": [], "concepts": [ "concept/algebraic-function", "concept/function", "concept/root" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-3adf3950b8", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "240", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int y\\, dx = \\frac{(ax + b)y}{2a} + \\frac{ac - b^{2}}{2a} \\int \\frac{dx}{y}", "name": null, "statement": "Integrating by parts reduces the integral of y dx, where y is the square root of a quadratic, to the integral of 1/y.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y", "meaning": "square root of ax^2 + 2bx + c" }, { "unit": null, "symbol": "a", "meaning": "coefficient of x^2 in the quadratic" }, { "unit": null, "symbol": "b", "meaning": "half the coefficient of x in the quadratic" }, { "unit": null, "symbol": "c", "meaning": "constant term of the quadratic" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/algebraic-function", "method/integration", "method/integration-by-parts" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-1f6b5848c7", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "242", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\frac{A + B\\sqrt{X}}{C + D\\sqrt{X}} = \\frac{(A + B\\sqrt{X})(C - D\\sqrt{X})}{C^{2} - D^{2}X} = E + F\\sqrt{X}", "name": null, "statement": "Multiplying numerator and denominator by C minus D sqrt(X) reduces a quotient of this form to E plus F sqrt(X), with E and F rational in x.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "A", "meaning": "rational function of x" }, { "unit": null, "symbol": "B", "meaning": "rational function of x" }, { "unit": null, "symbol": "C", "meaning": "rational function of x" }, { "unit": null, "symbol": "D", "meaning": "rational function of x" }, { "unit": null, "symbol": "E", "meaning": "rational function of x" }, { "unit": null, "symbol": "F", "meaning": "rational function of x" }, { "unit": null, "symbol": "X", "meaning": "the quadratic ax^2 + 2bx + c" } ], "sympy": "Eq((A + B*sqrt(X))/(C + D*sqrt(X)), E + F*sqrt(X))", "physics": false, "states": [], "concepts": [ "concept/algebraic-expression", "concept/rational-function", "concept/root" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-5012b5ce2b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "242", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int \\frac{G}{\\sqrt{X}}\\, dx", "name": null, "statement": "The one remaining type of integral, which can always be evaluated by splitting G into partial fractions; this is equation (2).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "G", "meaning": "a rational function of x" }, { "unit": null, "symbol": "X", "meaning": "the quadratic ax^2 + 2bx + c" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/rational-function", "method/integration", "method/partial-fractions" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-eb34c122ac", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "242", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int \\frac{x^{m}}{\\sqrt{X}}\\, dx", "name": null, "statement": "Type (i) integral (3): x to the power m, with m a positive integer, divided by sqrt(X).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "m", "meaning": "a positive integer" }, { "unit": null, "symbol": "X", "meaning": "the quadratic ax^2 + 2bx + c" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/algebraic-function", "concept/integer", "method/integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c21ae385c4", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "243", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\frac{d}{dx}(x^{m-1}\\sqrt{X}) = (m - 1)x^{m-2} \\sqrt{X} + \\frac{(ax + b) x^{m-1}}{\\sqrt{X}} = \\frac{\\alpha x^{m} + \\beta x^{m-1} + \\gamma x^{m-2}}{\\sqrt{X}}", "name": null, "statement": "Differentiating x^(m-1) sqrt(X) gives a combination of three successive terms over sqrt(X), so integrating yields a relation between three successive integrals of type (3).", "kind": "result", "symbols": [ { "unit": null, "symbol": "m", "meaning": "a positive integer" }, { "unit": null, "symbol": "X", "meaning": "the quadratic ax^2 + 2bx + c" }, { "unit": null, "symbol": "alpha", "meaning": "constant coefficient of x^m in the combined term" }, { "unit": null, "symbol": "beta", "meaning": "constant coefficient of x^(m-1)" }, { "unit": null, "symbol": "gamma", "meaning": "constant coefficient of x^(m-2)" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/algebraic-function", "concept/derivative", "method/differentiation", "method/integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-8fc3be7fa3", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "243", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int \\frac{dx}{(x - p)^{m}\\sqrt{X}}", "name": null, "statement": "Type (ii) integral (4), where p is real; the substitution x - p = 1/t reduces it to a type (3) integral in t.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "p", "meaning": "a real number" }, { "unit": null, "symbol": "m", "meaning": "a positive integer exponent" }, { "unit": null, "symbol": "X", "meaning": "the quadratic ax^2 + 2bx + c" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/algebraic-function", "method/integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-7b3d5bbf61", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "243", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int \\frac{Lx + M}{(Ax^{2} + 2Bx + C) \\sqrt{ax^{2} + 2bx + c}}\\, dx", "name": null, "statement": "Type (iii) integral (5), arising from a pair of conjugate complex roots of the denominator of G.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "L", "meaning": "constant in the numerator" }, { "unit": null, "symbol": "M", "meaning": "constant in the numerator" }, { "unit": null, "symbol": "A", "meaning": "coefficient in the quadratic factor of the denominator" }, { "unit": null, "symbol": "B", "meaning": "coefficient in the quadratic factor of the denominator" }, { "unit": null, "symbol": "C", "meaning": "constant term of the quadratic factor of the denominator" }, { "unit": null, "symbol": "a", "meaning": "coefficient of x^2 in the square root" }, { "unit": null, "symbol": "b", "meaning": "half the coefficient of x in the square root" }, { "unit": null, "symbol": "c", "meaning": "constant term in the square root" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/algebraic-function", "concept/complex-number", "concept/conjugate-complex-numbers", "method/integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-591c679177", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "243", "location": "DERIVATIVES AND INTEGRALS", "latex": "x = \\frac{\\mu t + \\nu}{t + 1}", "name": null, "statement": "The substitution used to evaluate integral (5), with mu and nu chosen to satisfy the two conditions given.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the original variable" }, { "unit": null, "symbol": "t", "meaning": "the new variable" }, { "unit": null, "symbol": "mu", "meaning": "a real constant chosen for the substitution" }, { "unit": null, "symbol": "nu", "meaning": "a real constant chosen for the substitution" } ], "sympy": "Eq(x, (mu*t + nu)/(t + 1))", "physics": false, "states": [], "concepts": [ "concept/variable", "method/integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-4cf186d8de", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "243", "location": "DERIVATIVES AND INTEGRALS", "latex": "a\\mu\\nu + b(\\mu + \\nu) + c = 0", "name": null, "statement": "First condition that fixes the constants mu and nu in the substitution for integral (5).", "kind": "result", "symbols": [ { "unit": null, "symbol": "mu", "meaning": "a constant of the substitution" }, { "unit": null, "symbol": "nu", "meaning": "a constant of the substitution" }, { "unit": null, "symbol": "a", "meaning": "coefficient of x^2 in the quadratic" }, { "unit": null, "symbol": "b", "meaning": "half the coefficient of x in the quadratic" }, { "unit": null, "symbol": "c", "meaning": "constant term of the quadratic" } ], "sympy": "Eq(a*mu*nu + b*(mu + nu) + c, 0)", "physics": false, "states": [], "concepts": [ "concept/equation", "method/integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e8f5a4d533", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "243", "location": "DERIVATIVES AND INTEGRALS", "latex": "A\\mu\\nu + B(\\mu + \\nu) + C = 0", "name": null, "statement": "Second condition fixing mu and nu, using the coefficients of the quadratic factor of the denominator.", "kind": "result", "symbols": [ { "unit": null, "symbol": "mu", "meaning": "a constant of the substitution" }, { "unit": null, "symbol": "nu", "meaning": "a constant of the substitution" }, { "unit": null, "symbol": "A", "meaning": "coefficient in the quadratic factor of the denominator" }, { "unit": null, "symbol": "B", "meaning": "coefficient in the quadratic factor of the denominator" }, { "unit": null, "symbol": "C", "meaning": "constant term of the quadratic factor of the denominator" } ], "sympy": "Eq(A*mu*nu + B*(mu + nu) + C, 0)", "physics": false, "states": [], "concepts": [ "concept/equation", "method/integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d16a30c26d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "243", "location": "DERIVATIVES AND INTEGRALS", "latex": "(aB - bA)\\xi^{2} - (cA - aC)\\xi + (bC - cB) = 0", "name": null, "statement": "Quadratic equation whose roots are mu and nu; the book notes it always has real roots.", "kind": "result", "symbols": [ { "unit": null, "symbol": "xi", "meaning": "unknown in the quadratic, whose roots are mu and nu" }, { "unit": null, "symbol": "a", "meaning": "coefficient of x^2 in the square root" }, { "unit": null, "symbol": "b", "meaning": "half the coefficient of x in the square root" }, { "unit": null, "symbol": "c", "meaning": "constant term in the square root" }, { "unit": null, "symbol": "A", "meaning": "coefficient in the quadratic factor of the denominator" }, { "unit": null, "symbol": "B", "meaning": "coefficient in the quadratic factor of the denominator" }, { "unit": null, "symbol": "C", "meaning": "constant term of the quadratic factor of the denominator" } ], "sympy": "Eq((a*B - b*A)*xi**2 - (c*A - a*C)*xi + (b*C - c*B), 0)", "physics": false, "states": [], "concepts": [ "concept/equation", "concept/real-number", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-7199c2f73d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "243", "location": "DERIVATIVES AND INTEGRALS", "latex": "H\\int \\frac{t\\, dt}{(\\alpha t^{2} + \\beta)\\sqrtp{\\gamma t^{2} + \\delta}} + K\\int \\frac{dt}{(\\alpha t^{2} + \\beta)\\sqrtp{\\gamma t^{2} + \\delta}}", "name": null, "statement": "After the substitution, integral (5) splits into H times one integral plus K times a second integral; this is equation (6).", "kind": "result", "symbols": [ { "unit": null, "symbol": "H", "meaning": "constant multiplying the first integral" }, { "unit": null, "symbol": "K", "meaning": "constant multiplying the second integral" }, { "unit": null, "symbol": "t", "meaning": "new variable after substitution" }, { "unit": null, "symbol": "alpha", "meaning": "constant in the transformed integrand" }, { "unit": null, "symbol": "beta", "meaning": "constant in the transformed integrand" }, { "unit": null, "symbol": "gamma", "meaning": "constant in the transformed square root" }, { "unit": null, "symbol": "delta", "meaning": "constant in the transformed square root" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/algebraic-function", "method/integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-1fd04971f7", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "243", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\frac{t}{\\sqrtp{\\gamma t^{2} + \\delta}} = u", "name": null, "statement": "Substitution that rationalises the second integral of equation (6).", "kind": "definition", "symbols": [ { "unit": null, "symbol": "t", "meaning": "the variable of the transformed integral" }, { "unit": null, "symbol": "u", "meaning": "new variable of the substitution" }, { "unit": null, "symbol": "gamma", "meaning": "constant in the square root" }, { "unit": null, "symbol": "delta", "meaning": "constant in the square root" } ], "sympy": "Eq(t/sqrt(gamma*t**2 + delta), u)", "physics": false, "states": [], "concepts": [ "concept/variable", "method/integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b48dc807c7", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "243", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int \\frac{dt}{(\\alpha t^{2} + \\beta) \\sqrtp{\\gamma t^{2} + \\delta}} = \\int \\frac{du}{\\beta + (\\alpha\\delta - \\beta\\gamma) u^{2}}", "name": null, "statement": "Under the substitution t/sqrt(gamma t^2 + delta) = u, the second integral in (6) becomes a rational integral in u.", "kind": "result", "symbols": [ { "unit": null, "symbol": "u", "meaning": "new variable t/sqrt(gamma t^2 + delta)" }, { "unit": null, "symbol": "alpha", "meaning": "constant in the integrand" }, { "unit": null, "symbol": "beta", "meaning": "constant in the integrand" }, { "unit": null, "symbol": "gamma", "meaning": "constant in the square root" }, { "unit": null, "symbol": "delta", "meaning": "constant in the square root" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/rational-function", "method/integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-97d11bbc04", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "247", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\cos x = \\frac{1 - t^{2}}{1 + t^{2}}", "name": null, "statement": "cos x expressed in terms of t, where t = tan(x/2).", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the angle" }, { "unit": null, "symbol": "t", "meaning": "tan(x/2)" } ], "sympy": "Eq(cos(x), (1 - t**2)/(1 + t**2))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/rational-function", "concept/tangent-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-de58b281bf", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "247", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\sin x = \\frac{2t}{1 + t^{2}}", "name": null, "statement": "sin x expressed in terms of t, where t = tan(x/2).", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the angle" }, { "unit": null, "symbol": "t", "meaning": "tan(x/2)" } ], "sympy": "Eq(sin(x), 2*t/(1 + t**2))", "physics": false, "states": [], "concepts": [ "concept/rational-function", "concept/sine", "concept/tangent-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-fc0d10a20e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "247", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\frac{dx}{dt} = \\frac{2}{1 + t^{2}}", "name": null, "statement": "The derivative of x with respect to t = tan(x/2), so that the substitution reduces the integral to a rational function of t.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the angle" }, { "unit": null, "symbol": "t", "meaning": "tan(x/2)" } ], "sympy": "Eq(Derivative(x, t), 2/(1 + t**2))", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/differential", "concept/rational-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b69b24503c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "248", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int \\phi(y)\\, dy = \\int xf'(x)\\, dx = xf(x) - \\int f(x)\\, dx", "name": null, "statement": "If y = f(x) and phi is the inverse of f, the integral of phi(y) equals x f(x) minus the integral of f(x).", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "the function inverse to f" }, { "unit": null, "symbol": "f", "meaning": "a function with y = f(x)" }, { "unit": null, "symbol": "x", "meaning": "the variable of f" }, { "unit": null, "symbol": "y", "meaning": "the variable with y = f(x)" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/inverse-function", "method/integration", "method/integration-by-parts" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ce11d5e7b0", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "249", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int x^{m}(\\log x)^{n}\\, dx = \\frac{x^{m+1} (\\log x)^{n}}{m + 1} - \\frac{n}{m + 1} \\int x^{m}(\\log x)^{n-1}\\, dx", "name": null, "statement": "Integration by parts reduces the power n of log x by one at each step, so the integral can be completed by repetition.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the variable" }, { "unit": null, "symbol": "m", "meaning": "a non-negative integer exponent of x" }, { "unit": null, "symbol": "n", "meaning": "a positive integer power of log x" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/logarithm", "method/formulae-of-reduction", "method/integration", "method/integration-by-parts" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c762f7884b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "250", "location": "DERIVATIVES AND INTEGRALS", "latex": "(PRP') + (NN'RP) = (NN'P'P)", "name": null, "statement": "Additivity of areas: the area PRP' plus the area NN'RP equals the area NN'P'P, taken as one of the common-sense properties of area assumed in the text.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "(PRP')", "meaning": "area of the region PRP'" }, { "unit": null, "symbol": "(NN'RP)", "meaning": "area of the region NN'RP" }, { "unit": null, "symbol": "(NN'P'P)", "meaning": "area of the region NN'P'P" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/area", "concept/perpendicular" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-0ac3ea7cfc", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "250", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\Phi(x + h) - \\Phi(x) = h\\{\\phi(x) + \\mu(h)\\}", "name": null, "statement": "The increase of the area function over an interval of length h equals h times the ordinate at x plus a small error mu(h).", "kind": "result", "symbols": [ { "unit": "area", "symbol": "Phi", "meaning": "area ONPP0 as a function of x" }, { "unit": null, "symbol": "phi", "meaning": "the ordinate of the curve y = phi(x)" }, { "unit": null, "symbol": "h", "meaning": "the increment of x" }, { "unit": null, "symbol": "mu", "meaning": "error term depending on h" } ], "sympy": "Eq(Phi(x + h) - Phi(x), h*(phi(x) + mu(h)))", "physics": false, "states": [], "concepts": [ "concept/area", "concept/function", "concept/increment", "concept/ordinate" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-78add9b1fb", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "250", "location": "DERIVATIVES AND INTEGRALS", "latex": "|\\mu(h)| < \\lambda(h)", "name": null, "statement": "The error mu(h) is bounded in absolute value by lambda(h), the greatest distance of any point of the arc from the chord line PR, and lambda(h) tends to 0 as h tends to 0.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "mu", "meaning": "error term in the area increment" }, { "unit": "length", "symbol": "lambda", "meaning": "greatest distance of the arc PP' from PR" }, { "unit": null, "symbol": "h", "meaning": "the increment of x" } ], "sympy": "Lt(Abs(mu(h)), lam(h))", "physics": false, "states": [], "concepts": [ "concept/area", "concept/inequality", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-cbf9e72a4c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "250", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\Phi'(x) = \\lim_{h \\to 0} \\frac{\\Phi(x + h) - \\Phi(x)}{h} = \\lim_{h \\to 0} \\{\\phi(x) + \\mu(h)\\} = \\phi(x)", "name": null, "statement": "The derivative of the area function is the ordinate: the ordinate of the curve is the derivative of the area, so the area is the integral of the ordinate.", "kind": "result", "symbols": [ { "unit": "area", "symbol": "Phi", "meaning": "area ONPP0 as a function of x" }, { "unit": null, "symbol": "phi", "meaning": "the ordinate of the curve" }, { "unit": null, "symbol": "h", "meaning": "the increment of x" }, { "unit": null, "symbol": "mu", "meaning": "error term tending to 0" } ], "sympy": "Eq(Derivative(Phi(x), x), phi(x))", "physics": false, "states": [], "concepts": [ "concept/area", "concept/derivative", "concept/limit", "concept/ordinate", "method/integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-8c8728dc78", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "251", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\{S(x + h) - S(x)\\}/h = \\{PP'\\}/h = (PP'/h) × (\\{PP'\\}/PP')", "name": null, "statement": "The increment of the arc length S over h, divided by h, equals the chord ratio PP'/h times the ratio of the arc to its chord.", "kind": "result", "symbols": [ { "unit": "length", "symbol": "S", "meaning": "length of the arc P0P as a function of x" }, { "unit": null, "symbol": "h", "meaning": "the increment of x" }, { "unit": "length", "symbol": "PP'", "meaning": "the chord joining P and P'" }, { "unit": "length", "symbol": "{PP'}", "meaning": "the arc whose chord is PP'" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/arc-of-a-curve", "concept/chord", "concept/increment", "quantity/length-of-a-curve" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-dc89131cfe", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "251", "location": "DERIVATIVES AND INTEGRALS", "latex": "PP' + \\sqrtp{PR^{2} + RP'^{2}} = h\\bigsqrtp{1 + \\frac{k^{2}}{h^{2}}}", "name": null, "statement": "As printed: a Pythagorean relation for the chord PP' with legs PR = h and RP' = k. The printed '+' between PP' and the root does not read as a standard identity and is flagged here as a possible transcription or printing error for '=', not corrected.", "kind": "result", "symbols": [ { "unit": "length", "symbol": "PP'", "meaning": "the chord joining P and P'" }, { "unit": "length", "symbol": "PR", "meaning": "horizontal step h" }, { "unit": "length", "symbol": "RP'", "meaning": "vertical step k" }, { "unit": null, "symbol": "h", "meaning": "the increment of x" }, { "unit": null, "symbol": "k", "meaning": "the increment of y = phi(x)" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/chord", "concept/parallel-lines", "quantity/length-of-a-curve" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-1b16a0d109", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "251", "location": "DERIVATIVES AND INTEGRALS", "latex": "k = \\phi(x + h) - \\phi(x) = h\\phi'(\\xi)", "name": null, "statement": "The increment k of the ordinate equals h times the derivative at some point xi between x and x + h.", "kind": "result", "symbols": [ { "unit": null, "symbol": "k", "meaning": "increment of the ordinate phi(x)" }, { "unit": null, "symbol": "phi", "meaning": "the function y = phi(x) whose graph is the curve" }, { "unit": null, "symbol": "h", "meaning": "the increment of x" }, { "unit": null, "symbol": "xi", "meaning": "a point between x and x + h" } ], "sympy": "Eq(k, phi(x + h) - phi(x))", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/function", "concept/increment" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-0506545013", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "251", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\lim (PP'/h) = \\lim \\sqrtb{1 + [\\phi'(\\xi)]^{2}} = \\sqrtb{1 + [\\phi'(x)]^{2}}", "name": null, "statement": "As h tends to 0 the ratio PP'/h tends to the square root of one plus the square of the derivative of phi at x.", "kind": "result", "symbols": [ { "unit": "length", "symbol": "PP'", "meaning": "the chord joining P and P'" }, { "unit": null, "symbol": "h", "meaning": "the increment of x" }, { "unit": null, "symbol": "phi", "meaning": "the function whose graph is the curve" }, { "unit": null, "symbol": "xi", "meaning": "a point between x and x + h" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/limit", "quantity/length-of-a-curve" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-737007c76b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "251", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\lim \\{PP'\\}/PP' = 1", "name": null, "statement": "Assumption that the arc PP' becomes indistinguishable from its chord as h tends to 0 (a hypothesis the text adopts, not proves).", "kind": "rule", "symbols": [ { "unit": "length", "symbol": "{PP'}", "meaning": "the arc whose chord is PP'" }, { "unit": "length", "symbol": "PP'", "meaning": "the chord" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/arc-of-a-curve", "concept/chord", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-2044544237", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "251", "location": "DERIVATIVES AND INTEGRALS", "latex": "S'(x) = \\lim \\{S(x + h) - S(x)\\}/h = \\sqrtb{1 + [\\phi'(x)]^{2}}", "name": null, "statement": "The derivative of the arc length with respect to x equals the square root of one plus the square of the derivative of the ordinate.", "kind": "result", "symbols": [ { "unit": "length", "symbol": "S", "meaning": "length of the arc P0P as a function of x" }, { "unit": null, "symbol": "phi", "meaning": "the function whose graph is the curve" }, { "unit": null, "symbol": "h", "meaning": "the increment of x" } ], "sympy": "Eq(Derivative(S(x), x), sqrt(1 + Derivative(phi(x), x)**2))", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/limit", "quantity/length-of-a-curve" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-541872891d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "251", "location": "DERIVATIVES AND INTEGRALS", "latex": "S(x) = \\int \\sqrtb{1 + [\\phi'(x)]^{2}}\\, dx", "name": null, "statement": "The arc length of the curve y = phi(x) from the origin is the integral of the square root of one plus the square of phi'(x).", "kind": "formula", "symbols": [ { "unit": "length", "symbol": "S", "meaning": "length of the arc P0P as a function of x" }, { "unit": null, "symbol": "phi", "meaning": "the function whose graph is the curve" }, { "unit": null, "symbol": "x", "meaning": "the abscissa" } ], "sympy": "Eq(S(x), Integral(sqrt(1 + Derivative(phi(x), x)**2), x))", "physics": false, "states": [], "concepts": [ "concept/arc-of-a-curve", "concept/derivative", "method/integration", "quantity/length-of-a-curve" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-89e230522a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "DERIVATIVES AND INTEGRALS", "latex": "u_{0}^{2} u_{3} - 3u_{0} u_{1} u_{2} + 2u_{1}^{3}", "name": null, "statement": "A combination of the u_r, where u_r are the successive derivative-type functions a, ax+b, ax^2+2bx+c, ..., which the exercise shows is independent of x.", "kind": "result", "symbols": [ { "unit": null, "symbol": "u_r", "meaning": "the successive functions a, ax + b, ax^2 + 2bx + c, ... (u_0, u_1, u_2, ...)" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/constant", "concept/function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-2148dba5f1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "DERIVATIVES AND INTEGRALS", "latex": "u_{0} u_{4} - 4u_{1} u_{3} + 3u_{2}^{2}", "name": null, "statement": "A second combination of the u_r which the exercise shows is independent of x.", "kind": "result", "symbols": [ { "unit": null, "symbol": "u_r", "meaning": "the successive functions a, ax + b, ax^2 + 2bx + c, ... (u_0, u_1, u_2, ...)" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/constant", "concept/function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-1f372e71dc", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "254", "location": "DERIVATIVES AND INTEGRALS", "latex": "U_{0}U_{2n} - 2nU_{1}U_{2n-1} + \\frac{2n(2n - 1)}{1·2} U_{2}U_{2n-2} - \\dots + U_{2n}U_{0}", "name": null, "statement": "An alternating binomial-weighted sum of products of the U_r which is independent of x.", "kind": "result", "symbols": [ { "unit": null, "symbol": "U_r", "meaning": "(a_0, a_1, ..., a_r, x, 1)^r, the r-th power of a linear form in the constants a_i and x" }, { "unit": null, "symbol": "n", "meaning": "half the number of constants a_0 ... a_{2n}" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/constant", "concept/formula" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a16f6f6ff3", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "254", "location": "DERIVATIVES AND INTEGRALS", "latex": "U_{r}' = rU_{r-1}", "name": null, "statement": "The derivative of U_r with respect to x equals r times U_{r-1}.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "U_r", "meaning": "the r-th power of the linear form in a_0, ..., a_r, x, 1" }, { "unit": null, "symbol": "r", "meaning": "integer index" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "method/differentiation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-9bf983ccb7", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "254", "location": "DERIVATIVES AND INTEGRALS", "latex": "y^{3} + 3yx + 2x^{3} = 0", "name": null, "statement": "The relation between x and y from which the second-derivative identity of Ex. 7 is deduced.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x", "meaning": "independent variable" }, { "unit": null, "symbol": "y", "meaning": "function of x defined by the relation" } ], "sympy": "Eq(y**3 + 3*y*x + 2*x**3, 0)", "physics": false, "states": [], "concepts": [ "concept/implicit-function", "concept/relation-between-variables" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-2553f6b481", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "254", "location": "DERIVATIVES AND INTEGRALS", "latex": "x^{2}(1 + x^{3})y'' - \\frac{3}{2}xy' + y = 0", "name": null, "statement": "Second-order differential equation satisfied by y when y^3 + 3yx + 2x^3 = 0.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y", "meaning": "function of x" }, { "unit": null, "symbol": "x", "meaning": "independent variable" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/differential-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-02e7c6bc5c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "254", "location": "DERIVATIVES AND INTEGRALS", "latex": "y = \\phi\\{\\psi(y_{1})\\} + \\phi\\{x - \\psi(y_{1})\\}", "name": null, "statement": "Differential equation of Ex. 8, where y_1 is the derivative of y and psi inverts phi', which the functions y = phi(c) + phi(x - c) and y = 2 phi(x/2) satisfy.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y", "meaning": "unknown function of x" }, { "unit": null, "symbol": "y_1", "meaning": "derivative of y with respect to x" }, { "unit": null, "symbol": "phi", "meaning": "arbitrary function" }, { "unit": null, "symbol": "psi", "meaning": "function inverse to phi'" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/differential-equation", "concept/function", "concept/inverse-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-27b62ea6eb", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "254", "location": "DERIVATIVES AND INTEGRALS", "latex": "y = \\phi(c) + \\phi(x - c)", "name": null, "statement": "A solution of the differential equation of Ex. 8, with c a constant.", "kind": "result", "symbols": [ { "unit": null, "symbol": "c", "meaning": "arbitrary constant" }, { "unit": null, "symbol": "phi", "meaning": "arbitrary function" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/constant", "concept/differential-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-4e72c182d1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "254", "location": "DERIVATIVES AND INTEGRALS", "latex": "y = 2\\phi(\\frac{1}{2}x)", "name": null, "statement": "A second solution of the differential equation of Ex. 8.", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "arbitrary function" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/differential-equation", "concept/function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-f16f7c3bed", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "254", "location": "DERIVATIVES AND INTEGRALS", "latex": "y = \\{x/\\psi(y_{1})\\} \\phi\\{\\psi(y_{1})\\}", "name": null, "statement": "Differential equation of Ex. 9, whose solutions include y = c phi(x/c) and y = beta x.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y", "meaning": "unknown function of x" }, { "unit": null, "symbol": "y_1", "meaning": "derivative of y" }, { "unit": null, "symbol": "phi", "meaning": "arbitrary function" }, { "unit": null, "symbol": "psi", "meaning": "function inverse to phi'" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/differential-equation", "concept/function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d2e2421fd9", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "254", "location": "DERIVATIVES AND INTEGRALS", "latex": "y = c\\phi(x/c)", "name": null, "statement": "A solution of the differential equation of Ex. 9, with c a constant.", "kind": "result", "symbols": [ { "unit": null, "symbol": "c", "meaning": "arbitrary constant" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/constant", "concept/differential-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-05ed1fd955", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "254", "location": "DERIVATIVES AND INTEGRALS", "latex": "y = \\beta x", "name": null, "statement": "A straight-line solution of the differential equation of Ex. 9.", "kind": "result", "symbols": [ { "unit": null, "symbol": "beta", "meaning": "constant equal to phi(alpha)/alpha" } ], "sympy": "Eq(y, beta*x)", "physics": false, "states": [], "concepts": [ "concept/constant", "concept/differential-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-3ff03767ed", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "254", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\beta = \\phi(\\alpha)/\\alpha", "name": null, "statement": "The constant beta is defined as phi(alpha) divided by alpha, where alpha is a root of phi(alpha) - alpha phi'(alpha) = 0.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "beta", "meaning": "constant" }, { "unit": null, "symbol": "alpha", "meaning": "a root of phi(alpha) - alpha phi'(alpha) = 0" }, { "unit": null, "symbol": "phi", "meaning": "arbitrary function" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/constant", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b7ee109918", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "254", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\phi(\\alpha) - \\alpha\\phi'(\\alpha) = 0", "name": null, "statement": "The equation whose root alpha determines the constant beta in the solution of Ex. 9.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "alpha", "meaning": "unknown root" }, { "unit": null, "symbol": "phi", "meaning": "arbitrary function" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b49ea70b07", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "254", "location": "DERIVATIVES AND INTEGRALS", "latex": "y_{2} = 0", "name": null, "statement": "The general differential equation of all straight lines ax + by + c = 0, with y_2 the second derivative of y with respect to x.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y_2", "meaning": "second derivative of y with respect to x" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/differential-equation", "concept/line" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a176985c56", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "254", "location": "DERIVATIVES AND INTEGRALS", "latex": "1 + y_{1}^{2} + yy_{2} = 0", "name": null, "statement": "The general differential equation of all circles with centres on the axis of x.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y_1", "meaning": "first derivative of y with respect to x" }, { "unit": null, "symbol": "y_2", "meaning": "second derivative of y with respect to x" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/circle", "concept/derivative", "concept/differential-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e55d32c919", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "254", "location": "DERIVATIVES AND INTEGRALS", "latex": "y_{1}^{2} + yy_{2} = 0", "name": null, "statement": "The general differential equation of all parabolas with axes along the axis of x.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y_1", "meaning": "first derivative of y" }, { "unit": null, "symbol": "y_2", "meaning": "second derivative of y" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/differential-equation", "concept/parabola" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-6e820aa4c1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "254", "location": "DERIVATIVES AND INTEGRALS", "latex": "y_{3} = 0", "name": null, "statement": "The general differential equation of all parabolas with axes parallel to the axis of y.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y_3", "meaning": "third derivative of y with respect to x" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/differential-equation", "concept/parabola" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-723cb15edc", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "254", "location": "DERIVATIVES AND INTEGRALS", "latex": "(1 + y_{1}^{2}) y_{3} = 3y_{1} y_{2}^{2}", "name": null, "statement": "The general differential equation of all circles.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y_1", "meaning": "first derivative of y" }, { "unit": null, "symbol": "y_2", "meaning": "second derivative of y" }, { "unit": null, "symbol": "y_3", "meaning": "third derivative of y" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/circle", "concept/derivative", "concept/differential-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-68422c1d2e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "254", "location": "DERIVATIVES AND INTEGRALS", "latex": "5y_{3}^{2} = 3y_{2} y_{4}", "name": null, "statement": "The general differential equation of all parabolas.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y_2", "meaning": "second derivative of y" }, { "unit": null, "symbol": "y_3", "meaning": "third derivative of y" }, { "unit": null, "symbol": "y_4", "meaning": "fourth derivative of y" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/differential-equation", "concept/parabola" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e4bbcb7afd", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "254", "location": "DERIVATIVES AND INTEGRALS", "latex": "9y_{2}^{2} y_{5} - 45y_{2} y_{3} y_{4} + 40y_{3}^{3} = 0", "name": null, "statement": "The general differential equation of all conics.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y_2", "meaning": "second derivative of y" }, { "unit": null, "symbol": "y_3", "meaning": "third derivative of y" }, { "unit": null, "symbol": "y_4", "meaning": "fourth derivative of y" }, { "unit": null, "symbol": "y_5", "meaning": "fifth derivative of y" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/conic-section", "concept/derivative", "concept/differential-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-1b308413ac", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "254", "location": "DERIVATIVES AND INTEGRALS", "latex": "D_{x}^{2} (y_{2}^{-2/3}) = 0", "name": null, "statement": "The general differential equation of all parabolas.", "kind": "result", "symbols": [ { "unit": null, "symbol": "D_x", "meaning": "differentiation with respect to x" }, { "unit": null, "symbol": "y_2", "meaning": "second derivative of y with respect to x" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/differential-equation", "concept/parabola", "method/differentiation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d0917f8c63", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "254", "location": "DERIVATIVES AND INTEGRALS", "latex": "D_{x}^{3} (y_{2}^{-2/3}) = 0", "name": null, "statement": "The general differential equation of all conics.", "kind": "result", "symbols": [ { "unit": null, "symbol": "D_x", "meaning": "differentiation with respect to x" }, { "unit": null, "symbol": "y_2", "meaning": "second derivative of y with respect to x" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/conic-section", "concept/differential-equation", "method/differentiation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-86888ad5a4", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "255", "location": "DERIVATIVES AND INTEGRALS", "latex": "y_{2} = ±(pr - q^{2})/(px^{2} + 2qx + r)^{3/2}", "name": null, "statement": "The second derivative of a conic written as y = ax + b ± sqrt(px^2 + 2qx + r).", "kind": "result", "symbols": [ { "unit": null, "symbol": "y_2", "meaning": "second derivative of y with respect to x" }, { "unit": null, "symbol": "p", "meaning": "coefficient in the conic form" }, { "unit": null, "symbol": "q", "meaning": "coefficient in the conic form" }, { "unit": null, "symbol": "r", "meaning": "coefficient in the conic form" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/conic-section", "concept/derivative" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-59b8baa10a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "255", "location": "DERIVATIVES AND INTEGRALS", "latex": "4ac - 5b^{2} = (4\\alpha\\gamma - 5\\beta^{2})/\\tau^{8}", "name": null, "statement": "Transformation relation between the coefficients a, b, c built from successive derivatives t, a, b, c and the reciprocal-derivative coefficients alpha, beta, gamma, tau.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "second coefficient 1/2! d^2y/dx^2" }, { "unit": null, "symbol": "b", "meaning": "third coefficient 1/3! d^3y/dx^3" }, { "unit": null, "symbol": "c", "meaning": "fourth coefficient 1/4! d^4y/dx^4" }, { "unit": null, "symbol": "tau", "meaning": "dx/dy" } ], "sympy": "Eq(4*a*c - 5*b**2, (4*alpha*gamma - 5*beta**2)/tau**8)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/derivative" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-df901e7a8f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "255", "location": "DERIVATIVES AND INTEGRALS", "latex": "bt - a^{2} = - (\\beta\\tau - \\alpha^{2})/\\tau^{6}", "name": null, "statement": "Second transformation identity relating the derivative coefficients in x and in y.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "t", "meaning": "dy/dx" }, { "unit": null, "symbol": "tau", "meaning": "dx/dy" }, { "unit": null, "symbol": "a", "meaning": "1/2! d^2y/dx^2" }, { "unit": null, "symbol": "b", "meaning": "1/3! d^3y/dx^3" } ], "sympy": "Eq(b*t - a**2, -(beta*tau - alpha**2)/tau**6)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/derivative" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-dc058a11b8", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "255", "location": "DERIVATIVES AND INTEGRALS", "latex": "(1 - x^{2})y_{k+2} - (2k + 1)xy_{k+1} + (n^{2} - k^{2})y_{k} = 0", "name": null, "statement": "Recurrence for the k-th derivatives of y = sin(n arcsin x).", "kind": "identity", "symbols": [ { "unit": null, "symbol": "y_k", "meaning": "k-th derivative of y = sin(n arcsin x)" }, { "unit": null, "symbol": "k", "meaning": "order of derivative" }, { "unit": null, "symbol": "n", "meaning": "constant in y = sin(n arcsin x)" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/differential-equation", "concept/inverse-circular-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-fc9a75e31b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "255", "location": "DERIVATIVES AND INTEGRALS", "latex": "vD_{x}^{n}u = D_{x}^{n}(uv) - nD_{x}^{n-1}(uD_{x}v)", "name": null, "statement": "The first two terms of the generalised Leibniz-type formula expressing v D_x^n u in terms of derivatives of the product uv, for positive integer n.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "D_x", "meaning": "differentiation with respect to x" }, { "unit": null, "symbol": "n", "meaning": "positive integer" }, { "unit": null, "symbol": "u", "meaning": "function of x" }, { "unit": null, "symbol": "v", "meaning": "function of x" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "method/differentiation", "theorem/general-leibniz-rule" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b5b713363c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "255", "location": "DERIVATIVES AND INTEGRALS", "latex": "x = a(2\\cos t + \\cos 2t)", "name": null, "statement": "Parametric equation of the curve of Ex. 15, x in terms of the parameter t.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a", "meaning": "constant length" }, { "unit": null, "symbol": "t", "meaning": "parameter" } ], "sympy": "Eq(x, a*(2*cos(t) + cos(2*t)))", "physics": false, "states": [], "concepts": [ "concept/curve", "concept/parameter" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-2b280c2b4a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "255", "location": "DERIVATIVES AND INTEGRALS", "latex": "y = a(2\\sin t - \\sin 2t)", "name": null, "statement": "Parametric equation of the curve of Ex. 15, y in terms of the parameter t.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a", "meaning": "constant length" }, { "unit": null, "symbol": "t", "meaning": "parameter" } ], "sympy": "Eq(y, a*(2*sin(t) - sin(2*t)))", "physics": false, "states": [], "concepts": [ "concept/curve", "concept/parameter" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-579f64ece5", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "255", "location": "DERIVATIVES AND INTEGRALS", "latex": "x\\sin \\tfrac{1}{2} t + y\\cos \\tfrac{1}{2} t = a\\sin \\tfrac{3}{2} t", "name": null, "statement": "Equation of the tangent at the point with parameter t on the curve of Ex. 15.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "coordinate of a point on the tangent" }, { "unit": null, "symbol": "y", "meaning": "coordinate of a point on the tangent" }, { "unit": null, "symbol": "t", "meaning": "parameter of the point P" }, { "unit": null, "symbol": "a", "meaning": "constant length" } ], "sympy": "Eq(x*sin(t/2) + y*cos(t/2), a*sin(3*t/2))", "physics": false, "states": [], "concepts": [ "concept/curve", "concept/tangent" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-1873cdb6a2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "255", "location": "DERIVATIVES AND INTEGRALS", "latex": "x\\cos \\tfrac{1}{2} t - y\\sin \\tfrac{1}{2} t = 3a\\cos \\tfrac{3}{2} t", "name": null, "statement": "Equation of the normal at the point with parameter t on the curve of Ex. 15.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "coordinate of a point on the normal" }, { "unit": null, "symbol": "y", "meaning": "coordinate of a point on the normal" }, { "unit": null, "symbol": "t", "meaning": "parameter of the point P" }, { "unit": null, "symbol": "a", "meaning": "constant length" } ], "sympy": "Eq(x*cos(t/2) - y*sin(t/2), 3*a*cos(3*t/2))", "physics": false, "states": [], "concepts": [ "concept/curve", "concept/normal" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-24271f3efa", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "255", "location": "DERIVATIVES AND INTEGRALS", "latex": "QR = 4a", "name": null, "statement": "The distance between the points Q and R where the tangent at P meets the curve is 4a.", "kind": "result", "symbols": [ { "unit": null, "symbol": "QR", "meaning": "distance between points Q and R on the curve" }, { "unit": null, "symbol": "a", "meaning": "constant length" } ], "sympy": "Eq(QR, 4*a)", "physics": false, "states": [], "concepts": [ "concept/curve", "quantity/distance" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-9cd3d4f531", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "255", "location": "DERIVATIVES AND INTEGRALS", "latex": "x^{2} + y^{2} = 9a^{2}", "name": null, "statement": "The circle on which the normals at P, Q and R intersect.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "constant length" } ], "sympy": "Eq(x**2 + y**2, 9*a**2)", "physics": false, "states": [], "concepts": [ "concept/circle", "concept/concurrent-lines", "concept/normal" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-39f550ba9a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "255", "location": "DERIVATIVES AND INTEGRALS", "latex": "(x^{2} + y^{2} + 12ax + 9a^{2})^{2} = 4a(2x + 3a)^{3}", "name": null, "statement": "The Cartesian equation of the curve parametrised in Ex. 15.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "Cartesian coordinate" }, { "unit": null, "symbol": "y", "meaning": "Cartesian coordinate" }, { "unit": null, "symbol": "a", "meaning": "constant length" } ], "sympy": "Eq((x**2 + y**2 + 12*a*x + 9*a**2)**2, 4*a*(2*x + 3*a)**3)", "physics": false, "states": [], "concepts": [ "concept/algebraic-form", "concept/curve" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a62ba4fe79", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "255", "location": "DERIVATIVES AND INTEGRALS", "latex": "u^{2}\\xi - u\\eta = a(u^{3} - 1)", "name": null, "statement": "Equation of the tangent at the point defined by u (Ex. 16), in complex coordinates xi = x + yi and eta = x - yi.", "kind": "result", "symbols": [ { "unit": null, "symbol": "u", "meaning": "Cis t, unit complex number" }, { "unit": null, "symbol": "xi", "meaning": "x + yi" }, { "unit": null, "symbol": "eta", "meaning": "x - yi" }, { "unit": null, "symbol": "a", "meaning": "constant length" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/tangent" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-0344bba0fa", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "255", "location": "DERIVATIVES AND INTEGRALS", "latex": "u^{2}\\xi + u\\eta = 3a(u^{3} + 1)", "name": null, "statement": "Equation of the normal at the point defined by u (Ex. 16), in complex coordinates.", "kind": "result", "symbols": [ { "unit": null, "symbol": "u", "meaning": "Cis t" }, { "unit": null, "symbol": "xi", "meaning": "x + yi" }, { "unit": null, "symbol": "eta", "meaning": "x - yi" }, { "unit": null, "symbol": "a", "meaning": "constant length" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/normal" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ca13778c8f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "255", "location": "DERIVATIVES AND INTEGRALS", "latex": "(p + q)^{2/3} - (p - q)^{2/3} = 1", "name": null, "statement": "The condition that x^4 + 4px^3 - 4qx - 1 = 0 should have equal roots.", "kind": "result", "symbols": [ { "unit": null, "symbol": "p", "meaning": "coefficient in the quartic" }, { "unit": null, "symbol": "q", "meaning": "coefficient in the quartic" } ], "sympy": "Eq((p + q)**(2/3) - (p - q)**(2/3), 1)", "physics": false, "states": [], "concepts": [ "concept/discriminant", "concept/equal-roots", "concept/equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-2784e7b875", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "257", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\begin{vmatrix} f(a) & \\phi(a) & \\psi(a)\\\\ f(b) & \\phi(b) & \\psi(b)\\\\ f'(\\xi) & \\phi'(\\xi) & \\psi'(\\xi) \\end{vmatrix} =0", "name": null, "statement": "Generalised mean value theorem: a determinant built from f, phi, psi at a and b and their derivatives at some xi between a and b vanishes.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f", "meaning": "function with derivative on [a, b]" }, { "unit": null, "symbol": "phi", "meaning": "function with derivative on [a, b]" }, { "unit": null, "symbol": "psi", "meaning": "function with derivative on [a, b]" }, { "unit": null, "symbol": "xi", "meaning": "a point between a and b" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/determinant", "theorem/mean-value-theorem", "theorem/work-done-in-charging-a-conductor" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-317adfc478", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "258", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\frac{f(b) - f(a)}{\\phi(b) - \\phi(a)} = \\frac{f'(\\xi)}{\\phi'(\\xi)}\\Add{.}", "name": null, "statement": "Cauchy-type quotient form of the mean value theorem deduced from Ex. 32, for some xi between a and b.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f", "meaning": "function" }, { "unit": null, "symbol": "phi", "meaning": "function" }, { "unit": null, "symbol": "xi", "meaning": "intermediate point between a and b" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/common-ratio", "concept/derivative", "theorem/mean-value-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-0afd7d6097", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "258", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\phi(x) - \\phi(x_{0}) = (x - x_{0})\\phi'(\\xi)", "name": null, "statement": "Mean value formula with x_0 < xi < x, used to prove the limit results of Ex. 34.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "function" }, { "unit": null, "symbol": "xi", "meaning": "intermediate point between x_0 and x" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/function", "theorem/mean-value-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-484e018461", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "257", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\phi(x) = 1/(1 + x^{2})", "name": null, "statement": "Definition of the function phi used in Ex. 31.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "function of x" }, { "unit": null, "symbol": "x", "meaning": "real variable" } ], "sympy": "Eq(phi(x), 1/(1 + x**2))", "physics": false, "states": [], "concepts": [ "concept/function", "concept/rational-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a3b0227f5a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "257", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\phi^{n} (x) = Q_{n}(x)/(1 + x^{2})^{n+1}", "name": null, "statement": "The n-th derivative of 1/(1+x^2) has the form Q_n(x) over (1+x^2)^(n+1), with Q_n a polynomial of degree n.", "kind": "result", "symbols": [ { "unit": null, "symbol": "Q_n", "meaning": "polynomial of degree n in x" }, { "unit": null, "symbol": "n", "meaning": "order of differentiation" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/polynomial" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-4e77b7e06a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "257", "location": "DERIVATIVES AND INTEGRALS", "latex": "Q_{n} = (-1)^{n} n!\\left\\{(n + 1)x^{n} - \\dfrac{(n + 1)n(n - 1)}{3!} x^{n-2} + \\dots\\right\\}", "name": null, "statement": "Leading terms of the polynomial Q_n in Ex. 31(iv).", "kind": "result", "symbols": [ { "unit": null, "symbol": "Q_n", "meaning": "polynomial of degree n" }, { "unit": null, "symbol": "n", "meaning": "positive integer" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/factor", "concept/polynomial" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b2ff7ce302", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "256", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\lambda(ax^{2} + bx + c) + \\mu(a'x^{2} + b'x + c') = 0", "name": null, "statement": "The combined quadratic whose roots, by choice of the ratio lambda:mu, can be made real with any difference, unless the roots of the two quadratics interlace.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "lambda", "meaning": "multiplier of the first quadratic" }, { "unit": null, "symbol": "mu", "meaning": "multiplier of the second quadratic" }, { "unit": null, "symbol": "a", "meaning": "coefficient of x^2 in the first quadratic" }, { "unit": null, "symbol": "b", "meaning": "coefficient of x in the first quadratic" }, { "unit": null, "symbol": "c", "meaning": "constant term of the first quadratic" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/quadratic-equation", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a0f44d1801", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "256", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\pi < \\frac{\\sin \\pi x}{x(1 - x)} \\leq 4", "name": null, "statement": "Bounds on sin(pi x)/(x(1-x)) for 0 < x < 1.", "kind": "approximation", "symbols": [ { "unit": null, "symbol": "x", "meaning": "real variable with 0 < x < 1" } ], "sympy": "And(Lt(pi, sin(pi*x)/(x*(1 - x))), Le(sin(pi*x)/(x*(1 - x)), 4))", "physics": false, "states": [], "concepts": [ "concept/function", "concept/inequality" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-080d8676a0", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "256", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\frac{dy}{dx} = \\frac{(6x^{2} + x - 1) (x - 1)^{2} (x + 1)^{3}}{x^{2}}", "name": null, "statement": "The derivative of y with respect to x, whose sign analysis gives the general graph form in Ex. 23.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "y", "meaning": "function of x" }, { "unit": null, "symbol": "x", "meaning": "independent variable" } ], "sympy": "Eq(Derivative(y, x), (6*x**2 + x - 1)*(x - 1)**2*(x + 1)**3/x**2)", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/graph-of-a-function", "concept/maximum", "concept/minimum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-5c8f8af742", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "256", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\arctan\\{(a^{2} - b^{2})/2ab\\}", "name": null, "statement": "The greatest acute angle at which the ellipse can be cut by a concentric circle.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "semi-axis of the ellipse along x" }, { "unit": null, "symbol": "b", "meaning": "semi-axis of the ellipse along y" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/acute-angle", "concept/concentric-circles", "concept/ellipse", "concept/maximum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e446461b37", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "256", "location": "DERIVATIVES AND INTEGRALS", "latex": "s(x - s) x^{2} + 4\\Delta^{2} = 0", "name": null, "statement": "Equation whose roots are the stationary values of one side of a triangle with fixed area and semi-perimeter.", "kind": "result", "symbols": [ { "unit": null, "symbol": "s", "meaning": "semi-perimeter of the triangle" }, { "unit": null, "symbol": "Delta", "meaning": "area of the triangle" }, { "unit": null, "symbol": "x", "meaning": "length of the side under consideration" } ], "sympy": "Eq(s*(x - s)*x**2 + 4*Delta**2, 0)", "physics": false, "states": [], "concepts": [ "concept/area", "concept/maximum", "concept/minimum", "concept/root-of-an-equation", "quantity/semi-perimeter" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-36370b0289", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "256", "location": "DERIVATIVES AND INTEGRALS", "latex": "s(s - a)(s - b)(s - c) = \\Delta^{2}", "name": null, "statement": "Heron's formula in the form used in Ex. 26, relating area Delta to the semi-perimeter s and sides a, b, c.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "s", "meaning": "semi-perimeter" }, { "unit": null, "symbol": "a", "meaning": "side of the triangle" }, { "unit": null, "symbol": "b", "meaning": "side of the triangle" }, { "unit": null, "symbol": "c", "meaning": "side of the triangle" }, { "unit": null, "symbol": "Delta", "meaning": "area of the triangle" } ], "sympy": "Eq(s*(s - a)*(s - b)*(s - c), Delta**2)", "physics": false, "states": [], "concepts": [ "concept/area", "concept/triangle", "quantity/semi-perimeter" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-abc3e60521", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "256", "location": "DERIVATIVES AND INTEGRALS", "latex": "a + b + c = 2s", "name": null, "statement": "The sum of the sides of a triangle is twice its semi-perimeter.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "side of the triangle" }, { "unit": null, "symbol": "b", "meaning": "side of the triangle" }, { "unit": null, "symbol": "c", "meaning": "side of the triangle" }, { "unit": null, "symbol": "s", "meaning": "semi-perimeter" } ], "sympy": "Eq(a + b + c, 2*s)", "physics": false, "states": [], "concepts": [ "concept/triangle", "quantity/perimeter", "quantity/semi-perimeter" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e916be595d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "257", "location": "DERIVATIVES AND INTEGRALS", "latex": "2\\Delta + \\frac{a^{2} + b^{2} + c^{2}}{2\\sqrt{3}}", "name": null, "statement": "The area of the greatest equilateral triangle with sides through three given points A, B, C.", "kind": "result", "symbols": [ { "unit": null, "symbol": "Delta", "meaning": "area of the triangle ABC" }, { "unit": null, "symbol": "a", "meaning": "side of ABC" }, { "unit": null, "symbol": "b", "meaning": "side of ABC" }, { "unit": null, "symbol": "c", "meaning": "side of ABC" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/area", "concept/equilateral-triangle", "concept/maximum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-dc076b8428", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "257", "location": "DERIVATIVES AND INTEGRALS", "latex": "256\\Delta\\Delta' = 25a^{4}\\sqrt{5}", "name": null, "statement": "Relation between the areas of the two maximum isosceles triangles on the cardioid r = a(1 + cos theta).", "kind": "result", "symbols": [ { "unit": null, "symbol": "Delta", "meaning": "area of one maximum isosceles triangle" }, { "unit": null, "symbol": "Delta_prime", "meaning": "area of the other maximum isosceles triangle" }, { "unit": null, "symbol": "a", "meaning": "constant in the cardioid r = a(1 + cos theta)" } ], "sympy": "Eq(256*Delta*Delta_prime, 25*a**4*sqrt(5))", "physics": false, "states": [], "concepts": [ "concept/area", "concept/cardioid", "concept/isosceles-triangle", "concept/maximum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-7f5b9564fa", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "257", "location": "DERIVATIVES AND INTEGRALS", "latex": "x^{2}y - 4x^{2} - 4xy + y^{2} + 16x - 2y - 7 = 0", "name": null, "statement": "The curve on which the point (x, y) approaches (2, 3) in Ex. 29.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x", "meaning": "coordinate of a point on the curve" }, { "unit": null, "symbol": "y", "meaning": "coordinate of a point on the curve" } ], "sympy": "Eq(x**2*y - 4*x**2 - 4*x*y + y**2 + 16*x - 2*y - 7, 0)", "physics": false, "states": [], "concepts": [ "concept/curve", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-cb382596c1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "257", "location": "DERIVATIVES AND INTEGRALS", "latex": "(x^{2} - 4y + 8)/(y^{2} - 6x + 3)", "name": null, "statement": "The function whose limiting values as (x, y) approaches (2, 3) on the curve are found in Ex. 29.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x", "meaning": "coordinate" }, { "unit": null, "symbol": "y", "meaning": "coordinate" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/function", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-bc56253f7a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "257", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\frac{d}{da}\\{\\lim_{x \\to a} f(x)\\} - \\lim_{x \\to a}f'(x) = \\tfrac{3}{4} \\sec^{3} a - \\tfrac{5}{12} \\sec a", "name": null, "statement": "The difference between the derivative in a of the limit of f and the limit of f' equals the stated secant expression.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f(x)", "meaning": "1/(sin x - sin a) - 1/((x - a) cos a)" }, { "unit": null, "symbol": "a", "meaning": "parameter" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/limit", "concept/secant" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-99d00f5968", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "258", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int \\frac{dx}{(1 + x^{2})^{3}}", "name": null, "statement": "Integral to be evaluated in Ex. 38 (a calculation exercise, listed only as an integral to compute).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable of integration" } ], "sympy": "Eq(Integral(1/(1 + x**2)**3, x), Integral(1/(1 + x**2)**3, x))", "physics": false, "states": [], "concepts": [ "concept/integral", "method/integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-4c0bb7150a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "259", "location": "DERIVATIVES AND INTEGRALS", "latex": "2(n - 1)(q - \\tfrac{1}{4}p^{2}) \\int \\frac{dx}{(x^{2} + px + q)^{n}} \\\\ = \\frac{x + \\frac{1}{2}p}{(x^{2} + px + q)^{n-1}} + (2n - 3) \\int \\frac{dx}{(x^{2} + px + q)^{n-1}}", "name": null, "statement": "Reduction formula expressing the integral of (x^2+px+q)^(-n) in terms of the integral with exponent n-1.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "n", "meaning": "positive integer exponent" }, { "unit": null, "symbol": "p", "meaning": "coefficient of x in the quadratic" }, { "unit": null, "symbol": "q", "meaning": "constant term of the quadratic" }, { "unit": null, "symbol": "x", "meaning": "variable of integration" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/integral", "method/formulae-of-reduction", "method/integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-bb37a6a8a5", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "259", "location": "DERIVATIVES AND INTEGRALS", "latex": "(p + 1) I_{p, q} = x^{p+1}(1 + x)^{q} - qI_{p+1, q-1}", "name": null, "statement": "Reduction formula for I_{p,q} = integral of x^p (1+x)^q dx.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "I_{p,q}", "meaning": "integral of x^p (1 + x)^q dx" }, { "unit": null, "symbol": "p", "meaning": "integer index" }, { "unit": null, "symbol": "q", "meaning": "integer index" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/integral", "method/formulae-of-reduction", "method/integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-8cf336b4da", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "259", "location": "DERIVATIVES AND INTEGRALS", "latex": "I_{p, q} = (-1)^{p+1} \\int y^{p} (1 + y)^{-p-q-2}\\, dy", "name": null, "statement": "Result of the substitution x = -y/(1+y) applied to I_{p,q}.", "kind": "result", "symbols": [ { "unit": null, "symbol": "I_{p,q}", "meaning": "integral of x^p (1 + x)^q dx" }, { "unit": null, "symbol": "y", "meaning": "new variable of integration, x = -y/(1 + y)" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/integral", "method/integration", "method/substitution" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c0a5643391", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "259", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int xX^{-1/3}\\, dx = -3(3a - 2bx) X^{2/3}/10b^{2}", "name": null, "statement": "Integral of x X^(-1/3) with X = a + bx.", "kind": "result", "symbols": [ { "unit": null, "symbol": "X", "meaning": "a + bx" }, { "unit": null, "symbol": "a", "meaning": "constant" }, { "unit": null, "symbol": "b", "meaning": "constant" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/integral", "method/integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-1ad2aa12d2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "259", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int x^{2}X^{-1/3}\\, dx = 3(9a^{2} - 6abx + 5b^{2}x^{2}) X^{2/3}/40b^{3}", "name": null, "statement": "Integral of x^2 X^(-1/3) with X = a + bx.", "kind": "result", "symbols": [ { "unit": null, "symbol": "X", "meaning": "a + bx" }, { "unit": null, "symbol": "a", "meaning": "constant" }, { "unit": null, "symbol": "b", "meaning": "constant" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/integral", "method/integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-fe30e3d75a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "259", "location": "DERIVATIVES AND INTEGRALS", "latex": "2(n - 1)I_{m, n} = -x^{m-1} (1 + x^{2})^{-(n-1)} + (m - 1)I_{m-2, n-1}", "name": null, "statement": "Reduction formula for I_{m,n} = integral of x^m/(1+x^2)^n dx.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "I_{m,n}", "meaning": "integral of x^m/(1 + x^2)^n dx" }, { "unit": null, "symbol": "m", "meaning": "integer index" }, { "unit": null, "symbol": "n", "meaning": "integer index" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/integral", "method/formulae-of-reduction", "method/integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-3e3d97e8eb", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "259", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\beta I_{n} = x^{n} \\sin\\beta x - nJ_{n-1}", "name": null, "statement": "Reduction formula for I_n = integral of x^n cos(beta x) dx, linking it to J_{n-1}.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "I_n", "meaning": "integral of x^n cos(beta x) dx" }, { "unit": null, "symbol": "J_n", "meaning": "integral of x^n sin(beta x) dx" }, { "unit": null, "symbol": "beta", "meaning": "constant" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "method/formulae-of-reduction", "method/integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-800f2708ac", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "259", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\beta J_{n} = -x^{n} \\cos\\beta x + nI_{n-1}", "name": null, "statement": "Reduction formula for J_n = integral of x^n sin(beta x) dx.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "J_n", "meaning": "integral of x^n sin(beta x) dx" }, { "unit": null, "symbol": "beta", "meaning": "constant" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "method/formulae-of-reduction", "method/integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d24f052b3e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "260", "location": "DERIVATIVES AND INTEGRALS", "latex": "nI_{n} = \\sin x\\cos^{n-1} x + (n - 1) I_{n-2}", "name": null, "statement": "Reduction formula for I_n = integral of cos^n x dx.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "I_n", "meaning": "integral of cos^n x dx" }, { "unit": null, "symbol": "n", "meaning": "positive integer" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "method/formulae-of-reduction", "method/integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a56ef41322", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "260", "location": "DERIVATIVES AND INTEGRALS", "latex": "nJ_{n} = -\\cos x\\sin^{n-1} x + (n - 1) J_{n-2}", "name": null, "statement": "Reduction formula for J_n = integral of sin^n x dx.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "J_n", "meaning": "integral of sin^n x dx" }, { "unit": null, "symbol": "n", "meaning": "positive integer" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "method/formulae-of-reduction", "method/integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-99c21aa1f2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "260", "location": "DERIVATIVES AND INTEGRALS", "latex": "(n - 1)(I_{n} + I_{n-2}) = \\tan^{n-1}x", "name": null, "statement": "Reduction formula for I_n = integral of tan^n x dx.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "I_n", "meaning": "integral of tan^n x dx" }, { "unit": null, "symbol": "n", "meaning": "integer index" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/tangent-function", "method/formulae-of-reduction", "method/integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-962ff1cc4a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "260", "location": "DERIVATIVES AND INTEGRALS", "latex": "(m+n)I_{m, n} = -\\cos^{m+1}x \\sin^{n-1}x + (n - 1) I_{m, n-2}", "name": null, "statement": "First reduction formula for I_{m,n} = integral of cos^m x sin^n x dx.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "I_{m,n}", "meaning": "integral of cos^m x sin^n x dx" }, { "unit": null, "symbol": "m", "meaning": "integer index" }, { "unit": null, "symbol": "n", "meaning": "integer index" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/sine", "method/formulae-of-reduction", "method/integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d52f5c5230", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "260", "location": "DERIVATIVES AND INTEGRALS", "latex": "(n - 1)(n - 2)I_{m, n} = (n - 2)^{2}I_{m, n-2} + m(m - 1)I_{m-2, n-2} \\\\ -x^{m-1} \\cosec^{n-1}x \\{m\\sin x + (n - 2) x\\cos x\\}", "name": null, "statement": "Reduction formula for I_{m,n} = integral of x^m cosec^n x dx.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "I_{m,n}", "meaning": "integral of x^m cosec^n x dx" }, { "unit": null, "symbol": "m", "meaning": "integer index" }, { "unit": null, "symbol": "n", "meaning": "integer index" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/cosecant", "method/formulae-of-reduction", "method/integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-bd21af0ad6", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "260", "location": "DERIVATIVES AND INTEGRALS", "latex": "(n - 1)(a^{2} - b^{2}) I_{n} = -b\\sin x (a + b\\cos x)^{-(n-1)} + (2n - 3)aI_{n-1} - (n - 2)I_{n-2}", "name": null, "statement": "Reduction formula for I_n = integral of (a + b cos x)^(-n) dx.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "I_n", "meaning": "integral of (a + b cos x)^(-n) dx" }, { "unit": null, "symbol": "a", "meaning": "constant" }, { "unit": null, "symbol": "b", "meaning": "constant" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/cosine", "method/formulae-of-reduction", "method/integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-5c45e2c263", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "260", "location": "DERIVATIVES AND INTEGRALS", "latex": "4n(n + 1)(ab - h^{2})I_{n+2} - 2n(2n + 1)(a + b)I_{n+1} + 4n^{2}I_{n} = -\\frac{d^{2} I_{n}}{dx^{2}}", "name": null, "statement": "Reduction identity for I_n = integral of (a cos^2 x + 2h cos x sin x + b sin^2 x)^(-n) dx, involving the second derivative of I_n.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "I_n", "meaning": "integral of (a cos^2 x + 2h cos x sin x + b sin^2 x)^(-n) dx" }, { "unit": null, "symbol": "a", "meaning": "constant" }, { "unit": null, "symbol": "b", "meaning": "constant" }, { "unit": null, "symbol": "h", "meaning": "constant" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "method/formulae-of-reduction", "theorem/discriminant-of-a-conic" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-dc9b228533", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "260", "location": "DERIVATIVES AND INTEGRALS", "latex": "(m + 1)I_{m, n} = x^{m+1}(\\log x)^{n} - nI_{m, n-1}", "name": null, "statement": "Reduction formula for I_{m,n} = integral of x^m (log x)^n dx.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "I_{m,n}", "meaning": "integral of x^m (log x)^n dx" }, { "unit": null, "symbol": "m", "meaning": "index" }, { "unit": null, "symbol": "n", "meaning": "positive integer index" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/logarithm", "method/formulae-of-reduction", "method/integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-8e6c2b61c3", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "260", "location": "DERIVATIVES AND INTEGRALS", "latex": "x^{m+1} \\left\\{\\frac{(\\log x)^{n}}{m + 1} - \\frac{n(\\log x)^{n-1}}{(m + 1)^{2}} + \\frac{n(n - 1)(\\log x)^{n-2}}{(m + 1)^{3}} - \\dots + \\frac{(-1)^{n}n!}{(m + 1)^{n+1}}\\right\\}", "name": null, "statement": "Closed form of the integral of x^m (log x)^n dx for positive integer n.", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "positive integer" }, { "unit": null, "symbol": "m", "meaning": "index" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/integral", "concept/logarithm", "method/integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d9c9f27216", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "260", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\phi'' + a^{2}\\phi = 0", "name": null, "statement": "Differential equation of simple harmonic type whose most general solution is A cos ax + B sin ax, or rho cos(ax + epsilon).", "kind": "law", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "unknown function of x" }, { "unit": null, "symbol": "a", "meaning": "constant" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/differential-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-52ce7faf3c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "261", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\phi'^{2} + a^{2}\\phi^{2} = a^{2}b^{2}", "name": null, "statement": "First integral of phi'' + a^2 phi = 0, with b a constant.", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "unknown function of x" }, { "unit": null, "symbol": "b", "meaning": "constant" }, { "unit": null, "symbol": "a", "meaning": "constant" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/constant", "concept/derivative", "concept/differential-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-043b847356", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "261", "location": "DERIVATIVES AND INTEGRALS", "latex": "y' + \\omega z = 0", "name": null, "statement": "First equation of the linear system whose most general solution y, z is sought in Ex. 42.", "kind": "law", "symbols": [ { "unit": null, "symbol": "y", "meaning": "unknown function of x" }, { "unit": null, "symbol": "z", "meaning": "unknown function of x" }, { "unit": null, "symbol": "omega", "meaning": "constant" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/constant", "concept/derivative", "concept/differential-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-83361a063b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "261", "location": "DERIVATIVES AND INTEGRALS", "latex": "z' - \\omega y = 0", "name": null, "statement": "Second equation of the linear system in Ex. 42.", "kind": "law", "symbols": [ { "unit": null, "symbol": "y", "meaning": "unknown function of x" }, { "unit": null, "symbol": "z", "meaning": "unknown function of x" }, { "unit": null, "symbol": "omega", "meaning": "constant" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/constant", "concept/derivative", "concept/differential-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-fc3da95a8c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "261", "location": "DERIVATIVES AND INTEGRALS", "latex": "x = \\cos\\phi + \\frac{\\sin\\alpha \\sin\\phi}{1 - \\cos^{2}\\alpha \\sin^{2}\\phi}", "name": null, "statement": "Parametric x-coordinate of the curve whose area is found in Ex. 43.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "parameter" }, { "unit": "degree of angle", "symbol": "alpha", "meaning": "positive acute angle" } ], "sympy": "Eq(x, cos(phi) + sin(alpha)*sin(phi)/(1 - cos(alpha)**2*sin(phi)**2))", "physics": false, "states": [], "concepts": [ "concept/area", "concept/curve", "concept/parameter" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-acf4b78adf", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "261", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\frac{1}{2}\\pi(1 + \\sin\\alpha)^{2}/\\sin\\alpha", "name": null, "statement": "Area enclosed by the curve of Ex. 43.", "kind": "result", "symbols": [ { "unit": "degree of angle", "symbol": "alpha", "meaning": "positive acute angle" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/area", "concept/curve" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-7ff7275368", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "261", "location": "DERIVATIVES AND INTEGRALS", "latex": "a^{2}(\\beta - \\cos\\beta\\sin\\beta)", "name": null, "statement": "Area of either loop of the locus of the middle point of the chord in Ex. 44.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "radius of the circle" }, { "unit": "degree of angle", "symbol": "beta", "meaning": "angle in the projection condition 2a cos beta" } ], "sympy": "Eq(A, a**2*(beta - cos(beta)*sin(beta)))", "physics": false, "states": [], "concepts": [ "concept/area", "concept/circle" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-1a1bce2514", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "261", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\pi(a^{2} + \\frac{1}{2}b^{2})", "name": null, "statement": "Area enclosed by the locus of the foot of the perpendicular from A to a tangent of the circle in Ex. 46.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "radius of the circle" }, { "unit": null, "symbol": "b", "meaning": "distance of point A from the centre" } ], "sympy": "Eq(A, pi*(a**2 + b**2/2))", "physics": false, "states": [], "concepts": [ "concept/area", "concept/circle", "concept/locus", "quantity/pi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-21f616cdf1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "261", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\int \\frac{dx}{(lx + my + n)(hx + by + f)} = \\alpha\\log \\frac{PT}{PT'} + \\beta", "name": null, "statement": "Integral along a conic expressed as a logarithm of ratio of tangent perpendiculars plus a constant.", "kind": "result", "symbols": [ { "unit": null, "symbol": "PT", "meaning": "perpendicular from P to a tangent at one end of the chord" }, { "unit": null, "symbol": "PT'", "meaning": "perpendicular from P to the tangent at the other end of the chord" }, { "unit": null, "symbol": "alpha", "meaning": "constant" }, { "unit": null, "symbol": "beta", "meaning": "constant" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/conic-section", "concept/integral", "concept/logarithm", "concept/tangent" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a85f5c7cc3", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "261", "location": "DERIVATIVES AND INTEGRALS", "latex": "\\alpha e + \\gamma = 0", "name": null, "statement": "Condition under which the integral of (alpha cos x + beta sin x + gamma)/(1 - e cos x)^2 is a rational function of cos x and sin x.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "alpha", "meaning": "coefficient of cos x in the numerator" }, { "unit": null, "symbol": "e", "meaning": "constant in the denominator 1 - e cos x" }, { "unit": null, "symbol": "gamma", "meaning": "constant term of the numerator" } ], "sympy": "Eq(alpha*e + gamma, 0)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/integral", "concept/rational-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-9ee587c987", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "271", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\lim \\frac{QR}{h^{n}} = \\frac{1}{n!}\\{\\phi^{(n)}(\\xi) - f^{(n)}(\\xi)\\}", "name": null, "statement": "When the first n-1 derivatives of the two curves agree at xi, the gap QR is of order n, with coefficient given by the difference of their n-th derivatives over n factorial.", "kind": "result", "symbols": [ { "unit": null, "symbol": "QR", "meaning": "difference of ordinates of the two curves at abscissa xi + h" }, { "unit": null, "symbol": "n", "meaning": "order of contact" }, { "unit": null, "symbol": "f", "meaning": "function defining the first curve" }, { "unit": null, "symbol": "\\phi", "meaning": "function defining the second curve" } ], "sympy": "Eq(Limit(QR/h**n, h, 0), (phi_n(xi) - f_n(xi))/factorial(n))", "physics": false, "states": [], "concepts": [ "concept/contact-of-the-nth-order", "concept/higher-order-derivative", "concept/limit", "concept/order-of-smallness" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-bb6f6a7c21", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "262", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "f(b) - f(a) = (b - a) f'(\\xi)", "name": "Mean Value Theorem", "statement": "For a function with a derivative on the interval from a to b, the change in f equals the interval length times the derivative at some point xi between a and b.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f", "meaning": "the function" }, { "unit": null, "symbol": "a", "meaning": "lower end of the interval" }, { "unit": null, "symbol": "b", "meaning": "upper end of the interval" }, { "unit": null, "symbol": "\\xi", "meaning": "a value of x with a < xi < b" } ], "sympy": "Eq(f(b) - f(a), (b - a)*fp(xi))", "physics": false, "states": [ "theorem/mean-value-theorem" ], "concepts": [ "concept/derivative", "concept/function", "concept/interval" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-1d94bdebf3", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "262", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "f(a + h) - f(a) = hf'(a + \\theta_{1} h)", "name": "Mean Value Theorem", "statement": "The increase of f over a step h equals h times the derivative at a point a + theta_1 h inside the step, with 0 < theta_1 < 1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f", "meaning": "the function" }, { "unit": null, "symbol": "a", "meaning": "starting point" }, { "unit": null, "symbol": "h", "meaning": "step length" }, { "unit": null, "symbol": "\\theta_{1}", "meaning": "fraction with 0 < theta_1 < 1" } ], "sympy": "Eq(f(a + h) - f(a), h*fp(a + theta1*h))", "physics": false, "states": [ "theorem/mean-value-theorem" ], "concepts": [ "concept/derivative", "concept/function", "concept/increment" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-cad52e3821", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "263", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "f(a + h) = f(a) + hf'(a) + \\tfrac{1}{2}h^{2} f''(a + \\theta_{2}h)", "name": "Mean Value Theorem of the second order", "statement": "The value of f at a + h equals its first-order expansion at a plus a second-order term with the second derivative at a point inside the step.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f", "meaning": "the function" }, { "unit": null, "symbol": "a", "meaning": "starting point" }, { "unit": null, "symbol": "h", "meaning": "step length" }, { "unit": null, "symbol": "\\theta_{2}", "meaning": "fraction with 0 < theta_2 < 1" } ], "sympy": "Eq(f(a + h), f(a) + h*fp(a) + Rational(1,2)*h**2*fpp(a + theta2*h))", "physics": false, "states": [ "theorem/mean-value-theorem" ], "concepts": [ "concept/derivative", "concept/higher-order-derivative", "theorem/taylor-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-449ff3d8ca", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "266", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "f(a + h) - S_{n} = R_{n}", "name": "Taylor's theorem", "statement": "The difference between f(a + h) and the partial sum S_n of its expansion equals the remainder R_n.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f", "meaning": "the function" }, { "unit": null, "symbol": "S_{n}", "meaning": "sum of the first n terms of the expansion, from nu = 0 to n-1" }, { "unit": null, "symbol": "R_{n}", "meaning": "remainder after n terms (Lagrange's form)" } ], "sympy": "Eq(f(a + h) - S_n, R_n)", "physics": false, "states": [ "theorem/taylor-s-theorem" ], "concepts": [ "concept/remainder", "concept/sum", "theorem/lagrange-s-form-of-the-remainder" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-8a0a2b6010", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "266", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "f(a + h) = \\lim_{n\\to\\infty} S_{n}", "name": "Taylor's series", "statement": "If the remainder tends to zero, f(a + h) equals the limit of the partial sums of its Taylor expansion.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f", "meaning": "the function" }, { "unit": null, "symbol": "S_{n}", "meaning": "partial sum of the expansion" }, { "unit": null, "symbol": "n", "meaning": "number of terms" } ], "sympy": "Eq(f(a + h), Limit(S_n, n, oo))", "physics": false, "states": [ "theorem/taylor-s-series" ], "concepts": [ "concept/infinite-sequence", "concept/limit", "theorem/taylor-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-60ff671089", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "266", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "R_{n} = \\frac{h^{n}}{n!} f^{(n)}(a + \\theta_{n} h)", "name": "Lagrange's form of the remainder", "statement": "The remainder after n terms of Taylor's expansion equals h^n over n factorial times the n-th derivative at a point inside the step.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "R_{n}", "meaning": "remainder after n terms" }, { "unit": null, "symbol": "h", "meaning": "step length" }, { "unit": null, "symbol": "n", "meaning": "order of the derivative and number of terms" }, { "unit": null, "symbol": "\\theta_{n}", "meaning": "fraction with 0 < theta_n < 1" } ], "sympy": "Eq(R_n, h**n/factorial(n)*f_n(a + theta_n*h))", "physics": false, "states": [ "theorem/lagrange-s-form-of-the-remainder" ], "concepts": [ "concept/factorial", "concept/higher-order-derivative", "concept/remainder", "theorem/taylor-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-98db441875", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "266", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "f(h) = f(0) + hf'(0) + \\frac{h^{2}}{2!} f''(0) + \\dots", "name": "Maclaurin's series", "statement": "Taylor's series with a = 0 expresses f(h) as a power series in h using the derivatives at zero.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f", "meaning": "the function" }, { "unit": null, "symbol": "h", "meaning": "variable of the expansion" } ], "sympy": null, "physics": false, "states": [ "concept/maclaurin-s-series" ], "concepts": [ "concept/derivative", "concept/higher-order-derivative", "theorem/taylor-s-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-0269f67e56", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "267", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "(1 + x)^{m} = 1 + \\binom{m}{1}x + \\binom{m}{2}x^{2} + \\dots", "name": "binomial series", "statement": "The power (1 + x) raised to any rational m is expanded as a series in x with binomial coefficients.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable" }, { "unit": null, "symbol": "m", "meaning": "any rational exponent" } ], "sympy": null, "physics": false, "states": [ "concept/binomial-series", "theorem/binomial-theorem" ], "concepts": [ "concept/binomial", "concept/binomial-coefficient", "concept/binomial-series", "concept/exponent", "concept/maclaurin-s-series" ], "pages": [ "267", "384" ], "chapters": [ "hardy-course-of-pure-mathematics-1921/ch-vii", "hardy-course-of-pure-mathematics-1921/ch-ix" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a7b003e918", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "267", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\sin h = h - \\frac{h^{3}}{3!} + \\frac{h^{5}}{5!} - \\dots", "name": null, "statement": "The sine of h is given by its alternating power series, valid for all values of h.", "kind": "result", "symbols": [ { "unit": null, "symbol": "h", "meaning": "angle in radians" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/infinite-sequence", "concept/maclaurin-s-series", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-9b36974cc6", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "265", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "x = \\xi - \\frac{f(\\xi)}{f'(\\xi)}", "name": "Newton's method", "statement": "Starting from an approximate root xi, the next approximation to a root of f(x) = 0 is xi minus f(xi) divided by f'(xi).", "kind": "rule", "symbols": [ { "unit": null, "symbol": "x", "meaning": "improved approximation to the root" }, { "unit": null, "symbol": "\\xi", "meaning": "current approximation to the root" }, { "unit": null, "symbol": "f", "meaning": "the function whose root is sought" } ], "sympy": "Eq(x, xi - f(xi)/fp(xi))", "physics": false, "states": [ "method/newton-s-method" ], "concepts": [ "concept/approximation", "concept/derivative", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-34e3509ef0", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "272", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "y - f(\\xi) = (x - \\xi)f'(\\xi)", "name": "equation of the tangent", "statement": "The tangent to y = f(x) at x = xi is the line through (xi, f(xi)) with slope f'(xi).", "kind": "result", "symbols": [ { "unit": null, "symbol": "y", "meaning": "ordinate of a point on the tangent" }, { "unit": null, "symbol": "x", "meaning": "abscissa of a point on the tangent" }, { "unit": null, "symbol": "\\xi", "meaning": "abscissa of the point of contact" } ], "sympy": "Eq(y - f(xi), (x - xi)*fp(xi))", "physics": false, "states": [ "theorem/equation-of-the-tangent" ], "concepts": [ "concept/abscissa", "concept/derivative", "concept/line", "concept/tangent" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-5051cb77e1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "268", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\psi(x) = f(x)/\\phi(x)", "name": null, "statement": "Psi is defined as the ratio of f to phi, a function that is not defined where phi vanishes.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "\\psi", "meaning": "the ratio function" }, { "unit": null, "symbol": "f", "meaning": "numerator function" }, { "unit": null, "symbol": "\\phi", "meaning": "denominator function" } ], "sympy": "Eq(psi(x), f(x)/phi(x))", "physics": false, "states": [], "concepts": [ "concept/function", "concept/limit", "concept/quotient" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b55d730a26", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "269", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "f(x)/\\phi(x) \\to f'(\\xi)/\\phi'(\\xi)", "name": null, "statement": "If f and phi vanish at xi and their first derivatives at xi are not zero, the ratio f/phi tends to the ratio of the derivatives.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f", "meaning": "numerator function" }, { "unit": null, "symbol": "\\phi", "meaning": "denominator function" }, { "unit": null, "symbol": "\\xi", "meaning": "point at which both functions vanish" } ], "sympy": "Eq(Limit(f(x)/phi(x), x, xi), fp(xi)/php(xi))", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/function", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-4c431ddcde", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "269", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "f(x)/\\phi(x) \\to f^{(p)}(\\xi)/\\phi^{(p)}(\\xi)", "name": null, "statement": "When the first non-vanishing derivatives of f and phi are of the same order p, the ratio f/phi tends to the ratio of those p-th derivatives at xi.", "kind": "result", "symbols": [ { "unit": null, "symbol": "p", "meaning": "common order of the first non-vanishing derivatives" }, { "unit": null, "symbol": "f", "meaning": "numerator function" }, { "unit": null, "symbol": "\\phi", "meaning": "denominator function" } ], "sympy": "Eq(Limit(f(x)/phi(x), x, xi), f_p(xi)/phi_p(xi))", "physics": false, "states": [], "concepts": [ "concept/higher-order-derivative", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-5416fc63cf", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "268", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\phi(\\xi + h) - \\phi(\\xi) = \\frac{h^{n}}{n!} \\phi^{(n)} (\\xi + \\theta_{n} h)", "name": null, "statement": "When the first n-1 derivatives of phi vanish at xi, the change in phi over a small step is h^n over n factorial times the n-th derivative at a point inside the step.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\phi", "meaning": "the function" }, { "unit": null, "symbol": "\\xi", "meaning": "point of the stationary test" }, { "unit": null, "symbol": "h", "meaning": "small step" }, { "unit": null, "symbol": "n", "meaning": "order of the first non-vanishing derivative" } ], "sympy": "Eq(phi(xi + h) - phi(xi), h**n/factorial(n)*phi_n(xi + theta_n*h))", "physics": false, "states": [], "concepts": [ "concept/higher-order-derivative", "concept/maximum", "concept/minimum", "theorem/taylor-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-28916bf52e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "271", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\lim \\frac{QR}{h^{2}} = \\tfrac{1}{2}\\{\\phi''(\\xi) - f''(\\xi)\\}", "name": null, "statement": "When two curves touch at xi, the gap QR between their ordinates is of second order in h, with coefficient half the difference of their second derivatives.", "kind": "result", "symbols": [ { "unit": null, "symbol": "QR", "meaning": "difference of ordinates of the two curves at abscissa xi + h" }, { "unit": null, "symbol": "h", "meaning": "small increment of abscissa" }, { "unit": null, "symbol": "f", "meaning": "function defining the first curve" }, { "unit": null, "symbol": "\\phi", "meaning": "function defining the second curve" } ], "sympy": "Eq(Limit(QR/h**2, h, 0), Rational(1,2)*(phi2(xi) - f2(xi)))", "physics": false, "states": [], "concepts": [ "concept/contact-of-the-nth-order", "concept/curve", "concept/higher-order-derivative", "concept/limit", "concept/order-of-smallness" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-728dc254aa", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "273", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "r = \\frac{(1 + \\eta_{1}^{2})^{3/2}}{\\eta_{2}}", "name": null, "statement": "The radius of curvature of y = f(x) at a point equals (1 + eta_1 squared) to the power 3/2 divided by eta_2, where eta_1 and eta_2 are the first and second derivatives there.", "kind": "result", "symbols": [ { "unit": null, "symbol": "r", "meaning": "radius of curvature" }, { "unit": null, "symbol": "\\eta_{1}", "meaning": "first derivative f'(xi)" }, { "unit": null, "symbol": "\\eta_{2}", "meaning": "second derivative f''(xi)" } ], "sympy": "Eq(r, (1 + eta1**2)**(Rational(3,2))/eta2)", "physics": false, "states": [], "concepts": [ "concept/circle-of-curvature", "concept/curvature", "concept/derivative", "concept/higher-order-derivative", "quantity/radius-of-curvature" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ba49ff925c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "273", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "(x - a)^{2} + (y - b)^{2} = r^{2}", "name": null, "statement": "A circle of centre (a, b) and radius r is the set of points (x, y) satisfying this equation.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a", "meaning": "abscissa of the centre" }, { "unit": null, "symbol": "b", "meaning": "ordinate of the centre" }, { "unit": null, "symbol": "r", "meaning": "radius" }, { "unit": null, "symbol": "x", "meaning": "abscissa of a point" }, { "unit": null, "symbol": "y", "meaning": "ordinate of a point" } ], "sympy": "Eq((x - a)**2 + (y - b)**2, r**2)", "physics": false, "states": [], "concepts": [ "concept/cartesian-coordinates", "concept/centre-of-a-circle", "concept/circle", "concept/radius", "theorem/equation-of-the-circle" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-1fbd60b077", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "266", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "f(x) = a_{0} + a_{1}x + a_{2}x^{2} + \\dots + (a_{n} + \\epsilon_{x}) x^{n}", "name": null, "statement": "If the n-th derivative of f is continuous at zero, f is a polynomial of degree n in x plus a remainder term that vanishes faster than x^n.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a_{r}", "meaning": "coefficient of x^r, equal to f^{(r)}(0)/r!" }, { "unit": null, "symbol": "\\epsilon_{x}", "meaning": "quantity tending to zero as x tends to zero" }, { "unit": null, "symbol": "n", "meaning": "order of the expansion" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/maclaurin-s-series", "concept/order-of-smallness", "concept/polynomial" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e75faa969a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "266", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "a_{r} = f^{(r)}(0)/r!", "name": null, "statement": "The coefficient of x to the power r in the expansion equals the r-th derivative of f at zero divided by r factorial.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a_{r}", "meaning": "coefficient of x^r" }, { "unit": null, "symbol": "f^{(r)}(0)", "meaning": "r-th derivative of f at zero" }, { "unit": null, "symbol": "r", "meaning": "power of x" } ], "sympy": "Eq(a_r, f_r(0)/factorial(r))", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/factorial", "concept/higher-order-derivative", "concept/maclaurin-s-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-57b2a49afd", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "270", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "(\\sin x\\arcsin x - x^{2})/x^{6} \\to \\frac{1}{18}", "name": null, "statement": "As x tends to zero, the expression (sin x arcsin x minus x squared) divided by x to the sixth tends to 1/18.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable tending to zero" } ], "sympy": "Eq(Limit((sin(x)*asin(x) - x**2)/x**6, x, 0), Rational(1,18))", "physics": false, "states": [], "concepts": [ "concept/inverse-circular-function", "concept/limit", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e50e76b6f2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "270", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\lim_{x \\to n} (x - n)\\cosec x\\pi = \\frac{(-1)^{n}}{\\pi}", "name": null, "statement": "For any integer n, (x - n) times cosec(pi x) tends to (-1) to the n over pi as x tends to n.", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "any integer" }, { "unit": null, "symbol": "x", "meaning": "variable tending to n" }, { "unit": null, "symbol": "\\pi", "meaning": "pi" } ], "sympy": "Eq(Limit((x - n)*csc(pi*x), x, n), (-1)**n/pi)", "physics": false, "states": [], "concepts": [ "concept/cosecant", "concept/integer", "concept/limit", "quantity/pi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a54aba2168", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "274", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\lim_{h\\to 0}\\frac{f(x + h, y) - f(x, y)}{h}", "name": null, "statement": "The partial derivative of f with respect to x is the limit of the difference quotient taken with y held fixed.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "f", "meaning": "function of two independent real variables x and y" }, { "unit": null, "symbol": "h", "meaning": "increment of x" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/function-of-two-variables", "concept/increment", "concept/limit", "concept/partial-derivative", "method/differentiation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ba21be7b42", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "275", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\frac{\\dd r}{\\dd x} = \\frac{x}{\\sqrtp{x^{2} + y^{2}}}", "name": null, "statement": "With x = r cos θ and y = r sin θ, the partial derivative of r with respect to x at fixed y equals x over r.", "kind": "result", "symbols": [ { "unit": null, "symbol": "r", "meaning": "polar distance of the point (x, y) from the origin" }, { "unit": null, "symbol": "x", "meaning": "Cartesian coordinate" }, { "unit": null, "symbol": "y", "meaning": "Cartesian coordinate" } ], "sympy": "Eq(Derivative(r, x), x/sqrt(x**2 + y**2))", "physics": false, "states": [], "concepts": [ "concept/cartesian-coordinates", "concept/partial-derivative", "concept/polar-coordinates" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-7016aea767", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "275", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\frac{\\dd \\theta}{\\dd x} = -\\frac{y}{x^{2} + y^{2}}", "name": null, "statement": "The partial derivative of the polar angle θ with respect to x at fixed y is minus y over x squared plus y squared.", "kind": "result", "symbols": [ { "unit": null, "symbol": "θ", "meaning": "polar angle of the point (x, y)" } ], "sympy": "Eq(Derivative(theta, x), -y/(x**2 + y**2))", "physics": false, "states": [], "concepts": [ "concept/partial-derivative", "concept/plane-angle", "concept/polar-coordinates" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-83a9184f5d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "275", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\frac{\\dd x}{\\dd r} = \\cos\\theta", "name": null, "statement": "With x and y regarded as functions of r and θ, the partial derivative of x with respect to r at fixed θ is cos θ.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "Cartesian coordinate, a function of r and θ" }, { "unit": null, "symbol": "r", "meaning": "polar distance" }, { "unit": null, "symbol": "θ", "meaning": "polar angle" } ], "sympy": "Eq(Derivative(x, r), cos(theta))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/partial-derivative", "concept/polar-coordinates" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-16465b396c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "275", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\frac{\\dd x}{\\dd \\theta} = -r\\sin\\theta", "name": null, "statement": "The partial derivative of x with respect to θ at fixed r is minus r times sin θ.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "Cartesian coordinate" }, { "unit": null, "symbol": "θ", "meaning": "polar angle" }, { "unit": null, "symbol": "r", "meaning": "polar distance" } ], "sympy": "Eq(Derivative(x, theta), -r*sin(theta))", "physics": false, "states": [], "concepts": [ "concept/partial-derivative", "concept/polar-coordinates", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-3d08d29cd8", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "276", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\lim (\\Delta r/\\Delta x) = \\lim (PP_{2}/PP_{1}) = \\sec\\theta", "name": null, "statement": "Along the other hypothesis (r increased with θ fixed), the ratio Δr/Δx tends to sec θ.", "kind": "result", "symbols": [ { "unit": null, "symbol": "Δr", "meaning": "increment of r with θ held constant" }, { "unit": null, "symbol": "Δx", "meaning": "corresponding increment of x" }, { "unit": null, "symbol": "θ", "meaning": "polar angle" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/increment", "concept/limit", "concept/secant" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-1d1f796a14", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "276", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\lim (\\delta r/\\Delta r) = \\cos^{2}\\theta", "name": null, "statement": "The two partial-derivative hypotheses give ratios whose quotient tends to cos squared θ, so dx/dr and dr/dx are not reciprocals.", "kind": "result", "symbols": [ { "unit": null, "symbol": "δr", "meaning": "increment of r with x varied" }, { "unit": null, "symbol": "Δr", "meaning": "increment of r with θ held constant" }, { "unit": null, "symbol": "θ", "meaning": "polar angle" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/increment", "concept/limit", "concept/partial-derivative" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-84c79d0bd2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "276", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "b(\\dd z/\\dd x) = a(\\dd z/\\dd y)", "name": null, "statement": "If z = f(ax + by), then b times the partial derivative of z with respect to x equals a times the partial derivative of z with respect to y.", "kind": "result", "symbols": [ { "unit": null, "symbol": "z", "meaning": "function f(ax + by)" }, { "unit": null, "symbol": "a", "meaning": "constant coefficient" }, { "unit": null, "symbol": "b", "meaning": "constant coefficient" } ], "sympy": "Eq(b*Derivative(z, x), a*Derivative(z, y))", "physics": false, "states": [], "concepts": [ "concept/function-of-two-variables", "concept/partial-derivative" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e0ba9a91dc", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "276", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\dd u/\\dd x = 1", "name": null, "statement": "For u = x + y + z with x, y, z independent, the partial derivative of u with respect to x is 1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "u", "meaning": "function x + y + z" }, { "unit": null, "symbol": "x", "meaning": "independent variable" } ], "sympy": "Eq(Derivative(u, x), 1)", "physics": false, "states": [], "concepts": [ "concept/partial-derivative", "concept/variable" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-480a2fa135", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "277", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\frac{df}{dt} = \\frac{\\dd f}{\\dd x}\\, \\frac{dx}{dt} + \\frac{\\dd f}{\\dd y}\\, \\frac{dy}{dt}", "name": "Theorem of the Total Differential Coefficient", "statement": "The derivative of f along a curve x = φ(t), y = ψ(t) is the sum of the partial derivatives of f times the derivatives of x and y with respect to t.", "kind": "law", "symbols": [ { "unit": null, "symbol": "f", "meaning": "function of two variables x and y" }, { "unit": null, "symbol": "x", "meaning": "variable, a function of t" }, { "unit": null, "symbol": "y", "meaning": "variable, a function of t" }, { "unit": null, "symbol": "t", "meaning": "third variable on which x and y depend" } ], "sympy": "Eq(Derivative(f, t), Derivative(f, x)*Derivative(x, t) + Derivative(f, y)*Derivative(y, t))", "physics": false, "states": [ "theorem/total-differential" ], "concepts": [ "concept/derivative", "concept/function-of-two-variables", "concept/partial-derivative", "concept/total-differential", "method/differentiation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-68ae94ca98", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "278", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\frac{df}{dx} = \\frac{\\dd f}{\\dd x} + \\frac{\\dd f}{\\dd y}\\, \\frac{dy}{dx}", "name": null, "statement": "When t is x, the total derivative of f{x, ψ(x)} equals the partial derivative in x plus the partial derivative in y times dy/dx.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f", "meaning": "function of two variables x and y" }, { "unit": null, "symbol": "y", "meaning": "function ψ(x) of x" } ], "sympy": "Eq(Derivative(f, x), Derivative(f, x) + Derivative(f, y)*Derivative(y, x))", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/partial-derivative", "concept/total-differential" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-6a225d3187", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "278", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\frac{\\dd f}{\\dd x}\\, \\frac{dx}{dt} + \\frac{\\dd f}{\\dd y}\\, \\frac{dy}{dt} = 0", "name": null, "statement": "If eliminating t between x = φ(t) and y = ψ(t) gives f(x, y) = 0, then the total derivative of f along the curve vanishes.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f", "meaning": "function whose zero set is the curve" }, { "unit": null, "symbol": "t", "meaning": "parameter" } ], "sympy": "Eq(Derivative(f, x)*Derivative(x, t) + Derivative(f, y)*Derivative(y, t), 0)", "physics": false, "states": [], "concepts": [ "concept/curve", "concept/implicit-function", "concept/partial-derivative", "concept/total-differential" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-8455a58188", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "278", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "r' = (xx' + yy')/r", "name": null, "statement": "The derivative of the polar distance r with respect to t is (x x' + y y') divided by r, where dashes denote d/dt.", "kind": "result", "symbols": [ { "unit": null, "symbol": "r", "meaning": "polar distance of (x, y)" }, { "unit": null, "symbol": "x'", "meaning": "derivative of x with respect to t" }, { "unit": null, "symbol": "y'", "meaning": "derivative of y with respect to t" } ], "sympy": "Eq(Derivative(r, t), (x*Derivative(x, t) + y*Derivative(y, t))/r)", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/polar-coordinates" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-491e7e0f0e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "278", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\theta' = (xy' - yx')/r^{2}", "name": null, "statement": "The derivative of the polar angle θ with respect to t is (x y' minus y x') divided by r squared.", "kind": "result", "symbols": [ { "unit": null, "symbol": "θ", "meaning": "polar angle of (x, y)" }, { "unit": null, "symbol": "r", "meaning": "polar distance" } ], "sympy": "Eq(Derivative(theta, t), (x*Derivative(y, t) - y*Derivative(x, t))/r**2)", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/plane-angle", "concept/polar-coordinates" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-2cf83046c4", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "278", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\phi(x + h) - \\phi(x) = hf'(x + \\theta h)", "name": "Mean Value Theorem", "statement": "The increment of φ over an interval h equals h times f' at some point between x and x + h.", "kind": "result", "symbols": [ { "unit": null, "symbol": "φ", "meaning": "function of x" }, { "unit": null, "symbol": "f'", "meaning": "derivative of φ" }, { "unit": null, "symbol": "h", "meaning": "increment of x" }, { "unit": null, "symbol": "θ", "meaning": "number between 0 and 1" } ], "sympy": null, "physics": false, "states": [ "theorem/mean-value-theorem" ], "concepts": [ "concept/derivative", "concept/increment" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a9a964bc9b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "278", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\delta z = f(x + h, y + k) - f(x, y)", "name": null, "statement": "The increment of z = f(x, y) when x and y receive increments h and k.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "z", "meaning": "function f(x, y)" }, { "unit": null, "symbol": "h", "meaning": "increment of x" }, { "unit": null, "symbol": "k", "meaning": "increment of y" } ], "sympy": "Eq(dz, f(x + h, y + k) - f(x, y))", "physics": false, "states": [], "concepts": [ "concept/function-of-two-variables", "concept/increment" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b028ca9177", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "279", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\delta z = (f_{x}' + \\epsilon)\\, \\delta x + (f_{y}' + \\eta)\\, \\delta y", "name": "Mean Value Theorem for functions of two variables", "statement": "The increment of z equals the partial derivatives times the increments, plus small corrections ε and η that vanish as the increments vanish.", "kind": "result", "symbols": [ { "unit": null, "symbol": "δz", "meaning": "increment of z" }, { "unit": null, "symbol": "f_x'", "meaning": "partial derivative of f with respect to x" }, { "unit": null, "symbol": "f_y'", "meaning": "partial derivative of f with respect to y" }, { "unit": null, "symbol": "ε", "meaning": "small quantity tending to zero with δx and δy" }, { "unit": null, "symbol": "η", "meaning": "small quantity tending to zero with δx and δy" } ], "sympy": null, "physics": false, "states": [ "theorem/mean-value-theorem-for-functions-of-two-variables" ], "concepts": [ "concept/approximation", "concept/increment", "concept/partial-derivative", "theorem/mean-value-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-5f75daaae5", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "279", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\delta z = f_{x}'\\, \\delta x + f_{y}'\\, \\delta y", "name": null, "statement": "The increment of z is approximately equal to the partial derivatives times the increments, with error small compared with the larger increment.", "kind": "approximation", "symbols": [ { "unit": null, "symbol": "δz", "meaning": "increment of z" }, { "unit": null, "symbol": "f_x'", "meaning": "partial derivative of f with respect to x" }, { "unit": null, "symbol": "f_y'", "meaning": "partial derivative of f with respect to y" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/function-of-two-variables", "concept/increment", "concept/partial-derivative" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-f5dd641ea0", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "280", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\delta y = f'(x)\\, \\delta x", "name": null, "statement": "For a function of one variable, the increment of y is approximately f'(x) times the increment of x.", "kind": "approximation", "symbols": [ { "unit": null, "symbol": "δy", "meaning": "increment of y = f(x)" }, { "unit": null, "symbol": "f'(x)", "meaning": "derivative of f at x" }, { "unit": null, "symbol": "δx", "meaning": "increment of x" } ], "sympy": "Eq(dy, Derivative(f(x), x)*dx)", "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/derivative", "concept/increment" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b681da5a98", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "280", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "dy = f'(x)\\, \\delta x", "name": null, "statement": "The differential dy is defined as f'(x) times the increment δx of x.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "dy", "meaning": "differential of y = f(x)" }, { "unit": null, "symbol": "f'(x)", "meaning": "derivative of f at x" }, { "unit": null, "symbol": "δx", "meaning": "increment of x" } ], "sympy": "Eq(dy, Derivative(f(x), x)*delta_x)", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/differential", "concept/increment" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-26c646e8b5", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "280", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "dx = \\delta x", "name": null, "statement": "The differential of the independent variable x equals its increment.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "dx", "meaning": "differential of x" }, { "unit": null, "symbol": "δx", "meaning": "increment of x" } ], "sympy": "Eq(dx, delta_x)", "physics": false, "states": [], "concepts": [ "concept/differential", "concept/increment" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-388103fd5d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "280", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "dy = f'(x)\\, dx", "name": null, "statement": "The differential of y equals the derivative of f times the differential of x.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "dy", "meaning": "differential of y = f(x)" }, { "unit": null, "symbol": "dx", "meaning": "differential of x" }, { "unit": null, "symbol": "f'(x)", "meaning": "derivative of f at x" } ], "sympy": "Eq(dy, Derivative(f(x), x)*dx)", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/differential" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e7c0e41e3a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "280", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\frac{dy}{dx} = f'(x)", "name": null, "statement": "The quotient of the differentials dy and dx equals the derivative of f.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "dy", "meaning": "differential of y" }, { "unit": null, "symbol": "dx", "meaning": "differential of x" }, { "unit": null, "symbol": "f'(x)", "meaning": "derivative of f" } ], "sympy": "Eq(dy/dx, Derivative(f(x), x))", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/differential", "concept/quotient" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c9133bfb30", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "280", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\lim \\frac{dy}{\\delta y} = 1", "name": null, "statement": "When f' is continuous, the ratio of dy to δy tends to 1 as δx tends to zero, so dy is the principal part of δy.", "kind": "result", "symbols": [ { "unit": null, "symbol": "dy", "meaning": "differential of y" }, { "unit": null, "symbol": "δy", "meaning": "increment of y" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/differential", "concept/increment", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c50f73a27b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "281", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "dz = f_{x}'\\, \\delta x + f_{y}'\\, \\delta y", "name": null, "statement": "The differential of z = f(x, y) is defined as the partial derivatives times the increments of x and y.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "dz", "meaning": "differential of z = f(x, y)" }, { "unit": null, "symbol": "f_x'", "meaning": "partial derivative of f with respect to x" }, { "unit": null, "symbol": "f_y'", "meaning": "partial derivative of f with respect to y" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/differential", "concept/function-of-two-variables", "concept/increment", "concept/partial-derivative" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c302768a00", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "281", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "dz = f_{x}'\\, dx + f_{y}'\\, dy", "name": null, "statement": "The differential of z equals the partial derivatives times the differentials of x and y, whether or not x and y are independent.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "dz", "meaning": "differential of z" }, { "unit": null, "symbol": "dx", "meaning": "differential of x" }, { "unit": null, "symbol": "dy", "meaning": "differential of y" }, { "unit": null, "symbol": "f_x'", "meaning": "partial derivative of f with respect to x" }, { "unit": null, "symbol": "f_y'", "meaning": "partial derivative of f with respect to y" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/differential", "concept/partial-derivative", "concept/total-differential" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-edbb136345", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "281", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "A = \\pi ab", "name": null, "statement": "The area of an ellipse with semiaxes a and b is π times a times b.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "A", "meaning": "area of the ellipse" }, { "unit": null, "symbol": "a", "meaning": "semiaxis" }, { "unit": null, "symbol": "b", "meaning": "semiaxis" } ], "sympy": "Eq(A, pi*a*b)", "physics": false, "states": [], "concepts": [ "concept/area", "concept/ellipse", "quantity/pi", "theorem/area-of-an-ellipse" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-4f588383f4", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "281", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\frac{dA}{A} = \\frac{da}{a} + \\frac{db}{b}", "name": null, "statement": "The relative change of the ellipse's area equals the sum of the relative changes of its semiaxes.", "kind": "result", "symbols": [ { "unit": null, "symbol": "A", "meaning": "area of the ellipse" }, { "unit": null, "symbol": "a", "meaning": "semiaxis" }, { "unit": null, "symbol": "b", "meaning": "semiaxis" } ], "sympy": "Eq(dA/A, da/a + db/b)", "physics": false, "states": [], "concepts": [ "concept/differential", "concept/ellipse", "theorem/area-of-an-ellipse" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-3c935922eb", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "282", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\frac{d\\Delta}{\\Delta} = \\cot A\\, dA + \\frac{db}{b} + \\frac{dc}{c}", "name": null, "statement": "The relative change of a triangle's area expressed through angle A and sides b and c.", "kind": "result", "symbols": [ { "unit": null, "symbol": "Δ", "meaning": "area of triangle ABC" }, { "unit": null, "symbol": "A", "meaning": "angle at vertex A" }, { "unit": null, "symbol": "b", "meaning": "side opposite B" }, { "unit": null, "symbol": "c", "meaning": "side opposite C" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/cotangent", "concept/differential", "concept/triangle", "theorem/area-of-a-triangle" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-7e3e0f2929", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "282", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\frac{d\\Delta}{\\Delta} = 2\\frac{da}{a} + \\frac{c\\, dB}{a\\sin B} + \\frac{b\\, dC}{a\\sin C}", "name": null, "statement": "The relative change of a triangle's area expressed through side a and angles B and C.", "kind": "result", "symbols": [ { "unit": null, "symbol": "Δ", "meaning": "area of triangle ABC" }, { "unit": null, "symbol": "a", "meaning": "side of the triangle" }, { "unit": null, "symbol": "B", "meaning": "angle at vertex B" }, { "unit": null, "symbol": "C", "meaning": "angle at vertex C" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/differential", "concept/sine", "concept/triangle", "theorem/area-of-a-triangle" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-eb080003f2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "282", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "d\\Delta = R(\\cos A\\, da + \\cos B\\, db + \\cos C\\, dc)", "name": null, "statement": "The differential of the triangle's area in terms of its sides, with R the circumradius.", "kind": "result", "symbols": [ { "unit": "square units", "symbol": "Δ", "meaning": "area of triangle ABC" }, { "unit": null, "symbol": "R", "meaning": "radius of the circumcircle" }, { "unit": null, "symbol": "a", "meaning": "side of the triangle" }, { "unit": null, "symbol": "b", "meaning": "side of the triangle" }, { "unit": null, "symbol": "c", "meaning": "side of the triangle" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/circumscribed-circle", "concept/cosine", "concept/differential", "concept/radius", "theorem/area-of-a-triangle" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-4d8f438300", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "282", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\frac{\\dd a}{\\dd b} = -\\frac{\\cos B}{\\cos A}", "name": null, "statement": "If the triangle's area stays constant, the partial derivative of a with respect to b is minus cos B over cos A.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "side of the triangle, a function of b and c" }, { "unit": null, "symbol": "b", "meaning": "side of the triangle" }, { "unit": null, "symbol": "A", "meaning": "angle at vertex A" }, { "unit": null, "symbol": "B", "meaning": "angle at vertex B" } ], "sympy": "Eq(Derivative(a, b), -cos(B)/cos(A))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/partial-derivative", "concept/triangle", "theorem/area-of-a-triangle" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-705207139e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "282", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\frac{\\dd a}{\\dd c} = -\\frac{\\cos C}{\\cos A}", "name": null, "statement": "If the triangle's area stays constant, the partial derivative of a with respect to c is minus cos C over cos A.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "side of the triangle, a function of b and c" }, { "unit": null, "symbol": "c", "meaning": "side of the triangle" }, { "unit": null, "symbol": "C", "meaning": "angle at vertex C" }, { "unit": null, "symbol": "A", "meaning": "angle at vertex A" } ], "sympy": "Eq(Derivative(a, c), -cos(C)/cos(A))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/partial-derivative", "concept/triangle", "theorem/area-of-a-triangle" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-98f4de2e4e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "282", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\frac{da}{\\cos A} + \\frac{db}{\\cos B} + \\frac{dc}{\\cos C} = 0", "name": null, "statement": "If the circumradius R stays constant, the sides of the triangle satisfy this linear relation among their differentials.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "side of the triangle" }, { "unit": null, "symbol": "b", "meaning": "side of the triangle" }, { "unit": null, "symbol": "c", "meaning": "side of the triangle" }, { "unit": null, "symbol": "A", "meaning": "angle at vertex A" } ], "sympy": "Eq(da/cos(A) + db/cos(B) + dc/cos(C), 0)", "physics": false, "states": [], "concepts": [ "concept/circumscribed-circle", "concept/cosine", "concept/differential", "concept/triangle" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-f9b130688c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "282", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\frac{\\dd a}{\\dd b} = -\\frac{\\cos A}{\\cos B}", "name": null, "statement": "With the circumradius constant, the partial derivative of a with respect to b is minus cos A over cos B.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "side of the triangle" }, { "unit": null, "symbol": "b", "meaning": "side of the triangle" }, { "unit": null, "symbol": "A", "meaning": "angle at vertex A" }, { "unit": null, "symbol": "B", "meaning": "angle at vertex B" } ], "sympy": "Eq(Derivative(a, b), -cos(A)/cos(B))", "physics": false, "states": [], "concepts": [ "concept/circumscribed-circle", "concept/cosine", "concept/partial-derivative", "concept/triangle" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-36195831b8", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "282", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\frac{\\dd z}{\\dd x} = \\frac{\\dd z}{\\dd u}\\, \\frac{\\dd u}{\\dd x} + \\frac{\\dd z}{\\dd v}\\, \\frac{\\dd v}{\\dd x}", "name": null, "statement": "The partial derivative of z with respect to x, where z depends on u and v which depend on x, is the sum of the chained partial derivatives through u and through v.", "kind": "result", "symbols": [ { "unit": null, "symbol": "z", "meaning": "function of u and v" }, { "unit": null, "symbol": "u", "meaning": "function of x and y" }, { "unit": null, "symbol": "v", "meaning": "function of x and y" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/function-of-two-variables", "concept/partial-derivative", "concept/total-differential" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b1608e240a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "283", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "P = -\\frac{a_{1}p + a_{2}q - a_{3}}{c_{1}p + c_{2}q - c_{3}}", "name": null, "statement": "After a linear change of variables, the partial derivative P of Z with respect to X is expressed in terms of p and q, the partial derivatives of z.", "kind": "result", "symbols": [ { "unit": null, "symbol": "P", "meaning": "partial derivative of Z with respect to X" }, { "unit": null, "symbol": "p", "meaning": "partial derivative of z with respect to x" }, { "unit": null, "symbol": "q", "meaning": "partial derivative of z with respect to y" }, { "unit": null, "symbol": "a_{1}", "meaning": "coefficient in the linear change of variables" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/function-of-two-variables", "concept/partial-derivative" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-64c18584e8", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "283", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\frac{dy}{dx} = -\\frac{f_{a}'}{f_{b}'}", "name": null, "statement": "For the implicit relation f(x, y) = 0 through (a, b), dy/dx equals minus the ratio of the partial derivatives of f at (a, b).", "kind": "result", "symbols": [ { "unit": null, "symbol": "f", "meaning": "function of x and y defining y implicitly" }, { "unit": null, "symbol": "f_a'", "meaning": "partial derivative of f with respect to x at (a, b)" }, { "unit": null, "symbol": "f_b'", "meaning": "partial derivative of f with respect to y at (a, b)" } ], "sympy": "Eq(dy/dx, -fa/fb)", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/implicit-function", "concept/partial-derivative", "method/implicit-differentiation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-41ce9a8bf5", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "283", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "(x - x_{0}) f_{x_{0}}'(x_{0}, y_{0}) + (y - y_{0}) f_{y_{0}}'(x_{0}, y_{0}) = 0", "name": "equation of the tangent", "statement": "The tangent to the curve f(x, y) = 0 at (x0, y0) is the line whose equation uses the partial derivatives of f at that point.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "f", "meaning": "function whose zero set is the curve" }, { "unit": null, "symbol": "x_0", "meaning": "x-coordinate of the point of tangency" }, { "unit": null, "symbol": "y_0", "meaning": "y-coordinate of the point of tangency" } ], "sympy": null, "physics": false, "states": [ "theorem/equation-of-the-tangent" ], "concepts": [ "concept/curve", "concept/line", "concept/partial-derivative", "concept/point", "concept/tangent" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-15e7a1924c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "284", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "F'(x) = f(x)", "name": null, "statement": "The derivative of the area function F(x) under the graph of f is f(x), now justified by the definition of area.", "kind": "result", "symbols": [ { "unit": null, "symbol": "F(x)", "meaning": "area under the graph of f from a to x" }, { "unit": null, "symbol": "f(x)", "meaning": "continuous function whose graph bounds the area" } ], "sympy": "Eq(Derivative(F(x), x), f(x))", "physics": false, "states": [], "concepts": [ "concept/area", "concept/definite-integral", "concept/derivative", "concept/function", "concept/indefinite-integral", "concept/integral" ], "pages": [ "284", "286" ], "chapters": [ "hardy-course-of-pure-mathematics-1921/ch-vii" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-f2767503d1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "284", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "s = m_{0}\\delta_{0} + m_{1}\\delta_{1} + \\dots + m_{n}\\delta_{n}", "name": null, "statement": "The lower sum s is the sum of each sub-interval length times the lower bound of f on that sub-interval.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "s", "meaning": "lower sum for a subdivision" }, { "unit": null, "symbol": "m_ν", "meaning": "lower bound of f on sub-interval δ_ν" }, { "unit": null, "symbol": "δ_ν", "meaning": "length of the ν-th sub-interval" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/interval", "concept/least-upper-bound", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-2bb53da838", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "284", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "s \\leq M(b - a)", "name": null, "statement": "Each lower sum is at most M times the length of the whole interval, where M is an upper bound of f.", "kind": "inequality", "symbols": [ { "unit": null, "symbol": "s", "meaning": "lower sum" }, { "unit": null, "symbol": "M", "meaning": "upper bound of f on [a, b]" }, { "unit": null, "symbol": "a", "meaning": "left end of the interval" }, { "unit": null, "symbol": "b", "meaning": "right end of the interval" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/bounded-function", "concept/interval", "concept/least-upper-bound" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-7e21b04086", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "284", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "S \\geq m(b - a)", "name": null, "statement": "Each upper sum is at least m times the length of the whole interval, where m is a lower bound of f.", "kind": "inequality", "symbols": [ { "unit": null, "symbol": "S", "meaning": "upper sum for a subdivision" }, { "unit": null, "symbol": "m", "meaning": "lower bound of f on [a, b]" }, { "unit": null, "symbol": "a", "meaning": "left end of the interval" }, { "unit": null, "symbol": "b", "meaning": "right end of the interval" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/bounded-function", "concept/interval", "concept/least-upper-bound" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e318568491", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "286", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "0 \\leq J - s < \\epsilon", "name": null, "statement": "For a fine enough subdivision, the lower sum lies within ε below the common limit J of the lower and upper sums.", "kind": "result", "symbols": [ { "unit": null, "symbol": "J", "meaning": "common limit of the lower and upper sums, the area" }, { "unit": null, "symbol": "s", "meaning": "lower sum" }, { "unit": null, "symbol": "ε", "meaning": "arbitrary positive number" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/area", "concept/definite-integral", "concept/least-upper-bound", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-62b2eed216", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "286", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "S - s = \\tsum (M_{\\nu} - m_{\\nu})\\, \\delta_{\\nu} < \\epsilon", "name": null, "statement": "The gap between the upper and lower sums can be made less than any ε by making every sub-interval short enough.", "kind": "result", "symbols": [ { "unit": null, "symbol": "S", "meaning": "upper sum" }, { "unit": null, "symbol": "s", "meaning": "lower sum" }, { "unit": null, "symbol": "M_ν", "meaning": "upper bound of f on δ_ν" }, { "unit": null, "symbol": "m_ν", "meaning": "lower bound of f on δ_ν" }, { "unit": null, "symbol": "δ_ν", "meaning": "length of the ν-th sub-interval" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/least-upper-bound", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-6cde32c20d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "286", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\sigma = \\tsum f_{\\nu}\\delta_{\\nu}", "name": null, "statement": "The sum σ takes f at any point of each sub-interval times the sub-interval length and lies between s and S.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "σ", "meaning": "sum of f values at chosen points times sub-interval lengths" }, { "unit": null, "symbol": "f_ν", "meaning": "value of f at a point of δ_ν" }, { "unit": null, "symbol": "δ_ν", "meaning": "length of the ν-th sub-interval" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/area", "concept/definite-integral", "concept/interval", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-fb6b9c4cac", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "286", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "F(x) = \\int f(x)\\, dx", "name": null, "statement": "The indefinite integral of f is written as the function F whose derivative is f.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "F", "meaning": "indefinite integral of f" }, { "unit": null, "symbol": "f", "meaning": "the function being integrated" } ], "sympy": "Eq(F(x), Integral(f(x), x))", "physics": false, "states": [], "concepts": [ "concept/function", "concept/indefinite-integral", "concept/integral" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ab1600dd79", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "286", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "(PpqQ) = \\int_{a}^{b} f(x)\\, dx", "name": null, "statement": "The area PpqQ under the curve y = f(x) between x = a and x = b is written as a definite integral.", "kind": "definition", "symbols": [ { "unit": "area", "symbol": "PpqQ", "meaning": "the region bounded by the curve y = f(x), the ordinates x = a and x = b, and the axis of x" }, { "unit": null, "symbol": "a", "meaning": "lower limit" }, { "unit": null, "symbol": "b", "meaning": "upper limit" }, { "unit": null, "symbol": "f", "meaning": "continuous function giving the curve" } ], "sympy": "Eq(Area_PpqQ, Integral(f(x), (x, a, b)))", "physics": false, "states": [], "concepts": [ "concept/area", "concept/definite-integral", "concept/function", "concept/integral", "concept/limits-of-integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a901ed35fb", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "287", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\ds\\int_{a}^{b} f(x)\\, dx = F(b) - F(a)", "name": null, "statement": "The definite integral of f from a to b equals the difference of any integral function F at the two limits.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "lower limit" }, { "unit": null, "symbol": "b", "meaning": "upper limit" }, { "unit": null, "symbol": "F", "meaning": "integral function of f" }, { "unit": null, "symbol": "f", "meaning": "integrand" } ], "sympy": "Eq(Integral(f(x), (x, a, b)), F(b) - F(a))", "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/indefinite-integral", "concept/integral", "concept/limits-of-integration", "theorem/fundamental-theorem-of-the-integral-calculus" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a3b02b7342", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "287", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "F(x) = F(a) + \\int_{a}^{x} f(t)\\, dt", "name": null, "statement": "Any indefinite integral F can be written as a constant F(a) plus a definite integral with variable upper limit.", "kind": "result", "symbols": [ { "unit": null, "symbol": "F", "meaning": "indefinite integral of f" }, { "unit": null, "symbol": "a", "meaning": "fixed lower limit" }, { "unit": null, "symbol": "t", "meaning": "dummy variable of integration" } ], "sympy": "Eq(F(x), F(a) + Integral(f(t), (t, a, x)))", "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/indefinite-integral", "concept/integral", "concept/variable" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-8de745a33e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "289", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\arctan m = \\int_{0}^{m} \\frac{dt}{1 + t^{2}}", "name": null, "statement": "The arctangent of m is defined as the integral of 1/(1+t^2) from 0 to m.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "m", "meaning": "a real number (slope of the line OP)" }, { "unit": null, "symbol": "t", "meaning": "dummy variable of integration" } ], "sympy": "Eq(atan(m), Integral(1/(1 + t**2), (t, 0, m)))", "physics": false, "states": [], "concepts": [ "concept/circular-function", "concept/definite-integral", "concept/inverse-circular-function", "concept/plane-angle" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a518e2622d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "289", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\phi(m) = \\tfrac{1}{2} m\\mu^{2} + \\int_{\\mu}^{1} \\sqrtp{1 - x^{2}}\\, dx", "name": null, "statement": "The area of the sector of the unit circle between OA and OP, as a function of the slope m, equals a triangle term plus an integral of the circle's height.", "kind": "result", "symbols": [ { "unit": "area", "symbol": "phi", "meaning": "area of the sector AOP of the unit circle, as a function of m" }, { "unit": null, "symbol": "m", "meaning": "slope of OP (m > 0)" }, { "unit": null, "symbol": "mu", "meaning": "x-coordinate of P, equal to (1 + m^2)^(-1/2)" }, { "unit": null, "symbol": "x", "meaning": "variable of integration" } ], "sympy": "Eq(phi(m), m*mu**2/2 + Integral(sqrt(1 - x**2), (x, mu, 1)))", "physics": false, "states": [], "concepts": [ "concept/area", "concept/circle", "concept/circular-measure", "concept/circular-sector", "concept/definite-integral", "concept/function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-bc98991963", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "289", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\phi'(m) = \\frac{1}{2(1 + m^{2})}", "name": null, "statement": "The derivative of the sector-area function with respect to m is 1/(2(1+m^2)).", "kind": "result", "symbols": [ { "unit": "area", "symbol": "phi", "meaning": "sector-area function of m" }, { "unit": null, "symbol": "m", "meaning": "slope of OP" } ], "sympy": "Eq(Derivative(phi(m), m), 1/(2*(1 + m**2)))", "physics": false, "states": [], "concepts": [ "concept/area", "concept/circular-sector", "concept/derivative" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-86fbfc1e0b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "289", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\phi(m) = \\tfrac{1}{2} \\int_{0}^{m} \\frac{dt}{1 + t^{2}}", "name": null, "statement": "The sector-area function equals half the integral of 1/(1+t^2) from 0 to m.", "kind": "result", "symbols": [ { "unit": "area", "symbol": "phi", "meaning": "sector-area function of m" }, { "unit": null, "symbol": "m", "meaning": "slope of OP" }, { "unit": null, "symbol": "t", "meaning": "dummy variable of integration" } ], "sympy": "Eq(phi(m), Integral(1/(1 + t**2), (t, 0, m))/2)", "physics": false, "states": [], "concepts": [ "concept/area", "concept/circular-sector", "concept/definite-integral" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ca4c2a3e11", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "291", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\ds\\int_{a}^{b} f(x)\\, dx = -\\int_{b}^{a} f(x)\\, dx", "name": null, "statement": "Interchanging the limits of a definite integral changes its sign.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "lower limit" }, { "unit": null, "symbol": "b", "meaning": "upper limit" }, { "unit": null, "symbol": "f", "meaning": "continuous integrand" } ], "sympy": "Eq(Integral(f(x), (x, a, b)), -Integral(f(x), (x, b, a)))", "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/integral", "concept/limits-of-integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-cc561c4ead", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "292", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\ds\\int_{a}^{a} f(x)\\, dx = 0", "name": null, "statement": "A definite integral over an interval of zero length is zero.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "a limit of integration (both limits equal)" }, { "unit": null, "symbol": "f", "meaning": "continuous integrand" } ], "sympy": "Eq(Integral(f(x), (x, a, a)), 0)", "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/limits-of-integration", "concept/zero" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-1eec198e49", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "292", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\ds\\int_{a}^{b}f(x)\\, dx + \\int_{b}^{c}f(x)\\, dx = \\int_{a}^{c}f(x)\\, dx", "name": null, "statement": "Definite integrals over adjacent intervals add to the integral over the combined interval.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "lower limit" }, { "unit": null, "symbol": "b", "meaning": "intermediate point" }, { "unit": null, "symbol": "c", "meaning": "upper limit" }, { "unit": null, "symbol": "f", "meaning": "continuous integrand" } ], "sympy": "Eq(Integral(f(x), (x, a, b)) + Integral(f(x), (x, b, c)), Integral(f(x), (x, a, c)))", "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/integral", "concept/limits-of-integration", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b81f10274f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "292", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\ds\\int_{a}^{b}kf(x)\\, dx = k \\int_{a}^{b}f(x)\\, dx", "name": null, "statement": "A constant factor can be taken outside a definite integral.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "k", "meaning": "constant factor" }, { "unit": null, "symbol": "f", "meaning": "continuous integrand" } ], "sympy": "Eq(Integral(k*f(x), (x, a, b)), k*Integral(f(x), (x, a, b)))", "physics": false, "states": [], "concepts": [ "concept/constant", "concept/definite-integral", "concept/integral" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-1d93993c19", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "292", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\ds\\int_{a}^{b}\\{f(x) + \\phi(x)\\}\\, dx = \\int_{a}^{b}f(x)\\, dx + \\int_{a}^{b}\\phi(x)\\, dx", "name": null, "statement": "The definite integral of a sum is the sum of the definite integrals.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "f", "meaning": "continuous integrand" }, { "unit": null, "symbol": "phi", "meaning": "second continuous integrand" } ], "sympy": "Eq(Integral(f(x) + phi(x), (x, a, b)), Integral(f(x), (x, a, b)) + Integral(phi(x), (x, a, b)))", "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/integral", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-03e1506c52", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "292", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\ds\\int_{a}^{b}f(x)\\, dx \\geq 0", "name": null, "statement": "If f(x) is nonnegative on [a, b], its definite integral over [a, b] is nonnegative.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f", "meaning": "continuous function, nonnegative on a <= x <= b" }, { "unit": null, "symbol": "a", "meaning": "lower limit" }, { "unit": null, "symbol": "b", "meaning": "upper limit" } ], "sympy": "Ge(Integral(f(x), (x, a, b)), 0)", "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/inequality", "concept/non-negative" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-fcbea156a4", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "292", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "H(b - a) \\leq \\int_{a}^{b}f(x)\\, dx \\leq K(b - a)", "name": null, "statement": "If f lies between constants H and K on [a, b], the integral of f lies between H(b-a) and K(b-a).", "kind": "result", "symbols": [ { "unit": null, "symbol": "H", "meaning": "lower bound of f on [a, b]" }, { "unit": null, "symbol": "K", "meaning": "upper bound of f on [a, b]" }, { "unit": null, "symbol": "a", "meaning": "lower limit" }, { "unit": null, "symbol": "b", "meaning": "upper limit" } ], "sympy": "And(Le(H*(b - a), Integral(f(x), (x, a, b))), Le(Integral(f(x), (x, a, b)), K*(b - a)))", "physics": false, "states": [], "concepts": [ "concept/bounded-function", "concept/definite-integral", "concept/inequality" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-8befb36c3f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "292", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\ds\\int_{a}^{b}f(x)\\, dx = (b-a)f(\\xi)", "name": "First Mean Value Theorem for Integrals", "statement": "For continuous f there is a point xi between a and b at which the integral equals (b - a) times f(xi).", "kind": "theorem", "symbols": [ { "unit": null, "symbol": "xi", "meaning": "a point lying between a and b" }, { "unit": null, "symbol": "a", "meaning": "lower limit" }, { "unit": null, "symbol": "b", "meaning": "upper limit" }, { "unit": null, "symbol": "f", "meaning": "continuous integrand" } ], "sympy": "Eq(Integral(f(x), (x, a, b)), (b - a)*f(xi))", "physics": false, "states": [ "theorem/first-mean-value-theorem-for-integrals" ], "concepts": [ "concept/continuous-function", "concept/definite-integral", "concept/value-of-a-function", "theorem/mean-value-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-15d99a5f83", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "292", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "F(b) - F(a) = (b - a)F'(\\xi)", "name": null, "statement": "Restated with an integral function F, the first mean value theorem for integrals is the ordinary mean value theorem of the differential calculus.", "kind": "result", "symbols": [ { "unit": null, "symbol": "F", "meaning": "integral function of f" }, { "unit": null, "symbol": "xi", "meaning": "a point between a and b" } ], "sympy": "Eq(F(b) - F(a), (b - a)*Subs(Derivative(F(x), x), x, xi))", "physics": false, "states": [], "concepts": [ "concept/derivative", "theorem/first-mean-value-theorem-for-integrals", "theorem/mean-value-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b1def553e1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "293", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "H\\int_{a}^{b} \\phi(x)\\, dx \\leq \\int_{a}^{b} f(x)\\phi(x)\\, dx \\leq K\\int_{a}^{b} \\phi(x)\\, dx", "name": "Generalised Mean Value Theorem for integrals (bounds)", "statement": "When phi is positive, the integral of f times phi lies between H and K times the integral of phi.", "kind": "theorem", "symbols": [ { "unit": null, "symbol": "H", "meaning": "lower bound of f on [a, b]" }, { "unit": null, "symbol": "K", "meaning": "upper bound of f on [a, b]" }, { "unit": null, "symbol": "phi", "meaning": "positive weight function phi(x)" } ], "sympy": "And(Le(H*Integral(phi(x), (x, a, b)), Integral(f(x)*phi(x), (x, a, b))), Le(Integral(f(x)*phi(x), (x, a, b)), K*Integral(phi(x), (x, a, b))))", "physics": false, "states": [ "theorem/generalised-mean-value-theorem-for-integrals-bounds" ], "concepts": [ "concept/bounded-function", "concept/definite-integral", "concept/inequality", "theorem/mean-value-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-697e376b03", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "293", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\int_{a}^{b} f(x)\\phi(x)\\, dx = f(\\xi) \\int_{a}^{b} \\phi(x)\\, dx", "name": "Generalised Mean Value Theorem for integrals", "statement": "For continuous f and positive phi, the integral of f times phi equals f at some point xi times the integral of phi.", "kind": "theorem", "symbols": [ { "unit": null, "symbol": "xi", "meaning": "a point between a and b" }, { "unit": null, "symbol": "phi", "meaning": "positive weight function phi(x)" } ], "sympy": "Eq(Integral(f(x)*phi(x), (x, a, b)), f(xi)*Integral(phi(x), (x, a, b)))", "physics": false, "states": [ "theorem/generalised-mean-value-theorem-for-integrals" ], "concepts": [ "concept/definite-integral", "concept/value-of-a-function", "theorem/mean-value-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-52f8a41000", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "293", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "F(x) = \\int_{a}^{x} f(t)\\, dt", "name": "Fundamental Theorem of the Integral Calculus", "statement": "The integral of f from a to x is a function of x whose derivative is f(x), and is therefore continuous.", "kind": "theorem", "symbols": [ { "unit": null, "symbol": "F", "meaning": "integral function of f with variable upper limit x" }, { "unit": null, "symbol": "t", "meaning": "dummy variable of integration" }, { "unit": null, "symbol": "a", "meaning": "fixed lower limit" } ], "sympy": "Eq(F(x), Integral(f(t), (t, a, x)))", "physics": false, "states": [ "theorem/fundamental-theorem-of-the-integral-calculus" ], "concepts": [ "concept/continuous-function", "concept/derivative", "concept/indefinite-integral", "concept/integral" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-35e3ffbda3", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "295", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\int_{a}^{b} f(x)\\phi'(x)\\, dx = f(b)\\phi(b) - f(a)\\phi(a) - \\int_{a}^{b} f'(x)\\phi(x)\\, dx", "name": "Integration by parts for a definite integral", "statement": "The integral of f times the derivative of phi equals the boundary terms minus the integral of f' times phi.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f", "meaning": "function with continuous derivative f'" }, { "unit": null, "symbol": "phi", "meaning": "function with continuous derivative phi'" }, { "unit": null, "symbol": "a", "meaning": "lower limit" }, { "unit": null, "symbol": "b", "meaning": "upper limit" } ], "sympy": "Eq(Integral(f(x)*Derivative(phi(x), x), (x, a, b)), f(b)*phi(b) - f(a)*phi(a) - Integral(Derivative(f(x), x)*phi(x), (x, a, b)))", "physics": false, "states": [ "theorem/integration-by-parts-for-a-definite-integral" ], "concepts": [ "concept/definite-integral", "concept/derivative", "method/differentiation", "method/integration-by-parts" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-494df6e75d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "295", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\int f\\{\\phi(x)\\}\\phi'(x)\\, dx = F\\{\\phi(x)\\}", "name": null, "statement": "If F is an integral function of f, then the integral of f(phi(x)) times phi'(x) is F(phi(x)); this is the rule for substitution in an indefinite integral.", "kind": "result", "symbols": [ { "unit": null, "symbol": "F", "meaning": "integral function of f" }, { "unit": null, "symbol": "phi", "meaning": "function of x used as the substitution" } ], "sympy": "Eq(Integral(f(phi(x))*Derivative(phi(x), x), x), F(phi(x)))", "physics": false, "states": [], "concepts": [ "concept/function", "concept/indefinite-integral", "method/integration", "method/substitution" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-bfb7a4880b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "295", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\int_{c}^{d} f(t)\\, dt = F(d) - F(c) = F\\{\\phi(b)\\} - F\\{\\phi(a)\\} = \\int_{a}^{b} f\\{\\phi(x)\\}\\phi'(x)\\, dx", "name": "Transformation of a definite integral by substitution", "statement": "With c = phi(a) and d = phi(b), a definite integral in t equals the definite integral in x after substituting t = phi(x).", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi(a) = c", "meaning": "substitution maps the lower limit a to c" }, { "unit": null, "symbol": "phi(b) = d", "meaning": "substitution maps the upper limit b to d" }, { "unit": null, "symbol": "F", "meaning": "integral function of f" } ], "sympy": "Eq(Integral(f(t), (t, c, d)), Integral(f(phi(x))*Derivative(phi(x), x), (x, a, b)))", "physics": false, "states": [ "theorem/transformation-of-a-definite-integral-by-substitution" ], "concepts": [ "concept/definite-integral", "concept/integral", "concept/limits-of-integration", "method/substitution" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-1399ff3efb", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "298", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "f(a + h) = f(a) + hf'(a) + \\dots + \\frac{h^{n-1}}{(n - 1)!} f^{(n-1)}(a) + R_{n}", "name": "Taylor's theorem", "statement": "A function with n continuous derivatives equals its Taylor polynomial of degree n-1 about a, plus a remainder R_n.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f", "meaning": "function whose first n derivatives are continuous" }, { "unit": null, "symbol": "a", "meaning": "point of expansion" }, { "unit": null, "symbol": "h", "meaning": "increment added to a" }, { "unit": null, "symbol": "n", "meaning": "order (number of derivatives used)" }, { "unit": null, "symbol": "R_n", "meaning": "remainder after n terms" } ], "sympy": null, "physics": false, "states": [ "theorem/taylor-s-theorem" ], "concepts": [ "concept/derivative", "concept/function", "concept/higher-order-derivative", "concept/remainder" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-63af1337f7", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "298", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "R_{n} = \\frac{h^{n}}{(n - 1)!} \\int_{0}^{1} (1 - t)^{n-1} f^{(n)}(a + th)\\, dt", "name": null, "statement": "The remainder after n terms of Taylor's expansion is an integral of the nth derivative, with the variable t running from 0 to 1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "R_n", "meaning": "remainder after n terms" }, { "unit": null, "symbol": "h", "meaning": "increment added to a" }, { "unit": null, "symbol": "n", "meaning": "order" }, { "unit": null, "symbol": "t", "meaning": "variable of integration from 0 to 1" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/higher-order-derivative", "concept/remainder", "theorem/taylor-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-342b9d1453", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "298", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "R_{n} = \\frac{(1 - \\theta)^{n-p} f^{(n)}(a + \\theta h)h^{n}}{p(n - 1)!}", "name": "Form of the remainder with parameter p", "statement": "The Taylor remainder can be written with a point a + theta h, 0 < theta < 1, and an integer p between 1 and n; p = n gives Lagrange's form.", "kind": "result", "symbols": [ { "unit": null, "symbol": "theta", "meaning": "number with 0 < theta < 1" }, { "unit": null, "symbol": "p", "meaning": "positive integer not greater than n" }, { "unit": null, "symbol": "n", "meaning": "order of the expansion" }, { "unit": null, "symbol": "h", "meaning": "increment added to a" } ], "sympy": null, "physics": false, "states": [ "theorem/form-of-the-remainder-with-parameter-p" ], "concepts": [ "concept/higher-order-derivative", "concept/remainder", "theorem/mean-value-theorem", "theorem/taylor-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-7cd0649a2f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "298", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "R_{n} = \\frac{(1 - \\theta)^{n-1} f^{(n)}(a + \\theta h) h^{n}}{(n - 1)!}", "name": "Cauchy's form of the remainder", "statement": "Taking p = 1 gives Cauchy's form of the Taylor remainder, with a point a + theta h, 0 < theta < 1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "theta", "meaning": "number with 0 < theta < 1" }, { "unit": null, "symbol": "n", "meaning": "order of the expansion" }, { "unit": null, "symbol": "h", "meaning": "increment added to a" } ], "sympy": null, "physics": false, "states": [ "theorem/cauchy-s-form-of-the-remainder" ], "concepts": [ "concept/higher-order-derivative", "concept/remainder", "theorem/taylor-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-455b947f75", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "299", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "R_{n} = \\frac{m(m - 1)\\dots (m - n + 1)}{1·2\\dots (n - 1)}\\, \\frac{(1 - \\theta )^{n-1} x^{n}}{(1 + \\theta x)^{n-m}}", "name": null, "statement": "For f(x) = (1 + x)^m, Cauchy's form of the remainder after n terms of the binomial series.", "kind": "result", "symbols": [ { "unit": null, "symbol": "m", "meaning": "exponent, not a positive integer" }, { "unit": null, "symbol": "x", "meaning": "variable with -1 < x < 1 for convergence" }, { "unit": null, "symbol": "theta", "meaning": "number with 0 < theta < 1" }, { "unit": null, "symbol": "n", "meaning": "number of terms" }, { "unit": null, "symbol": "R_n", "meaning": "remainder after n terms" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/binomial-series", "concept/remainder", "theorem/binomial-theorem", "theorem/taylor-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ec8d98bc9c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "299", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "|R_{n}| < K |m| \\left|\\binom{m - 1}{n - 1}\\right| |x^{n}| = \\rho_{n}", "name": null, "statement": "The binomial-series remainder is bounded by rho_n, which tends to zero as n tends to infinity, so the remainder tends to zero.", "kind": "result", "symbols": [ { "unit": null, "symbol": "R_n", "meaning": "remainder after n terms of the binomial series" }, { "unit": null, "symbol": "K", "meaning": "constant bounding (1 + theta x)^(m-1) for all n" }, { "unit": null, "symbol": "rho_n", "meaning": "bounding sequence tending to zero" }, { "unit": null, "symbol": "m", "meaning": "exponent" }, { "unit": null, "symbol": "x", "meaning": "variable with -1 < x < 1" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/binomial-series", "concept/inequality", "concept/limit", "concept/remainder", "theorem/binomial-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-991d059070", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "299", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\int_{a}^{b} f(x)\\, dx = \\int_{a}^{b} \\{\\phi(x) + i\\psi(x)\\}\\, dx = \\int_{a}^{b} \\phi(x)\\, dx + i \\int_{a}^{b} \\psi(x)\\, dx", "name": null, "statement": "The integral of a complex function of a real variable is defined as the integral of its real part plus i times the integral of its imaginary part.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "f", "meaning": "complex function f(x) = phi(x) + i psi(x) of the real variable x" }, { "unit": null, "symbol": "phi", "meaning": "real part of f" }, { "unit": null, "symbol": "psi", "meaning": "imaginary part of f (coefficient of i)" }, { "unit": null, "symbol": "a", "meaning": "lower limit of integration" }, { "unit": null, "symbol": "b", "meaning": "upper limit of integration" }, { "unit": null, "symbol": "x", "meaning": "real variable" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/definite-integral", "concept/integral", "concept/real-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-26b7f21b71", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "299", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\left|\\int_{a}^{b} f(x)\\, dx\\right| \\leq \\int_{a}^{b} |f(x)|\\, dx", "name": null, "statement": "The modulus of the integral of a complex function is at most the integral of its modulus.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "f", "meaning": "complex function of the real variable x" }, { "unit": null, "symbol": "a", "meaning": "lower limit" }, { "unit": null, "symbol": "b", "meaning": "upper limit" } ], "sympy": "Le(Abs(Integral(f(x), (x, a, b))), Integral(Abs(f(x)), (x, a, b)))", "physics": false, "states": [], "concepts": [ "concept/absolute-value", "concept/complex-number", "concept/definite-integral", "concept/inequality" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-9e9f1cbff1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "300", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "|\\tsum f_{\\nu}\\, \\delta_{\\nu}| \\leq \\tsum |f_{\\nu}|\\, \\delta_{\\nu}", "name": null, "statement": "The modulus of a finite sum of complex terms times increments is at most the sum of the moduli.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "f_nu", "meaning": "complex value of f at a point of the nu-th subinterval" }, { "unit": null, "symbol": "delta_nu", "meaning": "length of the nu-th subinterval" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/inequality", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-9a8d7025df", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "300", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\tan x &= x + \\tfrac{1}{3} x^{3} + \\tfrac{2}{15} x^{5} + \\dots", "name": null, "statement": "The first terms of the Taylor series of tan x about 0 (to be verified).", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "independent variable" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/tangent-function", "theorem/taylor-s-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d4d082d272", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "300", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\sec x &= 1 + \\tfrac{1}{2} x^{2} + \\tfrac{5}{24} x^{4} + \\dots", "name": null, "statement": "The first terms of the Taylor series of sec x about 0 (to be verified).", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "independent variable" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/secant", "theorem/taylor-s-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-0243a2624c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "300", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "x\\cosec x &= 1 + \\tfrac{1}{6} x^{2} + \\tfrac{7}{360} x^{4} + \\dots", "name": null, "statement": "The first terms of the Taylor series of x cosec x about 0 (to be verified).", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "independent variable" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/cosecant", "theorem/taylor-s-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-729e6f7e9a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "300", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "x\\cot x &= 1 - \\tfrac{1}{3} x^{2} - \\tfrac{1}{45} x^{4} - \\dots", "name": null, "statement": "The first terms of the Taylor series of x cot x about 0 (to be verified).", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "independent variable" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/cotangent", "theorem/taylor-s-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e969d9ad20", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "300", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\theta_{n} = \\frac{1}{n + 1} + \\frac{n}{2(n + 1)^{2}(n + 2)} \\left\\{\\frac{f^{(n+2)}(0)}{f^{(n+1)}(0)} + \\epsilon_{x}\\right\\}x", "name": null, "statement": "The value of theta_n in Lagrange's form of the remainder after n terms, expanded for small x, where epsilon_x tends to 0 as x tends to 0.", "kind": "result", "symbols": [ { "unit": null, "symbol": "theta_n", "meaning": "value of theta in Lagrange's form of the remainder after n terms of Taylor's series" }, { "unit": null, "symbol": "n", "meaning": "number of terms of Taylor's series" }, { "unit": null, "symbol": "f", "meaning": "function whose Taylor series is taken" }, { "unit": null, "symbol": "epsilon_x", "meaning": "quantity tending to 0 as x tends to 0" }, { "unit": null, "symbol": "x", "meaning": "variable tending to 0" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/limit", "concept/remainder", "theorem/lagrange-s-form-of-the-remainder", "theorem/taylor-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-7c5fd29dbb", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "300", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "f(b) = f(a) + \\tfrac{1}{2}(b - a) \\{f'(a) + f'(b)\\} - \\tfrac{1}{12}(b - a)^{3} f'''(\\alpha)", "name": null, "statement": "Expresses f(b) through f(a), f' at the ends and f''' at an intermediate point alpha, where a < alpha < b.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f", "meaning": "function with derivatives of the first three orders" }, { "unit": null, "symbol": "a", "meaning": "lower end point" }, { "unit": null, "symbol": "b", "meaning": "upper end point" }, { "unit": null, "symbol": "alpha", "meaning": "intermediate point with a < alpha < b" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/higher-order-derivative", "concept/interval", "theorem/taylor-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b6814eb243", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "300", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "f(b) = f(a) + (b - a) f'\\{\\tfrac{1}{2}(a + b)\\} + \\tfrac{1}{24}(b - a)^{3}f'''(\\alpha)", "name": null, "statement": "Expresses f(b) using the derivative at the midpoint of [a, b] plus a third-derivative remainder at an intermediate point alpha.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f", "meaning": "function with derivatives of the first three orders" }, { "unit": null, "symbol": "alpha", "meaning": "intermediate point between a and b" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/interval", "theorem/taylor-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ee4e71bd85", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "300", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "f(b) = f(a) + \\tfrac{1}{6}(b - a) [f'(a) + f'(b) + 4f'\\{\\tfrac{1}{2}(a + b)\\}] - \\tfrac{1}{2880}(b - a)^{5} f^{(5)}(\\DPtypo{a}{\\alpha})", "name": null, "statement": "Expresses f(b) through end-point and midpoint derivatives plus a fifth-derivative remainder at an intermediate point.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f", "meaning": "function with derivatives of the first five orders" }, { "unit": null, "symbol": "alpha", "meaning": "intermediate point between a and b" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/higher-order-derivative", "concept/interval", "theorem/taylor-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d0dbb97785", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "300", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "f(b) = f(a) + \\tfrac{1}{2}(b - a) \\{f'(a) + f'(b)\\} - \\tfrac{1}{12}(b - a)^{2} \\{f''(b) - f''(a)\\} + \\tfrac{1}{720}(b - a)^{5} f^{(5)}(\\alpha)", "name": null, "statement": "Expresses f(b) through first and second derivatives at the end points plus a fifth-derivative remainder at an intermediate point alpha.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f", "meaning": "function with derivatives of the first five orders" }, { "unit": null, "symbol": "alpha", "meaning": "intermediate point between a and b" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/higher-order-derivative", "concept/interval", "theorem/taylor-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e8695e73d5", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "301", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\begin{vmatrix} f(a) & f(b)\\\\ g(a) & g(b) \\end{vmatrix} = (b - a) \\begin{vmatrix} f(a) & f'(\\beta)\\\\ g(a) & g'(\\beta) \\end{vmatrix}", "name": null, "statement": "A second-order determinant of function values equals (b - a) times a determinant with f' at an intermediate point beta.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f", "meaning": "function" }, { "unit": null, "symbol": "g", "meaning": "second function" }, { "unit": null, "symbol": "beta", "meaning": "point lying between a and b" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/determinant", "concept/mean-value" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-0a53fa3f89", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "301", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\begin{vmatrix} f(a) & f(b) & f(c)\\\\ g(a) & g(b) & g(c)\\\\ h(a) & h(b) & h(c) \\end{vmatrix} = \\tfrac{1}{2} (b - c)(c - a)(a - b) \\begin{vmatrix} f(a) & f'(\\beta) & f''(\\gamma)\\\\ g(a) & g'(\\beta) & g''(\\gamma)\\\\ h(a) & h'(\\beta) & h''(\\gamma) \\end{vmatrix}", "name": null, "statement": "A third-order determinant of function values equals a product of differences times a determinant of f, f' and f'' at intermediate points beta and gamma.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f", "meaning": "function" }, { "unit": null, "symbol": "g", "meaning": "second function" }, { "unit": null, "symbol": "h", "meaning": "third function" }, { "unit": null, "symbol": "beta", "meaning": "point between least and greatest of a, b, c" }, { "unit": null, "symbol": "gamma", "meaning": "point between least and greatest of a, b, c" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/determinant", "concept/higher-order-derivative" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a8d1fa8567", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "301", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "A(x^{n}/n!) \\leq F(x) \\leq B(x^{n}/n!)", "name": null, "statement": "If the n-th derivative of F lies between A and B on [0, h] and the lower derivatives vanish at 0, then F lies between A and B times x^n/n!.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "F", "meaning": "function with continuous derivatives of the first n orders" }, { "unit": null, "symbol": "A", "meaning": "lower bound of F^(n)" }, { "unit": null, "symbol": "B", "meaning": "upper bound of F^(n)" }, { "unit": null, "symbol": "n", "meaning": "order of derivative" }, { "unit": null, "symbol": "x", "meaning": "variable on [0, h]" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/factor", "concept/higher-order-derivative", "concept/inequality" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-bec150828e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "301", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\Delta_{h}^{n}\\phi(x) = \\sum_{r=0}^{n}(-1)^{r} \\binom{n}{r} \\phi(x + rh) = (-h)^{n} \\phi^{(n)}(\\xi)", "name": null, "statement": "The n-th finite difference of phi equals a binomial-weighted sum, and also equals (-h)^n times the n-th derivative at an intermediate point xi.", "kind": "result", "symbols": [ { "unit": null, "symbol": "Delta_h", "meaning": "forward difference operator with step h" }, { "unit": null, "symbol": "phi", "meaning": "function with derivatives of the first n orders" }, { "unit": null, "symbol": "h", "meaning": "step size" }, { "unit": null, "symbol": "n", "meaning": "order of difference" }, { "unit": null, "symbol": "r", "meaning": "summation index" }, { "unit": null, "symbol": "xi", "meaning": "point between x and x + nh" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/binomial-coefficient", "concept/higher-order-derivative", "concept/sum", "theorem/taylor-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-0858e2a43c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "301", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\{\\Delta_{h}^{n}\\phi(x)\\}/h^{n} \\to (-1)^{n}\\phi^{(n)}(x)", "name": null, "statement": "As h tends to 0 the scaled n-th difference tends to (-1)^n times the n-th derivative, when that derivative is continuous.", "kind": "result", "symbols": [ { "unit": null, "symbol": "h", "meaning": "step size tending to 0" }, { "unit": null, "symbol": "phi^(n)", "meaning": "n-th derivative of phi" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/continuous-function", "concept/higher-order-derivative", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e62004f273", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "301", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "x^{n-m}\\, \\Delta_{h}^{n} x^{m} \\to m(m - 1) \\dots (m - n + 1)h^{n}", "name": null, "statement": "As x tends to infinity, x^(n-m) times the n-th difference of x^m tends to m(m-1)...(m-n+1) h^n.", "kind": "result", "symbols": [ { "unit": null, "symbol": "m", "meaning": "any rational number" }, { "unit": null, "symbol": "n", "meaning": "positive integer" }, { "unit": null, "symbol": "h", "meaning": "step size" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/infinity", "concept/limit", "concept/product", "concept/rational-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-23c0182ab2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "301", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "x\\sqrt{x} \\{\\sqrt{x} - 2\\sqrtp{x + 1} + \\sqrtp{x + 2}\\} \\to -\\tfrac{1}{4}", "name": null, "statement": "A specific instance: the scaled second difference of sqrt(x) tends to -1/4 as x tends to infinity.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable tending to infinity" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/infinity", "concept/limit", "concept/rational-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-3636549117", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "302", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "y = \\phi(x) = x + a_{2}x^{2} + a_{3}x^{3} + (a_{4} + \\epsilon_{x})x^{4}", "name": null, "statement": "Series expansion of y = phi(x) near x = 0 with phi(0) = 0 and phi'(0) = 1, where epsilon_x tends to 0.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y", "meaning": "dependent variable" }, { "unit": null, "symbol": "a_2", "meaning": "second coefficient of the expansion" }, { "unit": null, "symbol": "a_3", "meaning": "third coefficient of the expansion" }, { "unit": null, "symbol": "a_4", "meaning": "fourth coefficient of the expansion" }, { "unit": null, "symbol": "epsilon_x", "meaning": "quantity tending to 0 as x tends to 0" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/function", "concept/limit", "theorem/taylor-s-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-fa326fef07", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "302", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "x = \\psi(y) = y - a_{2}y^{2} + (2a_{2}^{2} - a_{3})y^{3} - (5a_{2}^{3} - 5a_{2}a_{3} + a_{4} + \\epsilon_{y})y^{4}", "name": null, "statement": "Series of the inverse function x = psi(y) for the branch vanishing with y, where epsilon_y tends to 0.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "inverse-function variable" }, { "unit": null, "symbol": "psi", "meaning": "inverse function of phi" }, { "unit": null, "symbol": "y", "meaning": "variable of phi" }, { "unit": null, "symbol": "epsilon_y", "meaning": "quantity tending to 0 as y tends to 0" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/inverse-function", "concept/limit", "theorem/taylor-s-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-62f12eea15", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "302", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\frac{\\phi(x)\\psi(x) - x^{2}}{x^{4}} \\to a_{2}^{2}", "name": null, "statement": "The limit of (phi(x) psi(x) - x^2)/x^4 as x tends to 0 equals a_2 squared.", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "function with phi(0)=0, phi'(0)=1" }, { "unit": null, "symbol": "psi", "meaning": "inverse function of phi" }, { "unit": null, "symbol": "a_2", "meaning": "second coefficient of the expansion of phi" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/inverse-function", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-0f2c31532a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "302", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "-(\\xi - x)/y' = (\\eta - y)/x' = (x'^{2} + y'^{2})/(x'y'' - x''y')", "name": null, "statement": "The coordinates (xi, eta) of the centre of curvature of a parametric curve at (x, y) satisfy this relation, with dashes denoting derivatives with respect to t.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "xi", "meaning": "x-coordinate of the centre of curvature" }, { "unit": null, "symbol": "eta", "meaning": "y-coordinate of the centre of curvature" }, { "unit": null, "symbol": "x", "meaning": "x-coordinate of the point on the curve" }, { "unit": null, "symbol": "y", "meaning": "y-coordinate of the point on the curve" }, { "unit": null, "symbol": "t", "meaning": "parameter of the curve" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/centre-of-curvature", "concept/curvature", "concept/curve", "concept/derivative" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-70b6709b57", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "302", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "(x'^{2} + y'^{2})^{3/2}/(x'y'' - x''y')", "name": null, "statement": "The radius of curvature of the parametric curve x = f(t), y = F(t) is this expression.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x'", "meaning": "first derivative of x with respect to t" }, { "unit": null, "symbol": "y'", "meaning": "first derivative of y with respect to t" }, { "unit": null, "symbol": "x''", "meaning": "second derivative of x with respect to t" }, { "unit": null, "symbol": "y''", "meaning": "second derivative of y with respect to t" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/curvature", "concept/derivative", "concept/higher-order-derivative", "quantity/radius-of-curvature" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-4dd0f77bfa", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "302", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "3a(\\xi + x) + 2x^{2} = 0", "name": null, "statement": "For the curve 27a y^2 = 4x^3, the x-coordinate of the centre of curvature satisfies this relation.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "constant in the curve 27a y^2 = 4x^3" }, { "unit": null, "symbol": "xi", "meaning": "x-coordinate of the centre of curvature" }, { "unit": null, "symbol": "x", "meaning": "x-coordinate of the point on the curve" } ], "sympy": "Eq(3*a*(xi + x) + 2*x**2, 0)", "physics": false, "states": [], "concepts": [ "concept/centre-of-curvature", "concept/curvature", "concept/curve" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c75ad4f187", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "302", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\eta = 4y + (9ay)/x.", "name": null, "statement": "For the curve 27a y^2 = 4x^3, the y-coordinate of the centre of curvature is given by this relation.", "kind": "result", "symbols": [ { "unit": null, "symbol": "eta", "meaning": "y-coordinate of the centre of curvature" }, { "unit": null, "symbol": "a", "meaning": "constant in the curve 27a y^2 = 4x^3" }, { "unit": null, "symbol": "x", "meaning": "x-coordinate of the point on the curve" }, { "unit": null, "symbol": "y", "meaning": "y-coordinate of the point on the curve" } ], "sympy": "Eq(eta, 4*y + 9*a*y/x)", "physics": false, "states": [], "concepts": [ "concept/centre-of-curvature", "concept/curvature", "concept/curve" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-65dbf7f428", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "302", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "(1 + y_{1}^{2})y_{3} = 3y_{1}y_{2}^{2}", "name": null, "statement": "Condition at a point for the circle of curvature to have contact of the third order with the curve.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y_1", "meaning": "first derivative of y with respect to x" }, { "unit": null, "symbol": "y_2", "meaning": "second derivative of y with respect to x" }, { "unit": null, "symbol": "y_3", "meaning": "third derivative of y with respect to x" } ], "sympy": "Eq((1 + y1**2)*y3, 3*y1*y2**2)", "physics": false, "states": [], "concepts": [ "concept/circle-of-curvature", "concept/contact-of-the-nth-order", "concept/derivative", "concept/higher-order-derivative" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-58f087dd73", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "302", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "a^{3}y = a^{4}x^{2} + a^{2}bxy + (ac - b^{2})y^{2}", "name": null, "statement": "The conic of closest contact with y = ax^2 + bx^3 + cx^4 + ... at the origin.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "coefficient of x^2 in the curve" }, { "unit": null, "symbol": "b", "meaning": "coefficient of x^3 in the curve" }, { "unit": null, "symbol": "c", "meaning": "coefficient of x^4 in the curve" } ], "sympy": "Eq(a**3*y, a**4*x**2 + a**2*b*x*y + (a*c - b**2)*y**2)", "physics": false, "states": [], "concepts": [ "concept/conic", "concept/contact-of-the-nth-order", "concept/curve" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-3fae3c1919", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "302", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "18\\eta_{2}^{3}T = 9\\eta_{2}^{4}(x - \\xi)^{2} + 6\\eta_{2}^{2}\\eta_{3}(x - \\xi)T + (3\\eta_{2}\\eta_{4} - 4\\eta_{3}^{2})T^{2}", "name": null, "statement": "The conic of closest contact at the point (xi, eta) of the curve y = f(x), where eta_k are derivatives of f and T = (y - eta) - eta_1 (x - xi).", "kind": "result", "symbols": [ { "unit": null, "symbol": "eta_k", "meaning": "k-th derivative of f at (xi, eta)" }, { "unit": null, "symbol": "T", "meaning": "(y - eta) - eta_1 (x - xi)" }, { "unit": null, "symbol": "xi", "meaning": "x-coordinate of the point of contact" }, { "unit": null, "symbol": "eta", "meaning": "y-coordinate of the point of contact" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/conic", "concept/contact-of-the-nth-order", "concept/derivative", "concept/higher-order-derivative" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-22117834f2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "302", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "T = (y - \\eta) - \\eta_{1}(x - \\xi)", "name": null, "statement": "Definition of T as the deviation of y from the tangent line at (xi, eta).", "kind": "definition", "symbols": [ { "unit": null, "symbol": "T", "meaning": "deviation of y from the tangent at (xi, eta)" }, { "unit": null, "symbol": "eta_1", "meaning": "first derivative of f at xi" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/tangent" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c8a9afde1e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "302", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "x\\frac{\\dd u}{\\dd x} + y\\frac{\\dd u}{\\dd y} + z\\frac{\\dd u}{\\dd z} + \\dots = nu", "name": "Euler's theorem on homogeneous functions", "statement": "For a homogeneous function u of degree n, the sum of each variable times its partial derivative equals n times u.", "kind": "result", "symbols": [ { "unit": null, "symbol": "u", "meaning": "homogeneous function of degree n" }, { "unit": null, "symbol": "n", "meaning": "degree of homogeneity" }, { "unit": null, "symbol": "x", "meaning": "first variable" }, { "unit": null, "symbol": "y", "meaning": "second variable" }, { "unit": null, "symbol": "z", "meaning": "third variable" } ], "sympy": null, "physics": false, "states": [ "theorem/euler-s-theorem-on-homogeneous-functions" ], "concepts": [ "concept/derivative", "concept/homogeneous-function", "concept/rule" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-00a79b78fd", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "302", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "u = x^{n} f(y/x, z/x, \\dots)", "name": null, "statement": "Definition of a homogeneous function u of degree n in the variables x, y, z.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "u", "meaning": "homogeneous function of degree n" }, { "unit": null, "symbol": "n", "meaning": "degree" }, { "unit": null, "symbol": "f", "meaning": "function of the ratios y/x, z/x" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/degree", "concept/function", "concept/homogeneous-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-2541e4e97b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "303", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "xF_{\\xi} + yF_{\\eta} + zF_{\\zeta} = 0", "name": null, "statement": "The equation of the tangent at (xi, eta) to the curve f(x, y) = 0, written from its homogeneous form F(x, y, z) = 0.", "kind": "result", "symbols": [ { "unit": null, "symbol": "F", "meaning": "homogenised form of f(x, y) = 0 with third variable z" }, { "unit": null, "symbol": "F_xi", "meaning": "value of partial derivative of F with respect to x at (xi, eta, 1)" }, { "unit": null, "symbol": "xi", "meaning": "x-coordinate of the point of tangency" }, { "unit": null, "symbol": "eta", "meaning": "y-coordinate of the point of tangency" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/homogeneous-function", "concept/tangent", "theorem/equation-of-the-tangent" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-60dd7b991c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "303", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "u_{x}v_{y} - u_{y}v_{x} = 0", "name": null, "statement": "The Jacobian of u and v vanishes; this is a necessary condition for u and v to be functionally dependent.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "u", "meaning": "first function of x and y" }, { "unit": null, "symbol": "v", "meaning": "second function of x and y" }, { "unit": null, "symbol": "u_x", "meaning": "partial derivative of u with respect to x" }, { "unit": null, "symbol": "v_y", "meaning": "partial derivative of v with respect to y" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/determinant", "concept/functional-relation", "concept/jacobian" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-4c279a68d5", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "303", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\frac{\\dd \\phi}{\\dd u}\\, \\frac{\\dd u}{\\dd x} + \\frac{\\dd \\phi}{\\dd v}\\, \\frac{\\dd v}{\\dd x} = 0", "name": null, "statement": "Differentiating the relation phi(u, v) = 0 with respect to x gives this equation.", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "function in the relation phi(u, v) = 0" }, { "unit": null, "symbol": "u", "meaning": "first function of x and y" }, { "unit": null, "symbol": "v", "meaning": "second function of x and y" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/dependent", "concept/derivative", "concept/functional-relation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-dda9f21a94", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "303", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "J = \\begin{vmatrix} u_{x} & u_{y}\\\\ v_{x} & v_{y} \\end{vmatrix} = u_{x}v_{y} - u_{y}v_{x} = 0", "name": null, "statement": "The Jacobian J of u and v is the determinant of their partial derivatives; it must vanish if u and v are dependent.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "J", "meaning": "Jacobian (functional determinant) of u and v" }, { "unit": null, "symbol": "u_x", "meaning": "partial derivative of u with respect to x" }, { "unit": null, "symbol": "v_y", "meaning": "partial derivative of v with respect to y" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/determinant", "concept/functional-relation", "concept/jacobian" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-832f7b3485", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "303", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "J = \\frac{\\dd(u, v)}{\\dd(x, y)}", "name": null, "statement": "Notation for the Jacobian of u and v with respect to x and y.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "J", "meaning": "Jacobian of u and v with respect to x and y" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/jacobian" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a5accd5bef", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "303", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "J = \\begin{vmatrix} u_{x} & u_{y} & u_{z}\\\\ v_{x} & v_{y} & v_{z}\\\\ w_{x} & w_{y} & w_{z} \\end{vmatrix} = \\frac{\\dd(u, v, w)}{\\dd(x, y, z)}", "name": null, "statement": "The Jacobian of three functions of three variables is the 3x3 determinant of their partial derivatives; it vanishes exactly when they are functionally dependent.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "J", "meaning": "Jacobian of u, v, w with respect to x, y, z" }, { "unit": null, "symbol": "u", "meaning": "first function" }, { "unit": null, "symbol": "v", "meaning": "second function" }, { "unit": null, "symbol": "w", "meaning": "third function" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/determinant", "concept/functional-relation", "concept/jacobian" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-715b9fd00e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "303", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "abc + 2fgh - af^{2} - bg^{2} - ch^{2} = 0", "name": null, "statement": "Condition for the quadratic form ax^2 + by^2 + cz^2 + 2fyz + 2gzx + 2hxy to factor into two linear functions.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "coefficient of x^2" }, { "unit": null, "symbol": "b", "meaning": "coefficient of y^2" }, { "unit": null, "symbol": "c", "meaning": "coefficient of z^2" }, { "unit": null, "symbol": "f", "meaning": "half coefficient of yz" }, { "unit": null, "symbol": "g", "meaning": "half coefficient of zx" }, { "unit": null, "symbol": "h", "meaning": "half coefficient of xy" } ], "sympy": "Eq(a*b*c + 2*f*g*h - a*f**2 - b*g**2 - c*h**2, 0)", "physics": false, "states": [], "concepts": [ "concept/determinant", "concept/factor", "concept/homogeneous-polynomial" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-abab22f7ee", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "304", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\frac{\\dd(u, v)}{\\dd(x, y)} = \\frac{\\dd(u, v)}{\\dd(\\xi, \\eta)}\\, \\frac{\\dd(\\xi, \\eta)}{\\dd(x, y)}", "name": null, "statement": "Chain rule for Jacobians when u, v depend on xi, eta which depend on x, y.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "xi", "meaning": "intermediate variable depending on x, y" }, { "unit": null, "symbol": "eta", "meaning": "intermediate variable depending on x, y" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/determinant", "concept/jacobian" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-1248d92da4", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "304", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "f(x) + f(y) = f(xy)", "name": null, "statement": "The functional equation satisfied by f when f' = 1/x and f(1) = 0.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f", "meaning": "function with derivative 1/x vanishing at x = 1" }, { "unit": null, "symbol": "x", "meaning": "variable" }, { "unit": null, "symbol": "y", "meaning": "variable" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/function", "concept/functional-relation", "concept/logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-15311498cb", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "304", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "f(x) + f(y) = f\\left(\\frac{x + y}{1 - xy}\\right)", "name": null, "statement": "The functional equation satisfied by f when f' = 1/(1 + x^2) and f(0) = 0.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f", "meaning": "function with derivative 1/(1 + x^2) vanishing at 0" }, { "unit": null, "symbol": "x", "meaning": "variable" }, { "unit": null, "symbol": "y", "meaning": "variable" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/function", "concept/functional-relation", "concept/inverse-circular-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-5cae4be440", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "304", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "f(x) = \\int_{0}^{x} \\frac{dt}{\\sqrtp{1 - t^{4}}}", "name": null, "statement": "Definition of f as an integral of 1/sqrt(1 - t^4) from 0 to x.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "f", "meaning": "function defined by the integral" }, { "unit": null, "symbol": "t", "meaning": "variable of integration" }, { "unit": null, "symbol": "x", "meaning": "upper limit" } ], "sympy": "Eq(f(x), Integral(1/sqrt(1 - t**4), (t, 0, x)))", "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e6d249137d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "304", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "f(x) + f(y) = f\\left\\{\\frac{x\\sqrtp{1 - y^{4}} + y\\sqrtp{1 - x^{4}}}{1 + x^{2}y^{2}}\\right\\}", "name": null, "statement": "The addition formula for the integral f(x) = integral of dt/sqrt(1 - t^4).", "kind": "result", "symbols": [ { "unit": null, "symbol": "f", "meaning": "integral function defined in Ex. 25" }, { "unit": null, "symbol": "x", "meaning": "variable" }, { "unit": null, "symbol": "y", "meaning": "variable" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/function", "concept/functional-relation", "concept/integral" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-bb8f38cfcf", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "304", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "f'(x)f'(y)f'(z) \\{f(y) - f(z)\\} \\{f(z) - f(x)\\} \\{f(x) - f(y)\\} = 0", "name": null, "statement": "The condition for a functional relation between u, v, w built from f(x), f(y), f(z); it forces f to be constant.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f", "meaning": "function of one variable" }, { "unit": null, "symbol": "x", "meaning": "variable" }, { "unit": null, "symbol": "y", "meaning": "variable" }, { "unit": null, "symbol": "z", "meaning": "variable" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/function", "concept/functional-relation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-42db0aa22a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "305", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\int_{x_{0}}^{x_{1}} \\frac{dx}{ax^{2} + 2bx + c} = \\frac{1}{\\sqrtp{ac - b^{2}}} \\arctan\\left\\{ \\frac{(x_{1} - x_{0}) \\sqrtp{ac - b^{2}}} {ax_{1}x_{0} + b(x_{1} + x_{0}) + c} \\right\\}", "name": null, "statement": "Evaluates the integral of 1/(ax^2 + 2bx + c) between x_0 and x_1 as an inverse tangent, for a > 0 and ac - b^2 > 0.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "coefficient of x^2, positive" }, { "unit": null, "symbol": "b", "meaning": "half coefficient of x" }, { "unit": null, "symbol": "c", "meaning": "constant term" }, { "unit": null, "symbol": "x_0", "meaning": "lower limit" }, { "unit": null, "symbol": "x_1", "meaning": "upper limit, greater than x_0" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/inverse-circular-function", "concept/inverse-tangent" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-4de17d44dc", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "305", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\int_{-1}^{1} \\frac{\\sin\\alpha\\, dx}{1 - 2x\\cos\\alpha + x^{2}}", "name": null, "statement": "The integral whose value, as a function of alpha, is discontinuous at multiples of pi (Ex. 34).", "kind": "definition", "symbols": [ { "unit": "radian", "symbol": "alpha", "meaning": "parameter of the integral" }, { "unit": null, "symbol": "x", "meaning": "variable of integration" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/definite-integral", "concept/discontinuous-function", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-3b87e537b5", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "305", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\int_{x_{0}}^{x_{1}} \\frac{dx}{y} = \\frac{1}{\\sqrt{a}} \\log \\frac{1 + X\\sqrt{a}}{1 - X\\sqrt{a}}", "name": null, "statement": "Evaluates integral of dx/y with y = sqrt(ax^2 + 2bx + c) as a logarithm when a is positive.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y", "meaning": "sqrt(ax^2 + 2bx + c)" }, { "unit": null, "symbol": "X", "meaning": "(x_1 - x_0)/(y_1 + y_0)" }, { "unit": null, "symbol": "a", "meaning": "coefficient of x^2, positive" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-9b886f3a07", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "305", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\int_{0}^{a} \\frac{dx}{x + \\sqrtp{a^{2} - x^{2}}} = \\tfrac{1}{4}\\pi", "name": null, "statement": "The definite integral from 0 to a of 1/(x + sqrt(a^2 - x^2)) equals pi/4.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "positive constant, upper limit" }, { "unit": null, "symbol": "x", "meaning": "variable of integration" } ], "sympy": "Eq(Integral(1/(x + sqrt(a**2 - x**2)), (x, 0, a)), pi/4)", "physics": false, "states": [], "concepts": [ "concept/definite-integral", "quantity/pi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-101a259fe1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "305", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\int_{-1}^{1} \\frac{\\sqrtp{1 - x^{2}}}{a - x}\\, dx = \\pi\\{a - \\sqrtp{a^{2} - 1}\\}", "name": null, "statement": "For a > 1 the integral of sqrt(1 - x^2)/(a - x) over [-1, 1] equals pi(a - sqrt(a^2 - 1)).", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "constant greater than 1" }, { "unit": null, "symbol": "x", "meaning": "variable of integration" } ], "sympy": "Eq(Integral(sqrt(1 - x**2)/(a - x), (x, -1, 1)), pi*(a - sqrt(a**2 - 1)))", "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/root", "quantity/pi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a4b3499791", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "306", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\int_{0}^{1} \\frac{dx}{\\sqrtbr{\\{1 + (p^{2} - 1)x\\}\\{1 - (1 - q^{2}) x\\}}} = \\frac{2\\omega}{(p + q)\\sin\\omega}", "name": null, "statement": "For p > 1 and 0 < q < 1 the integral equals 2 omega/((p+q) sin omega), where omega is the acute angle with cosine (1 + pq)/(p + q).", "kind": "result", "symbols": [ { "unit": null, "symbol": "p", "meaning": "constant greater than 1" }, { "unit": null, "symbol": "q", "meaning": "constant between 0 and 1" }, { "unit": "radian", "symbol": "omega", "meaning": "positive acute angle with cosine (1 + pq)/(p + q)" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/acute-angle", "concept/cosine", "concept/definite-integral", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-56a9fb91cb", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "306", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\int_{0}^{2\\pi} \\frac{\\sin^{2}\\theta\\, d\\theta}{a - b\\cos\\theta} = \\frac{2\\pi}{b^{2}} \\{a - \\sqrtp{a^{2} - b^{2}}\\}", "name": null, "statement": "For a > b > 0 the integral of sin^2 theta over (a - b cos theta) from 0 to 2pi equals (2pi/b^2)(a - sqrt(a^2 - b^2)).", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "constant greater than b" }, { "unit": null, "symbol": "b", "meaning": "positive constant" }, { "unit": "radian", "symbol": "theta", "meaning": "angle variable" } ], "sympy": "Eq(Integral(sin(theta)**2/(a - b*cos(theta)), (theta, 0, 2*pi)), 2*pi/b**2*(a - sqrt(a**2 - b**2)))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/definite-integral", "concept/sine", "quantity/pi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c554128653", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "306", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\int_{0}^{\\pi} \\frac{d\\theta}{a + b\\cos\\theta + c\\sin\\theta} = \\frac{2}{\\sqrtp{a^{2} - b^{2} - c^{2}}} \\arctan \\left\\{\\frac{\\sqrtp{a^{2} - b^{2} - c^{2}}}{c}\\right\\}", "name": null, "statement": "For a > sqrt(b^2 + c^2) the integral over [0, pi] equals an inverse tangent expression, with the inverse tangent between 0 and pi.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "constant greater than sqrt(b^2 + c^2)" }, { "unit": null, "symbol": "b", "meaning": "constant coefficient of cos theta" }, { "unit": null, "symbol": "c", "meaning": "constant coefficient of sin theta" }, { "unit": "radian", "symbol": "theta", "meaning": "angle variable" } ], "sympy": "Eq(Integral(1/(a + b*cos(theta) + c*sin(theta)), (theta, 0, pi)), 2/sqrt(a**2 - b**2 - c**2)*atan(sqrt(a**2 - b**2 - c**2)/c))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/definite-integral", "concept/inverse-tangent", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-8f0c9b4120", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "306", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\left(\\int_{a}^{b} \\phi\\psi\\, dx\\right)^{2} \\leq \\int_{a}^{b} \\phi^{2}\\, dx \\int_{a}^{b} \\psi^{2}\\, dx", "name": "Schwarz's inequality for integrals", "statement": "The square of the integral of phi psi is at most the product of the integrals of phi^2 and psi^2.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "first function on [a, b]" }, { "unit": null, "symbol": "psi", "meaning": "second function on [a, b]" }, { "unit": null, "symbol": "a", "meaning": "lower limit" }, { "unit": null, "symbol": "b", "meaning": "upper limit" } ], "sympy": "Le(Integral(phi(x)*psi(x), (x, a, b))**2, Integral(phi(x)**2, (x, a, b))*Integral(psi(x)**2, (x, a, b)))", "physics": false, "states": [ "theorem/schwarz-s-inequality-for-integrals" ], "concepts": [ "concept/definite-integral", "concept/inequality", "theorem/cauchy-schwarz-inequality" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-04df36e3b1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "306", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "P_{n}(x) = \\frac{1}{(\\beta - \\alpha)^{n} n!} \\left(\\frac{d}{dx}\\right)^{n} \\{(x - \\alpha)(\\beta - x)\\}^{n}", "name": null, "statement": "Definition of P_n(x) as an n-th derivative (Rodrigues-type formula) of (x - alpha)^n (beta - x)^n; it is a polynomial of degree n.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "P_n", "meaning": "polynomial of degree n defined by this formula" }, { "unit": null, "symbol": "n", "meaning": "degree" }, { "unit": null, "symbol": "alpha", "meaning": "lower end of the interval" }, { "unit": null, "symbol": "beta", "meaning": "upper end of the interval" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/factorial", "concept/higher-order-derivative", "concept/polynomial" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-658918f35b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "306", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\int_{\\alpha}^{\\beta} P_{n}(x)\\theta(x)\\, dx = 0", "name": null, "statement": "P_n is orthogonal to every polynomial theta of degree less than n on the interval from alpha to beta.", "kind": "result", "symbols": [ { "unit": null, "symbol": "P_n", "meaning": "polynomial defined in Ex. 43" }, { "unit": null, "symbol": "theta", "meaning": "any polynomial of degree less than n" }, { "unit": null, "symbol": "alpha", "meaning": "lower end of the interval" }, { "unit": null, "symbol": "beta", "meaning": "upper end of the interval" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/degree", "concept/polynomial" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-8d28dffc10", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "306", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\int_{\\alpha}^{\\beta} P_{m}(x) P_{n}(x)\\, dx = 0", "name": null, "statement": "Polynomials P_m and P_n are orthogonal on [alpha, beta] when m differs from n.", "kind": "result", "symbols": [ { "unit": null, "symbol": "P_m", "meaning": "polynomial of degree m" }, { "unit": null, "symbol": "P_n", "meaning": "polynomial of degree n" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/degree", "concept/polynomial" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-7be7687336", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "307", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\int_{\\alpha}^{\\beta} (Q_{n} - \\kappa P_{n})^{2}\\, dx = 0", "name": null, "statement": "Step in Ex. 45: the square of Q_n minus a multiple of P_n integrates to zero.", "kind": "result", "symbols": [ { "unit": null, "symbol": "Q_n", "meaning": "polynomial of degree n with the orthogonality property" }, { "unit": null, "symbol": "kappa", "meaning": "constant chosen so that Q_n - kappa P_n has degree n-1" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/degree", "concept/polynomial" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-686b64c6d4", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "307", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\int_{0}^{1} \\phi(x)\\, dx = \\tfrac{1}{18}\\{5\\phi(\\alpha) + 8\\phi(\\tfrac{1}{2}) + 5\\phi(\\beta)\\}", "name": null, "statement": "For a fifth-degree polynomial phi, the integral over [0, 1] is given exactly by a three-point rule with alpha and beta the roots of x^2 - x + 1/10 = 0.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "polynomial of the fifth degree" }, { "unit": null, "symbol": "alpha", "meaning": "root of x^2 - x + 1/10 = 0" }, { "unit": null, "symbol": "beta", "meaning": "root of x^2 - x + 1/10 = 0" } ], "sympy": "Eq(Integral(phi(x), (x, 0, 1)), (5*phi(alpha) + 8*phi(Rational(1, 2)) + 5*phi(beta))/18)", "physics": false, "states": [], "concepts": [ "concept/approximation", "concept/definite-integral", "concept/degree", "concept/polynomial" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b285f386eb", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "307", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "x^{2} - x + \\frac{1}{10} = 0", "name": null, "statement": "The quadratic whose roots alpha and beta are the three-point nodes of the rule in Ex. 47.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable" } ], "sympy": "Eq(x**2 - x + Rational(1, 10), 0)", "physics": false, "states": [], "concepts": [ "concept/polynomial", "concept/quadratic-equation", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-974dc8b535", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "307", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "\\tfrac{1}{4}\\pi = \\int_{0}^{1} \\frac{dx}{1 + x^{2}}", "name": null, "statement": "pi/4 equals the integral of 1/(1 + x^2) from 0 to 1, used in Ex. 48 to approximate pi with Simpson's Rule.", "kind": "result", "symbols": [ { "unit": null, "symbol": "pi", "meaning": "the ratio of circumference to diameter" }, { "unit": null, "symbol": "x", "meaning": "variable of integration" } ], "sympy": "Eq(pi/4, Integral(1/(1 + x**2), (x, 0, 1)))", "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/inverse-tangent", "quantity/pi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-bfa55a5ab6", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "307", "location": "ADDITIONAL THEOREMS IN THE DIFFERENTIAL AND \\\\ INTEGRAL CALCULUS", "latex": "8.9 < \\int_{3}^{5} \\sqrtp{4 + x^{2}}\\, dx < 9", "name": null, "statement": "Bounds on the integral of sqrt(4 + x^2) from 3 to 5.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable of integration" } ], "sympy": "And(Lt(Rational(89, 10), Integral(sqrt(4 + x**2), (x, 3, 5))), Lt(Integral(sqrt(4 + x**2), (x, 3, 5)), 9))", "physics": false, "states": [], "concepts": [ "concept/bounded-function", "concept/definite-integral", "concept/inequality" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-38834fd8b4", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "308", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "u_{m} + u_{m+1} + \\dots + u_{n} = \\sum_{m}^{n} \\phi(\\nu)", "name": null, "statement": "The sum of the terms from u_m to u_n is written as a sum of phi(nu) from m to n, which is the book's shorthand notation for partial sums. Flag: the right-hand summand is printed as phi(nu) although the left side is in u; this is the book's own notation and is reported as printed.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "u", "meaning": "term of the series" }, { "unit": null, "symbol": "m", "meaning": "first suffix of the partial sum" }, { "unit": null, "symbol": "n", "meaning": "last suffix of the partial sum" }, { "unit": null, "symbol": "phi", "meaning": "function whose value at nu is summed" }, { "unit": null, "symbol": "nu", "meaning": "summation index" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/function-notation", "concept/infinite-sequence", "concept/sum", "concept/term" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-902f750389", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "309", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "u_{0} + u_{1} + \\dots + u_{n} < K", "name": null, "statement": "A series of positive terms is convergent exactly when there is a number K that bounds all its partial sums from above.", "kind": "result", "symbols": [ { "unit": null, "symbol": "u", "meaning": "term of a series of positive terms" }, { "unit": null, "symbol": "n", "meaning": "index of the partial sum" }, { "unit": null, "symbol": "K", "meaning": "a bound that the partial sums do not exceed" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/convergent-series", "concept/infinite-sequence", "concept/positive-number", "concept/series-of-positive-terms", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e4d99a205a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "309", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "v_{n} \\leq Ku_{n}", "name": "comparison theorem", "statement": "If v_n is at most K times u_n for all sufficiently large n and the series of u_n converges, then the series of v_n converges (Theorem C).", "kind": "rule", "symbols": [ { "unit": null, "symbol": "v", "meaning": "term of the series being tested" }, { "unit": null, "symbol": "u", "meaning": "term of the comparison series" }, { "unit": null, "symbol": "K", "meaning": "a constant" }, { "unit": null, "symbol": "n", "meaning": "index" } ], "sympy": "Le(v_n, K*u_n)", "physics": false, "states": [ "theorem/comparison-theorem" ], "concepts": [ "concept/constant", "concept/convergent-series", "concept/sufficiently-large-values" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-3e788ab1b7", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "309", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\sum v_{n} \\leq K \\sum u_{n}", "name": "comparison theorem", "statement": "Under the comparison hypothesis, the sum of the v_n is at most K times the sum of the u_n (Theorem C).", "kind": "result", "symbols": [ { "unit": null, "symbol": "v", "meaning": "terms of the series being tested" }, { "unit": null, "symbol": "u", "meaning": "terms of the comparison series" }, { "unit": null, "symbol": "K", "meaning": "a constant" } ], "sympy": null, "physics": false, "states": [ "theorem/comparison-theorem" ], "concepts": [ "concept/constant", "concept/convergent-series", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e1b471951c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "310", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "v_{n} \\leq Kr^{n}", "name": null, "statement": "If v_n is at most K r^n for all sufficiently large n, with r less than 1, the series of v_n is convergent, by comparison with the geometric series.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "v", "meaning": "term of the series" }, { "unit": null, "symbol": "K", "meaning": "a constant" }, { "unit": null, "symbol": "r", "meaning": "a positive number less than 1 (the common ratio)" }, { "unit": null, "symbol": "n", "meaning": "index" } ], "sympy": "Le(v_n, K*r**n)", "physics": false, "states": [], "concepts": [ "concept/common-ratio", "concept/convergent-series", "concept/geometrical-progression", "concept/sufficiently-large-values", "theorem/comparison-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-f8d4257289", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "310", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "v_{n}^{1/n} \\leq r", "name": "Cauchy's test", "statement": "Cauchy's test: the series of positive terms v_n converges if the nth root of v_n is at most r, with r less than 1, for all sufficiently large n.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "v", "meaning": "term of a series of positive terms" }, { "unit": null, "symbol": "n", "meaning": "index" }, { "unit": null, "symbol": "r", "meaning": "a number less than 1" } ], "sympy": "Le(v_n**(1/n), r)", "physics": false, "states": [ "theorem/cauchy-s-root-test" ], "concepts": [ "concept/convergent-series", "concept/rational-number", "concept/series-of-positive-terms", "concept/sufficiently-large-values" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-8827d24aca", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "310", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "v_{n}^{1/n} \\geq 1", "name": null, "statement": "The divergence form of Cauchy's test: the series of v_n diverges if the nth root of v_n is at least 1 for an infinity of values of n.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "v", "meaning": "term of a series of positive terms" }, { "unit": null, "symbol": "n", "meaning": "index" } ], "sympy": "Ge(v_n**(1/n), 1)", "physics": false, "states": [], "concepts": [ "concept/divergent-series", "concept/infinity", "concept/series-of-positive-terms", "theorem/cauchy-s-root-test" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-6b719c60a1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "310", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "v_{n+1}/v_{n} \\leq r", "name": "d'Alembert's ratio test", "statement": "d'Alembert's ratio test: the series of v_n converges if the ratio of successive terms is at most r, with r less than 1, for all sufficiently large n.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "v", "meaning": "term of a series of positive terms" }, { "unit": null, "symbol": "n", "meaning": "index" }, { "unit": null, "symbol": "r", "meaning": "a number less than 1" } ], "sympy": "Le(v_np1/v_n, r)", "physics": false, "states": [ "theorem/d-alembert-s-ratio-test" ], "concepts": [ "concept/convergent-series", "concept/ratio", "concept/sufficiently-large-values" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-7ef7861f04", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "310", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "v_{n+1}/v_{n} \\geq r \\geq 1", "name": "d'Alembert's ratio test", "statement": "If the ratio of successive terms is at least r, with r at least 1, for all (or all sufficiently large) n, the series of v_n diverges. The book notes that the same bound holding only for an infinity of n does not suffice.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "v", "meaning": "term of a series of positive terms" }, { "unit": null, "symbol": "n", "meaning": "index" }, { "unit": null, "symbol": "r", "meaning": "a number not less than 1" } ], "sympy": "And(Ge(v_np1/v_n, r), Ge(r, 1))", "physics": false, "states": [ "theorem/d-alembert-s-ratio-test" ], "concepts": [ "concept/divergent-series", "concept/ratio" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-da7feb4b8f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "311", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "v_{n+1}/v_{n} \\to l", "name": null, "statement": "The footnote states that if the ratio v_{n+1}/v_n tends to l, then v_n^{1/n} also tends to l (proof deferred to a later chapter); the converse is false.", "kind": "result", "symbols": [ { "unit": null, "symbol": "v", "meaning": "term of a series of positive terms" }, { "unit": null, "symbol": "n", "meaning": "index" }, { "unit": null, "symbol": "l", "meaning": "the limit" } ], "sympy": "Eq(Limit(v_np1/v_n, n, oo), l)", "physics": false, "states": [], "concepts": [ "concept/limit", "concept/ratio", "concept/tends-to-infinity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-6ec870b328", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "314", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "u_{0} v_{0} + (u_{1} v_{0} + u_{0} v_{1}) + (u_{2} v_{0} + u_{1} v_{1} + u_{0} v_{2}) + \\dots", "name": null, "statement": "The product of two convergent series of positive terms, with sums s and t, is the convergent series whose terms are grouped by total suffix; its sum is st.", "kind": "result", "symbols": [ { "unit": null, "symbol": "u", "meaning": "terms of the first series" }, { "unit": null, "symbol": "v", "meaning": "terms of the second series" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/convergent-series", "concept/product-of-series", "concept/rearrangement-of-a-series", "concept/series-of-positive-terms" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ca4138217a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "314", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "(u_{0} + u_{1} + \\dots + u_{n})(v_{0} + v_{1} + \\dots + v_{n})", "name": null, "statement": "The sum of the first n+1 groups in the product arrangement equals the product of the two partial sums, and so tends to st.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "u", "meaning": "terms of the first series" }, { "unit": null, "symbol": "v", "meaning": "terms of the second series" }, { "unit": null, "symbol": "n", "meaning": "index of the last partial sum" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/limit", "concept/product-of-series", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b7b07b154f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "318", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "u_{n} = \\phi(n)", "name": null, "statement": "The term u_n of the series is written as phi(n), the value of a decreasing continuous function phi(x) at x = n.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "u", "meaning": "term of the series" }, { "unit": null, "symbol": "n", "meaning": "positive integer index" }, { "unit": null, "symbol": "phi", "meaning": "continuous steadily decreasing function of the continuous variable x" } ], "sympy": "Eq(u_n, phi(n))", "physics": false, "states": [], "concepts": [ "concept/continuous-function", "concept/function", "concept/function-of-a-positive-integer-variable", "concept/steadily-increasing-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-9362cab691", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "316", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "u_{n+1} \\leq u_{n}", "name": null, "statement": "The assumed condition that the terms decrease steadily: each term is at most the one before it, for all or all sufficiently large n.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "u", "meaning": "term of a series of positive terms" }, { "unit": null, "symbol": "n", "meaning": "index" } ], "sympy": "Le(u_np1, u_n)", "physics": false, "states": [], "concepts": [ "concept/infinite-sequence", "concept/steadily-increasing-function", "concept/sufficiently-large-values" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-03cac8156c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "316", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\lim nu_{n} = 0", "name": "Abel's (or Pringsheim's) theorem", "statement": "Abel's theorem: if the series of u_n is convergent with positive decreasing terms, then n u_n tends to zero. It is one-sided: it gives divergence tests but not convergence.", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "index" }, { "unit": null, "symbol": "u", "meaning": "term of a convergent series of positive decreasing terms" } ], "sympy": "Eq(Limit(n*u_n, n, oo), 0)", "physics": false, "states": [ "theorem/abel-s-theorem" ], "concepts": [ "concept/convergent-series", "concept/infinite-sequence", "concept/limit", "concept/series-of-positive-terms" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-6ab53686eb", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "318", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\phi(\\nu - 1) \\geq \\phi(x) \\geq \\phi(\\nu)", "name": null, "statement": "For x between nu-1 and nu, phi(x) lies between phi(nu-1) and phi(nu), because phi is steadily decreasing.", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "continuous steadily decreasing function of x" }, { "unit": null, "symbol": "nu", "meaning": "positive integer" }, { "unit": null, "symbol": "x", "meaning": "continuous variable with nu-1 <= x <= nu" } ], "sympy": "And(Ge(phi(nu - 1), phi(x)), Ge(phi(x), phi(nu)))", "physics": false, "states": [], "concepts": [ "concept/continuous-function", "concept/inequality", "concept/integer", "concept/steadily-increasing-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-29aa7fb076", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "318", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "v_{\\nu} = \\phi(\\nu - 1) - \\int_{\\nu-1}^{\\nu} \\phi(x)\\, dx", "name": null, "statement": "v_nu is defined as the difference between phi(nu-1) and the integral of phi over the interval from nu-1 to nu, used in the integral test.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "v", "meaning": "auxiliary positive term in the integral test" }, { "unit": null, "symbol": "nu", "meaning": "positive integer" }, { "unit": null, "symbol": "phi", "meaning": "continuous steadily decreasing function of x" }, { "unit": null, "symbol": "x", "meaning": "integration variable" } ], "sympy": "Eq(v_nu, phi(nu - 1) - Integral(phi(x), (x, nu - 1, nu)))", "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/function", "concept/interval", "theorem/integral-test" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e24481e948", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "318", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "0 \\leq v_{\\nu} \\leq \\phi(\\nu - 1) - \\phi(\\nu)", "name": null, "statement": "The auxiliary term v_nu is non-negative and at most phi(nu-1) minus phi(nu).", "kind": "result", "symbols": [ { "unit": null, "symbol": "v", "meaning": "auxiliary term in the integral test" }, { "unit": null, "symbol": "phi", "meaning": "continuous steadily decreasing function" }, { "unit": null, "symbol": "nu", "meaning": "positive integer" } ], "sympy": "And(Le(0, v_nu), Le(v_nu, phi(nu - 1) - phi(nu)))", "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/inequality", "concept/positive-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c17ffdc2e6", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "318", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "v_{2} + v_{3} + \\dots + v_{n} \\leq \\phi(1) - \\phi(n) \\leq \\phi(1)", "name": null, "statement": "The partial sums of the auxiliary series are bounded by phi(1), so that series converges.", "kind": "result", "symbols": [ { "unit": null, "symbol": "v", "meaning": "auxiliary term" }, { "unit": null, "symbol": "phi", "meaning": "continuous steadily decreasing function" }, { "unit": null, "symbol": "n", "meaning": "index" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/convergent-series", "concept/inequality", "concept/infinite-sequence", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-5b960b7ae6", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "318", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\Phi(\\xi) = \\int_{1}^{\\xi} \\phi(x)\\, dx", "name": null, "statement": "Phi(xi) is defined as the integral of phi from 1 to xi; it is continuous and steadily increasing in xi.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "Phi", "meaning": "integral of phi from 1 to xi, a function of xi" }, { "unit": null, "symbol": "xi", "meaning": "upper limit of integration (continuous variable)" }, { "unit": null, "symbol": "phi", "meaning": "continuous steadily decreasing function of x" }, { "unit": null, "symbol": "x", "meaning": "integration variable" } ], "sympy": "Eq(Phi(xi), Integral(phi(x), (x, 1, xi)))", "physics": false, "states": [], "concepts": [ "concept/continuous-function", "concept/definite-integral", "concept/function", "concept/limits-of-integration", "concept/steadily-increasing-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-8b11aaf72a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "318", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\sum_{1}^{n-1} \\phi(\\nu) - \\int_{1}^{n} \\phi(x)\\, dx", "name": null, "statement": "The difference between the partial sums of phi(nu) and the integral up to n tends to a positive limit as n tends to infinity.", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "continuous steadily decreasing function" }, { "unit": null, "symbol": "nu", "meaning": "summation index" }, { "unit": null, "symbol": "n", "meaning": "index" }, { "unit": null, "symbol": "x", "meaning": "integration variable" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/limit", "concept/sum", "theorem/integral-test" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b0736d8105", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "319", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "v_{\\nu} < \\phi(\\nu - 1) - \\phi(\\nu)", "name": null, "statement": "Remark: the strict form of the bound for the auxiliary term holds unless phi is constant on the interval, so the sum is strictly less than phi(1) + l.", "kind": "result", "symbols": [ { "unit": null, "symbol": "v", "meaning": "auxiliary term" }, { "unit": null, "symbol": "phi", "meaning": "continuous steadily decreasing function" }, { "unit": null, "symbol": "nu", "meaning": "positive integer" } ], "sympy": "Lt(v_nu, phi(nu - 1) - phi(nu))", "physics": false, "states": [], "concepts": [ "concept/constant", "concept/inequality", "theorem/integral-test" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-244c1cdb94", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "318", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\phi(1) + \\phi(2) + \\dots", "name": "integral test", "statement": "Maclaurin's (or Cauchy's) integral test: for a positive continuous decreasing phi, the series phi(1)+phi(2)+... converges if and only if the integral of phi from 1 to xi tends to a limit as xi tends to infinity; its sum is then at most phi(1) + l.", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "positive continuous steadily decreasing function of x for x > 1" }, { "unit": null, "symbol": "l", "meaning": "the limit of the integral as xi tends to infinity" } ], "sympy": null, "physics": false, "states": [ "theorem/integral-test" ], "concepts": [ "concept/convergent-series", "concept/definite-integral", "concept/divergent-series", "concept/limit", "concept/tends-to-infinity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-9d9f45da8e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "319", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\Phi(\\xi) = \\int_{1}^{\\xi} \\frac{dx}{x^{s}} = \\frac{\\xi^{1-s} - 1}{1 - s}", "name": null, "statement": "For s not equal to 1, the integral of x^(-s) from 1 to xi equals (xi^(1-s) - 1)/(1 - s).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "Phi", "meaning": "integral of x^(-s) from 1 to xi" }, { "unit": null, "symbol": "xi", "meaning": "upper limit of integration" }, { "unit": null, "symbol": "s", "meaning": "a rational number (the power)" }, { "unit": null, "symbol": "x", "meaning": "integration variable" } ], "sympy": "Eq(Phi(xi), (xi**(1 - s) - 1)/(1 - s))", "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/limits-of-integration", "concept/rational-number", "concept/series-of-n-s", "theorem/integral-test" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ae97425246", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "319", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\Phi(\\xi) \\to \\frac{1}{(s - 1)} = l", "name": null, "statement": "For s greater than 1, the integral tends to the limit 1/(s-1), so the series of n^(-s) converges with sum at most s/(s-1).", "kind": "result", "symbols": [ { "unit": null, "symbol": "Phi", "meaning": "integral of x^(-s) from 1 to xi" }, { "unit": null, "symbol": "s", "meaning": "rational number greater than 1" }, { "unit": null, "symbol": "l", "meaning": "the limit 1/(s-1)" } ], "sympy": "Eq(Limit(Phi(xi), xi, oo), 1/(s - 1))", "physics": false, "states": [], "concepts": [ "concept/convergent-series", "concept/limit", "concept/rational-number", "concept/tends-to-infinity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-567e7850a0", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "319", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\Phi(\\xi) = \\int_{1}^{\\xi} \\frac{dx}{x}", "name": null, "statement": "For s equal to 1, Phi(xi) is the integral of 1/x from 1 to xi, which tends to infinity, so the harmonic series diverges.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "Phi", "meaning": "integral of 1/x from 1 to xi" }, { "unit": null, "symbol": "xi", "meaning": "upper limit of integration" }, { "unit": null, "symbol": "x", "meaning": "integration variable" } ], "sympy": "Eq(Phi(xi), Integral(1/x, (x, 1, xi)))", "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/divergent-series", "concept/tends-to-infinity", "theorem/integral-test" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-03dd00067e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "320", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\ds\\Phi(\\xi) > n\\int_{1}^{2} \\frac{du}{u}", "name": null, "statement": "For xi greater than 2^n, the integral of 1/x from 1 to xi exceeds n times the integral of 1/u from 1 to 2, which shows it tends to infinity.", "kind": "result", "symbols": [ { "unit": null, "symbol": "Phi", "meaning": "integral of 1/x from 1 to xi" }, { "unit": null, "symbol": "n", "meaning": "positive integer" }, { "unit": null, "symbol": "u", "meaning": "substitution variable (x = 2^r u)" } ], "sympy": "Gt(Phi(xi), n*Integral(1/u, (u, 1, 2)))", "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/inequality", "concept/integer", "concept/tends-to-infinity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-fe79c967e6", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "320", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\int_{2^{r}}^{2^{r+1}} \\frac{dx}{x} = \\int_{1}^{2} \\frac{du}{u}", "name": null, "statement": "Substituting x = 2^r u shows that the integral of 1/x over each interval from 2^r to 2^(r+1) equals the same value as over 1 to 2.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "r", "meaning": "non-negative integer index of the dyadic interval" }, { "unit": null, "symbol": "x", "meaning": "integration variable" }, { "unit": null, "symbol": "u", "meaning": "substitution variable with x = 2^r u" } ], "sympy": "Eq(Integral(1/x, (x, 2**r, 2**(r + 1))), Integral(1/u, (u, 1, 2)))", "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/integer", "method/substitution" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c1f6796b61", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "321", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\lim_{x \\to \\infty} \\int_{1}^{x} \\phi(t)\\, dt = l", "name": null, "statement": "If the integral of a positive decreasing function from 1 up to x approaches a limit l as x grows without bound, the infinite integral is convergent with value l.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "function of t (positive and decreasing in the special case considered)" }, { "unit": null, "symbol": "l", "meaning": "the limit, and value of the infinite integral" }, { "unit": null, "symbol": "x", "meaning": "upper limit of integration, tending to infinity" }, { "unit": null, "symbol": "t", "meaning": "variable of integration" } ], "sympy": "Eq(Limit(Integral(phi(t), (t, 1, x)), x, oo), l)", "physics": false, "states": [], "concepts": [ "concept/convergent-series", "concept/definite-integral", "concept/function", "concept/infinite-integral", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-8a2990b4d2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "321", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\lim_{x \\to \\infty} \\int_{a}^{x} \\phi(t)\\, dt = l", "name": null, "statement": "The infinite integral from a to infinity of phi is defined as convergent with value l when the integral from a to x tends to l as x tends to infinity.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "function of t, continuous when t >= a" }, { "unit": null, "symbol": "a", "meaning": "lower limit of integration" }, { "unit": null, "symbol": "l", "meaning": "limit, and value of the infinite integral" }, { "unit": null, "symbol": "x", "meaning": "upper limit of integration, tending to infinity" } ], "sympy": "Eq(Limit(Integral(phi(t), (t, a, x)), x, oo), l)", "physics": false, "states": [], "concepts": [ "concept/continuous-function", "concept/convergent-series", "concept/definite-integral", "concept/infinite-integral", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-2d1e0b9082", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "322", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\int_{a}^{x}\\phi(t)\\, dt = \\Phi(x)", "name": null, "statement": "Writing Phi(x) for the integral of phi from a to x defines the integral function, and the infinite integral converges, diverges or oscillates according as Phi(x) tends to a limit, to infinity, or oscillates.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "Phi", "meaning": "integral function of phi, with upper limit x" }, { "unit": null, "symbol": "phi", "meaning": "function of t being integrated" }, { "unit": null, "symbol": "a", "meaning": "lower limit of integration" }, { "unit": null, "symbol": "x", "meaning": "upper limit of integration" } ], "sympy": "Eq(Integral(phi(t), (t, a, x)), Phi(x))", "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/function", "concept/infinite-integral", "concept/infinity", "concept/limit", "concept/oscillation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-f0fbadf928", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "323", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\int_{a}^{\\infty} \\phi(x)\\, dx = \\int_{a}^{b} \\phi(x)\\, dx + \\int_{b}^{\\infty}\\phi(x)\\, dx", "name": null, "statement": "If the infinite integral from a converges and b > a, the infinite integral splits into a finite integral from a to b plus the infinite integral from b.", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "function of x" }, { "unit": null, "symbol": "a", "meaning": "lower limit of the infinite integral" }, { "unit": null, "symbol": "b", "meaning": "intermediate point, b > a" } ], "sympy": "Eq(Integral(phi(x), (x, a, oo)), Integral(phi(x), (x, a, b)) + Integral(phi(x), (x, b, oo)))", "physics": false, "states": [], "concepts": [ "concept/convergent-series", "concept/definite-integral", "concept/function", "concept/infinite-integral" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-3735dd186f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "323", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\int_{a}^{x} \\phi(t)\\, dt < K", "name": null, "statement": "The infinite integral from a converges exactly when the integral from a to x stays below some fixed constant K for all x greater than a.", "kind": "result", "symbols": [ { "unit": null, "symbol": "K", "meaning": "a fixed constant bounding the integral" }, { "unit": null, "symbol": "phi", "meaning": "positive function of t" }, { "unit": null, "symbol": "a", "meaning": "lower limit of integration" }, { "unit": null, "symbol": "x", "meaning": "upper limit, greater than a" } ], "sympy": "Lt(Integral(phi(t), (t, a, x)), K)", "physics": false, "states": [], "concepts": [ "concept/constant", "concept/convergent-series", "concept/definite-integral", "concept/inequality", "concept/infinite-integral" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-cc20ee237d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "323", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\int_{a}^{\\infty} \\psi(x)\\, dx \\leq K\\int_{a}^{\\infty} \\phi(x)\\, dx", "name": null, "statement": "If psi(x) is at most K times phi(x) beyond a and the integral of phi converges, then the integral of psi converges and is at most K times the integral of phi.", "kind": "result", "symbols": [ { "unit": null, "symbol": "psi", "meaning": "function dominated by phi" }, { "unit": null, "symbol": "phi", "meaning": "convergent comparison function" }, { "unit": null, "symbol": "K", "meaning": "constant multiplier in the comparison" }, { "unit": null, "symbol": "a", "meaning": "lower limit of the infinite integrals" } ], "sympy": "Le(Integral(psi(x), (x, a, oo)), K*Integral(phi(x), (x, a, oo)))", "physics": false, "states": [], "concepts": [ "concept/constant", "concept/convergent-series", "concept/function", "concept/inequality", "concept/infinite-integral" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-460e8debee", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "324", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\lim x^{s}\\phi(x) = l", "name": null, "statement": "If x^s times phi(x) tends to a positive limit l, the infinite integral of phi is convergent when s > 1 and divergent when s <= 1.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "s", "meaning": "exponent of the comparison power x^(-s)" }, { "unit": null, "symbol": "phi", "meaning": "positive function of x" }, { "unit": null, "symbol": "l", "meaning": "positive limit" } ], "sympy": "Eq(Limit(x**s*phi(x), x, oo), l)", "physics": false, "states": [], "concepts": [ "concept/convergent-series", "concept/divergent-series", "concept/function", "concept/infinite-integral", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c0d5acee59", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "324", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\int_{0}^{\\xi} \\phi(x)\\, dx < \\sum_{0}^{\\infty} \\frac{1}{(n + 1)^{2}}", "name": null, "statement": "For the constructed function of the remark, the integral up to any xi is bounded by the convergent sum of 1/(n+1)^2, so the infinite integral converges.", "kind": "result", "symbols": [ { "unit": null, "symbol": "xi", "meaning": "any upper limit of integration" }, { "unit": null, "symbol": "n", "meaning": "summation index (positive integer)" }, { "unit": null, "symbol": "phi", "meaning": "the positive peaked function of the remark" } ], "sympy": "Lt(Integral(phi(x), (x, 0, xi)), Sum(1/(n + 1)**2, (n, 0, oo)))", "physics": false, "states": [], "concepts": [ "concept/convergent-series", "concept/inequality", "concept/infinite-integral", "concept/infinite-sequence", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b52333897a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "326", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\int_{a}^{\\xi} \\phi(x)\\, dx = \\int_{b}^{\\tau} \\phi\\{f(t)\\}f'(t)\\, dt", "name": null, "statement": "Under the substitution x = f(t), with a = f(b) and xi = f(tau), the integral over x equals the integral over t of phi{f(t)} f'(t).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "f", "meaning": "function giving the substitution x = f(t)" }, { "unit": null, "symbol": "phi", "meaning": "function of x being integrated" }, { "unit": null, "symbol": "a", "meaning": "lower limit in x, with a = f(b)" }, { "unit": null, "symbol": "b", "meaning": "lower limit in t" }, { "unit": null, "symbol": "xi", "meaning": "upper limit in x, with xi = f(tau)" }, { "unit": null, "symbol": "tau", "meaning": "upper limit in t" } ], "sympy": "Eq(Integral(phi(x), (x, a, xi)), Integral(phi(f(t))*Derivative(f(t), t), (t, b, tau)))", "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/derivative", "concept/function", "method/substitution" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-60dc235b0d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "331", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\int_{a}^{\\infty} \\phi(x)\\, dx = \\int_{b}^{c} \\phi\\{f(t)\\}f'(t)\\, dt", "name": null, "statement": "The infinite integral in x equals the integral in t from b to c, which is an infinite integral of the second kind when the integrand is infinite at t = c.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "b", "meaning": "lower limit in t" }, { "unit": null, "symbol": "c", "meaning": "upper limit in t, image of infinity under the substitution" }, { "unit": null, "symbol": "f", "meaning": "substitution function x = f(t)" }, { "unit": null, "symbol": "phi", "meaning": "function of x" } ], "sympy": "Eq(Integral(phi(x), (x, a, oo)), Integral(phi(f(t))*Derivative(f(t), t), (t, b, c)))", "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/derivative", "concept/function", "concept/infinite-integral", "method/substitution" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-9cde6cbf00", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "328", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\int_{a}^{\\xi} f(x)\\phi'(x)\\, dx = f(\\xi)\\phi(\\xi) - f(a)\\phi(a) - \\int_{a}^{\\xi} f'(x)\\phi(x)\\, dx", "name": null, "statement": "Integration by parts: the integral of f times the derivative of phi equals the boundary terms f phi evaluated at xi and a, minus the integral of f' times phi.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "f", "meaning": "first function in the product" }, { "unit": null, "symbol": "phi", "meaning": "second function, whose derivative phi' appears" }, { "unit": null, "symbol": "a", "meaning": "lower limit" }, { "unit": null, "symbol": "xi", "meaning": "upper limit" } ], "sympy": "Eq(Integral(f(x)*Derivative(phi(x), x), (x, a, xi)), f(xi)*phi(xi) - f(a)*phi(a) - Integral(Derivative(f(x), x)*phi(x), (x, a, xi)))", "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/derivative", "concept/function", "method/integration-by-parts" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-55c47f213c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "330", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\int_{a}^{A} (x - a)^{-s}\\, dx = \\lim_{\\epsilon\\to +0} \\int_{a+\\epsilon}^{A} (x - a)^{-s}\\, dx", "name": null, "statement": "An integral whose integrand tends to infinity at the lower limit a is defined as the limit, as epsilon tends to zero from above, of the integral starting at a + epsilon.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "s", "meaning": "positive exponent of the singularity" }, { "unit": null, "symbol": "epsilon", "meaning": "small positive quantity tending to zero" }, { "unit": null, "symbol": "a", "meaning": "lower limit, where the integrand is infinite" }, { "unit": null, "symbol": "A", "meaning": "upper limit of integration" } ], "sympy": "Eq(Integral((x - a)**(-s), (x, a, A)), Limit(Integral((x - a)**(-s), (x, a + epsilon, A)), epsilon, 0))", "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/infinite-integral-of-the-second-kind", "concept/infinity", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-fef21e88d9", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "330", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\int_{1/A}^{\\eta} y^{s-2}\\, dy = \\int_{1/\\eta}^{A} x^{-s}\\, dx", "name": null, "statement": "Under the substitution y = 1/x, the integral of y to the power s-2 from 1/A to eta equals the integral of x to the power -s from 1/eta to A.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y", "meaning": "substituted variable, y = 1/x" }, { "unit": null, "symbol": "x", "meaning": "original variable of integration" }, { "unit": null, "symbol": "s", "meaning": "exponent" }, { "unit": null, "symbol": "eta", "meaning": "upper limit tending to infinity" }, { "unit": null, "symbol": "A", "meaning": "upper limit in x" } ], "sympy": "Eq(Integral(y**(s - 2), (y, 1/A, eta)), Integral(x**(-s), (x, 1/eta, A)))", "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/exponent", "concept/power", "method/substitution" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-357ca4d096", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "326", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\int_{a}^{\\infty} \\phi(x)\\, dx = \\lim_{\\tau\\to c} \\int_{b}^{\\tau} \\phi\\{f(t)\\}f'(t)\\, dt", "name": null, "statement": "Equation (4) of the substitution rule: the infinite integral equals the limit of the transformed integral as tau approaches c.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "tau", "meaning": "upper limit in t, tending to c" }, { "unit": null, "symbol": "c", "meaning": "limit value of t" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/function", "concept/infinite-integral", "concept/limit", "method/substitution" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-54a817f896", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "334", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "J = \\int_{1}^{7} (x^{2} - 6x + 13)\\, dx", "name": null, "statement": "J is defined as the definite integral of x squared minus six x plus thirteen from 1 to 7.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "J", "meaning": "the definite integral of x^2 - 6x + 13 between x = 1 and x = 7" }, { "unit": null, "symbol": "x", "meaning": "the variable of integration" } ], "sympy": "Eq(J, Integral(x**2 - 6*x + 13, (x, 1, 7)))", "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a4d67b61ac", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "334", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "J = 48", "name": null, "statement": "Direct integration gives the value of the integral J as 48.", "kind": "result", "symbols": [ { "unit": null, "symbol": "J", "meaning": "the definite integral of x^2 - 6x + 13 from 1 to 7" } ], "sympy": "Eq(J, 48)", "physics": false, "states": [], "concepts": [ "concept/definite-integral" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-98ddd3ffc5", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "334", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "y = x^{2} - 6x + 13", "name": null, "statement": "The substitution sets y equal to x squared minus six x plus thirteen.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "y", "meaning": "the new variable given by the substitution" }, { "unit": null, "symbol": "x", "meaning": "the original variable" } ], "sympy": "Eq(y, x**2 - 6*x + 13)", "physics": false, "states": [], "concepts": [ "concept/function", "concept/variable", "method/substitution" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a6868aa802", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "334", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "x = 3 ± \\sqrtp{y - 4}", "name": null, "statement": "Solving the substitution for x gives two branches, x equal to 3 plus or minus the square root of y minus 4.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the original variable" }, { "unit": null, "symbol": "y", "meaning": "the substituted variable" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/function", "concept/inverse-circular-function", "concept/root" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-3a1c3fab01", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "335", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "J = \\int_{1}^{7} y\\, dx = \\int_{8}^{4} \\left\\{-\\frac{y}{2\\sqrtp{y - 4}}\\right\\} dy + \\int_{4}^{20} \\frac{y}{2\\sqrtp{y - 4}}\\, dy", "name": null, "statement": "The correct value of J is the sum of two integrals in y, one on each branch of the substitution, with the sign of dx/dy chosen on each branch.", "kind": "result", "symbols": [ { "unit": null, "symbol": "J", "meaning": "the definite integral of x^2 - 6x + 13 from 1 to 7" }, { "unit": null, "symbol": "y", "meaning": "the substituted variable y = x^2 - 6x + 13" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/zodiacal-sign", "method/substitution" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-76c13f95dc", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "335", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\int_{0}^{\\pi} dx = \\pi", "name": null, "statement": "The integral of 1 from 0 to pi equals pi.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the variable of integration" } ], "sympy": "Eq(Integral(1, (x, 0, pi)), pi)", "physics": false, "states": [], "concepts": [ "concept/definite-integral", "quantity/pi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-9f90ceb50b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "335", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "x = \\arcsin y", "name": null, "statement": "The substitution x equals the inverse sine of y.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the original variable" }, { "unit": null, "symbol": "y", "meaning": "the substituted variable" } ], "sympy": "Eq(x, asin(y))", "physics": false, "states": [], "concepts": [ "concept/inverse-circular-function", "concept/sine", "method/substitution" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-bdcc88f13c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "335", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "dx/dy = 1/\\sqrtp{1 - y^{2}}", "name": null, "statement": "When x is between 0 and a half pi, dx/dy equals one over the square root of one minus y squared.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the original variable, 0 <= x < pi/2" }, { "unit": null, "symbol": "y", "meaning": "the substituted variable" } ], "sympy": "Eq(Derivative(x, y), 1/sqrt(1 - y**2))", "physics": false, "states": [], "concepts": [ "concept/inverse-circular-function", "concept/rate-of-change", "concept/zodiacal-sign" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-99f13dd4e4", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "335", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "dx/dy = -1/\\sqrtp{1 - y^{2}}", "name": null, "statement": "When x is between a half pi and pi, dx/dy equals minus one over the square root of one minus y squared.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the original variable, pi/2 < x <= pi" }, { "unit": null, "symbol": "y", "meaning": "the substituted variable" } ], "sympy": "Eq(Derivative(x, y), -1/sqrt(1 - y**2))", "physics": false, "states": [], "concepts": [ "concept/inverse-circular-function", "concept/rate-of-change", "concept/zodiacal-sign" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-99c0850d64", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "336", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "|u_{n}| = \\alpha_{n}", "name": null, "statement": "alpha_n is defined as the modulus of u_n.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "u_n", "meaning": "the nth term of the series" }, { "unit": null, "symbol": "alpha_n", "meaning": "the modulus of the nth term" } ], "sympy": "Eq(Abs(u_n), alpha_n)", "physics": false, "states": [], "concepts": [ "concept/absolute-convergence", "concept/modulus-of-a-complex-number", "concept/term" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-02be3723c4", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "336", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "u_{n} = v_{n} - w_{n}", "name": null, "statement": "Each term u_n is the positive part v_n minus the negative part w_n.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "u_n", "meaning": "the nth term of the series" }, { "unit": null, "symbol": "v_n", "meaning": "u_n if u_n is positive, otherwise 0" }, { "unit": null, "symbol": "w_n", "meaning": "-u_n if u_n is negative, otherwise 0" } ], "sympy": "Eq(u_n, v_n - w_n)", "physics": false, "states": [], "concepts": [ "concept/absolute-convergence", "concept/series-of-positive-terms", "concept/term" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-5b57b6a962", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "336", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\alpha_{n} = v_{n} + w_{n}", "name": null, "statement": "The modulus alpha_n is the sum of the positive part and the negative part of u_n.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "alpha_n", "meaning": "the modulus of the nth term" }, { "unit": null, "symbol": "v_n", "meaning": "positive part of u_n" }, { "unit": null, "symbol": "w_n", "meaning": "negative part of u_n, taken positive" } ], "sympy": "Eq(alpha_n, v_n + w_n)", "physics": false, "states": [], "concepts": [ "concept/absolute-convergence", "concept/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-0ab8704e5e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "338", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\tsum u'_{n} = \\tsum v'_{n} - \\tsum w'_{n} = \\tsum v_{n} - \\tsum w_{n} = \\tsum u_{n}", "name": null, "statement": "A rearrangement of an absolutely convergent series has the same sum as the original series.", "kind": "result", "symbols": [ { "unit": null, "symbol": "u'_n", "meaning": "the rearranged terms" }, { "unit": null, "symbol": "u_n", "meaning": "the original terms" }, { "unit": null, "symbol": "v_n", "meaning": "positive part of u_n" }, { "unit": null, "symbol": "w_n", "meaning": "negative part of u_n taken positive" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/absolute-convergence", "concept/rearrangement-of-a-series", "concept/sum", "theorem/dirichlet-s-rearrangement-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-4f3e49adf5", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "338", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\sum_{0}^{N} u_{n} = \\sum_{0}^{N} v_{n} - \\sum_{0}^{N} w_{n}", "name": null, "statement": "The partial sum of u_n up to N equals the partial sum of v_n minus the partial sum of w_n.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "N", "meaning": "the upper limit of the partial sum" }, { "unit": null, "symbol": "u_n", "meaning": "the nth term of the series" }, { "unit": null, "symbol": "v_n", "meaning": "positive part of u_n" }, { "unit": null, "symbol": "w_n", "meaning": "negative part of u_n taken positive" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/conditionally-convergent-series", "concept/finite-set", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ceb2c24679", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "339", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "|v_{0}| + |v_{1}| + \\dots + |v_{n}| < |u_{0}| + \\dots + |u_{n}|", "name": null, "statement": "The sum of the moduli of the v terms is less than the sum of the moduli of the u terms.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "u_n", "meaning": "the nth term of the series" }, { "unit": null, "symbol": "v_n", "meaning": "positive part of u_n" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/absolute-convergence", "concept/conditionally-convergent-series", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-cfbb063153", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "340", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "s_{2n+1} - s_{2n-1} = \\phi_{2n} - \\phi_{2n+1}\\geq 0", "name": null, "statement": "The odd partial sums of an alternating series with steadily decreasing terms increase at each step.", "kind": "result", "symbols": [ { "unit": null, "symbol": "s_n", "meaning": "the nth partial sum phi_0 - phi_1 + ... + (-1)^n phi_n" }, { "unit": null, "symbol": "phi_n", "meaning": "the positive function of n tending steadily to zero" } ], "sympy": "Eq(s(2*n+1) - s(2*n-1), phi(2*n) - phi(2*n+1))", "physics": false, "states": [], "concepts": [ "concept/alternating-series", "concept/infinite-sequence", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d7dec9985a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "340", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "s_{2n} - s_{2n-2} = -(\\phi_{2n-1} - \\phi_{2n}) \\leq 0", "name": null, "statement": "The even partial sums of an alternating series with steadily decreasing terms decrease at each step.", "kind": "result", "symbols": [ { "unit": null, "symbol": "s_n", "meaning": "the nth partial sum of the alternating series" }, { "unit": null, "symbol": "phi_n", "meaning": "the positive function of n tending steadily to zero" } ], "sympy": "Eq(s(2*n) - s(2*n-2), -(phi(2*n-1) - phi(2*n)))", "physics": false, "states": [], "concepts": [ "concept/alternating-series", "concept/infinite-sequence" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-425f5305f1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "340", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "s_{n} = \\phi_{0} - \\phi_{1} + \\phi_{2} - \\dots + (-1)^{n}\\phi_{n}", "name": null, "statement": "s_n is defined as the nth partial sum of the alternating series of phi values.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "s_n", "meaning": "the nth partial sum of the alternating series" }, { "unit": null, "symbol": "phi_n", "meaning": "phi(n), a positive function of n tending steadily to zero" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/alternating-series", "concept/function-of-a-positive-integer-variable", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-8da8affd87", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "340", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\lim (s_{2n+1} - s_{2n}) = \\lim (-1)^{2n+1} \\phi_{2n+1} = 0", "name": null, "statement": "The difference between successive partial sums of an alternating series tends to zero.", "kind": "result", "symbols": [ { "unit": null, "symbol": "s_n", "meaning": "the nth partial sum of the alternating series" }, { "unit": null, "symbol": "phi_n", "meaning": "the positive function of n tending steadily to zero" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/alternating-series", "concept/convergent-series", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-0ea90775f0", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "340", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "s_{1} = \\phi_{0} - \\phi_{1}", "name": null, "statement": "The first partial sum of the alternating series is phi_0 minus phi_1.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "s_1", "meaning": "the first partial sum of the alternating series" }, { "unit": null, "symbol": "phi_0", "meaning": "phi(0)" }, { "unit": null, "symbol": "phi_1", "meaning": "phi(1)" } ], "sympy": "Eq(s1, phi0 - phi1)", "physics": false, "states": [], "concepts": [ "concept/alternating-series", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-dd4ebb813b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "342", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "a_{0}\\phi_{0} + a_{1}\\phi_{1} + \\dots + a_{n}\\phi_{n} = s_{0}(\\phi_{0} - \\phi_{1}) + s_{1}(\\phi_{1} - \\phi_{2}) + \\dots + s_{n-1}(\\phi_{n-1} - \\phi_{n}) + s_{n}\\phi_{n}", "name": "Abel's summation identity", "statement": "The partial sum of the products a_k phi_k can be rewritten using partial sums s_k of the a's and differences of the phi's.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a_n", "meaning": "the nth term of the series sum a_n" }, { "unit": null, "symbol": "phi_n", "meaning": "the positive function of n in Dirichlet's test" }, { "unit": null, "symbol": "s_n", "meaning": "a_0 + a_1 + ... + a_n" } ], "sympy": null, "physics": false, "states": [ "theorem/abel-s-summation-identity" ], "concepts": [ "concept/convergent-series", "concept/sum", "concept/term", "theorem/dirichlet-s-test" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-82b8dce2d0", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "342", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "s_{n} = a_{0} + a_{1} + \\dots + a_{n}", "name": null, "statement": "s_n is the nth partial sum of the series of a's.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "s_n", "meaning": "the nth partial sum of sum a_n" }, { "unit": null, "symbol": "a_n", "meaning": "the nth term of the series" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/infinite-sequence", "concept/power-series", "concept/sum" ], "pages": [ "342", "349" ], "chapters": [ "hardy-course-of-pure-mathematics-1921/ch-viii" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-35b8ac461a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "342", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "|s_{\\nu}| < K", "name": null, "statement": "The partial sums s_nu of a series that converges or oscillates finitely are bounded by a constant K.", "kind": "result", "symbols": [ { "unit": null, "symbol": "s_nu", "meaning": "partial sum of the a's" }, { "unit": null, "symbol": "K", "meaning": "a bounding constant" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/infinite-sequence", "concept/oscillation", "theorem/dirichlet-s-test" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b92057290c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "343", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "s_{m, \\nu} = a_{m} + a_{m+1} + \\dots + a_{\\nu}", "name": null, "statement": "s_{m,nu} is the partial sum of the a's from index m to index nu.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "s_{m, nu}", "meaning": "the sum of a_m through a_nu" }, { "unit": null, "symbol": "a_n", "meaning": "the nth term of the series" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/finite-set", "concept/sum", "theorem/dirichlet-s-test" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-523398b605", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "343", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "|a_{m}\\phi_{m} + a_{m+1}\\phi_{m+1} + \\dots + a_{n}\\phi_{n}| < \\DELTA \\phi_{m} \\leq \\DELTA \\phi_{1}", "name": null, "statement": "The modulus of a block of terms a_k phi_k is less than a positive number Delta times phi_1, when m is large enough.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\DELTA", "meaning": "an arbitrary positive number" }, { "unit": null, "symbol": "phi_n", "meaning": "the positive decreasing function of n" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/limit", "theorem/abel-s-test", "theorem/general-principle-of-convergence" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a6443df9f5", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "343", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "z = \\Cis\\theta", "name": null, "statement": "z is defined as cos theta plus i sin theta, so that the modulus of z is 1.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "z", "meaning": "a complex number on the unit circle" }, { "unit": "radian", "symbol": "theta", "meaning": "an angle" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/complex-variable", "concept/modulus-of-a-complex-number", "concept/plane-angle" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-1c0930c267", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "343", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "|s_{n} + it_{n}| = \\left|\\frac{1 - z^{n}}{1 - z}\\right| \\leq \\frac{1 + |z^{n}|}{|1 - z|} \\leq \\frac{2}{|1 - z|}", "name": null, "statement": "The partial sums of cos n theta and sin n theta are bounded by 2 over the modulus of 1 minus z, when z is not 1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "s_n", "meaning": "partial sum of the cosine series" }, { "unit": null, "symbol": "t_n", "meaning": "partial sum of the sine series" }, { "unit": null, "symbol": "z", "meaning": "Cis theta, with |z| = 1 and z not equal to 1" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/cosine", "concept/modulus-of-a-complex-number", "concept/oscillation", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-25f7810ef4", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "344", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\tsum u_{n} = \\tsum (v_{n} + iw_{n})", "name": null, "statement": "A complex series is written as the sum of its real parts v_n plus i times its imaginary parts w_n.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "u_n", "meaning": "the nth term of a complex series" }, { "unit": null, "symbol": "v_n", "meaning": "real part of u_n" }, { "unit": null, "symbol": "w_n", "meaning": "imaginary part of u_n" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/series-of-complex-terms", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-583d0c481d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "344", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "u_{n} = v_{n} + iw_{n}", "name": null, "statement": "The nth term of a complex series is its real part plus i times its imaginary part.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "u_n", "meaning": "the nth term of a complex series" }, { "unit": null, "symbol": "v_n", "meaning": "real part of u_n" }, { "unit": null, "symbol": "w_n", "meaning": "imaginary part of u_n" } ], "sympy": "Eq(u_n, v_n + I*w_n)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/term" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a56ee4ff4f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "344", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "|u_{n}| = \\sqrtp{v_{n}^{2} + w_{n}^{2}}", "name": null, "statement": "The modulus of a complex term is the square root of the sum of the squares of its real and imaginary parts.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "u_n", "meaning": "the nth complex term" }, { "unit": null, "symbol": "v_n", "meaning": "real part" }, { "unit": null, "symbol": "w_n", "meaning": "imaginary part" } ], "sympy": "Eq(Abs(u_n), sqrt(v_n**2 + w_n**2))", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a2093ee5ce", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "344", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "|u_{n}| = \\sqrtp{v_{n}^{2} + w_{n}^{2}} \\leq |v_{n}| + |w_{n}|", "name": null, "statement": "The modulus of a complex term is at most the sum of the moduli of its real and imaginary parts.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "u_n", "meaning": "the nth complex term" }, { "unit": null, "symbol": "v_n", "meaning": "real part" }, { "unit": null, "symbol": "w_n", "meaning": "imaginary part" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/absolute-convergence", "concept/complex-number", "concept/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d6d17a64c8", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "345", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "|v_{n}| \\leq \\sqrtp{v_{n}^{2} + w_{n}^{2}}", "name": null, "statement": "The modulus of the real part is at most the modulus of the complex term.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "v_n", "meaning": "real part of u_n" }, { "unit": null, "symbol": "w_n", "meaning": "imaginary part of u_n" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c06eef33f4", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "345", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "|u_{n+1}|/|u_{n}| = |z|/(n + 1) \\to 0", "name": null, "statement": "For the series of z^n over n factorial, the ratio of successive term moduli tends to zero for every z, giving convergence for all z.", "kind": "result", "symbols": [ { "unit": null, "symbol": "u_n", "meaning": "z^n / n!" }, { "unit": null, "symbol": "z", "meaning": "a complex variable" } ], "sympy": "Eq(Abs(u(n+1))/Abs(u(n)), Abs(z)/(n+1))", "physics": false, "states": [], "concepts": [ "concept/complex-variable", "concept/limit", "concept/power-series", "concept/tends-to-infinity", "theorem/d-alembert-s-ratio-test" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-3f2fc7bc64", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "345", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "|u_{n+1}|/|u_{n}| = (n + 1)|z|", "name": null, "statement": "For the series of n! z^n, the ratio of successive term moduli grows without limit unless z is zero.", "kind": "result", "symbols": [ { "unit": null, "symbol": "u_n", "meaning": "n! z^n" }, { "unit": null, "symbol": "z", "meaning": "a complex variable" } ], "sympy": "Eq(Abs(u(n+1))/Abs(u(n)), (n+1)*Abs(z))", "physics": false, "states": [], "concepts": [ "concept/power-series", "concept/tends-to-infinity", "theorem/d-alembert-s-ratio-test" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ae58b7fbd0", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "341", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\lim \\left[\\frac{1}{2n + 1} - \\frac{1}{2n + 2} + \\frac{1}{2n + 3} - \\dots + \\frac{1}{4n - 1} - \\frac{1}{4n}\\right] = 0", "name": null, "statement": "The alternating bracket of reciprocals tends to zero as n tends to infinity.", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "the index tending to infinity" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/alternating-series", "concept/limit", "concept/tends-to-infinity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d9f7f836c4", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "341", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\lim \\left(\\frac{1}{2n + 2} + \\frac{1}{2n + 4} + \\dots + \\frac{1}{4n}\\right) = \\tfrac{1}{2} \\lim \\frac{1}{n} \\sum_{r=1}^{n} \\frac{1}{1 + (r/n)} = \\tfrac{1}{2} \\int_{1}^{2} \\frac{dx}{x}", "name": null, "statement": "The limit of the sum of reciprocals from 2n+2 to 4n equals one half the integral of 1/x from 1 to 2.", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "the index tending to infinity" }, { "unit": null, "symbol": "r", "meaning": "summation index from 1 to n" }, { "unit": null, "symbol": "x", "meaning": "variable of integration" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/limit", "concept/logarithm", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-2ab1b94b1a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "341", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\lim t_{3n} = s + \\tfrac{1}{2} \\int_{1}^{2} \\frac{dx}{x}", "name": null, "statement": "The sum of the rearranged series is s plus one half the integral of 1/x from 1 to 2, not s.", "kind": "result", "symbols": [ { "unit": null, "symbol": "s", "meaning": "sum of 1 - 1/2 + 1/3 - 1/4 + ..." }, { "unit": null, "symbol": "t_{3n}", "meaning": "sum of the first 3n terms of the rearranged series" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/conditionally-convergent-series", "concept/definite-integral", "concept/rearrangement-of-a-series", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-6fc81f793d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "346", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\lim a_{n}z_{1}^{n} = 0", "name": null, "statement": "If the power series converges at z_1, its terms a_n z_1^n tend to zero.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a_n", "meaning": "coefficient of z^n in the power series" }, { "unit": null, "symbol": "z_1", "meaning": "a value of z at which the series converges" }, { "unit": null, "symbol": "n", "meaning": "index of the term" } ], "sympy": "Eq(Limit(a_n*z_1**n, n, oo), 0)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/convergent-series", "concept/limit", "concept/power-series", "concept/term" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-233bd69fb7", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "346", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "|a_{n}z^{n}| = |a_{n}z_{1}^{n}| \\left(\\frac{r}{r_{1}}\\right)^{n} < K \\left(\\frac{r}{r_{1}}\\right)^{n}", "name": null, "statement": "For |z| = r < r_1 the terms of the power series are bounded by a geometric term, so comparison with a convergent geometrical series gives absolute convergence.", "kind": "result", "symbols": [ { "unit": null, "symbol": "z", "meaning": "complex variable of the power series" }, { "unit": null, "symbol": "r", "meaning": "modulus |z|" }, { "unit": null, "symbol": "r_1", "meaning": "modulus |z_1| of the point where the series converges" }, { "unit": null, "symbol": "K", "meaning": "constant bounding |a_n z_1^n|" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/absolute-convergence", "concept/comparison", "concept/geometrical-progression", "concept/power-series", "quantity/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-12b1ed631e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "348", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "z = \\cos\\theta + i\\sin\\theta", "name": null, "statement": "A complex number of modulus one written in terms of its angle θ, cos θ + i sin θ.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "z", "meaning": "complex number on the circle of convergence" }, { "unit": null, "symbol": "theta", "meaning": "angle" }, { "unit": null, "symbol": "i", "meaning": "imaginary unit" } ], "sympy": "Eq(z, cos(theta) + I*sin(theta))", "physics": false, "states": [], "concepts": [ "concept/circle-of-convergence", "concept/complex-number", "concept/cosine", "concept/plane-angle", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-aa5bff62cf", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "348", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\lim |a_{n+1}z^{n+1}|/|a_{n}z^{n}| = \\lambda |z|", "name": null, "statement": "The ratio of successive absolute terms of a power series tends to λ|z|, so by the ratio test the series converges when |z| < 1/λ.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a_n", "meaning": "coefficient of z^n" }, { "unit": null, "symbol": "lambda", "meaning": "limit of |a_{n+1}|/|a_n|" }, { "unit": null, "symbol": "z", "meaning": "complex variable" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/limit", "concept/power-series", "concept/radius-of-convergence", "quantity/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-5b25e7fbcd", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "348", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\frac{|a_{n+1}|}{|a_{n}|} = \\frac{|m - n|}{n + 1} \\to 1", "name": null, "statement": "For the binomial series the ratio of successive absolute coefficients tends to 1, so its radius of convergence is unity.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a_n", "meaning": "n-th coefficient of the binomial series" }, { "unit": null, "symbol": "m", "meaning": "the exponent of the binomial series" }, { "unit": null, "symbol": "n", "meaning": "index" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/binomial-series", "concept/limit", "concept/radius-of-convergence" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c6213349b7", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "349", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "1/(1 - z)^{2} = 1 + 2z + 3z^{2} + \\dots", "name": null, "statement": "For |z| < 1 the square of the reciprocal of 1 - z expands as a power series with coefficients 1, 2, 3, ...", "kind": "result", "symbols": [ { "unit": null, "symbol": "z", "meaning": "complex variable with |z| < 1" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/expansion", "concept/power", "concept/power-series", "concept/rational-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b4aa3963e4", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "349", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\frac{1}{(1 - z)^{m}} = 1 + mz + \\frac{m(m + 1)}{1·2} z^{2} + \\dots", "name": "binomial theorem for a negative integral exponent", "statement": "For |z| < 1 and positive integer m, the reciprocal power 1/(1 - z)^m expands in an infinite power series.", "kind": "result", "symbols": [ { "unit": null, "symbol": "m", "meaning": "positive integer exponent" }, { "unit": null, "symbol": "z", "meaning": "complex variable with |z| < 1" } ], "sympy": null, "physics": false, "states": [ "theorem/binomial-theorem-for-a-negative-integral-exponent" ], "concepts": [ "concept/binomial-series", "concept/expansion", "concept/integer", "concept/power-series", "theorem/binomial-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-2cb05e003f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "349", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "f(m, z) = 1 + \\binom{m}{1} z + \\binom{m}{2} z^{2} + \\dots", "name": null, "statement": "Defines the binomial series f(m, z) as a power series in z with binomial coefficients.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "f(m, z)", "meaning": "binomial series as a function of m and z" }, { "unit": null, "symbol": "m", "meaning": "exponent" }, { "unit": null, "symbol": "z", "meaning": "complex variable" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/binomial", "concept/binomial-series", "concept/function-notation", "concept/power-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-5948eabcd0", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "349", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "f(m, z)f(m', z) = f(m + m', z)", "name": null, "statement": "Multiplying two binomial series with exponents m and m' gives the binomial series with exponent m + m' for |z| < 1; this is the basis of Euler's proof of the binomial theorem.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "f(m, z)", "meaning": "binomial series" }, { "unit": null, "symbol": "m", "meaning": "exponent" }, { "unit": null, "symbol": "m'", "meaning": "second exponent" }, { "unit": null, "symbol": "z", "meaning": "complex variable with |z| < 1" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/binomial-series", "concept/function-notation", "concept/product-of-series", "theorem/binomial-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-21408658da", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "350", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "f(z)f(z') = f(z + z')", "name": null, "statement": "The series f(z) = 1 + z + z^2/2! + ... satisfies the addition law f(z)f(z') = f(z + z') for all z, z'.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "f(z)", "meaning": "series 1 + z + z^2/2! + ..." }, { "unit": null, "symbol": "z", "meaning": "complex variable" }, { "unit": null, "symbol": "z'", "meaning": "second complex variable" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/absolute-convergence", "concept/function-notation", "concept/power-series", "concept/product-of-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-0dda8fc2a1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "350", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "C(z) = 1 - \\frac{z^{2}}{2!} + \\frac{z^{4}}{4!} - \\dots", "name": null, "statement": "Defines C(z) as the power series whose even-power pattern gives the cosine series.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "C(z)", "meaning": "cosine-type power series" }, { "unit": null, "symbol": "z", "meaning": "complex variable" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/function-notation", "concept/power-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-24514bc0e3", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "350", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "S(z) = z - \\frac{z^{3}}{3!} + \\frac{z^{5}}{5!} - \\dots", "name": null, "statement": "Defines S(z) as the power series whose odd-power pattern gives the sine series.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "S(z)", "meaning": "sine-type power series" }, { "unit": null, "symbol": "z", "meaning": "complex variable" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/function-notation", "concept/power-series", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-20a8bed8b7", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "350", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "C(z + z') = C(z)C(z') - S(z)S(z')", "name": null, "statement": "The series C satisfies the addition formula for the cosine.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "C", "meaning": "cosine-type series" }, { "unit": null, "symbol": "S", "meaning": "sine-type series" }, { "unit": null, "symbol": "z, z'", "meaning": "complex variables" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/product-of-series", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-4de0c56ec2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "350", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "S(z + z') = S(z)C(z') + C(z)S(z')", "name": null, "statement": "The series S satisfies the addition formula for the sine.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "S", "meaning": "sine-type series" }, { "unit": null, "symbol": "C", "meaning": "cosine-type series" }, { "unit": null, "symbol": "z, z'", "meaning": "complex variables" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/product-of-series", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-3a4f152e7a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "350", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\{C(z)\\}^{2} + \\{S(z)\\}^{2} = 1", "name": null, "statement": "The squares of the cosine-type and sine-type series sum to one.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "C(z)", "meaning": "cosine-type series" }, { "unit": null, "symbol": "S(z)", "meaning": "sine-type series" } ], "sympy": "Eq(C(z)**2 + S(z)**2, 1)", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/identity", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-57332ed8e5", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "350", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "u_{n} = v_{n} = \\frac{(-1)^{n}}{\\sqrtp{n + 1}}", "name": null, "statement": "The two series considered have terms (-1)^n over the square root of n+1; they are conditionally, not absolutely, convergent.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "u_n", "meaning": "n-th term of the first series" }, { "unit": null, "symbol": "v_n", "meaning": "n-th term of the second series" }, { "unit": null, "symbol": "n", "meaning": "index" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/absolute-convergence", "concept/convergent-series", "concept/root", "concept/term" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-85d8dfda6c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "350", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "w_{n} = (-1)^{n} \\sum_{r=0}^{n} \\frac{1}{\\sqrtb{(r + 1)(n + 1 - r)}}", "name": null, "statement": "The Cauchy product coefficient of the two series of example 9, which does not tend to zero and so the product series fails to converge.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "w_n", "meaning": "n-th coefficient of the product series" }, { "unit": null, "symbol": "r", "meaning": "summation index" }, { "unit": null, "symbol": "n", "meaning": "index" } ], "sympy": "Eq(w_n, (-1)**n*Sum(1/sqrt((r+1)*(n+1-r)), (r, 0, n)))", "physics": false, "states": [], "concepts": [ "concept/infinite-sequence", "concept/product-of-series", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-722c219ea4", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "349", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\sum u_{n} × \\sum v_{n} = \\sum w_{n}", "name": null, "statement": "The product of two absolutely convergent series equals the series of Cauchy-product terms w_n.", "kind": "result", "symbols": [ { "unit": null, "symbol": "u_n", "meaning": "terms of the first series" }, { "unit": null, "symbol": "v_n", "meaning": "terms of the second series" }, { "unit": null, "symbol": "w_n", "meaning": "product coefficients" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/absolute-convergence", "concept/convergent-series", "concept/product-of-series", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-510ee9f380", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "349", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "c_{n} = a_{0}b_{n} + a_{1}b_{n-1} + \\dots + a_{n}b_{0}", "name": null, "statement": "The coefficient of z^n in the product of two power series is this sum of products of coefficients.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "c_n", "meaning": "coefficient of z^n in the product series" }, { "unit": null, "symbol": "a_n, b_n", "meaning": "coefficients of the two power series" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/power-series", "concept/product-of-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-8e3f01e948", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "349", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "f(z)/(1 - z) = \\sum s_{n}z^{n}", "name": null, "statement": "Dividing a convergent power series by 1 - z gives the power series whose coefficients are the partial sums of the a_n.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f(z)", "meaning": "sum of the power series for |z| < R" }, { "unit": null, "symbol": "s_n", "meaning": "partial sum a_0 + ... + a_n" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/function-notation", "concept/power-series", "concept/radius-of-convergence", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-7edee13785", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "350", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\sum_{1}^{\\infty} \\frac{n^{2} + 9n + 5}{(n + 1)(2n + 3)(2n + 5)(n + 4)} = \\frac{5}{36}", "name": null, "statement": "The infinite series of this rational general term has sum 5/36.", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "summation index" } ], "sympy": "Eq(Sum((n**2 + 9*n + 5)/((n + 1)*(2*n + 3)*(2*n + 5)*(n + 4)), (n, 1, oo)), Rational(5, 36))", "physics": false, "states": [], "concepts": [ "concept/infinite-sequence", "concept/rational-function", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d6bd18ee9e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "352", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "u_{n} = \\frac{x^{n} - x^{-n-1}}{(x^{n} + x^{-n})(x^{n+1} + x^{-n-1}) }", "name": null, "statement": "The general term of the series of example 13.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "u_n", "meaning": "n-th term of the series" }, { "unit": null, "symbol": "x", "meaning": "real parameter" }, { "unit": null, "symbol": "n", "meaning": "index" } ], "sympy": "Eq(u_n, (x**n - x**(-n-1))/((x**n + x**(-n))*(x**(n+1) + x**(-n-1))))", "physics": false, "states": [], "concepts": [ "concept/rational-function", "concept/sum", "concept/term" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-820aca6aa2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "352", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\frac{x^{n} - x^{-n-1}}{(x^{n} + x^{-n})(x^{n+1} + x^{-n-1}) } = \\frac{1}{x - 1} \\left(\\frac{1}{x^{n} + x^{-n}} - \\frac{1}{x^{n+1} + x^{-n-1}}\\right)", "name": null, "statement": "The general term splits as a difference of two terms, which makes the series telescope.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "real parameter" }, { "unit": null, "symbol": "n", "meaning": "index" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/rational-function", "concept/sum", "concept/term", "method/partial-fractions" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-108ffc8534", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "352", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "a_{n} + p_{1}a_{n-1} + p_{2}a_{n-2} + \\dots + p_{k}a_{n-k} = 0", "name": null, "statement": "A recurring series has coefficients satisfying this linear relation with constant coefficients p_1,...,p_k.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a_n", "meaning": "coefficient of z^n in the recurring series" }, { "unit": null, "symbol": "p_1, ..., p_k", "meaning": "constants independent of n" }, { "unit": null, "symbol": "k", "meaning": "order of the relation" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/equation", "concept/linear", "concept/recurring-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a723c7ae56", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "352", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "(1 + p_{1}z + p_{2}z^{2} + \\dots + p_{k}z^{k})f(z) = P_{0} + P_{1}z + \\dots + P_{k-1}z^{k-1}", "name": null, "statement": "Multiplying a recurring series by its scale of relation gives a polynomial of degree k-1, so the series is a rational function.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f(z)", "meaning": "sum of the recurring series" }, { "unit": null, "symbol": "p_i", "meaning": "constants of the scale of relation" }, { "unit": null, "symbol": "P_i", "meaning": "constants" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/polynomial", "concept/rational-function", "concept/recurring-series", "concept/scale-of-relation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-3e3028563d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "353", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "a_{n} - a_{n-1} - 8a_{n-2} + 12a_{n-3} = 0", "name": null, "statement": "Example of a linear difference equation with constant coefficients.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "a_n", "meaning": "unknown sequence" }, { "unit": null, "symbol": "n", "meaning": "index" } ], "sympy": "Eq(a_n - a_(n-1) - 8*a_(n-2) + 12*a_(n-3), 0)", "physics": false, "states": [], "concepts": [ "concept/equation", "concept/linear", "concept/recurring-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b882dc77ab", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "353", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "a_{n} = 2^{n}\\{A_{1} + (n + 1) A_{2}\\} + (-3)^{n} B", "name": null, "statement": "The general solution of the example difference equation, with constants fixed by a_0, a_1, a_2.", "kind": "result", "symbols": [ { "unit": null, "symbol": "A_1, A_2, B", "meaning": "constants determined by a_0, a_1, a_2" }, { "unit": null, "symbol": "n", "meaning": "index" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/power", "concept/recurring-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ae71d4c304", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "353", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "u_{n} - 2\\cos\\theta u_{n-1} + u_{n-2} = 0", "name": null, "statement": "The difference equation whose solutions are sinusoidal in n.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "u_n", "meaning": "unknown sequence" }, { "unit": null, "symbol": "theta", "meaning": "angle" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/equation", "concept/linear" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ffb45f6b9d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "353", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "u_{n} = A\\cos n\\theta + B\\sin n\\theta", "name": null, "statement": "The general solution of the difference equation with cos θ, with arbitrary constants A and B.", "kind": "result", "symbols": [ { "unit": null, "symbol": "A, B", "meaning": "arbitrary constants" }, { "unit": null, "symbol": "theta", "meaning": "angle" }, { "unit": null, "symbol": "n", "meaning": "index" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/equation", "concept/plane-angle", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-84fcf7c305", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "353", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "f(n) + f(n - 1) + f(n - 2) = 0", "name": null, "statement": "The coefficients of z/(1 + z + z^2) satisfy this three-term relation.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f(n)", "meaning": "coefficient of z^n in z/(1 + z + z^2)" }, { "unit": null, "symbol": "n", "meaning": "index" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/equation", "concept/function-notation", "concept/recurring-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-5d4af092cf", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "353", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "f(n) = (\\omega_{3}^{n} - \\omega_{3}^{2n})/(\\omega_{3} - \\omega_{3}^{2})", "name": null, "statement": "Closed form for the coefficient f(n), using a complex cube root of unity ω_3.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f(n)", "meaning": "coefficient of z^n in z/(1 + z + z^2)" }, { "unit": null, "symbol": "omega_3", "meaning": "a complex cube root of unity" }, { "unit": null, "symbol": "n", "meaning": "index" } ], "sympy": "Eq(f(n), (omega3**n - omega3**(2*n))/(omega3 - omega3**2))", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/cube-root-of-unity", "concept/root-of-unity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c23c1d994f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "353", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "z/(1 + z + z^{2}) = z(1 - z)/(1 - z^{3})", "name": null, "statement": "The rational function equals z(1 - z)/(1 - z^3), used to verify the coefficients of f(n).", "kind": "identity", "symbols": [ { "unit": null, "symbol": "z", "meaning": "complex variable" } ], "sympy": "Eq(z/(1 + z + z**2), z*(1 - z)/(1 - z**3))", "physics": false, "states": [], "concepts": [ "concept/cube-root-of-unity", "concept/identity", "concept/rational-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-505d1c71d3", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "353", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "p_{n} = \\frac{1}{2} (p_{n-1} + p_{n-2})", "name": null, "statement": "The probability of exactly reaching total n satisfies this recurrence.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "p_n", "meaning": "probability of making exactly the total n" }, { "unit": null, "symbol": "n", "meaning": "score total" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/equation", "concept/probability" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-483582be78", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "353", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\frac{1}{3}\\{2 + (-\\frac{1}{2})^{n}\\}", "name": null, "statement": "The chance that the player's score makes exactly the total n.", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "target total of the score" } ], "sympy": "Eq(p_n, Rational(1, 3)*(2 + (-Rational(1, 2))**n))", "physics": false, "states": [], "concepts": [ "concept/power", "concept/probability", "concept/rational-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-8c8213b98a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "354", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\frac{1}{a + 1} + \\frac{1}{a + 2} + \\dots + \\frac{1}{a + n} = \\binom{n}{1}\\frac{1}{a + 1} - \\binom{n}{2}\\frac{1!}{(a + 1)(a + 2)} + \\dots", "name": null, "statement": "Identity expressing a sum of reciprocals as a sum of binomial-weighted partial-fraction terms, for positive integer n and a not in {-1,...,-n}.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "parameter not equal to -1,...,-n" }, { "unit": null, "symbol": "n", "meaning": "positive integer" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/binomial", "concept/identity", "concept/sum", "method/partial-fractions" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b105edb288", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "354", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\int_{0}^{1} x^{a}\\frac{1 - x^{n}}{1 - x}\\, dx = \\int_{0}^{1} (1 - x)^{a}\\{1 - (1 - x)^{n}\\}\\frac{dx}{x}", "name": null, "statement": "Two integral forms that give the same value, used to prove the preceding identity when a > -1.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "parameter with a > -1" }, { "unit": null, "symbol": "n", "meaning": "positive integer" }, { "unit": null, "symbol": "x", "meaning": "variable of integration" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/binomial", "concept/definite-integral", "concept/integral" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a30ed911b3", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "354", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\sum_{0}^{\\infty} \\frac{z^{n}}{n!} \\sum_{1}^{\\infty} \\frac{(-1)^{n-1}z^{n}}{n·n!} = \\sum_{1}^{\\infty} \\left(1 + \\frac{1}{2} + \\frac{1}{3} + \\dots + \\frac{1}{n}\\right) \\frac{z^{n}}{n!}", "name": null, "statement": "The product of the exponential-type series with the logarithm-type series equals a series with harmonic-number coefficients.", "kind": "result", "symbols": [ { "unit": null, "symbol": "z", "meaning": "complex variable" }, { "unit": null, "symbol": "n", "meaning": "index" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/power-series", "concept/product-of-series", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b78588f364", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "354", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "(A_{1}B_{n} + A_{2}B_{n-1} + \\dots + A_{n}B_{1})/n \\to AB", "name": null, "statement": "If A_n tends to A and B_n tends to B, the Cesàro-type mean of the convolution tends to AB.", "kind": "result", "symbols": [ { "unit": null, "symbol": "A_n, B_n", "meaning": "partial sums of the two series" }, { "unit": null, "symbol": "A, B", "meaning": "sums of the two series" }, { "unit": null, "symbol": "n", "meaning": "index" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/limit", "concept/product-of-series", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-585cf34373", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "354", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "c_{n} = a_{1}b_{n} + a_{2}b_{n-1} + \\dots + a_{n}b_{1}", "name": null, "statement": "Definition of the convolution terms c_n of two series.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "c_n", "meaning": "convolution term" }, { "unit": null, "symbol": "a_n, b_n", "meaning": "terms of the two series" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/product-of-series", "concept/sum", "concept/term" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-df75c33db6", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "354", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "C_{n} = a_{1}B_{n} + a_{2}B_{n-1} + \\dots + a_{n}B_{1}", "name": null, "statement": "The n-th partial sum of the convolution equals this sum of terms with partial sums of the b_n.", "kind": "result", "symbols": [ { "unit": null, "symbol": "C_n", "meaning": "partial sum of the c_n" }, { "unit": null, "symbol": "B_n", "meaning": "partial sum b_1 + ... + b_n" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/partial-sums", "concept/product-of-series", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-8368cf46ea", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "354", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "C_{1} + C_{2} + \\dots + C_{n} = A_{1}B_{n} + A_{2}B_{n-1} + \\dots + A_{n}B_{1}", "name": null, "statement": "The sum of the first n partial sums of the convolution equals the convolution of partial sums of the two series.", "kind": "result", "symbols": [ { "unit": null, "symbol": "C_n", "meaning": "partial sums of the c_n" }, { "unit": null, "symbol": "A_n, B_n", "meaning": "partial sums of the two series" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/partial-sums", "concept/product-of-series", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c261ad2262", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "354", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "(C_{1} + C_{2} + \\dots + C_{n})/n \\to AB", "name": "Abel's theorem on the multiplication of series", "statement": "The Cesàro mean of the partial sums of the product series tends to AB; hence if the product series converges its sum is AB.", "kind": "result", "symbols": [ { "unit": null, "symbol": "C_n", "meaning": "partial sums of the product series" }, { "unit": null, "symbol": "A, B", "meaning": "sums of the two series" }, { "unit": null, "symbol": "n", "meaning": "index" } ], "sympy": null, "physics": false, "states": [ "theorem/abel-s-theorem" ], "concepts": [ "concept/convergent-series", "concept/limit", "concept/product-of-series", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-3838bad0f2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "355", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\int_{-1}^{1} \\frac{dx}{(a - x) \\sqrtp{1 - x^{2}}} = \\frac{\\pi}{\\sqrtp{a^{2} - 1}}", "name": null, "statement": "Evaluates a definite integral with a convergent integrand for a > 1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "parameter with a > 1" }, { "unit": null, "symbol": "x", "meaning": "variable of integration" }, { "unit": null, "symbol": "pi", "meaning": "the ratio of circumference to diameter" } ], "sympy": "Eq(Integral(1/((a - x)*sqrt(1 - x**2)), (x, -1, 1)), pi/sqrt(a**2 - 1))", "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/integral", "concept/root", "quantity/pi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-37051b676b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "355", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\int_{0}^{\\infty} \\frac{dx}{\\{\\sqrtp{x^{2} + 1} + x\\}^{n}} = \\int_{0}^{\\infty} \\{\\sqrtp{x^{2} + 1} - x\\}^{n}\\, dx = \\frac{n}{n^{2} - 1}", "name": null, "statement": "The two infinite integrals are equal and have value n/(n^2 - 1) for n > 1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "integer exponent with n > 1" }, { "unit": null, "symbol": "x", "meaning": "variable of integration" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/infinite-integral", "concept/integer", "concept/integral", "concept/tends-to-infinity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-fde36f8348", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "355", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "2y = ax - (b/x)", "name": null, "statement": "Substitution relation between y and x, with a, b positive, which increases steadily from -infinity to infinity.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "y", "meaning": "dependent variable" }, { "unit": null, "symbol": "x", "meaning": "independent variable" }, { "unit": null, "symbol": "a, b", "meaning": "positive constants" } ], "sympy": "Eq(2*y, a*x - b/x)", "physics": false, "states": [], "concepts": [ "concept/infinity", "concept/steadily-increasing-function", "concept/variable" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a0913a27fe", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "355", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "2y = ax + (b/x)", "name": null, "statement": "Substitution relation between y and x, where two values of x correspond to each y greater than sqrt(ab).", "kind": "definition", "symbols": [ { "unit": null, "symbol": "y", "meaning": "dependent variable" }, { "unit": null, "symbol": "x", "meaning": "independent variable" }, { "unit": null, "symbol": "a, b", "meaning": "positive constants" } ], "sympy": "Eq(2*y, a*x + b/x)", "physics": false, "states": [], "concepts": [ "concept/relation-between-variables", "concept/root", "concept/variable" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-be6cfd16f8", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "356", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\int_{0}^{\\pi} f(\\sec\\tfrac{1}{2}x + \\tan\\tfrac{1}{2}x)\\frac{dx}{\\sqrtp{\\sin x}} = \\int_{0}^{\\pi} f(\\cosec x)\\frac{dx}{\\sqrtp{\\sin x}}", "name": null, "statement": "Transformation formula equating two integrals with the same weight 1/sqrt(sin x).", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable of integration" }, { "unit": null, "symbol": "f", "meaning": "arbitrary function" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/definite-integral", "concept/function", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-526c9f4a28", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "356", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\int_{0}^{\\infty} \\frac{dx}{(x^{2} + a^{2})(x^{2} + b^{2})} = \\frac{\\pi}{2ab(a + b)}", "name": null, "statement": "Evaluates an infinite integral for positive a and b.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a, b", "meaning": "positive constants" }, { "unit": null, "symbol": "x", "meaning": "variable of integration" } ], "sympy": "Eq(Integral(1/((x**2 + a**2)*(x**2 + b**2)), (x, 0, oo)), pi/(2*a*b*(a + b)))", "physics": false, "states": [], "concepts": [ "concept/infinite-integral", "concept/integral", "quantity/pi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-63dbe8d1d2", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "356", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\int_{0}^{\\infty} \\frac{x^{2}\\, dx}{(x^{2} + a^{2})(x^{2} + b^{2})} = \\frac{\\pi}{2(a + b)}", "name": null, "statement": "Evaluates an infinite integral for positive a and b.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a, b", "meaning": "positive constants" }, { "unit": null, "symbol": "x", "meaning": "variable of integration" } ], "sympy": "Eq(Integral(x**2/((x**2 + a**2)*(x**2 + b**2)), (x, 0, oo)), pi/(2*(a + b)))", "physics": false, "states": [], "concepts": [ "concept/infinite-integral", "concept/integral", "quantity/pi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ae8b9f040e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "356", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "A = \\beta + \\sqrtp{\\alpha\\gamma}", "name": null, "statement": "Defines the constant A used in the closed forms of the quartic-denominator integrals.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "A", "meaning": "constant beta + sqrt(alpha gamma)" }, { "unit": null, "symbol": "alpha, beta, gamma", "meaning": "positive constants with beta^2 >= alpha gamma" } ], "sympy": "Eq(A, beta + sqrt(alpha*gamma))", "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/root" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-5554e7def1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "356", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\int_{0}^{\\infty} \\frac{x^{2}\\, dx}{(x^{2} - a^{2})^{2} + b^{2}x^{2}} = \\frac{\\pi}{2b}", "name": null, "statement": "Evaluates an infinite integral for positive b.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a, b", "meaning": "constants with b positive" }, { "unit": null, "symbol": "x", "meaning": "variable of integration" } ], "sympy": "Eq(Integral(x**2/((x**2 - a**2)**2 + b**2*x**2), (x, 0, oo)), pi/(2*b))", "physics": false, "states": [], "concepts": [ "concept/infinite-integral", "concept/integral", "quantity/pi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-bd2cdd6e4f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "356", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\int_{0}^{\\infty} \\phi(x)\\, dx = \\sum_{0}^{\\infty} \\frac{1}{(n + 1)^{2}}", "name": null, "statement": "The integral of the function phi from the end of section 178 equals the sum of 1/(n+1)^2 over n.", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi(x)", "meaning": "the function considered at the end of section 178" }, { "unit": null, "symbol": "n", "meaning": "index" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/function-notation", "concept/infinite-integral", "concept/integer", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-47e45f8b14", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "356", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\int_{1}^{\\infty} dx \\left(\\int_{1}^{\\infty} \\frac{x - y}{(x + y)^{3}}\\, dy\\right) = -1", "name": null, "statement": "Iterated infinite integral with the outer variable x; its order-reversed value is 1, showing the order of integration matters.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x, y", "meaning": "variables of integration" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/infinite-integral", "concept/integral", "concept/order-of-integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-cf5734d245", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "356", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\int_{1}^{\\infty} dy \\left(\\int_{1}^{\\infty} \\frac{x - y}{(x + y)^{3}}\\, dx\\right) = 1", "name": null, "statement": "Iterated infinite integral with the outer variable y; its value differs in sign from the reversed order.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x, y", "meaning": "variables of integration" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/infinite-integral", "concept/integral", "concept/order-of-integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-38ff9778a9", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "356", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\int_{1}^{\\infty} dx \\left(\\int_{1}^{\\infty} \\frac{x^{2} - y^{2}}{(x^{2} + y^{2})^{2}}\\, dy\\right) = -\\tfrac{1}{4}\\pi", "name": null, "statement": "Iterated infinite integral with outer variable x; its reversed order gives +pi/4.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x, y", "meaning": "variables of integration" }, { "unit": null, "symbol": "pi", "meaning": "the ratio of circumference to diameter" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/infinite-integral", "concept/integral", "concept/order-of-integration", "quantity/pi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-975d6e2226", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "356", "location": "THE CONVERGENCE OF INFINITE SERIES AND \\\\ INFINITE INTEGRALS", "latex": "\\int_{1}^{\\infty} dy \\left(\\int_{1}^{\\infty} \\frac{x^{2} - y^{2}}{(x^{2} + y^{2})^{2}}\\, dx\\right) = \\tfrac{1}{4}\\pi", "name": null, "statement": "Iterated infinite integral with outer variable y; the two orders give different values.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x, y", "meaning": "variables of integration" }, { "unit": null, "symbol": "pi", "meaning": "the ratio of circumference to diameter" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/infinite-integral", "concept/integral", "concept/order-of-integration", "quantity/pi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-67071498ac", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "358", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "\\log x = \\int \\frac{dx}{x}", "name": null, "statement": "The logarithm of x is defined as the integral of 1/x, as already introduced in Chapter VI.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable (positive real number)" }, { "unit": null, "symbol": "\\log x", "meaning": "logarithm of x (natural, base e)" } ], "sympy": "Eq(log(x), Integral(1/x, x))", "physics": false, "states": [], "concepts": [ "concept/integral", "concept/logarithm", "concept/variable" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-8c323ef3cc", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "358", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "\\log x = \\int_{1}^{x} \\frac{dt}{t}", "name": null, "statement": "Definition of log x as the integral of dt/t from 1 to x, for positive x.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x", "meaning": "positive real variable" }, { "unit": null, "symbol": "t", "meaning": "variable of integration" }, { "unit": null, "symbol": "\\log x", "meaning": "the logarithm of x" } ], "sympy": "Eq(log(x), Integral(1/t, (t, 1, x)))", "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/integral", "concept/logarithm", "concept/variable" ], "pages": [ "358", "396" ], "chapters": [ "hardy-course-of-pure-mathematics-1921/ch-ix", "hardy-course-of-pure-mathematics-1921/ch-x" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c51a295a07", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "359", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "D_{x} \\log x = 1/x", "name": null, "statement": "The derivative of log x with respect to x is 1/x.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "positive real variable" }, { "unit": null, "symbol": "D_{x}", "meaning": "differentiation with respect to x" } ], "sympy": "Eq(Derivative(log(x), x), 1/x)", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/logarithm", "concept/reciprocal", "method/differentiation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-4654bde320", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "359", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "\\log x = \\int_{1}^{x} \\frac{dt}{t} = -\\int_{x}^{1} \\frac{dt}{t} < 0", "name": null, "statement": "For 0 < x < 1, log x is negative, since reversing the limits of the integral changes its sign.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "positive real variable less than 1" }, { "unit": null, "symbol": "t", "meaning": "variable of integration" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/inequality", "concept/logarithm", "concept/negative-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-05db6ca01f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "359", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "\\log x = \\int_{1}^{x} \\frac{dt}{t} = -\\int_{1}^{1/x} \\frac{du}{u} = -\\log(1/x)", "name": null, "statement": "Substituting t = 1/u shows that log x equals minus log(1/x).", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "positive real variable" }, { "unit": null, "symbol": "u", "meaning": "variable of integration after substitution t = 1/u" } ], "sympy": "Eq(log(x), -log(1/x))", "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/logarithm", "concept/reciprocal", "method/substitution" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-9112fa9327", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "360", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "f(xy) = f(x) + f(y)", "name": null, "statement": "The logarithm satisfies the functional equation: the function of a product is the sum of the functions of the factors.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "f", "meaning": "function satisfying the equation (log x is the solution treated)" }, { "unit": null, "symbol": "x", "meaning": "positive real variable" }, { "unit": null, "symbol": "y", "meaning": "positive real variable" } ], "sympy": "Eq(f(x*y), f(x) + f(y))", "physics": false, "states": [], "concepts": [ "concept/function", "concept/functional-equation", "concept/logarithm", "concept/product" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c67d6cb998", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "363", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "\\log x^{n} = n\\log x", "name": null, "statement": "The logarithm of a positive integer power of x is n times log x.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "n", "meaning": "positive integer" }, { "unit": null, "symbol": "x", "meaning": "positive real variable" } ], "sympy": "Eq(log(x**n), n*log(x))", "physics": false, "states": [], "concepts": [ "concept/integer", "concept/laws-of-indices", "concept/logarithm", "concept/power" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e10f0b5935", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "363", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "\\log e^{n} = n\\log e = n", "name": null, "statement": "The logarithm of e to a positive integer power n equals n, since log e = 1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "e", "meaning": "the number whose logarithm is 1" }, { "unit": null, "symbol": "n", "meaning": "positive integer" } ], "sympy": "Eq(log(exp(n)), n)", "physics": false, "states": [], "concepts": [ "concept/integer", "concept/logarithm", "concept/power", "concept/real-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ad054a0478", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "364", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "\\log e^{y} = y", "name": null, "statement": "The logarithm of e raised to y is y, for rational y (extended later to all real y).", "kind": "result", "symbols": [ { "unit": null, "symbol": "y", "meaning": "real (initially rational) exponent" }, { "unit": null, "symbol": "e", "meaning": "the number whose logarithm is 1" } ], "sympy": "Eq(log(exp(y)), y)", "physics": false, "states": [], "concepts": [ "concept/exponential-function", "concept/logarithm", "concept/power", "concept/rational-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-0e119c5541", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "364", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "y = \\log x,\\quad x = e^{y}", "name": null, "statement": "The equations y = log x and x = e^y are consequences of one another.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x", "meaning": "positive real variable" }, { "unit": null, "symbol": "y", "meaning": "logarithm of x, the exponent of e giving x" } ], "sympy": "Eq(y, log(x)) & Eq(x, exp(y))", "physics": false, "states": [], "concepts": [ "concept/exponential-function", "concept/inverse-function", "concept/logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-31586c938c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "363", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "1 = \\int_{1}^{e} \\frac{dt}{t}", "name": null, "statement": "Definition of the number e as the number whose logarithm is 1.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "e", "meaning": "the number whose logarithm is 1" }, { "unit": null, "symbol": "t", "meaning": "variable of integration" } ], "sympy": "Eq(1, Integral(1/t, (t, 1, E)))", "physics": false, "states": [], "concepts": [ "concept/constant", "concept/definite-integral", "concept/logarithm", "concept/real-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b0f6beedc7", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "365", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "dy/dx = 1/x", "name": null, "statement": "If x = e^y, the derivative of y = log x with respect to x is 1/x.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "positive real variable" }, { "unit": null, "symbol": "y", "meaning": "log x" } ], "sympy": "Eq(Derivative(y, x), 1/x)", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/logarithm", "concept/partial-derivative", "concept/reciprocal" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c3147220ec", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "365", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "\\frac{dx}{dy} = x = e^{y}", "name": null, "statement": "The derivative of the exponential function equals the function itself.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "e^y" }, { "unit": null, "symbol": "y", "meaning": "real exponent" } ], "sympy": "Eq(Derivative(x, y), x)", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/exponential-function", "concept/partial-derivative" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-388281d235", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "365", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "dx/dy = ae^{ay}", "name": null, "statement": "If x = e^{ay} then the derivative of x with respect to y is a e^{ay}.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "constant multiplier in the exponent" }, { "unit": null, "symbol": "x", "meaning": "e^{ay}" }, { "unit": null, "symbol": "y", "meaning": "real variable" } ], "sympy": "Eq(Derivative(x, y), a*exp(a*y))", "physics": false, "states": [], "concepts": [ "concept/constant", "concept/derivative", "concept/exponential-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-5151f6c424", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "365", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "f(y + z) = f(y)f(z)", "name": null, "statement": "The exponential function satisfies the functional equation that turns addition of exponents into multiplication.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "f", "meaning": "function (here the exponential function e^y)" }, { "unit": null, "symbol": "y", "meaning": "real variable" }, { "unit": null, "symbol": "z", "meaning": "real variable" } ], "sympy": "Eq(f(y + z), f(y)*f(z))", "physics": false, "states": [], "concepts": [ "concept/exponential-function", "concept/function", "concept/functional-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b2e7066e77", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "365", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "e^{-y} = 1/e^{y}", "name": null, "statement": "A negative exponent gives the reciprocal of the positive power.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "e", "meaning": "the number whose logarithm is 1" }, { "unit": null, "symbol": "y", "meaning": "real exponent" } ], "sympy": "Eq(exp(-y), 1/exp(y))", "physics": false, "states": [], "concepts": [ "concept/exponent", "concept/exponential-function", "concept/power", "concept/reciprocal" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-9fbd544332", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "366", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "\\lim y^{\\alpha}/e^{y} = \\lim e^{-y}y^{\\alpha} = 0", "name": null, "statement": "e^y tends to infinity faster than any power of y, so y^alpha / e^y tends to zero as y tends to infinity.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y", "meaning": "real variable tending to infinity" }, { "unit": null, "symbol": "\\alpha", "meaning": "any fixed real number (power)" } ], "sympy": "Eq(Limit(y**alpha/exp(y), y, oo), 0)", "physics": false, "states": [], "concepts": [ "concept/exponential-function", "concept/limit", "concept/order-of-greatness", "concept/sufficiently-large-values", "concept/tends-to-infinity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-3857592cf1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "361", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "\\frac{\\log x}{x^{\\alpha}} \\to 0", "name": null, "statement": "log x tends to infinity more slowly than any positive power of x, as x tends to infinity.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "real variable tending to infinity" }, { "unit": null, "symbol": "\\alpha", "meaning": "any positive 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"concept/inequality", "concept/logarithm", "concept/power" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-0e9b0c3fe7", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "362", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "(\\log\\log y)/(\\log y)^{\\alpha} = (\\log x)/x^{\\alpha} \\to 0", "name": null, "statement": "log log y tends to infinity more slowly than any power of log y as y tends to infinity.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y", "meaning": "real variable tending to infinity (x = log y)" }, { "unit": null, "symbol": "\\alpha", "meaning": "any positive number" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/limit", "concept/logarithm", "concept/scale-of-infinity", "concept/tends-to-infinity" ] }, { "id": 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"location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "\\log_{10} x = (\\log_{e} x)/(\\log_{e} 10)", "name": null, "statement": "The common logarithm of x equals the natural logarithm of x divided by the natural logarithm of 10.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "x", "meaning": "positive number" }, { "unit": null, "symbol": "\\log_{10} x", "meaning": "common logarithm of x" }, { "unit": null, "symbol": "\\log_{e} x", "meaning": "logarithm of x to base e" } ], "sympy": "Eq(log(x)/log(10), log(x, 10))", "physics": false, "states": [], "concepts": [ "concept/base-of-a-logarithm-system", "concept/common-logarithm", "concept/logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e90e79dd44", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "367", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "(a^{x})^{y} = a^{xy}", "name": null, "statement": "The power of a power law a^(xy) holds for all real exponents.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "positive number (base)" }, { "unit": null, "symbol": "x", "meaning": "real exponent" }, { "unit": null, "symbol": "y", "meaning": "real exponent" } ], "sympy": "Eq((a**x)**y, a**(x*y))", "physics": false, "states": [], "concepts": [ "concept/exponent", "concept/general-power", "concept/laws-of-indices" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-00692a1075", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "367", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "y = e^{x\\log a}", "name": null, "statement": "Definition of the general power a^x for irrational x as e^(x log a).", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a", "meaning": "positive number (base)" }, { "unit": null, "symbol": "x", "meaning": "real exponent" }, { "unit": null, "symbol": "y", "meaning": "a^x" } ], "sympy": "Eq(y, exp(x*log(a)))", "physics": false, "states": [], "concepts": [ "concept/exponential-function", "concept/general-power", "concept/logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-7099a1643a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "367", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "\\log a^{x} = x\\log a", "name": null, "statement": "The logarithm of a^x equals x times log a.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "positive number (base)" }, { "unit": null, "symbol": "x", "meaning": "real exponent" } ], "sympy": "Eq(log(a**x), x*log(a))", "physics": false, "states": [], "concepts": [ "concept/general-power", "concept/laws-of-indices", "concept/logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-fb3be39d7c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "367", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "a^{x} = e^{x\\log a} = e^{\\alpha x}", "name": null, "statement": "For a > 1, a^x equals e^(alpha x) with alpha positive, so a^x tends to infinity as x tends to infinity.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "number greater than 1" }, { "unit": null, "symbol": "\\alpha", "meaning": "positive constant equal to log a" }, { "unit": null, "symbol": "x", "meaning": "real variable" } ], "sympy": "Eq(a**x, exp(alpha*x))", "physics": false, "states": [], "concepts": [ 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"concept/tends-to-infinity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-0704c2859d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "367", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "D_{x} e^{x\\log a} = e^{x\\log a} \\log a = a^{x} \\log a", "name": null, "statement": "The derivative of a^x with respect to x is a^x times log a.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a", "meaning": "positive number (base)" }, { "unit": null, "symbol": "x", "meaning": "real variable" }, { "unit": null, "symbol": "D_{x}", "meaning": "differentiation with respect to x" } ], "sympy": "Eq(Derivative(a**x, x), a**x*log(a))", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/general-power", "concept/logarithm", "method/differentiation" ] }, { "id": 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FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "\\lim_{n\\to\\infty} \\left(1 + \\frac{x}{n}\\right)^{n} = \\lim_{n\\to\\infty} \\left(1 - \\frac{x}{n}\\right)^{-n} = e^{x}", "name": null, "statement": "The exponential e^x is the limit of (1 + x/n)^n and of (1 - x/n)^(-n) as n tends to infinity.", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "positive integer tending to infinity" }, { "unit": null, "symbol": "x", "meaning": "real variable" }, { "unit": null, "symbol": "e", "meaning": "the number whose logarithm is 1" } ], "sympy": "Eq(Limit((1 + x/n)**n, n, oo), exp(x))", "physics": false, "states": [], "concepts": [ "concept/exponential-function", "concept/limit", "concept/real-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-07ce19e33a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "368", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "\\lim_{\\xi\\to\\infty} \\left(1 + \\frac{x}{\\xi}\\right)^{\\xi} = \\lim_{\\xi\\to -\\infty} \\left(1 + \\frac{x}{\\xi}\\right)^{\\xi} = e^{x}", "name": null, "statement": "The generalisation of the limit representation of e^x to a continuous variable xi tending to plus or minus infinity.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\xi", "meaning": "continuous real variable tending to plus or minus infinity" }, { "unit": null, "symbol": "x", "meaning": "real variable" } ], "sympy": "Eq(Limit((1 + x/xi)**xi, xi, oo), exp(x))", "physics": false, "states": [], "concepts": [ "concept/exponential-function", "concept/limit", "concept/tends-to-infinity", "concept/variable" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-aa3e300157", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "368", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "n(1 - x^{-1/n}) < \\log x < n(x^{1/n} - 1)", "name": null, "statement": "For x > 1 and any positive integer n, log x lies between n(1 - x^(-1/n)) and n(x^(1/n) - 1).", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "positive integer" }, { "unit": null, "symbol": "x", "meaning": "real variable greater than 1" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/inequality", "concept/integer", "concept/logarithm", "concept/power", "concept/root" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-9dea1624e7", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "368", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "\\left(1 + \\frac{y}{n}\\right)^{n} < x < \\left(1 - \\frac{y}{n}\\right)^{-n}", "name": null, "statement": "With y = log x and x = e^y, x lies between (1 + y/n)^n and (1 - y/n)^(-n).", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "positive integer" }, { "unit": null, "symbol": "y", "meaning": "log x" }, { "unit": null, "symbol": "x", "meaning": "positive real variable" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/exponential-function", "concept/inequality", "concept/integer", "concept/logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-906a72d767", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "369", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "\\lim n(1 - x^{-1/n}) = \\lim n(x^{1/n} - 1) = \\log x", "name": null, "statement": "The logarithm of x is the limit of n(x^(1/n) - 1) and of n(1 - x^(-1/n)) as n tends to infinity.", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "positive integer tending to infinity" }, { "unit": null, "symbol": "x", "meaning": "positive real variable" } ], "sympy": "Eq(Limit(n*(x**(1/n) - 1), n, oo), log(x))", "physics": false, "states": [], "concepts": [ "concept/limit", "concept/logarithm", "concept/root" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a1914f46a6", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "375", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "D_{x}(\\log x)^{1-s} = \\frac{1 - s}{x(\\log x)^{s}}", "name": null, "statement": "The derivative of (log x) raised to the power 1 − s equals (1 − s) divided by x(log x)^s.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the real variable" }, { "unit": null, "symbol": "s", "meaning": "a constant exponent" } ], "sympy": "Eq(Derivative(log(x)**(1 - s), x), (1 - s)/(x*log(x)**s))", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/logarithm", "concept/power", "method/differentiation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-73521a181a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "375", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "D_{x}\\log\\log x = \\frac{1}{x\\log x}", "name": null, "statement": "The derivative of log log x with respect to x equals 1 divided by x log x.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the real variable" } ], "sympy": "Eq(Derivative(log(log(x)), x), 1/(x*log(x)))", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/logarithm", 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"states": [], "concepts": [ "concept/convergent-series", "concept/divergent-series", "concept/infinite-sequence", "concept/logarithm", "theorem/integral-test" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-fe15121998", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "378", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "e^{x} = 1 + x + \\frac{x^{2}}{2!} + \\dots + \\frac{x^{n-1}}{(n - 1)!} + \\frac{x^{n}}{n!} e^{\\theta x}", "name": null, "statement": "Taylor's theorem expansion of e^x with remainder term, where 0 < θ < 1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the real variable" }, { "unit": null, "symbol": "n", "meaning": "a positive integer order of the expansion" }, { "unit": null, "symbol": "θ", "meaning": "a number strictly between 0 and 1" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/exponential-function", "concept/power-series", "concept/remainder", "theorem/taylor-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e99d881ca5", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "378", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "x^{n}/n! \\to 0", "name": null, "statement": "x^n divided by n! tends to zero as n tends to infinity, whatever the value of x.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the real variable" }, { "unit": null, "symbol": "n", "meaning": "a positive integer index" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/factorial", "concept/limit", "concept/sufficiently-large-values", "concept/tends-to-infinity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-71fbae91db", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "378", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "e^{x} = 1 + x + \\frac{x^{2}}{2!} + \\dots + \\frac{x^{n}}{n!} + \\dots", "name": "exponential series", "statement": "e^x equals the infinite power series 1 + x + x²/2! + … ; this is the exponential series.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the real variable" }, { "unit": null, "symbol": "n", "meaning": "a non-negative integer index of the terms" } ], "sympy": null, "physics": false, "states": [ "theorem/exponential-series" ], "concepts": [ "concept/convergent-series", "concept/exponential-function", "concept/exponential-theorem", "concept/power-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-3afad81ab4", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "378", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "e = 1 + 1 + \\frac{1}{2!} + \\dots + \\frac{1}{n!} + \\dots", "name": null, "statement": "Euler's number e is the sum of the series 1 + 1 + 1/2! + 1/3! + …", "kind": "definition", "symbols": [ { "unit": null, "symbol": "e", "meaning": "the number whose logarithm is 1 (Euler's number)" }, { "unit": null, "symbol": "n", "meaning": "a non-negative integer index of the terms" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/euler-s-number", "concept/factorial", "concept/infinite-sequence", "theorem/exponential-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-1967672872", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "378", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "\\left(1 + 1 + \\frac{1}{2!} + \\dots + \\frac{1}{n!} + \\dots\\right)^{x} = 1 + x + \\frac{x^{2}}{2!} + \\dots + \\frac{x^{n}}{n!} + \\dots", "name": "exponential theorem", "statement": "The x-th power of the series for e equals the exponential series in x.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the real variable" }, { "unit": null, "symbol": "n", "meaning": "a non-negative integer index of the terms" } ], "sympy": null, "physics": false, "states": [ "concept/exponential-theorem" ], "concepts": [ "concept/exponential-function", "concept/power", "theorem/exponential-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-1a7c4ebb69", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "379", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "a^{x} = e^{x\\log a} = 1 + (x\\log a) + \\frac{(x\\log a)^{2}}{2!} + \\dots", "name": null, "statement": "a^x equals e^(x log a), and so equals the exponential series in x log a, for positive a.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "a", "meaning": "a positive number (the base)" }, { "unit": null, "symbol": "x", "meaning": "the real exponent" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/exponential-function", "concept/general-power", "concept/logarithm", "theorem/exponential-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c25c2f3685", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "379", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "\\left(1 + \\frac{x}{n}\\right)^{n} < E_{n}(x) < \\left(1 - \\frac{x}{n}\\right)^{-n}", "name": null, "statement": "For x > 0 and n > x, the partial sum E_n(x) of the exponential series lies between (1 + x/n)^n and (1 − x/n)^(−n).", "kind": "result", "symbols": [ { "unit": null, "symbol": "E_{n}(x)", "meaning": "the partial sum 1 + x + x²/2! + … + x^n/n!" }, { "unit": null, "symbol": "x", "meaning": "a positive real number" }, { "unit": null, "symbol": "n", "meaning": "a positive integer greater than x" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/exponential-function", "concept/inequality", "concept/limit", "theorem/binomial-theorem", "theorem/exponential-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d20dfde0cc", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "379", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "f(x)f(y) = f(x + y)", "name": null, "statement": "The exponential series satisfies the functional equation f(x)f(y) = f(x+y).", "kind": "identity", "symbols": [ { "unit": null, "symbol": "f", "meaning": "the exponential series as a function of x" }, { "unit": null, "symbol": "x", "meaning": "a real variable" }, { "unit": null, "symbol": "y", "meaning": "a real variable" } ], "sympy": "Eq(f(x)*f(y), f(x + y))", "physics": false, "states": [], "concepts": [ "concept/exponential-function", "concept/functional-equation", "theorem/exponential-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ce934c2ef1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "379", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "f(x)f(-x) = f(0) = 1", "name": null, "statement": "Since f(x)f(y) = f(x+y), f(x) 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"\\tfrac{1}{4}\\pi = 1 - \\tfrac{1}{3} + \\tfrac{1}{5} - \\dots", "name": null, "statement": "Setting x = 1 in the arctangent series gives pi/4 as the alternating series 1 - 1/3 + 1/5 - ...", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\pi", "meaning": "the ratio of a circle's circumference to its diameter" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/infinite-sequence", "concept/inverse-circular-function", "quantity/pi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-95310d5613", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "382", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "\\argtanh x = \\frac{1}{2} \\log\\left(\\dfrac{1 + x}{1 - x}\\right)", "name": null, "statement": "The inverse hyperbolic tangent of x equals half the logarithm of (1 + x)/(1 - x).", "kind": 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"D_{x} \\exp x = \\exp x", "name": null, "statement": "The derivative of exp x with respect to x is exp x itself.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "real variable" } ], "sympy": "Eq(Derivative(exp(x), x), exp(x))", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/exponential-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-248d05baef", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "386", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "\\frac{dy}{dx} = y", "name": null, "statement": "With y = exp x, the derivative dy/dx equals y.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y", "meaning": "exp x" }, { "unit": null, "symbol": "x", "meaning": "real variable" } ], "sympy": "Eq(Derivative(y, x), y)", "physics": false, "states": 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null, "symbol": "x", "meaning": "real variable (rational here, extended to all real x)" } ], "sympy": "Eq(exp(x), exp(1)**x)", "physics": false, "states": [], "concepts": [ "concept/euler-s-number", "concept/exponential-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-bae6802363", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS \\\\ OF A REAL VARIABLE", "latex": "e = \\exp 1 = 1 + 1 + \\frac{1}{2!} + \\frac{1}{3!} + \\dots", "name": null, "statement": "Euler's number e is exp 1, given by the sum of the exponential series at x = 1.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "e", "meaning": "Euler's number, exp 1" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/euler-s-number", "theorem/exponential-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-7412c75fbf", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "395", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "z = x + iy", "name": null, "statement": "The complex variable z is written as x plus i times y, with x and y real.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "z", "meaning": "complex variable" }, { "unit": null, "symbol": "x", "meaning": "real part of z" }, { "unit": null, "symbol": "y", "meaning": "coefficient of i in z (imaginary part)" } ], "sympy": "Eq(z, x + I*y)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/complex-variable" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a14adef8cc", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "395", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "|z| = \\sqrtp{x^{2} + y^{2}}", "name": "modulus", "statement": "The modulus of z is the square root of x squared plus y squared.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "|z|", "meaning": "modulus of z" }, { "unit": null, "symbol": "x", "meaning": "real part of z" }, { "unit": null, "symbol": "y", "meaning": "imaginary part of z (coefficient of i)" } ], "sympy": "Eq(Abs(z), sqrt(x**2 + y**2))", "physics": false, "states": [ "concept/congruence" ], "concepts": [ "quantity/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-93d92edb58", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "395", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\am z = \\arctan(y/x)", "name": null, "statement": "The amplitude of z is the arctangent of y over x.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "am z", "meaning": "amplitude of the complex variable z" }, { "unit": null, "symbol": "x", "meaning": "real part of z" }, { "unit": null, "symbol": "y", "meaning": "imaginary part of z" } ], "sympy": "Eq(am_z, atan(y/x))", "physics": false, "states": [], "concepts": [ "concept/circular-function", "quantity/amplitude-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-8324d8969c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "397", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\int_{C} \\{g(x, y)\\, dx + h(x, y)\\, dy\\}", "name": "curvilinear integral", "statement": "The real curvilinear integral of g dx + h dy along the path C is defined as the ordinary integral obtained by substituting the parametric equations of C.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "C", "meaning": "path of integration (arc of a curve from A to B)" }, { "unit": null, "symbol": "g", "meaning": "continuous function of x and y" }, { "unit": null, "symbol": "h", "meaning": "continuous function of x and y" }, { "unit": null, "symbol": "x", "meaning": "coordinate of a point on C" }, { "unit": null, "symbol": "y", "meaning": "coordinate of a point on C" } ], "sympy": null, "physics": false, "states": [ "concept/curvilinear-integral" ], "concepts": [ "concept/arc-of-a-curve", "concept/path-of-integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-1b9860d31e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "397", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\int_{C} f(z)\\, dz", "name": "complex curvilinear integral", "statement": "The integral of f(z) dz along C is defined as the real curvilinear integrals of (u dx - v dy) plus i times those of (v dx + u dy), where f = u + iv.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "C", "meaning": "path of integration in Argand's diagram" }, { "unit": null, "symbol": "f(z)", "meaning": "polynomial or rational function of z" }, { "unit": null, "symbol": "z", "meaning": "complex variable" } ], "sympy": null, "physics": false, "states": [ "theorem/complex-curvilinear-integral" ], "concepts": [ "concept/complex-variable", "concept/curvilinear-integral", "concept/path-of-integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d8c22cdfb0", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "397", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\Log \\zeta = \\int_{C} \\frac{dz}{z}", "name": "general logarithm", "statement": "The general logarithm of zeta is the integral of dz/z along any curve C from 1 to zeta that does not pass through the origin.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "\\Log \\zeta", "meaning": "general logarithm of zeta" }, { "unit": null, "symbol": "zeta", "meaning": "complex number (endpoint of the path)" }, { "unit": null, "symbol": "C", "meaning": "path of integration from 1 to zeta avoiding the origin" }, { "unit": null, "symbol": "z", "meaning": "complex variable on the path" } ], "sympy": null, "physics": false, "states": [ "concept/logarithm" ], "concepts": [ "concept/curvilinear-integral", "concept/origin", "concept/path-of-integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-8bd3c1fa57", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "399", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\zeta = \\rho(\\cos\\phi + i\\sin\\phi)", "name": null, "statement": "A complex number zeta is written in polar form with modulus rho and amplitude phi.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "zeta", "meaning": "complex number" }, { "unit": null, "symbol": "rho", "meaning": "modulus of zeta (positive)" }, { "unit": null, "symbol": "phi", "meaning": "amplitude of zeta" } ], "sympy": "Eq(zeta, rho*(cos(phi) + I*sin(phi)))", "physics": false, "states": [], "concepts": [ "concept/complex-number", "quantity/amplitude-of-a-complex-number", "quantity/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-3b3bf72f35", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "399", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\Log \\zeta = \\log \\rho + i\\phi", "name": "principal value of the logarithm", "statement": "When the path of integration is a straight line from 1 to zeta, the value of Log zeta is log rho plus i phi.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\Log \\zeta", "meaning": "general logarithm of zeta (here its principal value)" }, { "unit": null, "symbol": "rho", "meaning": "modulus of zeta" }, { "unit": null, "symbol": "phi", "meaning": "amplitude of zeta, between -pi and pi" } ], "sympy": null, "physics": false, "states": [ "theorem/principal-value-of-the-logarithm" ], "concepts": [ "concept/logarithm", "concept/principal-value-of-a-logarithm", "quantity/amplitude-of-a-complex-number", "quantity/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-164f5639da", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "400", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\log \\zeta = \\log \\rho + i\\phi", "name": "principal value of the logarithm", "statement": "The principal value of Log zeta, written log zeta, equals log rho plus i phi, with imaginary part between -pi and pi.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "\\log \\zeta", "meaning": "principal value of the general logarithm of zeta" }, { "unit": null, "symbol": "rho", "meaning": "modulus of zeta" }, { "unit": null, "symbol": "phi", "meaning": "principal amplitude of zeta" } ], "sympy": "Eq(log_zeta, log(rho) + I*phi)", "physics": false, "states": [ "theorem/principal-value-of-the-logarithm" ], "concepts": [ "concept/logarithm", "concept/principal-value-of-a-logarithm", "concept/principal-value-of-amplitude" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-96cede1da4", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "401", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\log |\\zeta| + i\\am \\zeta = \\log \\rho + i(2k\\pi + \\phi)", "name": null, "statement": "The general value of Log zeta is log of the modulus of zeta plus i times the general amplitude, where k is any integer fixed by the path.", "kind": "result", "symbols": [ { "unit": null, "symbol": "|\\zeta|", "meaning": "modulus of zeta (equal to rho)" }, { "unit": "radian", "symbol": "\\am \\zeta", "meaning": "general amplitude of zeta (2k pi + phi)" }, { "unit": null, "symbol": "rho", "meaning": "modulus of zeta" }, { "unit": "radian", "symbol": "phi", "meaning": "principal amplitude of zeta" }, { "unit": null, "symbol": "k", "meaning": "integer determined by the path of integration (winding number)" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/integer", "concept/logarithm", "concept/principal-value-of-a-logarithm", "quantity/amplitude-of-a-complex-number", "quantity/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-0db684eb56", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "402", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\Log z_{1} z_{2} = \\Log z_{1} + \\Log z_{2}", "name": null, "statement": "The general logarithm of a product equals the sum of the general logarithms; every value of each side is a value of the other, so the equation is completely true.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "z_{1}", "meaning": "complex number" }, { "unit": null, "symbol": "z_{2}", "meaning": "complex number" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/complete-equation", "concept/functional-equation", "concept/logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-8e827e52a7", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "402", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\log z_{1}z_{2} = \\log z_{1} + \\log z_{2}", "name": null, "statement": "The principal-value form of the logarithm of a product is not true in all circumstances; the book gives z1 = z2 = (-1 + i sqrt 3)/2 as a counterexample.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "z_{1}", "meaning": "complex number" }, { "unit": null, "symbol": "z_{2}", "meaning": "complex number" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/functional-equation", "concept/logarithm", "concept/principal-value-of-a-logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c9403a896c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "402", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\Log z^{m} = m\\Log z", "name": null, "statement": "For integer m, the general logarithm of z to the m equals m times the general logarithm of z; this is not completely true, since only one direction holds for all values.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "m", "meaning": "integer" }, { "unit": null, "symbol": "z", "meaning": "complex variable" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/complete-equation", "concept/integer", "concept/logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-808f2a413b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "402", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\Log (1/z) = -\\Log z", "name": null, "statement": "The general logarithm of the reciprocal of z is minus the general logarithm of z; this equation is completely true.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "z", "meaning": "complex variable" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/complete-equation", "concept/logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-4a15684938", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "407", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\Log e^{\\zeta} = (1 + 2m\\pi i)\\zeta + 2n\\pi i", "name": null, "statement": "The general logarithm of e to the zeta has values (1 + 2m pi i) zeta + 2n pi i, for integers m and n.", "kind": "result", "symbols": [ { "unit": null, "symbol": "zeta", "meaning": "complex number" }, { "unit": null, "symbol": "m", "meaning": "integer" }, { "unit": null, "symbol": "n", "meaning": "integer" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/exponential-function", "concept/integer", "concept/logarithm", "quantity/pi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e9b0e2db1d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "403", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "z = \\exp \\zeta", "name": "exponential function", "statement": "z is defined as the exponential of zeta when some value of Log z equals zeta.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "z", "meaning": "complex variable (value of the exponential)" }, { "unit": null, "symbol": "zeta", "meaning": "complex number (argument of the function)" } ], "sympy": "Eq(z, exp(zeta))", "physics": false, "states": [ "concept/exponential-function" ], "concepts": [ "concept/argument-of-a-function", "concept/inverse-function", "concept/logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-4e4c372aed", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "404", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\exp \\zeta = e^{\\zeta}", "name": null, "statement": "When zeta is real, the complex exponential exp zeta equals the real exponential e to the zeta.", "kind": "result", "symbols": [ { "unit": null, "symbol": "zeta", "meaning": "real number" }, { "unit": null, "symbol": "e", "meaning": "base of the natural logarithm" } ], "sympy": "Eq(exp(zeta), E**zeta)", "physics": false, "states": [], "concepts": [ "concept/exponential-function", "concept/logarithm", "concept/real-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b725935cc4", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "404", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\exp (\\xi + i\\eta) = e^{\\xi} (\\cos\\eta + i\\sin\\eta)", "name": null, "statement": "The exponential of xi plus i eta has modulus e to the xi and amplitude eta.", "kind": "result", "symbols": [ { "unit": null, "symbol": "xi", "meaning": "real part of zeta" }, { "unit": "radian", "symbol": "eta", "meaning": "imaginary part of zeta (amplitude of the result)" } ], "sympy": "Eq(exp(xi + I*eta), exp(xi)*(cos(eta) + I*sin(eta)))", "physics": false, "states": [], "concepts": [ "concept/circular-function", "concept/exponential-function", "quantity/amplitude-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ee592e04bf", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "404", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "f(\\zeta_{1} + \\zeta_{2}) = f(\\zeta_{1}) f(\\zeta_{2})", "name": "functional relation of the exponential", "statement": "The exponential function satisfies the functional relation f(zeta1 + zeta2) = f(zeta1) f(zeta2) for complex arguments.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "f", "meaning": "the exponential function" }, { "unit": null, "symbol": "\\zeta_{1}", "meaning": "complex number" }, { "unit": null, "symbol": "\\zeta_{2}", "meaning": "complex number" } ], "sympy": "Eq(f(zeta1 + zeta2), f(zeta1)*f(zeta2))", "physics": false, "states": [ "theorem/functional-relation-of-the-exponential" ], "concepts": [ "concept/exponential-function", "concept/functional-equation", "concept/functional-relation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d1b0d3cc98", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "405", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "a^{\\zeta} = e^{\\zeta\\log a}", "name": null, "statement": "For positive a and real zeta, the general power a to the zeta equals e to the zeta log a, the definition used in the earlier chapter.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a", "meaning": "positive real base" }, { "unit": null, "symbol": "zeta", "meaning": "real exponent" } ], "sympy": "Eq(a**zeta, exp(zeta*log(a)))", "physics": false, "states": [], "concepts": [ "concept/exponential-function", "concept/general-power", "concept/logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d853ad2904", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "405", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "a^{\\zeta} = \\exp (\\zeta\\Log a)", "name": null, "statement": "The general power a to the zeta is defined as exp of zeta times any value of the logarithm of a, for any nonzero complex a and zeta.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a", "meaning": "complex number, not zero" }, { "unit": null, "symbol": "zeta", "meaning": "complex exponent" }, { "unit": null, "symbol": "\\Log a", "meaning": "any value of the logarithm of a" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/argument-of-a-function", "concept/exponential-function", "concept/general-power", "concept/logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-66e4f1cbc8", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "406", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "|a^{\\zeta}| = e^{\\xi\\log \\sigma - \\eta(\\psi+2m\\pi)}", "name": null, "statement": "The modulus of a general power a to the zeta depends on the integer m unless eta is zero, so the general power has infinitely many values.", "kind": "result", "symbols": [ { "unit": null, "symbol": "sigma", "meaning": "modulus of a" }, { "unit": "radian", "symbol": "psi", "meaning": "principal amplitude of a" }, { "unit": null, "symbol": "xi", "meaning": "real part of zeta" }, { "unit": null, "symbol": "eta", "meaning": "imaginary part of zeta" }, { "unit": null, "symbol": "m", "meaning": "integer labelling the branch of Log a" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/general-power", "quantity/amplitude-of-a-complex-number", "quantity/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-8a98e6d22d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "407", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "e^{\\zeta} = e^{\\xi-2m\\pi\\eta} \\{\\cos(\\eta + 2m\\pi\\xi) + i\\sin(\\eta + 2m\\pi\\xi)\\}", "name": null, "statement": "The general value of e to the zeta, with integer m, is given in terms of xi and eta; its principal value is exp zeta.", "kind": "result", "symbols": [ { "unit": null, "symbol": "xi", "meaning": "real part of zeta" }, { "unit": "radian", "symbol": "eta", "meaning": "imaginary part of zeta" }, { "unit": null, "symbol": "m", "meaning": "integer" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/exponential-function", "concept/general-power", "concept/integer" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-6d645de494", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "408", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\zeta = \\Log_{e} z = \\frac{\\log |z| + (\\am z + 2m\\pi)i}{1 + 2n\\pi i}", "name": null, "statement": "On the second definition the logarithm to base e of z is doubly infinitely many-valued, given by this formula for integers m and n.", "kind": "result", "symbols": [ { "unit": null, "symbol": "z", "meaning": "complex variable" }, { "unit": null, "symbol": "m", "meaning": "integer" }, { "unit": null, "symbol": "n", "meaning": "integer" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/base-of-a-logarithm-system", "concept/integer", "concept/logarithm", "quantity/amplitude-of-a-complex-number", "quantity/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d29a40bc99", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "407", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "a^{\\zeta} × b^{\\zeta} = (ab)^{\\zeta}", "name": null, "statement": "The product of a to the zeta and b to the zeta equals (ab) to the zeta; this is completely true but not always true of principal values.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "complex number, not zero" }, { "unit": null, "symbol": "b", "meaning": "complex number, not zero" }, { "unit": null, "symbol": "zeta", "meaning": "complex exponent" } ], "sympy": "Eq(a**zeta*b**zeta, (a*b)**zeta)", "physics": false, "states": [], "concepts": [ "concept/complete-equation", "concept/general-power", "concept/principal-value-of-a-root" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-912bd9770e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "407", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "a^{\\zeta} × a^{\\zeta'} = a^{\\zeta+\\zeta'}", "name": null, "statement": "The product of a to the zeta and a to the zeta-prime equals a to the zeta plus zeta-prime; this is not completely true but holds for principal values.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a", "meaning": "complex number, not zero" }, { "unit": null, "symbol": "zeta", "meaning": "complex exponent" }, { "unit": null, "symbol": "zeta'", "meaning": "complex exponent" } ], "sympy": "Eq(a**zeta*a**zetaprime, a**(zeta + zetaprime))", "physics": false, "states": [], "concepts": [ "concept/complete-equation", "concept/general-power", "concept/principal-value-of-a-root" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-fb756c9603", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "402", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\pi f(x) = p\\pi + (q - p)\\Imag(\\log x)", "name": null, "statement": "The function f of the real variable x equals p for positive x and q for negative x, since Im(log x) is 0 or pi.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f(x)", "meaning": "function of the real variable x" }, { "unit": null, "symbol": "p", "meaning": "constant value of f for positive x" }, { "unit": null, "symbol": "q", "meaning": "constant value of f for negative x" }, { "unit": null, "symbol": "x", "meaning": "real variable" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/function", "concept/logarithm", "quantity/pi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-1c6f8cf66f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "409", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\exp (\\xi + i\\eta) = \\exp \\xi(\\cos\\eta + i\\sin\\eta)", "name": null, "statement": "The exponential of a complex sum equals e^xi times (cos eta + i sin eta).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "xi", "meaning": "real part of the complex argument" }, { "unit": null, "symbol": "eta", "meaning": "imaginary part of the complex argument" } ], "sympy": "Eq(exp(xi + I*eta), exp(xi)*(cos(eta) + I*sin(eta)))", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/exponential-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c3b7035d32", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "409", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\exp (i\\eta) = \\cos\\eta + i\\sin\\eta", "name": "Euler's formula", "statement": "The exponential of i times eta equals cos eta plus i sin eta.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "eta", "meaning": "real angle" } ], "sympy": "Eq(exp(I*eta), cos(eta) + I*sin(eta))", "physics": false, "states": [ "theorem/euler-s-formula" ], "concepts": [ "concept/complex-number", "concept/cosine", "concept/exponential-function", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c478b4d4c3", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "409", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\cos\\eta = \\tfrac{1}{2} \\{\\exp (i\\eta) + \\exp (-i\\eta)\\}", "name": null, "statement": "Cosine of a real angle written as the average of exp(i eta) and exp(-i eta).", "kind": "identity", "symbols": [ { "unit": null, "symbol": "eta", "meaning": "real angle" } ], "sympy": "Eq(cos(eta), Rational(1,2)*(exp(I*eta) + exp(-I*eta)))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/exponential-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-2a9420c7ae", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "409", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\sin\\eta = -\\tfrac{1}{2}i\\{\\exp (i\\eta) - \\exp (-i\\eta)\\}", "name": null, "statement": "Sine of a real angle written in terms of exp(i eta) and exp(-i eta).", "kind": "identity", "symbols": [ { "unit": null, "symbol": "eta", "meaning": "real angle" } ], "sympy": "Eq(sin(eta), -Rational(1,2)*I*(exp(I*eta) - exp(-I*eta)))", "physics": false, "states": [], "concepts": [ "concept/exponential-function", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-993e5f0a44", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "409", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\cos\\zeta = \\tfrac{1}{2} \\{\\exp (i\\zeta) + \\exp (-i\\zeta)\\}", "name": null, "statement": "Defines cos zeta for every complex zeta by this exponential expression; it agrees with the elementary cosine for real zeta.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "zeta", "meaning": "argument of the trigonometrical function, real or complex" } ], "sympy": "Eq(cos(zeta), Rational(1,2)*(exp(I*zeta) + exp(-I*zeta)))", "physics": false, "states": [], "concepts": [ "concept/complex-variable", "concept/cosine", "concept/exponential-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b6fa0df6d8", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "409", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\sin\\zeta = -\\tfrac{1}{2}i \\{\\exp (i\\zeta) - \\exp (-i\\zeta)\\}", "name": null, "statement": "Defines sin zeta for every complex zeta by this exponential expression; it agrees with the elementary sine for real zeta.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "zeta", "meaning": "argument of the trigonometrical function, real or complex" } ], "sympy": "Eq(sin(zeta), -Rational(1,2)*I*(exp(I*zeta) - exp(-I*zeta)))", "physics": false, "states": [], "concepts": [ "concept/complex-variable", "concept/exponential-function", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-9021b427be", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "409", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\tan \\zeta = \\frac{\\sin \\zeta}{\\cos \\zeta}", "name": null, "statement": "Tangent defined as sine over cosine, for complex argument.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "zeta", "meaning": "argument of the function" } ], "sympy": "Eq(tan(zeta), sin(zeta)/cos(zeta))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/sine", "concept/tangent-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b22a4f7b65", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "409", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\cot \\zeta = \\frac{\\cos \\zeta}{\\sin \\zeta}", "name": null, "statement": "Cotangent defined as cosine over sine, for complex argument.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "zeta", "meaning": "argument of the function" } ], "sympy": "Eq(cot(zeta), cos(zeta)/sin(zeta))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/cotangent", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-f1650ef6ad", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "409", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\sec \\zeta = \\frac{1}{\\cos \\zeta}", "name": null, "statement": "Secant defined as the reciprocal of cosine.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "zeta", "meaning": "argument of the function" } ], "sympy": "Eq(sec(zeta), 1/cos(zeta))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/secant" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-4265396288", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "409", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\cosec \\zeta = \\frac{1}{\\sin \\zeta}", "name": null, "statement": "Cosecant defined as the reciprocal of sine.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "zeta", "meaning": "argument of the function" } ], "sympy": "Eq(csc(zeta), 1/sin(zeta))", "physics": false, "states": [], "concepts": [ "concept/cosecant", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-2d612aa7bd", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "409", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\cos \\zeta = \\tfrac{1}{2} \\{t + (1/t)\\}", "name": null, "statement": "Cosine written in terms of t = exp(i zeta).", "kind": "identity", "symbols": [ { "unit": null, "symbol": "t", "meaning": "exp(i zeta)" }, { "unit": null, "symbol": "zeta", "meaning": "argument" } ], "sympy": "Eq(cos(zeta), Rational(1,2)*(t + 1/t))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/exponential-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-eba14f6f25", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "409", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\sin \\zeta = -\\tfrac{1}{2}i \\{t - (1/t)\\}", "name": null, "statement": "Sine written in terms of t = exp(i zeta).", "kind": "identity", "symbols": [ { "unit": null, "symbol": "t", "meaning": "exp(i zeta)" }, { "unit": null, "symbol": "zeta", "meaning": "argument" } ], "sympy": "Eq(sin(zeta), -Rational(1,2)*I*(t - 1/t))", "physics": false, "states": [], "concepts": [ "concept/exponential-function", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-8f5d09ead1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "409", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\cos^{2} \\zeta + \\sin^{2} \\zeta = \\tfrac{1}{4}[\\{t + (1/t)\\}^{2} - \\{t - (1/t)\\}^{2}] = 1", "name": null, "statement": "The sum of the squares of cosine and sine is 1 for all complex zeta.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "t", "meaning": "exp(i zeta)" }, { "unit": null, "symbol": "zeta", "meaning": "argument" } ], "sympy": "Eq(cos(zeta)**2 + sin(zeta)**2, 1)", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/sine", "concept/trigonometrical-identity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-399708c107", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "410", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\sin (\\zeta + \\zeta') = \\sin\\zeta \\cos\\zeta' + \\cos\\zeta \\sin\\zeta'", "name": null, "statement": "Sine of a sum of two complex arguments, in the same form as elementary trigonometry.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "zeta", "meaning": "first argument" }, { "unit": null, "symbol": "zeta'", "meaning": "second argument" } ], "sympy": "Eq(sin(zeta + zeta2), sin(zeta)*cos(zeta2) + cos(zeta)*sin(zeta2))", "physics": false, "states": [], "concepts": [ "concept/addition-formulae", "concept/cosine", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-de6211a0a0", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "410", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\cos(\\zeta + \\tfrac{1}{2}\\pi) = -\\sin\\zeta", "name": null, "statement": "Shifting the argument by a right angle turns cosine into minus sine.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "zeta", "meaning": "argument" } ], "sympy": "Eq(cos(zeta + pi/2), -sin(zeta))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/sine", "quantity/pi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-2dbdc7923d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "410", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\sin(\\zeta + \\tfrac{1}{2}\\pi) = \\cos\\zeta", "name": null, "statement": "Shifting the argument by a right angle turns sine into cosine.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "zeta", "meaning": "argument" } ], "sympy": "Eq(sin(zeta + pi/2), cos(zeta))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/sine", "quantity/pi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-57fbf9c8a9", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "410", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\cosh\\zeta = \\tfrac{1}{2} \\{\\exp \\zeta + \\exp (-\\zeta)\\}", "name": null, "statement": "Defines the hyperbolic cosine for all complex zeta by this exponential expression.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "zeta", "meaning": "argument, real or complex" } ], "sympy": "Eq(cosh(zeta), Rational(1,2)*(exp(zeta) + exp(-zeta)))", "physics": false, "states": [], "concepts": [ "concept/exponential-function", "concept/hyperbolic-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-dd4c6df1c5", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "410", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\sinh\\zeta = \\tfrac{1}{2} \\{\\exp \\zeta - \\exp (-\\zeta)\\}", "name": null, "statement": "Defines the hyperbolic sine for all complex zeta by this exponential expression.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "zeta", "meaning": "argument, real or complex" } ], "sympy": "Eq(sinh(zeta), Rational(1,2)*(exp(zeta) - exp(-zeta)))", "physics": false, "states": [], "concepts": [ "concept/exponential-function", "concept/hyperbolic-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c91df48771", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "410", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\cos i\\zeta = \\cosh \\zeta", "name": null, "statement": "Cosine of i zeta equals the hyperbolic cosine of zeta.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "zeta", "meaning": "argument" } ], "sympy": "Eq(cos(I*zeta), cosh(zeta))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/hyperbolic-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-f782912cfa", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "410", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\sin i\\zeta = i\\sinh \\zeta", "name": null, "statement": "Sine of i zeta equals i times the hyperbolic sine of zeta.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "zeta", "meaning": "argument" } ], "sympy": "Eq(sin(I*zeta), I*sinh(zeta))", "physics": false, "states": [], "concepts": [ "concept/hyperbolic-function", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c7b8d250ed", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "410", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\cosh 2\\zeta = \\cosh^{2} \\zeta + \\sinh^{2} \\zeta", "name": null, "statement": "Double-argument identity for the hyperbolic cosine, obtained by transforming the cosine double-angle formula.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "zeta", "meaning": "argument" } ], "sympy": "Eq(cosh(2*zeta), cosh(zeta)**2 + sinh(zeta)**2)", "physics": false, "states": [], "concepts": [ "concept/hyperbolic-function", "concept/trigonometrical-identity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-0c0df7759e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "411", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\zeta = 2k\\pi ± \\arccos a", "name": null, "statement": "Solutions of cos zeta = a for real a with -1 <= a <= 1 (real branch).", "kind": "result", "symbols": [ { "unit": null, "symbol": "k", "meaning": "integer" }, { "unit": null, "symbol": "a", "meaning": "real constant with -1 <= a <= 1" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/integer", "concept/inverse-circular-function", "concept/root-of-an-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-3b352f7d86", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "413", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\int \\frac{dx}{x^{2} + \\alpha} = \\frac{1}{\\sqrt{\\alpha}} \\arctan \\frac{x}{\\sqrt{\\alpha}}", "name": null, "statement": "Integral of 1/(x^2 + alpha) for alpha > 0 is an inverse tangent.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable of integration" }, { "unit": null, "symbol": "alpha", "meaning": "positive constant" } ], "sympy": "Eq(Integral(1/(x**2 + alpha), x), atan(x/sqrt(alpha))/sqrt(alpha))", "physics": false, "states": [], "concepts": [ "concept/integral", "concept/inverse-circular-function", "concept/inverse-tangent" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-5c2d2a2d7d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "413", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\int \\frac{dx}{x^{2} + \\alpha} = \\frac{1}{2\\sqrtp{-\\alpha}} \\log \\left|\\frac{x - \\sqrtp{-\\alpha}}{x + \\sqrtp{-\\alpha}}\\right|", "name": null, "statement": "Integral of 1/(x^2 + alpha) for alpha < 0 is a logarithm.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable of integration" }, { "unit": null, "symbol": "alpha", "meaning": "negative constant" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/integral", "concept/logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a12ab9760f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "414", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\arctan \\left(\\frac{x}{\\alpha}\\right) = \\frac{1}{2i} \\log\\left(\\frac{x - i\\alpha}{x + i\\alpha}\\right) + C", "name": null, "statement": "The formula the book proposes by analogy (i alpha written for alpha), with C a constant; the book then tests whether it holds.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "x", "meaning": "real variable" }, { "unit": null, "symbol": "alpha", "meaning": "real constant" }, { "unit": null, "symbol": "C", "meaning": "constant of integration" } ], "sympy": "Eq(atan(x/alpha), log((x - I*alpha)/(x + I*alpha))/(2*I) + C)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/inverse-circular-function", "concept/logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a94084b540", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "414", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\arctan x = \\frac{1}{2i} \\Log\\left(\\frac{1 + ix}{1 - ix}\\right)", "name": null, "statement": "Standard connection between the inverse tangent and the principal logarithm, true for real x; checked by putting x = tan y.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "real number" } ], "sympy": "Eq(atan(x), log((1 + I*x)/(1 - I*x))/(2*I))", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/inverse-circular-function", "concept/logarithm", "concept/principal-value-of-a-logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-93d72b1554", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "414", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\exp z = 1 + z + \\frac{z^{2}}{2!} + \\dots", "name": null, "statement": "The exponential function equals its power series for all complex z.", "kind": "result", "symbols": [ { "unit": null, "symbol": "z", "meaning": "complex variable" } ], "sympy": "Eq(exp(z), Sum(z**n/factorial(n), (n, 0, oo)))", "physics": false, "states": [], "concepts": [ "concept/absolute-convergence", "concept/convergent-series", "concept/exponential-function", "concept/power-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-06ee5ba57a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "415", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "F(z) F(h) = F(z + h)", "name": null, "statement": "The series sum F satisfies the functional equation of the exponential.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "F(z)", "meaning": "sum of the exponential power series" }, { "unit": null, "symbol": "z", "meaning": "complex variable" }, { "unit": null, "symbol": "h", "meaning": "complex increment" } ], "sympy": "Eq(F(z)*F(h), F(z + h))", "physics": false, "states": [], "concepts": [ "concept/exponential-function", "concept/functional-equation", "concept/power-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-2da0fde5dd", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "415", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "f'(y) = \\lim_{k \\to 0} \\frac{f(y + k) - f(y)}{k} = if(y)", "name": null, "statement": "Differential equation satisfied by f(y) = F(iy): its derivative is i times f.", "kind": "result", "symbols": [ { "unit": null, "symbol": "f(y)", "meaning": "F(iy), with y real" }, { "unit": null, "symbol": "y", "meaning": "real variable" }, { "unit": null, "symbol": "k", "meaning": "small real increment" } ], "sympy": "Eq(Derivative(f(y), y), I*f(y))", "physics": false, "states": [], "concepts": [ "concept/derivative", "concept/exponential-function", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e058ba38ee", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "415", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "f(y) = \\cos Y + i \\sin Y", "name": null, "statement": "f(y) has modulus 1 and so can be written as cos Y + i sin Y for some angle function Y.", "kind": "result", "symbols": [ { "unit": null, "symbol": "Y", "meaning": "real angle, a function of y, with -pi < Y <= pi" } ], "sympy": "Eq(f(y), cos(Y) + I*sin(Y))", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/cosine", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-4bc326b487", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "416", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "F(iy) = \\cos y + i\\sin y", "name": null, "statement": "The exponential series at a pure imaginary argument gives cos y + i sin y for all real y.", "kind": "result", "symbols": [ { "unit": null, "symbol": "y", "meaning": "real number" } ], "sympy": "Eq(F(I*y), cos(y) + I*sin(y))", "physics": false, "states": [], "concepts": [ "concept/complex-variable", "concept/cosine", "concept/exponential-function", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-82757cef79", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "416", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "F(x + iy) = F(x) F(iy) = \\exp x(\\cos y + i\\sin y) = \\exp(x + iy)", "name": null, "statement": "The exponential of a complex number x + iy is exp x times (cos y + i sin y).", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "real part" }, { "unit": null, "symbol": "y", "meaning": "imaginary part" } ], "sympy": "Eq(F(x + I*y), exp(x + I*y))", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/exponential-function", "concept/functional-equation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-426d320adc", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "416", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\cos z = 1 - \\frac{z^{2}}{2!} + \\frac{z^{4}}{4!} - \\dots", "name": null, "statement": "Power series of cosine, valid for all complex z.", "kind": "result", "symbols": [ { "unit": null, "symbol": "z", "meaning": "complex variable" } ], "sympy": "Eq(cos(z), Sum((-1)**n*z**(2*n)/factorial(2*n), (n, 0, oo)))", "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/power-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-34403013fa", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "416", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\sin z = z - \\frac{z^{3}}{3!} + \\frac{z^{5}}{5!} - \\dots", "name": null, "statement": "Power series of sine, valid for all complex z.", "kind": "result", "symbols": [ { "unit": null, "symbol": "z", "meaning": "complex variable" } ], "sympy": "Eq(sin(z), Sum((-1)**n*z**(2*n+1)/factorial(2*n+1), (n, 0, oo)))", "physics": false, "states": [], "concepts": [ "concept/power-series", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-9ffde08d95", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "417", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\log(1 + z) = z - \\tfrac{1}{2} z^{2} + \\tfrac{1}{3} z^{3} - \\dots", "name": null, "statement": "The principal logarithm of 1 + z equals its logarithmic series for |z| <= 1 except z = -1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "z", "meaning": "complex number with |z| <= 1, z not equal to -1" } ], "sympy": "Eq(log(1 + z), Sum((-1)**(n+1)*z**n/n, (n, 1, oo)))", "physics": false, "states": [], "concepts": [ "concept/convergent-series", "concept/logarithm", "concept/logarithmic-series", "concept/principal-value-of-a-logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-4da4219942", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "419", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\log \\left(\\frac{1}{1 - z}\\right) = -\\log(1 - z) = z + \\tfrac{1}{2} z^{2} + \\tfrac{1}{3} z^{3} + \\dots", "name": null, "statement": "Logarithmic series obtained by replacing z with -z in the previous series.", "kind": "result", "symbols": [ { "unit": null, "symbol": "z", "meaning": "complex number, |z| <= 1, z not equal to 1" } ], "sympy": "Eq(log(1/(1 - z)), Sum(z**n/n, (n, 1, oo)))", "physics": false, "states": [], "concepts": [ "concept/logarithm", "concept/logarithmic-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-75f34e8be4", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "418", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\log(1 + z) = \\int_{C} \\frac{du}{u}", "name": null, "statement": "The principal logarithm of 1 + z as an integral over the straight line C from 1 to 1 + z.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "u", "meaning": "complex variable of integration" }, { "unit": null, "symbol": "C", "meaning": "straight line from 1 to 1 + z" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/complex-variable", "concept/integral", "concept/logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-7e3caa33a7", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "420", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\arctan z = z - \\tfrac{1}{3}z^{3} + \\tfrac{1}{5}z^{5} - \\dots", "name": null, "statement": "Power series of the inverse tangent for |z| < 1, obtained from the logarithmic series.", "kind": "result", "symbols": [ { "unit": null, "symbol": "z", "meaning": "complex number, |z| < 1" } ], "sympy": "Eq(atan(z), Sum((-1)**n*z**(2*n+1)/(2*n+1), (n, 0, oo)))", "physics": false, "states": [], "concepts": [ "concept/inverse-circular-function", "concept/logarithmic-series", "concept/power-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-becab479c4", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "419", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\cos\\theta - \\tfrac{1}{2} \\cos 2\\theta + \\tfrac{1}{3} \\cos 3\\theta - \\dots = \\tfrac{1}{2} \\log(4\\cos^{2} \\tfrac{1}{2}\\theta)", "name": null, "statement": "Sum of the cosine series for log(1 + e^(i theta)) on the unit circle, valid for theta not an odd multiple of pi.", "kind": "result", "symbols": [ { "unit": null, "symbol": "theta", "meaning": "real angle" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/logarithm", "concept/logarithmic-series", "concept/periodic-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-4b6ee51cca", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "419", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\sin\\theta - \\tfrac{1}{2} \\sin 2\\theta + \\tfrac{1}{3} \\sin 3\\theta - \\dots = \\tfrac{1}{2} \\theta", "name": null, "statement": "Sum of the sine series for -pi < theta < pi; the sum is a periodic, discontinuous function of theta.", "kind": "result", "symbols": [ { "unit": null, "symbol": "theta", "meaning": "real angle with -pi < theta < pi" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/discontinuous-function", "concept/logarithmic-series", "concept/periodic-function", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-31beedaa72", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "421", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\log(1 + hz) = hz - \\tfrac{1}{2}(hz)^{2} + \\tfrac{1}{3}(hz)^{3} - \\dots", "name": "logarithmic series", "statement": "For |hz| < 1, the logarithm of 1 + hz is given by its power series in hz.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "log", "meaning": "logarithm (principal value)" }, { "unit": null, "symbol": "z", "meaning": "any complex number" }, { "unit": null, "symbol": "h", "meaning": "a real number small enough that |hz| < 1" } ], "sympy": null, "physics": false, "states": [ "concept/logarithmic-series" ], "concepts": [ "concept/complex-number", "concept/convergent-series", "concept/infinite-sequence", "concept/logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c585cceb31", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "421", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\frac{\\log(1 + hz)}{h} = z + \\phi(h, z)", "name": null, "statement": "The quotient log(1 + hz)/h equals z plus a remainder term phi(h, z) that tends to zero as h tends to zero.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "h", "meaning": "a real number small enough that |hz| < 1" }, { "unit": null, "symbol": "z", "meaning": "any complex number" }, { "unit": null, "symbol": "phi(h, z)", "meaning": "remainder term, -1/2 hz^2 + 1/3 h^2 z^3 - ..." } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/limit", "concept/logarithm", "concept/remainder" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b2270fa7eb", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "421", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\lim_{h\\to 0} \\frac{\\log(1 + hz)}{h} = z", "name": null, "statement": "The limit as h tends to zero of log(1 + hz) divided by h is z.", "kind": "result", "symbols": [ { "unit": null, "symbol": "h", "meaning": "a real number tending to zero" }, { "unit": null, "symbol": "z", "meaning": "any complex number" } ], "sympy": "Eq(Limit(log(1 + h*z)/h, h, 0), z)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/limit", "concept/logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e9359426a0", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "421", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\lim_{n\\to \\infty} n\\log \\left(1 + \\frac{z}{n}\\right) = z", "name": null, "statement": "Taking h = 1/n, n times the logarithm of (1 + z/n) tends to z as n tends to infinity.", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "positive integer" }, { "unit": null, "symbol": "z", "meaning": "any complex number" } ], "sympy": "Eq(Limit(n*log(1 + z/n), n, oo), z)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/integer", "concept/limit", "concept/logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-2625fde525", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "421", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\lim_{n\\to \\infty} \\left(1 + \\frac{z}{n}\\right)^{n} = \\lim_{n\\to \\infty} \\exp\\left\\{n\\log\\left(1 + \\frac{z}{n}\\right)\\right\\} = \\exp z", "name": "exponential limit", "statement": "The limit of (1 + z/n) to the power n as n tends to infinity is the exponential exp z, a generalisation of the real case.", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "positive integer" }, { "unit": null, "symbol": "z", "meaning": "any complex number" }, { "unit": null, "symbol": "exp", "meaning": "exponential function" } ], "sympy": "Eq(Limit((1 + z/n)**n, n, oo), exp(z))", "physics": false, "states": [ "theorem/exponential-limit" ], "concepts": [ "concept/complex-number", "concept/exponential-function", "concept/limit", "concept/logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-56c058468e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "421", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\frac{d}{dt} \\{\\log(1 + tz)\\} = \\frac{z}{1 + tz}", "name": null, "statement": "The derivative with respect to t of log(1 + tz) is z divided by 1 + tz.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "t", "meaning": "real variable" }, { "unit": null, "symbol": "z", "meaning": "any complex number" } ], "sympy": "Eq(Derivative(log(1 + t*z), t), z/(1 + t*z))", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/derivative", "concept/logarithm", "concept/variable" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-0918f60e32", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "422", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "(\\psi' + i\\chi') \\exp(\\psi + i\\chi) = \\phi' \\exp\\phi", "name": null, "statement": "For a complex function phi = psi + i chi of a real variable, the derivative of exp(phi) has the same form as for real phi: phi' exp(phi).", "kind": "rule", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "complex function of t, psi + i chi" }, { "unit": null, "symbol": "psi", "meaning": "real part of phi" }, { "unit": null, "symbol": "chi", "meaning": "imaginary part of phi" }, { "unit": null, "symbol": "exp", "meaning": "exponential function" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/complex-variable", "concept/derivative", "concept/exponential-function", "concept/function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d2ced67269", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "423", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\phi^{(n)}(t) = m(m - 1) \\dots (m - n + 1)z^{n} (1 + tz)^{m-n}", "name": null, "statement": "The nth derivative of (1 + tz)^m with respect to t is m(m-1)...(m-n+1) z^n (1 + tz)^(m-n).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "the function (1 + tz)^m of t" }, { "unit": null, "symbol": "n", "meaning": "order of differentiation (positive integer)" }, { "unit": null, "symbol": "m", "meaning": "exponent, real or complex" }, { "unit": null, "symbol": "z", "meaning": "any real or complex number" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/binomial", "concept/derivative", "concept/integer", "concept/power" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c5aaff6190", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "423", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\frac{\\phi^{n}(0)}{n!} = \\binom{m}{n} z^{n}", "name": null, "statement": "The nth derivative at t = 0, divided by n factorial, gives the binomial coefficient times z^n. (The book writes phi^n(0) for the nth derivative.)", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "the function (1 + tz)^m of t" }, { "unit": null, "symbol": "n", "meaning": "non-negative integer, order of derivative" }, { "unit": null, "symbol": "m", "meaning": "exponent, real or complex" }, { "unit": null, "symbol": "z", "meaning": "any real or complex number" } ], "sympy": "Eq(phi_n_0/factorial(n), binomial(m, n)*z**n)", "physics": false, "states": [], "concepts": [ "concept/binomial", "concept/binomial-series", "concept/coefficient", "concept/derivative", "concept/integer" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-7a0455f910", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "423", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\phi(1) = \\phi(0) + \\phi'(0) + \\frac{\\phi''(0)}{2!} + \\dots + \\frac{\\phi^{(n-1)}(0)}{(n - 1)!} + R_{n}", "name": "Taylor's theorem with remainder", "statement": "The value of phi at 1 equals its Taylor polynomial of degree n - 1 at 0 plus a remainder R_n.", "kind": "result", "symbols": [ { "unit": null, "symbol": "phi", "meaning": "the function (1 + tz)^m of t" }, { "unit": null, "symbol": "R_n", "meaning": "remainder after n terms" }, { "unit": null, "symbol": "n", "meaning": "positive integer" } ], "sympy": null, "physics": false, "states": [ "theorem/taylor-s-theorem-with-remainder" ], "concepts": [ "concept/derivative", "concept/function", "concept/infinite-sequence", "concept/remainder" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-8e25ded8b4", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "423", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "R_{n} = \\frac{1}{(n - 1)!}\\int_{0}^{1} (1 - t)^{n-1} \\phi^{(n)}(t)\\, dt", "name": null, "statement": "The remainder R_n is given by an integral of the nth derivative against (1 - t)^(n-1).", "kind": "formula", "symbols": [ { "unit": null, "symbol": "R_n", "meaning": "remainder after n terms" }, { "unit": null, "symbol": "t", "meaning": "real variable of integration" }, { "unit": null, "symbol": "phi", "meaning": "the function (1 + tz)^m of t" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/definite-integral", "concept/derivative", "concept/remainder" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d4b87310a5", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "423", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "|1 + tz| = \\sqrtp{1 + 2tr\\cos\\theta + t^{2}r^{2}} \\geq 1 - tr", "name": null, "statement": "For z = r(cos theta + i sin theta) and 0 <= t <= 1, the modulus |1 + tz| is at least 1 - tr.", "kind": "result", "symbols": [ { "unit": null, "symbol": "z", "meaning": "complex variable z = r(cos theta + i sin theta)" }, { "unit": null, "symbol": "r", "meaning": "modulus of z" }, { "unit": "radian", "symbol": "theta", "meaning": "amplitude of z" }, { "unit": null, "symbol": "t", "meaning": "real variable between 0 and 1" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/inequality", "quantity/amplitude-of-a-complex-number", "quantity/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-73b6dda4b1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "422", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "(1 + z)^{m} = \\exp\\{m\\log(1 + z)\\}", "name": "binomial theorem (general form)", "statement": "For all real m and real z between -1 and 1, (1 + z)^m equals exp of m times log(1 + z); the general form extends this to complex m and z with |z| < 1, using the principal value of the logarithm.", "kind": "result", "symbols": [ { "unit": null, "symbol": "m", "meaning": "exponent, real or complex" }, { "unit": null, "symbol": "z", "meaning": "real or complex number, |z| < 1 in the general theorem" }, { "unit": null, "symbol": "exp", "meaning": "exponential function" }, { "unit": null, "symbol": "log", "meaning": "logarithm, principal value" } ], "sympy": "Eq((1 + z)**m, exp(m*log(1 + z)))", "physics": false, "states": [ "theorem/binomial-theorem-general-form" ], "concepts": [ "concept/binomial-series", "concept/exponential-function", "concept/logarithm", "concept/power", "concept/principal-value-of-logarithm", "theorem/binomial-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-412cfdb012", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "422", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\left|\\frac{a_{n+1}}{a_{n}}\\right| = \\left|\\frac{m - n}{n + 1}\\right| \\to 1", "name": null, "statement": "The ratio of successive absolute coefficients of the binomial series tends to 1, whether m is real or complex, so the series converges for |z| < 1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "a_n", "meaning": "coefficient of z^n in the binomial series, binom(m, n)" }, { "unit": null, "symbol": "m", "meaning": "exponent, real or complex" }, { "unit": null, "symbol": "n", "meaning": "non-negative integer" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/binomial-series", "concept/coefficient", "concept/convergent-series", "concept/integer", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-8954d0489f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "422", "location": "THE GENERAL THEORY OF THE LOGARITHMIC, EXPONENTIAL, AND CIRCULAR FUNCTIONS", "latex": "\\frac{d}{dt}(1 + tz)^{m} = mz(1 + tz)^{m-1}", "name": null, "statement": "Restated for the binomial series argument: derivative of (1 + tz)^m with respect to real t equals m z (1 + tz)^(m-1), each side with its principal value.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "t", "meaning": "real variable" }, { "unit": null, "symbol": "m", "meaning": "exponent, real or complex" }, { "unit": null, "symbol": "z", "meaning": "real or complex number" } ], "sympy": "Eq(Derivative((1 + t*z)**m, t), m*z*(1 + t*z)**(m - 1))", "physics": false, "states": [], "concepts": [ "concept/binomial-series", "concept/complex-number", "concept/derivative", "concept/power", "concept/principal-value-of-logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ec86ce3556", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "433", "location": "The Proof that every Equation has a Root", "latex": "Z = P(z) = \\alpha_{0} z^{n} + \\alpha_{1} z^{n-1} + \\dots + \\alpha_{n}", "name": null, "statement": "The polynomial Z = P(z) is a sum of powers of z with constant coefficients, the highest power being z^n.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "Z", "meaning": "value of the polynomial P(z), a point in the Z-plane" }, { "unit": null, "symbol": "z", "meaning": "independent complex variable, a point in the z-plane" }, { "unit": null, "symbol": "\\alpha_{k}", "meaning": "coefficient of the power z^{n-k} in the polynomial, real or complex" }, { "unit": null, "symbol": "n", "meaning": "degree of the polynomial" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/complex-number", "concept/degree", "concept/polynomial", "concept/power", "concept/variable" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ac0e59712b", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "436", "location": "The Proof that every Equation has a Root", "latex": "|Z| = |P(x + iy)|", "name": null, "statement": "The modulus of Z is the modulus of the polynomial evaluated at x + iy, and it is a positive continuous function of x and y.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "|Z|", "meaning": "modulus of the complex number Z" }, { "unit": null, "symbol": "x", "meaning": "real part of z" }, { "unit": null, "symbol": "y", "meaning": "imaginary part of z" } ], "sympy": "Eq(Abs(Z), Abs(P(x + I*y)))", "physics": false, "states": [], "concepts": [ "concept/absolute-value", "concept/continuous-function", "concept/polynomial", "quantity/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-361981b4fc", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "435", "location": "The Proof that every Equation has a Root", "latex": "P(x_{0} + iy_{0}) = 0", "name": null, "statement": "The point x_0 + i y_0 is a root of the polynomial P, so P has a root in the complex plane.", "kind": "result", "symbols": [ { "unit": null, "symbol": "P", "meaning": "the polynomial in z with complex coefficients" }, { "unit": null, "symbol": "x_{0}", "meaning": "real part of the point where the minimum of |P| is attained (the root)" }, { "unit": null, "symbol": "y_{0}", "meaning": "imaginary part of that point" } ], "sympy": "Eq(P(x0 + I*y0), 0)", "physics": false, "states": [], "concepts": [ "concept/imaginary-root", "concept/polynomial", "concept/root-of-an-equation", "theorem/fundamental-theorem-of-algebra" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-42539b078a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "435", "location": "The Proof that every Equation has a Root", "latex": "|P(x + iy) - P(x_{0} + iy_{0})| < \\tfrac{1}{2}\\rho", "name": null, "statement": "Near the point x_0 + i y_0, the value of P stays within half of rho of its value at that point, by continuity of P.", "kind": "approximation", "symbols": [ { "unit": null, "symbol": "\\rho", "meaning": "modulus |a| of the nonzero value a = P(x_0 + iy_0) in the contradiction argument" }, { "unit": null, "symbol": "a", "meaning": "the assumed nonzero value P(x_0 + iy_0)" } ], "sympy": "Lt(Abs(P(x + I*y) - P(x0 + I*y0)), rho/2)", "physics": false, "states": [], "concepts": [ "concept/continuous-function", "concept/limit", "quantity/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-2e9a31a790", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "435", "location": "The Proof that every Equation has a Root", "latex": "P(x + iy) = a + \\phi", "name": null, "statement": "Near x_0 + i y_0, P(x + iy) equals the constant a plus a small correction phi.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "a", "meaning": "the nonzero value P(x_0 + iy_0)" }, { "unit": null, "symbol": "\\phi", "meaning": "small correction term with |phi| < rho/2 inside the square" } ], "sympy": "Eq(P(x + I*y), a + phi)", "physics": false, "states": [], "concepts": [ "concept/complex-number", "concept/continuous-function", "concept/polynomial" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-94cd082b71", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "435", "location": "The Proof that every Equation has a Root", "latex": "\\delta_{m} = \\delta_{1}/2^{m-1}", "name": null, "statement": "One admissible choice of the decreasing sequence of side lengths delta_m is halving at each step.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "\\delta_{m}", "meaning": "side length of the m-th auxiliary square, decreasing to zero" }, { "unit": null, "symbol": "m", "meaning": "index of the step" } ], "sympy": "Eq(delta_m, delta_1/2**(m-1))", "physics": false, "states": [], "concepts": [ "concept/infinite-sequence", "concept/limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-c0bd5e2e79", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "437", "location": "The Proof that every Equation has a Root", "latex": "z = z_{0} + \\zeta", "name": null, "statement": "The variable z is written as the fixed point z_0 plus a displacement zeta.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "z_{0}", "meaning": "the point at which P vanishes or is minimal modulus" }, { "unit": null, "symbol": "\\zeta", "meaning": "displacement of z from z_0" } ], "sympy": "Eq(z, z0 + zeta)", "physics": false, "states": [], "concepts": [ "concept/point", "concept/polynomial", "concept/variable" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-8696acd0cb", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "437", "location": "The Proof that every Equation has a Root", "latex": "|\\zeta| = \\rho", "name": null, "statement": "The displacement zeta is taken to have modulus rho, so z moves on a circle of radius rho about z_0.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "\\rho", "meaning": "radius of the circle about z_0 on which z moves" } ], "sympy": "Eq(Abs(zeta), rho)", "physics": false, "states": [], "concepts": [ "concept/circle", "concept/radius", "quantity/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-7a49549e21", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "437", "location": "The Proof that every Equation has a Root", "latex": "P(z) = P(z_{0}) + A_{1}\\zeta + A_{2}\\zeta^{2} + \\dots + A_{n}\\zeta^{n}", "name": null, "statement": "Expanding P in powers of zeta about z_0 gives the polynomial with coefficients A_k.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "A_{k}", "meaning": "coefficient of zeta^k in the expansion of P about z_0" }, { "unit": null, "symbol": "\\zeta", "meaning": "displacement z - z_0" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/polynomial", "concept/power", "concept/variable" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-2fe6ce0428", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "437", "location": "The Proof that every Equation has a Root", "latex": "|A_{k}| = \\mu", "name": null, "statement": "The modulus of the first nonvanishing expansion coefficient A_k is named mu.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "A_{k}", "meaning": "first coefficient in the expansion of P about z_0 that does not vanish" }, { "unit": null, "symbol": "\\mu", "meaning": "modulus of A_k" } ], "sympy": "Eq(Abs(A_k), mu)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/multiple-root", "quantity/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-5d7a79bef0", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "437", "location": "The Proof that every Equation has a Root", "latex": "|A_{k+1}|\\rho + |A_{k+2}|\\rho^{2} + \\dots + |A_{n}|\\rho^{n-k} < \\tfrac{1}{2}\\mu", "name": null, "statement": "The radius rho can be chosen small enough that the higher-order terms of the expansion total less than half of mu.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "\\rho", "meaning": "radius of the small circle about z_0" }, { "unit": null, "symbol": "\\mu", "meaning": "modulus of the first nonvanishing coefficient A_k" } ], "sympy": "Lt(Abs(A_k1)*rho + Abs(A_k2)*rho**2 + Abs(An)*rho**(n-k), mu/2)", "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/limit", "quantity/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ec63881b2a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "437", "location": "The Proof that every Equation has a Root", "latex": "|P(z) - P(z_{0}) - A_{k}\\zeta^{k}| < \\tfrac{1}{2}\\mu\\rho^{k}", "name": null, "statement": "The terms of P beyond the k-th differ from zero by less than half of mu rho^k on the small circle.", "kind": "approximation", "symbols": [ { "unit": null, "symbol": "\\mu\\rho^{k}", "meaning": "modulus of A_k zeta^k on the circle of radius rho" } ], "sympy": "Lt(Abs(P(z) - P(z0) - A_k*zeta**k), mu*rho**k/2)", "physics": false, "states": [], "concepts": [ "concept/limit", "concept/power", "quantity/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d269f8b6ff", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "437", "location": "The Proof that every Equation has a Root", "latex": "|P(z)| < |P(z_{0} + A_{k}\\zeta^{k}| + \\tfrac{1}{2}\\mu\\rho^{k}", "name": null, "statement": "Bounds |P(z)| by the modulus of P(z_0) + A_k zeta^k plus half of mu rho^k. FLAG: the source has an unclosed modulus bar, |P(z_{0} + A_{k}\\zeta^{k}|, which is probably a Gutenberg transcription or printing slip for |P(z_{0}) + A_{k}\\zeta^{k}|; the intended form is inferred, not checked against the printed page.", "kind": "approximation", "symbols": [ { "unit": null, "symbol": "\\mu\\rho^{k}", "meaning": "modulus of A_k zeta^k on the circle of radius rho" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/inequality", "concept/limit", "quantity/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-6354c2b223", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "437", "location": "The Proof that every Equation has a Root", "latex": "|P(z_{0}) + A_{k}\\zeta^{k}| = |P(z_{0})| - \\mu\\rho^{k}", "name": null, "statement": "At k points on the circle, the modulus of P(z_0) + A_k zeta^k equals |P(z_0)| minus mu rho^k, because that circle passes through the origin-side point.", "kind": "result", "symbols": [ { "unit": null, "symbol": "\\mu\\rho^{k}", "meaning": "radius of the circle traced by A_k zeta^k" } ], "sympy": "Eq(Abs(P(z0) + A_k*zeta**k), Abs(P(z0)) - mu*rho**k)", "physics": false, "states": [], "concepts": [ "concept/multiple-root", "quantity/amplitude-of-a-complex-number", "quantity/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e8ec2a6e1c", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "437", "location": "The Proof that every Equation has a Root", "latex": "|P(z)| < |P(z_{0})| - \\mu\\rho^{k} + \\tfrac{1}{2}\\mu\\rho^{k}", "name": null, "statement": "Combining the estimates, |P(z)| is less than |P(z_0)| minus half of mu rho^k at some point of the circle, contradicting that m is the lower bound.", "kind": "result", "symbols": [ { "unit": null, "symbol": "m", "meaning": "least value of |P(z)| on and inside the contour" } ], "sympy": "Lt(Abs(P(z)), Abs(P(z0)) - mu*rho**k + mu*rho**k/2)", "physics": false, "states": [], "concepts": [ "concept/least-upper-bound", "quantity/modulus-of-a-complex-number", "theorem/fundamental-theorem-of-algebra" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-310b9aa22d", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "436", "location": "The Proof that every Equation has a Root", "latex": "|P(z)| \\to \\infty", "name": null, "statement": "The modulus of P(z) grows without bound as |z| grows without bound.", "kind": "result", "symbols": [ { "unit": null, "symbol": "|z|", "meaning": "modulus of the complex variable z" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/infinity", "concept/limit", "concept/polynomial", "quantity/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e4579af174", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "436", "location": "The Proof that every Equation has a Root", "latex": "Z = a_{0} z^{n} \\left(1 + \\frac{a_{1}}{a_{0}z} + \\frac{a_{2}}{a_{0} z^{2}} + \\dots + \\frac{a_{n}}{a_{0} z^{n}}\\right)", "name": null, "statement": "For large |z| the polynomial is a_0 z^n times a factor close to 1, which is the form used to find a circle on which the amplitude of Z increases by 2n pi.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "a_{k}", "meaning": "coefficient of z^{n-k} in the polynomial Z" }, { "unit": null, "symbol": "a_{0}", "meaning": "leading coefficient of the polynomial" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/polynomial", "concept/power", "quantity/amplitude-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a78410e544", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "436", "location": "The Proof that every Equation has a Root", "latex": "Z = a_{0} z^{n} (1 + \\rho)", "name": null, "statement": "On the circle of radius R, Z equals a_0 z^n times (1 + rho), with rho a small correction.", "kind": "formula", "symbols": [ { "unit": null, "symbol": "\\rho", "meaning": "small complex correction with |rho| < delta on the circle" }, { "unit": null, "symbol": "a_{0}", "meaning": "leading coefficient of the polynomial" } ], "sympy": "Eq(Z, a0*z**n*(1 + rho))", "physics": false, "states": [], "concepts": [ "concept/limit", "concept/polynomial", "quantity/amplitude-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-59c507e61e", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "436", "location": "The Proof that every Equation has a Root", "latex": "\\frac{|a_{1}|}{|a_{0}| R} + \\frac{|a_{2}|}{|a_{0}| R^{2}} + \\dots + \\frac{|a_{n}|}{|a_{0}| R^{n}} < \\delta", "name": null, "statement": "Choosing R large enough makes the sum of scaled coefficient moduli smaller than any positive delta.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "R", "meaning": "radius of the circle centred at the origin" }, { "unit": null, "symbol": "\\delta", "meaning": "any positive number, however small" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/coefficient", "concept/infinity", "concept/limit", "concept/sufficiently-large-values" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-5f1a29f57f", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "438", "location": "The Proof that every Equation has a Root", "latex": "f'(z) = f(z) \\left(\\frac{1}{z - z_{1}} + \\frac{1}{z - z_{2}} + \\frac{1}{z - z_{3}}\\right)", "name": null, "statement": "The derivative of a cubic f equals f times the sum of reciprocals of z minus each root, used as a hint in Exercise 8.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "f(z)", "meaning": "cubic polynomial with roots z_1, z_2, z_3" }, { "unit": null, "symbol": "z_{1}, z_{2}, z_{3}", "meaning": "roots of f(z) = 0" } ], "sympy": "Eq(Derivative(f(z), z), f(z)*(1/(z - z1) + 1/(z - z2) + 1/(z - z3)))", "physics": false, "states": [], "concepts": [ "concept/cubic-equation", "concept/polynomial", "concept/root-of-an-equation", "concept/variable" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e3813bf344", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "439", "location": "A Note on Double Limit Problems", "latex": "-1 < x \\leq 1", "name": null, "statement": "The range of x for which the logarithmic series holds, as proved in section 213.", "kind": "rule", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the variable in the logarithmic series" } ], "sympy": "(-1 < x) & (x <= 1)", "physics": false, "states": [], "concepts": [ "concept/infinite-sequence", "concept/real-number", "concept/variable" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-9552626931", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "439", "location": "A Note on Double Limit Problems", "latex": "\\log(1 + x) = x - \\tfrac{1}{2}x^{2} + \\tfrac{1}{3}x^{3} - \\dots", "name": null, "statement": "The logarithm of 1 + x equals an infinite alternating series in x, valid for -1 < x ≤ 1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the variable" }, { "unit": null, "symbol": "log", "meaning": "logarithm" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/infinite-sequence", "concept/logarithm", "concept/sum", "concept/term" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-a840f7a8bb", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "439", "location": "A Note on Double Limit Problems", "latex": "1/(1 + t) = 1 - t + t^{2} - \\dots", "name": null, "statement": "The function 1/(1 + t) is expanded as an infinite series in powers of t.", "kind": "result", "symbols": [ { "unit": null, "symbol": "t", "meaning": "the variable" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/function", "concept/infinite-sequence", "concept/sum", "concept/term" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-4f42c1d0ed", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "439", "location": "A Note on Double Limit Problems", "latex": "\\int_{0}^{x} \\frac{dt}{1 + t} = \\int_{0}^{x} dt - \\int_{0}^{x} t\\, dt + \\int_{0}^{x} t^{2}\\, dt - \\dots", "name": null, "statement": "Integrating the series term by term between 0 and x gives the integral of 1/(1 + t) as the sum of the integrals of its terms.", "kind": "result", "symbols": [ { "unit": null, "symbol": "t", "meaning": "the variable of integration" }, { "unit": null, "symbol": "x", "meaning": "the upper limit of integration" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/commutativity-of-operations", "concept/definite-integral", "concept/infinite-sequence", "concept/integral", "concept/limits-of-integration" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-7cb5f84a9a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "439", "location": "A Note on Double Limit Problems", "latex": "D_{x} \\left(1 + x + \\frac{x^{2}}{2!} + \\dots\\right) = D_{x}1 + D_{x}x + D_{x} \\frac{x^{2}}{2!} + \\dots", "name": null, "statement": "The derivative of the exponential series equals the series of the derivatives of its terms, so the operations of differentiation and summation commute here.", "kind": "result", "symbols": [ { "unit": null, "symbol": "D_{x}", "meaning": "differentiation with respect to x (differential coefficient)" }, { "unit": null, "symbol": "x", "meaning": "the variable" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/commutativity-of-operations", "concept/derivative", "concept/infinite-sequence", "concept/sum", "method/differentiation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-bb81891aa4", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "440", "location": "A Note on Double Limit Problems", "latex": "\\lim_{x\\to\\xi} \\left(1 + x + \\frac{x^{2}}{2!} + \\dots\\right) = 1 + \\xi + \\frac{\\xi^{2}}{2!} + \\dots = \\lim_{x\\to\\xi} 1 + \\lim_{x\\to\\xi} x + \\lim_{x\\to\\xi} \\frac{x^{2}}{2!} + \\dots", "name": null, "statement": "The limit of the exponential series as x tends to ξ equals the sum of the termwise limits, so exp x is continuous at x = ξ.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "the variable tending to ξ" }, { "unit": null, "symbol": "ξ", "meaning": "a fixed value of x" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/commutativity-of-operations", "concept/continuous-function", "concept/infinite-sequence", "concept/limit", "theorem/limit-of-a-sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-975d2fe3d7", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "440", "location": "A Note on Double Limit Problems", "latex": "\\lim_{x\\to 0} \\{\\lim_{y\\to 0} (x + y)\\} = \\lim_{x\\to 0} x = 0", "name": null, "statement": "Taking the limit in y first and then in x gives 0 for the function x + y.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "a variable tending to 0" }, { "unit": null, "symbol": "y", "meaning": "a variable tending to 0" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/commutativity-of-operations", "concept/limit", "concept/limit-operation", "concept/zero" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-e1867ae145", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "440", "location": "A Note on Double Limit Problems", "latex": "\\lim_{y\\to 0} \\{\\lim_{x\\to 0} (x + y)\\} = \\lim_{y\\to 0} y = 0", "name": null, "statement": "Taking the limits in the opposite order also gives 0 for x + y.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "a variable tending to 0" }, { "unit": null, "symbol": "y", "meaning": "a variable tending to 0" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/commutativity-of-operations", "concept/limit", "concept/limit-operation", "concept/zero" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-673073db4a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "441", "location": "A Note on Double Limit Problems", "latex": "\\lim_{x\\to 0} \\left(\\lim_{y\\to 0} \\frac{x - y}{x + y}\\right) &= \\lim_{x\\to 0} \\frac{x}{x} &&= \\lim_{x\\to 0} 1 = 1", "name": null, "statement": "Taking the limit in y first and then in x gives 1 for (x - y)/(x + y).", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "a variable tending to 0" }, { "unit": null, "symbol": "y", "meaning": "a variable tending to 0" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/commutativity-of-operations", "concept/limit", "concept/limit-operation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-8f955076f9", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "441", "location": "A Note on Double Limit Problems", "latex": "\\lim_{y\\to 0} \\left(\\lim_{x\\to 0} \\frac{x - y}{x + y}\\right) &= \\lim_{y\\to 0}\\frac{-y}{y} &&= \\lim_{y\\to 0} (-1) = -1", "name": null, "statement": "Taking the limit in x first and then in y gives -1 for (x - y)/(x + y), so the two iterated limits differ.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "a variable tending to 0" }, { "unit": null, "symbol": "y", "meaning": "a variable tending to 0" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/commutativity-of-operations", "concept/limit", "concept/limit-operation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-7ac4f47ec0", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "441", "location": "A Note on Double Limit Problems", "latex": "\\lim_{x\\to 1} \\left\\{\\sum_{1}^{\\infty} \\frac{(-1)^{n}}{n}x^{n}\\right\\} &= \\lim_{x\\to 1}\\log(1 + x) &&= \\log 2", "name": null, "statement": "Summing the series first and then letting x tend to 1 from below gives log 2.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable tending to 1 from below" }, { "unit": null, "symbol": "n", "meaning": "index of summation" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/commutativity-of-operations", "concept/infinite-sequence", "concept/limit", "concept/logarithm", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ed12986fe8", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "441", "location": "A Note on Double Limit Problems", "latex": "\\sum_{1}^{\\infty} \\left\\{\\lim_{x\\to 1} \\frac{(-1)^{n}}{n}x^{n}\\right\\} &= \\quad \\sum_{1}^{\\infty} \\frac{(-1)^{n}}{n} &&= \\log 2", "name": null, "statement": "Taking the limit of each term first and then summing also gives log 2 in this case, so the operations commute here.", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "index of summation" }, { "unit": null, "symbol": "x", "meaning": "variable tending to 1 from below" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/commutativity-of-operations", "concept/infinite-sequence", "concept/limit", "concept/logarithm", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-5612e88b46", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "441", "location": "A Note on Double Limit Problems", "latex": "\\lim_{x\\to 1} \\left\\{\\sum_{1}^{\\infty} (x^{n} - x^{n+1})\\right\\} &= \\lim_{x\\to 1} \\{(1 - x) + (x - x^{2}) + \\dots\\}", "name": null, "statement": "Summing the telescoping series first and then taking the limit as x tends to 1 is set up as an iterated limit.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable tending to 1" }, { "unit": null, "symbol": "n", "meaning": "index of summation" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/commutativity-of-operations", "concept/infinite-sequence", "concept/limit", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b99bf33261", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "441", "location": "A Note on Double Limit Problems", "latex": "\\lim_{x\\to 1} \\{(1 - x) + (x - x^{2}) + \\dots\\} = \\lim_{x\\to 1} 1 = 1", "name": null, "statement": "The telescoping series sums to 1 for x < 1, so its limit as x tends to 1 is 1.", "kind": "result", "symbols": [ { "unit": null, "symbol": "x", "meaning": "variable tending to 1" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/infinite-sequence", "concept/limit", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-5ea86a1826", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "441", "location": "A Note on Double Limit Problems", "latex": "\\sum_{1}^{\\infty} \\left\\{\\lim_{x\\to 1} (x^{n} - x^{n+1})\\right\\} &= \\sum_{1}^{\\infty} (1 - 1) = 0 + 0 + 0 + \\dots = 0", "name": null, "statement": "Taking the limit of each term first and then summing gives 0, which differs from the limit of the sum (1), so these operations do not commute.", "kind": "result", "symbols": [ { "unit": null, "symbol": "n", "meaning": "index of summation" }, { "unit": null, "symbol": "x", "meaning": "variable tending to 1" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/commutativity-of-operations", "concept/infinite-sequence", "concept/limit", "concept/sum", "concept/zero" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-628a00c9db", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "443", "location": "The circular functions", "latex": "y = y(x) = \\arctan x = \\ds\\int_{0}^{x} \\frac{dt}{1 + t^{2}}", "name": null, "statement": "Defines y as the arctangent of x, given as the integral of 1/(1+t^2) from 0 to x.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "y", "meaning": "the value of the arctangent function at x" }, { "unit": null, "symbol": "x", "meaning": "real variable" }, { "unit": null, "symbol": "t", "meaning": "variable of integration" } ], "sympy": "Eq(y, Integral(1/(1 + t**2), (t, 0, x)))", "physics": false, "states": [], "concepts": [ "concept/function", "concept/integral", "concept/inverse-circular-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-43716da505", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "443", "location": "The circular functions", "latex": "x = x(y) = \\tan y", "name": null, "statement": "Defines the tangent of y as the inverse function of the arctangent, so x equals tan y.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x", "meaning": "real variable, the inverse function of y" }, { "unit": null, "symbol": "y", "meaning": "angle variable" } ], "sympy": "Eq(x, tan(y))", "physics": false, "states": [], "concepts": [ "concept/circular-function", "concept/inverse-function", "concept/tangent-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-b02e3eacc3", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "443", "location": "The circular functions", "latex": "\\frac{1}{2}\\pi = \\ds\\int_{0}^{\\infty} \\frac{dt}{1 + t^{2}}", "name": null, "statement": "Defines pi as twice the integral of 1/(1+t^2) from 0 to infinity.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "\\pi", "meaning": "the constant pi, defined by this integral" }, { "unit": null, "symbol": "t", "meaning": "variable of integration" } ], "sympy": "Eq(pi/2, Integral(1/(1 + t**2), (t, 0, oo)))", "physics": false, "states": [], "concepts": [ "concept/infinity", "concept/integral", "quantity/pi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-40ff3920ff", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "443", "location": "The circular functions", "latex": "\\cos y = \\dfrac{1}{\\sqrt{1 + x^{2}}}", "name": null, "statement": "Defines the cosine of y in terms of x, with the square root taken positive.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "\\cos y", "meaning": "cosine of y" }, { "unit": null, "symbol": "x", "meaning": "tangent of y" } ], "sympy": "Eq(cos(y), 1/sqrt(1 + x**2))", "physics": false, "states": [], "concepts": [ "concept/circular-function", "concept/cosine", "concept/root" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ccb9ff2d51", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "443", "location": "The circular functions", "latex": "\\sin y = \\dfrac{x}{\\sqrt{1 + x^{2}}}", "name": null, "statement": "Defines the sine of y in terms of x, with the square root taken positive.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "\\sin y", "meaning": "sine of y" }, { "unit": null, "symbol": "x", "meaning": "tangent of y" } ], "sympy": "Eq(sin(y), x/sqrt(1 + x**2))", "physics": false, "states": [], "concepts": [ "concept/circular-function", "concept/root", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-8d95712e4a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "443", "location": "The circular functions", "latex": "\\tan(y + \\pi) &= &&\\tan y", "name": null, "statement": "Extends the tangent beyond the basic interval by making it periodic with period pi.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "y", "meaning": "angle variable" }, { "unit": null, "symbol": "\\pi", "meaning": "the constant pi" } ], "sympy": "Eq(tan(y + pi), tan(y))", "physics": false, "states": [], "concepts": [ "concept/circular-function", "concept/tangent-function", "quantity/pi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-d50680d562", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "443", "location": "The circular functions", "latex": "\\cos(y + \\pi) &= -&&\\cos y", "name": null, "statement": "Extends the cosine beyond the basic interval so that shifting y by pi changes its sign.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "y", "meaning": "angle variable" }, { "unit": null, "symbol": "\\pi", "meaning": "the constant pi" } ], "sympy": "Eq(cos(y + pi), -cos(y))", "physics": false, "states": [], "concepts": [ "concept/circular-function", "concept/cosine", "quantity/pi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-6979a154b1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "443", "location": "The circular functions", "latex": "\\sin(y + \\pi) &= -&&\\sin y", "name": null, "statement": "Extends the sine beyond the basic interval so that shifting y by pi changes its sign.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "y", "meaning": "angle variable" }, { "unit": null, "symbol": "\\pi", "meaning": "the constant pi" } ], "sympy": "Eq(sin(y + pi), -sin(y))", "physics": false, "states": [], "concepts": [ "concept/circular-function", "concept/sine", "quantity/pi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-f9d9ba64c3", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "444", "location": "The circular functions", "latex": "\\tan (y_{1} + y_{2}) = \\dfrac{\\tan y_{1} + \\tan y_{2}}{1 - \\tan y_{1}\\tan y_{2}}", "name": null, "statement": "The tangent of a sum of two angles equals the sum of their tangents divided by one minus their product.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "y_{1}", "meaning": "first angle" }, { "unit": null, "symbol": "y_{2}", "meaning": "second angle" } ], "sympy": "Eq(tan(y1 + y2), (tan(y1) + tan(y2))/(1 - tan(y1)*tan(y2)))", "physics": false, "states": [], "concepts": [ "concept/addition-formulae", "concept/circular-function", "concept/tangent-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-cbbc63a009", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "444", "location": "The circular functions", "latex": "\\cos(y_{1} + y_{2}) = ±(\\cos y_{1}\\cos y_{2} - \\sin y_{1}\\sin y_{2})", "name": null, "statement": "The cosine of a sum of two angles, with the sign of the right side fixed as positive by a continuity argument.", "kind": "identity", "symbols": [ { "unit": null, "symbol": "y_{1}", "meaning": "first angle" }, { "unit": null, "symbol": "y_{2}", "meaning": "second angle" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/addition-formulae", "concept/cosine", "concept/zodiacal-sign" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-2f20c83664", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-iii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "444", "location": "The circular functions", "latex": "\\cos x = 1 - \\frac{x^{2}}{2!} + \\frac{x^{4}}{4!} - \\dots", "name": null, "statement": "In the alternative infinite-series theory, cos x is defined by this power series.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "x", "meaning": "real variable" } ], "sympy": null, "physics": false, "states": [], "concepts": [ "concept/cosine", "concept/infinite-sequence", "concept/power-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-3e84a1d0ec", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "445", "location": "The infinite in analysis and geometry", "latex": "ax + by + c=0", "name": null, "statement": "In common Cartesian geometry a line is the set of points (x, y) whose coordinates satisfy this linear relation, with a and b not both zero.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a", "meaning": "coefficient in the linear relation of a line (a and b not both zero)" }, { "unit": null, "symbol": "b", "meaning": "coefficient in the linear relation of a line (a and b not both zero)" }, { "unit": null, "symbol": "c", "meaning": "constant term in the linear relation of a line" }, { "unit": null, "symbol": "x", "meaning": "first real coordinate of a point (x, y)" }, { "unit": null, "symbol": "y", "meaning": "second real coordinate of a point (x, y)" } ], "sympy": "Eq(a*x + b*y + c, 0)", "physics": false, "states": [], "concepts": [ "concept/coordinate-geometry", "concept/line", "concept/linear-relation", "concept/point", "concept/real-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-47e4ba5c86", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "445", "location": "The infinite in analysis and geometry", "latex": "ax + by + cz = 0", "name": null, "statement": "In a system of real homogeneous geometry a line is the class of points (x, y, z) satisfying this linear relation, where a, b, c are not all zero.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "a", "meaning": "coefficient in the homogeneous linear relation of a line (a, b, c not all zero)" }, { "unit": null, "symbol": "b", "meaning": "coefficient in the homogeneous linear relation of a line (a, b, c not all zero)" }, { "unit": null, "symbol": "c", "meaning": "coefficient in the homogeneous linear relation of a line (a, b, c not all zero)" }, { "unit": null, "symbol": "x", "meaning": "first constituent of a homogeneous triad defining a point" }, { "unit": null, "symbol": "y", "meaning": "second constituent of a homogeneous triad defining a point" }, { "unit": null, "symbol": "z", "meaning": "third constituent of a homogeneous triad defining a point (not all zero)" } ], "sympy": "Eq(a*x + b*y + c*z, 0)", "physics": false, "states": [], "concepts": [ "concept/homogeneous-geometry", "concept/homogeneous-linear-equations", "concept/line", "concept/linear-relation", "concept/point", "concept/triad" ] }, { "id": "hardy-course-of-pure-mathematics-1921/eq-ea427caa6a", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-iv", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "445", "location": "The infinite in analysis and geometry", "latex": "z = 0", "name": "line at infinity", "statement": "The special points of real homogeneous Cartesian geometry are those with z = 0, and the single special line z = 0 is called the line at infinity.", "kind": "definition", "symbols": [ { "unit": null, "symbol": "z", "meaning": "third constituent of a homogeneous triad (point coordinate) in real homogeneous Cartesian geometry" } ], "sympy": "Eq(z, 0)", "physics": false, "states": [ "concept/line-at-infinity" ], "concepts": [ "concept/homogeneous-geometry", "concept/infinity", "concept/line", "concept/point" ] } ], "exercise_sets": [ { "id": "hardy-course-of-pure-mathematics-1921/ex-i", "set": "I", "page": "1", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "practices": [ "concept/decimal-fraction", "concept/infinite-sequence", "concept/integer", "concept/rational-number", "concept/right-triangle", "theorem/pythagorean-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-l", "set": "L", "page": "244", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "practices": [ "concept/algebraic-function", "concept/conic", "concept/rational-function", "method/integration", "method/integration-of-algebraic-functions", "method/partial-fractions", "method/substitution" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-v", "set": "V", "page": "17", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "practices": [ "concept/absolute-value", "concept/negative-number", "concept/real-number", "method/addition", "method/subtraction" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-x", "set": "X", "page": "39", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "practices": [ "concept/constant", "concept/domain-of-definition", "concept/function", "concept/one-valued-function", "concept/variable", "law/boyle-s-law", "law/van-der-waals-law" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-ii", "set": "II", "page": "6", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "practices": [ "concept/integer", "concept/irrational-number", "concept/perfect-cube", "concept/rational-number", "concept/rational-root", "concept/root-of-an-equation", "theorem/irrationality-of-square-root-of-2", "theorem/rational-root-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-iv", "set": "IV", "page": "16", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "practices": [ "concept/absolute-value", "concept/equality", "concept/inequality", "concept/irrational-number", "concept/negative-number", "concept/positive-number", "concept/real-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-ix", "set": "IX", "page": "30", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "practices": [ "concept/infinite-sequence", "concept/integer", "concept/origin", "concept/point-of-accumulation", "concept/rational-number", "concept/set" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-li", "set": "LI", "page": "246", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "practices": [ "concept/arbitrary-constant-of-integration", "concept/cosine", "concept/polynomial", "concept/sine", "concept/transcendental-function", "method/integration", "method/integration-of-polynomials-in-cosines-and-sines-of-multiples-of-x", "method/substitution" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lv", "set": "LV", "page": "264", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "practices": [ "concept/approximation", "concept/continuity", "concept/cosine", "concept/higher-order-derivative", "concept/order-of-smallness", "concept/polynomial", "concept/root-of-an-equation", "concept/sine", "method/newton-s-method", "theorem/binomial-theorem", "theorem/mean-value-theorem", "theorem/taylor-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lx", "set": "LX", "page": "275", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "practices": [ "concept/derivative", "concept/function-of-several-variables", "concept/function-of-two-variables", "concept/inverse-circular-function", "concept/partial-derivative", "concept/polar-coordinates" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-vi", "set": "VI", "page": "19", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "practices": [ "concept/absolute-value", "concept/real-number", "concept/reciprocal", "method/division", "method/multiplication" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xc", "set": "XC", "page": "379", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "practices": [ "concept/euler-s-number", "concept/exponential-function", "concept/exponential-theorem", "concept/graph-of-a-function", "concept/hyperbolic-function", "concept/integer-part-function", "concept/limit", "concept/polynomial", "concept/power-series", "concept/sum", "theorem/exponential-series", "theorem/irrationality-of-e", "theorem/taylor-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xi", "set": "XI", "page": "46", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "practices": [ "concept/graph-of-a-function", "concept/parabola", "concept/polynomial", "concept/power", "concept/relative-rate-of-growth" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xl", "set": "XL", "page": "206", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "practices": [ "concept/derivative", "method/differentiation", "theorem/power-rule", "theorem/power-rule-for-derivatives", "theorem/product-rule", "theorem/product-rule-for-derivatives" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv", "set": "XV", "page": "53", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "practices": [ "concept/algebraic-function", "concept/circular-function", "concept/cosecant", "concept/cosine", "concept/cotangent", "concept/graph-of-a-function", "concept/inverse-circular-function", "concept/inverse-function", "concept/periodic-function", "concept/rational-function", "concept/secant", "concept/sine", "concept/tangent-function", "concept/transcendental-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xx", "set": "XX", "page": "73", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "practices": [ "concept/centre-of-gravity", "concept/collinear-points", "concept/displacement", "concept/parallelogram", "law/associative-law", "law/commutative-law", "law/distributive-law", "method/addition-of-displacements", "method/multiplication-of-displacements-by-numbers" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-iii", "set": "III", "page": "11", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "practices": [ "concept/approximation", "concept/dedekind-section", "concept/irrational-number", "concept/rational-number", "concept/root", "concept/square-root-of-2" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lii", "set": "LII", "page": "247", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "practices": [ "concept/cosine", "concept/polynomial", "concept/sine", "concept/transcendental-function", "method/integration", "method/integration-by-parts", "method/integration-of-polynomials-in-cosines-and-sines-of-multiples-of-x" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-liv", "set": "LIV", "page": "251", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "practices": [ "concept/area", "concept/ellipse", "concept/one-valued-function", "concept/parameter", "concept/polar-coordinates", "method/integration", "quantity/area", "quantity/eccentricity", "quantity/length-of-a-curve" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lix", "set": "LIX", "page": "272", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "practices": [ "concept/circle-of-curvature", "concept/conic-section", "concept/contact-of-the-nth-order", "concept/curvature", "concept/curve", "concept/line", "concept/point-of-inflexion", "concept/tangent", "quantity/radius-of-curvature" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lvi", "set": "LVI", "page": "267", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "practices": [ "concept/binomial-series", "concept/cosine", "concept/maclaurin-s-series", "concept/sine", "theorem/binomial-theorem", "theorem/lagrange-s-form-of-the-remainder", "theorem/taylor-s-series", "theorem/taylor-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxi", "set": "LXI", "page": "277", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "practices": [ "concept/derivative", "concept/function-of-two-variables", "concept/implicit-function", "concept/partial-derivative", "concept/polar-coordinates", "theorem/total-differential" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxv", "set": "LXV", "page": "293", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "practices": [ "concept/continuous-function", "concept/cosine", "concept/definite-integral", "concept/inequality", "concept/properties-of-the-definite-integral", "concept/sine", "quantity/pi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxx", "set": "LXX", "page": "319", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "practices": [ "concept/convergent-series", "concept/definite-integral", "concept/p-series", "concept/steadily-increasing-function", "concept/sum", "theorem/integral-test" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-vii", "set": "VII", "page": "20", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "practices": [ "concept/pure-quadratic-surd", "concept/quadratic-equation", "concept/quadratic-surd", "concept/rational-number", "concept/root", "method/euclidean-construction" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xci", "set": "XCI", "page": "382", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "practices": [ "concept/approximation", "concept/inverse-circular-function", "concept/inverse-hyperbolic-function", "concept/logarithm", "concept/logarithmic-series", "method/taylor-s-theorem", "quantity/pi", "theorem/inverse-tangent-series", "theorem/logarithmic-series", "theorem/taylor-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcv", "set": "XCV", "page": "411", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "practices": [ "concept/complex-number", "concept/cosecant", "concept/cosine", "concept/cotangent", "concept/exponential-function", "concept/hyperbolic-function", "concept/infinity", "concept/inverse-circular-function", "concept/logarithm", "concept/modulus-of-a-complex-number", "concept/secant", "concept/sine", "concept/solution", "concept/tangent-function", "concept/tends-to-infinity", "method/equating-real-and-imaginary-parts" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xii", "set": "XII", "page": "48", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "practices": [ "concept/domain-of-definition", "concept/graph-of-a-function", "concept/rational-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xiv", "set": "XIV", "page": "51", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "practices": [ "concept/algebraic-function", "concept/domain-of-definition", "concept/explicit-function", "concept/function", "concept/quadratic-equation", "concept/rational-function", "concept/root-of-an-equation", "method/quadratic-formula" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xix", "set": "XIX", "page": "62", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "practices": [ "concept/cartesian-coordinates", "concept/cone", "concept/contour-line", "concept/curve", "concept/cylinder", "concept/cylindrical-surface", "concept/function-of-two-variables", "concept/linear-equation", "concept/locus", "concept/plane", "concept/right-circular-cone", "concept/ruled-surface", "concept/surface", "concept/surface-of-a-polyhedron", "concept/surface-of-revolution" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xli", "set": "XLI", "page": "208", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "practices": [ "concept/coefficient", "concept/cubic-equation", "concept/derivative", "concept/discriminant", "concept/divisibility", "concept/highest-common-factor", "concept/multiple-root", "concept/polynomial", "concept/root-of-an-equation", "theorem/rolle-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlv", "set": "XLV", "page": "215", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "practices": [ "concept/binomial-coefficient", "concept/complex-number", "concept/cosine", "concept/determinant", "concept/higher-order-derivative", "concept/inverse-circular-function", "concept/polynomial", "concept/power", "concept/sine", "method/mathematical-induction", "method/partial-fractions", "theorem/general-leibniz-rule", "unit/degree-of-angle" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xvi", "set": "XVI", "page": "55", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "practices": [ "concept/continuous-function", "concept/denominator", "concept/discontinuous-function", "concept/function", "concept/graph-of-a-function", "concept/integer-part-function", "concept/irrational-number", "concept/prime-factor", "concept/rational-number", "concept/relatively-prime" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxi", "set": "XXI", "page": "89", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "practices": [ "concept/amplitude-of-a-complex-number", "concept/bilinear-transformation", "concept/centre-of-gravity", "concept/circle", "concept/circum-centre", "concept/coaxal-circles", "concept/complex-number", "concept/conic-section", "concept/conjugate-complex-numbers", "concept/cotangent", "concept/cross-ratio", "concept/cubic-equation", "concept/equal-roots", "concept/harmonic-points", "concept/imaginary-root", "concept/linear-equation", "concept/linear-transformation", "concept/magnification", "concept/modulus-of-a-complex-number", "concept/parabola", "concept/parallel-lines", "concept/perpendicular", "concept/quadratic-equation", "concept/rational-function", "concept/real-root", "concept/rotation", "concept/secant", "concept/tangent-function", "concept/transformation", "concept/translation", "concept/triangle", "method/equating-real-and-imaginary-parts", "method/mathematical-induction", "quantity/amplitude-of-a-complex-number", "quantity/imaginary-part-of-a-complex-number", "quantity/modulus-of-a-complex-number", "quantity/real-part-of-a-complex-number", "theorem/de-moivre-s-theorem", "theorem/triangle-inequality-for-complex-numbers" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxv", "set": "XXV", "page": "124", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "practices": [ "concept/absolute-value", "concept/finite-oscillation", "concept/function-of-a-positive-integer-variable", "concept/inequality", "concept/infinite-oscillation", "concept/integer-part", "concept/limit", "concept/oscillation", "concept/sufficiently-large-values", "concept/tends-to-infinity", "theorem/limit-of-a-shifted-sequence" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxx", "set": "XXX", "page": "145", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "practices": [ "concept/convergent-series", "concept/divergent-series", "concept/geometrical-progression", "concept/harmonic-series", "concept/limit", "concept/series-of-positive-terms", "concept/sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-liii", "set": "LIII", "page": "247", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "practices": [ "concept/cosine", "concept/logarithm", "concept/rational-function", "concept/sine", "concept/tangent-function", "concept/transcendental-function", "method/integration", "method/substitution", "method/tangent-half-angle-substitution" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lvii", "set": "LVII", "page": "268", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "practices": [ "concept/cosine", "concept/higher-order-derivative", "concept/maximum", "concept/minimum", "concept/polynomial", "concept/sine", "method/higher-derivative-test-for-maxima-and-minima" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxii", "set": "LXII", "page": "281", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "practices": [ "concept/approximation", "concept/cosine", "concept/differential", "concept/implicit-function", "concept/partial-derivative", "concept/tangent", "method/implicit-differentiation", "theorem/equation-of-the-tangent", "theorem/total-differential" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxiv", "set": "LXIV", "page": "290", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "practices": [ "concept/definite-integral", "concept/integral", "concept/limit", "concept/sum", "method/evaluation-of-a-definite-integral-as-the-limit-of-a-sum" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxix", "set": "LXIX", "page": "317", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "practices": [ "concept/convergent-series", "concept/converse-of-a-theorem", "concept/harmonic-series", "concept/infinite-sequence", "concept/limit", "concept/series-of-positive-terms", "theorem/abel-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxvi", "set": "LXVI", "page": "295", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "practices": [ "concept/definite-integral", "concept/inverse-circular-function", "method/integration-by-parts", "method/substitution", "quantity/pi", "theorem/bonnet-s-form-of-the-second-mean-value-theorem", "theorem/second-mean-value-theorem-for-integrals" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxi", "set": "LXXI", "page": "320", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "practices": [ "concept/convergent-series", "concept/divergent-series", "concept/limit", "concept/p-series", "theorem/comparison-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxv", "set": "LXXV", "page": "328", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "practices": [ "concept/infinite-integral", "method/integration-by-parts" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxx", "set": "LXXX", "page": "347", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "practices": [ "concept/absolute-convergence", "concept/binomial-series", "concept/circle-of-convergence", "concept/complex-number", "concept/convergent-series", "concept/divergent-series", "concept/geometrical-progression", "concept/logarithmic-series", "concept/oscillation", "concept/power-series", "concept/radius-of-convergence", "theorem/cauchy-s-root-test", "theorem/d-alembert-s-ratio-test" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-viii", "set": "VIII", "page": "22", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "practices": [ "concept/mixed-quadratic-surd", "concept/polynomial", "concept/pure-quadratic-surd", "concept/quadratic-equation", "concept/quadratic-surd", "concept/rational-expression", "concept/rational-number", "concept/root", "concept/root-of-an-equation", "concept/similar-surds" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcii", "set": "XCII", "page": "385", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "practices": [ "concept/approximation", "concept/binomial-coefficient", "concept/binomial-series", "concept/quadratic-surd", "concept/root", "concept/surd", "method/approximation-of-surds-by-the-binomial-series", "theorem/binomial-series", "theorem/binomial-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xciv", "set": "XCIV", "page": "407", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "practices": [ "concept/amplitude-of-a-complex-number", "concept/argand-diagram", "concept/base-of-a-logarithm-system", "concept/complete-equation", "concept/complex-number", "concept/equiangular-spiral", "concept/exponential-function", "concept/functional-equation", "concept/general-power", "concept/logarithm", "concept/logarithm-to-any-base", "concept/principal-value-of-a-logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcvi", "set": "XCVI", "page": "416", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "practices": [ "concept/cosine", "concept/exponential-function", "concept/hyperbolic-function", "concept/power-series", "concept/sine", "theorem/binomial-theorem", "theorem/power-series-for-the-exponential-function", "theorem/power-series-for-the-sine-and-cosine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xiii", "set": "XIII", "page": "50", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "practices": [ "concept/domain-of-definition", "concept/explicit-function", "concept/graph-of-a-function", "concept/one-valued-function" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlii", "set": "XLII", "page": "210", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "practices": [ "concept/denominator", "concept/derivative", "concept/divisibility", "concept/factor", "concept/lowest-terms", "concept/rational-function", "method/differentiation" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xliv", "set": "XLIV", "page": "212", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "practices": [ "concept/circle", "concept/complex-number", "concept/cosine", "concept/cotangent", "concept/discontinuous-derivative", "concept/discontinuous-function", "concept/ellipse", "concept/hyperbola", "concept/inverse-circular-function", "concept/normal", "concept/real-root", "concept/root-of-an-equation", "concept/secant", "concept/sine", "concept/tangent", "concept/tangent-function", "method/differentiation", "theorem/chain-rule", "theorem/chain-rule", "theorem/derivative-of-inverse-circular-function", "theorem/derivative-of-trigonometric-functions" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlix", "set": "XLIX", "page": "240", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "practices": [ "concept/algebraic-function", "concept/derivative", "concept/higher-order-derivative", "concept/integral", "concept/inverse-circular-function", "concept/logarithm", "concept/rational-function", "method/integration-by-parts", "method/integration-of-algebraic-functions", "method/substitution" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "set": "XLVI", "page": "222", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "practices": [ "concept/cosecant", "concept/cosine", "concept/cotangent", "concept/derivative", "concept/ellipse", "concept/hypotenuse", "concept/maximum", "concept/minimum", "concept/polynomial", "concept/power", "concept/rational-function", "concept/right-triangle", "concept/secant", "concept/sign-of-the-derivative", "concept/sine", "concept/steadily-increasing-function", "concept/tangent", "concept/tangent-function", "method/second-derivative-test", "theorem/increasing-function-criterion", "theorem/rolle-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xvii", "set": "XVII", "page": "58", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "practices": [ "concept/graph-of-a-function", "concept/quadratic-equation", "concept/root-of-an-equation", "concept/sine", "method/graphical-solution-of-an-equation", "method/solving-an-equation-graphically", "quantity/pi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxii", "set": "XXII", "page": "99", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "practices": [ "concept/complex-number", "concept/cube-root-of-unity", "concept/factor", "concept/factored-form", "concept/principal-value-of-a-root", "concept/quadratic-surd", "concept/root-of-a-complex-number", "concept/root-of-an-equation", "concept/root-of-unity", "theorem/de-moivre-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiv", "set": "XXIV", "page": "122", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "practices": [ "concept/cosine", "concept/factorial", "concept/finite-oscillation", "concept/function-of-a-positive-integer-variable", "concept/infinite-oscillation", "concept/infinity", "concept/integer-part", "concept/irrational-number", "concept/limit", "concept/oscillation", "concept/prime-factor", "concept/rational-number", "concept/sine", "concept/tends-to-infinity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxix", "set": "XXIX", "page": "143", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "practices": [ "concept/decimal", "concept/denary-scale-of-notation", "concept/divisibility", "concept/geometrical-progression", "concept/irrational-number", "concept/prime-number", "concept/proper-algebraic-fraction", "concept/rational-number", "concept/recurring-decimal", "concept/relatively-prime" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxvi", "set": "XXVI", "page": "131", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "practices": [ "concept/function", "concept/function-of-a-positive-integer-variable", "concept/infinite-sequence", "concept/infinity", "concept/limit", "concept/oscillation", "concept/rational-function", "concept/tends-to-infinity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxi", "set": "XXXI", "page": "148", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "practices": [ "concept/cosine", "concept/domain-of-definition", "concept/function", "concept/function-of-a-positive-integer-variable", "concept/inverse-circular-function", "concept/limit", "concept/representation-of-a-function-by-a-limit", "concept/sine", "concept/value-of-a-function", "concept/variable", "quantity/pi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxv", "set": "XXXV", "page": "168", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "practices": [ "concept/limit", "concept/one-sided-limit", "concept/polynomial", "concept/rational-function", "concept/tends-to-infinity", "theorem/limit-of-a-product", "theorem/limit-of-a-quotient" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-app-i", "set": "App-I", "page": "437", "chapter": "hardy-course-of-pure-mathematics-1921/ch-appendix-i", "practices": [ "concept/absolute-value", "concept/amplitude-of-a-complex-number", "concept/closed-contour", "concept/complex-number", "concept/cubic-equation", "concept/ellipse", "concept/focus", "concept/increment", "concept/logarithm", "concept/modulus-of-a-complex-number", "concept/multiple-root", "concept/polynomial", "concept/root-of-an-equation", "theorem/argument-principle" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lviii", "set": "LVIII", "page": "270", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "practices": [ "concept/cosine", "concept/indeterminate-form", "concept/infinity", "concept/limit", "concept/sine", "method/evaluating-limits-by-derivatives", "quantity/pi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxiii", "set": "LXIII", "page": "289", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "practices": [ "concept/circular-function", "concept/cosine", "concept/definite-integral", "concept/indefinite-integral", "concept/integral", "concept/inverse-circular-function", "concept/sine", "quantity/area", "quantity/pi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxvii", "set": "LXVII", "page": "311", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "practices": [ "concept/convergent-series", "concept/divergent-series", "concept/geometrical-progression", "concept/harmonic-series", "concept/limit", "concept/p-series", "concept/series-of-positive-terms", "concept/sufficiently-large-values", "concept/tends-to-infinity", "theorem/cauchy-s-root-test", "theorem/comparison-theorem", "theorem/d-alembert-s-ratio-test" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxii", "set": "LXXII", "page": "321", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "practices": [ "concept/convergent-series", "theorem/abel-s-theorem", "theorem/cauchy-s-condensation-test" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxiv", "set": "LXXIV", "page": "327", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "practices": [ "concept/infinite-integral", "concept/limit", "method/substitution", "theorem/integral-test" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxix", "set": "LXXIX", "page": "343", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "practices": [ "concept/absolute-convergence", "concept/conditionally-convergent-series", "concept/convergent-series", "concept/oscillation", "theorem/abel-s-test", "theorem/dirichlet-s-test", "theorem/general-principle-of-convergence" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxvi", "set": "LXXVI", "page": "331", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "practices": [ "concept/infinite-integral", "concept/infinite-integral-of-the-second-kind", "method/integration-by-parts", "method/substitution", "theorem/comparison-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxi", "set": "LXXXI", "page": "349", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "practices": [ "concept/absolute-convergence", "concept/binomial-series", "concept/convergent-series", "concept/divergent-series", "concept/geometrical-progression", "concept/power-series", "concept/product-of-series", "theorem/binomial-theorem", "theorem/binomial-theorem-for-a-negative-integral-exponent" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxv", "set": "LXXXV", "page": "366", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "practices": [ "concept/derivative", "concept/exponential-function", "concept/functional-equation", "method/differentiation", "theorem/mean-value-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xciii", "set": "XCIII", "page": "401", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "practices": [ "concept/amplitude-of-a-complex-number", "concept/complete-equation", "concept/complex-number", "concept/function", "concept/functional-equation", "concept/graph-of-a-function", "concept/logarithm", "concept/many-valued-function", "concept/principal-value-of-a-logarithm", "quantity/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcvii", "set": "XCVII", "page": "420", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "practices": [ "concept/cotangent", "concept/inverse-circular-function", "concept/logarithm", "concept/logarithmic-series", "concept/principal-value-of-a-logarithm", "method/equating-real-and-imaginary-parts", "theorem/logarithmic-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xliii", "set": "XLIII", "page": "211", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "practices": [ "concept/derivative", "concept/power", "concept/rational-function", "concept/root", "method/differentiation", "method/implicit-differentiation", "theorem/chain-rule", "theorem/power-rule" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvii", "set": "XLVII", "page": "227", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "practices": [ "concept/chord", "concept/curve", "concept/derivative", "concept/limit", "theorem/mean-value-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xviii", "set": "XVIII", "page": "61", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "practices": [ "concept/curve", "concept/equation", "concept/graph-of-a-function", "concept/intersection-of-two-curves", "concept/locus", "concept/parametric-representation-of-a-curve", "concept/simultaneous-equations", "concept/standard-form" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiii", "set": "XXIII", "page": "120", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "practices": [ "concept/function-of-a-positive-integer-variable", "concept/integer", "concept/integer-part", "concept/limit", "concept/oscillation", "concept/prime-number", "concept/rational-number", "concept/sine", "concept/tends-to-infinity", "theorem/infinitude-of-primes" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxvii", "set": "XXVII", "page": "135", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "practices": [ "concept/function-of-a-positive-integer-variable", "concept/limit", "concept/steadily-increasing-function", "concept/sufficiently-large-values", "concept/tends-to-infinity", "theorem/limit-of-the-nth-root-of-a-positive-number", "theorem/limit-of-x-n", "theorem/ratio-test-for-sequences" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxii", "set": "XXXII", "page": "151", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "practices": [ "concept/bounded-function", "concept/function-of-a-positive-integer-variable", "concept/least-upper-bound", "concept/limit", "concept/limit-inferior", "concept/limit-superior", "concept/oscillation", "concept/sine", "quantity/pi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxiv", "set": "XXXIV", "page": "164", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "practices": [ "concept/function-of-a-positive-integer-variable", "concept/graph-of-a-function", "concept/integer-part-function", "concept/limit", "concept/oscillation", "concept/sine", "concept/tangent-function", "concept/tends-to-infinity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxix", "set": "XXXIX", "page": "201", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "practices": [ "concept/constant", "concept/cosine", "concept/derivative", "concept/line", "concept/normal", "concept/parabola", "concept/sine", "concept/tangent", "method/differentiation", "theorem/power-rule", "theorem/power-rule-for-derivatives" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvi", "set": "XXXVI", "page": "171", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "practices": [ "concept/cosecant", "concept/cosine", "concept/cotangent", "concept/domain-of-definition", "concept/indeterminate-form", "concept/integer-part-function", "concept/inverse-circular-function", "concept/limit", "concept/one-sided-limit", "concept/order-of-greatness", "concept/order-of-smallness", "concept/rational-function", "concept/sine", "concept/tangent-function", "theorem/limit-of-sin-x-over-x" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxviii", "set": "LXVIII", "page": "315", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "practices": [ "concept/convergent-series", "concept/divergent-series", "concept/geometrical-progression", "concept/product-of-series", "concept/rearrangement-of-a-series", "concept/series-of-positive-terms", "concept/sum", "theorem/dirichlet-s-rearrangement-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxiii", "set": "LXXIII", "page": "324", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "practices": [ "concept/definite-integral", "concept/infinite-integral", "concept/oscillation", "theorem/abel-s-theorem", "theorem/comparison-theorem", "theorem/integral-test" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxvii", "set": "LXXVII", "page": "337", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "practices": [ "concept/absolute-convergence", "concept/convergent-series", "concept/series-of-positive-terms", "theorem/general-principle-of-convergence" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxii", "set": "LXXXII", "page": "359", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "practices": [ "concept/definite-integral", "concept/inequality", "concept/limit", "concept/logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxiv", "set": "LXXXIV", "page": "362", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "practices": [ "concept/derivative", "concept/logarithm", "concept/order-of-greatness", "concept/scale-of-infinity", "concept/tends-to-infinity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxix", "set": "LXXXIX", "page": "377", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "practices": [ "concept/convergent-series", "concept/divergent-series", "concept/infinite-sequence", "concept/limit", "concept/logarithm", "concept/oscillation", "concept/series-of-positive-terms", "concept/sum", "quantity/euler-s-constant", "theorem/logarithmic-test-of-convergence" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxvi", "set": "LXXXVI", "page": "369", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "practices": [ "concept/e", "concept/euler-s-number", "concept/exponential-function", "concept/limit", "concept/logarithm", "concept/tends-to-infinity", "theorem/exponential-as-a-limit" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-i", "set": "Misc-I", "page": "31", "chapter": "hardy-course-of-pure-mathematics-1921/ch-i", "practices": [ "concept/algebraical-number", "concept/arithmetical-mean", "concept/continued-fraction", "concept/determinant", "concept/geometrical-mean", "concept/golden-section", "concept/identity", "concept/inequality", "concept/irrational-number", "concept/linear-equation", "concept/minor-arc", "concept/polynomial", "concept/quadratic-equation", "concept/rational-number", "concept/rational-root", "concept/root", "concept/root-of-an-equation", "concept/similar-surds", "concept/surd", "theorem/arithmetic-geometric-mean-inequality", "theorem/cauchy-schwarz-inequality", "theorem/inequality-of-arithmetic-and-geometric-means" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-v", "set": "Misc-V", "page": "194", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "practices": [ "concept/continuous-function", "concept/cosine", "concept/discontinuous-function", "concept/function", "concept/implicit-function", "concept/integer-part-function", "concept/inverse-circular-function", "concept/inverse-function", "concept/irrational-number", "concept/limit", "concept/order-of-smallness", "concept/polynomial", "concept/rational-function", "concept/root", "concept/secant", "concept/sine", "concept/steadily-increasing-function", "concept/sufficiently-large-values", "concept/tangent-function", "theorem/inverse-function-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "set": "Misc-X", "page": "425", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "practices": [ "concept/circular-function", "concept/complex-number", "concept/complex-variable", "concept/cosine", "concept/derivative", "concept/equiangular-spiral", "concept/exponential-function", "concept/hyperbolic-function", "concept/level-curve", "concept/limit", "concept/logarithm", "concept/logarithmic-series", "concept/mercator-s-projection", "concept/root-of-an-equation", "concept/sine", "concept/stereographic-projection", "concept/transformation", "quantity/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcviii", "set": "XCVIII", "page": "424", "chapter": "hardy-course-of-pure-mathematics-1921/ch-x", "practices": [ "concept/binomial-series", "concept/complex-number", "concept/cosine", "concept/exponential-function", "concept/inverse-circular-function", "concept/logarithm", "concept/principal-value-of-a-logarithm", "concept/sine", "concept/tangent-function", "theorem/binomial-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlviii", "set": "XLVIII", "page": "235", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "practices": [ "concept/complex-number", "concept/conic-section", "concept/conjugate-complex-numbers", "concept/inverse-circular-function", "concept/logarithm", "concept/multiple-root", "concept/rational-function", "method/integration", "method/integration-of-rational-functions", "method/partial-fractions" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxviii", "set": "XXVIII", "page": "139", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "practices": [ "concept/exponent", "concept/inequality", "concept/limit", "concept/rational-number", "theorem/power-difference-inequality" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxiii", "set": "XXXIII", "page": "157", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "practices": [ "concept/complex-number", "concept/convergent-series", "concept/cosine", "concept/function-of-a-positive-integer-variable", "concept/geometrical-progression", "concept/limit", "concept/modulus-of-a-complex-number", "concept/oscillation", "concept/polar-form-of-a-complex-number", "concept/sine", "concept/tends-to-infinity", "quantity/modulus-of-a-complex-number", "quantity/pi", "theorem/limit-of-z-n" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvii", "set": "XXXVII", "page": "176", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "practices": [ "concept/continuous-function", "concept/discontinuous-function", "concept/infinity", "concept/integer-part-function", "concept/limit", "concept/oscillatory-discontinuity", "concept/polynomial", "concept/rational-function", "concept/simple-discontinuity", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxviii", "set": "LXXVIII", "page": "340", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "practices": [ "concept/absolute-convergence", "concept/alternating-series", "concept/conditionally-convergent-series", "concept/divergent-series", "concept/oscillation", "concept/rearrangement-of-a-series", "concept/sum-to-infinity" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxiii", "set": "LXXXIII", "page": "360", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "practices": [ "concept/derivative", "concept/functional-equation", "concept/inverse-circular-function", "concept/logarithm" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxvii", "set": "LXXXVII", "page": "370", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "practices": [ "concept/approximation", "concept/definite-integral", "concept/derivative", "concept/euler-s-number", "concept/exponential-function", "concept/geometrical-progression", "concept/graph-of-a-function", "concept/hyperbolic-function", "concept/inverse-hyperbolic-function", "concept/limit", "concept/logarithm", "concept/maximum", "concept/minimum", "concept/point-of-inflexion", "concept/root-of-an-equation", "method/differentiation", "method/integration", "method/integration-by-parts" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ii", "set": "Misc-II", "page": "65", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ii", "practices": [ "concept/algebraic-function", "concept/circle", "concept/cissoid-of-diocles", "concept/cosecant", "concept/duplication-of-the-cube", "concept/even-function", "concept/function", "concept/graph-of-a-function", "concept/integer-part-function", "concept/inverse-circular-function", "concept/inverse-function", "concept/irrational-number", "concept/locus", "concept/odd-function", "concept/parabola", "concept/polynomial", "concept/quadratic-surd", "concept/rational-function", "concept/rational-number", "concept/root-of-an-equation", "concept/secant", "concept/squaring-the-circle", "method/euclidean-construction", "method/graphical-solution-of-an-equation", "method/polynomial-interpolation", "method/solving-an-equation-graphically", "quantity/pi" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-iv", "set": "Misc-IV", "page": "157", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iv", "practices": [ "concept/arithmetical-mean", "concept/conjugate-complex-numbers", "concept/convergent-series", "concept/euler-s-number", "concept/expansion", "concept/function-of-a-positive-integer-variable", "concept/geometrical-progression", "concept/graph-of-a-function", "concept/infinite-sequence", "concept/irrational-number", "concept/limit", "concept/oscillation", "concept/rational-number", "concept/real-number", "concept/root-of-an-equation", "concept/sum", "concept/tends-to-infinity", "method/square-root-iteration", "theorem/limit-of-the-arithmetic-mean-of-a-sequence" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "set": "Misc-IX", "page": "387", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "practices": [ "concept/continuous-function", "concept/definite-integral", "concept/derivative", "concept/euler-s-number", "concept/exponential-function", "concept/hyperbolic-function", "concept/integral", "concept/inverse-circular-function", "concept/inverse-hyperbolic-function", "concept/limit", "concept/logarithm", "concept/logarithmic-series", "concept/root-of-an-equation", "concept/steadily-increasing-function", "concept/tends-to-infinity", "method/integration", "method/partial-fractions", "quantity/euler-s-constant", "theorem/exponential-series", "theorem/inverse-tangent-series", "theorem/logarithmic-series" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "set": "Misc-VI", "page": "253", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vi", "practices": [ "concept/area", "concept/cardioid", "concept/circle", "concept/complex-number", "concept/conic-section", "concept/continuity", "concept/cosecant", "concept/cosine", "concept/derivative", "concept/determinant", "concept/differential-equation", "concept/ellipse", "concept/equal-roots", "concept/graph-of-a-function", "concept/integral", "concept/inverse-circular-function", "concept/limit", "concept/locus", "concept/logarithm", "concept/maximum", "concept/minimum", "concept/normal", "concept/parabola", "concept/parameter", "concept/perimeter", "concept/rational-function", "concept/real-root", "concept/root-of-an-equation", "concept/sine", "concept/tangent", "concept/tangent-function", "method/differentiation", "method/formulae-of-reduction", "method/integration", "method/integration-by-parts", "theorem/general-leibniz-rule", "theorem/generalised-mean-value-theorem", "theorem/mean-value-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxviii", "set": "XXXVIII", "page": "184", "chapter": "hardy-course-of-pure-mathematics-1921/ch-v", "practices": [ "concept/bounded-function", "concept/continuous-function", "concept/discontinuous-function", "concept/integer-part-function", "concept/least-upper-bound", "concept/rational-number", "concept/sine" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxviii", "set": "LXXXVIII", "page": "375", "chapter": "hardy-course-of-pure-mathematics-1921/ch-ix", "practices": [ "concept/convergent-series", "concept/definite-integral", "concept/divergent-series", "concept/exponential-function", "concept/limit", "concept/logarithm", "concept/series-of-positive-terms", "theorem/comparison-theorem", "theorem/integral-test", "theorem/logarithmic-test-of-convergence" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-iii", "set": "Misc-III", "page": "101", "chapter": "hardy-course-of-pure-mathematics-1921/ch-iii", "practices": [ "concept/argand-diagram", "concept/cardioid", "concept/circle", "concept/complex-number", "concept/conjugate-complex-numbers", "concept/cube-root-of-unity", "concept/ellipse", "concept/equilateral-triangle", "concept/fixed-point-of-a-transformation", "concept/harmonic-points", "concept/imaginary-straight-line", "concept/root-of-an-equation", "concept/root-of-unity", "concept/transformation", "person/ptolemy", "quantity/modulus-of-a-complex-number" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vii", "set": "Misc-VII", "page": "300", "chapter": "hardy-course-of-pure-mathematics-1921/ch-vii", "practices": [ "concept/centre-of-curvature", "concept/circle-of-curvature", "concept/contact-of-the-nth-order", "concept/definite-integral", "concept/dependent", "concept/determinant", "concept/functional-relation", "concept/higher-order-derivative", "concept/homogeneous-function", "concept/jacobian", "method/simpson-s-rule", "quantity/radius-of-curvature", "theorem/euler-s-theorem-on-homogeneous-functions", "theorem/lagrange-s-form-of-the-remainder", "theorem/schwarz-s-inequality-for-integrals", "theorem/taylor-s-theorem" ] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-viii", "set": "Misc-VIII", "page": "350", "chapter": "hardy-course-of-pure-mathematics-1921/ch-viii", "practices": [ "concept/convergent-series", "concept/cube-root-of-unity", "concept/definite-integral", "concept/divergent-series", "concept/infinite-integral", "concept/linear-difference-equation", "concept/power-series", "concept/product-of-series", "concept/rational-function", "concept/recurring-series", "concept/scale-of-relation", "method/partial-fractions", "theorem/abel-s-theorem", "theorem/binomial-theorem", "theorem/cauchy-schwarz-inequality" ] } ], "problems": [ { "id": "hardy-course-of-pure-mathematics-1921/ex-app-i/1", "set": "hardy-course-of-pure-mathematics-1921/ex-app-i", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "437", "location": "Exercise App-I, problem 1", "problem_latex": "Show that the number of roots of $f(z) = 0$ which lie within a closed\ncontour which does not pass through any root is equal to the increment of\n\\[\n\\{\\log f(z)\\}/2\\pi i\n\\]\nwhen $z$~describes the contour.", "markdown": "Show that the number of roots of $f(z) = 0$ which lie within a closed contour which does not pass through any root is equal to the increment of f(z)/2i when $z$ describes the contour.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-app-i/10", "set": "hardy-course-of-pure-mathematics-1921/ex-app-i", "number": 10, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "437", "location": "Exercise App-I, problem 10", "problem_latex": "Extend the result of Ex.~8 to equations of any degree.", "markdown": "Extend the result of Ex. 8 to equations of any degree.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-app-i/11", "set": "hardy-course-of-pure-mathematics-1921/ex-app-i", "number": 11, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "437", "location": "Exercise App-I, problem 11", "problem_latex": "If $f(z)$ and~$\\phi(z)$ are two polynomials in~$z$, and $\\gamma$~is a contour which\ndoes not pass through any root of~$f(z)$, and $|\\phi(z)| < |f(z)|$ at all points on~$\\gamma$,\nthen the numbers of the roots of the equations\n\\[\nf(z) = 0,\\quad\nf(z) + \\phi(z) = 0\n\\]\nwhich lie inside~$\\gamma$ are the same.", "markdown": "If $f(z)$ and $\\phi(z)$ are two polynomials in $z$, and $\\gamma$ is a contour which does not pass through any root of $f(z)$, and $|\\phi(z)| < |f(z)|$ at all points on $\\gamma$, then the numbers of the roots of the equations f(z) = 0,0pt minus 3ptf(z) + (z) = 0 which lie inside $\\gamma$ are the same.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-app-i/12", "set": "hardy-course-of-pure-mathematics-1921/ex-app-i", "number": 12, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "437", "location": "Exercise App-I, problem 12", "problem_latex": "Show that the equations\n\\[\ne^{z} = az,\\quad\ne^{z} = az^{2},\\quad\ne^{z} = az^{3},\n\\]\nwhere $a > e$, have respectively (i)~one positive root (ii)~one positive and one\nnegative root and (iii)~one positive and two complex roots within the circle\n$|z| = 1$.", "markdown": "Show that the equations e^z = az,0pt minus 3pte^z = az^2,0pt minus 3pte^z = az^3, where $a > e$, have respectively (i) one positive root (ii) one positive and one negative root and (iii) one positive and two complex roots within the circle $|z| = 1$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-app-i/2", "set": "hardy-course-of-pure-mathematics-1921/ex-app-i", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "437", "location": "Exercise App-I, problem 2", "problem_latex": "Show that if $R$~is any number such that\n\\[\n\\frac{|a_{1}|}{R} + \\frac{|a_{2}|}{R^{2}} + \\dots + \\frac{|a_{n}|}{R^{n}} < 1,\n\\]\nthen all the roots of $z^{n} + a_{1}z^{n-1} + \\dots + a_{n} = 0$ are in absolute value less than~$R$.\nIn particular show that all the roots of $z^{5} - 13z -7 = 0$ are in absolute\nvalue less than~$2\\frac{1}{67}$.", "markdown": "Show that if $R$ is any number such that |a_1|R + |a_2|R^2 + …+ |a_n|R^n < 1, then all the roots of $z^{n} + a_{1}z^{n-1} + \\dots + a_{n} = 0$ are in absolute value less than $R$. In particular show that all the roots of $z^{5} - 13z -7 = 0$ are in absolute value less than $2\\frac{1}{67}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-app-i/3", "set": "hardy-course-of-pure-mathematics-1921/ex-app-i", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "437", "location": "Exercise App-I, problem 3", "problem_latex": "Determine the numbers of the roots of the equation $z^{2p} + az + b = 0$\nwhere $a$~and~$b$ are real and $p$~odd, which have their real parts positive and\nnegative. Show that if $a > 0$, $b > 0$ then the numbers are $p - 1$ and $p + 1$; if\n$a < 0$, $b > 0$ they are $p + 1$ and $p - 1$; and if $b < 0$ they are $p$~and~$p$. Discuss\nthe particular cases in which $a = 0$ or $b = 0$. Verify the results when $p = 1$.\n\n[Trace the variation of $\\am(z^{2p} + az + b)$ as $z$~describes the contour formed\nby a large semicircle whose centre is the origin and whose radius is~$R$, and\nthe part of the imaginary axis intercepted by the semicircle.]", "markdown": "Determine the numbers of the roots of the equation $z^{2p} + az + b = 0$ where $a$ and $b$ are real and $p$ odd, which have their real parts positive and negative. Show that if $a > 0$, $b > 0$ then the numbers are $p - 1$ and $p + 1$; if $a < 0$, $b > 0$ they are $p + 1$ and $p - 1$; and if $b < 0$ they are $p$ and $p$. Discuss the particular cases in which $a = 0$ or $b = 0$. Verify the results when $p = 1$. [Trace the variation of $\\am(z^{2p} + az + b)$ as $z$ describes the contour formed by a large semicircle whose centre is the origin and whose radius is $R$, and the part of the imaginary axis intercepted by the semicircle.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-app-i/4", "set": "hardy-course-of-pure-mathematics-1921/ex-app-i", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "437", "location": "Exercise App-I, problem 4", "problem_latex": "Consider similarly the equations\n\\[\nz^{4q} + az + b = 0,\\quad\nz^{4q-1} + az + b = 0,\\quad\nz^{4q+1} + az + b = 0.\n\\]", "markdown": "Consider similarly the equations z^4q + az + b = 0,0pt minus 3ptz^4q-1 + az + b = 0,0pt minus 3ptz^4q+1 + az + b = 0.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-app-i/5", "set": "hardy-course-of-pure-mathematics-1921/ex-app-i", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "437", "location": "Exercise App-I, problem 5", "problem_latex": "Show that if $\\alpha$~and~$\\beta$ are real then the numbers of the roots of the\nequation $z^{2n} + \\alpha^{2} z^{2n-1} + \\beta^{2} = 0$ which have their real parts positive and\nnegative are $n - 1$ and $n + 1$, or $n$~and~$n$, according as $n$~is odd or even.", "markdown": "Show that if $\\alpha$ and $\\beta$ are real then the numbers of the roots of the equation $z^{2n} + \\alpha^{2} z^{2n-1} + \\beta^{2} = 0$ which have their real parts positive and negative are $n - 1$ and $n + 1$, or $n$ and $n$, according as $n$ is odd or even.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-app-i/6", "set": "hardy-course-of-pure-mathematics-1921/ex-app-i", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "437", "location": "Exercise App-I, problem 6", "problem_latex": "Show that when $z$~moves along the straight line joining the points\n$z = z_{1}$, $z = z_{2}$, from a point near~$z_{1}$ to a point near~$z_{2}$, the increment of\n\\[\n\\am \\left(\\frac{1}{z - z_{1}} + \\frac{1}{z - z_{2}}\\right)\n\\]\nis nearly equal to~$\\pi$.", "markdown": "Show that when $z$ moves along the straight line joining the points $z = z_{1}$, $z = z_{2}$, from a point near $z_{1}$ to a point near $z_{2}$, the increment of (1z - z_1 + 1z - z_2) is nearly equal to $\\pi$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-app-i/7", "set": "hardy-course-of-pure-mathematics-1921/ex-app-i", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "437", "location": "Exercise App-I, problem 7", "problem_latex": "A contour enclosing the three points $z = z_{1}$, $z = z_{2}$, $z = z_{3}$ is defined by\nparts of the sides of the triangle formed by $z_{1}$,~$z_{2}$,~$z_{3}$, and the parts exterior\nto the triangle of three small circles with their centres at those points.\nShow that when $z$~describes the contour the increment of\n\\[\n\\am \\left(\\frac{1}{z - z_{1}} + \\frac{1}{z - z_{2}} + \\frac{1}{z - z_{3}}\\right)\n\\]\nis equal to~$-2\\pi$.", "markdown": "A contour enclosing the three points $z = z_{1}$, $z = z_{2}$, $z = z_{3}$ is defined by parts of the sides of the triangle formed by $z_{1}$, $z_{2}$, $z_{3}$, and the parts exterior to the triangle of three small circles with their centres at those points. Show that when $z$ describes the contour the increment of (1z - z_1 + 1z - z_2 + 1z - z_3) is equal to $-2\\pi$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-app-i/8", "set": "hardy-course-of-pure-mathematics-1921/ex-app-i", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "437", "location": "Exercise App-I, problem 8", "problem_latex": "Prove that a closed oval path which surrounds all the roots of a cubic\nequation $f(z) = 0$ also surrounds those of the derived equation $f'(z) = 0$. [Use\nthe equation\n\\[\nf'(z) = f(z) \\left(\n \\frac{1}{z - z_{1}} + \\frac{1}{z - z_{2}} + \\frac{1}{z - z_{3}}\n\\right),\n\\]\nwhere $z_{1}$,~$z_{2}$,~$z_{3}$ are the roots of $f(z) = 0$, and the result of Ex.~7.]", "markdown": "Prove that a closed oval path which surrounds all the roots of a cubic equation $f(z) = 0$ also surrounds those of the derived equation $f'(z) = 0$. [Use the equation f’(z) = f(z) ( 1z - z_1 + 1z - z_2 + 1z - z_3 ), where $z_{1}$, $z_{2}$, $z_{3}$ are the roots of $f(z) = 0$, and the result of Ex. 7.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-app-i/9", "set": "hardy-course-of-pure-mathematics-1921/ex-app-i", "number": 9, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "437", "location": "Exercise App-I, problem 9", "problem_latex": "Show that the roots of $f'(z) = 0$ are the foci of the ellipse which touches\nthe sides of the triangle $(z_{1}, z_{2}, z_{3})$ at their middle points. [For a proof see\nCesàro's \\textit{Elementares Lehrbuch der algebraischen Analysis}, p.~352.]", "markdown": "Show that the roots of $f'(z) = 0$ are the foci of the ellipse which touches the sides of the triangle $(z_{1}, z_{2}, z_{3})$ at their middle points. [For a proof see Cesàro’s *Elementares Lehrbuch der algebraischen Analysis*, p. 352.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-i/1", "set": "hardy-course-of-pure-mathematics-1921/ex-i", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "1", "location": "Exercise I, problem 1", "problem_latex": "If $r$~and~$s$ are rational numbers, then $r + s$, $r - s$, $rs$, and\n$r/s$ are rational numbers, unless in the last case $s = 0$ (when $r/s$~is of course\nmeaningless).", "markdown": "If $r$ and $s$ are rational numbers, then $r + s$, $r - s$, $rs$, and $r/s$ are rational numbers, unless in the last case $s = 0$ (when $r/s$ is of course meaningless).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-i/2", "set": "hardy-course-of-pure-mathematics-1921/ex-i", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "1", "location": "Exercise I, problem 2", "problem_latex": "{\\Loosen If $\\lambda$,~$m$, and~$n$ are positive rational numbers, and $m > n$, then\n$\\lambda(m^{2} - n^{2})$, $2\\lambda mn$, and $\\lambda(m^{2} + n^{2})$ are positive rational numbers. Hence show\nhow to determine any number of right-angled triangles the lengths of all of\nwhose sides are rational.}", "markdown": "0.375em plus 0.75em minus 0.25emIf $\\lambda$, $m$, and $n$ are positive rational numbers, and $m > n$, then $\\lambda(m^{2} - n^{2})$, $2\\lambda mn$, and $\\lambda(m^{2} + n^{2})$ are positive rational numbers. Hence show how to determine any number of right-angled triangles the lengths of all of whose sides are rational.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-i/3", "set": "hardy-course-of-pure-mathematics-1921/ex-i", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "1", "location": "Exercise I, problem 3", "problem_latex": "Any terminated decimal represents a rational number whose denominator\ncontains no factors other than $2$~or~$5$. Conversely, any such rational\nnumber can be expressed, and in one way only, as a terminated decimal.", "markdown": "Any terminated decimal represents a rational number whose denominator contains no factors other than $2$ or $5$. Conversely, any such rational number can be expressed, and in one way only, as a terminated decimal.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-i/4", "set": "hardy-course-of-pure-mathematics-1921/ex-i", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "1", "location": "Exercise I, problem 4", "problem_latex": "The positive rational numbers may be arranged in the form of a simple\nseries as follows:\n\\[\n\\tfrac{1}{1},\\quad\n\\tfrac{2}{1},\\quad\n\\tfrac{1}{2},\\quad\n\\tfrac{3}{1},\\quad\n\\tfrac{2}{2},\\quad\n\\tfrac{1}{3},\\quad\n\\tfrac{4}{1},\\quad\n\\tfrac{3}{2},\\quad\n\\tfrac{2}{3},\\quad\n\\tfrac{1}{4},\\ \\dots.\n\\]\n\nShow that $p/q$ is the $[\\frac{1}{2}(p + q - 1)(p + q - 2) + q]$th term of the series.", "markdown": "The positive rational numbers may be arranged in the form of a simple series as follows: 11,0pt minus 3pt21,0pt minus 3pt12,0pt minus 3pt31,0pt minus 3pt22,0pt minus 3pt13,0pt minus 3pt41,0pt minus 3pt32,0pt minus 3pt23,0pt minus 3pt14, …. Show that $p/q$ is the $[\\frac{1}{2}(p + q - 1)(p + q - 2) + q]$th term of the series.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-ii/1", "set": "hardy-course-of-pure-mathematics-1921/ex-ii", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "6", "location": "Exercise II, problem 1", "problem_latex": "Show that no rational number can have its cube equal\n to~$2$.", "markdown": "Show that no rational number can have its cube equal to $2$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-ii/2", "set": "hardy-course-of-pure-mathematics-1921/ex-ii", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "6", "location": "Exercise II, problem 2", "problem_latex": "Prove generally that a rational fraction~$p/q$ in its lowest terms cannot\nbe the cube of a rational number unless $p$~and~$q$ are both perfect cubes.", "markdown": "Prove generally that a rational fraction $p/q$ in its lowest terms cannot be the cube of a rational number unless $p$ and $q$ are both perfect cubes.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-ii/3", "set": "hardy-course-of-pure-mathematics-1921/ex-ii", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "6", "location": "Exercise II, problem 3", "problem_latex": "A more general proposition, which is due to Gauss and includes those\nwhich precede as particular cases, is the following: \\emph{an algebraical equation\n\\[\nx^{n} + p_{1}x^{n-1} + p_{2}x^{n-2} + \\dots + p_{n} = 0,\n\\]\nwith integral coefficients, cannot have a rational but non-integral root}.\n\n[For suppose that the equation has a root~$a/b$, where $a$~and~$b$ are integers\n\\PageSep{7}\nwithout a common factor, and $b$~is positive. Writing~$a/b$ for~$x$, and multiplying\nby~$b^{n-1}$, we obtain\n\\[\n-\\frac{a^{n}}{b} = p_{1}a^{n-1} + p_{2}a^{n-2}b + \\dots + p_{n}b^{n-1},\n\\]\na fraction in its lowest terms equal to an integer, which is absurd. Thus $b = 1$,\nand the root is~$a$. It is evident that $a$~must be a divisor of~$p_{n}$.]", "markdown": "A more general proposition, which is due to Gauss and includes those which precede as particular cases, is the following: *an algebraical equation x^n + p_1x^n-1 + p_2x^n-2 + …+ p_n = 0, with integral coefficients, cannot have a rational but non-integral root*. [For suppose that the equation has a root $a/b$, where $a$ and $b$ are integers [pg]7 without a common factor, and $b$ is positive. Writing $a/b$ for $x$, and multiplying by $b^{n-1}$, we obtain -a^nb = p_1a^n-1 + p_2a^n-2b + …+ p_nb^n-1, a fraction in its lowest terms equal to an integer, which is absurd. Thus $b = 1$, and the root is $a$. It is evident that $a$ must be a divisor of $p_{n}$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-ii/4", "set": "hardy-course-of-pure-mathematics-1921/ex-ii", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "6", "location": "Exercise II, problem 4", "problem_latex": "Show that if $p_{n} = 1$ and neither of\n\\[\n1 + p_{1} + p_{2} + p_{3} + \\dots,\\quad\n1 - p_{1} + p_{2} - p_{3} + \\dots\n\\]\nis zero, then the equation cannot have a rational root.", "markdown": "Show that if $p_{n} = 1$ and neither of 1 + p_1 + p_2 + p_3 + …,0pt minus 3pt1 - p_1 + p_2 - p_3 + … is zero, then the equation cannot have a rational root.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-ii/5", "set": "hardy-course-of-pure-mathematics-1921/ex-ii", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "6", "location": "Exercise II, problem 5", "problem_latex": "Find the rational roots (if any) of\n\\[\nx^{4} - 4x^{3} - 8x^{2} + 13x + 10 = 0.\n\\]\n\n[The roots can only be integral, and so $±1$, $±2$, $±5$, $±10$ are the only\npossibilities: whether these are roots can be determined by trial. It is clear\nthat we can in this way determine the rational roots of any such equation.]", "markdown": "Find the rational roots (if any) of x^4 - 4x^3 - 8x^2 + 13x + 10 = 0. [The roots can only be integral, and so $±1$, $±2$, $±5$, $±10$ are the only possibilities: whether these are roots can be determined by trial. It is clear that we can in this way determine the rational roots of any such equation.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "solve", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "Eq(x**4 - 4*x**3 - 8*x**2 + 13*x + 10, 0)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "solve: Eq(x**4 - 4*x**3 - 8*x**2 + 13*x + 10, 0)" ], "shape": [ "solve: Eq(N*x + 2*N*x**N + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-iii/1", "set": "hardy-course-of-pure-mathematics-1921/ex-iii", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "11", "location": "Exercise III, problem 1", "problem_latex": "Find the difference between~$2$ and the squares of the\ndecimals given in \\SecNo[§]{4} as approximations to~$\\sqrt{2}$.", "markdown": "Find the difference between $2$ and the squares of the decimals given in [§]4 as approximations to $\\sqrt{2}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-iii/2", "set": "hardy-course-of-pure-mathematics-1921/ex-iii", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "11", "location": "Exercise III, problem 2", "problem_latex": "Find the differences between~$2$ and the squares of\n\\[\n\\tfrac{1}{1},\\quad\n\\tfrac{3}{2},\\quad\n\\tfrac{7}{5},\\quad\n\\tfrac{17}{12},\\quad\n\\tfrac{41}{29},\\quad\n\\tfrac{99}{70}.\n\\]", "markdown": "Find the differences between $2$ and the squares of 11,0pt minus 3pt32,0pt minus 3pt75,0pt minus 3pt1712,0pt minus 3pt4129,0pt minus 3pt9970.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.frac" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-iii/3", "set": "hardy-course-of-pure-mathematics-1921/ex-iii", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "11", "location": "Exercise III, problem 3", "problem_latex": "Show that if $m/n$ is a good approximation to~$\\sqrt{2}$, then~$(m + 2n)/(m + n)$\nis a better one, and that the errors in the two cases are in opposite directions.\nApply this result to continue the series of approximations in the last\nexample.", "markdown": "Show that if $m/n$ is a good approximation to $\\sqrt{2}$, then $(m + 2n)/(m + n)$ is a better one, and that the errors in the two cases are in opposite directions. Apply this result to continue the series of approximations in the last example.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-iii/4", "set": "hardy-course-of-pure-mathematics-1921/ex-iii", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "11", "location": "Exercise III, problem 4", "problem_latex": "If $x$~and~$y$ are approximations to~$\\sqrt{2}$, by defect and by excess respectively,\nand $2 - x^{2} < \\delta$, $y^{2} - 2 < \\delta$, then $y - x < \\delta$.", "markdown": "If $x$ and $y$ are approximations to $\\sqrt{2}$, by defect and by excess respectively, and $2 - x^{2} < \\delta$, $y^{2} - 2 < \\delta$, then $y - x < \\delta$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-iii/5", "set": "hardy-course-of-pure-mathematics-1921/ex-iii", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "11", "location": "Exercise III, problem 5", "problem_latex": "The equation $x^{2} = 4$ is satisfied by $x = 2$. Examine how far the argument\nof the preceding sections applies to this equation (writing~$4$ for~$2$\nthroughout). [If we define the classes $L$,~$R$ as before, they do not include \\emph{all}\nrational numbers. The rational number~$2$ is an exception, since~$2^{2}$ is neither\nless than or greater than~$4$.]", "markdown": "The equation $x^{2} = 4$ is satisfied by $x = 2$. Examine how far the argument of the preceding sections applies to this equation (writing $4$ for $2$ throughout). [If we define the classes $L$, $R$ as before, they do not include *all* rational numbers. The rational number $2$ is an exception, since $2^{2}$ is neither less than or greater than $4$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-iv/1", "set": "hardy-course-of-pure-mathematics-1921/ex-iv", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "16", "location": "Exercise IV, problem 1", "problem_latex": "Prove that $0 = -0$.", "markdown": "Prove that $0 = -0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-iv/2", "set": "hardy-course-of-pure-mathematics-1921/ex-iv", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "16", "location": "Exercise IV, problem 2", "problem_latex": "Prove that $\\beta = \\alpha$, $\\beta < \\alpha$, or $\\beta > \\alpha$ according as $\\alpha = \\beta$, $\\alpha > \\beta$, or $\\alpha < \\beta$.", "markdown": "Prove that $\\beta = \\alpha$, $\\beta < \\alpha$, or $\\beta > \\alpha$ according as $\\alpha = \\beta$, $\\alpha > \\beta$, or $\\alpha < \\beta$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-iv/3", "set": "hardy-course-of-pure-mathematics-1921/ex-iv", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "16", "location": "Exercise IV, problem 3", "problem_latex": "If $\\alpha = \\beta$ and $\\beta = \\gamma$, then $\\alpha = \\gamma$.", "markdown": "If $\\alpha = \\beta$ and $\\beta = \\gamma$, then $\\alpha = \\gamma$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-iv/4", "set": "hardy-course-of-pure-mathematics-1921/ex-iv", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "16", "location": "Exercise IV, problem 4", "problem_latex": "If $\\alpha \\leq \\beta$, $\\beta < \\gamma$, or $\\alpha < \\beta$, $\\beta \\leq \\gamma$, then $\\alpha < \\gamma$.", "markdown": "If $\\alpha \\leq \\beta$, $\\beta < \\gamma$, or $\\alpha < \\beta$, $\\beta \\leq \\gamma$, then $\\alpha < \\gamma$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-iv/5", "set": "hardy-course-of-pure-mathematics-1921/ex-iv", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "16", "location": "Exercise IV, problem 5", "problem_latex": "Prove that $-\\beta = -\\alpha$, $-\\beta < -\\alpha$, or $-\\beta > -\\alpha$, according as $\\alpha = \\beta$, $\\alpha < \\beta$,\nor $\\alpha > \\beta$.", "markdown": "Prove that $-\\beta = -\\alpha$, $-\\beta < -\\alpha$, or $-\\beta > -\\alpha$, according as $\\alpha = \\beta$, $\\alpha < \\beta$, or $\\alpha > \\beta$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-iv/6", "set": "hardy-course-of-pure-mathematics-1921/ex-iv", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "16", "location": "Exercise IV, problem 6", "problem_latex": "Prove that $\\alpha > 0$ if $\\alpha$~is positive, and $\\alpha < 0$ if $\\alpha$~is negative.", "markdown": "Prove that $\\alpha > 0$ if $\\alpha$ is positive, and $\\alpha < 0$ if $\\alpha$ is negative.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-iv/7", "set": "hardy-course-of-pure-mathematics-1921/ex-iv", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "16", "location": "Exercise IV, problem 7", "problem_latex": "Prove that $\\alpha \\leq |\\alpha|$.", "markdown": "Prove that $\\alpha \\leq |\\alpha|$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-iv/8", "set": "hardy-course-of-pure-mathematics-1921/ex-iv", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "16", "location": "Exercise IV, problem 8", "problem_latex": "Prove that $1 < \\sqrt{2} < \\sqrt{3} < 2$.", "markdown": "Prove that $1 < \\sqrt{2} < \\sqrt{3} < 2$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-iv/9", "set": "hardy-course-of-pure-mathematics-1921/ex-iv", "number": 9, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "16", "location": "Exercise IV, problem 9", "problem_latex": "Prove that, if $\\alpha$~and~$\\beta$ are two different real numbers, we can always\nfind an infinity of rational numbers lying between $\\alpha$~and~$\\beta$.", "markdown": "Prove that, if $\\alpha$ and $\\beta$ are two different real numbers, we can always find an infinity of rational numbers lying between $\\alpha$ and $\\beta$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-ix/1", "set": "hardy-course-of-pure-mathematics-1921/ex-ix", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "30", "location": "Exercise IX, problem 1", "problem_latex": "If $S$~consists of the points corresponding to the\npositive integers, or all the integers, there are no points of accumulation.", "markdown": "If $S$ consists of the points corresponding to the positive integers, or all the integers, there are no points of accumulation.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-ix/2", "set": "hardy-course-of-pure-mathematics-1921/ex-ix", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "30", "location": "Exercise IX, problem 2", "problem_latex": "If $S$~consists of all the rational points, every point of the line is a\npoint of accumulation.", "markdown": "If $S$ consists of all the rational points, every point of the line is a point of accumulation.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-ix/3", "set": "hardy-course-of-pure-mathematics-1921/ex-ix", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "30", "location": "Exercise IX, problem 3", "problem_latex": "If $S$~consists of the points $1$, $\\frac{1}{2}$, $\\frac{1}{3}, \\dots$, there is one point of accumulation,\nviz.\\ the origin.", "markdown": "If $S$ consists of the points $1$, $\\frac{1}{2}$, $\\frac{1}{3}, \\dots$, there is one point of accumulation, viz. the origin.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-ix/4", "set": "hardy-course-of-pure-mathematics-1921/ex-ix", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "30", "location": "Exercise IX, problem 4", "problem_latex": "If $S$~consists of all the positive rational points, the points of accumulation\nare the origin and all positive points of the line.", "markdown": "If $S$ consists of all the positive rational points, the points of accumulation are the origin and all positive points of the line.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-l/10", "set": "hardy-course-of-pure-mathematics-1921/ex-l", "number": 10, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "244", "location": "Exercise L, problem 10", "problem_latex": "Show that the integral $\\ds\\int R(x, y)\\, dx$, where $y^{2} = ax^{2} + 2bx + c$, is rationalised\nby the substitution $t = (x - p)/(y + q)$, where $(p, q)$~is any point on the\nconic $y^{2} = ax^{2} + 2bx + c$. [The integral is of course also rationalised by the\nsubstitution $t = (x - p)/(y - q)$: cf.~\\SecNo[§]{134}.]", "markdown": "Show that the integral $\\ds\\int R(x, y)\\, dx$, where $y^{2} = ax^{2} + 2bx + c$, is rationalised by the substitution $t = (x - p)/(y + q)$, where $(p, q)$ is any point on the conic $y^{2} = ax^{2} + 2bx + c$. [The integral is of course also rationalised by the substitution $t = (x - p)/(y - q)$: cf. [§]134.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-l/1a", "set": "hardy-course-of-pure-mathematics-1921/ex-l", "number": 1, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "244", "location": "Exercise L, problem 1a", "problem_latex": "Evaluate\n\\[\n\\int \\frac{dx}{x \\sqrtp{x^{2} + 2x + 3}},\\quad\n\\int \\frac{dx}{(x - 1) \\sqrtp{x^{2} + 1}},\\quad\n\\int \\frac{dx}{(x + 1) \\sqrtp{1 + 2x - x^{2}}}.\n\\]", "markdown": "Evaluate dxx x^2 + 2x + 3,0pt minus 3ptdx(x - 1) x^2 + 1,0pt minus 3ptdx(x + 1) 1 + 2x - x^2.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "1/(x*sqrt(x**2 + 2*x + 3))", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: 1/(x*sqrt(x**2 + 2*x + 3))" ], "shape": [ "integrate: (N*x + N + x**N)**N/x" ], "same_problem_in": [], "needs": [ "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-l/1b", "set": "hardy-course-of-pure-mathematics-1921/ex-l", "number": 1, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "244", "location": "Exercise L, problem 1b", "problem_latex": "Evaluate\n\\[\n\\int \\frac{dx}{x \\sqrtp{x^{2} + 2x + 3}},\\quad\n\\int \\frac{dx}{(x - 1) \\sqrtp{x^{2} + 1}},\\quad\n\\int \\frac{dx}{(x + 1) \\sqrtp{1 + 2x - x^{2}}}.\n\\]", "markdown": "Evaluate dxx x^2 + 2x + 3,0pt minus 3ptdx(x - 1) x^2 + 1,0pt minus 3ptdx(x + 1) 1 + 2x - x^2.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "1/((x - 1)*sqrt(x**2 + 1))", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: 1/((x - 1)*sqrt(x**2 + 1))" ], "shape": [ "integrate: (x**N + 1)**N/(x - 1)" ], "same_problem_in": [], "needs": [ "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-l/1c", "set": "hardy-course-of-pure-mathematics-1921/ex-l", "number": 1, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "244", "location": "Exercise L, problem 1c", "problem_latex": "Evaluate\n\\[\n\\int \\frac{dx}{x \\sqrtp{x^{2} + 2x + 3}},\\quad\n\\int \\frac{dx}{(x - 1) \\sqrtp{x^{2} + 1}},\\quad\n\\int \\frac{dx}{(x + 1) \\sqrtp{1 + 2x - x^{2}}}.\n\\]", "markdown": "Evaluate dxx x^2 + 2x + 3,0pt minus 3ptdx(x - 1) x^2 + 1,0pt minus 3ptdx(x + 1) 1 + 2x - x^2.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "1/((x + 1)*sqrt(1 + 2*x - x**2))", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: 1/((x + 1)*sqrt(-x**2 + 2*x + 1))" ], "shape": [ "integrate: (N*x - x**N + 1)**N/(x + 1)" ], "same_problem_in": [], "needs": [ "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-l/2", "set": "hardy-course-of-pure-mathematics-1921/ex-l", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "244", "location": "Exercise L, problem 2", "problem_latex": "Prove that\n\\[\n\\int \\frac{dx}{(x - p) \\sqrtb{(x - p) (x - q)}}\n = \\frac{2}{q - p} \\bigsqrtp{\\frac{x - q}{x - p}}.\n\\]", "markdown": "Prove that dx(x - p) (x - p) (x - q) = 2q - p x - qx - p.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-l/3", "set": "hardy-course-of-pure-mathematics-1921/ex-l", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "244", "location": "Exercise L, problem 3", "problem_latex": "If $ag^{2} + ch^{2} = -\\nu < 0$ then\n\\[\n\\int \\frac{dx}{(hx + g) \\sqrtp{ax^{2} + c}}\n = -\\frac{1}{\\sqrt{\\nu}} \\arctan\\left[\n \\frac{\\sqrtb{\\nu(ax^{2} + c)}}{ch - agx}\n \\right].\n\\]", "markdown": "If $ag^{2} + ch^{2} = -\\nu < 0$ then dx(hx + g) ax^2 + c = -1 [ (ax^2 + c)ch - agx ].", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-l/4", "set": "hardy-course-of-pure-mathematics-1921/ex-l", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "244", "location": "Exercise L, problem 4", "problem_latex": "Show that $\\ds\\int \\frac{dx}{(x - x_{0})y}$, where $y^{2} = ax^{2} + 2bx + c$, may be expressed in one\nor other of the forms\n\\[\n-\\frac{1}{y_{0}} \\log\\left|\n \\frac{axx_{0} + b(x + x_{0}) + c + yy_{0}}{x - x_{0}}\n\\right|,\\quad\n\\frac{1}{z_{0}} \\arctan \\left\\{\n \\frac{axx_{0} + b(x + x_{0}) + c}{yz_{0}}\n\\right\\},\n\\]\naccording as $ax_{0}^{2} + 2bx_{0} + c$ is positive and equal to~$y_{0}^{2}$ or negative and equal\nto~$-z_{0}^{2}$.", "markdown": "Show that $\\ds\\int \\frac{dx}{(x - x_{0})y}$, where $y^{2} = ax^{2} + 2bx + c$, may be expressed in one or other of the forms -1y_0 | axx_0 + b(x + x_0) + c + yy_0x - x_0 |,0pt minus 3pt1z_0 axx_0 + b(x + x_0) + cyz_0 , according as $ax_{0}^{2} + 2bx_{0} + c$ is positive and equal to $y_{0}^{2}$ or negative and equal to $-z_{0}^{2}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-l/5", "set": "hardy-course-of-pure-mathematics-1921/ex-l", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "244", "location": "Exercise L, problem 5", "problem_latex": "Show by means of the substitution $y = \\sqrtp{ax^{2} + 2bx + c}/(x - p)$ that\n\\[\n\\int \\frac{dx}{(x - p) \\sqrtp{ax^{2} + 2bx + c}}\n = \\int \\frac{dy}{\\sqrtp{\\lambda y^{2} - \\mu}},\n\\]\nwhere $\\lambda = ap^{2} + 2bp + c$, $\\mu = ac - b^{2}$. [This method of reduction is elegant but\nless straightforward than that explained in \\SecNo[§]{139}.]", "markdown": "Show by means of the substitution $y = \\sqrtp{ax^{2} + 2bx + c}/(x - p)$ that dx(x - p) ax^2 + 2bx + c = dyy^2 - , where $\\lambda = ap^{2} + 2bp + c$, $\\mu = ac - b^{2}$. [This method of reduction is elegant but less straightforward than that explained in [§]139.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-l/6", "set": "hardy-course-of-pure-mathematics-1921/ex-l", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "244", "location": "Exercise L, problem 6", "problem_latex": "Show that the integral\n\\[\n\\int \\frac{dx}{x \\sqrtp{3x^{2} + 2x + 1}}\n\\]\nis rationalised by the substitution $x = (1 + y^{2})/(3 - y^{2})$. \\MathTrip{1911.}", "markdown": "Show that the integral dxx 3x^2 + 2x + 1 is rationalised by the substitution $x = (1 + y^{2})/(3 - y^{2})$. % [0]% (*Math. Trip.* 1911.)% [1]%", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-l/7", "set": "hardy-course-of-pure-mathematics-1921/ex-l", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "244", "location": "Exercise L, problem 7", "problem_latex": "Calculate\n\\[\n\\int \\frac{(x + 1)\\, dx}{(x^{2} + 4) \\sqrtp{x^{2} + 9}}.\n\\]\n\\PageSep{245}", "markdown": "Calculate (x + 1)  dx(x^2 + 4) x^2 + 9. [pg]245", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "(x + 1)/((x**2 + 4)*sqrt(x**2 + 9))", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: (x + 1)/((x**2 + 4)*sqrt(x**2 + 9))" ], "shape": [ "integrate: (N + x**N)**N*(x + 1)/(N + x**N)" ], "same_problem_in": [], "needs": [ "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-l/8", "set": "hardy-course-of-pure-mathematics-1921/ex-l", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "244", "location": "Exercise L, problem 8", "problem_latex": "Calculate\n\\[\n\\int \\frac{dx}{(5x^{2} + 12x + 8) \\sqrtp{5x^{2} + 2x - 7}}.\n\\]", "markdown": "Calculate dx(5x^2 + 12x + 8) 5x^2 + 2x - 7.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "1/((5*x**2 + 12*x + 8)*sqrt(5*x**2 + 2*x - 7))", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: 1/(sqrt(5*x**2 + 2*x - 7)*(5*x**2 + 12*x + 8))" ], "shape": [ "integrate: (N*x + N*x**N + N)**N/(N*x + N*x**N + N)" ], "same_problem_in": [], "needs": [ "cas.integrate", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-l/9a", "set": "hardy-course-of-pure-mathematics-1921/ex-l", "number": 9, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "244", "location": "Exercise L, problem 9a", "problem_latex": "Calculate\n\\[\n\\int \\frac{(x + 1)\\, dx}{(2x^{2} - 2x + 1) \\sqrtp{3x^{2} - 2x + 1}},\\quad\n\\int \\frac{(x - 1)\\, dx}{(2x^{2} - 6x + 5) \\sqrtp{7x^{2} - 22x + 19}}.\n\\]\n\\MathTrip{1911.}", "markdown": "Calculate (x + 1)  dx(2x^2 - 2x + 1) 3x^2 - 2x + 1,0pt minus 3pt(x - 1)  dx(2x^2 - 6x + 5) 7x^2 - 22x + 19. % [0]% (*Math. Trip.* 1911.)% [1]%", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "(x + 1)/((2*x**2 - 2*x + 1)*sqrt(3*x**2 - 2*x + 1))", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: (x + 1)/((2*x**2 - 2*x + 1)*sqrt(3*x**2 - 2*x + 1))" ], "shape": [ "integrate: (x + 1)*(N*x + N*x**N + 1)**N/(N*x + N*x**N + 1)" ], "same_problem_in": [], "needs": [ "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-l/9b", "set": "hardy-course-of-pure-mathematics-1921/ex-l", "number": 9, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "244", "location": "Exercise L, problem 9b", "problem_latex": "Calculate\n\\[\n\\int \\frac{(x + 1)\\, dx}{(2x^{2} - 2x + 1) \\sqrtp{3x^{2} - 2x + 1}},\\quad\n\\int \\frac{(x - 1)\\, dx}{(2x^{2} - 6x + 5) \\sqrtp{7x^{2} - 22x + 19}}.\n\\]\n\\MathTrip{1911.}", "markdown": "Calculate (x + 1)  dx(2x^2 - 2x + 1) 3x^2 - 2x + 1,0pt minus 3pt(x - 1)  dx(2x^2 - 6x + 5) 7x^2 - 22x + 19. % [0]% (*Math. Trip.* 1911.)% [1]%", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "(x - 1)/((2*x**2 - 6*x + 5)*sqrt(7*x**2 - 22*x + 19))", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: (x - 1)/((2*x**2 - 6*x + 5)*sqrt(7*x**2 - 22*x + 19))" ], "shape": [ "integrate: (x - 1)*(N*x + N*x**N + N)**N/(N*x + N*x**N + N)" ], "same_problem_in": [], "needs": [ "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-li/1", "set": "hardy-course-of-pure-mathematics-1921/ex-li", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "246", "location": "Exercise LI, problem 1", "problem_latex": "Integrate $\\sin^{3} x \\cos^{2} 2x$.", "markdown": "Integrate $\\sin^{3} x \\cos^{2} 2x$.", "answer_latex": [ "- \\tfrac{7}{16}\\cos x + \\tfrac{5}{48}\\cos 3x\n - \\tfrac{3}{80}\\cos 5x + \\tfrac{1}{112}\\cos 7x." ], "answer_markdown": [ "- 716 x + 5483x - 3805x + 11127x." ], "checks": [ { "task": "integrate", "verdict": "PASS", "judge_why": null, "problem_expr": "sin(x)**3*cos(2*x)**2", "answer_expr": "-7*cos(x)/16 + 5*cos(3*x)/48 - 3*cos(5*x)/80 + cos(7*x)/112" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "integrate: sin(x)**3*cos(2*x)**2" ], "shape": [ "integrate: sin(x)**N*cos(N*x)**N" ], "same_problem_in": [], "needs": [ "cas.integrate", "cas.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-li/2a", "set": "hardy-course-of-pure-mathematics-1921/ex-li", "number": 2, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "246", "location": "Exercise LI, problem 2a", "problem_latex": "Integrate by any method $\\cos ax \\cos bx$, $\\sin ax \\sin bx$, $\\cos ax \\sin bx$,\n$\\cos^{2}x$, $\\sin^{3}x$, $\\cos^{4}x$, $\\cos x \\cos 2x \\cos 3x$, $\\cos^{3}2x \\sin^{2}3x$, $\\cos^{5}x \\sin^{7}x$. [In cases of\nthis kind it is sometimes convenient to use a formula of reduction (\\MiscEx{VI}~39).]", "markdown": "Integrate by any method $\\cos ax \\cos bx$, $\\sin ax \\sin bx$, $\\cos ax \\sin bx$, $\\cos^{2}x$, $\\sin^{3}x$, $\\cos^{4}x$, $\\cos x \\cos 2x \\cos 3x$, $\\cos^{3}2x \\sin^{2}3x$, $\\cos^{5}x \\sin^{7}x$. [In cases of this kind it is sometimes convenient to use a formula of reduction ([misc:VI]Misc. Ex. 39).]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "cos(a*x)*cos(b*x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: cos(a*x)*cos(b*x)" ], "shape": [ "integrate: cos(a*x)*cos(b*x)" ], "same_problem_in": [], "needs": [ "cas.integrate", "cas.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-li/2b", "set": "hardy-course-of-pure-mathematics-1921/ex-li", "number": 2, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "246", "location": "Exercise LI, problem 2b", "problem_latex": "Integrate by any method $\\cos ax \\cos bx$, $\\sin ax \\sin bx$, $\\cos ax \\sin bx$,\n$\\cos^{2}x$, $\\sin^{3}x$, $\\cos^{4}x$, $\\cos x \\cos 2x \\cos 3x$, $\\cos^{3}2x \\sin^{2}3x$, $\\cos^{5}x \\sin^{7}x$. [In cases of\nthis kind it is sometimes convenient to use a formula of reduction (\\MiscEx{VI}~39).]", "markdown": "Integrate by any method $\\cos ax \\cos bx$, $\\sin ax \\sin bx$, $\\cos ax \\sin bx$, $\\cos^{2}x$, $\\sin^{3}x$, $\\cos^{4}x$, $\\cos x \\cos 2x \\cos 3x$, $\\cos^{3}2x \\sin^{2}3x$, $\\cos^{5}x \\sin^{7}x$. [In cases of this kind it is sometimes convenient to use a formula of reduction ([misc:VI]Misc. Ex. 39).]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "sin(a*x)*sin(b*x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: sin(a*x)*sin(b*x)" ], "shape": [ "integrate: sin(a*x)*sin(b*x)" ], "same_problem_in": [], "needs": [ "cas.integrate", "cas.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-li/2c", "set": "hardy-course-of-pure-mathematics-1921/ex-li", "number": 2, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "246", "location": "Exercise LI, problem 2c", "problem_latex": "Integrate by any method $\\cos ax \\cos bx$, $\\sin ax \\sin bx$, $\\cos ax \\sin bx$,\n$\\cos^{2}x$, $\\sin^{3}x$, $\\cos^{4}x$, $\\cos x \\cos 2x \\cos 3x$, $\\cos^{3}2x \\sin^{2}3x$, $\\cos^{5}x \\sin^{7}x$. [In cases of\nthis kind it is sometimes convenient to use a formula of reduction (\\MiscEx{VI}~39).]", "markdown": "Integrate by any method $\\cos ax \\cos bx$, $\\sin ax \\sin bx$, $\\cos ax \\sin bx$, $\\cos^{2}x$, $\\sin^{3}x$, $\\cos^{4}x$, $\\cos x \\cos 2x \\cos 3x$, $\\cos^{3}2x \\sin^{2}3x$, $\\cos^{5}x \\sin^{7}x$. [In cases of this kind it is sometimes convenient to use a formula of reduction ([misc:VI]Misc. Ex. 39).]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "cos(a*x)*sin(b*x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: sin(b*x)*cos(a*x)" ], "shape": [ "integrate: sin(b*x)*cos(a*x)" ], "same_problem_in": [ "hardy-course-of-pure-mathematics-1921/ex-lxiii/10c", "hardy-course-of-pure-mathematics-1921/ex-lxiii/10d" ], "needs": [ "cas.integrate", "cas.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-li/2d", "set": "hardy-course-of-pure-mathematics-1921/ex-li", "number": 2, "part": "d", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "246", "location": "Exercise LI, problem 2d", "problem_latex": "Integrate by any method $\\cos ax \\cos bx$, $\\sin ax \\sin bx$, $\\cos ax \\sin bx$,\n$\\cos^{2}x$, $\\sin^{3}x$, $\\cos^{4}x$, $\\cos x \\cos 2x \\cos 3x$, $\\cos^{3}2x \\sin^{2}3x$, $\\cos^{5}x \\sin^{7}x$. [In cases of\nthis kind it is sometimes convenient to use a formula of reduction (\\MiscEx{VI}~39).]", "markdown": "Integrate by any method $\\cos ax \\cos bx$, $\\sin ax \\sin bx$, $\\cos ax \\sin bx$, $\\cos^{2}x$, $\\sin^{3}x$, $\\cos^{4}x$, $\\cos x \\cos 2x \\cos 3x$, $\\cos^{3}2x \\sin^{2}3x$, $\\cos^{5}x \\sin^{7}x$. [In cases of this kind it is sometimes convenient to use a formula of reduction ([misc:VI]Misc. Ex. 39).]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "cos(x)**2", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: cos(x)**2" ], "shape": [ "integrate: cos(x)**N" ], "same_problem_in": [], "needs": [ "cas.integrate", "cas.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-li/2e", "set": "hardy-course-of-pure-mathematics-1921/ex-li", "number": 2, "part": "e", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "246", "location": "Exercise LI, problem 2e", "problem_latex": "Integrate by any method $\\cos ax \\cos bx$, $\\sin ax \\sin bx$, $\\cos ax \\sin bx$,\n$\\cos^{2}x$, $\\sin^{3}x$, $\\cos^{4}x$, $\\cos x \\cos 2x \\cos 3x$, $\\cos^{3}2x \\sin^{2}3x$, $\\cos^{5}x \\sin^{7}x$. [In cases of\nthis kind it is sometimes convenient to use a formula of reduction (\\MiscEx{VI}~39).]", "markdown": "Integrate by any method $\\cos ax \\cos bx$, $\\sin ax \\sin bx$, $\\cos ax \\sin bx$, $\\cos^{2}x$, $\\sin^{3}x$, $\\cos^{4}x$, $\\cos x \\cos 2x \\cos 3x$, $\\cos^{3}2x \\sin^{2}3x$, $\\cos^{5}x \\sin^{7}x$. [In cases of this kind it is sometimes convenient to use a formula of reduction ([misc:VI]Misc. Ex. 39).]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "sin(x)**3", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: sin(x)**3" ], "shape": [ "integrate: sin(x)**N" ], "same_problem_in": [], "needs": [ "cas.integrate", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-li/2f", "set": "hardy-course-of-pure-mathematics-1921/ex-li", "number": 2, "part": "f", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "246", "location": "Exercise LI, problem 2f", "problem_latex": "Integrate by any method $\\cos ax \\cos bx$, $\\sin ax \\sin bx$, $\\cos ax \\sin bx$,\n$\\cos^{2}x$, $\\sin^{3}x$, $\\cos^{4}x$, $\\cos x \\cos 2x \\cos 3x$, $\\cos^{3}2x \\sin^{2}3x$, $\\cos^{5}x \\sin^{7}x$. [In cases of\nthis kind it is sometimes convenient to use a formula of reduction (\\MiscEx{VI}~39).]", "markdown": "Integrate by any method $\\cos ax \\cos bx$, $\\sin ax \\sin bx$, $\\cos ax \\sin bx$, $\\cos^{2}x$, $\\sin^{3}x$, $\\cos^{4}x$, $\\cos x \\cos 2x \\cos 3x$, $\\cos^{3}2x \\sin^{2}3x$, $\\cos^{5}x \\sin^{7}x$. [In cases of this kind it is sometimes convenient to use a formula of reduction ([misc:VI]Misc. Ex. 39).]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "cos(x)**4", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: cos(x)**4" ], "shape": [ "integrate: cos(x)**N" ], "same_problem_in": [], "needs": [ "cas.expand", "cas.integrate", "cas.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-li/2g", "set": "hardy-course-of-pure-mathematics-1921/ex-li", "number": 2, "part": "g", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "246", "location": "Exercise LI, problem 2g", "problem_latex": "Integrate by any method $\\cos ax \\cos bx$, $\\sin ax \\sin bx$, $\\cos ax \\sin bx$,\n$\\cos^{2}x$, $\\sin^{3}x$, $\\cos^{4}x$, $\\cos x \\cos 2x \\cos 3x$, $\\cos^{3}2x \\sin^{2}3x$, $\\cos^{5}x \\sin^{7}x$. [In cases of\nthis kind it is sometimes convenient to use a formula of reduction (\\MiscEx{VI}~39).]", "markdown": "Integrate by any method $\\cos ax \\cos bx$, $\\sin ax \\sin bx$, $\\cos ax \\sin bx$, $\\cos^{2}x$, $\\sin^{3}x$, $\\cos^{4}x$, $\\cos x \\cos 2x \\cos 3x$, $\\cos^{3}2x \\sin^{2}3x$, $\\cos^{5}x \\sin^{7}x$. [In cases of this kind it is sometimes convenient to use a formula of reduction ([misc:VI]Misc. Ex. 39).]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "cos(x)*cos(2*x)*cos(3*x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: cos(x)*cos(2*x)*cos(3*x)" ], "shape": [ "integrate: cos(x)*cos(N*x)**2" ], "same_problem_in": [], "needs": [ "cas.integrate", "cas.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-li/2h", "set": "hardy-course-of-pure-mathematics-1921/ex-li", "number": 2, "part": "h", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "246", "location": "Exercise LI, problem 2h", "problem_latex": "Integrate by any method $\\cos ax \\cos bx$, $\\sin ax \\sin bx$, $\\cos ax \\sin bx$,\n$\\cos^{2}x$, $\\sin^{3}x$, $\\cos^{4}x$, $\\cos x \\cos 2x \\cos 3x$, $\\cos^{3}2x \\sin^{2}3x$, $\\cos^{5}x \\sin^{7}x$. [In cases of\nthis kind it is sometimes convenient to use a formula of reduction (\\MiscEx{VI}~39).]", "markdown": "Integrate by any method $\\cos ax \\cos bx$, $\\sin ax \\sin bx$, $\\cos ax \\sin bx$, $\\cos^{2}x$, $\\sin^{3}x$, $\\cos^{4}x$, $\\cos x \\cos 2x \\cos 3x$, $\\cos^{3}2x \\sin^{2}3x$, $\\cos^{5}x \\sin^{7}x$. [In cases of this kind it is sometimes convenient to use a formula of reduction ([misc:VI]Misc. Ex. 39).]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "cos(2*x)**3*sin(3*x)**2", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: sin(3*x)**2*cos(2*x)**3" ], "shape": [ "integrate: sin(N*x)**N*cos(N*x)**N" ], "same_problem_in": [], "needs": [ "cas.integrate", "cas.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-li/2i", "set": "hardy-course-of-pure-mathematics-1921/ex-li", "number": 2, "part": "i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "246", "location": "Exercise LI, problem 2i", "problem_latex": "Integrate by any method $\\cos ax \\cos bx$, $\\sin ax \\sin bx$, $\\cos ax \\sin bx$,\n$\\cos^{2}x$, $\\sin^{3}x$, $\\cos^{4}x$, $\\cos x \\cos 2x \\cos 3x$, $\\cos^{3}2x \\sin^{2}3x$, $\\cos^{5}x \\sin^{7}x$. [In cases of\nthis kind it is sometimes convenient to use a formula of reduction (\\MiscEx{VI}~39).]", "markdown": "Integrate by any method $\\cos ax \\cos bx$, $\\sin ax \\sin bx$, $\\cos ax \\sin bx$, $\\cos^{2}x$, $\\sin^{3}x$, $\\cos^{4}x$, $\\cos x \\cos 2x \\cos 3x$, $\\cos^{3}2x \\sin^{2}3x$, $\\cos^{5}x \\sin^{7}x$. [In cases of this kind it is sometimes convenient to use a formula of reduction ([misc:VI]Misc. Ex. 39).]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "cos(x)**5*sin(x)**7", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: sin(x)**7*cos(x)**5" ], "shape": [ "integrate: sin(x)**N*cos(x)**N" ], "same_problem_in": [], "needs": [ "cas.integrate", "cas.integrate.parts", "cas.integrate.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lii/1a", "set": "hardy-course-of-pure-mathematics-1921/ex-lii", "number": 1, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "247", "location": "Exercise LII, problem 1a", "problem_latex": "Integrate $x\\sin x$, $x^{2}\\cos x$, $x^{2}\\cos^{2}x$, $x^{2}\\sin^{2}x \\sin^{2} 2x$,\n$x\\sin^{2}x \\cos^{4}x$, $x^{3}\\sin^{3}\\frac{1}{3}x$.", "markdown": "Integrate $x\\sin x$, $x^{2}\\cos x$, $x^{2}\\cos^{2}x$, $x^{2}\\sin^{2}x \\sin^{2} 2x$, $x\\sin^{2}x \\cos^{4}x$, $x^{3}\\sin^{3}\\frac{1}{3}x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "x*sin(x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: x*sin(x)" ], "shape": [ "integrate: x*sin(x)" ], "same_problem_in": [], "needs": [ "cas.integrate.parts" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lii/1b", "set": "hardy-course-of-pure-mathematics-1921/ex-lii", "number": 1, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "247", "location": "Exercise LII, problem 1b", "problem_latex": "Integrate $x\\sin x$, $x^{2}\\cos x$, $x^{2}\\cos^{2}x$, $x^{2}\\sin^{2}x \\sin^{2} 2x$,\n$x\\sin^{2}x \\cos^{4}x$, $x^{3}\\sin^{3}\\frac{1}{3}x$.", "markdown": "Integrate $x\\sin x$, $x^{2}\\cos x$, $x^{2}\\cos^{2}x$, $x^{2}\\sin^{2}x \\sin^{2} 2x$, $x\\sin^{2}x \\cos^{4}x$, $x^{3}\\sin^{3}\\frac{1}{3}x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "x**2*cos(x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: x**2*cos(x)" ], "shape": [ "integrate: x**N*cos(x)" ], "same_problem_in": [], "needs": [ "cas.integrate.parts" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lii/1c", "set": "hardy-course-of-pure-mathematics-1921/ex-lii", "number": 1, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "247", "location": "Exercise LII, problem 1c", "problem_latex": "Integrate $x\\sin x$, $x^{2}\\cos x$, $x^{2}\\cos^{2}x$, $x^{2}\\sin^{2}x \\sin^{2} 2x$,\n$x\\sin^{2}x \\cos^{4}x$, $x^{3}\\sin^{3}\\frac{1}{3}x$.", "markdown": "Integrate $x\\sin x$, $x^{2}\\cos x$, $x^{2}\\cos^{2}x$, $x^{2}\\sin^{2}x \\sin^{2} 2x$, $x\\sin^{2}x \\cos^{4}x$, $x^{3}\\sin^{3}\\frac{1}{3}x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "x**2*cos(x)**2", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: x**2*cos(x)**2" ], "shape": [ "integrate: x**N*cos(x)**N" ], "same_problem_in": [], "needs": [ "cas.integrate.parts", "cas.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lii/1d", "set": "hardy-course-of-pure-mathematics-1921/ex-lii", "number": 1, "part": "d", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "247", "location": "Exercise LII, problem 1d", "problem_latex": "Integrate $x\\sin x$, $x^{2}\\cos x$, $x^{2}\\cos^{2}x$, $x^{2}\\sin^{2}x \\sin^{2} 2x$,\n$x\\sin^{2}x \\cos^{4}x$, $x^{3}\\sin^{3}\\frac{1}{3}x$.", "markdown": "Integrate $x\\sin x$, $x^{2}\\cos x$, $x^{2}\\cos^{2}x$, $x^{2}\\sin^{2}x \\sin^{2} 2x$, $x\\sin^{2}x \\cos^{4}x$, $x^{3}\\sin^{3}\\frac{1}{3}x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "x**2*sin(x)**2*sin(2*x)**2", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: x**2*sin(x)**2*sin(2*x)**2" ], "shape": [ "integrate: x**N*sin(x)**N*sin(N*x)**N" ], "same_problem_in": [], "needs": [ "cas.integrate.parts", "cas.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lii/1e", "set": "hardy-course-of-pure-mathematics-1921/ex-lii", "number": 1, "part": "e", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "247", "location": "Exercise LII, problem 1e", "problem_latex": "Integrate $x\\sin x$, $x^{2}\\cos x$, $x^{2}\\cos^{2}x$, $x^{2}\\sin^{2}x \\sin^{2} 2x$,\n$x\\sin^{2}x \\cos^{4}x$, $x^{3}\\sin^{3}\\frac{1}{3}x$.", "markdown": "Integrate $x\\sin x$, $x^{2}\\cos x$, $x^{2}\\cos^{2}x$, $x^{2}\\sin^{2}x \\sin^{2} 2x$, $x\\sin^{2}x \\cos^{4}x$, $x^{3}\\sin^{3}\\frac{1}{3}x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "x*sin(x)**2*cos(x)**4", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: x*sin(x)**2*cos(x)**4" ], "shape": [ "integrate: x*sin(x)**N*cos(x)**N" ], "same_problem_in": [], "needs": [ "cas.integrate.parts", "cas.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lii/1f", "set": "hardy-course-of-pure-mathematics-1921/ex-lii", "number": 1, "part": "f", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "247", "location": "Exercise LII, problem 1f", "problem_latex": "Integrate $x\\sin x$, $x^{2}\\cos x$, $x^{2}\\cos^{2}x$, $x^{2}\\sin^{2}x \\sin^{2} 2x$,\n$x\\sin^{2}x \\cos^{4}x$, $x^{3}\\sin^{3}\\frac{1}{3}x$.", "markdown": "Integrate $x\\sin x$, $x^{2}\\cos x$, $x^{2}\\cos^{2}x$, $x^{2}\\sin^{2}x \\sin^{2} 2x$, $x\\sin^{2}x \\cos^{4}x$, $x^{3}\\sin^{3}\\frac{1}{3}x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "x**3*sin(x/3)**3", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: x**3*sin(x/3)**3" ], "shape": [ "integrate: x**N*sin(N*x)**N" ], "same_problem_in": [], "needs": [ "cas.integrate.parts", "cas.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lii/2", "set": "hardy-course-of-pure-mathematics-1921/ex-lii", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "247", "location": "Exercise LII, problem 2", "problem_latex": "Find polynomials $P$~and~$Q$ such that\n\\[\n\\int\\{(3x - 1)\\cos x + (1 - 2x)\\sin x\\}\\, dx = P\\cos x + Q\\sin x.\n\\]", "markdown": "Find polynomials $P$ and $Q$ such that (3x - 1)x + (1 - 2x)x  dx = Px + Qx.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate.parts", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lii/3", "set": "hardy-course-of-pure-mathematics-1921/ex-lii", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "247", "location": "Exercise LII, problem 3", "problem_latex": "Prove that $\\ds\\int x^{n}\\cos x\\, dx = P_{n}\\cos x + Q_{n}\\sin x$, where\n\\[\nP_{n} = nx^{n-1} - n(n - 1)(n - 2) x^{n-3} + \\dots,\\quad\nQ_{n} = x^{n} - n(n - 1) x^{n-2} + \\dots.\n\\]", "markdown": "Prove that $\\ds\\int x^{n}\\cos x\\, dx = P_{n}\\cos x + Q_{n}\\sin x$, where P_n = nx^n-1 - n(n - 1)(n - 2) x^n-3 + …,0pt minus 3ptQ_n = x^n - n(n - 1) x^n-2 + ….", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.integrate.parts" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-liii/1a", "set": "hardy-course-of-pure-mathematics-1921/ex-liii", "number": 1, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "247", "location": "Exercise LIII, problem 1a", "problem_latex": "Prove that\n\\[\n\\int \\sec x\\, dx = \\log |\\sec x + \\tan x|,\\quad\n\\int \\cosec x\\, dx = \\log |\\tan \\tfrac{1}{2}x|.\n\\]", "markdown": "Prove that x  dx = |x + x|,0pt minus 3ptx  dx = |12x|.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-liii/1b", "set": "hardy-course-of-pure-mathematics-1921/ex-liii", "number": 1, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "247", "location": "Exercise LIII, problem 1b", "problem_latex": "Prove that\n\\[\n\\int \\sec x\\, dx = \\log |\\sec x + \\tan x|,\\quad\n\\int \\cosec x\\, dx = \\log |\\tan \\tfrac{1}{2}x|.\n\\]", "markdown": "Prove that x  dx = |x + x|,0pt minus 3ptx  dx = |12x|.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-liii/2a", "set": "hardy-course-of-pure-mathematics-1921/ex-liii", "number": 2, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "247", "location": "Exercise LIII, problem 2a", "problem_latex": "$\\ds\\int \\tan x\\, dx = -\\log |\\cos x|$, $\\ds\\int \\cot x\\, dx = \\log |\\sin x|$, $\\ds\\int\\sec^{2} x\\, dx = \\tan x$,\n$\\ds\\int \\cosec^{2} x\\, dx = -\\cot x$, $\\ds\\int \\tan x\\sec x\\, dx = \\sec x$, $\\ds\\int \\cot x \\cosec x\\, dx = -\\cosec x$.", "markdown": "$\\ds\\int \\tan x\\, dx = -\\log |\\cos x|$, $\\ds\\int \\cot x\\, dx = \\log |\\sin x|$, $\\ds\\int\\sec^{2} x\\, dx = \\tan x$, $\\ds\\int \\cosec^{2} x\\, dx = -\\cot x$, $\\ds\\int \\tan x\\sec x\\, dx = \\sec x$, $\\ds\\int \\cot x \\cosec x\\, dx = -\\cosec x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "tan(x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: tan(x)" ], "shape": [ "integrate: tan(x)" ], "same_problem_in": [], "needs": [ "cas.integrate", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-liii/2b", "set": "hardy-course-of-pure-mathematics-1921/ex-liii", "number": 2, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "247", "location": "Exercise LIII, problem 2b", "problem_latex": "$\\ds\\int \\tan x\\, dx = -\\log |\\cos x|$, $\\ds\\int \\cot x\\, dx = \\log |\\sin x|$, $\\ds\\int\\sec^{2} x\\, dx = \\tan x$,\n$\\ds\\int \\cosec^{2} x\\, dx = -\\cot x$, $\\ds\\int \\tan x\\sec x\\, dx = \\sec x$, $\\ds\\int \\cot x \\cosec x\\, dx = -\\cosec x$.", "markdown": "$\\ds\\int \\tan x\\, dx = -\\log |\\cos x|$, $\\ds\\int \\cot x\\, dx = \\log |\\sin x|$, $\\ds\\int\\sec^{2} x\\, dx = \\tan x$, $\\ds\\int \\cosec^{2} x\\, dx = -\\cot x$, $\\ds\\int \\tan x\\sec x\\, dx = \\sec x$, $\\ds\\int \\cot x \\cosec x\\, dx = -\\cosec x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "cot(x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: cot(x)" ], "shape": [ "integrate: cot(x)" ], "same_problem_in": [], "needs": [ "cas.integrate", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-liii/2c", "set": "hardy-course-of-pure-mathematics-1921/ex-liii", "number": 2, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "247", "location": "Exercise LIII, problem 2c", "problem_latex": "$\\ds\\int \\tan x\\, dx = -\\log |\\cos x|$, $\\ds\\int \\cot x\\, dx = \\log |\\sin x|$, $\\ds\\int\\sec^{2} x\\, dx = \\tan x$,\n$\\ds\\int \\cosec^{2} x\\, dx = -\\cot x$, $\\ds\\int \\tan x\\sec x\\, dx = \\sec x$, $\\ds\\int \\cot x \\cosec x\\, dx = -\\cosec x$.", "markdown": "$\\ds\\int \\tan x\\, dx = -\\log |\\cos x|$, $\\ds\\int \\cot x\\, dx = \\log |\\sin x|$, $\\ds\\int\\sec^{2} x\\, dx = \\tan x$, $\\ds\\int \\cosec^{2} x\\, dx = -\\cot x$, $\\ds\\int \\tan x\\sec x\\, dx = \\sec x$, $\\ds\\int \\cot x \\cosec x\\, dx = -\\cosec x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "sec(x)**2", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: sec(x)**2" ], "shape": [ "integrate: sec(x)**N" ], "same_problem_in": [], "needs": [ "cas.integrate", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-liii/2d", "set": "hardy-course-of-pure-mathematics-1921/ex-liii", "number": 2, "part": "d", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "247", "location": "Exercise LIII, problem 2d", "problem_latex": "$\\ds\\int \\tan x\\, dx = -\\log |\\cos x|$, $\\ds\\int \\cot x\\, dx = \\log |\\sin x|$, $\\ds\\int\\sec^{2} x\\, dx = \\tan x$,\n$\\ds\\int \\cosec^{2} x\\, dx = -\\cot x$, $\\ds\\int \\tan x\\sec x\\, dx = \\sec x$, $\\ds\\int \\cot x \\cosec x\\, dx = -\\cosec x$.", "markdown": "$\\ds\\int \\tan x\\, dx = -\\log |\\cos x|$, $\\ds\\int \\cot x\\, dx = \\log |\\sin x|$, $\\ds\\int\\sec^{2} x\\, dx = \\tan x$, $\\ds\\int \\cosec^{2} x\\, dx = -\\cot x$, $\\ds\\int \\tan x\\sec x\\, dx = \\sec x$, $\\ds\\int \\cot x \\cosec x\\, dx = -\\cosec x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "csc(x)**2", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: csc(x)**2" ], "shape": [ "integrate: csc(x)**N" ], "same_problem_in": [], "needs": [ "cas.integrate", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-liii/2e", "set": "hardy-course-of-pure-mathematics-1921/ex-liii", "number": 2, "part": "e", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "247", "location": "Exercise LIII, problem 2e", "problem_latex": "$\\ds\\int \\tan x\\, dx = -\\log |\\cos x|$, $\\ds\\int \\cot x\\, dx = \\log |\\sin x|$, $\\ds\\int\\sec^{2} x\\, dx = \\tan x$,\n$\\ds\\int \\cosec^{2} x\\, dx = -\\cot x$, $\\ds\\int \\tan x\\sec x\\, dx = \\sec x$, $\\ds\\int \\cot x \\cosec x\\, dx = -\\cosec x$.", "markdown": "$\\ds\\int \\tan x\\, dx = -\\log |\\cos x|$, $\\ds\\int \\cot x\\, dx = \\log |\\sin x|$, $\\ds\\int\\sec^{2} x\\, dx = \\tan x$, $\\ds\\int \\cosec^{2} x\\, dx = -\\cot x$, $\\ds\\int \\tan x\\sec x\\, dx = \\sec x$, $\\ds\\int \\cot x \\cosec x\\, dx = -\\cosec x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "tan(x)*sec(x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: tan(x)*sec(x)" ], "shape": [ "integrate: tan(x)*sec(x)" ], "same_problem_in": [], "needs": [ "cas.integrate", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-liii/2f", "set": "hardy-course-of-pure-mathematics-1921/ex-liii", "number": 2, "part": "f", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "247", "location": "Exercise LIII, problem 2f", "problem_latex": "$\\ds\\int \\tan x\\, dx = -\\log |\\cos x|$, $\\ds\\int \\cot x\\, dx = \\log |\\sin x|$, $\\ds\\int\\sec^{2} x\\, dx = \\tan x$,\n$\\ds\\int \\cosec^{2} x\\, dx = -\\cot x$, $\\ds\\int \\tan x\\sec x\\, dx = \\sec x$, $\\ds\\int \\cot x \\cosec x\\, dx = -\\cosec x$.", "markdown": "$\\ds\\int \\tan x\\, dx = -\\log |\\cos x|$, $\\ds\\int \\cot x\\, dx = \\log |\\sin x|$, $\\ds\\int\\sec^{2} x\\, dx = \\tan x$, $\\ds\\int \\cosec^{2} x\\, dx = -\\cot x$, $\\ds\\int \\tan x\\sec x\\, dx = \\sec x$, $\\ds\\int \\cot x \\cosec x\\, dx = -\\cosec x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "cot(x)*csc(x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: cot(x)*csc(x)" ], "shape": [ "integrate: cot(x)*csc(x)" ], "same_problem_in": [], "needs": [ "cas.integrate", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-liii/3", "set": "hardy-course-of-pure-mathematics-1921/ex-liii", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "247", "location": "Exercise LIII, problem 3", "problem_latex": "Show that the integral of $1/(a + b\\cos x)$, where $a + b$~is positive, may\nbe expressed in one or other of the forms\n\\[\n\\frac{2}{\\sqrtp{a^{2} - b^{2}}}\n \\arctan \\left\\{t\\bigsqrtp{\\frac{a - b}{a + b}}\\right\\},\\quad\n\\frac{1}{\\sqrtp{b^{2} - a^{2}}}\n \\log \\left|\\frac{\\sqrtp{b + a} + t\\sqrtp{b - a}}\n {\\sqrtp{b + a} - t\\sqrtp{b - a}}\\right|,\n\\]\nwhere $t = \\tan\\frac{1}{2}x$, according as $a^{2} > b^{2}$ or $a^{2} < b^{2}$. If $a^{2} = b^{2}$ then the integral\nreduces to a constant multiple of that of $\\sec^{2}\\frac{1}{2}x$ or $\\cosec^{2}\\frac{1}{2}x$, and its value\nmay at once be written down. Deduce the forms of the integral when $a + b$\nis negative.", "markdown": "Show that the integral of $1/(a + b\\cos x)$, where $a + b$ is positive, may be expressed in one or other of the forms 2a^2 - b^2 ta - ba + b,0pt minus 3pt1b^2 - a^2 |b + a + tb - a b + a - tb - a|, where $t = \\tan\\frac{1}{2}x$, according as $a^{2} > b^{2}$ or $a^{2} < b^{2}$. If $a^{2} = b^{2}$ then the integral reduces to a constant multiple of that of $\\sec^{2}\\frac{1}{2}x$ or $\\cosec^{2}\\frac{1}{2}x$, and its value may at once be written down. Deduce the forms of the integral when $a + b$ is negative.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate", "cas.subst", "core.frac", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-liii/4", "set": "hardy-course-of-pure-mathematics-1921/ex-liii", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "247", "location": "Exercise LIII, problem 4", "problem_latex": "Show that if $y$~is defined in terms of~$x$ by means of the equation\n\\\\[\n(a + b\\\\cos x)(a - b\\\\cos y) = a^{2} - b^{2},\n\\\\]\nwhere $a$~is positive and $a^{2} > b^{2}$, then as $x$~varies from $0$~to~$\\\\pi$ one value of~$y$\nalso varies from $0$~to~$\\\\pi$. Show also that\n\\\\[\n\\\\sin x = \\\\frac{\\\\sqrtp{a^{2} - b^{2}} \\\\sin y}{a - b\\\\cos y},\\\\quad\n\\\\frac{\\\\sin x}{a + b\\\\cos x}\\\\, \\\\frac{dx}{dy} = \\\\frac{\\\\sin y}{a - b\\\\cos y};\n\\\\PageSep{248}\nand deduce that if $0 < x < \\\\pi$ then\n\\\\[\n\\\\int \\\\frac{dx}{a + b\\\\cos x}\n = \\\\frac{1}{\\\\sqrtp{a^{2} - b^{2}}}\n \\\\arccos \\\\left(\\\\frac{a\\\\cos x + b}{a + b\\\\cos x}\\\\right).\n\\\\]\n\nShow that this result agrees with that of Ex.~3.", "markdown": "Show that if $y$ is defined in terms of $x$ by means of the equation [ (a + bcos x)(a - bcos y) = a^2 - b^2, ] where $a$ is positive and $a^{2} > b^{2}$, then as $x$ varies from $0$ to $\\\\pi$ one value of $y$ also varies from $0$ to $\\\\pi$. Show also that [ sin x = fracsqrtpa^2 - b^2 sin ya - bcos y,quad fracsin xa + bcos x, fracdxdy = fracsin ya - bcos y; PageSep248 and deduce that if $0 < x < \\\\pi$ then [ int fracdxa + bcos x = frac1sqrtpa^2 - b^2 arccos left(fracacos x + ba + bcos xright). ] Show that this result agrees with that of Ex. 3.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "FLAG-EXTRACTION", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.simplify", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-liii/5", "set": "hardy-course-of-pure-mathematics-1921/ex-liii", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "247", "location": "Exercise LIII, problem 5", "problem_latex": "Show how to integrate $1/(a + b\\cos x + c\\sin x)$.", "markdown": "Show how to integrate $1/(a + b\\cos x + c\\sin x)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-liii/6", "set": "hardy-course-of-pure-mathematics-1921/ex-liii", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "247", "location": "Exercise LIII, problem 6", "problem_latex": "Integrate $(a + b\\cos x + c\\sin x)/(\\alpha + \\beta\\cos x + \\gamma\\sin x)$\\Add{.}", "markdown": "Integrate $(a + b\\cos x + c\\sin x)/(\\alpha + \\beta\\cos x + \\gamma\\sin x)$", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "(a + b*cos(x) + c*sin(x))/(alpha + beta*cos(x) + gamma*sin(x))", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: (a + c*cos(x) + e*sin(x))/(b + d*cos(x) + f*sin(x))" ], "shape": [ "integrate: (a + c*cos(x) + e*sin(x))/(b + d*cos(x) + f*sin(x))" ], "same_problem_in": [], "needs": [ "cas.integrate", "core.linsys", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-liii/7", "set": "hardy-course-of-pure-mathematics-1921/ex-liii", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "247", "location": "Exercise LIII, problem 7", "problem_latex": "Integrate $1/(a\\cos^{2} x + 2b\\cos x\\sin x + c\\sin^{2} x)$.", "markdown": "Integrate $1/(a\\cos^{2} x + 2b\\cos x\\sin x + c\\sin^{2} x)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "1/(a*cos(x)**2 + 2*b*cos(x)*sin(x) + c*sin(x)**2)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: 1/(a*cos(x)**2 + 2*b*sin(x)*cos(x) + c*sin(x)**2)" ], "shape": [ "integrate: 1/(N*b*sin(x)*cos(x) + a*cos(x)**N + c*sin(x)**N)" ], "same_problem_in": [], "needs": [ "cas.integrate", "cas.subst", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-liv/1", "set": "hardy-course-of-pure-mathematics-1921/ex-liv", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "251", "location": "Exercise LIV, problem 1", "problem_latex": "Calculate the area of the segment cut off from the\nparabola $y = x^{2}/4a$ by the ordinate $x = \\xi$, and the length of the arc which\nbounds it.", "markdown": "Calculate the area of the segment cut off from the parabola $y = x^{2}/4a$ by the ordinate $x = \\xi$, and the length of the arc which bounds it.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-liv/10", "set": "hardy-course-of-pure-mathematics-1921/ex-liv", "number": 10, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "251", "location": "Exercise LIV, problem 10", "problem_latex": "Find the area of the loop of the curve $x^{5} + y^{5} = 5ax^{2}y^{2}$.", "markdown": "Find the area of the loop of the curve $x^{5} + y^{5} = 5ax^{2}y^{2}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-liv/11", "set": "hardy-course-of-pure-mathematics-1921/ex-liv", "number": 11, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "251", "location": "Exercise LIV, problem 11", "problem_latex": "Prove that the area of a loop of the curve $x = a\\sin 2t$, $y = a\\sin t$ is~$\\frac{4}{3}a^{2}$. \\MathTrip{1908.}", "markdown": "Prove that the area of a loop of the curve $x = a\\sin 2t$, $y = a\\sin t$ is $\\frac{4}{3}a^{2}$. % [0]% (*Math. Trip.* 1908.)% [1]%", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-liv/12", "set": "hardy-course-of-pure-mathematics-1921/ex-liv", "number": 12, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "251", "location": "Exercise LIV, problem 12", "problem_latex": "The arc of the ellipse given by $x = a\\cos t$, $y = b\\sin t$, between the\npoints $t = t_{1}$ and $t = t_{2}$, is $F(t_{2}) - F(t_{1})$, where\n\\[\nF(t) = a\\int \\sqrtp{1 - e^{2}\\sin^{2} t}\\, dt,\n\\]\n$e$~being the eccentricity. [This integral cannot however be evaluated in\nterms of such functions as are at present at our disposal.]", "markdown": "The arc of the ellipse given by $x = a\\cos t$, $y = b\\sin t$, between the points $t = t_{1}$ and $t = t_{2}$, is $F(t_{2}) - F(t_{1})$, where F(t) = a1 - e^2^2 t  dt, $e$ being the eccentricity. [This integral cannot however be evaluated in terms of such functions as are at present at our disposal.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-liv/13a", "set": "hardy-course-of-pure-mathematics-1921/ex-liv", "number": 13, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "251", "location": "Exercise LIV, problem 13a", "problem_latex": "\\Topic{Polar coordinates.} Show that the area bounded by the curve\n$r = f(\\theta)$, where $f(\\theta)$~is a one-valued function of~$\\theta$, and the radii $\\theta = \\theta_{1}$, $\\theta = \\theta_{2}$, is\n$F(\\theta_{2}) - F(\\theta_{1})$, where $\\ds F(\\theta) = \\tfrac{1}{2} \\int r^{2}\\, d\\theta$. And the length of the corresponding\narc of the curve is $\\Phi(\\theta_{2}) - \\Phi(\\theta_{1})$, where\n\\[\n\\Phi(\\theta)\n = \\bigint \\bigsqrtb{r^{2} + \\biggl(\\frac{dr}{d\\theta}\\biggr)^{2}}\\, d\\theta.\n\\]\n\nHence determine (i)~the area and perimeter of the circle $r = 2a\\sin\\theta$;\n(ii)~the area between the parabola $r = \\frac{1}{2}l\\sec^{2} \\frac{1}{2}\\theta$ and its latus rectum, and the\nlength of the corresponding arc of the parabola; (iii)~the area of the limaçon\n$r = a + b\\cos\\theta$, distinguishing the cases in which $a > b$, $a = b$, and $a < b$;\nand (iv)~the areas of the ellipses $1/r^{2} = a\\cos^{2} \\theta + 2h\\cos\\theta\\sin\\theta + b\\sin^{2} \\theta$ and\n$l/r = 1 + e\\cos\\theta$. [In the last case we are led to the integral $\\ds \\int \\frac{d\\theta}{(1 + e\\cos\\theta)^{2}}$,\nwhich may be calculated (cf.\\ \\Ex{liii}.~4) by the help of the substitution\n\\[\n(1 + e\\cos\\theta) (1 - e\\cos\\phi) = 1 - e^{2}.]\n\\]", "markdown": "**coordinates.** Show that the area bounded by the curve $r = f(\\theta)$, where $f(\\theta)$ is a one-valued function of $\\theta$, and the radii $\\theta = \\theta_{1}$, $\\theta = \\theta_{2}$, is $F(\\theta_{2}) - F(\\theta_{1})$, where $\\ds F(\\theta) = \\tfrac{1}{2} \\int r^{2}\\, d\\theta$. And the length of the corresponding arc of the curve is $\\Phi(\\theta_{2}) - \\Phi(\\theta_{1})$, where () = r^2 + (drd)^2  d. Hence determine (i) the area and perimeter of the circle $r = 2a\\sin\\theta$; (ii) the area between the parabola $r = \\frac{1}{2}l\\sec^{2} \\frac{1}{2}\\theta$ and its latus rectum, and the length of the corresponding arc of the parabola; (iii) the area of the limaçon $r = a + b\\cos\\theta$, distinguishing the cases in which $a > b$, $a = b$, and $a < b$; and (iv) the areas of the ellipses $1/r^{2} = a\\cos^{2} \\theta + 2h\\cos\\theta\\sin\\theta + b\\sin^{2} \\theta$ and $l/r = 1 + e\\cos\\theta$. [In the last case we are led to the integral $\\ds \\int \\frac{d\\theta}{(1 + e\\cos\\theta)^{2}}$, which may be calculated (cf. % [examples:liii]Ex. liii%. 4) by the help of the substitution (1 + e) (1 - e) = 1 - e^2.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-liv/13b", "set": "hardy-course-of-pure-mathematics-1921/ex-liv", "number": 13, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "251", "location": "Exercise LIV, problem 13b", "problem_latex": "\\Topic{Polar coordinates.} Show that the area bounded by the curve\n$r = f(\\theta)$, where $f(\\theta)$~is a one-valued function of~$\\theta$, and the radii $\\theta = \\theta_{1}$, $\\theta = \\theta_{2}$, is\n$F(\\theta_{2}) - F(\\theta_{1})$, where $\\ds F(\\theta) = \\tfrac{1}{2} \\int r^{2}\\, d\\theta$. And the length of the corresponding\narc of the curve is $\\Phi(\\theta_{2}) - \\Phi(\\theta_{1})$, where\n\\[\n\\Phi(\\theta)\n = \\bigint \\bigsqrtb{r^{2} + \\biggl(\\frac{dr}{d\\theta}\\biggr)^{2}}\\, d\\theta.\n\\]\n\nHence determine (i)~the area and perimeter of the circle $r = 2a\\sin\\theta$;\n(ii)~the area between the parabola $r = \\frac{1}{2}l\\sec^{2} \\frac{1}{2}\\theta$ and its latus rectum, and the\nlength of the corresponding arc of the parabola; (iii)~the area of the limaçon\n$r = a + b\\cos\\theta$, distinguishing the cases in which $a > b$, $a = b$, and $a < b$;\nand (iv)~the areas of the ellipses $1/r^{2} = a\\cos^{2} \\theta + 2h\\cos\\theta\\sin\\theta + b\\sin^{2} \\theta$ and\n$l/r = 1 + e\\cos\\theta$. [In the last case we are led to the integral $\\ds \\int \\frac{d\\theta}{(1 + e\\cos\\theta)^{2}}$,\nwhich may be calculated (cf.\\ \\Ex{liii}.~4) by the help of the substitution\n\\[\n(1 + e\\cos\\theta) (1 - e\\cos\\phi) = 1 - e^{2}.]\n\\]", "markdown": "**coordinates.** Show that the area bounded by the curve $r = f(\\theta)$, where $f(\\theta)$ is a one-valued function of $\\theta$, and the radii $\\theta = \\theta_{1}$, $\\theta = \\theta_{2}$, is $F(\\theta_{2}) - F(\\theta_{1})$, where $\\ds F(\\theta) = \\tfrac{1}{2} \\int r^{2}\\, d\\theta$. And the length of the corresponding arc of the curve is $\\Phi(\\theta_{2}) - \\Phi(\\theta_{1})$, where () = r^2 + (drd)^2  d. Hence determine (i) the area and perimeter of the circle $r = 2a\\sin\\theta$; (ii) the area between the parabola $r = \\frac{1}{2}l\\sec^{2} \\frac{1}{2}\\theta$ and its latus rectum, and the length of the corresponding arc of the parabola; (iii) the area of the limaçon $r = a + b\\cos\\theta$, distinguishing the cases in which $a > b$, $a = b$, and $a < b$; and (iv) the areas of the ellipses $1/r^{2} = a\\cos^{2} \\theta + 2h\\cos\\theta\\sin\\theta + b\\sin^{2} \\theta$ and $l/r = 1 + e\\cos\\theta$. [In the last case we are led to the integral $\\ds \\int \\frac{d\\theta}{(1 + e\\cos\\theta)^{2}}$, which may be calculated (cf. % [examples:liii]Ex. liii%. 4) by the help of the substitution (1 + e) (1 - e) = 1 - e^2.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-liv/13c", "set": "hardy-course-of-pure-mathematics-1921/ex-liv", "number": 13, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "251", "location": "Exercise LIV, problem 13c", "problem_latex": "\\Topic{Polar coordinates.} Show that the area bounded by the curve\n$r = f(\\theta)$, where $f(\\theta)$~is a one-valued function of~$\\theta$, and the radii $\\theta = \\theta_{1}$, $\\theta = \\theta_{2}$, is\n$F(\\theta_{2}) - F(\\theta_{1})$, where $\\ds F(\\theta) = \\tfrac{1}{2} \\int r^{2}\\, d\\theta$. And the length of the corresponding\narc of the curve is $\\Phi(\\theta_{2}) - \\Phi(\\theta_{1})$, where\n\\[\n\\Phi(\\theta)\n = \\bigint \\bigsqrtb{r^{2} + \\biggl(\\frac{dr}{d\\theta}\\biggr)^{2}}\\, d\\theta.\n\\]\n\nHence determine (i)~the area and perimeter of the circle $r = 2a\\sin\\theta$;\n(ii)~the area between the parabola $r = \\frac{1}{2}l\\sec^{2} \\frac{1}{2}\\theta$ and its latus rectum, and the\nlength of the corresponding arc of the parabola; (iii)~the area of the limaçon\n$r = a + b\\cos\\theta$, distinguishing the cases in which $a > b$, $a = b$, and $a < b$;\nand (iv)~the areas of the ellipses $1/r^{2} = a\\cos^{2} \\theta + 2h\\cos\\theta\\sin\\theta + b\\sin^{2} \\theta$ and\n$l/r = 1 + e\\cos\\theta$. [In the last case we are led to the integral $\\ds \\int \\frac{d\\theta}{(1 + e\\cos\\theta)^{2}}$,\nwhich may be calculated (cf.\\ \\Ex{liii}.~4) by the help of the substitution\n\\[\n(1 + e\\cos\\theta) (1 - e\\cos\\phi) = 1 - e^{2}.]\n\\]", "markdown": "**coordinates.** Show that the area bounded by the curve $r = f(\\theta)$, where $f(\\theta)$ is a one-valued function of $\\theta$, and the radii $\\theta = \\theta_{1}$, $\\theta = \\theta_{2}$, is $F(\\theta_{2}) - F(\\theta_{1})$, where $\\ds F(\\theta) = \\tfrac{1}{2} \\int r^{2}\\, d\\theta$. And the length of the corresponding arc of the curve is $\\Phi(\\theta_{2}) - \\Phi(\\theta_{1})$, where () = r^2 + (drd)^2  d. Hence determine (i) the area and perimeter of the circle $r = 2a\\sin\\theta$; (ii) the area between the parabola $r = \\frac{1}{2}l\\sec^{2} \\frac{1}{2}\\theta$ and its latus rectum, and the length of the corresponding arc of the parabola; (iii) the area of the limaçon $r = a + b\\cos\\theta$, distinguishing the cases in which $a > b$, $a = b$, and $a < b$; and (iv) the areas of the ellipses $1/r^{2} = a\\cos^{2} \\theta + 2h\\cos\\theta\\sin\\theta + b\\sin^{2} \\theta$ and $l/r = 1 + e\\cos\\theta$. [In the last case we are led to the integral $\\ds \\int \\frac{d\\theta}{(1 + e\\cos\\theta)^{2}}$, which may be calculated (cf. % [examples:liii]Ex. liii%. 4) by the help of the substitution (1 + e) (1 - e) = 1 - e^2.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-liv/13d", "set": "hardy-course-of-pure-mathematics-1921/ex-liv", "number": 13, "part": "d", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "251", "location": "Exercise LIV, problem 13d", "problem_latex": "\\Topic{Polar coordinates.} Show that the area bounded by the curve\n$r = f(\\theta)$, where $f(\\theta)$~is a one-valued function of~$\\theta$, and the radii $\\theta = \\theta_{1}$, $\\theta = \\theta_{2}$, is\n$F(\\theta_{2}) - F(\\theta_{1})$, where $\\ds F(\\theta) = \\tfrac{1}{2} \\int r^{2}\\, d\\theta$. And the length of the corresponding\narc of the curve is $\\Phi(\\theta_{2}) - \\Phi(\\theta_{1})$, where\n\\[\n\\Phi(\\theta)\n = \\bigint \\bigsqrtb{r^{2} + \\biggl(\\frac{dr}{d\\theta}\\biggr)^{2}}\\, d\\theta.\n\\]\n\nHence determine (i)~the area and perimeter of the circle $r = 2a\\sin\\theta$;\n(ii)~the area between the parabola $r = \\frac{1}{2}l\\sec^{2} \\frac{1}{2}\\theta$ and its latus rectum, and the\nlength of the corresponding arc of the parabola; (iii)~the area of the limaçon\n$r = a + b\\cos\\theta$, distinguishing the cases in which $a > b$, $a = b$, and $a < b$;\nand (iv)~the areas of the ellipses $1/r^{2} = a\\cos^{2} \\theta + 2h\\cos\\theta\\sin\\theta + b\\sin^{2} \\theta$ and\n$l/r = 1 + e\\cos\\theta$. [In the last case we are led to the integral $\\ds \\int \\frac{d\\theta}{(1 + e\\cos\\theta)^{2}}$,\nwhich may be calculated (cf.\\ \\Ex{liii}.~4) by the help of the substitution\n\\[\n(1 + e\\cos\\theta) (1 - e\\cos\\phi) = 1 - e^{2}.]\n\\]", "markdown": "**coordinates.** Show that the area bounded by the curve $r = f(\\theta)$, where $f(\\theta)$ is a one-valued function of $\\theta$, and the radii $\\theta = \\theta_{1}$, $\\theta = \\theta_{2}$, is $F(\\theta_{2}) - F(\\theta_{1})$, where $\\ds F(\\theta) = \\tfrac{1}{2} \\int r^{2}\\, d\\theta$. And the length of the corresponding arc of the curve is $\\Phi(\\theta_{2}) - \\Phi(\\theta_{1})$, where () = r^2 + (drd)^2  d. Hence determine (i) the area and perimeter of the circle $r = 2a\\sin\\theta$; (ii) the area between the parabola $r = \\frac{1}{2}l\\sec^{2} \\frac{1}{2}\\theta$ and its latus rectum, and the length of the corresponding arc of the parabola; (iii) the area of the limaçon $r = a + b\\cos\\theta$, distinguishing the cases in which $a > b$, $a = b$, and $a < b$; and (iv) the areas of the ellipses $1/r^{2} = a\\cos^{2} \\theta + 2h\\cos\\theta\\sin\\theta + b\\sin^{2} \\theta$ and $l/r = 1 + e\\cos\\theta$. [In the last case we are led to the integral $\\ds \\int \\frac{d\\theta}{(1 + e\\cos\\theta)^{2}}$, which may be calculated (cf. % [examples:liii]Ex. liii%. 4) by the help of the substitution (1 + e) (1 - e) = 1 - e^2.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-liv/14", "set": "hardy-course-of-pure-mathematics-1921/ex-liv", "number": 14, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "251", "location": "Exercise LIV, problem 14", "problem_latex": "Trace the curve $2\\theta = (a/r) + (r/a)$, and show that the area bounded\nby the radius vector $\\theta = \\beta$, and the two branches which touch at the point\n$r = a$, $\\theta = 1$, is $\\frac{2}{3} a^{2}(\\beta^{2} - 1)^{3/2}$. \\MathTrip{1900.}", "markdown": "Trace the curve $2\\theta = (a/r) + (r/a)$, and show that the area bounded by the radius vector $\\theta = \\beta$, and the two branches which touch at the point $r = a$, $\\theta = 1$, is $\\frac{2}{3} a^{2}(\\beta^{2} - 1)^{3/2}$. % [0]% (*Math. Trip.* 1900.)% [1]%", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-liv/15", "set": "hardy-course-of-pure-mathematics-1921/ex-liv", "number": 15, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "251", "location": "Exercise LIV, problem 15", "problem_latex": "A curve is given by an equation $p = f(r)$, $r$~being the radius vector\nand $p$~the perpendicular from the origin on to the tangent. Show that the\ncalculation of the area of the region bounded by an arc of the curve and two\nradii vectores depends upon that of the integral $\\frac{1}{2} \\ds \\int \\frac{pr\\, dr}{\\sqrtp{r^{2} - p^{2}}}$.", "markdown": "A curve is given by an equation $p = f(r)$, $r$ being the radius vector and $p$ the perpendicular from the origin on to the tangent. Show that the calculation of the area of the region bounded by an arc of the curve and two radii vectores depends upon that of the integral $\\frac{1}{2} \\ds \\int \\frac{pr\\, dr}{\\sqrtp{r^{2} - p^{2}}}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-liv/2", "set": "hardy-course-of-pure-mathematics-1921/ex-liv", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "251", "location": "Exercise LIV, problem 2", "problem_latex": "Answer the same questions for the curve $ay^{2} = x^{3}$, showing that the\nlength of the arc is\n\\[\n\\frac{8a}{27} \\left\\{\\left(1 + \\frac{9\\xi}{4a}\\right)^{3/2} - 1\\right\\}.\n\\]", "markdown": "Answer the same questions for the curve $ay^{2} = x^{3}$, showing that the length of the arc is 8a27 (1 + 94a)^3/2 - 1.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-liv/3", "set": "hardy-course-of-pure-mathematics-1921/ex-liv", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "251", "location": "Exercise LIV, problem 3", "problem_latex": "Calculate the areas and lengths of the circles $x^{2} + y^{2} = a^{2}$, $x^{2} + y^{2} = 2ax$\nby means of the formulae of \\SecNo[§§]{145}--\\SecNo{146}.", "markdown": "Calculate the areas and lengths of the circles $x^{2} + y^{2} = a^{2}$, $x^{2} + y^{2} = 2ax$ by means of the formulae of [§§]145--146.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-liv/4", "set": "hardy-course-of-pure-mathematics-1921/ex-liv", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "251", "location": "Exercise LIV, problem 4", "problem_latex": "Show that the area of the ellipse $(x^{2}/a^{2}) + (y^{2}/b^{2}) = 1$ is~$\\pi ab$.", "markdown": "Show that the area of the ellipse $(x^{2}/a^{2}) + (y^{2}/b^{2}) = 1$ is $\\pi ab$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-liv/5", "set": "hardy-course-of-pure-mathematics-1921/ex-liv", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "251", "location": "Exercise LIV, problem 5", "problem_latex": "Find the area bounded by the curve $y = \\sin x$ and the segment of the\naxis of~$x$ from $x = 0$ to $x = 2\\pi$. [Here $\\Phi(x) = -\\cos x$, and the difference\nbetween the values of $-\\cos x$ for $x = 0$ and $x = 2\\pi$ is zero. The explanation of\nthis is of course that between $x = \\pi$ and $x = 2\\pi$ the curve lies below the axis\nof~$x$, and so the corresponding part of the area is counted negative in applying\nthe method. The area from $x = 0$ to $x = \\pi$ is $-\\cos \\pi + \\cos 0 = 2$; and the\nwhole area required, when every part is counted positive, is twice this,\n\\ie\\ is~$4$.]", "markdown": "Find the area bounded by the curve $y = \\sin x$ and the segment of the axis of $x$ from $x = 0$ to $x = 2\\pi$. [Here $\\Phi(x) = -\\cos x$, and the difference between the values of $-\\cos x$ for $x = 0$ and $x = 2\\pi$ is zero. The explanation of this is of course that between $x = \\pi$ and $x = 2\\pi$ the curve lies below the axis of $x$, and so the corresponding part of the area is counted negative in applying the method. The area from $x = 0$ to $x = \\pi$ is $-\\cos \\pi + \\cos 0 = 2$; and the whole area required, when every part is counted positive, is twice this, *i.e.* is $4$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.defint", "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-liv/6", "set": "hardy-course-of-pure-mathematics-1921/ex-liv", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "251", "location": "Exercise LIV, problem 6", "problem_latex": "Suppose that the coordinates of any point on a curve are expressed\nas functions of a parameter~$t$ by equations of the type $x = \\phi(t)$, $y = \\psi(t)$,\n$\\phi$~and~$\\psi$ being functions of~$t$ with continuous derivatives. Prove that\nif $x$~steadily increases as $t$~varies from $t_{0}$ to~$t_{1}$, then the area of the region\nbounded by the corresponding portion of the curve, the axis of~$x$, and the two\nordinates corresponding to $t_{0}$ and~$t_{1}$, is, apart from sign, $A(t_{1}) - A(t_{0})$, where\n\\[\nA(t) = \\int \\psi(t)\\phi'(t)\\, dt = \\int y \\frac{dx}{dt}\\, dt.\n\\]", "markdown": "Suppose that the coordinates of any point on a curve are expressed as functions of a parameter $t$ by equations of the type $x = \\phi(t)$, $y = \\psi(t)$, $\\phi$ and $\\psi$ being functions of $t$ with continuous derivatives. Prove that if $x$ steadily increases as $t$ varies from $t_{0}$ to $t_{1}$, then the area of the region bounded by the corresponding portion of the curve, the axis of $x$, and the two ordinates corresponding to $t_{0}$ and $t_{1}$, is, apart from sign, $A(t_{1}) - A(t_{0})$, where A(t) = (t)’(t)  dt = y dxdt  dt.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-liv/7", "set": "hardy-course-of-pure-mathematics-1921/ex-liv", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "251", "location": "Exercise LIV, problem 7", "problem_latex": "Suppose that $C$~is a closed curve formed of a single loop and not\nmet by any parallel to either axis in more than two points. And suppose\nthat the coordinates of any point~$P$ on the curve can be expressed as in Ex.~6\nin terms of~$t$, and that, as $t$~varies from $t_{0}$ to~$t_{1}$, $P$~moves in the same\ndirection round the curve and returns after a single circuit to its original\nposition. Show that the area of the loop is equal to the difference of the\ninitial and final values of any one of the integrals\n\\[\n-\\int y \\frac{dx}{dt}\\, dt,\\quad\n \\int x \\frac{dy}{dt}\\, dt,\\quad\n\\tfrac{1}{2} \\int \\left(x \\frac{dy}{dt} - y \\frac{dx}{dt}\\right) dt,\n\\]\nthis difference being of course taken positively.", "markdown": "Suppose that $C$ is a closed curve formed of a single loop and not met by any parallel to either axis in more than two points. And suppose that the coordinates of any point $P$ on the curve can be expressed as in Ex. 6 in terms of $t$, and that, as $t$ varies from $t_{0}$ to $t_{1}$, $P$ moves in the same direction round the curve and returns after a single circuit to its original position. Show that the area of the loop is equal to the difference of the initial and final values of any one of the integrals -y dxdt  dt,0pt minus 3pt x dydt  dt,0pt minus 3pt12 (x dydt - y dxdt) dt, this difference being of course taken positively.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-liv/8a", "set": "hardy-course-of-pure-mathematics-1921/ex-liv", "number": 8, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "251", "location": "Exercise LIV, problem 8a", "problem_latex": "Apply the result of Ex.~7 to determine the areas of the curves\ngiven by\n\\[\n\\Itemp{(i)}\n\\frac{x}{a} = \\frac{1 - t^{2}}{1 + t^{2}},\\quad\n\\frac{y}{a} = \\frac{2t}{1 + t^{2}},\\qquad\n\\Itemp{(ii)}\nx = a\\cos^{3} t,\\quad\ny = b\\sin^{3} t.\n\\]", "markdown": "Apply the result of Ex. 7 to determine the areas of the curves given by % [2.25em][l](i)% [2.25em][l](i)% % xa = 1 - t^21 + t^2,0pt minus 3ptya = 2t1 + t^2, % [2.25em][l](ii)% [2.25em][l](ii)% % x = a^3 t,0pt minus 3pty = b^3 t.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-liv/8b", "set": "hardy-course-of-pure-mathematics-1921/ex-liv", "number": 8, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "251", "location": "Exercise LIV, problem 8b", "problem_latex": "Apply the result of Ex.~7 to determine the areas of the curves\ngiven by\n\\[\n\\Itemp{(i)}\n\\frac{x}{a} = \\frac{1 - t^{2}}{1 + t^{2}},\\quad\n\\frac{y}{a} = \\frac{2t}{1 + t^{2}},\\qquad\n\\Itemp{(ii)}\nx = a\\cos^{3} t,\\quad\ny = b\\sin^{3} t.\n\\]", "markdown": "Apply the result of Ex. 7 to determine the areas of the curves given by % [2.25em][l](i)% [2.25em][l](i)% % xa = 1 - t^21 + t^2,0pt minus 3ptya = 2t1 + t^2, % [2.25em][l](ii)% [2.25em][l](ii)% % x = a^3 t,0pt minus 3pty = b^3 t.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-liv/9", "set": "hardy-course-of-pure-mathematics-1921/ex-liv", "number": 9, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "251", "location": "Exercise LIV, problem 9", "problem_latex": "Find the area of the loop of the curve $x^{3} + y^{3} = 3axy$. [Putting\n$y = tx$ we obtain $x = 3at/(1 + t^{3})$, $y = 3at^{2}/(1 + t^{3})$. As $t$~varies from~$0$ towards~$\\infty$\nthe loop is described once. Also\n\\[\n\\tfrac{1}{2} \\int \\left(y \\frac{dx}{dt} - x \\frac{dy}{dt}\\right)\\, dt\n = -\\tfrac{1}{2} \\int x^{2} \\frac{d}{dt}\\left(\\frac{y}{x}\\right)\\, dt\n = -\\tfrac{1}{2} \\int \\frac{9a^{2}t^{2}}{(1 + t^{3})^{2}}\\, dt\n = \\frac{3a^{2}}{2(1 + t^{3})},\n\\]\nwhich tends to~$0$ as $t \\to \\infty$. Thus the area of the loop is~$\\frac{3}{2}a^{2}$.]", "markdown": "Find the area of the loop of the curve $x^{3} + y^{3} = 3axy$. [Putting $y = tx$ we obtain $x = 3at/(1 + t^{3})$, $y = 3at^{2}/(1 + t^{3})$. As $t$ varies from $0$ towards $\\infty$ the loop is described once. Also 12 (y dxdt - x dydt)  dt = -12 x^2 ddt(yx)  dt = -12 9a^2t^2(1 + t^3)^2  dt = 3a^22(1 + t^3), which tends to $0$ as $t \\to \\infty$. Thus the area of the loop is $\\frac{3}{2}a^{2}$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.cancel", "cas.derive", "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lix/1", "set": "hardy-course-of-pure-mathematics-1921/ex-lix", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "272", "location": "Exercise LIX, problem 1", "problem_latex": "Let $\\phi(x) = ax + b$, so that $y = \\phi(x)$ is a straight line.\nThe conditions for contact at the point for which $x = \\xi$ are $f(\\xi) = a\\xi + b$,\n$f'(\\xi) = a$. If we determine $a$~and~$b$ so as to satisfy these equations we find\n$a = f'(\\xi)$, $b = f(\\xi) - \\xi f'(\\xi)$, and the equation of the tangent to $y = f(x)$ at the\npoint $x = \\xi$ is\n\\[\ny = xf'(\\xi) + \\{f(\\xi) - \\xi f'(\\xi)\\},\n\\]\nor $y - f(\\xi) = (x - \\xi)f'(\\xi)$. Cf.\\ \\Ex{xxxix}.~5.", "markdown": "Let $\\phi(x) = ax + b$, so that $y = \\phi(x)$ is a straight line. The conditions for contact at the point for which $x = \\xi$ are $f(\\xi) = a\\xi + b$, $f'(\\xi) = a$. If we determine $a$ and $b$ so as to satisfy these equations we find $a = f'(\\xi)$, $b = f(\\xi) - \\xi f'(\\xi)$, and the equation of the tangent to $y = f(x)$ at the point $x = \\xi$ is y = xf’() + f() - f’(), or $y - f(\\xi) = (x - \\xi)f'(\\xi)$. Cf. % [examples:xxxix]Ex. xxxix%. 5.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lix/10", "set": "hardy-course-of-pure-mathematics-1921/ex-lix", "number": 10, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "272", "location": "Exercise LIX, problem 10", "problem_latex": "Verify that the curvature of a circle is constant and equal to the\nreciprocal of the radius; and show that the circle is the only curve whose\ncurvature is constant.", "markdown": "Verify that the curvature of a circle is constant and equal to the reciprocal of the radius; and show that the circle is the only curve whose curvature is constant.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lix/11", "set": "hardy-course-of-pure-mathematics-1921/ex-lix", "number": 11, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "272", "location": "Exercise LIX, problem 11", "problem_latex": "{\\Loosen Find the centre and radius of curvature at any point of the conics\n$y^{2} = 4ax$, $(x/a)^{2} + (y/b)^{2} = 1$.}", "markdown": "0.375em plus 0.75em minus 0.25emFind the centre and radius of curvature at any point of the conics $y^{2} = 4ax$, $(x/a)^{2} + (y/b)^{2} = 1$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lix/12", "set": "hardy-course-of-pure-mathematics-1921/ex-lix", "number": 12, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "272", "location": "Exercise LIX, problem 12", "problem_latex": "In an ellipse the radius of curvature at~$P$ is~$CD^{3}/ab$, where $CD$~is\nthe semi-diameter conjugate to~$CP$.", "markdown": "In an ellipse the radius of curvature at $P$ is $CD^{3}/ab$, where $CD$ is the semi-diameter conjugate to $CP$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lix/13", "set": "hardy-course-of-pure-mathematics-1921/ex-lix", "number": 13, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "272", "location": "Exercise LIX, problem 13", "problem_latex": "Show that in general a conic can be drawn to have contact of the\nfourth order with the curve $y = f(x)$ at a given point~$P$.\n\n[Take the general equation of a conic, viz.\n\\[\nax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0,\n\\]\nand differentiate four times with respect to~$x$. Using suffixes to denote\ndifferentiation we obtain\n\\begin{align*}\nax + hy + g + (hx + by + f) y_{1} &= 0,\\\\\na + 2hy_{1} + by_{1}^{2} + (hx + by + f) y_{2} &= 0,\\\\\n3(h + by_{1}) y_{2} + (hx + by + f) y_{3} &= 0,\\\\\n4(h + by_{1}) y_{3} +3by_{2}^{2} + (hx + by + f) y_{4} &= 0.\n\\end{align*}\nIf the conic has contact of the fourth order, then these five equations must\nbe satisfied by writing $\\xi$, $\\eta$, $\\eta_{1}$, $\\eta_{2}$, $\\eta_{3}$, $\\eta_{4}$, for $x$, $y$, $y_{1}$, $y_{2}$, $y_{3}$, $y_{4}$. We have thus\njust enough equations to determine the ratios $a : b : c : f : g : h$.]", "markdown": "Show that in general a conic can be drawn to have contact of the fourth order with the curve $y = f(x)$ at a given point $P$. [Take the general equation of a conic, viz. ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0, and differentiate four times with respect to $x$. Using suffixes to denote differentiation we obtain align* ax + hy + g + (hx + by + f) y_1 &= 0, a + 2hy_1 + by_1^2 + (hx + by + f) y_2 &= 0, 3(h + by_1) y_2 + (hx + by + f) y_3 &= 0, 4(h + by_1) y_3 +3by_2^2 + (hx + by + f) y_4 &= 0. align* If the conic has contact of the fourth order, then these five equations must be satisfied by writing $\\xi$, $\\eta$, $\\eta_{1}$, $\\eta_{2}$, $\\eta_{3}$, $\\eta_{4}$, for $x$, $y$, $y_{1}$, $y_{2}$, $y_{3}$, $y_{4}$. We have thus just enough equations to determine the ratios $a : b : c : f : g : h$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lix/14", "set": "hardy-course-of-pure-mathematics-1921/ex-lix", "number": 14, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "272", "location": "Exercise LIX, problem 14", "problem_latex": "An infinity of conics can be drawn having contact of the third order\nwith the curve at~$P$. Show that their centres all lie on a straight line.\n\n[Take the tangent and normal as axes. Then the equation of the conic is\nof the form $2y = ax^{2} + 2hxy + by^{2}$, and when $x$~is small one value of~$y$ may be\nexpressed (\\okrickRef{Ch.}{V}, \\MiscEx{V}~22) in the form\n\\[\ny = \\tfrac{1}{2}ax^{2} + \\left(\\tfrac{1}{2}ah + \\epsilon_{x}\\right) x^{3},\n\\]\nwhere $\\epsilon_{x} \\to 0$ with~$x$. But this expression must be the same as\n\\[\ny = \\tfrac{1}{2}f''(0) x^{2} + \\{\\tfrac{1}{6}f'''(0) + \\epsilon'_{x}\\} x^{3},\n\\]\nwhere $\\epsilon'_{x} \\to 0$ with~$x$, and so $a = f''(0)$, $h = f'''(0)/3f''(0)$, in virtue of the result\nof \\Ex{lv}.~15. But the centre lies on the line $ax + hy = 0$.]", "markdown": "An infinity of conics can be drawn having contact of the third order with the curve at $P$. Show that their centres all lie on a straight line. [Take the tangent and normal as axes. Then the equation of the conic is of the form $2y = ax^{2} + 2hxy + by^{2}$, and when $x$ is small one value of $y$ may be expressed (Ch.V, [misc:V]Misc. Ex. 22) in the form y = 12ax^2 + (12ah + _x) x^3, where $\\epsilon_{x} \\to 0$ with $x$. But this expression must be the same as y = 12f”(0) x^2 + 16f”’(0) + ’_x x^3, where $\\epsilon'_{x} \\to 0$ with $x$, and so $a = f''(0)$, $h = f'''(0)/3f''(0)$, in virtue of the result of % [examples:lv]Ex. lv%. 15. But the centre lies on the line $ax + hy = 0$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.series" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lix/15", "set": "hardy-course-of-pure-mathematics-1921/ex-lix", "number": 15, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "272", "location": "Exercise LIX, problem 15", "problem_latex": "Determine a parabola which has contact of the third order with the\nellipse $(x/a)^{2} + (y/b)^{2} = 1$ at the extremity of the major axis.", "markdown": "Determine a parabola which has contact of the third order with the ellipse $(x/a)^{2} + (y/b)^{2} = 1$ at the extremity of the major axis.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lix/16", "set": "hardy-course-of-pure-mathematics-1921/ex-lix", "number": 16, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "272", "location": "Exercise LIX, problem 16", "problem_latex": "The locus of the centres of conics which have contact of the third\norder with the ellipse $(x/a)^{2} + (y/b)^{2} = 1$ at the point $(a\\cos\\alpha, b\\sin\\alpha)$ is the\ndiameter $x/(a\\cos\\alpha) = y/(b\\sin\\alpha)$. [For the ellipse itself is one such conic.]", "markdown": "The locus of the centres of conics which have contact of the third order with the ellipse $(x/a)^{2} + (y/b)^{2} = 1$ at the point $(a\\cos\\alpha, b\\sin\\alpha)$ is the diameter $x/(a\\cos\\alpha) = y/(b\\sin\\alpha)$. [For the ellipse itself is one such conic.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lix/2", "set": "hardy-course-of-pure-mathematics-1921/ex-lix", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "272", "location": "Exercise LIX, problem 2", "problem_latex": "The fact that the line is to have simple contact with the curve\ncompletely determines the line. In order that the tangent should have\n\\emph{contact of the second order} with the curve we must have $f''(\\xi) = \\phi''(\\xi)$, \\ie\\\n$f''(\\xi) = 0$. A point at which the tangent to a curve has contact of the\nsecond order is called a \\Emph{point of inflexion}.\n%[** TN: Differs from the modern definition]", "markdown": "The fact that the line is to have simple contact with the curve completely determines the line. In order that the tangent should have *contact of the second order* with the curve we must have $f''(\\xi) = \\phi''(\\xi)$, *i.e.* $f''(\\xi) = 0$. A point at which the tangent to a curve has contact of the second order is called a **of inflexion**. %[** TN: Differs from the modern definition]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lix/3", "set": "hardy-course-of-pure-mathematics-1921/ex-lix", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "272", "location": "Exercise LIX, problem 3", "problem_latex": "Find the points of inflexion on the graphs of the functions $3x^{4} - 6x^{3} + 1$,\n$2x/(1 + x^{2})$, $\\sin x$, $a\\cos^{2}x + b\\sin^{2}x$, $\\tan x$, $\\arctan x$.", "markdown": "Find the points of inflexion on the graphs of the functions $3x^{4} - 6x^{3} + 1$, $2x/(1 + x^{2})$, $\\sin x$, $a\\cos^{2}x + b\\sin^{2}x$, $\\tan x$, $\\arctan x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.solve.poly", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lix/4", "set": "hardy-course-of-pure-mathematics-1921/ex-lix", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "272", "location": "Exercise LIX, problem 4", "problem_latex": "Show that the conic $ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0$ cannot have a\npoint of inflexion. [Here $ax + hy + g + (hx + by + f)y_{1} = 0$ and\n\\[\na + 2hy_{1} + by_{1}^{2} + (hx + by + f)y_{2} = 0,\n\\]\nsuffixes denoting differentiations. Thus at a point of inflexion\n\\[\na + 2hy_{1} + by_{1}^{2} = 0,\n\\]\nor\n\\[\na(hx + by + f)^{2} - 2h(ax + hy + g)(hx + by + f) + b(ax + hy + g)^{2} = 0,\n\\]\nor\n\\[\n(ab - h^{2})\\{ax^{2} + 2hxy + by^{2} + 2gx + 2fy\\} + af^{2} - 2fgh + bg^{2} = 0.\n\\]\nBut this is inconsistent with the equation of the conic unless\n\\[\naf^{2} - 2fgh + bg^{2} = c(ab - h^{2})\n\\]\nor $abc + 2fgh - af^{2} - bg^{2} - ch^{2} = 0$; and this is the condition that the conic\nshould degenerate into two straight lines.]", "markdown": "Show that the conic $ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0$ cannot have a point of inflexion. [Here $ax + hy + g + (hx + by + f)y_{1} = 0$ and a + 2hy_1 + by_1^2 + (hx + by + f)y_2 = 0, suffixes denoting differentiations. Thus at a point of inflexion a + 2hy_1 + by_1^2 = 0, or a(hx + by + f)^2 - 2h(ax + hy + g)(hx + by + f) + b(ax + hy + g)^2 = 0, or (ab - h^2)ax^2 + 2hxy + by^2 + 2gx + 2fy + af^2 - 2fgh + bg^2 = 0. But this is inconsistent with the equation of the conic unless af^2 - 2fgh + bg^2 = c(ab - h^2) or $abc + 2fgh - af^{2} - bg^{2} - ch^{2} = 0$; and this is the condition that the conic should degenerate into two straight lines.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lix/5", "set": "hardy-course-of-pure-mathematics-1921/ex-lix", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "272", "location": "Exercise LIX, problem 5", "problem_latex": "The curve $y = (ax^{2} + 2bx + c)/(\\alpha x^{2} + 2\\beta x + \\gamma)$ has one or three points of\ninflexion according as the roots of $\\alpha x^{2} + 2\\beta x + \\gamma = 0$ are real or complex.\n\n[The equation of the curve can, by a change of origin (cf.\\ \\Ex{xlvi}.~15), be\nreduced to the form\n\\[\n\\eta = \\xi/(A\\xi^{2} + 2B\\xi + C) = \\xi/\\{A(\\xi - p)(\\xi - q)\\},\n\\]\nwhere $p$,~$q$ are real or conjugate. The condition for a point of inflexion will\nbe found to be $\\xi^{3} - 3pq\\xi + pq(p + q) = 0$, which has one or three real roots\naccording as $\\DPtypo{\\{pq(p - q)\\}}{\\{pq(p - q)\\}^{2}}$ is positive or negative, \\ie\\ according as $p$~and~$q$ are\nreal or conjugate.]\n\\PageSep{273}", "markdown": "The curve $y = (ax^{2} + 2bx + c)/(\\alpha x^{2} + 2\\beta x + \\gamma)$ has one or three points of inflexion according as the roots of $\\alpha x^{2} + 2\\beta x + \\gamma = 0$ are real or complex. [The equation of the curve can, by a change of origin (cf. % [examples:xlvi]Ex. xlvi%. 15), be reduced to the form = /(A^2 + 2B+ C) = /A(- p)(- q), where $p$, $q$ are real or conjugate. The condition for a point of inflexion will be found to be $\\xi^{3} - 3pq\\xi + pq(p + q) = 0$, which has one or three real roots according as $\\DPtypo{\\{pq(p - q)\\}}{\\{pq(p - q)\\}^{2}}$ is positive or negative, *i.e.* according as $p$ and $q$ are real or conjugate.] [pg]273", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lix/6", "set": "hardy-course-of-pure-mathematics-1921/ex-lix", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "272", "location": "Exercise LIX, problem 6", "problem_latex": "Discuss in particular the curves $y = (1 - x)/(1 + x^{2})$, $y = (1 - x^{2})/(1 + x^{2})$,\n$y = (1 + x^{2})/(1 - x^{2})$.", "markdown": "Discuss in particular the curves $y = (1 - x)/(1 + x^{2})$, $y = (1 - x^{2})/(1 + x^{2})$, $y = (1 + x^{2})/(1 - x^{2})$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lix/7", "set": "hardy-course-of-pure-mathematics-1921/ex-lix", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "272", "location": "Exercise LIX, problem 7", "problem_latex": "Show that when the curve of Ex.~5 has three points of inflexion, they\nlie on a straight line. [The equation $\\xi^{3} - 3pq\\xi + pq(p + q) = 0$ can be put in\nthe form $(\\xi - p)(\\xi - q)(\\xi + p + q) + (p - q)^{2}\\xi = 0$, so that the points of inflexion\nlie on the line $\\xi + A(p - q)^{2}\\eta + p + q = 0$ or $A\\xi - 4(AC - B^{2})\\eta = 2B$.]", "markdown": "Show that when the curve of Ex. 5 has three points of inflexion, they lie on a straight line. [The equation $\\xi^{3} - 3pq\\xi + pq(p + q) = 0$ can be put in the form $(\\xi - p)(\\xi - q)(\\xi + p + q) + (p - q)^{2}\\xi = 0$, so that the points of inflexion lie on the line $\\xi + A(p - q)^{2}\\eta + p + q = 0$ or $A\\xi - 4(AC - B^{2})\\eta = 2B$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lix/8", "set": "hardy-course-of-pure-mathematics-1921/ex-lix", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "272", "location": "Exercise LIX, problem 8", "problem_latex": "Show that the curves $y = x\\sin x$, $y = (\\sin x)/x$ have each infinitely\nmany points of inflexion.", "markdown": "Show that the curves $y = x\\sin x$, $y = (\\sin x)/x$ have each infinitely many points of inflexion.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lix/9", "set": "hardy-course-of-pure-mathematics-1921/ex-lix", "number": 9, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "272", "location": "Exercise LIX, problem 9", "problem_latex": "\\Topic{Contact of a circle with a curve. Curvature.\\footnote\n {A much fuller discussion of the theory of curvature will be found in Mr~Fowler's\n%[** TN: Reference on page 272 of orig. points to page 266.]\n tract referred to on \\PageRef{p.}{\\DPchg{272}{266}}.}}\nThe general\nequation of a circle, viz.\n\\[\n(x - a)^{2} + (y - b)^{2} = r^{2},\n\\Tag{(1)}\n\\]\ncontains three arbitrary constants. Let us attempt to determine them so\nthat the circle has contact of as high an order as possible with the curve\n$y = f(x)$ at the point $(\\xi, \\eta)$, where $\\eta = f(\\xi)$. We write $\\eta_{1}$,~$\\eta_{2}$ for $f'(\\xi)$,~$f''(\\xi)$.\nDifferentiating the equation of the circle twice we obtain\n\\begin{align}\n(x - a) + (y - b)y_{1} &= 0,\n\\Tag{(2)}\\\\\n1 + y_{1}^{2} + (y - b)y_{2} &= 0.\n\\Tag{(3)}\n\\end{align}\n\nIf the circle touches the curve then the equations \\Eq{(1)}~and~\\Eq{(2)} are satisfied\nwhen $x = \\xi$, $y = \\eta$, $y_{1} = \\eta_{1}$. This gives $(\\xi - a)/\\eta_{1} = -(\\eta - b) = r/\\sqrtp{1 + \\eta_{1}^{2}}$. If\nthe contact is of the second order then the equation~\\Eq{(3)} must also be satisfied\nwhen $y_{2} = \\eta_{2}$. Thus $b = \\eta + \\{(1 + \\eta_{1}^{2})/\\eta_{2}\\}$; and hence we find\n\\[\na = \\xi - \\frac{\\eta_{1}(1 + \\eta_{1}^{2})}{\\eta_{2}},\\quad\nb = \\eta + \\frac{1 + \\eta_{1}^{2}}{\\eta_{2}},\\quad\nr = \\frac{(1 + \\eta_{1}^{2})^{3/2}}{\\eta_{2}}.\n\\]\n\nThe circle which has contact of the second order with the curve at the point\n$(\\xi, \\eta)$ is called the \\Emph{circle of curvature}, and its radius the \\Emph{radius of curvature}.\nThe \\Emph{measure of curvature} (or simply the \\emph{curvature}) is the reciprocal of the\nradius: thus the measure of curvature is $f''(\\xi)/\\{1 + [f'(\\xi)]^{2}\\}^{3/2}$, or\n\\[\n\\frac{d^{2}\\eta}{d\\xi^{2}} \\bigg/\n \\biggl\\{1 + \\biggl(\\frac{d\\eta}{d\\xi}\\biggr)^{2}\\biggr\\}^{3/2}.\n\\]", "markdown": "**of a circle with a curve. Curvature. A much fuller discussion of the theory of curvature will be found in Mr Fowler’s %[** TN: Reference on page 272 of orig. points to page 266.] tract referred to on p.272266.** The general equation of a circle, viz. (x - a)^2 + (y - b)^2 = r^2, (1) contains three arbitrary constants. Let us attempt to determine them so that the circle has contact of as high an order as possible with the curve $y = f(x)$ at the point $(\\xi, \\eta)$, where $\\eta = f(\\xi)$. We write $\\eta_{1}$, $\\eta_{2}$ for $f'(\\xi)$, $f''(\\xi)$. Differentiating the equation of the circle twice we obtain align (x - a) + (y - b)y_1 &= 0, (2) 1 + y_1^2 + (y - b)y_2 &= 0. (3) align If the circle touches the curve then the equations (1) and (2) are satisfied when $x = \\xi$, $y = \\eta$, $y_{1} = \\eta_{1}$. This gives $(\\xi - a)/\\eta_{1} = -(\\eta - b) = r/\\sqrtp{1 + \\eta_{1}^{2}}$. If the contact is of the second order then the equation (3) must also be satisfied when $y_{2} = \\eta_{2}$. Thus $b = \\eta + \\{(1 + \\eta_{1}^{2})/\\eta_{2}\\}$; and hence we find a = - _1(1 + _1^2)_2,0pt minus 3ptb = + 1 + _1^2_2,0pt minus 3ptr = (1 + _1^2)^3/2_2. The circle which has contact of the second order with the curve at the point $(\\xi, \\eta)$ is called the **of curvature**, and its radius the **of curvature**. The **of curvature** (or simply the *curvature*) is the reciprocal of the radius: thus the measure of curvature is $f''(\\xi)/\\{1 + [f'(\\xi)]^{2}\\}^{3/2}$, or d^2d^2 / 1 + (dd)^2^3/2.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lv/1", "set": "hardy-course-of-pure-mathematics-1921/ex-lv", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "264", "location": "Exercise LV, problem 1", "problem_latex": "Suppose that $f(x)$~is a polynomial of degree~$r$.\nThen $f^{(n)}(x)$~is identically zero when $n > r$, and the theorem leads to the\nalgebraical identity\n\\[\nf(a + h) = f(a) + hf'(a) + \\frac{h^{2}}{2!} f''(a) + \\dots\n + \\frac{h^{r}}{r!} f^{(r)}(a).\n\\]", "markdown": "Suppose that $f(x)$ is a polynomial of degree $r$. Then $f^{(n)}(x)$ is identically zero when $n > r$, and the theorem leads to the algebraical identity f(a + h) = f(a) + hf’(a) + h^22! f”(a) + … + h^rr! f^(r)(a).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lv/10", "set": "hardy-course-of-pure-mathematics-1921/ex-lv", "number": 10, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "264", "location": "Exercise LV, problem 10", "problem_latex": "Show that the error in taking the root to be $\\xi - (f/f') - \\frac{1}{2}(f^{2}f''/f'^{3})$,\nwhere $\\xi$~is the argument of every function, is in general of the third order.", "markdown": "Show that the error in taking the root to be $\\xi - (f/f') - \\frac{1}{2}(f^{2}f''/f'^{3})$, where $\\xi$ is the argument of every function, is in general of the third order.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lv/11", "set": "hardy-course-of-pure-mathematics-1921/ex-lv", "number": 11, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "264", "location": "Exercise LV, problem 11", "problem_latex": "The equation $\\sin x = \\alpha x$, where $\\alpha$~is small, has a root nearly equal to~$\\pi$.\nShow that $(1 - \\alpha)\\pi$~is a better approximation, and $(1 - \\alpha + \\alpha^{2})\\pi$ a better\nstill. [The method of Exs.~7--10 does not depend on $f(x) = 0$ being an\nalgebraical equation, so long as $f'$~and~$f''$ are continuous.]", "markdown": "The equation $\\sin x = \\alpha x$, where $\\alpha$ is small, has a root nearly equal to $\\pi$. Show that $(1 - \\alpha)\\pi$ is a better approximation, and $(1 - \\alpha + \\alpha^{2})\\pi$ a better still. [The method of Exs. 7--10 does not depend on $f(x) = 0$ being an algebraical equation, so long as $f'$ and $f''$ are continuous.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.solve.num", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lv/12", "set": "hardy-course-of-pure-mathematics-1921/ex-lv", "number": 12, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "264", "location": "Exercise LV, problem 12", "problem_latex": "Show that the limit when $h \\to 0$ of the number~$\\theta_{n}$ which occurs in\nthe general Mean Value Theorem is~$1/(n + 1)$, provided that $f^{(n+1)}(x)$~is\ncontinuous.\n\n[For $f(x + h)$~is equal to each of\n\\[\nf(x) + \\dots + \\frac{h^{n}}{n!} f^{(n)}(x + \\theta_{n}h),\\quad\nf(x) + \\dots + \\frac{h^{n}}{n!} f^{(n)}(x)\n + \\frac{h^{n+1}}{(n + 1)!} f^{(n+1)}(x + \\theta_{n+1}h),\n\\]\nwhere $\\theta_{n+1}$ as well as~$\\theta_{n}$ lies between $0$~and~$1$. Hence\n\\[\nf^{(n)}(x + \\theta_{n}h)\n = f^{(n)}(x) + \\frac{hf^{(n+1)}(x + \\theta_{n+1}h)}{n + 1}\\Add{.}\n\\]\nBut if we apply the original Mean Value Theorem to the function~$f^{(n)}(x)$,\ntaking $\\theta_{n}h$ in place of~$h$, we find\n\\[\nf^{(n)}(x + \\theta_{n}h)\n = f^{(n)}(x) + \\theta_{n}hf^{(n+1)}(x + \\theta\\theta_{n}h),\n\\]\n\\PageSep{266}\nwhere $\\theta$ also lies between $0$~and~$1$. Hence\n\\[\n\\theta_{n} f^{(n+1)}(x + \\theta\\theta_{n} h)\n = \\frac{f^{(n+1)}(x + \\theta_{n+1} h)}{n + 1},\n\\]\nfrom which the result follows, since $f^{(n+1)}(x + \\theta\\theta_{n} h)$ and $f^{(n+1)}(x + \\theta_{n+1} h)$ tend\nto the same limit~$f^{(n+1)}(x)$ as $h \\to 0$.]", "markdown": "Show that the limit when $h \\to 0$ of the number $\\theta_{n}$ which occurs in the general Mean Value Theorem is $1/(n + 1)$, provided that $f^{(n+1)}(x)$ is continuous. [For $f(x + h)$ is equal to each of f(x) + …+ h^nn! f^(n)(x + _nh),0pt minus 3ptf(x) + …+ h^nn! f^(n)(x) + h^n+1(n + 1)! f^(n+1)(x + _n+1h), where $\\theta_{n+1}$ as well as $\\theta_{n}$ lies between $0$ and $1$. Hence f^(n)(x + _nh) = f^(n)(x) + hf^(n+1)(x + _n+1h)n + 1 But if we apply the original Mean Value Theorem to the function $f^{(n)}(x)$, taking $\\theta_{n}h$ in place of $h$, we find f^(n)(x + _nh) = f^(n)(x) + _nhf^(n+1)(x + _nh), [pg]266 where $\\theta$ also lies between $0$ and $1$. Hence _n f^(n+1)(x + _n h) = f^(n+1)(x + _n+1 h)n + 1, from which the result follows, since $f^{(n+1)}(x + \\theta\\theta_{n} h)$ and $f^{(n+1)}(x + \\theta_{n+1} h)$ tend to the same limit $f^{(n+1)}(x)$ as $h \\to 0$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lv/13", "set": "hardy-course-of-pure-mathematics-1921/ex-lv", "number": 13, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "264", "location": "Exercise LV, problem 13", "problem_latex": "Prove that $\\{f(x + 2h) - 2f(x + h) + f(x)\\}/h^{2} \\to f''(x)$ as $h \\to 0$, provided\nthat $f''(x)$~is continuous. [Use equation~\\Eq{(2)} of~\\SecNo[§]{147}.]", "markdown": "Prove that $\\{f(x + 2h) - 2f(x + h) + f(x)\\}/h^{2} \\to f''(x)$ as $h \\to 0$, provided that $f''(x)$ is continuous. [Use equation (2) of [§]147.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lv/14", "set": "hardy-course-of-pure-mathematics-1921/ex-lv", "number": 14, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "264", "location": "Exercise LV, problem 14", "problem_latex": "Show that, if the $f^{(n)}(x)$ is continuous for $x = 0$, then\n\\[\nf(x) = a_{0} + a_{1}x + a_{2}x^{2} + \\dots + (a_{n} + \\epsilon_{x}) x^{n},\n\\]\nwhere $a_{r} = f^{(r)}(0)/r!$ and $\\epsilon_{x} \\to 0$ as $x \\to 0$.\\footnote\n {It is in fact sufficient to suppose that \\emph{$f^{(n)}(0)$~exists}. See R.~H. Fowler, ``The\n elementary differential geometry of plane curves'' (\\textit{Cambridge Tracts in Mathematics},\n No.~20, p.~104).\\PageLabel{266}}", "markdown": "Show that, if the $f^{(n)}(x)$ is continuous for $x = 0$, then f(x) = a_0 + a_1x + a_2x^2 + …+ (a_n + _x) x^n, where $a_{r} = f^{(r)}(0)/r!$ and $\\epsilon_{x} \\to 0$ as $x \\to 0$. It is in fact sufficient to suppose that *$f^{(n)}(0)$ exists*. See R. H. Fowler, “The elementary differential geometry of plane curves” (*Cambridge Tracts in Mathematics*, No. 20, p. 104).266", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lv/15", "set": "hardy-course-of-pure-mathematics-1921/ex-lv", "number": 15, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "264", "location": "Exercise LV, problem 15", "problem_latex": "Show that if\n\\[\na_{0} + a_{1}x + a_{2}x^{2} + \\dots + (a_{n} + \\epsilon_{x}) x^{n} =\nb_{0} + b_{1}x + b_{2}x^{2} + \\dots + (b_{n} + \\eta_{x}) x^{n},\n\\]\nwhere $\\epsilon_{x}$ and~$\\eta_{x}$ tend to zero as $x \\to 0$, then $a_{0} = b_{0}$, $a_{1} = b_{1}$,~\\dots, $a_{n} = b_{n}$. [Making\n$x \\to 0$ we see that $a_{0} = b_{0}$. Now divide by~$x$ and afterwards make $x \\to 0$.\nWe thus obtain $a_{1} = b_{1}$; and this process may be repeated as often as is\nnecessary. It follows that if $f(x) = a_{0} + a_{1}x + a_{2}x^{2} + \\dots + (a_{n} + \\epsilon_{x}) x^{n}$, and the\nfirst~$n$ derivatives of~$f(x)$ are continuous, then $a_{r} = f^{(r)}(0)/r!$.]", "markdown": "Show that if a_0 + a_1x + a_2x^2 + …+ (a_n + _x) x^n = b_0 + b_1x + b_2x^2 + …+ (b_n + _x) x^n, where $\\epsilon_{x}$ and $\\eta_{x}$ tend to zero as $x \\to 0$, then $a_{0} = b_{0}$, $a_{1} = b_{1}$, …, $a_{n} = b_{n}$. [Making $x \\to 0$ we see that $a_{0} = b_{0}$. Now divide by $x$ and afterwards make $x \\to 0$. We thus obtain $a_{1} = b_{1}$; and this process may be repeated as often as is necessary. It follows that if $f(x) = a_{0} + a_{1}x + a_{2}x^{2} + \\dots + (a_{n} + \\epsilon_{x}) x^{n}$, and the first $n$ derivatives of $f(x)$ are continuous, then $a_{r} = f^{(r)}(0)/r!$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.limit.num" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lv/2", "set": "hardy-course-of-pure-mathematics-1921/ex-lv", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "264", "location": "Exercise LV, problem 2", "problem_latex": "By applying the theorem to $f(x) = 1/x$, and supposing $x$ and~$x + h$\npositive, obtain the result\n\\[\n\\frac{1}{x + h} = \\frac{1}{x} - \\frac{h}{x^{2}} + \\frac{h^{2}}{x^{3}} - \\dots\n + \\frac{(-1)^{n-1} h^{n-1}}{x^{n}}\n + \\frac{(-1)^{n} h^{n}}{(x + \\theta_{n} h)^{n+1}}.\n\\]\n\n[Since\n\\[\n\\frac{1}{x + h} = \\frac{1}{x} - \\frac{h}{x^{2}} + \\frac{h^{2}}{x^{3}} - \\dots\n + \\frac{(-1)^{n-1} h^{n-1}}{x^{n}}\n + \\frac{(-1)^{n} h^{n}}{x^{n}(x + h)},\\quad%[** TN: Quick spacing hack]\n\\]\nwe can verify the result by showing that $x^{n}(x + h)$ can be put in the form\n$(x + \\theta_{n}h)^{n+1}$, or that $x^{n+1} < x^{n}(x + h) < (x + h)^{n+1}$, as is evidently the case.]", "markdown": "By applying the theorem to $f(x) = 1/x$, and supposing $x$ and $x + h$ positive, obtain the result 1x + h = 1x - hx^2 + h^2x^3 - … + (-1)^n-1 h^n-1x^n + (-1)^n h^n(x + _n h)^n+1. [Since 1x + h = 1x - hx^2 + h^2x^3 - … + (-1)^n-1 h^n-1x^n + (-1)^n h^nx^n(x + h),0pt minus 3pt%[** TN: Quick spacing hack] we can verify the result by showing that $x^{n}(x + h)$ can be put in the form $(x + \\theta_{n}h)^{n+1}$, or that $x^{n+1} < x^{n}(x + h) < (x + h)^{n+1}$, as is evidently the case.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.arith" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lv/3", "set": "hardy-course-of-pure-mathematics-1921/ex-lv", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "264", "location": "Exercise LV, problem 3", "problem_latex": "Obtain the formula\n\\begin{multline*}\n\\sin(x + h)\n = \\sin x + h\\cos x - \\frac{h^{2}}{2!}\\sin x - h^{3}\\frac{3!}\\cos x + \\dots\\\\\n + (-1)^{n-1}\\frac{h^{2n-1}}{(2n - 1)!}\\cos x\n + (-1)^{n} h^{2n}\\frac{2n!}\\sin(x + \\theta_{2n} h),\n\\end{multline*}\nthe corresponding formula for $\\cos(x + h)$, and similar formulae involving\npowers of~$h$ extending up to~$h^{2n+1}$.", "markdown": "Obtain the formula multline* (x + h) = x + hx - h^22!x - h^33!x + … + (-1)^n-1h^2n-1(2n - 1)!x + (-1)^n h^2n2n!(x + _2n h), multline* the corresponding formula for $\\cos(x + h)$, and similar formulae involving powers of $h$ extending up to $h^{2n+1}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lv/4", "set": "hardy-course-of-pure-mathematics-1921/ex-lv", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "264", "location": "Exercise LV, problem 4", "problem_latex": "Show that if $m$~is a positive integer, and $n$~a positive integer not\ngreater than~$m$, then\n\\[\n(x + h)^{m} = x^{m} + \\binom{m}{1}x^{m-1} h + \\dots\n + \\binom{m}{n - 1}x^{m-n+1} h^{n-1}\n + \\binom{m}{n}(x + \\theta_{n} h)^{m-n} h^{n}.\n\\]\nShow also that, if the interval $\\DPmod{(x, x + h)}{[x, x + h]}$ does not include $x = 0$, the formula\nholds for all real values of~$m$ and all positive integral values of~$n$; and that,\neven if $x < 0 < x + h$ or $x + h < 0 < x$, the formula still holds if $m - n$~is\npositive.", "markdown": "Show that if $m$ is a positive integer, and $n$ a positive integer not greater than $m$, then (x + h)^m = x^m + m1x^m-1 h + … + mn - 1x^m-n+1 h^n-1 + mn(x + _n h)^m-n h^n. Show also that, if the interval $\\DPmod{(x, x + h)}{[x, x + h]}$ does not include $x = 0$, the formula holds for all real values of $m$ and all positive integral values of $n$; and that, even if $x < 0 < x + h$ or $x + h < 0 < x$, the formula still holds if $m - n$ is positive.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lv/5", "set": "hardy-course-of-pure-mathematics-1921/ex-lv", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "264", "location": "Exercise LV, problem 5", "problem_latex": "The formula $f(x + h) = f(x) + hf'(x + \\theta_{1}h)$ is not true if $f(x) = 1/x$ and\n$x < 0 < x + h$. [For $f(x + h) - f(x) > 0$ and $hf'(x + \\theta_{1} h) = -h/(x + \\theta_{1} h)^{2} < 0$; it\nis evident that the conditions for the truth of the Mean Value Theorem are\nnot satisfied.]", "markdown": "The formula $f(x + h) = f(x) + hf'(x + \\theta_{1}h)$ is not true if $f(x) = 1/x$ and $x < 0 < x + h$. [For $f(x + h) - f(x) > 0$ and $hf'(x + \\theta_{1} h) = -h/(x + \\theta_{1} h)^{2} < 0$; it is evident that the conditions for the truth of the Mean Value Theorem are not satisfied.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.arith" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lv/6", "set": "hardy-course-of-pure-mathematics-1921/ex-lv", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "264", "location": "Exercise LV, problem 6", "problem_latex": "If $x = -a$, $h = 2a$, $f(x) = x^{1/3}$, then the equation\n\\[\nf(x + h) = f(x) + hf'(x + \\theta_{1} h)\n\\]\nis satisfied by $\\theta_{1} = \\frac{1}{2} ± \\frac{1}{18}\\sqrt{3}$. [This example shows that the result of the\ntheorem may hold even if the conditions under which it was proved are\nnot satisfied.]", "markdown": "If $x = -a$, $h = 2a$, $f(x) = x^{1/3}$, then the equation f(x + h) = f(x) + hf’(x + _1 h) is satisfied by $\\theta_{1} = \\frac{1}{2} ± \\frac{1}{18}\\sqrt{3}$. [This example shows that the result of the theorem may hold even if the conditions under which it was proved are not satisfied.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.arith" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lv/7", "set": "hardy-course-of-pure-mathematics-1921/ex-lv", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "264", "location": "Exercise LV, problem 7", "problem_latex": "\\Topic{Newton's method of approximation to the roots of equations.} Let\n$\\xi$~be an approximation to a root of an algebraical equation $f(x) = 0$, the actual\nroot being~$\\xi + h$. Then\n\\[\n0 = f(\\xi + h) = f(\\xi) + hf'(\\xi) + \\tfrac{1}{2} h^{2}f''(\\xi + \\theta_{2}h),\n\\]\nso that\n\\[\nh = -\\frac{f(\\xi)}{f'(\\xi)}\n - \\tfrac{1}{2} h^{2} \\frac{f''(\\xi + \\theta_{2}h)}{f'(\\xi)}.\n\\]\n\nIt follows that in general a better approximation than $x = \\xi$ is\n\\[\nx = \\xi - \\frac{f(\\xi)}{f'(\\xi)}.\n\\]\nIf the root is a simple root, so that $f'(\\xi + h) \\neq 0$, we can, when $h$~is small\nenough, find a positive constant~$K$ such that $|f'(x)| > K$ for all the values of~$x$\nwhich we are considering, and then, if $h$~is regarded as of the first order of\nsmallness, $f(\\xi)$~is of the first order of smallness, and the error in taking\n$\\xi - \\{f(\\xi)/f'(\\xi)\\}$ as the root is of the second order.", "markdown": "**’s method of approximation to the roots of equations.** Let $\\xi$ be an approximation to a root of an algebraical equation $f(x) = 0$, the actual root being $\\xi + h$. Then 0 = f(+ h) = f() + hf’() + 12 h^2f”(+ _2h), so that h = -f()f’() - 12 h^2 f”(+ _2h)f’(). It follows that in general a better approximation than $x = \\xi$ is x = - f()f’(). If the root is a simple root, so that $f'(\\xi + h) \\neq 0$, we can, when $h$ is small enough, find a positive constant $K$ such that $|f'(x)| > K$ for all the values of $x$ which we are considering, and then, if $h$ is regarded as of the first order of smallness, $f(\\xi)$ is of the first order of smallness, and the error in taking $\\xi - \\{f(\\xi)/f'(\\xi)\\}$ as the root is of the second order.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lv/8", "set": "hardy-course-of-pure-mathematics-1921/ex-lv", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "264", "location": "Exercise LV, problem 8", "problem_latex": "Apply this process to the equation $x^{2} = 2$, taking $\\xi = 3/2$ as the first\napproximation. [We find $h = -1/12$, $\\xi + h = 17/12 = 1.417\\dots$, which is quite a\ngood approximation, in spite of the roughness of the first. If now we repeat\nthe process, taking $\\xi = 17/12$, we obtain $\\xi + h = 577/408 = 1.414\\MS215\\dots$, which\nis correct to $5$~places of decimals.\\Add{]}", "markdown": "Apply this process to the equation $x^{2} = 2$, taking $\\xi = 3/2$ as the first approximation. [We find $h = -1/12$, $\\xi + h = 17/12 = 1.417\\dots$, which is quite a good approximation, in spite of the roughness of the first. If now we repeat the process, taking $\\xi = 17/12$, we obtain $\\xi + h = 577/408 = 1.414\\MS215\\dots$, which is correct to $5$ places of decimals.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.frac", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lv/9", "set": "hardy-course-of-pure-mathematics-1921/ex-lv", "number": 9, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "264", "location": "Exercise LV, problem 9", "problem_latex": "By considering in this way the equation $x^{2} - 1 - y = 0$, where $y$~is\nsmall, show that $\\sqrtp{1 + y} = 1 + \\frac{1}{2} y - \\{\\frac{1}{4}y^{2}/(2 + y)\\}$ approximately, the error being\nof the fourth order.", "markdown": "By considering in this way the equation $x^{2} - 1 - y = 0$, where $y$ is small, show that $\\sqrtp{1 + y} = 1 + \\frac{1}{2} y - \\{\\frac{1}{4}y^{2}/(2 + y)\\}$ approximately, the error being of the fourth order.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lvi/1", "set": "hardy-course-of-pure-mathematics-1921/ex-lvi", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "267", "location": "Exercise LVI, problem 1", "problem_latex": "Let $f(x) = \\sin x$. Then all the derivatives of~$f(x)$\nare continuous for all values of~$x$. Also $|f^{n}(x)| \\leq 1$ for all values of $x$~and~$n$.\nHence in this case $|R_{n}| \\leq h^{n}/n!$, which tends to zero as $n \\to \\infty$ (\\Ex{xxvii}.~12)\nwhatever value $h$ may have. It follows that\n\\[\n\\sin(x + h) = \\sin x + h\\cos x - \\frac{h^{2}}{2!}\\sin x\n - \\frac{h^{3}}{3!}\\cos x + \\frac{h^{4}}{4!}\\sin x + \\dots,\n\\]\nfor all values of $x$~and~$h$. In particular\n\\[\n\\sin h = h - \\frac{h^{3}}{3!} + \\frac{h^{5}}{5!} - \\dots,\n\\]\nfor all values of~$h$. Similarly we can prove that\n\\[\n\\cos(x + h) = \\cos x - h\\sin x - \\frac{h^{2}}{2!}\\cos x\n + \\frac{h^{3}}{3!} \\sin x + \\dots,\\quad\n\\cos h = 1 - \\frac{h^{2}}{2!} + \\frac{h^{4}}{4!} - \\dots.", "markdown": "Let $f(x) = \\sin x$. Then all the derivatives of $f(x)$ are continuous for all values of $x$. Also $|f^{n}(x)| \\leq 1$ for all values of $x$ and $n$. Hence in this case $|R_{n}| \\leq h^{n}/n!$, which tends to zero as $n \\to \\infty$ (% [examples:xxvii]Ex. xxvii%. 12) whatever value $h$ may have. It follows that (x + h) = x + hx - h^22!x - h^33!x + h^44!x + …, for all values of $x$ and $h$. In particular h = h - h^33! + h^55! - …, for all values of $h$. Similarly we can prove that (x + h) = x - hx - h^22!x + h^33! x + …,0pt minus 3pth = 1 - h^22! + h^44! - ….", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.limit", "other:taylor_remainder_bound" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lvi/2", "set": "hardy-course-of-pure-mathematics-1921/ex-lvi", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "267", "location": "Exercise LVI, problem 2", "problem_latex": "\\Topic{The Binomial Series.} Let $f(x) = (1 + x)^{m}$, where $m$~is any rational\nnumber, positive or negative. Then $f^{(n)}(x) = m(m - 1) \\dots (m - n + 1) (1 + x)^{m-n}$\nand Maclaurin's Series takes the form\n\\[\n(1 + x)^{m} = 1 + \\binom{m}{1}x + \\binom{m}{2}x^{2} + \\dots.\n\\]\n\nWhen $m$~is a positive integer the series terminates, and we obtain the\nordinary formula for the Binomial Theorem with a positive integral exponent.\nIn the general case\n\\[\nR_{n} = \\frac{x^{n}}{n!} f^{(n)}(\\theta_{n}x)\n = \\binom{m}{n}x^{n}(1 + \\theta_{n}x)^{m-n},\n\\]\nand in order to show that Maclaurin's Series really represents $(1 + x)^{m}$ for\nany range of values of~$x$ when $m$~is not a positive integer, we must show that\n$R_{n} \\to 0$ for every value of~$x$ in that range. This is so in fact if $-1 < x < 1$,\nand may be proved, when $0\\leq x < 1$, by means of the expression given above\nfor~$R_{n}$, since $(1 + \\theta_{n}x)^{m-n} < 1$ if $n > m$, and $\\dbinom{m}{n} x^{n} \\to 0$ as $n \\to \\infty$ (\\Ex{xxvii}.~13).\nBut a difficulty arises if $-1 < x < 0$, since $1 + \\theta_{n}x < 1$ and $(1 + \\theta_{n}x)^{m-n} > 1$\nif $n > m$; knowing only that $0 < \\theta_{n} < 1$, we cannot be assured that $1 + \\theta_{n}x$~is not\nquite small and $(1 + \\theta _{n}x)^{m-n}$ quite large.\n\nIn fact, in order to prove the Binomial Theorem by means of Taylor's\nTheorem, we need some different form for~$R_{n}$, such as will be given later~(\\SecNo[§]{162}).", "markdown": "**Binomial Series.** Let $f(x) = (1 + x)^{m}$, where $m$ is any rational number, positive or negative. Then $f^{(n)}(x) = m(m - 1) \\dots (m - n + 1) (1 + x)^{m-n}$ and Maclaurin’s Series takes the form (1 + x)^m = 1 + m1x + m2x^2 + …. When $m$ is a positive integer the series terminates, and we obtain the ordinary formula for the Binomial Theorem with a positive integral exponent. In the general case R_n = x^nn! f^(n)(_nx) = mnx^n(1 + _nx)^m-n, and in order to show that Maclaurin’s Series really represents $(1 + x)^{m}$ for any range of values of $x$ when $m$ is not a positive integer, we must show that $R_{n} \\to 0$ for every value of $x$ in that range. This is so in fact if $-1 < x < 1$, and may be proved, when $0\\leq x < 1$, by means of the expression given above for $R_{n}$, since $(1 + \\theta_{n}x)^{m-n} < 1$ if $n > m$, and $\\dbinom{m}{n} x^{n} \\to 0$ as $n \\to \\infty$ (% [examples:xxvii]Ex. xxvii%. 13). But a difficulty arises if $-1 < x < 0$, since $1 + \\theta_{n}x < 1$ and $(1 + \\theta_{n}x)^{m-n} > 1$ if $n > m$; knowing only that $0 < \\theta_{n} < 1$, we cannot be assured that $1 + \\theta_{n}x$ is not quite small and $(1 + \\theta _{n}x)^{m-n}$ quite large. In fact, in order to prove the Binomial Theorem by means of Taylor’s Theorem, we need some different form for $R_{n}$, such as will be given later ([§]162).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.limit", "core.arith", "other:taylor_remainder_bound" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lvii/1", "set": "hardy-course-of-pure-mathematics-1921/ex-lvii", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "268", "location": "Exercise LVII, problem 1", "problem_latex": "Verify the result when $\\phi(x) = (x - a)^{m}$, $m$~being a\npositive integer, and $\\xi = a$.", "markdown": "Verify the result when $\\phi(x) = (x - a)^{m}$, $m$ being a positive integer, and $\\xi = a$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lvii/2", "set": "hardy-course-of-pure-mathematics-1921/ex-lvii", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "268", "location": "Exercise LVII, problem 2", "problem_latex": "Test the function $(x - a)^{m} (x - b)^{n}$, where $m$~and~$n$ are positive integers,\nfor maxima and minima at the points $x = a$, $x = b$. Draw graphs of the\ndifferent possible forms of the curve $y = (x - a)^{m} (x - b)^{n}$.", "markdown": "Test the function $(x - a)^{m} (x - b)^{n}$, where $m$ and $n$ are positive integers, for maxima and minima at the points $x = a$, $x = b$. Draw graphs of the different possible forms of the curve $y = (x - a)^{m} (x - b)^{n}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lvii/3", "set": "hardy-course-of-pure-mathematics-1921/ex-lvii", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "268", "location": "Exercise LVII, problem 3", "problem_latex": "Test the functions $\\sin x - x$, $\\sin x - x + \\dfrac{x^{3}}{6}$,\n$\\sin x - x + \\dfrac{x^{3}}{6} - \\dfrac{x^{5}}{120}$,~\\dots,\n$\\cos x - 1$, $\\cos x - 1 + \\dfrac{x^{2}}{2}$, $\\cos x - 1 + \\dfrac{x^{2}}{2} - \\dfrac{x^{4}}{24}$,~\\dots\\\nfor maxima or minima at $x = 0$.", "markdown": "Test the functions $\\sin x - x$, $\\sin x - x + \\dfrac{x^{3}}{6}$, $\\sin x - x + \\dfrac{x^{3}}{6} - \\dfrac{x^{5}}{120}$, …, $\\cos x - 1$, $\\cos x - 1 + \\dfrac{x^{2}}{2}$, $\\cos x - 1 + \\dfrac{x^{2}}{2} - \\dfrac{x^{4}}{24}$, … for maxima or minima at $x = 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lviii/1", "set": "hardy-course-of-pure-mathematics-1921/ex-lviii", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "270", "location": "Exercise LVIII, problem 1", "problem_latex": "Find the limit of\n\\[\n\\{x - (n + 1)x^{n+1} + nx^{n+2}\\}/(1 - x)^{2},\n\\]\nas $x \\to 1$. [Here the functions and their first derivatives vanish for $x = 1$,\nand $f''(1) = n(n + 1)$, $\\phi''(1) = 2$.]", "markdown": "Find the limit of x - (n + 1)x^n+1 + nx^n+2/(1 - x)^2, as $x \\to 1$. [Here the functions and their first derivatives vanish for $x = 1$, and $f''(1) = n(n + 1)$, $\\phi''(1) = 2$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(x - (n + 1)*x**(n + 1) + n*x**(n + 2))/(1 - x)**2", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lviii/2a", "set": "hardy-course-of-pure-mathematics-1921/ex-lviii", "number": 2, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "270", "location": "Exercise LVIII, problem 2a", "problem_latex": "Find the limits as $x \\to 0$ of\n\\[\n(\\tan x - x)/(x - \\sin x),\\quad\n(\\tan nx - n\\tan x)/(n\\sin x - \\sin nx).\n\\]", "markdown": "Find the limits as $x \\to 0$ of (x - x)/(x - x),0pt minus 3pt(nx - nx)/(nx - nx).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(tan(x) - x)/(x - sin(x))", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "cas.series" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lviii/2b", "set": "hardy-course-of-pure-mathematics-1921/ex-lviii", "number": 2, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "270", "location": "Exercise LVIII, problem 2b", "problem_latex": "Find the limits as $x \\to 0$ of\n\\[\n(\\tan x - x)/(x - \\sin x),\\quad\n(\\tan nx - n\\tan x)/(n\\sin x - \\sin nx).\n\\]", "markdown": "Find the limits as $x \\to 0$ of (x - x)/(x - x),0pt minus 3pt(nx - nx)/(nx - nx).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(tan(n*x) - n*tan(x))/(n*sin(x) - sin(n*x))", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "cas.series" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lviii/3", "set": "hardy-course-of-pure-mathematics-1921/ex-lviii", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "270", "location": "Exercise LVIII, problem 3", "problem_latex": "Find the limit of $x\\{\\sqrtp{x^{2} + a^{2}} - x\\}$ as $x \\to \\infty$. [Put $x = 1/y$.]", "markdown": "Find the limit of $x\\{\\sqrtp{x^{2} + a^{2}} - x\\}$ as $x \\to \\infty$. [Put $x = 1/y$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "x*(sqrt(x**2 + a**2) - x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lviii/4a", "set": "hardy-course-of-pure-mathematics-1921/ex-lviii", "number": 4, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "270", "location": "Exercise LVIII, problem 4a", "problem_latex": "Prove that\n\\[\n\\lim_{x \\to n} (x - n)\\cosec x\\pi = \\frac{(-1)^{n}}{\\pi},\\quad\n\\lim_{x \\to n} \\frac{1}{x - n} \\left\\{\n \\cosec x\\pi - \\frac{(-1)^{n}}{(x - n)\\pi}\n\\right\\} = \\frac{(-1)^{n}\\pi}{6},\n\\]\n$n$~being any integer; and evaluate the corresponding limits involving $\\cot x\\pi$.", "markdown": "Prove that _x n (x - n)x= (-1)^n,0pt minus 3pt_x n 1x - n x- (-1)^n(x - n) = (-1)^n6, $n$ being any integer; and evaluate the corresponding limits involving $\\cot x\\pi$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(x - n)*csc(pi*x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lviii/4b", "set": "hardy-course-of-pure-mathematics-1921/ex-lviii", "number": 4, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "270", "location": "Exercise LVIII, problem 4b", "problem_latex": "Prove that\n\\[\n\\lim_{x \\to n} (x - n)\\cosec x\\pi = \\frac{(-1)^{n}}{\\pi},\\quad\n\\lim_{x \\to n} \\frac{1}{x - n} \\left\\{\n \\cosec x\\pi - \\frac{(-1)^{n}}{(x - n)\\pi}\n\\right\\} = \\frac{(-1)^{n}\\pi}{6},\n\\]\n$n$~being any integer; and evaluate the corresponding limits involving $\\cot x\\pi$.", "markdown": "Prove that _x n (x - n)x= (-1)^n,0pt minus 3pt_x n 1x - n x- (-1)^n(x - n) = (-1)^n6, $n$ being any integer; and evaluate the corresponding limits involving $\\cot x\\pi$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(1/(x - n))*(csc(pi*x) - (-1)**n/((x - n)*pi))", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "cas.series", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lviii/4c", "set": "hardy-course-of-pure-mathematics-1921/ex-lviii", "number": 4, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "270", "location": "Exercise LVIII, problem 4c", "problem_latex": "Prove that\n\\[\n\\lim_{x \\to n} (x - n)\\cosec x\\pi = \\frac{(-1)^{n}}{\\pi},\\quad\n\\lim_{x \\to n} \\frac{1}{x - n} \\left\\{\n \\cosec x\\pi - \\frac{(-1)^{n}}{(x - n)\\pi}\n\\right\\} = \\frac{(-1)^{n}\\pi}{6},\n\\]\n$n$~being any integer; and evaluate the corresponding limits involving $\\cot x\\pi$.", "markdown": "Prove that _x n (x - n)x= (-1)^n,0pt minus 3pt_x n 1x - n x- (-1)^n(x - n) = (-1)^n6, $n$ being any integer; and evaluate the corresponding limits involving $\\cot x\\pi$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(x - n)*cot(pi*x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lviii/5a", "set": "hardy-course-of-pure-mathematics-1921/ex-lviii", "number": 5, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "270", "location": "Exercise LVIII, problem 5a", "problem_latex": "Find the limits as $x \\to 0$ of\n\\[\n\\frac{1}{x^{3}}\\left(\\cosec x - \\frac{1}{x} - \\frac{x}{6}\\right),\\quad\n\\frac{1}{x^{3}}\\left(\\cot x - \\frac{1}{x} + \\frac{x}{3}\\right).\n\\]", "markdown": "Find the limits as $x \\to 0$ of 1x^3(x - 1x - x6),0pt minus 3pt1x^3(x - 1x + x3).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(1/x**3)*(csc(x) - 1/x - x/6)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "cas.series", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lviii/5b", "set": "hardy-course-of-pure-mathematics-1921/ex-lviii", "number": 5, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "270", "location": "Exercise LVIII, problem 5b", "problem_latex": "Find the limits as $x \\to 0$ of\n\\[\n\\frac{1}{x^{3}}\\left(\\cosec x - \\frac{1}{x} - \\frac{x}{6}\\right),\\quad\n\\frac{1}{x^{3}}\\left(\\cot x - \\frac{1}{x} + \\frac{x}{3}\\right).\n\\]", "markdown": "Find the limits as $x \\to 0$ of 1x^3(x - 1x - x6),0pt minus 3pt1x^3(x - 1x + x3).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(1/x**3)*(cot(x) - 1/x + x/3)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "cas.series", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lviii/6a", "set": "hardy-course-of-pure-mathematics-1921/ex-lviii", "number": 6, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "270", "location": "Exercise LVIII, problem 6a", "problem_latex": "$(\\sin x\\arcsin x - x^{2})/x^{6} \\to \\frac{1}{18}$, $(\\tan x\\arctan x - x^{2})/x^{6} \\to \\frac{2}{9}$, as $x \\to 0$.", "markdown": "$(\\sin x\\arcsin x - x^{2})/x^{6} \\to \\frac{1}{18}$, $(\\tan x\\arctan x - x^{2})/x^{6} \\to \\frac{2}{9}$, as $x \\to 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(sin(x)*asin(x) - x**2)/x**6", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "cas.series", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lviii/6b", "set": "hardy-course-of-pure-mathematics-1921/ex-lviii", "number": 6, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "270", "location": "Exercise LVIII, problem 6b", "problem_latex": "$(\\sin x\\arcsin x - x^{2})/x^{6} \\to \\frac{1}{18}$, $(\\tan x\\arctan x - x^{2})/x^{6} \\to \\frac{2}{9}$, as $x \\to 0$.", "markdown": "$(\\sin x\\arcsin x - x^{2})/x^{6} \\to \\frac{1}{18}$, $(\\tan x\\arctan x - x^{2})/x^{6} \\to \\frac{2}{9}$, as $x \\to 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(tan(x)*atan(x) - x**2)/x**6", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "cas.series", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lx/1", "set": "hardy-course-of-pure-mathematics-1921/ex-lx", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "275", "location": "Exercise LX, problem 1", "problem_latex": " {\\Loosen Prove that if $x = r\\cos\\theta$, $y = r\\sin\\theta$, so that $r = \\sqrtp{x^{2} + y^{2}}$,\n$\\theta = \\arctan(y/x)$, then}\n\\begin{align*}\n \\frac{\\dd r}{\\dd x} &= \\frac{x}{\\sqrtp{x^{2} + y^{2}}},\n&\\frac{\\dd r}{\\dd y} &= \\frac{y}{\\sqrtp{x^{2} + y^{2}}},\n&\\frac{\\dd \\theta}{\\dd x} &= -\\frac{y}{x^{2} + y^{2}},\n&\\frac{\\dd \\theta}{\\dd y} &= \\frac{x}{x^{2} + y^{2}},\\\\\n%\n \\frac{\\dd x}{\\dd r} &= \\cos\\theta,\n&\\frac{\\dd y}{\\dd r} &= \\sin\\theta,\n&\\frac{\\dd x}{\\dd \\theta} &= -r\\sin\\theta,\n&\\frac{\\dd y}{\\dd \\theta} &= r\\cos\\theta.\n\\end{align*}", "markdown": "0.375em plus 0.75em minus 0.25emProve that if $x = r\\cos\\theta$, $y = r\\sin\\theta$, so that $r = \\sqrtp{x^{2} + y^{2}}$, $\\theta = \\arctan(y/x)$, then align* rx &= xx^2 + y^2, &ry &= yx^2 + y^2, &x &= -yx^2 + y^2, &y &= xx^2 + y^2, % xr &= , &yr &= , &x &= -r, &y &= r. align*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lx/2", "set": "hardy-course-of-pure-mathematics-1921/ex-lx", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "275", "location": "Exercise LX, problem 2", "problem_latex": " Account for the fact that\n$\\dfrac{\\dd r}{\\dd x}\\neq 1\\bigg/\\biggl(\\dfrac{\\dd x}{\\dd r}\\biggr)$ and\n$\\dfrac{\\dd \\theta}{\\dd x}\\neq 1\\bigg/\\biggl(\\dfrac{\\dd x}{\\dd \\theta}\\biggr)$. [When\nwe were considering a function~$y$ of one variable~$x$ it followed from the\ndefinitions that $dy/dx$ and~$dx/dy$ were reciprocals. This is no longer the\n\\PageSep{276}\ncase when we are dealing with functions of two variables. Let $P$ (\\Fig{46})\nbe the point $(x, y)$ or $(r, \\theta)$. To find $\\dd r/\\dd x$ we must increase~$x$, say by an\nincrement $MM_{1} = \\delta x$, while keeping $y$~constant. This brings~$P$ to~$P_{1}$. If\nalong~$OP_{1}$ we take $OP' = OP$, the increment of~$r$ is $P'P_{1} = \\delta r$, say; and\n$\\dd r/\\dd x = \\lim(\\delta r/\\delta x)$. If on the other hand we want to calculate $\\dd x/\\dd r$, $x$~and~$y$\n%[Illustration: Fig. 46.]\n\\Figure[2.25in]{46}{p276}\nbeing now regarded as functions of $r$~and~$\\theta$,\nwe must increase~$r$ by~$\\Delta r$, say,\nkeeping $\\theta$~constant. This brings~$P$ to~$P_{2}$,\nwhere $PP_{2} = \\Delta r$: the corresponding\nincrement of~$x$ is $MM_{1} = \\Delta x$, say; and\n\\[\n\\dd x/\\dd r = \\lim(\\Delta x/\\Delta r).\n\\]\nNow $\\Delta x = \\delta x$:\\footnote\n {Of course the fact that $\\Delta x = \\delta x$ is due merely to the particular value of~$\\Delta r$\n that we have chosen (viz.~$PP_{2}$). Any other choice would give us values of $\\Delta x$,~$\\Delta r$\n proportional to those used here.}\nbut $\\Delta r \\neq \\delta r$. Indeed it is\neasy to see from the figure that\n\\[\n\\lim (\\delta r/\\delta x) = \\lim (P'P_{1}/PP_{1}) = \\cos\\theta,\n\\]\nbut\n\\[\n\\lim (\\Delta r/\\Delta x) = \\lim (PP_{2}/PP_{1}) = \\sec\\theta,\n\\]\nso that\n\\[\n\\lim (\\delta r/\\Delta r) = \\cos^{2}\\theta.\n\\]\n\nThe fact is of course that \\emph{$\\dd x/\\dd r$ and\n$\\dd r/\\dd x$ are not formed upon the same hypothesis as to the variation of~$P$.}]", "markdown": "Account for the fact that $\\dfrac{\\dd r}{\\dd x}\\neq 1\\bigg/\\biggl(\\dfrac{\\dd x}{\\dd r}\\biggr)$ and $\\dfrac{\\dd \\theta}{\\dd x}\\neq 1\\bigg/\\biggl(\\dfrac{\\dd x}{\\dd \\theta}\\biggr)$. [When we were considering a function $y$ of one variable $x$ it followed from the definitions that $dy/dx$ and $dx/dy$ were reciprocals. This is no longer the [pg]276 case when we are dealing with functions of two variables. Let $P$ ([fig:46]Fig. 46) be the point $(x, y)$ or $(r, \\theta)$. To find $\\dd r/\\dd x$ we must increase $x$, say by an increment $MM_{1} = \\delta x$, while keeping $y$ constant. This brings $P$ to $P_{1}$. If along $OP_{1}$ we take $OP' = OP$, the increment of $r$ is $P'P_{1} = \\delta r$, say; and $\\dd r/\\dd x = \\lim(\\delta r/\\delta x)$. If on the other hand we want to calculate $\\dd x/\\dd r$, $x$ and $y$ %[Illustration: Fig. 46.] [2.25in]46p276 being now regarded as functions of $r$ and $\\theta$, we must increase $r$ by $\\Delta r$, say, keeping $\\theta$ constant. This brings $P$ to $P_{2}$, where $PP_{2} = \\Delta r$: the corresponding increment of $x$ is $MM_{1} = \\Delta x$, say; and x/r = (x/r). Now $\\Delta x = \\delta x$: Of course the fact that $\\Delta x = \\delta x$ is due merely to the particular value of $\\Delta r$ that we have chosen (viz. $PP_{2}$). Any other choice would give us values of $\\Delta x$, $\\Delta r$ proportional to those used here. but $\\Delta r \\neq \\delta r$. Indeed it is easy to see from the figure that (r/x) = (P’P_1/PP_1) = , but (r/x) = (PP_2/PP_1) = , so that (r/r) = ^2. The fact is of course that *$\\dd x/\\dd r$ and $\\dd r/\\dd x$ are not formed upon the same hypothesis as to the variation of $P$.*]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lx/3", "set": "hardy-course-of-pure-mathematics-1921/ex-lx", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "275", "location": "Exercise LX, problem 3", "problem_latex": " Prove that if $z = f(ax + by)$ then $b(\\dd z/\\dd x) = a(\\dd z/\\dd y)$.", "markdown": "Prove that if $z = f(ax + by)$ then $b(\\dd z/\\dd x) = a(\\dd z/\\dd y)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lx/4", "set": "hardy-course-of-pure-mathematics-1921/ex-lx", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "275", "location": "Exercise LX, problem 4", "problem_latex": " Find $\\dd X/\\dd x$, $\\dd X/\\dd y$,~\\dots\\ when $X + Y = x$, $Y = xy$. Express $x$,~$y$ as\nfunctions of $X$,~$Y$ and find $\\dd x/\\dd X$, $\\dd x/\\dd Y$,~\\dots.", "markdown": "Find $\\dd X/\\dd x$, $\\dd X/\\dd y$, … when $X + Y = x$, $Y = xy$. Express $x$, $y$ as functions of $X$, $Y$ and find $\\dd x/\\dd X$, $\\dd x/\\dd Y$, ….", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.implicit", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lx/5", "set": "hardy-course-of-pure-mathematics-1921/ex-lx", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "275", "location": "Exercise LX, problem 5", "problem_latex": " Find $\\dd X/\\dd x$,~\\dots\\ when $X + Y + Z = x$, $Y + Z = xy$, $Z = xyz$; express\n$x$,~$y$,~$z$ in terms of $X$,~$Y$,~$Z$ and find $\\dd x/\\dd X$,~\\dots.", "markdown": "Find $\\dd X/\\dd x$, … when $X + Y + Z = x$, $Y + Z = xy$, $Z = xyz$; express $x$, $y$, $z$ in terms of $X$, $Y$, $Z$ and find $\\dd x/\\dd X$, ….", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.implicit", "core.linsys" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxi/1", "set": "hardy-course-of-pure-mathematics-1921/ex-lxi", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "277", "location": "Exercise LXI, problem 1", "problem_latex": "Suppose $\\phi(t) = (1 - t^{2})/(1 + t^{2})$, $\\psi(t) = 2t/(1 + t^{2})$, so\nthat the locus of~$(x, y)$ is the circle $x^{2} + y^{2} = 1$. Then\n\\begin{align*}\n\\phi'(t) &= -4t/(1 + t^{2})^{2},\\quad \\psi'(t) = 2(1 - t^{2})/(1 + t^{2})^{2},\\\\\nF'(t) &= \\{-4t/(1 + t^{2})^{2}\\}f_{x}' + \\{2(1 - t^{2})/(1 + t^{2})^{2}\\}f_{y}',\n\\end{align*}\nwhere $x$~and~$y$ are to be put equal to $(1 - t^{2})/(1 + t^{2})$ and $2t/(1 + t^{2})$ after\ncarrying out the differentiations.\n\\PageSep{278}\n\n{\\Loosen We can easily verify this formula in particular cases. Suppose, \\eg,\nthat $f(x, y) = x^{2} + y^{2}$. Then $f_{x}' = 2x$, $f_{y}' = 2y$, and it is easily verified that\n$F'(t) = 2x\\phi'(t) + 2y\\psi'(t) = 0$, which is obviously correct, since $F(t) = 1$.}", "markdown": "Suppose $\\phi(t) = (1 - t^{2})/(1 + t^{2})$, $\\psi(t) = 2t/(1 + t^{2})$, so that the locus of $(x, y)$ is the circle $x^{2} + y^{2} = 1$. Then align* ’(t) &= -4t/(1 + t^2)^2,0pt minus 3pt’(t) = 2(1 - t^2)/(1 + t^2)^2, F’(t) &= -4t/(1 + t^2)^2f_x’ + 2(1 - t^2)/(1 + t^2)^2f_y’, align* where $x$ and $y$ are to be put equal to $(1 - t^{2})/(1 + t^{2})$ and $2t/(1 + t^{2})$ after carrying out the differentiations. [pg]278 0.375em plus 0.75em minus 0.25emWe can easily verify this formula in particular cases. Suppose, *e.g.*, that $f(x, y) = x^{2} + y^{2}$. Then $f_{x}' = 2x$, $f_{y}' = 2y$, and it is easily verified that $F'(t) = 2x\\phi'(t) + 2y\\psi'(t) = 0$, which is obviously correct, since $F(t) = 1$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxi/2a", "set": "hardy-course-of-pure-mathematics-1921/ex-lxi", "number": 2, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "277", "location": "Exercise LXI, problem 2a", "problem_latex": "Verify the theorem in the same way when (\\ia)~$x = t^{m}$, $y = 1 - t^{m}$,\n$f(x, y) = x + y$; (\\ib)~$x = a\\cos t$, $y = a\\sin t$, $f(x, y) = x^{2} + y^{2}$.", "markdown": "Verify the theorem in the same way when (*a*) $x = t^{m}$, $y = 1 - t^{m}$, $f(x, y) = x + y$; (*b*) $x = a\\cos t$, $y = a\\sin t$, $f(x, y) = x^{2} + y^{2}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.simplify", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxi/2b", "set": "hardy-course-of-pure-mathematics-1921/ex-lxi", "number": 2, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "277", "location": "Exercise LXI, problem 2b", "problem_latex": "Verify the theorem in the same way when (\\ia)~$x = t^{m}$, $y = 1 - t^{m}$,\n$f(x, y) = x + y$; (\\ib)~$x = a\\cos t$, $y = a\\sin t$, $f(x, y) = x^{2} + y^{2}$.", "markdown": "Verify the theorem in the same way when (*a*) $x = t^{m}$, $y = 1 - t^{m}$, $f(x, y) = x + y$; (*b*) $x = a\\cos t$, $y = a\\sin t$, $f(x, y) = x^{2} + y^{2}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.simplify", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxi/3", "set": "hardy-course-of-pure-mathematics-1921/ex-lxi", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "277", "location": "Exercise LXI, problem 3", "problem_latex": "One of the most important cases is that in which $t$ is $x$~itself. We\nthen obtain\n\\[\nD_{x}f\\{x, \\psi(x)\\} = D_{x}f(x, y) + D_{y}f(x, y)\\psi'(x).\n\\]\nwhere $y$~is to be replaced by~$\\psi(x)$ after differentiation.\n\nIt was this case which led to the introduction of the notation $\\dd f/\\dd x$, $\\dd f/\\dd y$.\nFor it would seem natural to use the notation~$df/dx$ for \\emph{either} of the functions\n$D_{x}f\\{x, \\psi(x)\\}$ and $D_{x}f(x, y)$, in one of which $y$~is put equal to~$\\psi(x)$ before\nand in the other after differentiation. Suppose for example that $y = 1 - x$\nand $f(x, y) = x + y$. Then $D_{x}f(x, 1 - x) = D_{x}1 = 0$, but $D_{x}f(x, y) = 1$.\n\nThe distinction between the two functions is adequately shown by\ndenoting the first by~$df/dx$ and the second by~$\\dd f/\\dd x$, in which case the\ntheorem takes the form\n\\[\n\\frac{df}{dx} = \\frac{\\dd f}{\\dd x} + \\frac{\\dd f}{\\dd y}\\, \\frac{dy}{dx};\n\\]\nthough this notation is also open to objection, in that it is a little misleading\nto denote the functions $f\\{x, \\psi(x)\\}$ and $f(x, y)$, whose forms as functions of~$x$\nare quite different from one another, by the same letter~$f$ in $df/dx$ and~$\\dd f/\\dd x$.", "markdown": "One of the most important cases is that in which $t$ is $x$ itself. We then obtain D_xfx, (x) = D_xf(x, y) + D_yf(x, y)’(x). where $y$ is to be replaced by $\\psi(x)$ after differentiation. It was this case which led to the introduction of the notation $\\dd f/\\dd x$, $\\dd f/\\dd y$. For it would seem natural to use the notation $df/dx$ for *either* of the functions $D_{x}f\\{x, \\psi(x)\\}$ and $D_{x}f(x, y)$, in one of which $y$ is put equal to $\\psi(x)$ before and in the other after differentiation. Suppose for example that $y = 1 - x$ and $f(x, y) = x + y$. Then $D_{x}f(x, 1 - x) = D_{x}1 = 0$, but $D_{x}f(x, y) = 1$. The distinction between the two functions is adequately shown by denoting the first by $df/dx$ and the second by $\\dd f/\\dd x$, in which case the theorem takes the form dfdx = fx + fy  dydx; though this notation is also open to objection, in that it is a little misleading to denote the functions $f\\{x, \\psi(x)\\}$ and $f(x, y)$, whose forms as functions of $x$ are quite different from one another, by the same letter $f$ in $df/dx$ and $\\dd f/\\dd x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxi/4", "set": "hardy-course-of-pure-mathematics-1921/ex-lxi", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "277", "location": "Exercise LXI, problem 4", "problem_latex": "If the result of eliminating~$t$ between $x = \\phi(t)$, $y = \\psi(t)$ is $f(x, y) = 0$,\nthen\n\\[\n\\frac{\\dd f}{\\dd x}\\, \\frac{dx}{dt} + \\frac{\\dd f}{\\dd y}\\, \\frac{dy}{dt} = 0.\n\\]", "markdown": "If the result of eliminating $t$ between $x = \\phi(t)$, $y = \\psi(t)$ is $f(x, y) = 0$, then fx  dxdt + fy  dydt = 0.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxi/5", "set": "hardy-course-of-pure-mathematics-1921/ex-lxi", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "277", "location": "Exercise LXI, problem 5", "problem_latex": "If $x$~and~$y$ are functions of~$t$, and $r$~and~$\\theta$ are the polar coordinates of\n$(x, y)$, then $r' = (xx' + yy')/r$, $\\theta' = (xy' - yx')/r^{2}$, dashes denoting differentiations\nwith respect to~$t$.", "markdown": "If $x$ and $y$ are functions of $t$, and $r$ and $\\theta$ are the polar coordinates of $(x, y)$, then $r' = (xx' + yy')/r$, $\\theta' = (xy' - yx')/r^{2}$, dashes denoting differentiations with respect to $t$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxii/1", "set": "hardy-course-of-pure-mathematics-1921/ex-lxii", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "281", "location": "Exercise LXII, problem 1", "problem_latex": "The area of an ellipse is given by $A = \\pi ab$, where\n$a$,~$b$ are the semiaxes. Prove that\n\\[\n\\frac{dA}{A} = \\frac{da}{a} + \\frac{db}{b},\n\\]\nand state the corresponding approximate equation connecting the increments\nof the axes and the area.", "markdown": "The area of an ellipse is given by $A = \\pi ab$, where $a$, $b$ are the semiaxes. Prove that dAA = daa + dbb, and state the corresponding approximate equation connecting the increments of the axes and the area.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxii/2", "set": "hardy-course-of-pure-mathematics-1921/ex-lxii", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "281", "location": "Exercise LXII, problem 2", "problem_latex": "Express $\\Delta$, the area of a triangle~$ABC$, as a function of (i)~$a$, $B$,~$C$,\n(ii)~$A$, $b$,~$c$, and (iii)~$a$, $b$,~$c$, and establish the formulae\n\\begin{gather*}\n\\frac{d\\Delta}{\\Delta}\n = 2\\frac{da}{a} + \\frac{c\\, dB}{a\\sin B} + \\frac{b\\, dC}{a\\sin C},\\quad\n\\frac{d\\Delta}{\\Delta}\n = \\cot A\\, dA + \\frac{db}{b} + \\frac{dc}{c},\\\\\nd\\Delta = R(\\cos A\\, da + \\cos B\\, db + \\cos C\\, dc),\n\\end{gather*}\n%[** TN: Sole instance of circumcircle, not hyphenated in the original]\nwhere $R$~is the radius of the circumcircle.", "markdown": "Express $\\Delta$, the area of a triangle $ABC$, as a function of (i) $a$, $B$, $C$, (ii) $A$, $b$, $c$, and (iii) $a$, $b$, $c$, and establish the formulae gather* d = 2daa + c  dBaB + b  dCaC,0pt minus 3ptd = A  dA + dbb + dcc, d= R(A  da + B  db + C  dc), gather* %[** TN: Sole instance of circumcircle, not hyphenated in the original] where $R$ is the radius of the circumcircle.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxii/3", "set": "hardy-course-of-pure-mathematics-1921/ex-lxii", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "281", "location": "Exercise LXII, problem 3", "problem_latex": "The sides of a triangle vary in such a way that the area remains\nconstant, so that $a$~may be regarded as a function of $b$~and~$c$. Prove that\n\\[\n\\frac{\\dd a}{\\dd b} = -\\frac{\\cos B}{\\cos A},\\quad\n\\frac{\\dd a}{\\dd c} = -\\frac{\\cos C}{\\cos A}.\n\\]\n\n[This follows from the equations\n\\[\nda = \\frac{\\dd a}{\\dd b}\\, db + \\frac{\\dd a}{\\dd c}\\, dc,\\quad\n\\cos A\\, da + \\cos B\\, db + \\cos C\\, dc = 0.\\Add{]}\n\\]", "markdown": "The sides of a triangle vary in such a way that the area remains constant, so that $a$ may be regarded as a function of $b$ and $c$. Prove that ab = -BA,0pt minus 3ptac = -CA. [This follows from the equations da = ab  db + ac  dc,0pt minus 3ptA  da + B  db + C  dc = 0.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxii/4", "set": "hardy-course-of-pure-mathematics-1921/ex-lxii", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "281", "location": "Exercise LXII, problem 4", "problem_latex": "If $a$,~$b$,~$c$ vary so that $R$~remains constant, then\n\\[\n\\frac{da}{\\cos A} + \\frac{db}{\\cos B} + \\frac{dc}{\\cos C} = 0,\n\\]\nand so\n\\[\n\\frac{\\dd a}{\\dd b} = -\\frac{\\cos A}{\\cos B},\\quad\n\\frac{\\dd a}{\\dd c} = -\\frac{\\cos A}{\\cos C}.\n\\]\n\n[Use the formulae $a = 2R\\sin A$,~\\dots, and the facts that $R$ and $A + B + C$ are\nconstant.]", "markdown": "If $a$, $b$, $c$ vary so that $R$ remains constant, then daA + dbB + dcC = 0, and so ab = -AB,0pt minus 3ptac = -AC. [Use the formulae $a = 2R\\sin A$, …, and the facts that $R$ and $A + B + C$ are constant.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxii/5", "set": "hardy-course-of-pure-mathematics-1921/ex-lxii", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "281", "location": "Exercise LXII, problem 5", "problem_latex": "If $z$~is a function of $u$~and~$v$, which are functions of $x$~and~$y$, then\n\\[\n\\frac{\\dd z}{\\dd x} = \\frac{\\dd z}{\\dd u}\\, \\frac{\\dd u}{\\dd x}\n + \\frac{\\dd z}{\\dd v}\\, \\frac{\\dd v}{\\dd x},\\quad\n\\frac{\\dd z}{\\dd y} = \\frac{\\dd z}{\\dd u}\\, \\frac{\\dd u}{\\dd y}\n + \\frac{\\dd z}{\\dd v}\\, \\frac{\\dd v}{\\dd y}.\n\\]\n\n[We have\n\\[\ndz = \\frac{\\dd z}{\\dd u}\\, du + \\frac{\\dd z}{\\dd v}\\, dv,\\quad\ndu = \\frac{\\dd u}{\\dd x}\\, dx + \\frac{\\dd u}{\\dd y}\\, dy,\\quad\ndv = \\frac{\\dd v}{\\dd x}\\, dx + \\frac{\\dd v}{\\dd y}\\, dy.\n\\]\nSubstitute for $du$~and~$dv$ in the first equation and compare the result with\nthe equation\n\\[\ndz = \\frac{\\dd z}{\\dd x}\\, dx + \\frac{\\dd z}{\\dd y}\\, dy.]\n\\]", "markdown": "If $z$ is a function of $u$ and $v$, which are functions of $x$ and $y$, then zx = zu  ux + zv  vx,0pt minus 3ptzy = zu  uy + zv  vy. [We have dz = zu  du + zv  dv,0pt minus 3ptdu = ux  dx + uy  dy,0pt minus 3ptdv = vx  dx + vy  dy. Substitute for $du$ and $dv$ in the first equation and compare the result with the equation dz = zx  dx + zy  dy.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxii/6", "set": "hardy-course-of-pure-mathematics-1921/ex-lxii", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "281", "location": "Exercise LXII, problem 6", "problem_latex": "Let $z$~be a function of $x$~and~$y$, and let $X$,~$Y$,~$Z$ be defined by the\nequations\n\\[\nx = a_{1} X + b_{1} Y + c_{1} Z,\\quad\ny = a_{2} X + b_{2} Y + c_{2} Z,\\quad\nz = a_{3} X + b_{3} Y + c_{3} Z.\n\\]\nThen $Z$~may be expressed as a function of $X$~and~$Y$. Express $\\dd Z/\\dd X$,\n$\\dd Z/\\dd Y$ in terms of $\\dd z/\\dd x$, $\\dd z/\\dd y$. [Let these differential coefficients be denoted\nby $P$,~$Q$ and $p$,~$q$. Then $dz - p\\, dx - q\\, dy = 0$, or\n\\[\n(c_{1} p + c_{2} q - c_{3})\\, dZ +\n(a_{1} p + a_{2} q - a_{3})\\, dX +\n(b_{1} p + b_{2} q - b_{3})\\, dY = 0.\n\\]\n\\PageSep{283}\nComparing this equation with $dZ - P\\, dX - Q\\, dY = 0$ we see that\n\\[\nP = -\\frac{a_{1}p + a_{2}q - a_{3}}{c_{1}p + c_{2}q - c_{3}},\\quad\nQ = -\\frac{b_{1}p + b_{2}q - b_{3}}{c_{1}p + c_{2}q - c_{3}}.]\n\\]", "markdown": "Let $z$ be a function of $x$ and $y$, and let $X$, $Y$, $Z$ be defined by the equations x = a_1 X + b_1 Y + c_1 Z,0pt minus 3pty = a_2 X + b_2 Y + c_2 Z,0pt minus 3ptz = a_3 X + b_3 Y + c_3 Z. Then $Z$ may be expressed as a function of $X$ and $Y$. Express $\\dd Z/\\dd X$, $\\dd Z/\\dd Y$ in terms of $\\dd z/\\dd x$, $\\dd z/\\dd y$. [Let these differential coefficients be denoted by $P$, $Q$ and $p$, $q$. Then $dz - p\\, dx - q\\, dy = 0$, or (c_1 p + c_2 q - c_3)  dZ + (a_1 p + a_2 q - a_3)  dX + (b_1 p + b_2 q - b_3)  dY = 0. [pg]283 Comparing this equation with $dZ - P\\, dX - Q\\, dY = 0$ we see that P = -a_1p + a_2q - a_3c_1p + c_2q - c_3,0pt minus 3ptQ = -b_1p + b_2q - b_3c_1p + c_2q - c_3.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxii/7", "set": "hardy-course-of-pure-mathematics-1921/ex-lxii", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "281", "location": "Exercise LXII, problem 7", "problem_latex": "If\n\\[\n(a_{1} x + b_{1} y + c_{1} z)p + (a_{2} x + b_{2} y + c_{2} z)q\n = a_{3} x + b_{3} y + c_{3} z,\n\\]\nthen\n\\[\n(a_{1} X + b_{1} Y + c_{1} Z) P + (a_{2} X + b_{2} Y + c_{2} Z) Q\n = a_{3} X + b_{3} Y + c_{3} Z.\n\\]\n\\MathTrip{1899.}", "markdown": "If (a_1 x + b_1 y + c_1 z)p + (a_2 x + b_2 y + c_2 z)q = a_3 x + b_3 y + c_3 z, then (a_1 X + b_1 Y + c_1 Z) P + (a_2 X + b_2 Y + c_2 Z) Q = a_3 X + b_3 Y + c_3 Z. % [0]% (*Math. Trip.* 1899.)% [1]%", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxii/8", "set": "hardy-course-of-pure-mathematics-1921/ex-lxii", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "281", "location": "Exercise LXII, problem 8", "problem_latex": "\\Topic{Differentiation of implicit functions.} Suppose that $f(x, y)$ and its\nderivative $f_{y}'(x, y)$ are continuous in the neighbourhood of the point $(a, b)$,\nand that\n\\[\nf(a, b) = 0,\\quad\nf_{b}'(a, b) \\neq 0.\n\\]\nThen we can find a neighbourhood of~$(a, b)$ throughout which $f_{y}'(x, y)$ has\nalways the same sign. Let us suppose, for example, that $f_{y}'(x, y)$~is positive\nnear $(a, b)$. Then $f(x, y)$~is, for any value of~$x$ sufficiently near to~$a$, and for\nvalues of~$y$ sufficiently near to~$b$, an increasing function of~$y$ in the stricter\nsense of \\SecNo[§]{95}. It follows, by the theorem of \\SecNo[§]{108}, that there is a unique\ncontinuous function~$y$ which is equal to~$b$ when $x = a$ and which satisfies the\nequation $f(x, y) = 0$ for all values of~$x$ sufficiently near to~$a$.\n\nLet us now suppose that $f(x, y)$ possesses a derivative $f_{x}'(x, y)$ which is\nalso continuous near $(a, b)$. If $f(x, y) = 0$, $x = a + h$, $y = b + k$, we have\n\\[\n0 = f(x, y) - f(a, b) = (f_{a}' + \\epsilon) h + (f_{b}' + \\eta) k,\n\\]\nwhere $\\DPtypo{}{\\epsilon}$ and~$\\eta$ tend to zero with $h$~and~$k$. Thus\n\\[\n\\frac{k}{h} = -\\frac{f_{a}' + \\epsilon}{f_{b}' + \\eta} \\to -\\frac{f_{a}'}{f_{b}'},\n\\]\nor\n\\[\n\\frac{dy}{dx} = -\\frac{f_{a}'}{f_{b}'}.", "markdown": "**of implicit functions.** Suppose that $f(x, y)$ and its derivative $f_{y}'(x, y)$ are continuous in the neighbourhood of the point $(a, b)$, and that f(a, b) = 0,0pt minus 3ptf_b’(a, b) 0. Then we can find a neighbourhood of $(a, b)$ throughout which $f_{y}'(x, y)$ has always the same sign. Let us suppose, for example, that $f_{y}'(x, y)$ is positive near $(a, b)$. Then $f(x, y)$ is, for any value of $x$ sufficiently near to $a$, and for values of $y$ sufficiently near to $b$, an increasing function of $y$ in the stricter sense of [§]95. It follows, by the theorem of [§]108, that there is a unique continuous function $y$ which is equal to $b$ when $x = a$ and which satisfies the equation $f(x, y) = 0$ for all values of $x$ sufficiently near to $a$. Let us now suppose that $f(x, y)$ possesses a derivative $f_{x}'(x, y)$ which is also continuous near $(a, b)$. If $f(x, y) = 0$, $x = a + h$, $y = b + k$, we have 0 = f(x, y) - f(a, b) = (f_a’ + ) h + (f_b’ + ) k, where $\\DPtypo{}{\\epsilon}$ and $\\eta$ tend to zero with $h$ and $k$. Thus kh = -f_a’ + f_b’ + -f_a’f_b’, or dydx = -f_a’f_b’.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.implicit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxii/9", "set": "hardy-course-of-pure-mathematics-1921/ex-lxii", "number": 9, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "281", "location": "Exercise LXII, problem 9", "problem_latex": "The equation of the tangent to the curve $f(x, y) = 0$, at the point\n$x_{0}$,~$y_{0}$, is\n\\[\n(x - x_{0}) f_{x_{0}}'(x_{0}, y_{0}) + (y - y_{0}) f_{y_{0}}'(x_{0}, y_{0}) = 0.", "markdown": "The equation of the tangent to the curve $f(x, y) = 0$, at the point $x_{0}$, $y_{0}$, is (x - x_0) f_x_0’(x_0, y_0) + (y - y_0) f_y_0’(x_0, y_0) = 0.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.implicit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxiii/10a", "set": "hardy-course-of-pure-mathematics-1921/ex-lxiii", "number": 10, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "289", "location": "Exercise LXIII, problem 10a", "problem_latex": "Prove that $\\ds\\int_{0}^{\\pi} \\cos mx \\cos nx\\, dx$ and $\\ds\\int_{0}^{\\pi} \\sin mx \\sin nx\\, dx$ are each equal\nto zero except when $m = n$, when each is equal to~$\\frac{1}{2}\\pi$; and that\n\\[\n\\int_{0}^{\\pi} \\cos mx \\sin nx\\, dx = \\frac{2n}{n^{2} - m^{2}},\\quad\n\\int_{0}^{\\pi} \\cos mx \\sin nx\\, dx = 0,\n\\]\naccording as $n - m$~is odd or even.", "markdown": "Prove that $\\ds\\int_{0}^{\\pi} \\cos mx \\cos nx\\, dx$ and $\\ds\\int_{0}^{\\pi} \\sin mx \\sin nx\\, dx$ are each equal to zero except when $m = n$, when each is equal to $\\frac{1}{2}\\pi$; and that _0^ mx nx  dx = 2nn^2 - m^2,0pt minus 3pt_0^ mx nx  dx = 0, according as $n - m$ is odd or even.", "answer_latex": [ "to zero except when $m = n$, when each is equal to~$\\frac{1}{2}\\pi$" ], "answer_markdown": [ "to zero except when $m = n$, when each is equal to $\\frac{1}{2}\\pi$" ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.defint", "cas.integrate", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxiii/10b", "set": "hardy-course-of-pure-mathematics-1921/ex-lxiii", "number": 10, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "289", "location": "Exercise LXIII, problem 10b", "problem_latex": "Prove that $\\ds\\int_{0}^{\\pi} \\cos mx \\cos nx\\, dx$ and $\\ds\\int_{0}^{\\pi} \\sin mx \\sin nx\\, dx$ are each equal\nto zero except when $m = n$, when each is equal to~$\\frac{1}{2}\\pi$; and that\n\\[\n\\int_{0}^{\\pi} \\cos mx \\sin nx\\, dx = \\frac{2n}{n^{2} - m^{2}},\\quad\n\\int_{0}^{\\pi} \\cos mx \\sin nx\\, dx = 0,\n\\]\naccording as $n - m$~is odd or even.", "markdown": "Prove that $\\ds\\int_{0}^{\\pi} \\cos mx \\cos nx\\, dx$ and $\\ds\\int_{0}^{\\pi} \\sin mx \\sin nx\\, dx$ are each equal to zero except when $m = n$, when each is equal to $\\frac{1}{2}\\pi$; and that _0^ mx nx  dx = 2nn^2 - m^2,0pt minus 3pt_0^ mx nx  dx = 0, according as $n - m$ is odd or even.", "answer_latex": [ "to zero except when $m = n$, when each is equal to~$\\frac{1}{2}\\pi$" ], "answer_markdown": [ "to zero except when $m = n$, when each is equal to $\\frac{1}{2}\\pi$" ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.defint", "cas.integrate", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxiii/10c", "set": "hardy-course-of-pure-mathematics-1921/ex-lxiii", "number": 10, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "289", "location": "Exercise LXIII, problem 10c", "problem_latex": "Prove that $\\ds\\int_{0}^{\\pi} \\cos mx \\cos nx\\, dx$ and $\\ds\\int_{0}^{\\pi} \\sin mx \\sin nx\\, dx$ are each equal\nto zero except when $m = n$, when each is equal to~$\\frac{1}{2}\\pi$; and that\n\\[\n\\int_{0}^{\\pi} \\cos mx \\sin nx\\, dx = \\frac{2n}{n^{2} - m^{2}},\\quad\n\\int_{0}^{\\pi} \\cos mx \\sin nx\\, dx = 0,\n\\]\naccording as $n - m$~is odd or even.", "markdown": "Prove that $\\ds\\int_{0}^{\\pi} \\cos mx \\cos nx\\, dx$ and $\\ds\\int_{0}^{\\pi} \\sin mx \\sin nx\\, dx$ are each equal to zero except when $m = n$, when each is equal to $\\frac{1}{2}\\pi$; and that _0^ mx nx  dx = 2nn^2 - m^2,0pt minus 3pt_0^ mx nx  dx = 0, according as $n - m$ is odd or even.", "answer_latex": [ "\\int_{0}^{\\pi} \\cos mx \\sin nx\\, dx = \\frac{2n}{n^{2} - m^{2}}" ], "answer_markdown": [ "_0^ mx nx  dx = 2nn^2 - m^2" ], "checks": [ { "task": "integrate", "verdict": "FLAG-MISMATCH", "judge_why": [ "0.0", "0.38743449927936839534" ], "problem_expr": "cos(m*x)*sin(n*x)", "answer_expr": "2*n/(n**2 - m**2)" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "integrate: sin(b*x)*cos(a*x)" ], "shape": [ "integrate: sin(b*x)*cos(a*x)" ], "same_problem_in": [ "hardy-course-of-pure-mathematics-1921/ex-li/2c", "hardy-course-of-pure-mathematics-1921/ex-lxiii/10d" ], "needs": [ "cas.defint", "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxiii/10d", "set": "hardy-course-of-pure-mathematics-1921/ex-lxiii", "number": 10, "part": "d", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "289", "location": "Exercise LXIII, problem 10d", "problem_latex": "Prove that $\\ds\\int_{0}^{\\pi} \\cos mx \\cos nx\\, dx$ and $\\ds\\int_{0}^{\\pi} \\sin mx \\sin nx\\, dx$ are each equal\nto zero except when $m = n$, when each is equal to~$\\frac{1}{2}\\pi$; and that\n\\[\n\\int_{0}^{\\pi} \\cos mx \\sin nx\\, dx = \\frac{2n}{n^{2} - m^{2}},\\quad\n\\int_{0}^{\\pi} \\cos mx \\sin nx\\, dx = 0,\n\\]\naccording as $n - m$~is odd or even.", "markdown": "Prove that $\\ds\\int_{0}^{\\pi} \\cos mx \\cos nx\\, dx$ and $\\ds\\int_{0}^{\\pi} \\sin mx \\sin nx\\, dx$ are each equal to zero except when $m = n$, when each is equal to $\\frac{1}{2}\\pi$; and that _0^ mx nx  dx = 2nn^2 - m^2,0pt minus 3pt_0^ mx nx  dx = 0, according as $n - m$ is odd or even.", "answer_latex": [ "\\int_{0}^{\\pi} \\cos mx \\sin nx\\, dx = 0" ], "answer_markdown": [ "_0^ mx nx  dx = 0" ], "checks": [ { "task": "integrate", "verdict": "FLAG-MISMATCH", "judge_why": [ "0.0", "0.58915377667236274882" ], "problem_expr": "cos(m*x)*sin(n*x)", "answer_expr": "0" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "integrate: sin(b*x)*cos(a*x)" ], "shape": [ "integrate: sin(b*x)*cos(a*x)" ], "same_problem_in": [ "hardy-course-of-pure-mathematics-1921/ex-li/2c", "hardy-course-of-pure-mathematics-1921/ex-lxiii/10c" ], "needs": [ "cas.defint", "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxiii/1a", "set": "hardy-course-of-pure-mathematics-1921/ex-lxiii", "number": 1, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "289", "location": "Exercise LXIII, problem 1a", "problem_latex": "Show that\n\\[\n\\int_{a}^{b} x^{n}\\, dx = \\frac{b^{n+1} - a^{n+1}}{n + 1},\n\\]\nand in particular that\n\\[\n\\int_{0}^{1} x^{n}\\, dx = \\frac{1}{n + 1}.\n\\]", "markdown": "Show that _a^b x^n  dx = b^n+1 - a^n+1n + 1, and in particular that _0^1 x^n  dx = 1n + 1.", "answer_latex": [ "\\int_{a}^{b} x^{n}\\, dx = \\frac{b^{n+1} - a^{n+1}}{n + 1}" ], "answer_markdown": [ "_a^b x^n  dx = b^n+1 - a^n+1n + 1" ], "checks": [ { "task": "integrate", "verdict": "FLAG-MISMATCH", "judge_why": [ "0.0", "3.7073594834912028426" ], "problem_expr": "x**n", "answer_expr": "(b**(n+1) - a**(n+1))/(n + 1)" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "integrate: x**a" ], "shape": [ "integrate: x**a" ], "same_problem_in": [ "hardy-course-of-pure-mathematics-1921/ex-lxiii/1b", "hardy-course-of-pure-mathematics-1921/ex-lxiv/3b" ], "needs": [ "cas.defint", "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxiii/1b", "set": "hardy-course-of-pure-mathematics-1921/ex-lxiii", "number": 1, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "289", "location": "Exercise LXIII, problem 1b", "problem_latex": "Show that\n\\[\n\\int_{a}^{b} x^{n}\\, dx = \\frac{b^{n+1} - a^{n+1}}{n + 1},\n\\]\nand in particular that\n\\[\n\\int_{0}^{1} x^{n}\\, dx = \\frac{1}{n + 1}.\n\\]", "markdown": "Show that _a^b x^n  dx = b^n+1 - a^n+1n + 1, and in particular that _0^1 x^n  dx = 1n + 1.", "answer_latex": [ "\\int_{0}^{1} x^{n}\\, dx = \\frac{1}{n + 1}" ], "answer_markdown": [ "_0^1 x^n  dx = 1n + 1" ], "checks": [ { "task": "integrate", "verdict": "FLAG-MISMATCH", "judge_why": [ "0.0", "8.9005344079969473749" ], "problem_expr": "x**n", "answer_expr": "1/(n + 1)" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "integrate: x**a" ], "shape": [ "integrate: x**a" ], "same_problem_in": [ "hardy-course-of-pure-mathematics-1921/ex-lxiii/1a", "hardy-course-of-pure-mathematics-1921/ex-lxiv/3b" ], "needs": [ "cas.defint", "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxiii/2a", "set": "hardy-course-of-pure-mathematics-1921/ex-lxiii", "number": 2, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "289", "location": "Exercise LXIII, problem 2a", "problem_latex": "$\\ds\\int_{a}^{b} \\cos mx\\, dx = \\frac{\\sin mb - \\sin ma}{m}$,\n$\\ds\\int_{a}^{b} \\sin mx\\, dx = \\frac{\\cos ma - \\cos mb}{m}$.", "markdown": "$\\ds\\int_{a}^{b} \\cos mx\\, dx = \\frac{\\sin mb - \\sin ma}{m}$, $\\ds\\int_{a}^{b} \\sin mx\\, dx = \\frac{\\cos ma - \\cos mb}{m}$.", "answer_latex": [ "\\ds\\int_{a}^{b} \\cos mx\\, dx = \\frac{\\sin mb - \\sin ma}{m}" ], "answer_markdown": [ "_a^b mx  dx = mb - mam" ], "checks": [ { "task": "integrate", "verdict": "FLAG-EXTRACTION", "judge_why": [ "0.0", "-0.99996465847134196163" ], "problem_expr": "cos(m*x)", "answer_expr": "(sin(m*b) - sin(m*a))/m" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [ "integrate: cos(a*x)" ], "shape": [ "integrate: cos(a*x)" ], "same_problem_in": [ "hardy-course-of-pure-mathematics-1921/ex-lxiv/4a" ], "needs": [ "cas.defint", "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxiii/2b", "set": "hardy-course-of-pure-mathematics-1921/ex-lxiii", "number": 2, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "289", "location": "Exercise LXIII, problem 2b", "problem_latex": "$\\ds\\int_{a}^{b} \\cos mx\\, dx = \\frac{\\sin mb - \\sin ma}{m}$,\n$\\ds\\int_{a}^{b} \\sin mx\\, dx = \\frac{\\cos ma - \\cos mb}{m}$.", "markdown": "$\\ds\\int_{a}^{b} \\cos mx\\, dx = \\frac{\\sin mb - \\sin ma}{m}$, $\\ds\\int_{a}^{b} \\sin mx\\, dx = \\frac{\\cos ma - \\cos mb}{m}$.", "answer_latex": [ "\\ds\\int_{a}^{b} \\sin mx\\, dx = \\frac{\\cos ma - \\cos mb}{m}" ], "answer_markdown": [ "_a^b mx  dx = ma - mbm" ], "checks": [ { "task": "integrate", "verdict": "FLAG-EXTRACTION", "judge_why": [ "0.0", "-0.060925330445990543209" ], "problem_expr": "sin(m*x)", "answer_expr": "(cos(m*a) - cos(m*b))/m" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [ "integrate: sin(a*x)" ], "shape": [ "integrate: sin(a*x)" ], "same_problem_in": [ "hardy-course-of-pure-mathematics-1921/ex-lxiv/4b" ], "needs": [ "cas.defint", "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxiii/3a", "set": "hardy-course-of-pure-mathematics-1921/ex-lxiii", "number": 3, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "289", "location": "Exercise LXIII, problem 3a", "problem_latex": "$\\ds\\int_{a}^{b}\\frac{dx}{1 + x^{2}} = \\arctan b - \\arctan a$,\n$\\ds\\int_{0}^{1}\\frac{dx}{1 + x^{2}} = \\tfrac{1}{4}\\pi$.\n\n[There is an apparent difficulty here owing to the fact that $\\arctan x$~is a\nmany valued function. The difficulty may be avoided by observing that, in\nthe equation\n\\[\n\\int_{0}^{x} \\frac{dt}{1 + t^{2}} = \\arctan x,\n\\]\n$\\arctan x$ must denote an angle lying between $-\\frac{1}{2}\\pi$ and~$\\frac{1}{2}\\pi$. For the integral\nvanishes when $x = 0$ and increases steadily and continuously as $x$~increases.\nThus the same is true of~$\\arctan x$, which therefore tends to~$\\tfrac{1}{2}\\pi$ as $x \\to \\infty$.\nIn the same way we can show that $\\arctan x \\to -\\frac{1}{2}\\pi$ as $x \\to -\\infty$. Similarly,\nin the equation\n\\[\n\\int_{0}^{x} \\frac{dt}{\\sqrtp{1 - t^{2}}} = \\arcsin x,\n\\]\nwhere $-1 < x < 1$, $\\arcsin x$ denotes an angle lying between $-\\frac{1}{2}\\pi$ and $\\frac{1}{2}\\pi$.\nThus, if $a$~and~$b$ are both numerically less than unity, we have\n\\[\n\\int_{a}^{b} \\frac{dx}{\\sqrtp{1 - x^{2}}} = \\arcsin b - \\arcsin a.]\n\\]", "markdown": "$\\ds\\int_{a}^{b}\\frac{dx}{1 + x^{2}} = \\arctan b - \\arctan a$, $\\ds\\int_{0}^{1}\\frac{dx}{1 + x^{2}} = \\tfrac{1}{4}\\pi$. [There is an apparent difficulty here owing to the fact that $\\arctan x$ is a many valued function. The difficulty may be avoided by observing that, in the equation _0^x dt1 + t^2 = x, $\\arctan x$ must denote an angle lying between $-\\frac{1}{2}\\pi$ and $\\frac{1}{2}\\pi$. For the integral vanishes when $x = 0$ and increases steadily and continuously as $x$ increases. Thus the same is true of $\\arctan x$, which therefore tends to $\\tfrac{1}{2}\\pi$ as $x \\to \\infty$. In the same way we can show that $\\arctan x \\to -\\frac{1}{2}\\pi$ as $x \\to -\\infty$. Similarly, in the equation _0^x dt1 - t^2 = x, where $-1 < x < 1$, $\\arcsin x$ denotes an angle lying between $-\\frac{1}{2}\\pi$ and $\\frac{1}{2}\\pi$. Thus, if $a$ and $b$ are both numerically less than unity, we have _a^b dx1 - x^2 = b - a.]", "answer_latex": [ "\\int_{a}^{b}\\frac{dx}{1 + x^{2}} = \\arctan b - \\arctan a" ], "answer_markdown": [ "_a^bdx1 + x^2 = b - a" ], "checks": [ { "task": "integrate", "verdict": "FLAG-EXTRACTION", "judge_why": [ "0.0", "0.10341525694222789658" ], "problem_expr": "1/(1 + x**2)", "answer_expr": "atan(b) - atan(a)" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [ "integrate: 1/(x**2 + 1)" ], "shape": [ "integrate: 1/(x**N + 1)" ], "same_problem_in": [ "hardy-course-of-pure-mathematics-1921/ex-lxiii/3b" ], "needs": [ "cas.defint", "cas.integrate", "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxiii/3b", "set": "hardy-course-of-pure-mathematics-1921/ex-lxiii", "number": 3, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "289", "location": "Exercise LXIII, problem 3b", "problem_latex": "$\\ds\\int_{a}^{b}\\frac{dx}{1 + x^{2}} = \\arctan b - \\arctan a$,\n$\\ds\\int_{0}^{1}\\frac{dx}{1 + x^{2}} = \\tfrac{1}{4}\\pi$.\n\n[There is an apparent difficulty here owing to the fact that $\\arctan x$~is a\nmany valued function. The difficulty may be avoided by observing that, in\nthe equation\n\\[\n\\int_{0}^{x} \\frac{dt}{1 + t^{2}} = \\arctan x,\n\\]\n$\\arctan x$ must denote an angle lying between $-\\frac{1}{2}\\pi$ and~$\\frac{1}{2}\\pi$. For the integral\nvanishes when $x = 0$ and increases steadily and continuously as $x$~increases.\nThus the same is true of~$\\arctan x$, which therefore tends to~$\\tfrac{1}{2}\\pi$ as $x \\to \\infty$.\nIn the same way we can show that $\\arctan x \\to -\\frac{1}{2}\\pi$ as $x \\to -\\infty$. Similarly,\nin the equation\n\\[\n\\int_{0}^{x} \\frac{dt}{\\sqrtp{1 - t^{2}}} = \\arcsin x,\n\\]\nwhere $-1 < x < 1$, $\\arcsin x$ denotes an angle lying between $-\\frac{1}{2}\\pi$ and $\\frac{1}{2}\\pi$.\nThus, if $a$~and~$b$ are both numerically less than unity, we have\n\\[\n\\int_{a}^{b} \\frac{dx}{\\sqrtp{1 - x^{2}}} = \\arcsin b - \\arcsin a.]\n\\]", "markdown": "$\\ds\\int_{a}^{b}\\frac{dx}{1 + x^{2}} = \\arctan b - \\arctan a$, $\\ds\\int_{0}^{1}\\frac{dx}{1 + x^{2}} = \\tfrac{1}{4}\\pi$. [There is an apparent difficulty here owing to the fact that $\\arctan x$ is a many valued function. The difficulty may be avoided by observing that, in the equation _0^x dt1 + t^2 = x, $\\arctan x$ must denote an angle lying between $-\\frac{1}{2}\\pi$ and $\\frac{1}{2}\\pi$. For the integral vanishes when $x = 0$ and increases steadily and continuously as $x$ increases. Thus the same is true of $\\arctan x$, which therefore tends to $\\tfrac{1}{2}\\pi$ as $x \\to \\infty$. In the same way we can show that $\\arctan x \\to -\\frac{1}{2}\\pi$ as $x \\to -\\infty$. Similarly, in the equation _0^x dt1 - t^2 = x, where $-1 < x < 1$, $\\arcsin x$ denotes an angle lying between $-\\frac{1}{2}\\pi$ and $\\frac{1}{2}\\pi$. Thus, if $a$ and $b$ are both numerically less than unity, we have _a^b dx1 - x^2 = b - a.]", "answer_latex": [ "\\int_{0}^{1}\\frac{dx}{1 + x^{2}} = \\tfrac{1}{4}\\pi" ], "answer_markdown": [ "_0^1dx1 + x^2 = 14" ], "checks": [ { "task": "integrate", "verdict": "FLAG-EXTRACTION", "judge_why": [ "0.0", "0.083059636992221261884" ], "problem_expr": "1/(1 + x**2)", "answer_expr": "pi/4" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [ "integrate: 1/(x**2 + 1)" ], "shape": [ "integrate: 1/(x**N + 1)" ], "same_problem_in": [ "hardy-course-of-pure-mathematics-1921/ex-lxiii/3a" ], "needs": [ "cas.defint", "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxiii/4a", "set": "hardy-course-of-pure-mathematics-1921/ex-lxiii", "number": 4, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "289", "location": "Exercise LXIII, problem 4a", "problem_latex": "$\\ds\\int_{0}^{1} \\frac{dx}{1 - x + x^{2}} = \\frac{2\\pi}{3\\sqrt3}$,\n$\\ds\\int_{0}^{1} \\frac{dx}{1 + x + x^{2}} = \\frac{\\pi}{3\\sqrt3}$\\Add{.}", "markdown": "$\\ds\\int_{0}^{1} \\frac{dx}{1 - x + x^{2}} = \\frac{2\\pi}{3\\sqrt3}$, $\\ds\\int_{0}^{1} \\frac{dx}{1 + x + x^{2}} = \\frac{\\pi}{3\\sqrt3}$", "answer_latex": [ "\\ds\\int_{0}^{1} \\frac{dx}{1 - x + x^{2}} = \\frac{2\\pi}{3\\sqrt3}" ], "answer_markdown": [ "_0^1 dx1 - x + x^2 = 233" ], "checks": [ { "task": "integrate", "verdict": "FLAG-EXTRACTION", "judge_why": [ "0.0", "0.36719400499583680266" ], "problem_expr": "1/(1 - x + x**2)", "answer_expr": "2*pi/(3*sqrt(3))" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [ "integrate: 1/(x**2 - x + 1)" ], "shape": [ "integrate: 1/(-x + x**N + 1)" ], "same_problem_in": [], "needs": [ "cas.defint", "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxiii/4b", "set": "hardy-course-of-pure-mathematics-1921/ex-lxiii", "number": 4, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "289", "location": "Exercise LXIII, problem 4b", "problem_latex": "$\\ds\\int_{0}^{1} \\frac{dx}{1 - x + x^{2}} = \\frac{2\\pi}{3\\sqrt3}$,\n$\\ds\\int_{0}^{1} \\frac{dx}{1 + x + x^{2}} = \\frac{\\pi}{3\\sqrt3}$\\Add{.}", "markdown": "$\\ds\\int_{0}^{1} \\frac{dx}{1 - x + x^{2}} = \\frac{2\\pi}{3\\sqrt3}$, $\\ds\\int_{0}^{1} \\frac{dx}{1 + x + x^{2}} = \\frac{\\pi}{3\\sqrt3}$", "answer_latex": [ "\\ds\\int_{0}^{1} \\frac{dx}{1 + x + x^{2}} = \\frac{\\pi}{3\\sqrt3}" ], "answer_markdown": [ "_0^1 dx1 + x + x^2 = 33" ], "checks": [ { "task": "integrate", "verdict": "FLAG-EXTRACTION", "judge_why": [ "0.0", "0.20163030448333732918" ], "problem_expr": "1/(1 + x + x**2)", "answer_expr": "pi/(3*sqrt(3))" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [ "integrate: 1/(x**2 + x + 1)" ], "shape": [ "integrate: 1/(x + x**N + 1)" ], "same_problem_in": [], "needs": [ "cas.defint", "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxiii/5", "set": "hardy-course-of-pure-mathematics-1921/ex-lxiii", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "289", "location": "Exercise LXIII, problem 5", "problem_latex": "$\\ds\\int_{0}^{1} \\frac{dx}{1 + 2x\\cos\\alpha + x^{2}} = \\frac{\\alpha}{2\\sin\\alpha}$ if $-\\pi < \\alpha < \\pi$, except when $\\alpha = 0$, when the\nvalue of the integral is~$\\frac{1}{2}$, which is the limit of~$\\frac{1}{2}\\alpha\\cosec\\alpha$ as $\\alpha \\to 0$.", "markdown": "$\\ds\\int_{0}^{1} \\frac{dx}{1 + 2x\\cos\\alpha + x^{2}} = \\frac{\\alpha}{2\\sin\\alpha}$ if $-\\pi < \\alpha < \\pi$, except when $\\alpha = 0$, when the value of the integral is $\\frac{1}{2}$, which is the limit of $\\frac{1}{2}\\alpha\\cosec\\alpha$ as $\\alpha \\to 0$.", "answer_latex": [ "\\frac{\\alpha}{2\\sin\\alpha}" ], "answer_markdown": [ "2" ], "checks": [ { "task": "integrate", "verdict": "FLAG-MISMATCH", "judge_why": [ "0.0", "2.961594990271258346" ], "problem_expr": "1/(1 + 2*x*cos(alpha) + x**2)", "answer_expr": "alpha/(2*sin(alpha))" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "integrate: 1/(x**2 + 2*x*cos(a) + 1)" ], "shape": [ "integrate: 1/(N*x*cos(a) + x**N + 1)" ], "same_problem_in": [], "needs": [ "cas.defint", "cas.integrate", "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxiii/6a", "set": "hardy-course-of-pure-mathematics-1921/ex-lxiii", "number": 6, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "289", "location": "Exercise LXIII, problem 6a", "problem_latex": "$\\ds\\int_{0}^{\\DPtypo{}{1}} \\sqrtp{1 - x^{2}}\\, dx = \\tfrac{1}{4}\\pi$,\n$\\ds\\int_{0}^{a} \\sqrtp{a^{2} - x^{2}}\\, dx = \\tfrac{1}{4}\\pi a^{2}$\\quad $(a > 0)$.", "markdown": "$\\ds\\int_{0}^{\\DPtypo{}{1}} \\sqrtp{1 - x^{2}}\\, dx = \\tfrac{1}{4}\\pi$, $\\ds\\int_{0}^{a} \\sqrtp{a^{2} - x^{2}}\\, dx = \\tfrac{1}{4}\\pi a^{2}$0pt minus 3pt $(a > 0)$.", "answer_latex": [ "\\ds\\int_{0}^{\\DPtypo{}{1}} \\sqrtp{1 - x^{2}}\\, dx = \\tfrac{1}{4}\\pi" ], "answer_markdown": [ "_0^ 1 - x^2  dx = 14" ], "checks": [ { "task": "integrate", "verdict": "FLAG-EXTRACTION", "judge_why": [ "0.0", "0.56196576181437244797" ], "problem_expr": "sqrt(1 - x**2)", "answer_expr": "pi/4" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [ "integrate: sqrt(-x**2 + 1)" ], "shape": [ "integrate: (-x**N + 1)**N" ], "same_problem_in": [], "needs": [ "cas.defint", "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxiii/6b", "set": "hardy-course-of-pure-mathematics-1921/ex-lxiii", "number": 6, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "289", "location": "Exercise LXIII, problem 6b", "problem_latex": "$\\ds\\int_{0}^{\\DPtypo{}{1}} \\sqrtp{1 - x^{2}}\\, dx = \\tfrac{1}{4}\\pi$,\n$\\ds\\int_{0}^{a} \\sqrtp{a^{2} - x^{2}}\\, dx = \\tfrac{1}{4}\\pi a^{2}$\\quad $(a > 0)$.", "markdown": "$\\ds\\int_{0}^{\\DPtypo{}{1}} \\sqrtp{1 - x^{2}}\\, dx = \\tfrac{1}{4}\\pi$, $\\ds\\int_{0}^{a} \\sqrtp{a^{2} - x^{2}}\\, dx = \\tfrac{1}{4}\\pi a^{2}$0pt minus 3pt $(a > 0)$.", "answer_latex": [ "\\ds\\int_{0}^{a} \\sqrtp{a^{2} - x^{2}}\\, dx = \\tfrac{1}{4}\\pi a^{2}" ], "answer_markdown": [ "_0^a a^2 - x^2  dx = 14a^2" ], "checks": [ { "task": "integrate", "verdict": "FLAG-EXTRACTION", "judge_why": [ "0.0", "2.5238572902012061165" ], "problem_expr": "sqrt(a**2 - x**2)", "answer_expr": "pi*a**2/4" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [ "integrate: sqrt(a**2 - x**2)" ], "shape": [ "integrate: (a**N - x**N)**N" ], "same_problem_in": [ "hardy-course-of-pure-mathematics-1921/ex-xlix/2b", "thompson-calculus-made-easy-1914/ex-xix/1" ], "needs": [ "cas.defint", "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxiii/7", "set": "hardy-course-of-pure-mathematics-1921/ex-lxiii", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "289", "location": "Exercise LXIII, problem 7", "problem_latex": "$\\ds\\int_{0}^{\\pi} \\frac{dx}{a + b\\cos x} = \\frac{\\pi}{\\sqrt{a^{2} - b^{2}}}$, if $a > |b|$. [For the form of the indefinite\nintegral see \\Exs{liii}.\\ 3,~4. If $|a| < |b|$ then the subject of integration has an\ninfinity between $0$ and~$\\pi$. What is the value of the integral when $a$~is\nnegative and $-a > |b|$?]", "markdown": "$\\ds\\int_{0}^{\\pi} \\frac{dx}{a + b\\cos x} = \\frac{\\pi}{\\sqrt{a^{2} - b^{2}}}$, if $a > |b|$. [For the form of the indefinite integral see liii. 3, 4. If $|a| < |b|$ then the subject of integration has an infinity between $0$ and $\\pi$. What is the value of the integral when $a$ is negative and $-a > |b|$?]", "answer_latex": [ "\\frac{\\pi}{\\sqrt{a^{2} - b^{2}}}" ], "answer_markdown": [ "a^2 - b^2" ], "checks": [ { "task": "integrate", "verdict": "FLAG-EXTRACTION", "judge_why": [ "0.0", "1.4251971058517250935" ], "problem_expr": "1/(a + b*cos(x))", "answer_expr": "pi/sqrt(a**2 - b**2)" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [ "integrate: 1/(a + b*cos(x))" ], "shape": [ "integrate: 1/(a + b*cos(x))" ], "same_problem_in": [], "needs": [ "cas.defint", "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxiii/8", "set": "hardy-course-of-pure-mathematics-1921/ex-lxiii", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "289", "location": "Exercise LXIII, problem 8", "problem_latex": "$\\ds\\int_{0}^{\\frac{1}{2}\\pi} \\frac{dx}{a^{2}\\cos^{2}x + b^{2}\\sin^{2}x} = \\frac{\\pi}{2ab}$, if $a$~and~$b$ are positive. What is the\nvalue of the integral when $a$~and~$b$ have opposite signs, or when both are\nnegative?", "markdown": "$\\ds\\int_{0}^{\\frac{1}{2}\\pi} \\frac{dx}{a^{2}\\cos^{2}x + b^{2}\\sin^{2}x} = \\frac{\\pi}{2ab}$, if $a$ and $b$ are positive. What is the value of the integral when $a$ and $b$ have opposite signs, or when both are negative?", "answer_latex": [ "\\frac{\\pi}{2ab}" ], "answer_markdown": [ "2ab" ], "checks": [ { "task": "integrate", "verdict": "FLAG-MISMATCH", "judge_why": [ "0.0", "0.11906029221901489387" ], "problem_expr": "1/(a**2*cos(x)**2 + b**2*sin(x)**2)", "answer_expr": "pi/(2*a*b)" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISMATCH" ] }, "form": [ "integrate: 1/(a**2*cos(x)**2 + b**2*sin(x)**2)" ], "shape": [ "integrate: 1/(a**N*cos(x)**N + b**N*sin(x)**N)" ], "same_problem_in": [], "needs": [ "cas.defint", "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxiii/9a", "set": "hardy-course-of-pure-mathematics-1921/ex-lxiii", "number": 9, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "289", "location": "Exercise LXIII, problem 9a", "problem_latex": "\\Topic{Fourier's integrals.} Prove that if $m$~and~$n$ are positive integers then\n\\[\n\\int_{0}^{2\\pi} \\cos mx \\sin nx\\, dx\n\\]\nis always equal to zero, and\n\\[\n\\int_{0}^{2\\pi} \\cos mx \\cos nx\\, dx,\\quad\n\\int_{0}^{2\\pi} \\sin mx \\sin nx\\, dx\n\\]\nare equal to zero unless $m = n$, when each is equal to~$\\pi$.", "markdown": "**’s integrals.** Prove that if $m$ and $n$ are positive integers then _0^2 mx nx  dx is always equal to zero, and _0^2 mx nx  dx,0pt minus 3pt_0^2 mx nx  dx are equal to zero unless $m = n$, when each is equal to $\\pi$.", "answer_latex": [ "is always equal to zero" ], "answer_markdown": [ "is always equal to zero" ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.defint", "cas.integrate", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxiii/9b", "set": "hardy-course-of-pure-mathematics-1921/ex-lxiii", "number": 9, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "289", "location": "Exercise LXIII, problem 9b", "problem_latex": "\\Topic{Fourier's integrals.} Prove that if $m$~and~$n$ are positive integers then\n\\[\n\\int_{0}^{2\\pi} \\cos mx \\sin nx\\, dx\n\\]\nis always equal to zero, and\n\\[\n\\int_{0}^{2\\pi} \\cos mx \\cos nx\\, dx,\\quad\n\\int_{0}^{2\\pi} \\sin mx \\sin nx\\, dx\n\\]\nare equal to zero unless $m = n$, when each is equal to~$\\pi$.", "markdown": "**’s integrals.** Prove that if $m$ and $n$ are positive integers then _0^2 mx nx  dx is always equal to zero, and _0^2 mx nx  dx,0pt minus 3pt_0^2 mx nx  dx are equal to zero unless $m = n$, when each is equal to $\\pi$.", "answer_latex": [ "\\int_{0}^{2\\pi} \\cos mx \\cos nx\\, dx" ], "answer_markdown": [ "_0^2 mx nx  dx" ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.defint", "cas.integrate", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxiii/9c", "set": "hardy-course-of-pure-mathematics-1921/ex-lxiii", "number": 9, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "289", "location": "Exercise LXIII, problem 9c", "problem_latex": "\\Topic{Fourier's integrals.} Prove that if $m$~and~$n$ are positive integers then\n\\[\n\\int_{0}^{2\\pi} \\cos mx \\sin nx\\, dx\n\\]\nis always equal to zero, and\n\\[\n\\int_{0}^{2\\pi} \\cos mx \\cos nx\\, dx,\\quad\n\\int_{0}^{2\\pi} \\sin mx \\sin nx\\, dx\n\\]\nare equal to zero unless $m = n$, when each is equal to~$\\pi$.", "markdown": "**’s integrals.** Prove that if $m$ and $n$ are positive integers then _0^2 mx nx  dx is always equal to zero, and _0^2 mx nx  dx,0pt minus 3pt_0^2 mx nx  dx are equal to zero unless $m = n$, when each is equal to $\\pi$.", "answer_latex": [ "\\int_{0}^{2\\pi} \\sin mx \\sin nx\\, dx" ], "answer_markdown": [ "_0^2 mx nx  dx" ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.defint", "cas.integrate", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxiv/1", "set": "hardy-course-of-pure-mathematics-1921/ex-lxiv", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "290", "location": "Exercise LXIV, problem 1", "problem_latex": "Evaluate $\\ds\\int_{a}^{b} x\\, dx$ by dividing $\\DPmod{(a, b)}{[a, b]}$ into $n$~equal\nparts by the points of division $a = x_{0}$, $x_{1}$, $x_{2}$,~\\dots, $x_{n} = b$, and calculating the\nlimit as $n \\to \\infty$ of\n\\[\n(x_{1} - x_{0})f(x_{0}) + (x_{2} - x_{1})f(x_{1}) + \\dots + (x_{n} - x_{n-1})f(x_{n-1}).\n\\]", "markdown": "Evaluate $\\ds\\int_{a}^{b} x\\, dx$ by dividing $\\DPmod{(a, b)}{[a, b]}$ into $n$ equal parts by the points of division $a = x_{0}$, $x_{1}$, $x_{2}$, …, $x_{n} = b$, and calculating the limit as $n \\to \\infty$ of (x_1 - x_0)f(x_0) + (x_2 - x_1)f(x_1) + …+ (x_n - x_n-1)f(x_n-1).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "cas.sum" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxiv/2", "set": "hardy-course-of-pure-mathematics-1921/ex-lxiv", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "290", "location": "Exercise LXIV, problem 2", "problem_latex": "Calculate $\\ds\\int_{a}^{b} x^{2}\\, dx$ in the same way.", "markdown": "Calculate $\\ds\\int_{a}^{b} x^{2}\\, dx$ in the same way.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "x**2", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: x**2" ], "shape": [ "integrate: x**N" ], "same_problem_in": [], "needs": [ "cas.defint" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxiv/3a", "set": "hardy-course-of-pure-mathematics-1921/ex-lxiv", "number": 3, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "290", "location": "Exercise LXIV, problem 3a", "problem_latex": "Calculate $\\ds\\int_{a}^{b} x\\, dx$, where $0 < a < b$, by dividing $\\DPmod{(a, b)}{[a, b]}$ into $n$~parts by\nthe points of division $a$, $ar$, $ar^{2}$,~\\dots\\Add{,} $ar^{n-1}$, $ar^{n}$, where $r^{n} = b/a$. Apply the same\nmethod to the more general integral $\\ds\\int_{a}^{b} x^{m}\\, dx$.", "markdown": "Calculate $\\ds\\int_{a}^{b} x\\, dx$, where $0 < a < b$, by dividing $\\DPmod{(a, b)}{[a, b]}$ into $n$ parts by the points of division $a$, $ar$, $ar^{2}$, … $ar^{n-1}$, $ar^{n}$, where $r^{n} = b/a$. Apply the same method to the more general integral $\\ds\\int_{a}^{b} x^{m}\\, dx$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "x", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: x" ], "shape": [ "integrate: x" ], "same_problem_in": [], "needs": [ "cas.defint" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxiv/3b", "set": "hardy-course-of-pure-mathematics-1921/ex-lxiv", "number": 3, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "290", "location": "Exercise LXIV, problem 3b", "problem_latex": "Calculate $\\ds\\int_{a}^{b} x\\, dx$, where $0 < a < b$, by dividing $\\DPmod{(a, b)}{[a, b]}$ into $n$~parts by\nthe points of division $a$, $ar$, $ar^{2}$,~\\dots\\Add{,} $ar^{n-1}$, $ar^{n}$, where $r^{n} = b/a$. Apply the same\nmethod to the more general integral $\\ds\\int_{a}^{b} x^{m}\\, dx$.", "markdown": "Calculate $\\ds\\int_{a}^{b} x\\, dx$, where $0 < a < b$, by dividing $\\DPmod{(a, b)}{[a, b]}$ into $n$ parts by the points of division $a$, $ar$, $ar^{2}$, … $ar^{n-1}$, $ar^{n}$, where $r^{n} = b/a$. Apply the same method to the more general integral $\\ds\\int_{a}^{b} x^{m}\\, dx$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "x**m", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: x**a" ], "shape": [ "integrate: x**a" ], "same_problem_in": [ "hardy-course-of-pure-mathematics-1921/ex-lxiii/1a", "hardy-course-of-pure-mathematics-1921/ex-lxiii/1b" ], "needs": [ "cas.defint" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxiv/4a", "set": "hardy-course-of-pure-mathematics-1921/ex-lxiv", "number": 4, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "290", "location": "Exercise LXIV, problem 4a", "problem_latex": "Calculate $\\ds\\int_{a}^{b}\\cos mx\\, dx$ and $\\ds\\int_{a}^{b}\\sin mx\\, dx$ by the method of Ex.~1.", "markdown": "Calculate $\\ds\\int_{a}^{b}\\cos mx\\, dx$ and $\\ds\\int_{a}^{b}\\sin mx\\, dx$ by the method of Ex. 1.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "cos(m*x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: cos(a*x)" ], "shape": [ "integrate: cos(a*x)" ], "same_problem_in": [ "hardy-course-of-pure-mathematics-1921/ex-lxiii/2a" ], "needs": [ "cas.defint", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxiv/4b", "set": "hardy-course-of-pure-mathematics-1921/ex-lxiv", "number": 4, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "290", "location": "Exercise LXIV, problem 4b", "problem_latex": "Calculate $\\ds\\int_{a}^{b}\\cos mx\\, dx$ and $\\ds\\int_{a}^{b}\\sin mx\\, dx$ by the method of Ex.~1.", "markdown": "Calculate $\\ds\\int_{a}^{b}\\cos mx\\, dx$ and $\\ds\\int_{a}^{b}\\sin mx\\, dx$ by the method of Ex. 1.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "sin(m*x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: sin(a*x)" ], "shape": [ "integrate: sin(a*x)" ], "same_problem_in": [ "hardy-course-of-pure-mathematics-1921/ex-lxiii/2b" ], "needs": [ "cas.defint", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxiv/5", "set": "hardy-course-of-pure-mathematics-1921/ex-lxiv", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "290", "location": "Exercise LXIV, problem 5", "problem_latex": "Prove that $n\\sum\\limits_{r=0}^{n-1} \\dfrac{1}{n^{2} + r^{2}} \\to \\tfrac{1}{4}\\pi$ as $n \\to \\infty$.", "markdown": "Prove that $n\\sum\\limits_{r=0}^{n-1} \\dfrac{1}{n^{2} + r^{2}} \\to \\tfrac{1}{4}\\pi$ as $n \\to \\infty$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "cas.sum" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxiv/6", "set": "hardy-course-of-pure-mathematics-1921/ex-lxiv", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "290", "location": "Exercise LXIV, problem 6", "problem_latex": "Prove that $\\dfrac{1}{n^{2}} \\sum\\limits_{r=0}^{n-1} \\sqrtp{n^{2} - r^{2}} \\to \\tfrac{1}{4}\\pi$.", "markdown": "Prove that $\\dfrac{1}{n^{2}} \\sum\\limits_{r=0}^{n-1} \\sqrtp{n^{2} - r^{2}} \\to \\tfrac{1}{4}\\pi$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "cas.sum" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxix/1", "set": "hardy-course-of-pure-mathematics-1921/ex-lxix", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "317", "location": "Exercise LXIX, problem 1", "problem_latex": "Use Abel's theorem to show that $\\sum (1/n)$ and\n$\\sum \\{1/(an + b)\\}$ are divergent. [Here $nu_{n} \\to 1$ or $nu_{n} \\to 1/a$.]", "markdown": "Use Abel’s theorem to show that $\\sum (1/n)$ and $\\sum \\{1/(an + b)\\}$ are divergent. [Here $nu_{n} \\to 1$ or $nu_{n} \\to 1/a$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:series.convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxix/2", "set": "hardy-course-of-pure-mathematics-1921/ex-lxix", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "317", "location": "Exercise LXIX, problem 2", "problem_latex": "Show that Abel's theorem is not true if we omit the condition that $u_{n}$~decreases\nas $n$~increases. [The series\n\\[\n1 + \\frac{1}{2^{2}} + \\frac{1}{3^{2}}\n + \\frac{1}{4}\n + \\frac{1}{5^{2}} + \\frac{1}{6^{2}} + \\frac{1}{7^{2}} + \\frac{1}{8^{2}}\n + \\frac{1}{9}\n + \\frac{1}{10^{2}} + \\dots,\n\\]\nin which $u_{n} = 1/n$ or $1/n^{2}$, according as $n$~is or is not a perfect square, is\nconvergent, since it may be rearranged in the form\n\\[\n\\frac{1}{2^{2}} + \\frac{1}{3^{2}}\n + \\frac{1}{5^{2}} + \\frac{1}{6^{2}} + \\frac{1}{7^{2}} + \\frac{1}{8^{2}}\n + \\frac{1}{10^{2}}\n + \\dots + \\left(1 + \\frac{1}{4} + \\frac{1}{9} + \\dots\\right),\n\\]\nand each of these series is convergent. But, since $nu_{n} = 1$ whenever $\\DPtypo{u}{n}$~is a\nperfect square, it is clearly not true that $nu_{n} \\to 0$.]", "markdown": "Show that Abel’s theorem is not true if we omit the condition that $u_{n}$ decreases as $n$ increases. [The series 1 + 12^2 + 13^2 + 14 + 15^2 + 16^2 + 17^2 + 18^2 + 19 + 110^2 + …, in which $u_{n} = 1/n$ or $1/n^{2}$, according as $n$ is or is not a perfect square, is convergent, since it may be rearranged in the form 12^2 + 13^2 + 15^2 + 16^2 + 17^2 + 18^2 + 110^2 + …+ (1 + 14 + 19 + …), and each of these series is convergent. But, since $nu_{n} = 1$ whenever $\\DPtypo{u}{n}$ is a perfect square, it is clearly not true that $nu_{n} \\to 0$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:series.convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxix/3", "set": "hardy-course-of-pure-mathematics-1921/ex-lxix", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "317", "location": "Exercise LXIX, problem 3", "problem_latex": "\\emph{The converse of Abel's theorem is not true}, \\ie\\ it is not true that, if $u_{n}$~decreases\nwith~$n$ and $\\lim nu_{n} = 0$, then $\\sum u_{n}$~is convergent.\n\n[Take the series $\\sum(1/n)$ and multiply the first term by~$1$, the second by~$\\frac{1}{2}$,\nthe next two by~$\\frac{1}{3}$, the next four by~$\\frac{1}{4}$, the next eight by~$\\frac{1}{5}$, and so on. On\ngrouping in brackets the terms of the new series thus formed we obtain\n\\[\n1 + \\tfrac{1}{2} · \\tfrac{1}{2}\n + \\tfrac{1}{3} \\left(\\tfrac{1}{3} + \\tfrac{1}{4}\\right)\n + \\tfrac{1}{4} \\left(\\tfrac{1}{5} + \\tfrac{1}{6} + \\tfrac{1}{7} + \\tfrac{1}{8}\\right) + \\dots;\n\\]\nand this series is divergent, since its terms are greater than those of\n\\[\n1 + \\tfrac{1}{2} · \\tfrac{1}{2}\n + \\tfrac{1}{3} · \\tfrac{1}{2}\n + \\tfrac{1}{4} · \\tfrac{1}{2} + \\dots,\n\\]\nwhich is divergent. But it is easy to see that the terms of the series\n\\[\n1 + \\tfrac{1}{2} · \\tfrac{1}{2}\n + \\tfrac{1}{3} · \\tfrac{1}{3}\n + \\tfrac{1}{3} · \\tfrac{1}{4}\n + \\tfrac{1}{4} · \\tfrac{1}{5}\n + \\tfrac{1}{4} · \\tfrac{1}{6} + \\dots\n\\]\nsatisfy the condition that $nu_{n} \\to 0$. In fact $nu_{n} = 1/\\nu$ if $2^{\\nu-2} < n \\leq 2^{\\nu-1}$, and\n$\\nu \\to \\infty$ as $n \\to \\infty$.]", "markdown": "*The converse of Abel’s theorem is not true*, *i.e.* it is not true that, if $u_{n}$ decreases with $n$ and $\\lim nu_{n} = 0$, then $\\sum u_{n}$ is convergent. [Take the series $\\sum(1/n)$ and multiply the first term by $1$, the second by $\\frac{1}{2}$, the next two by $\\frac{1}{3}$, the next four by $\\frac{1}{4}$, the next eight by $\\frac{1}{5}$, and so on. On grouping in brackets the terms of the new series thus formed we obtain 1 + 12 · 12 + 13 (13 + 14) + 14 (15 + 16 + 17 + 18) + …; and this series is divergent, since its terms are greater than those of 1 + 12 · 12 + 13 · 12 + 14 · 12 + …, which is divergent. But it is easy to see that the terms of the series 1 + 12 · 12 + 13 · 13 + 13 · 14 + 14 · 15 + 14 · 16 + … satisfy the condition that $nu_{n} \\to 0$. In fact $nu_{n} = 1/\\nu$ if $2^{\\nu-2} < n \\leq 2^{\\nu-1}$, and $\\nu \\to \\infty$ as $n \\to \\infty$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:series.convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxv/10", "set": "hardy-course-of-pure-mathematics-1921/ex-lxv", "number": 10, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "293", "location": "Exercise LXV, problem 10", "problem_latex": "Prove that\n\\\\[\n\\\\tfrac{1}{2} < \\\\int_{0}^{1} \\\\frac{dx}{\\\\sqrtp{4 - x^{2} + x^{3}}}\n < \\\\tfrac{1}{6}\\\\pi.\n\\\\]", "markdown": "Prove that [ tfrac12 < int_0^1 fracdxsqrtp4 - x^2 + x^3 < tfrac16pi. ]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "FLAG-EXTRACTION", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxv/11", "set": "hardy-course-of-pure-mathematics-1921/ex-lxv", "number": 11, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "293", "location": "Exercise LXV, problem 11", "problem_latex": "Prove that $(3x + 8)/16 < 1/\\\\sqrtp{4 - 3x + x^{3}} < 1/\\\\sqrtp{4 - 3x}$ if $0 < x < 1$,\nand hence that\n\\\\[\n\\\\tfrac{19}{32} < \\\\int_{0}^{1} \\\\frac{dx}{\\\\sqrtp{4 - 3x + x^{3}}} < \\\\tfrac{2}{3}.\n\\\\]", "markdown": "Prove that $(3x + 8)/16 < 1/\\\\sqrtp{4 - 3x + x^{3}} < 1/\\\\sqrtp{4 - 3x}$ if $0 < x < 1$, and hence that [ tfrac1932 < int_0^1 fracdxsqrtp4 - 3x + x^3 < tfrac23. ]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "FLAG-EXTRACTION", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxv/12", "set": "hardy-course-of-pure-mathematics-1921/ex-lxv", "number": 12, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "293", "location": "Exercise LXV, problem 12", "problem_latex": "Prove that\n\\\\[\n.573 < \\\\int_{1}^{2} \\\\frac{dx}{\\\\sqrtp{4 - 3x + x^{3}}} < .595.\n\\\\]", "markdown": "Prove that [ .573 < int_1^2 fracdxsqrtp4 - 3x + x^3 < .595. ]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "FLAG-EXTRACTION", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxv/13", "set": "hardy-course-of-pure-mathematics-1921/ex-lxv", "number": 13, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "293", "location": "Exercise LXV, problem 13", "problem_latex": "If $\\alpha$~and~$\\phi$ are positive acute angles then\n\\[\n\\phi < \\int_{0}^{\\phi} \\frac{dx}{\\sqrtp{1 - \\sin^{2}\\alpha \\sin^{2} x}}\n < \\frac{\\phi}{\\sqrtp{1 - \\sin^{2}\\alpha \\sin^{2}\\phi}}.\n\\]\nIf $\\alpha = \\phi = \\frac{1}{6}\\pi$, then the integral lies between $.523$ and~$.541$.", "markdown": "If $\\alpha$ and $\\phi$ are positive acute angles then < _0^ dx1 - ^2^2 x < 1 - ^2^2. If $\\alpha = \\phi = \\frac{1}{6}\\pi$, then the integral lies between $.523$ and $.541$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxv/14", "set": "hardy-course-of-pure-mathematics-1921/ex-lxv", "number": 14, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "293", "location": "Exercise LXV, problem 14", "problem_latex": "Prove that\n\\\\[\n\\\\left|\\\\int_{a}^{b} f(x)\\\\, dx\\\\right| \\\\leq \\\\int_{a}^{b}|f(x)|\\\\, dx.\n\\\\]", "markdown": "Prove that [ left|int_a^b f(x), dxright| leq int_a^b|f(x)|, dx. ]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "FLAG-EXTRACTION", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxv/15", "set": "hardy-course-of-pure-mathematics-1921/ex-lxv", "number": 15, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "293", "location": "Exercise LXV, problem 15", "problem_latex": "If $|f(x)| \\\\leq M$, then\n\\\\[\n\\\\left|\\\\int_{a}^{b} f(x)\\\\phi(x)\\\\, dx\\\\right| \\\\leq M\\\\int_{a}^{b}|\\\\phi(x)|\\\\, dx.\n\\\\]", "markdown": "If $|f(x)| \\\\leq M$, then [ left|int_a^b f(x)phi(x), dxright| leq Mint_a^b|phi(x)|, dx. ]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "FLAG-EXTRACTION", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxv/1a", "set": "hardy-course-of-pure-mathematics-1921/ex-lxv", "number": 1, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "293", "location": "Exercise LXV, problem 1a", "problem_latex": "Show, by means of the direct definition of the\ndefinite integral, and equations \\\\Eq{(1)}--\\\\Eq{(5)} above, that\n\\\\CenterLine{\\\\Itemp{(i)}}{$\\\\ds\\\\int_{-a}^{a} \\\\phi(x^{2})\\\\, dx = 2\\\\int_{0}^{a} \\\\phi(x^{2})\\\\, dx$,\\\\quad\n$\\\\ds\\\\int_{-a}^{a} x\\\\phi(x^{2})\\\\, dx = 0$;}\n\\\\CenterLine{\\\\Itemp{(ii)}}{$\\\\ds\\\\int_{0}^{\\\\frac{1}{2}\\\\pi} \\\\phi(\\\\cos x)\\\\, dx\n= \\\\int_{0}^{\\\\frac{1}{2} \\\\pi} \\\\phi(\\\\sin x)\\\\, dx\n= \\\\tfrac{1}{2} \\\\int_{0}^{\\\\pi} \\\\phi(\\\\sin x)\\\\, dx$;}\n\\\\CenterLine{\\\\Itemp{(iii)}}{$\\\\ds\\\\int_{0}^{m\\\\pi} \\\\phi(\\\\cos^{2} x)\\\\, dx = m\\\\int_{0}^{\\\\pi} \\\\phi(\\\\cos^{2} x)\\\\, dx$,}\n$m$~being an integer.", "markdown": "Show, by means of the direct definition of the definite integral, and equations Eq(1)--Eq(5) above, that CenterLineItemp(i)$\\\\ds\\\\int_{-a}^{a} \\\\phi(x^{2})\\\\, dx = 2\\\\int_{0}^{a} \\\\phi(x^{2})\\\\, dx$,quad $\\\\ds\\\\int_{-a}^{a} x\\\\phi(x^{2})\\\\, dx = 0$; CenterLineItemp(ii)$\\\\ds\\\\int_{0}^{\\\\frac{1}{2}\\\\pi} \\\\phi(\\\\cos x)\\\\, dx = \\\\int_{0}^{\\\\frac{1}{2} \\\\pi} \\\\phi(\\\\sin x)\\\\, dx = \\\\tfrac{1}{2} \\\\int_{0}^{\\\\pi} \\\\phi(\\\\sin x)\\\\, dx$; CenterLineItemp(iii)$\\\\ds\\\\int_{0}^{m\\\\pi} \\\\phi(\\\\cos^{2} x)\\\\, dx = m\\\\int_{0}^{\\\\pi} \\\\phi(\\\\cos^{2} x)\\\\, dx$, $m$ being an integer.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "FLAG-EXTRACTION", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxv/1b", "set": "hardy-course-of-pure-mathematics-1921/ex-lxv", "number": 1, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "293", "location": "Exercise LXV, problem 1b", "problem_latex": "Show, by means of the direct definition of the\ndefinite integral, and equations \\\\Eq{(1)}--\\\\Eq{(5)} above, that\n\\\\CenterLine{\\\\Itemp{(i)}}{$\\\\ds\\\\int_{-a}^{a} \\\\phi(x^{2})\\\\, dx = 2\\\\int_{0}^{a} \\\\phi(x^{2})\\\\, dx$,\\\\quad\n$\\\\ds\\\\int_{-a}^{a} x\\\\phi(x^{2})\\\\, dx = 0$;}\n\\\\CenterLine{\\\\Itemp{(ii)}}{$\\\\ds\\\\int_{0}^{\\\\frac{1}{2}\\\\pi} \\\\phi(\\\\cos x)\\\\, dx\n= \\\\int_{0}^{\\\\frac{1}{2} \\\\pi} \\\\phi(\\\\sin x)\\\\, dx\n= \\\\tfrac{1}{2} \\\\int_{0}^{\\\\pi} \\\\phi(\\\\sin x)\\\\, dx$;}\n\\\\CenterLine{\\\\Itemp{(iii)}}{$\\\\ds\\\\int_{0}^{m\\\\pi} \\\\phi(\\\\cos^{2} x)\\\\, dx = m\\\\int_{0}^{\\\\pi} \\\\phi(\\\\cos^{2} x)\\\\, dx$,}\n$m$~being an integer.", "markdown": "Show, by means of the direct definition of the definite integral, and equations Eq(1)--Eq(5) above, that CenterLineItemp(i)$\\\\ds\\\\int_{-a}^{a} \\\\phi(x^{2})\\\\, dx = 2\\\\int_{0}^{a} \\\\phi(x^{2})\\\\, dx$,quad $\\\\ds\\\\int_{-a}^{a} x\\\\phi(x^{2})\\\\, dx = 0$; CenterLineItemp(ii)$\\\\ds\\\\int_{0}^{\\\\frac{1}{2}\\\\pi} \\\\phi(\\\\cos x)\\\\, dx = \\\\int_{0}^{\\\\frac{1}{2} \\\\pi} \\\\phi(\\\\sin x)\\\\, dx = \\\\tfrac{1}{2} \\\\int_{0}^{\\\\pi} \\\\phi(\\\\sin x)\\\\, dx$; CenterLineItemp(iii)$\\\\ds\\\\int_{0}^{m\\\\pi} \\\\phi(\\\\cos^{2} x)\\\\, dx = m\\\\int_{0}^{\\\\pi} \\\\phi(\\\\cos^{2} x)\\\\, dx$, $m$ being an integer.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "FLAG-EXTRACTION", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxv/1c", "set": "hardy-course-of-pure-mathematics-1921/ex-lxv", "number": 1, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "293", "location": "Exercise LXV, problem 1c", "problem_latex": "Show, by means of the direct definition of the\ndefinite integral, and equations \\\\Eq{(1)}--\\\\Eq{(5)} above, that\n\\\\CenterLine{\\\\Itemp{(i)}}{$\\\\ds\\\\int_{-a}^{a} \\\\phi(x^{2})\\\\, dx = 2\\\\int_{0}^{a} \\\\phi(x^{2})\\\\, dx$,\\\\quad\n$\\\\ds\\\\int_{-a}^{a} x\\\\phi(x^{2})\\\\, dx = 0$;}\n\\\\CenterLine{\\\\Itemp{(ii)}}{$\\\\ds\\\\int_{0}^{\\\\frac{1}{2}\\\\pi} \\\\phi(\\\\cos x)\\\\, dx\n= \\\\int_{0}^{\\\\frac{1}{2} \\\\pi} \\\\phi(\\\\sin x)\\\\, dx\n= \\\\tfrac{1}{2} \\\\int_{0}^{\\\\pi} \\\\phi(\\\\sin x)\\\\, dx$;}\n\\\\CenterLine{\\\\Itemp{(iii)}}{$\\\\ds\\\\int_{0}^{m\\\\pi} \\\\phi(\\\\cos^{2} x)\\\\, dx = m\\\\int_{0}^{\\\\pi} \\\\phi(\\\\cos^{2} x)\\\\, dx$,}\n$m$~being an integer.", "markdown": "Show, by means of the direct definition of the definite integral, and equations Eq(1)--Eq(5) above, that CenterLineItemp(i)$\\\\ds\\\\int_{-a}^{a} \\\\phi(x^{2})\\\\, dx = 2\\\\int_{0}^{a} \\\\phi(x^{2})\\\\, dx$,quad $\\\\ds\\\\int_{-a}^{a} x\\\\phi(x^{2})\\\\, dx = 0$; CenterLineItemp(ii)$\\\\ds\\\\int_{0}^{\\\\frac{1}{2}\\\\pi} \\\\phi(\\\\cos x)\\\\, dx = \\\\int_{0}^{\\\\frac{1}{2} \\\\pi} \\\\phi(\\\\sin x)\\\\, dx = \\\\tfrac{1}{2} \\\\int_{0}^{\\\\pi} \\\\phi(\\\\sin x)\\\\, dx$; CenterLineItemp(iii)$\\\\ds\\\\int_{0}^{m\\\\pi} \\\\phi(\\\\cos^{2} x)\\\\, dx = m\\\\int_{0}^{\\\\pi} \\\\phi(\\\\cos^{2} x)\\\\, dx$, $m$ being an integer.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "FLAG-EXTRACTION", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxv/2", "set": "hardy-course-of-pure-mathematics-1921/ex-lxv", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "293", "location": "Exercise LXV, problem 2", "problem_latex": "Prove that $\\ds\\int_{0}^{\\pi} \\frac{\\sin nx}{\\sin x}\\, dx$ is equal to~$\\pi$ or to~$0$ according as $n$~is odd or\nor even.", "markdown": "Prove that $\\ds\\int_{0}^{\\pi} \\frac{\\sin nx}{\\sin x}\\, dx$ is equal to $\\pi$ or to $0$ according as $n$ is odd or or even.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxv/3", "set": "hardy-course-of-pure-mathematics-1921/ex-lxv", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "293", "location": "Exercise LXV, problem 3", "problem_latex": "Prove that $\\ds\\int_{0}^{\\pi} \\sin nx \\cot x\\, dx$ is equal to~$0$ or to~$\\pi$ according as $n$~is odd\nor even.", "markdown": "Prove that $\\ds\\int_{0}^{\\pi} \\sin nx \\cot x\\, dx$ is equal to $0$ or to $\\pi$ according as $n$ is odd or even.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxv/4", "set": "hardy-course-of-pure-mathematics-1921/ex-lxv", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "293", "location": "Exercise LXV, problem 4", "problem_latex": "If $\\phi(x) = a_{0} + a_{1}\\cos x + b_{1}\\sin x + a_{2}\\cos 2x + \\dots + a_{n}\\cos nx + b_{n}\\sin nx$,\nand $k$~is a positive integer not greater than~$n$, then\n\\[\n\\int_{0}^{2\\pi} \\phi(x)\\, dx = 2\\pi a_{0},\\quad\n\\int_{0}^{2\\pi} \\cos kx \\phi(x)\\, dx = \\pi a_{k},\\quad\n\\int_{0}^{2\\pi} \\sin kx \\phi(x)\\, dx = \\pi b_{k}.\n\\]\nIf $k > n$ then the value of each of the last two integrals is zero.", "markdown": "If $\\phi(x) = a_{0} + a_{1}\\cos x + b_{1}\\sin x + a_{2}\\cos 2x + \\dots + a_{n}\\cos nx + b_{n}\\sin nx$, and $k$ is a positive integer not greater than $n$, then _0^2 (x)  dx = 2a_0,0pt minus 3pt_0^2 kx (x)  dx = a_k,0pt minus 3pt_0^2 kx (x)  dx = b_k. If $k > n$ then the value of each of the last two integrals is zero.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxv/5", "set": "hardy-course-of-pure-mathematics-1921/ex-lxv", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "293", "location": "Exercise LXV, problem 5", "problem_latex": "If $\\phi(x) = a_{0} + a_{1} \\cos x + a_{2}\\cos 2x + \\dots + a_{n}\\cos nx$, and $k$~is a positive\ninteger not greater than~$n$, then\n\\[\n\\int_{0}^{\\pi} \\phi(x)\\, dx = \\pi a_{0},\\quad\n\\int_{0}^{\\pi} \\cos kx \\phi(x)\\, dx = \\tfrac{1}{2}\\pi a_{k}.\n\\]\nIf $k > n$ then the value of the last integral is zero.", "markdown": "If $\\phi(x) = a_{0} + a_{1} \\cos x + a_{2}\\cos 2x + \\dots + a_{n}\\cos nx$, and $k$ is a positive integer not greater than $n$, then _0^ (x)  dx = a_0,0pt minus 3pt_0^ kx (x)  dx = 12a_k. If $k > n$ then the value of the last integral is zero.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxv/6", "set": "hardy-course-of-pure-mathematics-1921/ex-lxv", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "293", "location": "Exercise LXV, problem 6", "problem_latex": "Prove that if $a$ and~$b$ are positive then\n\\\\[\n\\\\int_{0}^{2\\\\pi} \\\\frac{dx}{a^{2}\\\\cos^{2} x + b^{2}\\\\sin^{2} x} = \\\\frac{2\\\\pi}{ab}.\n\\\\]", "markdown": "Prove that if $a$ and $b$ are positive then [ int_0^2pi fracdxa^2cos^2 x + b^2sin^2 x = frac2piab. ]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "FLAG-EXTRACTION", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxv/7", "set": "hardy-course-of-pure-mathematics-1921/ex-lxv", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "293", "location": "Exercise LXV, problem 7", "problem_latex": "If $f(x) \\leq \\phi(x)$ when $a \\leq x \\leq b$, then\n$\\ds\\int_{a}^{b} f\\, dx \\leq \\int_{a}^{b}\\phi\\, dx$.", "markdown": "If $f(x) \\leq \\phi(x)$ when $a \\leq x \\leq b$, then $\\ds\\int_{a}^{b} f\\, dx \\leq \\int_{a}^{b}\\phi\\, dx$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxv/8", "set": "hardy-course-of-pure-mathematics-1921/ex-lxv", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "293", "location": "Exercise LXV, problem 8", "problem_latex": "Prove that\n\\\\begin{alignat*}{2}\n0 &< \\\\int_{0}^{\\\\frac{1}{2}\\\\pi} \\\\sin^{n+1}x\\\\, dx\n &&< \\\\int_{0}^{\\\\frac{1}{2}\\\\pi} \\\\sin^{n}x\\\\, dx,\\\\\\\\\n0 &< \\\\int_{0}^{\\\\frac{1}{4}\\\\pi} \\\\tan^{n+1}x\\\\, dx\n &&< \\\\int_{0}^{\\\\frac{1}{4}\\\\pi} \\\\tan^{n}x\\\\, dx.\n\\\\end{alignat*}", "markdown": "Prove that beginalignat*2 0 &< int_0^frac12pi sin^n+1x, dx &&< int_0^frac12pi sin^nx, dx, 0 &< int_0^frac14pi tan^n+1x, dx &&< int_0^frac14pi tan^nx, dx. endalignat*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "FLAG-EXTRACTION", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxv/9", "set": "hardy-course-of-pure-mathematics-1921/ex-lxv", "number": 9, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "293", "location": "Exercise LXV, problem 9", "problem_latex": "If $n > 1$ then\n\\\\[\n.5 < \\\\int_{0}^{\\\\frac{1}{2}} \\\\frac{dx}{\\\\sqrtp{1 - x^{2n}}} < .524.\n\\\\]", "markdown": "If $n > 1$ then [ .5 < int_0^frac12 fracdxsqrtp1 - x^2n < .524. ]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "FLAG-EXTRACTION", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxvi/1", "set": "hardy-course-of-pure-mathematics-1921/ex-lxvi", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "295", "location": "Exercise LXVI, problem 1", "problem_latex": "Prove that\n\\[\n\\int_{a}^{b} x f''(x)\\, dx = \\{bf'(b) - f(b)\\} - \\{af'(a) - f(a)\\}.\n\\]", "markdown": "Prove that _a^b x f”(x)  dx = bf’(b) - f(b) - af’(a) - f(a).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate.parts" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxvi/10", "set": "hardy-course-of-pure-mathematics-1921/ex-lxvi", "number": 10, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "295", "location": "Exercise LXVI, problem 10", "problem_latex": "Deduce that $u_{n}$~is equal to\n\\[\n\\frac{2·4·6 \\dots (n - 1)}{3·5·7 \\dots n},\\quad\n\\tfrac{1}{2}\\pi \\frac{1·3·5 \\dots (n - 1)}{2·4·6 \\dots n},\n\\]\naccording as $n$~is odd or even.", "markdown": "Deduce that $u_{n}$ is equal to 2·4·6 …(n - 1)3·5·7 …n,0pt minus 3pt121·3·5 …(n - 1)2·4·6 …n, according as $n$ is odd or even.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxvi/11", "set": "hardy-course-of-pure-mathematics-1921/ex-lxvi", "number": 11, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "295", "location": "Exercise LXVI, problem 11", "problem_latex": "\\Topic{The Second Mean Value Theorem.} If $f(x)$~is a function of~$x$\nwhich has a differential coefficient of constant sign for all values of~$x$ from\n$x = a$ to $x = b$, then there is a number~$\\xi$ between $a$~and~$b$ such that\n\\[\n\\int_{a}^{b} f(x)\\phi(x)\\, dx\n = f(a) \\int_{a}^{\\xi} \\phi(x)\\, dx\n + f(b) \\int_{\\xi}^{b} \\phi(x)\\, dx.\n\\]", "markdown": "**Second Mean Value Theorem.** If $f(x)$ is a function of $x$ which has a differential coefficient of constant sign for all values of $x$ from $x = a$ to $x = b$, then there is a number $\\xi$ between $a$ and $b$ such that _a^b f(x)(x)  dx = f(a) _a^ (x)  dx + f(b) _^b (x)  dx.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate.parts" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxvi/12a", "set": "hardy-course-of-pure-mathematics-1921/ex-lxvi", "number": 12, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "295", "location": "Exercise LXVI, problem 12a", "problem_latex": "\\Topic{Bonnet's form of the Second Mean Value Theorem.} If $f'(x)$~is\nof constant sign, and $f(b)$ and $f(a) - f(b)$ have the same sign, then\n\\[\n\\int_{a}^{b} f(x)\\phi(x)\\, dx = f(a) \\int_{a}^{X} \\phi(x)\\, dx,\n\\]\nwhere $X$~lies between $a$ and~$b$.", "markdown": "**’s form of the Second Mean Value Theorem.** If $f'(x)$ is of constant sign, and $f(b)$ and $f(a) - f(b)$ have the same sign, then _a^b f(x)(x)  dx = f(a) _a^X (x)  dx, where $X$ lies between $a$ and $b$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate.parts" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxvi/12b", "set": "hardy-course-of-pure-mathematics-1921/ex-lxvi", "number": 12, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "295", "location": "Exercise LXVI, problem 12b", "problem_latex": "Prove similarly that if $f(a)$ and $f(b) - f(a)$ have the same sign, then\n\\[\n\\int_{a}^{b} f(x)\\phi(x)\\, dx = f(b) \\int_{X}^{b} \\phi(x)\\, dx,\n\\]\nwhere $X$~lies between $a$ and~$b$.", "markdown": "Prove similarly that if $f(a)$ and $f(b) - f(a)$ have the same sign, then _a^b f(x)(x)  dx = f(b) _X^b (x)  dx, where $X$ lies between $a$ and $b$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate.parts" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxvi/13", "set": "hardy-course-of-pure-mathematics-1921/ex-lxvi", "number": 13, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "295", "location": "Exercise LXVI, problem 13", "problem_latex": "Prove that\n\\[\n\\left|\\int_{X}^{X'} \\frac{\\sin x}{x}\\, dx\\right| < \\frac{2}{X}\n\\]\nif $X' > X > 0$.", "markdown": "Prove that |_X^X’ xx  dx| < 2X if $X' > X > 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "FLAG-EXTRACTION", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate.parts" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxvi/14", "set": "hardy-course-of-pure-mathematics-1921/ex-lxvi", "number": 14, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "295", "location": "Exercise LXVI, problem 14", "problem_latex": "Establish the results of \\Ex{lxv}.~1 by means of the rule for substitution.", "markdown": "Establish the results of % [examples:lxv]Ex. lxv%. 1 by means of the rule for substitution.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxvi/15", "set": "hardy-course-of-pure-mathematics-1921/ex-lxvi", "number": 15, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "295", "location": "Exercise LXVI, problem 15", "problem_latex": "Prove that\n\\[\n\\int_{a}^{b} F(x)\\, dx = \\int_{a}^{b} F(a + b - x)\\, dx.\n\\]", "markdown": "Prove that _a^b F(x)  dx = _a^b F(a + b - x)  dx.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxvi/16", "set": "hardy-course-of-pure-mathematics-1921/ex-lxvi", "number": 16, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "295", "location": "Exercise LXVI, problem 16", "problem_latex": "Prove that\n\\[\n\\int_{0}^{\\frac{1}{2}\\pi} \\cos^{m} x\\sin^{m} x\\, dx\n = 2^{-m} \\int_{0}^{\\frac{1}{2}\\pi} \\cos^{m} x\\, dx.\n\\]", "markdown": "Prove that _0^12 ^m x^m x  dx = 2^-m _0^12 ^m x  dx.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.subst", "cas.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxvi/17", "set": "hardy-course-of-pure-mathematics-1921/ex-lxvi", "number": 17, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "295", "location": "Exercise LXVI, problem 17", "problem_latex": "Prove that\n\\[\n\\int_{0}^{\\pi} x\\phi(\\sin x)\\, dx\n = \\tfrac{1}{2}\\pi \\int_{0}^{\\pi} \\phi(\\sin x)\\, dx.\n\\]", "markdown": "Prove that _0^ x(x)  dx = 12_0^ (x)  dx.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxvi/18", "set": "hardy-course-of-pure-mathematics-1921/ex-lxvi", "number": 18, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "295", "location": "Exercise LXVI, problem 18", "problem_latex": "Prove that\n\\[\n\\int_{0}^{\\pi} \\frac{x\\sin x}{1 + \\cos^{2} x}\\, dx = \\tfrac{1}{4}\\pi^{2}.\n\\]", "markdown": "Prove that _0^ xx1 + ^2 x  dx = 14^2.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxvi/19", "set": "hardy-course-of-pure-mathematics-1921/ex-lxvi", "number": 19, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "295", "location": "Exercise LXVI, problem 19", "problem_latex": "Show by means of the transformation $x = a\\cos^{2}\\theta + b\\sin^{2}\\theta$ that\n\\[\n\\int_{a}^{b} \\sqrtb{(x - a)(b - x)}\\, dx = \\tfrac{1}{8}\\pi (b - a)^{2}.\n\\]", "markdown": "Show by means of the transformation $x = a\\cos^{2}\\theta + b\\sin^{2}\\theta$ that _a^b (x - a)(b - x)  dx = 18(b - a)^2.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.subst", "cas.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxvi/2", "set": "hardy-course-of-pure-mathematics-1921/ex-lxvi", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "295", "location": "Exercise LXVI, problem 2", "problem_latex": "More generally,\n\\[\n\\int_{a}^{b} x^{m} f^{(m+1)}(x)\\, dx = F(b) - F(a),\n\\]\nwhere\n\\begin{multline*}\nF(x) = x^{m} f^{(m)}(x)\n - mx^{m-1} f^{(m-1)}\\DPtypo{x}{(x)} \\\\\n + m(m - 1)x^{m-2} f^{(m-2)}\\DPtypo{x}{(x)} - \\dots\n + (-1)^{m} m!\\, f(x).\n\\end{multline*}", "markdown": "More generally, _a^b x^m f^(m+1)(x)  dx = F(b) - F(a), where multline* F(x) = x^m f^(m)(x) - mx^m-1 f^(m-1)x + m(m - 1)x^m-2 f^(m-2)x - … + (-1)^m m!  f(x). multline*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "FLAG-EXTRACTION", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.integrate.parts" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxvi/20", "set": "hardy-course-of-pure-mathematics-1921/ex-lxvi", "number": 20, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "295", "location": "Exercise LXVI, problem 20", "problem_latex": "Show by means of the substitution $(a + b\\cos x) (a - b\\cos y) = a^{2} - b^{2}$\nthat\n\\[\n\\int_{0}^{\\pi} (a + b\\cos x)^{-n}\\, dx\n = (a^{2} - b^{2})^{-(n - \\frac{1}{2})} \\int_{0}^{\\pi} (a - b\\cos y)^{n-1}\\, dy,\n\\]\nwhen $n$~is a positive integer and $a > |b|$, and evaluate the integral when\n$n = 1$, $2$,~$3$.", "markdown": "Show by means of the substitution $(a + b\\cos x) (a - b\\cos y) = a^{2} - b^{2}$ that _0^ (a + bx)^-n  dx = (a^2 - b^2)^-(n - 12) _0^ (a - by)^n-1  dy, when $n$ is a positive integer and $a > |b|$, and evaluate the integral when $n = 1$, $2$, $3$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxvi/21", "set": "hardy-course-of-pure-mathematics-1921/ex-lxvi", "number": 21, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "295", "location": "Exercise LXVI, problem 21", "problem_latex": "If $m$~and~$n$ are positive integers then\n\\[\n\\int_{a}^{b} (x - a)^{m} (b - x)^{n}\\, dx\n = (b - a)^{m+n+1} \\frac{m!\\, n!}{(m + n + 1)!}.\n\\]", "markdown": "If $m$ and $n$ are positive integers then _a^b (x - a)^m (b - x)^n  dx = (b - a)^m+n+1 m!  n!(m + n + 1)!.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxvi/3a", "set": "hardy-course-of-pure-mathematics-1921/ex-lxvi", "number": 3, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "295", "location": "Exercise LXVI, problem 3a", "problem_latex": "Prove that\n\\[\n\\int_{0}^{1} \\arcsin x\\, dx = \\tfrac{1}{2}\\pi - 1,\\quad\n\\int_{0}^{1}x\\arctan x\\, dx = \\tfrac{1}{4}\\pi - \\tfrac{1}{2}.\n\\]", "markdown": "Prove that _0^1 x  dx = 12- 1,0pt minus 3pt_0^1xx  dx = 14- 12.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate.parts" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxvi/3b", "set": "hardy-course-of-pure-mathematics-1921/ex-lxvi", "number": 3, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "295", "location": "Exercise LXVI, problem 3b", "problem_latex": "Prove that\n\\[\n\\int_{0}^{1} \\arcsin x\\, dx = \\tfrac{1}{2}\\pi - 1,\\quad\n\\int_{0}^{1}x\\arctan x\\, dx = \\tfrac{1}{4}\\pi - \\tfrac{1}{2}.\n\\]", "markdown": "Prove that _0^1 x  dx = 12- 1,0pt minus 3pt_0^1xx  dx = 14- 12.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate.parts" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxvi/4", "set": "hardy-course-of-pure-mathematics-1921/ex-lxvi", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "295", "location": "Exercise LXVI, problem 4", "problem_latex": "Prove that if $a$~and~$b$ are positive then\n\\[\n\\int_{0}^{\\frac{1}{2}\\pi}\n \\frac{x\\cos x\\sin x\\, dx}{(a^{2}\\cos^{2}x + b^{2}\\sin^{2}x)^{2}}\n = \\frac{\\pi}{4ab^{2}(a + b)}.\n\\]", "markdown": "Prove that if $a$ and $b$ are positive then _0^12 xxx  dx(a^2^2x + b^2^2x)^2 = 4ab^2(a + b).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate.parts" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxvi/5", "set": "hardy-course-of-pure-mathematics-1921/ex-lxvi", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "295", "location": "Exercise LXVI, problem 5", "problem_latex": "If\n\\[\nf_{1}(x) = \\int_{0}^{x}f(t)\\, dt,\\quad\nf_{2}(x) = \\int_{0}^{x}f_{1}(t)\\, dt,\\ \\dots,\\quad\nf_{k}(x) = \\int_{0}^{x} f_{k-1}(t)\\, dt,\n\\]\nthen\n\\[\nf_{k}(x) = \\frac{1}{(k - 1)!} \\int_{0}^{x} f(t)(x - t)^{k-1}\\, dt.\n\\]", "markdown": "If f_1(x) = _0^xf(t)  dt,0pt minus 3ptf_2(x) = _0^xf_1(t)  dt, …,0pt minus 3ptf_k(x) = _0^x f_k-1(t)  dt, then f_k(x) = 1(k - 1)! _0^x f(t)(x - t)^k-1  dt.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate.parts" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxvi/6", "set": "hardy-course-of-pure-mathematics-1921/ex-lxvi", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "295", "location": "Exercise LXVI, problem 6", "problem_latex": "Prove by integration by parts that if\n\\[\nu_{m, n} = \\int_{0}^{1} x^{m} (1 - x)^{n}\\, dx,\n\\]\nwhere $m$~and~$n$ are positive integers, then $(m + n + 1) u_{m, n} = nu_{m, n-1}$, and deduce that\n\\[\nu_{m, n} = \\frac{m!\\, n!}{(m + n + 1)!}.\n\\]", "markdown": "Prove by integration by parts that if u_m, n = _0^1 x^m (1 - x)^n  dx, where $m$ and $n$ are positive integers, then $(m + n + 1) u_{m, n} = nu_{m, n-1}$, and deduce that u_m, n = m!  n!(m + n + 1)!.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "FLAG-EXTRACTION", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate.parts" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxvi/7", "set": "hardy-course-of-pure-mathematics-1921/ex-lxvi", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "295", "location": "Exercise LXVI, problem 7", "problem_latex": "Prove that if\n\\[\nu_{n} = \\int_{0}^{\\frac{1}{4}\\pi} \\tan^{n}x\\, dx\n\\]\nthen $u_{n} + u_{n-2} = 1/(n - 1)$. Hence\nevaluate the integral for all positive integral values of~$n$.", "markdown": "Prove that if u_n = _0^14 ^nx  dx then $u_{n} + u_{n-2} = 1/(n - 1)$. Hence evaluate the integral for all positive integral values of $n$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "FLAG-EXTRACTION", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate.parts", "cas.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxvi/8", "set": "hardy-course-of-pure-mathematics-1921/ex-lxvi", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "295", "location": "Exercise LXVI, problem 8", "problem_latex": "Deduce from the last example that $u_{n}$~lies between $1/\\{2(n - 1)\\}$ and\n$1/\\{2(n + 1)\\}$.", "markdown": "Deduce from the last example that $u_{n}$ lies between $1/\\{2(n - 1)\\}$ and $1/\\{2(n + 1)\\}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxvi/9", "set": "hardy-course-of-pure-mathematics-1921/ex-lxvi", "number": 9, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "295", "location": "Exercise LXVI, problem 9", "problem_latex": "Prove that if\n\\[\nu_{n} = \\int_{0}^{\\frac{1}{2}\\pi} \\sin^{n} x\\, dx\n\\]\nthen $u_{n} = \\{(n - 1)/n\\} u_{n-2}$. [Write\n$\\sin^{n-1}x\\sin x$ for $\\sin^{n}x$ and integrate by parts.]", "markdown": "Prove that if u_n = _0^12 ^n x  dx then $u_{n} = \\{(n - 1)/n\\} u_{n-2}$. [Write $\\sin^{n-1}x\\sin x$ for $\\sin^{n}x$ and integrate by parts.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "FLAG-EXTRACTION", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate.parts" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxvii/1", "set": "hardy-course-of-pure-mathematics-1921/ex-lxvii", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "311", "location": "Exercise LXVII, problem 1", "problem_latex": "Apply Cauchy's and d'Alembert's tests (as\\PageLabel{311}\nspecialised in 4~above) to the series $\\sum n^{k} r^{n}$, where $k$~is a positive rational\nnumber.", "markdown": "Apply Cauchy’s and d’Alembert’s tests (as311 specialised in 4 above) to the series $\\sum n^{k} r^{n}$, where $k$ is a positive rational number.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxvii/10", "set": "hardy-course-of-pure-mathematics-1921/ex-lxvii", "number": 10, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "311", "location": "Exercise LXVII, problem 10", "problem_latex": "The series $1 + r + \\dfrac{r^{2}}{2!} + \\dfrac{r^{3}}{3!} + \\dots$ and $1 + r + \\dfrac{r^{2}}{2^{2}} + \\dfrac{r^{3}}{3^{3}} + \\dots$ are convergent\nfor all positive values of~$r$.", "markdown": "The series $1 + r + \\dfrac{r^{2}}{2!} + \\dfrac{r^{3}}{3!} + \\dots$ and $1 + r + \\dfrac{r^{2}}{2^{2}} + \\dfrac{r^{3}}{3^{3}} + \\dots$ are convergent for all positive values of $r$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxvii/11", "set": "hardy-course-of-pure-mathematics-1921/ex-lxvii", "number": 11, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "311", "location": "Exercise LXVII, problem 11", "problem_latex": "If $\\sum u_{n}$~is convergent then so are $\\sum u_{n}^{2}$ and $\\sum u_{n}/(1 + u_{n})$.", "markdown": "If $\\sum u_{n}$ is convergent then so are $\\sum u_{n}^{2}$ and $\\sum u_{n}/(1 + u_{n})$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxvii/12", "set": "hardy-course-of-pure-mathematics-1921/ex-lxvii", "number": 12, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "311", "location": "Exercise LXVII, problem 12", "problem_latex": "If $\\sum u_{n}^{2}$~is convergent then so is $\\sum u_{n}/n$.", "markdown": "If $\\sum u_{n}^{2}$ is convergent then so is $\\sum u_{n}/n$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxvii/13", "set": "hardy-course-of-pure-mathematics-1921/ex-lxvii", "number": 13, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "311", "location": "Exercise LXVII, problem 13", "problem_latex": "Show that\n\\[\n%[** TN: In-line in the original]\n1 + \\frac{1}{3^{2}} + \\frac{1}{5^{2}} + \\dots\n = \\frac{3}{4}\\left(1 + \\frac{1}{2^{2}} + \\frac{1}{3^{2}} + \\dots \\right)\n\\]\nand\n\\[\n1 + \\frac{1}{2^{2}} + \\frac{1}{3^{2}}\n + \\frac{1}{5^{2}} + \\frac{1}{6^{2}}\n + \\frac{1}{7^{2}} + \\frac{1}{9^{2}} + \\dots\n = \\frac{15}{16} \\left(1 + \\frac{1}{2^{2}} + \\frac{1}{3^{2}} + \\dots\\right).\n\\]", "markdown": "Show that %[** TN: In-line in the original] 1 + 13^2 + 15^2 + … = 34(1 + 12^2 + 13^2 + …) and 1 + 12^2 + 13^2 + 15^2 + 16^2 + 17^2 + 19^2 + … = 1516 (1 + 12^2 + 13^2 + …).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxvii/14", "set": "hardy-course-of-pure-mathematics-1921/ex-lxvii", "number": 14, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "311", "location": "Exercise LXVII, problem 14", "problem_latex": "Prove by a \\textit{reductio ad absurdum} that $\\sum (1/n)$~is divergent.", "markdown": "Prove by a *reductio ad absurdum* that $\\sum (1/n)$ is divergent.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxvii/2", "set": "hardy-course-of-pure-mathematics-1921/ex-lxvii", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "311", "location": "Exercise LXVII, problem 2", "problem_latex": "Consider the series $\\sum(An^{k} + Bn^{k-1} + \\dots + K) r^{n}$.", "markdown": "Consider the series $\\sum(An^{k} + Bn^{k-1} + \\dots + K) r^{n}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxvii/3", "set": "hardy-course-of-pure-mathematics-1921/ex-lxvii", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "311", "location": "Exercise LXVII, problem 3", "problem_latex": "Consider\n\\[\n\\sum \\frac{An^{k} + Bn^{k-1} + \\dots + K}\n {\\alpha n^{l} + \\beta n^{l-1} + \\dots + \\kappa} r^{n}\\quad\n(A > 0,\\ \\alpha > 0).\n\\]", "markdown": "Consider An^k + Bn^k-1 + …+ K n^l + n^l-1 + …+ r^n0pt minus 3pt(A > 0, > 0).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxvii/4", "set": "hardy-course-of-pure-mathematics-1921/ex-lxvii", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "311", "location": "Exercise LXVII, problem 4", "problem_latex": "We have seen (\\okrickRef{Ch.}{IV}, \\MiscEx{IV}~17) that the series\n\\[\n\\sum \\frac{1}{n(n + 1)},\\quad\n\\sum \\frac{1}{n(n + 1)\\dots (n + p)}\n\\]\nare convergent. Show that Cauchy's and d'Alembert's tests both fail when\napplied to them.", "markdown": "We have seen (Ch.IV, [misc:IV]Misc. Ex. 17) that the series 1n(n + 1),0pt minus 3pt1n(n + 1)…(n + p) are convergent. Show that Cauchy’s and d’Alembert’s tests both fail when applied to them.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxvii/5", "set": "hardy-course-of-pure-mathematics-1921/ex-lxvii", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "311", "location": "Exercise LXVII, problem 5", "problem_latex": "Show that the series~$\\sum n^{-p}$, where $p$~is an integer not less than~$2$, is\nconvergent.", "markdown": "Show that the series $\\sum n^{-p}$, where $p$ is an integer not less than $2$, is convergent.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxvii/6", "set": "hardy-course-of-pure-mathematics-1921/ex-lxvii", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "311", "location": "Exercise LXVII, problem 6", "problem_latex": "Show that the series\n\\[\n\\sum \\frac{An^{k} + Bn^{k-1} + \\dots + K}\n {\\alpha n^{l} + \\beta n^{l-1} + \\dots + \\kappa}\n\\]\nis convergent if $l > k + 1$ and divergent if $l \\leq k + 1$.", "markdown": "Show that the series An^k + Bn^k-1 + …+ K n^l + n^l-1 + …+ is convergent if $l > k + 1$ and divergent if $l \\leq k + 1$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxvii/7", "set": "hardy-course-of-pure-mathematics-1921/ex-lxvii", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "311", "location": "Exercise LXVII, problem 7", "problem_latex": "If $m_{n}$~is a positive integer, and $m_{n+1} > m_{n}$, then the series $\\sum r^{m_{n}}$ is convergent\nif $r < 1$ and divergent if $r \\geq 1$. For example the series $1 + r + r^{4} + r^{9} + \\dots$\nis convergent if $r < 1$ and divergent if $r \\geq 1$.", "markdown": "If $m_{n}$ is a positive integer, and $m_{n+1} > m_{n}$, then the series $\\sum r^{m_{n}}$ is convergent if $r < 1$ and divergent if $r \\geq 1$. For example the series $1 + r + r^{4} + r^{9} + \\dots$ is convergent if $r < 1$ and divergent if $r \\geq 1$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxvii/8", "set": "hardy-course-of-pure-mathematics-1921/ex-lxvii", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "311", "location": "Exercise LXVII, problem 8", "problem_latex": "Sum the series $1 + 2r + 2r^{4} + \\dots$ to $24$~places of decimals when $r = .1$\nand to $2$~places when $r = .9$.", "markdown": "Sum the series $1 + 2r + 2r^{4} + \\dots$ to $24$ places of decimals when $r = .1$ and to $2$ places when $r = .9$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxvii/9", "set": "hardy-course-of-pure-mathematics-1921/ex-lxvii", "number": 9, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "311", "location": "Exercise LXVII, problem 9", "problem_latex": "If $0 < a < b < 1$, then the series $a + b + a^{2} + b^{2} + a^{3} + \\dots$ is convergent.\nShow that Cauchy's test may be applied to this series, but that d'Alembert's\ntest fails.", "markdown": "If $0 < a < b < 1$, then the series $a + b + a^{2} + b^{2} + a^{3} + \\dots$ is convergent. Show that Cauchy’s test may be applied to this series, but that d’Alembert’s test fails.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxviii/1", "set": "hardy-course-of-pure-mathematics-1921/ex-lxviii", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "315", "location": "Exercise LXVIII, problem 1", "problem_latex": " Verify that if $r < 1$ then\n\\[\n1 + r^{2} + r + r^{4} + r^{6} + r^{3} + \\dots\n = 1 + r + r^{3} + r^{2} + r^{5} + r^{7} + \\dots\n = 1/(1 - r).\n\\]", "markdown": "Verify that if $r < 1$ then 1 + r^2 + r + r^4 + r^6 + r^3 + … = 1 + r + r^3 + r^2 + r^5 + r^7 + … = 1/(1 - r).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxviii/2", "set": "hardy-course-of-pure-mathematics-1921/ex-lxviii", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "315", "location": "Exercise LXVIII, problem 2", "problem_latex": "\nIf either of the series $u_{0} + u_{1} + \\dots$, $v_{0} + v_{1} + \\dots$ is divergent, then so is\nthe series $u_{0}v_{0} + (u_{1}v_{0} + u_{0}v_{1}) + (u_{2}v_{0} + u_{1}v_{1} + u_{0}v_{2}) + \\dots$, except in the trivial\ncase in which every term of one series is zero.", "markdown": "If either of the series $u_{0} + u_{1} + \\dots$, $v_{0} + v_{1} + \\dots$ is divergent, then so is the series $u_{0}v_{0} + (u_{1}v_{0} + u_{0}v_{1}) + (u_{2}v_{0} + u_{1}v_{1} + u_{0}v_{2}) + \\dots$, except in the trivial case in which every term of one series is zero.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxviii/3", "set": "hardy-course-of-pure-mathematics-1921/ex-lxviii", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "315", "location": "Exercise LXVIII, problem 3", "problem_latex": " If the series $u_{0} + u_{1} + \\dots$, $v_{0} + v_{1} + \\dots$, $w_{0} + w_{1} + \\dots$ converge to sums\n$r$,~$s$,~$t$, then the series $\\sum \\lambda_{k}$, where $\\lambda_{k} = \\sum u_{m}v_{n}w_{p}$, the summation being extended\nto all sets of values of $m$,~$n$,~$p$ such that $m + n + p = k$, converges to the\nsum~$rst$.", "markdown": "If the series $u_{0} + u_{1} + \\dots$, $v_{0} + v_{1} + \\dots$, $w_{0} + w_{1} + \\dots$ converge to sums $r$, $s$, $t$, then the series $\\sum \\lambda_{k}$, where $\\lambda_{k} = \\sum u_{m}v_{n}w_{p}$, the summation being extended to all sets of values of $m$, $n$, $p$ such that $m + n + p = k$, converges to the sum $rst$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxviii/4", "set": "hardy-course-of-pure-mathematics-1921/ex-lxviii", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "315", "location": "Exercise LXVIII, problem 4", "problem_latex": " If $\\sum u_{n}$ and~$\\sum v_{n}$ converge to sums $s$ and~$t$, then the series~$\\sum w_{n}$, where\n$w_{n} = \\sum u_{l} v_{m}$, the summation extending to all pairs $l$,~$m$ for which $lm = n$,\nconverges to the sum~$st$.", "markdown": "If $\\sum u_{n}$ and $\\sum v_{n}$ converge to sums $s$ and $t$, then the series $\\sum w_{n}$, where $w_{n} = \\sum u_{l} v_{m}$, the summation extending to all pairs $l$, $m$ for which $lm = n$, converges to the sum $st$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxx/1", "set": "hardy-course-of-pure-mathematics-1921/ex-lxx", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "319", "location": "Exercise LXX, problem 1", "problem_latex": " Prove that\n\\[\n\\sum_{1}^{\\infty} \\frac{1}{n^{2} + 1} < \\tfrac{1}{2} + \\tfrac{1}{4}\\pi\\Add{.}\n\\]", "markdown": "Prove that _1^ 1n^2 + 1 < 12 + 14", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.sum", "core.const" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxx/2", "set": "hardy-course-of-pure-mathematics-1921/ex-lxx", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "319", "location": "Exercise LXX, problem 2", "problem_latex": " Prove that\n\\[\n-\\tfrac{1}{2} \\pi < \\sum_{1}^{\\infty} \\frac{a}{a^{2} + n^{2}} < \\tfrac{1}{2} \\pi.\n\\]", "markdown": "Prove that -12 < _1^ aa^2 + n^2 < 12 .", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.sum", "core.const", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxx/3", "set": "hardy-course-of-pure-mathematics-1921/ex-lxx", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "319", "location": "Exercise LXX, problem 3", "problem_latex": " Prove that if $m > 0$ then\n\\[\n\\frac{1}{m^{2}} + \\frac{1}{(m + 1)^{2}} + \\frac{1}{(m + 2)^{2}} + \\dots\n < \\frac{m + 1}{m}\\Add{.}\n\\]", "markdown": "Prove that if $m > 0$ then 1m^2 + 1(m + 1)^2 + 1(m + 2)^2 + … < m + 1m", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.sum" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxi/1", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxi", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "320", "location": "Exercise LXXI, problem 1", "problem_latex": "Prove by an argument similar to that used above,\nand without integration, that $\\ds\\Phi(\\xi) = \\int_{1}^{\\xi} \\frac{dx}{x^{s}}$, where $s < 1$, tends to infinity with~$\\xi$.", "markdown": "Prove by an argument similar to that used above, and without integration, that $\\ds\\Phi(\\xi) = \\int_{1}^{\\xi} \\frac{dx}{x^{s}}$, where $s < 1$, tends to infinity with $\\xi$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:convergence-proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxi/2", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxi", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "320", "location": "Exercise LXXI, problem 2", "problem_latex": "The series $\\sum n^{-2}$, $\\sum n^{-3/2}$, $\\sum n^{-11/10}$ are convergent, and their sums are\nnot greater than $2$,~$3$,~$11$ respectively. The series $\\sum n^{-1/2}$, $\\sum n^{-10/11}$ are\ndivergent.", "markdown": "The series $\\sum n^{-2}$, $\\sum n^{-3/2}$, $\\sum n^{-11/10}$ are convergent, and their sums are not greater than $2$, $3$, $11$ respectively. The series $\\sum n^{-1/2}$, $\\sum n^{-10/11}$ are divergent.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxi/3", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxi", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "320", "location": "Exercise LXXI, problem 3", "problem_latex": "The series $\\sum n^{s}/(n^{t} + a)$, where $a > 0$, is convergent or divergent according\nas $t > 1 + s$ or $t \\leq 1 + s$. [Compare with~$\\sum n^{s-t}$.]", "markdown": "The series $\\sum n^{s}/(n^{t} + a)$, where $a > 0$, is convergent or divergent according as $t > 1 + s$ or $t \\leq 1 + s$. [Compare with $\\sum n^{s-t}$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "n**s/(n**t + a)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:convergence-proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxi/4", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxi", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "320", "location": "Exercise LXXI, problem 4", "problem_latex": "Discuss the convergence or divergence of the series\n\\[\n\\tsum(a_{1}n^{s_{1}} + a_{2}n^{s_{2}} + \\dots + a_{k}n^{s_{k}})/\n (b_{1}n^{t_{1}} + b_{2}n^{t_{2}} + \\dots + b_{l}n^{t_{l}}),\n\\]\nwhere all the letters denote positive numbers and the $s$'s and~$t$'s are rational\nand arranged in descending order of magnitude.", "markdown": "Discuss the convergence or divergence of the series (a_1n^s_1 + a_2n^s_2 + …+ a_kn^s_k)/ (b_1n^t_1 + b_2n^t_2 + …+ b_ln^t_l), where all the letters denote positive numbers and the $s$’s and $t$’s are rational and arranged in descending order of magnitude.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:convergence-proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxi/5", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxi", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "320", "location": "Exercise LXXI, problem 5", "problem_latex": "Prove that\n\\begin{gather*}\n2\\sqrt{n} - 2\n < \\frac{1}{\\sqrt{1}} + \\frac{1}{\\sqrt{2}} + \\dots + \\frac{1}{\\sqrt{n}}\n < 2\\sqrt{n} - 1, \\\\\n\\tfrac{1}{2} \\pi\n < \\frac{1}{2\\sqrt{1}} + \\frac{1}{3\\sqrt{2}} + \\frac{1}{4\\sqrt{3}} + \\dots\n < \\tfrac{1}{2}(\\pi + 1).\n\\end{gather*}\n\\MathTrip{1911.}", "markdown": "Prove that gather* 2n - 2 < 11 + 12 + …+ 1n < 2n - 1, 12 < 121 + 132 + 143 + … < 12(+ 1). gather* % [0]% (*Math. Trip.* 1911.)% [1]%", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.const" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxi/6", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxi", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "320", "location": "Exercise LXXI, problem 6", "problem_latex": "If $\\phi(n) \\to l > 1$ then the series $\\sum n^{-\\phi(n)}$ is convergent. If $\\phi(n) \\to l < 1$\nthen it is divergent.", "markdown": "If $\\phi(n) \\to l > 1$ then the series $\\sum n^{-\\phi(n)}$ is convergent. If $\\phi(n) \\to l < 1$ then it is divergent.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "other:convergence-proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxii/1", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxii", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "321", "location": "Exercise LXXII, problem 1", "problem_latex": "Show that if $a$~is any positive integer greater\nthan~$1$ then $\\sum \\phi(n)$~is convergent or divergent according as $\\sum a^{n}\\phi(a^{n})$ is\nconvergent or divergent. [Use the same arguments as above, taking groups\nof $a$,~$a^{2}$, $a^{3}$,~\\dots\\ terms.]", "markdown": "Show that if $a$ is any positive integer greater than $1$ then $\\sum \\phi(n)$ is convergent or divergent according as $\\sum a^{n}\\phi(a^{n})$ is convergent or divergent. [Use the same arguments as above, taking groups of $a$, $a^{2}$, $a^{3}$, … terms.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxii/2", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxii", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "321", "location": "Exercise LXXII, problem 2", "problem_latex": "If $\\sum 2^{n}\\phi(2^{n})$ converges then it is obvious that $\\lim 2^{n}\\phi(2^{n}) = 0$. Hence\ndeduce Abel's Theorem of~\\SecNo[§]{173}.", "markdown": "If $\\sum 2^{n}\\phi(2^{n})$ converges then it is obvious that $\\lim 2^{n}\\phi(2^{n}) = 0$. Hence deduce Abel’s Theorem of [§]173.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxiii/1", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxiii", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "324", "location": "Exercise LXXIII, problem 1", "problem_latex": "The integral\n\\[\n\\int_{a}^{\\infty} \\frac{\\alpha x^{r} + \\beta x^{r-1} + \\dots + \\lambda}\n {Ax^{s} + Bx^{s-1} + \\dots + L}\\, dx,\n\\]\nwhere $\\alpha$ and~$A$ are positive and $a$~is greater than the greatest root of the\ndenominator, is convergent if $s > r + 1$ and otherwise divergent.", "markdown": "The integral _a^ x^r + x^r-1 + …+ Ax^s + Bx^s-1 + …+ L  dx, where $\\alpha$ and $A$ are positive and $a$ is greater than the greatest root of the denominator, is convergent if $s > r + 1$ and otherwise divergent.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:improper_integral_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxiii/10a", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxiii", "number": 10, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "324", "location": "Exercise LXXIII, problem 10a", "problem_latex": "\\Topic{Analogue of Abel's Theorem of \\SecNo[§]{173}.} \\emph{If $\\phi(x)$~is positive and\nsteadily decreases, and $\\ds\\int_{a}^{\\infty} \\phi(x)\\, dx$ is convergent, then $x\\phi(x) \\to 0$.} Prove this\n(\\ia)~by means of Abel's Theorem and the Integral Test and (\\ib)~directly, by\narguments analogous to those of~\\SecNo[§]{173}.", "markdown": "**of Abel’s Theorem of [§]173.** *If $\\phi(x)$ is positive and steadily decreases, and $\\ds\\int_{a}^{\\infty} \\phi(x)\\, dx$ is convergent, then $x\\phi(x) \\to 0$.* Prove this (*a*) by means of Abel’s Theorem and the Integral Test and (*b*) directly, by arguments analogous to those of [§]173.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:abel_theorem_integral_test" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxiii/10b", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxiii", "number": 10, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "324", "location": "Exercise LXXIII, problem 10b", "problem_latex": "\\Topic{Analogue of Abel's Theorem of \\SecNo[§]{173}.} \\emph{If $\\phi(x)$~is positive and\nsteadily decreases, and $\\ds\\int_{a}^{\\infty} \\phi(x)\\, dx$ is convergent, then $x\\phi(x) \\to 0$.} Prove this\n(\\ia)~by means of Abel's Theorem and the Integral Test and (\\ib)~directly, by\narguments analogous to those of~\\SecNo[§]{173}.", "markdown": "**of Abel’s Theorem of [§]173.** *If $\\phi(x)$ is positive and steadily decreases, and $\\ds\\int_{a}^{\\infty} \\phi(x)\\, dx$ is convergent, then $x\\phi(x) \\to 0$.* Prove this (*a*) by means of Abel’s Theorem and the Integral Test and (*b*) directly, by arguments analogous to those of [§]173.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:abel_theorem_integral_test" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxiii/11", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxiii", "number": 11, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "324", "location": "Exercise LXXIII, problem 11", "problem_latex": "If $a = x_{0} < x_{1} < x_{2} < \\dots$ and $x_{n} \\to \\infty$, and $\\ds u_{n}= \\int_{x_{n}}^{x_{n+1}} \\phi(x)\\, dx$, then the\nconvergence of $\\ds\\int_{a}^{\\infty} \\phi(x)\\, dx$ involves that of $\\sum u_{n}$. If $\\phi(x)$~is always positive\nthe converse statement is also true. [That the converse is not true in\ngeneral is shown by the example in which $\\phi(x) = \\cos x$, $x_{n} = n\\pi$.]", "markdown": "If $a = x_{0} < x_{1} < x_{2} < \\dots$ and $x_{n} \\to \\infty$, and $\\ds u_{n}= \\int_{x_{n}}^{x_{n+1}} \\phi(x)\\, dx$, then the convergence of $\\ds\\int_{a}^{\\infty} \\phi(x)\\, dx$ involves that of $\\sum u_{n}$. If $\\phi(x)$ is always positive the converse statement is also true. [That the converse is not true in general is shown by the example in which $\\phi(x) = \\cos x$, $x_{n} = n\\pi$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:series_integral_comparison" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxiii/2", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxiii", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "324", "location": "Exercise LXXIII, problem 2", "problem_latex": "Which of the integrals\n%[** TN: All are displayed on one line in the original]\n$\\ds\\int_{a}^{\\infty} \\frac{dx}{\\sqrt{x}}$,\n$\\ds\\int_{a}^{\\infty} \\frac{dx}{x^{4/3}}$,\n\\[\n\\int_{a}^{\\infty} \\frac{dx}{c^{2} + x^{2}},\\quad\n\\int_{a}^{\\infty} \\frac{x\\, dx}{c^{2} + x^{2}},\\quad\n\\int_{a}^{\\infty} \\frac{x^{2}\\, dx}{c^{2} + x^{2}},\\quad\n\\int_{a}^{\\infty} \\frac{x^{2}\\, dx}{\\alpha + 2\\beta x^{2} + \\gamma x^{4}}\n\\]\nare convergent? In the first two integrals it is supposed that $a > 0$, and\nin the last that $a$~is greater than the greatest root (if any) of the denominator.", "markdown": "Which of the integrals %[** TN: All are displayed on one line in the original] $\\ds\\int_{a}^{\\infty} \\frac{dx}{\\sqrt{x}}$, $\\ds\\int_{a}^{\\infty} \\frac{dx}{x^{4/3}}$, _a^ dxc^2 + x^2,0pt minus 3pt_a^ x  dxc^2 + x^2,0pt minus 3pt_a^ x^2  dxc^2 + x^2,0pt minus 3pt_a^ x^2  dx+ 2x^2 + x^4 are convergent? In the first two integrals it is supposed that $a > 0$, and in the last that $a$ is greater than the greatest root (if any) of the denominator.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:improper_integral_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxiii/3", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxiii", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "324", "location": "Exercise LXXIII, problem 3", "problem_latex": "The integrals\n\\[\n\\int_{a}^{\\xi} \\cos x\\, dx,\\quad\n\\int_{a}^{\\xi} \\sin x\\, dx,\\quad\n\\int_{a}^{\\xi} \\cos(\\alpha x + \\beta)\\, dx\n\\]\noscillate finitely as $\\xi \\to \\infty$.", "markdown": "The integrals _a^ x  dx,0pt minus 3pt_a^ x  dx,0pt minus 3pt_a^ (x + )  dx oscillate finitely as $\\xi \\to \\infty$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate", "cas.limit", "other:oscillation_behaviour" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxiii/4", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxiii", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "324", "location": "Exercise LXXIII, problem 4", "problem_latex": "The integrals\n\\[\n\\int_{a}^{\\xi} x\\cos x\\, dx,\\quad\n\\int_{a}^{\\xi} x^{2}\\sin x\\, dx\\quad\n\\int_{a}^{\\xi} x^{n} \\cos(\\alpha x + \\beta)\\, dx,\n\\]\nwhere $n$~is any positive integer, oscillate infinitely as $\\xi \\to \\infty$.", "markdown": "The integrals _a^ xx  dx,0pt minus 3pt_a^ x^2x  dx0pt minus 3pt_a^ x^n (x + )  dx, where $n$ is any positive integer, oscillate infinitely as $\\xi \\to \\infty$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate", "cas.limit", "other:oscillation_behaviour" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxiii/5", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxiii", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "324", "location": "Exercise LXXIII, problem 5", "problem_latex": "\\Topic{Integrals to~$-\\infty$.} If $\\ds\\int_{\\xi}^{a} \\phi(x)\\, dx$ tends to a limit~$l$ as $\\xi \\to -\\infty$, then we\nsay that $\\ds\\int_{-\\infty}^{a} \\phi(x)\\, dx$ is convergent and equal to~$l$. Such integrals possess\nproperties in every respect analogous to those of the integrals discussed in the\npreceding sections: the reader will find no difficulty in formulating them.", "markdown": "**to $-\\infty$.** If $\\ds\\int_{\\xi}^{a} \\phi(x)\\, dx$ tends to a limit $l$ as $\\xi \\to -\\infty$, then we say that $\\ds\\int_{-\\infty}^{a} \\phi(x)\\, dx$ is convergent and equal to $l$. Such integrals possess properties in every respect analogous to those of the integrals discussed in the preceding sections: the reader will find no difficulty in formulating them.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxiii/6", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxiii", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "324", "location": "Exercise LXXIII, problem 6", "problem_latex": "\\Topic{Integrals from~$-\\infty$ to~$+\\infty$.} If the integrals\n\\[\n\\int_{-\\infty}^{a} \\phi(x)\\, dx,\\quad\n\\int_{a}^{\\infty} \\phi(x)\\, dx\n\\]\nare both convergent, and have the values $k$,~$l$ respectively, then we say that\n\\[\n\\int_{-\\infty}^{\\infty} \\phi(x)\\, dx\n\\]\nis convergent and has the value $k + l$.", "markdown": "**from $-\\infty$ to $+\\infty$.** If the integrals _-^a (x)  dx,0pt minus 3pt_a^ (x)  dx are both convergent, and have the values $k$, $l$ respectively, then we say that _-^ (x)  dx is convergent and has the value $k + l$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxiii/7", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxiii", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "324", "location": "Exercise LXXIII, problem 7", "problem_latex": "Prove that\n\\[\n\\int_{-\\infty}^{0} \\frac{dx}{1 + x^{2}}\n = \\int_{0}^{\\infty} \\frac{dx}{1 + x^{2}}\n = \\tfrac{1}{2} \\int_{-\\infty}^{\\infty} \\frac{dx}{1 + x^{2}}\n = \\tfrac{1}{2}\\pi.\n\\]", "markdown": "Prove that _-^0 dx1 + x^2 = _0^ dx1 + x^2 = 12 _-^ dx1 + x^2 = 12.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.defint", "cas.limit", "core.const" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxiii/8", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxiii", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "324", "location": "Exercise LXXIII, problem 8", "problem_latex": "Prove generally that\n\\[\n\\int_{-\\infty}^{\\infty} \\phi(x^{2})\\, dx = 2\\int_{0}^{\\infty} \\phi(x^{2})\\, dx,\n\\]\nprovided that the integral $\\ds\\int_{0}^{\\infty} \\phi(x^{2})\\, dx$ is convergent.", "markdown": "Prove generally that _-^ (x^2)  dx = 2_0^ (x^2)  dx, provided that the integral $\\ds\\int_{0}^{\\infty} \\phi(x^{2})\\, dx$ is convergent.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxiii/9", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxiii", "number": 9, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "324", "location": "Exercise LXXIII, problem 9", "problem_latex": "Prove that if $\\ds\\int_{0}^{\\infty} x\\phi(x^{2})\\, dx$ is convergent then $\\ds\\int_{-\\infty}^{\\infty} x\\phi(x^{2})\\, dx = 0$.", "markdown": "Prove that if $\\ds\\int_{0}^{\\infty} x\\phi(x^{2})\\, dx$ is convergent then $\\ds\\int_{-\\infty}^{\\infty} x\\phi(x^{2})\\, dx = 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxiv/1", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxiv", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "327", "location": "Exercise LXXIV, problem 1", "problem_latex": "Show, by means of the substitution $x = t^{\\alpha}$,\nthat if $s > 1$ and $\\alpha >0$ then\n\\[\n\\int_{1}^{\\infty} x^{-s}\\, dx = \\alpha\\int_{1}^{\\infty} t^{\\alpha(1-s) - 1}\\, dt;\n\\]\nand verify the result by calculating the value of each integral directly.", "markdown": "Show, by means of the substitution $x = t^{\\alpha}$, that if $s > 1$ and $\\alpha >0$ then _1^ x^-s  dx = _1^ t^(1-s) - 1  dt; and verify the result by calculating the value of each integral directly.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.defint", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxiv/2", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxiv", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "327", "location": "Exercise LXXIV, problem 2", "problem_latex": "If $\\ds\\int_{a}^{\\infty} \\phi(x)\\, dx$ is convergent then it is equal to one or other of\n\\[\n \\alpha\\int_{(a-\\beta)/\\alpha}^{\\infty} \\phi(\\alpha t + \\beta)\\, dt,\\quad\n-\\alpha\\int_{-\\infty}^{(a-\\beta)/\\alpha} \\phi(\\alpha t + \\beta)\\, dt,\n\\]\naccording as $\\alpha$~is positive or negative.", "markdown": "If $\\ds\\int_{a}^{\\infty} \\phi(x)\\, dx$ is convergent then it is equal to one or other of _(a-)/^ (t + )  dt,0pt minus 3pt-_-^(a-)/ (t + )  dt, according as $\\alpha$ is positive or negative.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxiv/3", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxiv", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "327", "location": "Exercise LXXIV, problem 3", "problem_latex": "If $\\phi(x)$~is a positive and steadily decreasing function of~$x$, and $\\alpha$~and~$\\beta$\nare any positive numbers, then the convergence of the series $\\sum \\phi(n)$ implies\nand is implied by that of the series $\\sum \\phi(\\alpha n + \\beta)$.", "markdown": "If $\\phi(x)$ is a positive and steadily decreasing function of $x$, and $\\alpha$ and $\\beta$ are any positive numbers, then the convergence of the series $\\sum \\phi(n)$ implies and is implied by that of the series $\\sum \\phi(\\alpha n + \\beta)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.subst", "other:integral_test" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxiv/4", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxiv", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "327", "location": "Exercise LXXIV, problem 4", "problem_latex": "Show that\n\\[\n%[** TN: In-line in the original]\n\\int_{1}^{\\infty} \\frac{dx}{(1 + x)\\sqrt{x}} = \\tfrac{1}{2} \\pi.\n\\]", "markdown": "Show that %[** TN: In-line in the original] _1^ dx(1 + x)x = 12 .", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "1/((1 + x)*sqrt(x))", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: 1/(sqrt(x)*(x + 1))" ], "shape": [ "integrate: x**N/(x + 1)" ], "same_problem_in": [], "needs": [ "cas.defint", "cas.integrate", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxiv/5", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxiv", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "327", "location": "Exercise LXXIV, problem 5", "problem_latex": "Show that\n\\[\n\\int_{0}^{\\infty} \\frac{\\sqrt{x}}{(1 + x)^{2}}\\, dx = \\tfrac{1}{2}\\pi.\n\\]", "markdown": "Show that _0^ x(1 + x)^2  dx = 12.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "sqrt(x)/(1 + x)**2", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: sqrt(x)/(x + 1)**2" ], "shape": [ "integrate: x**N*(x + 1)**N" ], "same_problem_in": [], "needs": [ "cas.defint", "cas.integrate.parts", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxiv/6", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxiv", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "327", "location": "Exercise LXXIV, problem 6", "problem_latex": "If $\\phi(x) \\to h$ as $x \\to \\infty$, and $\\phi(x) \\to k$ as $x \\to -\\infty$, then\n\\[\n\\int_{-\\infty}^{\\infty} \\{\\phi(x - a) - \\phi(x - b)\\}\\, dx = -(a - b)(h - k).\n\\]", "markdown": "If $\\phi(x) \\to h$ as $x \\to \\infty$, and $\\phi(x) \\to k$ as $x \\to -\\infty$, then _-^ (x - a) - (x - b)  dx = -(a - b)(h - k).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.defint", "cas.subst", "other:limit_argument" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxix/1", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxix", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "343", "location": "Exercise LXXIX, problem 1", "problem_latex": "Dirichlet's and Abel's Tests may also be established\nby means of the general principle of convergence (\\SecNo[§]{84}). Let us suppose,\nfor example, that the conditions of Abel's Test are satisfied. We have\nidentically\n{\\setlength{\\multlinegap}{0pt}\n\\begin{multline*}\na_{m}\\phi_{m} + a_{m+1}\\phi_{m+1} + \\dots + a_{n}\\phi_{n}\n = s_{m, m}(\\phi_{m} - \\phi_{m+1}) + s_{m, m+1}(\\phi_{m+1} - \\phi_{m+2})\\\\\n + \\dots + s_{m, n-1}(\\phi_{n-1} - \\phi_{n}) + s_{m, n}\\phi_{n}\\dots,\n\\Tag{(1)}\n\\end{multline*}}%\nwhere\n\\[\ns_{m, \\nu} = a_{m} + a_{m+1} + \\dots + a_{\\nu}.\n\\]\n\nThe left-hand side of~\\Eq{(1)} therefore lies between $h\\phi_{m}$ and~$H\\phi_{m}$, where $h$~and~$H$\nare the algebraically least and greatest of $s_{m, m}$, $s_{m, m+1}$,~\\dots, $s_{m, n}$. But,\ngiven any positive number~$\\DELTA$, we can choose~$m_{0}$ so that $|s_{m, \\nu}| < \\DELTA$ when $m \\geq m_{0}$,\nand so\n\\[\n|a_{m}\\phi_{m} + a_{m+1}\\phi_{m+1} + \\dots + a_{n}\\phi_{n}|\n < \\DELTA \\phi_{m} \\leq \\DELTA \\phi_{1}\n\\]\nwhen $n > m \\geq m_{0}$. Thus the series $\\sum a_{n}\\phi_{n}$ is convergent.", "markdown": "Dirichlet’s and Abel’s Tests may also be established by means of the general principle of convergence ([§]84). Let us suppose, for example, that the conditions of Abel’s Test are satisfied. We have identically 0pt multline* a_m_m + a_m+1_m+1 + …+ a_n_n = s_m, m(_m - _m+1) + s_m, m+1(_m+1 - _m+2) + …+ s_m, n-1(_n-1 - _n) + s_m, n_n…, (1) multline*% where s_m, = a_m + a_m+1 + …+ a_. The left-hand side of (1) therefore lies between $h\\phi_{m}$ and $H\\phi_{m}$, where $h$ and $H$ are the algebraically least and greatest of $s_{m, m}$, $s_{m, m+1}$, …, $s_{m, n}$. But, given any positive number $\\DELTA$, we can choose $m_{0}$ so that $|s_{m, \\nu}| < \\DELTA$ when $m \\geq m_{0}$, and so |a_m_m + a_m+1_m+1 + …+ a_n_n| < _m _1 when $n > m \\geq m_{0}$. Thus the series $\\sum a_{n}\\phi_{n}$ is convergent.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxix/2", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxix", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "343", "location": "Exercise LXXIX, problem 2", "problem_latex": "The series $\\sum \\cos n\\theta$ and $\\sum \\sin n\\theta$ oscillate finitely when $\\theta$~is not a\nmultiple of~$\\pi$. For, if we denote the sums of the first $n$ terms of the two\nseries by $s_{n}$ and~$t_{n}$, and write $z = \\Cis\\theta$, so that $|z| = 1$ and $z \\neq 1$, we have\n\\[\n|s_{n} + it_{n}|\n = \\left|\\frac{1 - z^{n}}{1 - z}\\right|\n \\leq \\frac{1 + |z^{n}|}{|1 - z|}\n \\leq \\frac{2}{|1 - z|};\n\\]\nand so $|s_{n}|$ and~$|t_{n}|$ are also not greater than~$2/|1 - z|$. That the series are\nnot actually convergent follows from the fact that their $n$th~terms do not tend\nto zero (\\Exs{xxiv}.~7,~8).\n\nThe sine series converges to zero if $\\theta$~is a multiple of~$\\pi$. The cosine\nseries oscillates finitely if $\\theta$~is an odd multiple of~$\\pi$ and diverges if $\\theta$~is an\neven multiple of~$\\pi$.\n\nIt follows that \\emph{if $\\theta_{n}$~is a positive function of~$n$ which tends steadily to\nzero as $n \\to \\infty$, then the series\n\\[\n\\tsum \\phi_{n} \\cos n\\theta,\\quad\n\\tsum \\phi_{n} \\sin n\\theta\n\\]\nare convergent}, except perhaps the first series when $\\theta$~is a multiple of~$2\\pi$. In\nthis case the first series reduces to~$\\sum \\phi_{n}$, which may or may not be convergent:\nthe second series vanishes identically. If $\\sum \\phi_{n}$~is convergent then both\nseries are absolutely convergent (\\Ex{lxxvii}.~4) for all values of~$\\theta$, and the\nwhole interest of the result lies in its application to the case in which\n$\\sum \\phi_{n}$~is divergent. And in this case the series above written are conditionally\nand \\emph{not} absolutely convergent, as will be proved in \\Ex{lxxix}.~6.\nIf we put $\\theta = \\pi$ in the cosine series we are led back to the result of \\SecNo[§]{188},\nsince $\\cos n\\pi = (-1)^{n}$.", "markdown": "The series $\\sum \\cos n\\theta$ and $\\sum \\sin n\\theta$ oscillate finitely when $\\theta$ is not a multiple of $\\pi$. For, if we denote the sums of the first $n$ terms of the two series by $s_{n}$ and $t_{n}$, and write $z = \\Cis\\theta$, so that $|z| = 1$ and $z \\neq 1$, we have |s_n + it_n| = |1 - z^n1 - z| 1 + |z^n||1 - z| 2|1 - z|; and so $|s_{n}|$ and $|t_{n}|$ are also not greater than $2/|1 - z|$. That the series are not actually convergent follows from the fact that their $n$th terms do not tend to zero (xxiv. 7, 8). The sine series converges to zero if $\\theta$ is a multiple of $\\pi$. The cosine series oscillates finitely if $\\theta$ is an odd multiple of $\\pi$ and diverges if $\\theta$ is an even multiple of $\\pi$. It follows that *if $\\theta_{n}$ is a positive function of $n$ which tends steadily to zero as $n \\to \\infty$, then the series _n n,0pt minus 3pt_n n are convergent*, except perhaps the first series when $\\theta$ is a multiple of $2\\pi$. In this case the first series reduces to $\\sum \\phi_{n}$, which may or may not be convergent: the second series vanishes identically. If $\\sum \\phi_{n}$ is convergent then both series are absolutely convergent (% [examples:lxxvii]Ex. lxxvii%. 4) for all values of $\\theta$, and the whole interest of the result lies in its application to the case in which $\\sum \\phi_{n}$ is divergent. And in this case the series above written are conditionally and *not* absolutely convergent, as will be proved in % [examples:lxxix]Ex. lxxix%. 6. If we put $\\theta = \\pi$ in the cosine series we are led back to the result of [§]188, since $\\cos n\\pi = (-1)^{n}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxix/3", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxix", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "343", "location": "Exercise LXXIX, problem 3", "problem_latex": "The series $\\sum n^{-s} \\cos n\\theta$, $\\sum n^{-s} \\sin n\\theta$ are convergent if $s > 0$, unless (in\nthe case of the first series) $\\theta$~is a multiple of~$2\\pi$ and $0 < s \\leq 1$.", "markdown": "The series $\\sum n^{-s} \\cos n\\theta$, $\\sum n^{-s} \\sin n\\theta$ are convergent if $s > 0$, unless (in the case of the first series) $\\theta$ is a multiple of $2\\pi$ and $0 < s \\leq 1$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxix/4", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxix", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "343", "location": "Exercise LXXIX, problem 4", "problem_latex": "The series of Ex.~3 are in general absolutely convergent if $s > 1$,\nconditionally convergent if $0 < s \\leq 1$, and oscillatory if $s \\leq 0$ (finitely if $s = 0$\nand infinitely if $s < 0$). Mention any exceptional cases.", "markdown": "The series of Ex. 3 are in general absolutely convergent if $s > 1$, conditionally convergent if $0 < s \\leq 1$, and oscillatory if $s \\leq 0$ (finitely if $s = 0$ and infinitely if $s < 0$). Mention any exceptional cases.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxix/5", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxix", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "343", "location": "Exercise LXXIX, problem 5", "problem_latex": "If $\\sum a_{n}n^{-s}$~is convergent or oscillates finitely, then $\\sum a_{n}n^{-t}$~is convergent\nwhen $t > s$.", "markdown": "If $\\sum a_{n}n^{-s}$ is convergent or oscillates finitely, then $\\sum a_{n}n^{-t}$ is convergent when $t > s$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxix/6", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxix", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "343", "location": "Exercise LXXIX, problem 6", "problem_latex": "If $\\phi_{n}$~is a positive function of~$n$ which tends steadily to~$0$ as $n \\to \\infty$,\nand $\\sum \\phi_{n}$~is divergent, then the series $\\sum \\phi_{n} \\cos n\\theta$, $\\sum \\phi_{n} \\sin n\\theta$ are \\emph{not} absolutely\nconvergent, except the sine-series when $\\theta$~is a multiple of~$\\pi$. [For suppose,\n\\eg, that $\\sum \\phi_{n} |\\cos n\\theta|$ is convergent. Since $\\cos^{2} n\\theta \\leq |\\cos n\\theta|$, it follows that\n$\\sum \\phi_{n} \\cos^{2} n\\theta$ or\n\\[\n\\tfrac{1}{2} \\tsum \\phi_{n} (1 + \\cos 2n\\theta)\n\\]\nis convergent. But this is impossible, since $\\sum \\phi_{n}$~is divergent and $\\sum \\phi_{n} \\cos 2n\\theta$,\nby Dirichlet's Test, convergent, unless $\\theta$~is a multiple of~$\\pi$. And in\nthis case it is obvious that $\\sum \\phi_{n} |\\cos n\\theta|$ is divergent. The reader should write\nout the corresponding argument for the sine-series, noting where it fails\nwhen $\\theta$~is a multiple of~$\\pi$.]", "markdown": "If $\\phi_{n}$ is a positive function of $n$ which tends steadily to $0$ as $n \\to \\infty$, and $\\sum \\phi_{n}$ is divergent, then the series $\\sum \\phi_{n} \\cos n\\theta$, $\\sum \\phi_{n} \\sin n\\theta$ are *not* absolutely convergent, except the sine-series when $\\theta$ is a multiple of $\\pi$. [For suppose, *e.g.*, that $\\sum \\phi_{n} |\\cos n\\theta|$ is convergent. Since $\\cos^{2} n\\theta \\leq |\\cos n\\theta|$, it follows that $\\sum \\phi_{n} \\cos^{2} n\\theta$ or 12 _n (1 + 2n) is convergent. But this is impossible, since $\\sum \\phi_{n}$ is divergent and $\\sum \\phi_{n} \\cos 2n\\theta$, by Dirichlet’s Test, convergent, unless $\\theta$ is a multiple of $\\pi$. And in this case it is obvious that $\\sum \\phi_{n} |\\cos n\\theta|$ is divergent. The reader should write out the corresponding argument for the sine-series, noting where it fails when $\\theta$ is a multiple of $\\pi$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxv/1", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxv", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "328", "location": "Exercise LXXV, problem 1", "problem_latex": " Show that\n\\[\n%[** TN: In-line in the original]\n\\int_{0}^{\\infty} \\frac{x}{(1 + x)^{3}}\\, dx\n = \\tfrac{1}{2} \\int_{0}^{\\infty} \\frac{dx}{(1 + x)^{2}}\n = \\tfrac{1}{2}.\n\\]", "markdown": "Show that %[** TN: In-line in the original] _0^ x(1 + x)^3  dx = 12 _0^ dx(1 + x)^2 = 12.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.defint", "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxv/2", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxv", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "328", "location": "Exercise LXXV, problem 2", "problem_latex": " $\\ds\\int_{0}^{\\infty} \\frac{x^{2}}{(1 + x)^{4}}\\, dx = \\tfrac{2}{3} \\int_{0}^{\\infty} \\frac{x}{(1 + x)^{3}}\\, dx = \\tfrac{1}{3}$.", "markdown": "$\\ds\\int_{0}^{\\infty} \\frac{x^{2}}{(1 + x)^{4}}\\, dx = \\tfrac{2}{3} \\int_{0}^{\\infty} \\frac{x}{(1 + x)^{3}}\\, dx = \\tfrac{1}{3}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.defint", "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxv/3", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxv", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "328", "location": "Exercise LXXV, problem 3", "problem_latex": " If $m$ and~$n$ are positive integers, and\n\\[\n%[** TN: Two equations not displayed in the original]\nI_{m, n} = \\int_{0}^{\\infty} \\frac{x^{m}\\, dx}{(1 + x)^{m+n}},\n\\]\nthen\n\\[\nI_{m, n} = \\{m/(m + n - 1)\\} I_{m-1, n}.\n\\]\nHence prove that $I_{m, n} = m!\\, (n - 2)!/(m + n - 1)!$.", "markdown": "If $m$ and $n$ are positive integers, and %[** TN: Two equations not displayed in the original] I_m, n = _0^ x^m  dx(1 + x)^m+n, then I_m, n = m/(m + n - 1) I_m-1, n. Hence prove that $I_{m, n} = m!\\, (n - 2)!/(m + n - 1)!$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.defint", "cas.integrate.parts", "core.arith" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxv/4", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxv", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "328", "location": "Exercise LXXV, problem 4", "problem_latex": " Show similarly that if\n\\[\n%[** TN: Not displayed in the original]\nI_{m, n} = \\int_{0}^{\\infty} \\frac{x^{2m+1}\\, dx}{(1 + x^{2})^{m+n}}\n\\]\nthen\n\\[\nI_{m, n} = \\{m/(m + n - 1)\\} I_{m-1, n},\\quad\n2I_{m, n} = m!\\, (n - 2)!/(m + n - 1)!.\n\\]\nVerify the result by applying the substitution $x = t^{2}$ to the result of Ex.~3.", "markdown": "Show similarly that if %[** TN: Not displayed in the original] I_m, n = _0^ x^2m+1  dx(1 + x^2)^m+n then I_m, n = m/(m + n - 1) I_m-1, n,0pt minus 3pt2I_m, n = m!  (n - 2)!/(m + n - 1)!. Verify the result by applying the substitution $x = t^{2}$ to the result of Ex. 3.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.defint", "cas.integrate.parts", "cas.integrate.subst", "core.arith" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxvi/1", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxvi", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "331", "location": "Exercise LXXVI, problem 1", "problem_latex": "If $\\phi(x)$~is continuous except for $x = a$, while\n$\\phi(x) \\to \\infty$ as $x \\to a$, then the necessary and sufficient condition that $\\ds\\int_{a}^{A} \\phi(x)\\, dx$\nshould be convergent is that we can find a constant~$K$ such that\n\\[\n\\int_{a+\\epsilon}^{A} \\phi(x)\\, dx < K\n\\]\nfor all values of~$\\epsilon$, however small (cf.~\\SecNo[§]{178}).\n\nIt is clear that we can choose a number~$A'$ between $a$ and~$A$, such that\n$\\phi(x)$~is positive throughout $\\DPmod{(a, A')}{[a, A']}$. If $\\phi(x)$~is positive throughout the\nwhole interval $\\DPmod{(a, A)}{[a, A]}$ then we can of course identify $A'$ and~$A$. Now\n\\[\n\\int_{a-\\epsilon}^{A} \\phi(x)\\, dx\n = \\int_{a-\\epsilon}^{A'} \\phi(x)\\, dx + \\int_{A'}^{A} \\phi(x)\\, dx.\n\\]\nThe first integral on the right-hand side of the above equation increases\nas $\\epsilon$~decreases, and therefore tends to a limit or to~$\\infty$; and the truth of the\nresult stated becomes evident.\n\nIf the condition is not satisfied then $\\ds\\int_{a-\\epsilon}^{A} \\phi(x)\\, dx \\to \\infty$. We shall then say\nthat the integral $\\ds\\int_{a}^{A} \\phi(x)\\, dx$ \\Emph{diverges} to~$\\infty$. It is clear that, if $\\phi(x) \\to \\infty$\nas $x \\to a + 0$, then convergence and divergence to~$\\infty$ are the only alternatives\nfor the integral. We may discuss similarly the case in which $\\phi(x) \\to -\\infty$.", "markdown": "If $\\phi(x)$ is continuous except for $x = a$, while $\\phi(x) \\to \\infty$ as $x \\to a$, then the necessary and sufficient condition that $\\ds\\int_{a}^{A} \\phi(x)\\, dx$ should be convergent is that we can find a constant $K$ such that _a+^A (x)  dx < K for all values of $\\epsilon$, however small (cf. [§]178). It is clear that we can choose a number $A'$ between $a$ and $A$, such that $\\phi(x)$ is positive throughout $\\DPmod{(a, A')}{[a, A']}$. If $\\phi(x)$ is positive throughout the whole interval $\\DPmod{(a, A)}{[a, A]}$ then we can of course identify $A'$ and $A$. Now _a-^A (x)  dx = _a-^A’ (x)  dx + _A’^A (x)  dx. The first integral on the right-hand side of the above equation increases as $\\epsilon$ decreases, and therefore tends to a limit or to $\\infty$; and the truth of the result stated becomes evident. If the condition is not satisfied then $\\ds\\int_{a-\\epsilon}^{A} \\phi(x)\\, dx \\to \\infty$. We shall then say that the integral $\\ds\\int_{a}^{A} \\phi(x)\\, dx$ **** to $\\infty$. It is clear that, if $\\phi(x) \\to \\infty$ as $x \\to a + 0$, then convergence and divergence to $\\infty$ are the only alternatives for the integral. We may discuss similarly the case in which $\\phi(x) \\to -\\infty$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxvi/10", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxvi", "number": 10, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "331", "location": "Exercise LXXVI, problem 10", "problem_latex": "Show that\n\\[\n\\int_{0}^{h} \\frac{\\sin x}{x^{p}}\\, dx,\n\\]\nwhere $0 < p < 2$, attains its greatest value\nwhen $h = \\pi$. \\MathTrip{1911.}", "markdown": "Show that _0^h xx^p  dx, where $0 < p < 2$, attains its greatest value when $h = \\pi$. % [0]% (*Math. Trip.* 1911.)% [1]%", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxvi/11", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxvi", "number": 11, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "331", "location": "Exercise LXXVI, problem 11", "problem_latex": "The integral\n\\[\n\\int_{0}^{\\frac{1}{2} \\pi}(\\cos x)^{l}(\\sin x)^{m}\\, dx\n\\]\nis convergent if and only if $l > -1$,\n$m > -1$.", "markdown": "The integral _0^12 (x)^l(x)^m  dx is convergent if and only if $l > -1$, $m > -1$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxvi/12", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxvi", "number": 12, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "331", "location": "Exercise LXXVI, problem 12", "problem_latex": "Such an integral as\n\\[\n\\int_{0}^{\\infty} \\frac{x^{s-1}\\, dx}{1 + x},\n\\]\nwhere $s < 1$, does not fall directly\nunder any of our previous definitions. For the range of integration is infinite\n\\PageSep{333}\nand the subject of integration tends to~$\\infty$ as $x \\to +0$. It is natural to\ndefine this integral as being equal to the sum\n\\[\n\\int_{0}^{1} \\frac{x^{s-1}\\, dx}{1 + x}\n + \\int_{1}^{\\infty} \\frac{x^{s-1}\\, dx}{1 + x},\n\\]\nprovided that these two integrals are both convergent.\n\n{\\Loosen The first integral is a convergent infinite integral of the second kind\nif $0 < s < 1$. The second is a convergent infinite integral of the first kind if\n$s < 1$. It should be noted that when $s > 1$ the first integral is an ordinary\nfinite integral; but then the second is divergent. Thus the integral from~$0$ to~$\\infty$\nis convergent if and only if $0 < s < 1$.}", "markdown": "Such an integral as _0^ x^s-1  dx1 + x, where $s < 1$, does not fall directly under any of our previous definitions. For the range of integration is infinite [pg]333 and the subject of integration tends to $\\infty$ as $x \\to +0$. It is natural to define this integral as being equal to the sum _0^1 x^s-1  dx1 + x + _1^ x^s-1  dx1 + x, provided that these two integrals are both convergent. 0.375em plus 0.75em minus 0.25emThe first integral is a convergent infinite integral of the second kind if $0 < s < 1$. The second is a convergent infinite integral of the first kind if $s < 1$. It should be noted that when $s > 1$ the first integral is an ordinary finite integral; but then the second is divergent. Thus the integral from $0$ to $\\infty$ is convergent if and only if $0 < s < 1$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxvi/13", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxvi", "number": 13, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "331", "location": "Exercise LXXVI, problem 13", "problem_latex": "Prove that\n\\[\n\\int_{0}^{\\infty} \\frac{x^{s-1}}{1 + x^{t}}\\, dx\n\\]\nis convergent if and only if $0 < s < t$.", "markdown": "Prove that _0^ x^s-11 + x^t  dx is convergent if and only if $0 < s < t$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxvi/14", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxvi", "number": 14, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "331", "location": "Exercise LXXVI, problem 14", "problem_latex": "The integral\n\\[\n\\int_{0}^{\\infty} \\frac{x^{s-1} - x^{t-1}}{1 - x}\\, dx\n\\]\nis convergent if and only if $0 < s < 1$,\n$0 < t < 1$. [It should be noticed that the subject of integration is undefined\nwhen $x = 1$; but $(x^{s-1} - x^{t-1})/(1 - x) \\to t - s$ as $x \\to 1$ from either side; so that\nthe subject of integration becomes a continuous function of~$x$ if we assign to it\nthe value $t - s$ when $x = 1$.\n\nIt often happens that the subject of integration has a discontinuity which\nis due simply to a failure in its definition at a particular point in the range\nof integration, and can be removed by attaching a particular value to it at\nthat point. In this case it is usual to suppose the definition of the subject\nof integration completed in this way. Thus the integrals\n\\[\n\\int_{0}^{\\frac{1}{2} \\pi} \\frac{\\sin mx}{x}\\, dx,\\quad\n\\int_{0}^{\\frac{1}{2} \\pi} \\frac{\\sin mx}{\\sin x}\\, dx\n\\]\nare ordinary finite integrals, if the subjects of integration are regarded as\nhaving the value~$m$ when $x = 0$.]", "markdown": "The integral _0^ x^s-1 - x^t-11 - x  dx is convergent if and only if $0 < s < 1$, $0 < t < 1$. [It should be noticed that the subject of integration is undefined when $x = 1$; but $(x^{s-1} - x^{t-1})/(1 - x) \\to t - s$ as $x \\to 1$ from either side; so that the subject of integration becomes a continuous function of $x$ if we assign to it the value $t - s$ when $x = 1$. It often happens that the subject of integration has a discontinuity which is due simply to a failure in its definition at a particular point in the range of integration, and can be removed by attaching a particular value to it at that point. In this case it is usual to suppose the definition of the subject of integration completed in this way. Thus the integrals _0^12 mxx  dx,0pt minus 3pt_0^12 mxx  dx are ordinary finite integrals, if the subjects of integration are regarded as having the value $m$ when $x = 0$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxvi/15", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxvi", "number": 15, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "331", "location": "Exercise LXXVI, problem 15", "problem_latex": "\\Topic{Substitution and integration by parts.} The formulae for transformation\nby substitution and integration by parts may of course be extended\nto infinite integrals of the second as well as of the first kind. The reader\nshould formulate the general theorems for himself, on the lines of~\\SecNo[§]{179}.", "markdown": "**and integration by parts.** The formulae for transformation by substitution and integration by parts may of course be extended to infinite integrals of the second as well as of the first kind. The reader should formulate the general theorems for himself, on the lines of [§]179.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxvi/16", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxvi", "number": 16, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "331", "location": "Exercise LXXVI, problem 16", "problem_latex": "Prove by integration by parts that if $s > 0$, $t > 1$, then\n\\[\n\\int_{0}^{1} x^{s-1}(1 - x)^{t-1}\\, dx\n = \\frac{t - 1}{s} \\int_{0}^{1} x^{s} (1 - x)^{t-2}\\, dx.\n\\]", "markdown": "Prove by integration by parts that if $s > 0$, $t > 1$, then _0^1 x^s-1(1 - x)^t-1  dx = t - 1s _0^1 x^s (1 - x)^t-2  dx.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate.parts" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxvi/17", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxvi", "number": 17, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "331", "location": "Exercise LXXVI, problem 17", "problem_latex": "If $s > 0$ then\n\\[\n\\int_{0}^{1} \\frac{x^{s-1}\\, dx}{1 + x}\n = \\int_{1}^{\\infty} \\frac{t^{-s}\\, dt}{1 + t}.\n\\]\n\n%[** TN: Added paragraph break]\n[Put $x = 1/t$.]", "markdown": "If $s > 0$ then _0^1 x^s-1  dx1 + x = _1^ t^-s  dt1 + t. %[** TN: Added paragraph break] [Put $x = 1/t$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxvi/18", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxvi", "number": 18, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "331", "location": "Exercise LXXVI, problem 18", "problem_latex": "If $0 < s < 1$ then\n\\[\n\\int_{0}^{1} \\frac{x^{s-1} + x^{-s}}{1 + x}\\, dx\n = \\int_{0}^{\\infty} \\frac{t^{-s}\\, dt}{1 + t}\n = \\int_{0}^{\\infty} \\frac{t^{s-1}\\, dt}{1 + t}.\n\\]", "markdown": "If $0 < s < 1$ then _0^1 x^s-1 + x^-s1 + x  dx = _0^ t^-s  dt1 + t = _0^ t^s-1  dt1 + t.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxvi/19", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxvi", "number": 19, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "331", "location": "Exercise LXXVI, problem 19", "problem_latex": "If $a + b > 0$ then\n\\[\n\\int_{b}^{\\infty} \\frac{dx}{(x + a)\\sqrtp{x - b}} = \\frac{\\pi}{\\sqrtp{a + b}}.\n\\]\n\\MathTrip{1909.}", "markdown": "If $a + b > 0$ then _b^ dx(x + a)x - b = a + b. % [0]% (*Math. Trip.* 1909.)% [1]%", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxvi/2", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxvi", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "331", "location": "Exercise LXXVI, problem 2", "problem_latex": "Prove that\n\\[\n\\int_{a}^{A} (x - a)^{-s}\\, dx = \\frac{(A - a)^{1-s}}{1 - s}\n\\]\nif $s < 1$, while the integral is divergent if $s \\geq 1$.", "markdown": "Prove that _a^A (x - a)^-s  dx = (A - a)^1-s1 - s if $s < 1$, while the integral is divergent if $s \\geq 1$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxvi/20", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxvi", "number": 20, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "331", "location": "Exercise LXXVI, problem 20", "problem_latex": "Show, by means of the substitution $x = t/(1 - t)$, that if $l$~and~$m$ are\nboth positive then\n\\[\n\\int_{0}^{\\infty} \\frac{x^{l-1}}{(1 + x)^{l+m}}\\, dx\n = \\int_{0}^{1} t^{l-1} (1 - t)^{m-1}\\, dt.\n\\]", "markdown": "Show, by means of the substitution $x = t/(1 - t)$, that if $l$ and $m$ are both positive then _0^ x^l-1(1 + x)^l+m  dx = _0^1 t^l-1 (1 - t)^m-1  dt.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxvi/21", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxvi", "number": 21, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "331", "location": "Exercise LXXVI, problem 21", "problem_latex": "Show, by means of the substitution $x = pt/(p + 1 - t)$, that if $l$,~$m$, and~$p$\nare all positive then\n\\[\n\\int_{0}^{1} x^{l-1} (1 - x)^{m-1}\\, \\frac{dx}{(x + p)^{l + m}}\n = \\frac{1}{(1 + p)^{l} p^{m}} \\int_{0}^{1} t^{l-1} (1 - t)^{m-1}\\, dt.\n\\]", "markdown": "Show, by means of the substitution $x = pt/(p + 1 - t)$, that if $l$, $m$, and $p$ are all positive then _0^1 x^l-1 (1 - x)^m-1  dx(x + p)^l + m = 1(1 + p)^l p^m _0^1 t^l-1 (1 - t)^m-1  dt.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxvi/22", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxvi", "number": 22, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "331", "location": "Exercise LXXVI, problem 22", "problem_latex": "Prove that\n\\[\n\\int_{a}^{b} \\frac{dx}{\\sqrtb{(x - a)(b - x)}} = \\pi\\quad\\text{and}\\quad\n\\int_{a}^{b} \\frac{x\\, dx}{\\sqrtb{(x - a)(b - x)}} = \\tfrac{1}{2} \\pi (a + b),\n\\]\n(i)~by means of the substitution $x = a + (b - a)t^{2}$, (ii)~by means of the substitution\n$(b - x)/(x - a) = t$, and (iii)~by means of the substitution $x = a\\cos^{2} t + b\\sin^{2} t$.", "markdown": "Prove that _a^b dx(x - a)(b - x) = 0pt minus 3ptand0pt minus 3pt_a^b x  dx(x - a)(b - x) = 12 (a + b), (i) by means of the substitution $x = a + (b - a)t^{2}$, (ii) by means of the substitution $(b - x)/(x - a) = t$, and (iii) by means of the substitution $x = a\\cos^{2} t + b\\sin^{2} t$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "FLAG-EXTRACTION", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxvi/23", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxvi", "number": 23, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "331", "location": "Exercise LXXVI, problem 23", "problem_latex": "If $s > -1$ then\n\\[\n\\int_{0}^{\\frac{1}{2} \\pi} (\\sin\\theta)^{s}\\, d\\theta\n = \\int_{0}^{1} \\frac{x^{s}\\, dx}{\\sqrtp{1 - x^{2}}}\n = \\tfrac{1}{2} \\int_{0}^{1} \\frac{x^{\\frac{1}{2}(s-1)}\\, dx}{\\sqrtp{1 - x}}\n = \\tfrac{1}{2} \\int_{0}^{1} (1 - x)^{\\frac{1}{2}(s-1)} \\frac{dx}{\\sqrt{x}}.\n\\]", "markdown": "If $s > -1$ then _0^12 ()^s  d = _0^1 x^s  dx1 - x^2 = 12 _0^1 x^12(s-1)  dx1 - x = 12 _0^1 (1 - x)^12(s-1) dxx.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxvi/24", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxvi", "number": 24, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "331", "location": "Exercise LXXVI, problem 24", "problem_latex": "Establish the formulae\n\\begin{align*}\n&\\int_{0}^{1} \\frac{f(x)\\, dx}{\\sqrtp{1 - x^{2}}}\n = \\int_{0}^{\\frac{1}{2}\\pi} f(\\sin\\theta)\\, d\\theta,\\\\\n%\n&\\int_{a}^{b} \\frac{f(x)\\, dx}{\\sqrtb{(x - a)(b - x)}}\n = 2\\int_{0}^{\\frac{1}{2}\\pi} f(a\\cos^{2}\\theta + b\\sin^{2}\\theta)\\, d\\theta,\\\\\n%\n&\\int_{-a}^{a} f\\left\\{\\bigsqrtp{\\frac{a - x}{a + x}}\\right\\} dx\n = 4a\\int_{0}^{\\frac{1}{2}\\pi} f(\\tan\\theta) \\cos\\theta \\sin\\theta\\, d\\theta\\Add{.}\n\\end{align*}", "markdown": "Establish the formulae align* &_0^1 f(x)  dx1 - x^2 = _0^12 f()  d, % &_a^b f(x)  dx(x - a)(b - x) = 2_0^12 f(a^2+ b^2)  d, % &_-a^a fa - xa + x dx = 4a_0^12 f()   d align*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxvi/25", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxvi", "number": 25, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "331", "location": "Exercise LXXVI, problem 25", "problem_latex": "Prove that\n\\[\n\\int_{0}^{1} \\frac{dx}{(1 + x)(2 + x) \\sqrtb{x(1 - x)}}\n = \\pi\\left(\\frac{1}{\\sqrt{2}} - \\frac{1}{\\sqrt{6}}\\right)\\Add{.}\n\\]\n\n%[** Added paragraph break]\n[Put $x = \\sin^{2}\\theta$ and use \\Ex{lxiii}.~8.] \\MathTrip{1912.} %[** TN: Dot added after \"Math\"]", "markdown": "Prove that _0^1 dx(1 + x)(2 + x) x(1 - x) = (12 - 16) %[** Added paragraph break] [Put $x = \\sin^{2}\\theta$ and use % [examples:lxiii]Ex. lxiii%. 8.] % [0]% (*Math. Trip.* 1912.)% [1]% %[** TN: Dot added after \"Math\"]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxvi/3", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxvi", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "331", "location": "Exercise LXXVI, problem 3", "problem_latex": "If $\\phi(x) \\to \\infty$ as $x \\to a + 0$ and $\\phi(x) < K(x - a)^{-s}$, where $s < 1$, then\n$\\ds\\int_{a}^{A} \\phi(x)\\, dx$ is convergent; and if $\\phi(x) > K(x - a)^{-s}$, where $s \\geq 1$, then the\nintegral is divergent. [This is merely a particular case of a general comparison\ntheorem analogous to that stated in~\\SecNo[§]{178}.]", "markdown": "If $\\phi(x) \\to \\infty$ as $x \\to a + 0$ and $\\phi(x) < K(x - a)^{-s}$, where $s < 1$, then $\\ds\\int_{a}^{A} \\phi(x)\\, dx$ is convergent; and if $\\phi(x) > K(x - a)^{-s}$, where $s \\geq 1$, then the integral is divergent. [This is merely a particular case of a general comparison theorem analogous to that stated in [§]178.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxvi/4a", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxvi", "number": 4, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "331", "location": "Exercise LXXVI, problem 4a", "problem_latex": "Are the integrals\n\\begin{gather*}\n\\int_{a}^{A} \\frac{dx}{\\sqrtb{(x - a)(A - x)}},\\quad\n\\int_{a}^{A} \\frac{dx}{(A - x)\\sqrtp[3]{x - a}},\\quad\n\\int_{a}^{A} \\frac{dx}{(A - x)\\sqrtp[3]{A - x}},\\\\\n\\int_{a}^{A} \\frac{dx}{\\sqrtp{x^{2} - a^{2}}},\\quad\n\\int_{a}^{A} \\frac{dx}{\\sqrtp[3]{A^{3} - x^{3}}},\\quad\n\\int_{a}^{A} \\frac{dx}{x^{2} - a^{2}},\\quad\n\\int_{a}^{A} \\frac{dx}{A^{3} - x^{3}}\n\\end{gather*}\nconvergent or divergent?", "markdown": "Are the integrals gather* _a^A dx(x - a)(A - x),0pt minus 3pt_a^A dx(A - x)[3]x - a,0pt minus 3pt_a^A dx(A - x)[3]A - x, _a^A dxx^2 - a^2,0pt minus 3pt_a^A dx[3]A^3 - x^3,0pt minus 3pt_a^A dxx^2 - a^2,0pt minus 3pt_a^A dxA^3 - x^3 gather* convergent or divergent?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxvi/4b", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxvi", "number": 4, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "331", "location": "Exercise LXXVI, problem 4b", "problem_latex": "Are the integrals\n\\begin{gather*}\n\\int_{a}^{A} \\frac{dx}{\\sqrtb{(x - a)(A - x)}},\\quad\n\\int_{a}^{A} \\frac{dx}{(A - x)\\sqrtp[3]{x - a}},\\quad\n\\int_{a}^{A} \\frac{dx}{(A - x)\\sqrtp[3]{A - x}},\\\\\n\\int_{a}^{A} \\frac{dx}{\\sqrtp{x^{2} - a^{2}}},\\quad\n\\int_{a}^{A} \\frac{dx}{\\sqrtp[3]{A^{3} - x^{3}}},\\quad\n\\int_{a}^{A} \\frac{dx}{x^{2} - a^{2}},\\quad\n\\int_{a}^{A} \\frac{dx}{A^{3} - x^{3}}\n\\end{gather*}\nconvergent or divergent?", "markdown": "Are the integrals gather* _a^A dx(x - a)(A - x),0pt minus 3pt_a^A dx(A - x)[3]x - a,0pt minus 3pt_a^A dx(A - x)[3]A - x, _a^A dxx^2 - a^2,0pt minus 3pt_a^A dx[3]A^3 - x^3,0pt minus 3pt_a^A dxx^2 - a^2,0pt minus 3pt_a^A dxA^3 - x^3 gather* convergent or divergent?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxvi/4c", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxvi", "number": 4, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "331", "location": "Exercise LXXVI, problem 4c", "problem_latex": "Are the integrals\n\\begin{gather*}\n\\int_{a}^{A} \\frac{dx}{\\sqrtb{(x - a)(A - x)}},\\quad\n\\int_{a}^{A} \\frac{dx}{(A - x)\\sqrtp[3]{x - a}},\\quad\n\\int_{a}^{A} \\frac{dx}{(A - x)\\sqrtp[3]{A - x}},\\\\\n\\int_{a}^{A} \\frac{dx}{\\sqrtp{x^{2} - a^{2}}},\\quad\n\\int_{a}^{A} \\frac{dx}{\\sqrtp[3]{A^{3} - x^{3}}},\\quad\n\\int_{a}^{A} \\frac{dx}{x^{2} - a^{2}},\\quad\n\\int_{a}^{A} \\frac{dx}{A^{3} - x^{3}}\n\\end{gather*}\nconvergent or divergent?", "markdown": "Are the integrals gather* _a^A dx(x - a)(A - x),0pt minus 3pt_a^A dx(A - x)[3]x - a,0pt minus 3pt_a^A dx(A - x)[3]A - x, _a^A dxx^2 - a^2,0pt minus 3pt_a^A dx[3]A^3 - x^3,0pt minus 3pt_a^A dxx^2 - a^2,0pt minus 3pt_a^A dxA^3 - x^3 gather* convergent or divergent?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxvi/4d", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxvi", "number": 4, "part": "d", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "331", "location": "Exercise LXXVI, problem 4d", "problem_latex": "Are the integrals\n\\begin{gather*}\n\\int_{a}^{A} \\frac{dx}{\\sqrtb{(x - a)(A - x)}},\\quad\n\\int_{a}^{A} \\frac{dx}{(A - x)\\sqrtp[3]{x - a}},\\quad\n\\int_{a}^{A} \\frac{dx}{(A - x)\\sqrtp[3]{A - x}},\\\\\n\\int_{a}^{A} \\frac{dx}{\\sqrtp{x^{2} - a^{2}}},\\quad\n\\int_{a}^{A} \\frac{dx}{\\sqrtp[3]{A^{3} - x^{3}}},\\quad\n\\int_{a}^{A} \\frac{dx}{x^{2} - a^{2}},\\quad\n\\int_{a}^{A} \\frac{dx}{A^{3} - x^{3}}\n\\end{gather*}\nconvergent or divergent?", "markdown": "Are the integrals gather* _a^A dx(x - a)(A - x),0pt minus 3pt_a^A dx(A - x)[3]x - a,0pt minus 3pt_a^A dx(A - x)[3]A - x, _a^A dxx^2 - a^2,0pt minus 3pt_a^A dx[3]A^3 - x^3,0pt minus 3pt_a^A dxx^2 - a^2,0pt minus 3pt_a^A dxA^3 - x^3 gather* convergent or divergent?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxvi/4e", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxvi", "number": 4, "part": "e", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "331", "location": "Exercise LXXVI, problem 4e", "problem_latex": "Are the integrals\n\\begin{gather*}\n\\int_{a}^{A} \\frac{dx}{\\sqrtb{(x - a)(A - x)}},\\quad\n\\int_{a}^{A} \\frac{dx}{(A - x)\\sqrtp[3]{x - a}},\\quad\n\\int_{a}^{A} \\frac{dx}{(A - x)\\sqrtp[3]{A - x}},\\\\\n\\int_{a}^{A} \\frac{dx}{\\sqrtp{x^{2} - a^{2}}},\\quad\n\\int_{a}^{A} \\frac{dx}{\\sqrtp[3]{A^{3} - x^{3}}},\\quad\n\\int_{a}^{A} \\frac{dx}{x^{2} - a^{2}},\\quad\n\\int_{a}^{A} \\frac{dx}{A^{3} - x^{3}}\n\\end{gather*}\nconvergent or divergent?", "markdown": "Are the integrals gather* _a^A dx(x - a)(A - x),0pt minus 3pt_a^A dx(A - x)[3]x - a,0pt minus 3pt_a^A dx(A - x)[3]A - x, _a^A dxx^2 - a^2,0pt minus 3pt_a^A dx[3]A^3 - x^3,0pt minus 3pt_a^A dxx^2 - a^2,0pt minus 3pt_a^A dxA^3 - x^3 gather* convergent or divergent?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxvi/4f", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxvi", "number": 4, "part": "f", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "331", "location": "Exercise LXXVI, problem 4f", "problem_latex": "Are the integrals\n\\begin{gather*}\n\\int_{a}^{A} \\frac{dx}{\\sqrtb{(x - a)(A - x)}},\\quad\n\\int_{a}^{A} \\frac{dx}{(A - x)\\sqrtp[3]{x - a}},\\quad\n\\int_{a}^{A} \\frac{dx}{(A - x)\\sqrtp[3]{A - x}},\\\\\n\\int_{a}^{A} \\frac{dx}{\\sqrtp{x^{2} - a^{2}}},\\quad\n\\int_{a}^{A} \\frac{dx}{\\sqrtp[3]{A^{3} - x^{3}}},\\quad\n\\int_{a}^{A} \\frac{dx}{x^{2} - a^{2}},\\quad\n\\int_{a}^{A} \\frac{dx}{A^{3} - x^{3}}\n\\end{gather*}\nconvergent or divergent?", "markdown": "Are the integrals gather* _a^A dx(x - a)(A - x),0pt minus 3pt_a^A dx(A - x)[3]x - a,0pt minus 3pt_a^A dx(A - x)[3]A - x, _a^A dxx^2 - a^2,0pt minus 3pt_a^A dx[3]A^3 - x^3,0pt minus 3pt_a^A dxx^2 - a^2,0pt minus 3pt_a^A dxA^3 - x^3 gather* convergent or divergent?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxvi/4g", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxvi", "number": 4, "part": "g", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "331", "location": "Exercise LXXVI, problem 4g", "problem_latex": "Are the integrals\n\\begin{gather*}\n\\int_{a}^{A} \\frac{dx}{\\sqrtb{(x - a)(A - x)}},\\quad\n\\int_{a}^{A} \\frac{dx}{(A - x)\\sqrtp[3]{x - a}},\\quad\n\\int_{a}^{A} \\frac{dx}{(A - x)\\sqrtp[3]{A - x}},\\\\\n\\int_{a}^{A} \\frac{dx}{\\sqrtp{x^{2} - a^{2}}},\\quad\n\\int_{a}^{A} \\frac{dx}{\\sqrtp[3]{A^{3} - x^{3}}},\\quad\n\\int_{a}^{A} \\frac{dx}{x^{2} - a^{2}},\\quad\n\\int_{a}^{A} \\frac{dx}{A^{3} - x^{3}}\n\\end{gather*}\nconvergent or divergent?", "markdown": "Are the integrals gather* _a^A dx(x - a)(A - x),0pt minus 3pt_a^A dx(A - x)[3]x - a,0pt minus 3pt_a^A dx(A - x)[3]A - x, _a^A dxx^2 - a^2,0pt minus 3pt_a^A dx[3]A^3 - x^3,0pt minus 3pt_a^A dxx^2 - a^2,0pt minus 3pt_a^A dxA^3 - x^3 gather* convergent or divergent?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxvi/5", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxvi", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "331", "location": "Exercise LXXVI, problem 5", "problem_latex": "The integrals\n\\[\n\\int_{-1}^{1}\\frac{dx}{\\sqrt[3]{x}},\\quad\n\\int_{a-1}^{a+1} \\frac{dx}{\\sqrtp[3]{x - a}}\n\\]\nare convergent, and the value of\neach is zero.", "markdown": "The integrals _-1^1dx[3]x,0pt minus 3pt_a-1^a+1 dx[3]x - a are convergent, and the value of each is zero.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "FLAG-EXTRACTION", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxvi/6", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxvi", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "331", "location": "Exercise LXXVI, problem 6", "problem_latex": "The integral\n\\[\n\\int_{0}^{\\pi} \\frac{dx}{\\sqrtp{\\sin x}}\n\\]\nis convergent. [The subject of integration\ntends to~$\\infty$ as $x$~tends to either limit.]", "markdown": "The integral _0^ dxx is convergent. [The subject of integration tends to $\\infty$ as $x$ tends to either limit.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxvi/7", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxvi", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "331", "location": "Exercise LXXVI, problem 7", "problem_latex": "The integral\n\\[\n\\int_{0}^{\\pi} \\frac{dx}{(\\sin x)^{s}}\n\\]\nis convergent if and only if $s < 1$.", "markdown": "The integral _0^ dx(x)^s is convergent if and only if $s < 1$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxvi/8", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxvi", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "331", "location": "Exercise LXXVI, problem 8", "problem_latex": "The integral\n\\[\n\\int_{0}^{\\frac{1}{2}\\pi} \\frac{x^{s}}{(\\sin x)^{t}}\\, dx\n\\]\nis convergent if $t < s + 1$.", "markdown": "The integral _0^12 x^s(x)^t  dx is convergent if $t < s + 1$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxvi/9", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxvi", "number": 9, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "331", "location": "Exercise LXXVI, problem 9", "problem_latex": "Show that\n\\[\n\\int_{0}^{h} \\frac{\\sin x}{x^{p}}\\, dx,\n\\]\nwhere $h > 0$, is convergent if $p < 2$. Show also\nthat, if $0 < p < 2$, the integrals\n\\[\n\\int_{0}^{\\pi} \\frac{\\sin x}{x^{p}} dx,\\quad\n\\int_{\\pi}^{2\\pi} \\frac{\\sin x}{x^{p}}\\, dx,\\quad\n\\int_{2\\pi}^{3\\pi} \\frac{\\sin x}{x^{p}}\\, dx,\\ \\dots\n\\]\nalternate in sign and steadily decrease in absolute value. [Transform the\nintegral whose limits are $k\\pi$ and~$(k + 1)\\pi$ by the substitution $x = k\\pi + y$.]", "markdown": "Show that _0^h xx^p  dx, where $h > 0$, is convergent if $p < 2$. Show also that, if $0 < p < 2$, the integrals _0^ xx^p dx,0pt minus 3pt_^2 xx^p  dx,0pt minus 3pt_2^3 xx^p  dx, … alternate in sign and steadily decrease in absolute value. [Transform the integral whose limits are $k\\pi$ and $(k + 1)\\pi$ by the substitution $x = k\\pi + y$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxvii/1", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxvii", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "337", "location": "Exercise LXXVII, problem 1", "problem_latex": "Employ the `general principle of convergence'\n(\\SecNo[§]{84}) to prove the theorem that an absolutely convergent series is convergent.\n[Since $\\sum |u_{n}|$ is convergent, we can, when any positive number~$\\DELTA$ is\nassigned, choose~$n_{0}$ so that\n\\[\n|u_{n_{1}+1}| + |u_{n_{1}+2}| + \\dots + |u_{n_{2}}| < \\DELTA\n\\]\nwhen $n_{2} > n_{1} \\geq n_{0}$. \\textit{A~fortiori}\n\\[\n|u_{n_{1}+1} + u_{n_{1}+2} + \\dots + u_{n_{2}}| < \\DELTA,\n\\]\nand therefore $\\sum u_{n}$~is convergent.]", "markdown": "Employ the ‘general principle of convergence’ ([§]84) to prove the theorem that an absolutely convergent series is convergent. [Since $\\sum |u_{n}|$ is convergent, we can, when any positive number $\\DELTA$ is assigned, choose $n_{0}$ so that |u_n_1+1| + |u_n_1+2| + …+ |u_n_2| < when $n_{2} > n_{1} \\geq n_{0}$. *A fortiori* |u_n_1+1 + u_n_1+2 + …+ u_n_2| < , and therefore $\\sum u_{n}$ is convergent.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:general_principle_of_convergence", "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxvii/2", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxvii", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "337", "location": "Exercise LXXVII, problem 2", "problem_latex": "If $\\sum a_{n}$ is a convergent series of positive terms, and $|b_{n}|\\leq Ka_{n}$, then\n$\\sum b_{n}$ is absolutely convergent.", "markdown": "If $\\sum a_{n}$ is a convergent series of positive terms, and $|b_{n}|\\leq Ka_{n}$, then $\\sum b_{n}$ is absolutely convergent.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:comparison_test", "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxvii/3", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxvii", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "337", "location": "Exercise LXXVII, problem 3", "problem_latex": "If $\\sum a_{n}$ is a convergent series of positive terms, then the series $\\sum a_{n}x^{n}$ is\nabsolutely convergent when $-1 \\leq x \\leq 1$.", "markdown": "If $\\sum a_{n}$ is a convergent series of positive terms, then the series $\\sum a_{n}x^{n}$ is absolutely convergent when $-1 \\leq x \\leq 1$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:comparison_test", "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxvii/4", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxvii", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "337", "location": "Exercise LXXVII, problem 4", "problem_latex": "If $\\sum a_{n}$ is a convergent series of positive terms, then the series $\\sum a_{n} \\cos n\\theta$,\n$\\sum a_{n}\\sin n\\theta$ are absolutely convergent for all values of~$\\theta$. [Examples are\nafforded by the series $\\sum r^{n}\\cos n\\theta$, $\\sum r^{n}\\sin n\\theta$ of~\\SecNo[§]{88}.]", "markdown": "If $\\sum a_{n}$ is a convergent series of positive terms, then the series $\\sum a_{n} \\cos n\\theta$, $\\sum a_{n}\\sin n\\theta$ are absolutely convergent for all values of $\\theta$. [Examples are afforded by the series $\\sum r^{n}\\cos n\\theta$, $\\sum r^{n}\\sin n\\theta$ of [§]88.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:comparison_test", "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxvii/5", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxvii", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "337", "location": "Exercise LXXVII, problem 5", "problem_latex": "Any series selected from the terms of an absolutely convergent series\nis absolutely convergent. [For the series of the moduli of its terms is a\nselection from the series of the moduli of the terms of the original series.]", "markdown": "Any series selected from the terms of an absolutely convergent series is absolutely convergent. [For the series of the moduli of its terms is a selection from the series of the moduli of the terms of the original series.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof", "other:subseries" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxvii/6", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxvii", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "337", "location": "Exercise LXXVII, problem 6", "problem_latex": "Prove that if $\\sum |u_{n}|$~is convergent then\n\\[\n|\\tsum u_{n}| \\leq \\tsum |u_{n}|,\n\\]\nand that the only case to which the sign of equality can apply is that in\nwhich every term has the same sign.", "markdown": "Prove that if $\\sum |u_{n}|$ is convergent then |u_n| |u_n|, and that the only case to which the sign of equality can apply is that in which every term has the same sign.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof", "other:triangle_inequality" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxviii/1", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxviii", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "340", "location": "Exercise LXXVIII, problem 1", "problem_latex": "The series\n\\begin{gather*}\n1 - \\frac{1}{2} + \\frac{1}{3} - \\frac{1}{4} + \\dots,\\quad\n1 - \\frac{1}{\\sqrt{2}} + \\frac{1}{\\sqrt{3}} - \\frac{1}{\\sqrt{4}} + \\dots,\\\\\n\\sum \\frac{(-1)^{n}}{(n + a)},\\quad\n\\sum \\frac{(-1)^{n}}{\\sqrtp{n + a}},\\quad\n\\sum \\frac{(-1)^{n}}{(\\sqrt{n} + \\sqrt{a})},\\quad\n\\sum \\frac{(-1)^{n}}{(\\sqrt{n} + \\sqrt{a})^{2}},\n\\end{gather*}\nwhere $a > 0$, are conditionally convergent.", "markdown": "The series gather* 1 - 12 + 13 - 14 + …,0pt minus 3pt1 - 12 + 13 - 14 + …, (-1)^n(n + a),0pt minus 3pt(-1)^nn + a,0pt minus 3pt(-1)^n(n + a),0pt minus 3pt(-1)^n(n + a)^2, gather* where $a > 0$, are conditionally convergent.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxviii/2", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxviii", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "340", "location": "Exercise LXXVIII, problem 2", "problem_latex": "The series $\\sum(-1)^{n}(n + a)^{-s}$, where $a > 0$, is absolutely convergent if\n$s > 1$, conditionally convergent if $0 < s \\leq 1$, and oscillatory if $s \\leq 0$.", "markdown": "The series $\\sum(-1)^{n}(n + a)^{-s}$, where $a > 0$, is absolutely convergent if $s > 1$, conditionally convergent if $0 < s \\leq 1$, and oscillatory if $s \\leq 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxviii/3", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxviii", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "340", "location": "Exercise LXXVIII, problem 3", "problem_latex": "The sum of the series of \\SecNo[§]{188} lies between $s_{n}$ and~$s_{n+1}$ for all values\nof~$n$; and the error committed by taking the sum of the first $n$ terms instead\nof the sum of the whole series is numerically not greater than the modulus of\nthe $(n + 1)$th~term.", "markdown": "The sum of the series of [§]188 lies between $s_{n}$ and $s_{n+1}$ for all values of $n$; and the error committed by taking the sum of the first $n$ terms instead of the sum of the whole series is numerically not greater than the modulus of the $(n + 1)$th term.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxviii/4", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxviii", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "340", "location": "Exercise LXXVIII, problem 4", "problem_latex": "Consider the series\n\\[\n\\sum \\frac{(-1)^{n}}{\\sqrt{n} + (-1)^{n}},\n\\]\nwhich we suppose to begin with the term for which $n = 2$, to avoid any\ndifficulty as to the definitions of the first few terms. This series may be\nwritten in the form\n\\[\n\\sum \\left[\\left\\{\n \\frac{(-1)^{n}}{\\sqrt{n} + (-1)^{n}}\n - \\frac{(-1)^{n}}{\\sqrt{n}}\\right\\}\n + \\frac{(-1)^{n}}{\\sqrt{n}}\\right]\n\\]\nor\n\\[\n\\sum \\left\\{\\frac{(-1)^{n}}{\\sqrt{n}} - \\frac{1}{n + (-1)^{n}\\sqrt{n}}\\right\\}\n = \\sum (\\psi_{n} - \\chi_{n}),\n\\]\nsay. The series $\\sum \\psi_{n}$ is convergent; but $\\sum \\chi_{n}$~is divergent, as all its terms are\npositive, and $\\lim n\\chi_{n} = 1$. Hence the original series is divergent, although it\nis of the form $\\phi_{2} - \\phi_{3} + \\phi_{4} - \\dots$, where $\\phi_{n} \\to 0$. This example shows that the\ncondition that $\\phi_{n}$~should tend \\emph{steadily} to zero is essential to the truth of the\ntheorem. The reader will easily verify that $\\sqrtp{2n + 1} - 1 < \\sqrtp{2n} + 1$, so that\nthis condition is not satisfied.", "markdown": "Consider the series (-1)^nn + (-1)^n, which we suppose to begin with the term for which $n = 2$, to avoid any difficulty as to the definitions of the first few terms. This series may be written in the form [ (-1)^nn + (-1)^n - (-1)^nn + (-1)^nn] or (-1)^nn - 1n + (-1)^nn = (_n - _n), say. The series $\\sum \\psi_{n}$ is convergent; but $\\sum \\chi_{n}$ is divergent, as all its terms are positive, and $\\lim n\\chi_{n} = 1$. Hence the original series is divergent, although it is of the form $\\phi_{2} - \\phi_{3} + \\phi_{4} - \\dots$, where $\\phi_{n} \\to 0$. This example shows that the condition that $\\phi_{n}$ should tend *steadily* to zero is essential to the truth of the theorem. The reader will easily verify that $\\sqrtp{2n + 1} - 1 < \\sqrtp{2n} + 1$, so that this condition is not satisfied.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxviii/5", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxviii", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "340", "location": "Exercise LXXVIII, problem 5", "problem_latex": "If the conditions of \\SecNo[§]{188} are satisfied except that $\\phi_{n}$~tends steadily\nto a positive limit~$l$, then the series $\\sum (-1)^{n}\\phi_{n}$ oscillates finitely.", "markdown": "If the conditions of [§]188 are satisfied except that $\\phi_{n}$ tends steadily to a positive limit $l$, then the series $\\sum (-1)^{n}\\phi_{n}$ oscillates finitely.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxviii/6", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxviii", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "340", "location": "Exercise LXXVIII, problem 6", "problem_latex": "\\Topic{Alteration of the sum of a conditionally convergent series by\nrearrangement of the terms.} Let $s$~be the sum of the series $1 - \\frac{1}{2} + \\frac{1}{3} - \\frac{1}{4} + \\dots$,\nand $s_{2n}$~the sum of its first $2n$ terms, so that $\\lim s_{2n} = s$.\n\nNow consider the series\n\\[\n1 + \\tfrac{1}{3} - \\tfrac{1}{2} + \\tfrac{1}{5} + \\tfrac{1}{7} - \\tfrac{1}{4} + \\dots\n\\Tag{(1)}\n\\]\nin which two positive terms are followed by one negative term, and let $t_{3n}$\ndenote the sum of the first $3n$~terms. Then\n\\begin{align*}\nt_{3n}\n &= 1 + \\frac{1}{3} + \\dots + \\frac{1}{4n-1}\n - \\frac{1}{2} - \\frac{1}{4} - \\dots - \\frac{1}{2n}\\\\\n &= s_{2n} + \\frac{1}{2n + 1} + \\frac{1}{2n + 3} + \\dots + \\frac{1}{4n - 1}.\n\\end{align*}\n\nNow\n\\[\n\\lim \\left[\\frac{1}{2n + 1} - \\frac{1}{2n + 2} + \\frac{1}{2n + 3} - \\dots\n + \\frac{1}{4n - 1} - \\frac{1}{4n}\\right] = 0,\n\\]\n{\\Loosen since the sum of the terms inside the bracket is clearly less than\n$n/(2n + 1)(2n + 2)$; and}\n\\[\n\\lim \\left(\\frac{1}{2n + 2} + \\frac{1}{2n + 4} + \\dots + \\frac{1}{4n}\\right)\n = \\tfrac{1}{2} \\lim \\frac{1}{n} \\sum_{r=1}^{n} \\frac{1}{1 + (r/n)}\n = \\tfrac{1}{2} \\int_{1}^{2} \\frac{dx}{x},\n\\]\nby \\SecNo[§§]{156} and~\\SecNo{158}. Hence\n\\[\n\\lim t_{3n} = s + \\tfrac{1}{2} \\int_{1}^{2} \\frac{dx}{x},\n\\]\n\\PageSep{342}\nand it follows that the sum of the series~\\Eq{(1)} is not~$s$, but the right-hand side of\nthe last equation. Later on we shall give the actual values of the sums of the\ntwo series: see \\SecNo[§]{213} and \\okrickRef{Ch.}{IX}, \\MiscEx{IX}~19.\n\nIt can indeed be proved that a conditionally convergent series can always\nbe so rearranged as to converge to any sum whatever, or to diverge to~$\\infty$ or\nto~$-\\infty$. For a proof we may refer to Bromwich's \\textit{Infinite Series}, p.~68.", "markdown": "**of the sum of a conditionally convergent series by rearrangement of the terms.** Let $s$ be the sum of the series $1 - \\frac{1}{2} + \\frac{1}{3} - \\frac{1}{4} + \\dots$, and $s_{2n}$ the sum of its first $2n$ terms, so that $\\lim s_{2n} = s$. Now consider the series 1 + 13 - 12 + 15 + 17 - 14 + …(1) in which two positive terms are followed by one negative term, and let $t_{3n}$ denote the sum of the first $3n$ terms. Then align* t_3n &= 1 + 13 + …+ 14n-1 - 12 - 14 - …- 12n &= s_2n + 12n + 1 + 12n + 3 + …+ 14n - 1. align* Now [12n + 1 - 12n + 2 + 12n + 3 - … + 14n - 1 - 14n] = 0, 0.375em plus 0.75em minus 0.25emsince the sum of the terms inside the bracket is clearly less than $n/(2n + 1)(2n + 2)$; and (12n + 2 + 12n + 4 + …+ 14n) = 12 1n _r=1^n 11 + (r/n) = 12 _1^2 dxx, by [§§]156 and 158. Hence t_3n = s + 12 _1^2 dxx, [pg]342 and it follows that the sum of the series (1) is not $s$, but the right-hand side of the last equation. Later on we shall give the actual values of the sums of the two series: see [§]213 and Ch.IX, [misc:IX]Misc. Ex. 19. It can indeed be proved that a conditionally convergent series can always be so rearranged as to converge to any sum whatever, or to diverge to $\\infty$ or to $-\\infty$. For a proof we may refer to Bromwich’s *Infinite Series*, p. 68.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxviii/7", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxviii", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "340", "location": "Exercise LXXVIII, problem 7", "problem_latex": "The series\n\\[\n1 + \\frac{1}{\\sqrt{3}} - \\frac{1}{\\sqrt{2}}\n + \\frac{1}{\\sqrt{5}} + \\frac{1}{\\sqrt{7}} - \\frac{1}{\\sqrt{4}} + \\dots\n\\]\ndiverges to~$\\infty$. [Here\n\\[\nt_{3n} = s_{2n} + \\frac{1}{\\sqrtp{2n + 1}} + \\frac{1}{\\sqrtp{2n + 3}} + \\dots\n + \\frac{1}{\\sqrtp{4n - 1}}\n > s_{2n} + \\frac{n}{\\sqrtp{4n - 1}},\n\\]\nwhere $s_{2n} = 1 - \\dfrac{1}{\\sqrt{2}} + \\dots - \\dfrac{1}{\\DPtypo{\\sqrt{2n}}{\\sqrtp{2n}}}$, which tends to a limit as $n \\to \\infty$.]", "markdown": "The series 1 + 13 - 12 + 15 + 17 - 14 + … diverges to $\\infty$. [Here t_3n = s_2n + 12n + 1 + 12n + 3 + … + 14n - 1 > s_2n + n4n - 1, where $s_{2n} = 1 - \\dfrac{1}{\\sqrt{2}} + \\dots - \\dfrac{1}{\\DPtypo{\\sqrt{2n}}{\\sqrtp{2n}}}$, which tends to a limit as $n \\to \\infty$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxx/1", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxx", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "347", "location": "Exercise LXXX, problem 1", "problem_latex": "The series $1 + az + a^{2}z^{2} + \\dots$, where $a > 0$, has a\nradius of convergence equal to~$1/a$. It does not converge anywhere on its\ncircle of convergence, diverging when $z = 1/a$ and oscillating finitely at all other\npoints on the circle.", "markdown": "The series $1 + az + a^{2}z^{2} + \\dots$, where $a > 0$, has a radius of convergence equal to $1/a$. It does not converge anywhere on its circle of convergence, diverging when $z = 1/a$ and oscillating finitely at all other points on the circle.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxx/2", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxx", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "347", "location": "Exercise LXXX, problem 2", "problem_latex": "The series $\\dfrac{z}{1^{2}} + \\dfrac{z^{2}}{2^{2}} + \\dfrac{z^{3}}{3^{2}} + \\dots$ has its radius of convergence equal to~$1$;\nit converges absolutely at all points on its circle of convergence.", "markdown": "The series $\\dfrac{z}{1^{2}} + \\dfrac{z^{2}}{2^{2}} + \\dfrac{z^{3}}{3^{2}} + \\dots$ has its radius of convergence equal to $1$; it converges absolutely at all points on its circle of convergence.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxx/3", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxx", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "347", "location": "Exercise LXXX, problem 3", "problem_latex": "More generally, if $|a_{n+1}|/|a_{n}| \\to \\lambda$, or $|a_{n}|^{1/n} \\to \\lambda$, as $n \\to \\infty$, then the\nseries $a_{0} + a_{1}z + a_{2}z^{2} + \\dots$ has $1/\\lambda$ as its radius of convergence. In the first case\n\\[\n\\lim |a_{n+1}z^{n+1}|/|a_{n}z^{n}| = \\lambda |z|,\n\\]\nwhich is less or greater than unity according as $|z|$~is less or greater than~$1/\\lambda$,\nso that we can use \\DPchg{D'Alembert's}{d'Alembert's} Test (\\SecNo[§]{168},~3). In the second case we\ncan use Cauchy's Test (\\SecNo[§]{168},~2) similarly.", "markdown": "More generally, if $|a_{n+1}|/|a_{n}| \\to \\lambda$, or $|a_{n}|^{1/n} \\to \\lambda$, as $n \\to \\infty$, then the series $a_{0} + a_{1}z + a_{2}z^{2} + \\dots$ has $1/\\lambda$ as its radius of convergence. In the first case |a_n+1z^n+1|/|a_nz^n| = |z|, which is less or greater than unity according as $|z|$ is less or greater than $1/\\lambda$, so that we can use D’Alembert’sd’Alembert’s Test ([§]168, 3). In the second case we can use Cauchy’s Test ([§]168, 2) similarly.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxx/4", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxx", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "347", "location": "Exercise LXXX, problem 4", "problem_latex": "\\Topic{The logarithmic series.} The series\n\\[\nz - \\tfrac{1}{2} z^{2} + \\tfrac{1}{3} z^{3} - \\dots\n\\]\nis called (for reasons which will appear later) the `logarithmic' series. It\nfollows from Ex.~3 that its radius of convergence is unity.\n\nWhen $z$~is on the circle of convergence we may write $z = \\cos\\theta + i\\sin\\theta$,\nand the series assumes the form\n\\[\n \\cos\\theta - \\tfrac{1}{2} \\cos 2\\theta + \\tfrac{1}{3} \\cos 3\\theta - \\dots\n+ i(\\sin\\theta - \\tfrac{1}{2} \\sin 2\\theta + \\tfrac{1}{3} \\sin 3\\theta - \\dots).\n\\]\n\nThe real and imaginary parts are both convergent, though not absolutely\nconvergent, unless $\\theta$~is an odd multiple of~$\\pi$ (\\Exs{lxxix}.~3,~4). If $\\theta$~is an odd\nmultiple of~$\\pi$ then $z = -1$, and the series assumes the form $-1 - \\frac{1}{2} - \\frac{1}{3} - \\dots$,\nand so diverges to~$-\\infty$. Thus the logarithmic series converges at all points\nof its circle of convergence except the point $z = -1$.", "markdown": "**logarithmic series.** The series z - 12 z^2 + 13 z^3 - … is called (for reasons which will appear later) the ‘logarithmic’ series. It follows from Ex. 3 that its radius of convergence is unity. When $z$ is on the circle of convergence we may write $z = \\cos\\theta + i\\sin\\theta$, and the series assumes the form - 12 2+ 13 3- …+ i(- 12 2+ 13 3- …). The real and imaginary parts are both convergent, though not absolutely convergent, unless $\\theta$ is an odd multiple of $\\pi$ (lxxix. 3, 4). If $\\theta$ is an odd multiple of $\\pi$ then $z = -1$, and the series assumes the form $-1 - \\frac{1}{2} - \\frac{1}{3} - \\dots$, and so diverges to $-\\infty$. Thus the logarithmic series converges at all points of its circle of convergence except the point $z = -1$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxx/5", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxx", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "347", "location": "Exercise LXXX, problem 5", "problem_latex": "\\Topic{The binomial series.} Consider the series\n\\[\n1 + mz + \\frac{m(m - 1)}{2!} z^{2} + \\frac{m(m - 1)(m - 2)}{3!} z^{3} + \\dots\\Add{.}\n\\]\nIf $m$~is a positive integer then the series terminates. In general\n\\[\n\\frac{|a_{n+1}|}{|a_{n}|} = \\frac{|m - n|}{n + 1} \\to 1,\n\\]\nso that the radius of convergence is unity. We shall not discuss here the\nquestion of its convergence on the circle, which is a little more difficult.\\footnote\n {See Bromwich, \\textit{Infinite Series}, pp.~225 \\textit{et~seq.}; Hobson, \\textit{Plane Trigonometry}\n (3rd~edition), pp.~268~\\textit{et~seq.}}", "markdown": "**binomial series.** Consider the series 1 + mz + m(m - 1)2! z^2 + m(m - 1)(m - 2)3! z^3 + … If $m$ is a positive integer then the series terminates. In general |a_n+1||a_n| = |m - n|n + 1 1, so that the radius of convergence is unity. We shall not discuss here the question of its convergence on the circle, which is a little more difficult. See Bromwich, *Infinite Series*, pp. 225 *et seq.*; Hobson, *Plane Trigonometry* (3rd edition), pp. 268 *et seq.*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxi/1", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxi", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "349", "location": "Exercise LXXXI, problem 1", "problem_latex": "If $|z|$~is less than the radius of convergence\nof either of the series $\\sum a_{n}z^{n}$, $\\sum b_{n}z^{n}$, then the product of the two series is\n$\\sum c_{n}z^{n}$, where $c_{n} = a_{0}b_{n} + a_{1}b_{n-1} + \\dots + a_{n}b_{0}$.", "markdown": "If $|z|$ is less than the radius of convergence of either of the series $\\sum a_{n}z^{n}$, $\\sum b_{n}z^{n}$, then the product of the two series is $\\sum c_{n}z^{n}$, where $c_{n} = a_{0}b_{n} + a_{1}b_{n-1} + \\dots + a_{n}b_{0}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxi/2", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxi", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "349", "location": "Exercise LXXXI, problem 2", "problem_latex": "{\\Loosen If the radius of convergence of $\\sum a_{n}z^{n}$ is~$R$, and $f(z)$~is the sum of\nthe series when $|z| < R$, and $|z|$~is less than either $R$ or unity, then\n$f(z)/(1 - z) = \\sum s_{n}z^{n}$, where $s_{n} = a_{0} + a_{1} + \\dots + a_{n}$.}", "markdown": "0.375em plus 0.75em minus 0.25emIf the radius of convergence of $\\sum a_{n}z^{n}$ is $R$, and $f(z)$ is the sum of the series when $|z| < R$, and $|z|$ is less than either $R$ or unity, then $f(z)/(1 - z) = \\sum s_{n}z^{n}$, where $s_{n} = a_{0} + a_{1} + \\dots + a_{n}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxi/3", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxi", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "349", "location": "Exercise LXXXI, problem 3", "problem_latex": "Prove, by squaring the series for $1/(1 - z)$, that $1/(1 - z)^{2} = 1 + 2z + 3z^{2} + \\dots$\nif $|z| < 1$.", "markdown": "Prove, by squaring the series for $1/(1 - z)$, that $1/(1 - z)^{2} = 1 + 2z + 3z^{2} + \\dots$ if $|z| < 1$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxi/4", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxi", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "349", "location": "Exercise LXXXI, problem 4", "problem_latex": "Prove similarly that $1/(1 - z)^{3} = 1 + 3z + 6z^{2} + \\dots$, the general term\nbeing $\\frac{1}{2}(n + 1)(n + 2)z^{n}$.", "markdown": "Prove similarly that $1/(1 - z)^{3} = 1 + 3z + 6z^{2} + \\dots$, the general term being $\\frac{1}{2}(n + 1)(n + 2)z^{n}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxi/5", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxi", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "349", "location": "Exercise LXXXI, problem 5", "problem_latex": "\\Topic{The Binomial Theorem for a negative integral exponent.} If\n$|z| < 1$, and $m$~is a positive integer, then\n\\[\n\\frac{1}{(1 - z)^{m}}\n = 1 + mz + \\frac{m(m + 1)}{1·2} z^{2} + \\dots\n + \\frac{m(m + 1) \\dots (m + n - 1)}{1·2 \\dots n} z^{n} + \\dots.\n\\]\n\n[Assume the truth of the theorem for all indices up to~$m$. Then, by Ex.~2,\n$1/(1 - z)^{m+1} = \\sum s_{n}z^{n}$, where\n\\begin{align*}\n%[** TN: Set on a single line in the original]\ns_{n}\n &= 1 + m + \\frac{m(m + 1)}{1·2} + \\dots\n + \\frac{m(m + 1) \\dots (m + n - 1)}{1·2 \\dots n} \\\\\n &= \\frac{(m + 1)(m + 2) \\dots (m + n)}{1·2 \\dots n},\n\\end{align*}\nas is easily proved by induction.]", "markdown": "**Binomial Theorem for a negative integral exponent.** If $|z| < 1$, and $m$ is a positive integer, then 1(1 - z)^m = 1 + mz + m(m + 1)1·2 z^2 + … + m(m + 1) …(m + n - 1)1·2 …n z^n + …. [Assume the truth of the theorem for all indices up to $m$. Then, by Ex. 2, $1/(1 - z)^{m+1} = \\sum s_{n}z^{n}$, where align* %[** TN: Set on a single line in the original] s_n &= 1 + m + m(m + 1)1·2 + … + m(m + 1) …(m + n - 1)1·2 …n &= (m + 1)(m + 2) …(m + n)1·2 …n, align* as is easily proved by induction.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxi/6", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxi", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "349", "location": "Exercise LXXXI, problem 6", "problem_latex": "Prove by multiplication of series that if\n\\[\nf(m, z) = 1 + \\binom{m}{1} z + \\binom{m}{2} z^{2} + \\dots,\n\\]\nand $|z| < 1$, then $f(m, z)f(m', z) = f(m + m', z)$. [This equation forms the basis of\nEuler's proof of the Binomial Theorem. The coefficient of~$z^{n}$ in the product\nseries is\n\\[\n\\binom{m'}{n}\n + \\binom{m}{1} \\binom{m'}{n - 1}\n + \\binom{m}{2} \\binom{m'}{n - 2} + \\dots\n + \\binom{m}{n - 1} \\binom{m'}{1}\n + \\binom{m}{n}.\n\\]\n\\PageSep{350}\n\nThis is a polynomial in $m$~and~$m'$: but when $m$~and~$m'$ are positive\nintegers this polynomial must reduce to $\\dbinom{m + m'}{k}$ in virtue of the Binomial\nTheorem for a positive integral exponent, and if two such polynomials are\nequal for all positive integral values of $m$~and~$m'$ then they must be equal\nidentically.]", "markdown": "Prove by multiplication of series that if f(m, z) = 1 + m1 z + m2 z^2 + …, and $|z| < 1$, then $f(m, z)f(m', z) = f(m + m', z)$. [This equation forms the basis of Euler’s proof of the Binomial Theorem. The coefficient of $z^{n}$ in the product series is m’n + m1 m’n - 1 + m2 m’n - 2 + … + mn - 1 m’1 + mn. [pg]350 This is a polynomial in $m$ and $m'$: but when $m$ and $m'$ are positive integers this polynomial must reduce to $\\dbinom{m + m'}{k}$ in virtue of the Binomial Theorem for a positive integral exponent, and if two such polynomials are equal for all positive integral values of $m$ and $m'$ then they must be equal identically.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxi/7", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxi", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "349", "location": "Exercise LXXXI, problem 7", "problem_latex": "If $f(z) = 1 + z + \\dfrac{z^{2}}{2!} + \\dots$ then $f(z)f(z') = f(z + z')$. [For the series for~$f(z)$\nis absolutely convergent for all values of~$z$: and it is easy to see that if\n$u_{n} = \\dfrac{z^{n}}{n!}$, $v_{n} = \\dfrac{z'^{n}}{n!}$, then $w_{n} = \\dfrac{(z + z')^{n}}{n!}$.]", "markdown": "If $f(z) = 1 + z + \\dfrac{z^{2}}{2!} + \\dots$ then $f(z)f(z') = f(z + z')$. [For the series for $f(z)$ is absolutely convergent for all values of $z$: and it is easy to see that if $u_{n} = \\dfrac{z^{n}}{n!}$, $v_{n} = \\dfrac{z'^{n}}{n!}$, then $w_{n} = \\dfrac{(z + z')^{n}}{n!}$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxi/8", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxi", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "349", "location": "Exercise LXXXI, problem 8", "problem_latex": "If\n\\[\nC(z) = 1 - \\frac{z^{2}}{2!} + \\frac{z^{4}}{4!} - \\dots,\\quad\nS(z) = z - \\frac{z^{3}}{3!} + \\frac{z^{5}}{5!} - \\dots,\n\\]\nthen\n\\[\nC(z + z') = C(z)C(z') - S(z)S(z'),\\quad\nS(z + z') = S(z)C(z') + C(z)S(z'),\n\\]\nand\n\\[\n\\{C(z)\\}^{2} + \\{S(z)\\}^{2} = 1.\n\\]", "markdown": "If C(z) = 1 - z^22! + z^44! - …,0pt minus 3ptS(z) = z - z^33! + z^55! - …, then C(z + z’) = C(z)C(z’) - S(z)S(z’),0pt minus 3ptS(z + z’) = S(z)C(z’) + C(z)S(z’), and C(z)^2 + S(z)^2 = 1.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxi/9", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxi", "number": 9, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "349", "location": "Exercise LXXXI, problem 9", "problem_latex": "\\Topic{Failure of the Multiplication Theorem.} That the theorem is not\nalways true when $\\sum u_{n}$ and $\\sum v_{n}$ are not \\emph{absolutely} convergent may be seen by\nconsidering the case in which\n\\[\nu_{n} = v_{n} = \\frac{(-1)^{n}}{\\sqrtp{n + 1}}.\n\\]\nThen\n\\[\nw_{n} = (-1)^{n} \\sum_{r=0}^{n} \\frac{1}{\\sqrtb{(r + 1)(n + 1 - r)}}.\n\\]\nBut $\\sqrtb{(r + 1)(n + 1 - r)} \\leq \\frac{1}{2}(n + 2)$, and so $|w_{n}| > (2n + 2)/(n + 2)$, which tends\nto~$2$; so that $\\sum w_{n}$~is certainly not convergent.", "markdown": "**of the Multiplication Theorem.** That the theorem is not always true when $\\sum u_{n}$ and $\\sum v_{n}$ are not *absolutely* convergent may be seen by considering the case in which u_n = v_n = (-1)^nn + 1. Then w_n = (-1)^n _r=0^n 1(r + 1)(n + 1 - r). But $\\sqrtb{(r + 1)(n + 1 - r)} \\leq \\frac{1}{2}(n + 2)$, and so $|w_{n}| > (2n + 2)/(n + 2)$, which tends to $2$; so that $\\sum w_{n}$ is certainly not convergent.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxii/1", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxii", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "359", "location": "Exercise LXXXII, problem 1", "problem_latex": "Prove from the definition that if $u > 0$ then\n\\[\nu/(1 + u) < \\log(1 + u) < u.\n\\]", "markdown": "Prove from the definition that if $u > 0$ then u/(1 + u) < (1 + u) < u.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxii/2", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxii", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "359", "location": "Exercise LXXXII, problem 2", "problem_latex": "Prove that $\\log(1 + u)$ lies between $u - \\dfrac{u^{2}}{2}$ and $u - \\dfrac{u^{2}}{2(1 + u)}$ when $u$~is\npositive.", "markdown": "Prove that $\\log(1 + u)$ lies between $u - \\dfrac{u^{2}}{2}$ and $u - \\dfrac{u^{2}}{2(1 + u)}$ when $u$ is positive.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxii/3", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxii", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "359", "location": "Exercise LXXXII, problem 3", "problem_latex": "If $0 < u < 1$ then $u < -\\log(1 - u) < u/(1 - u)$.", "markdown": "If $0 < u < 1$ then $u < -\\log(1 - u) < u/(1 - u)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxii/4", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxii", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "359", "location": "Exercise LXXXII, problem 4", "problem_latex": "Prove that\n\\[\n\\lim_{x\\to 1} \\frac{\\log x}{x - 1} = \\lim_{t\\to 0} \\frac{\\log (1 + t)}{t} = 1.\n\\]", "markdown": "Prove that _x1 xx - 1 = _t0 (1 + t)t = 1.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxiii/1", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxiii", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "360", "location": "Exercise LXXXIII, problem 1", "problem_latex": "It can be shown that there is no solution of\nthe equation~\\Eq{(1)} which possesses a differential coefficient and is fundamentally\ndistinct from $\\log x$. For when we differentiate the functional equation, first\nwith respect to~$x$ and then with respect to~$y$, we obtain the two equations\n\\[\nyf'(xy) = f'(x),\\quad\nxf'(xy) = f'(y);\n\\]\nand so, eliminating $f'(xy)$, $xf'(x) = yf'(y)$. But if this is true for every pair\nof values of $x$~and~$y$, then we must have $xf'(x) = C$, or $f'(x) = C/x$, where $C$~is\na constant. Hence\n\\[\nf(x) = \\int \\frac{C}{x}\\, dx + C' = C\\log x + C',\n\\]\nand it is easy to see that $C' = 0$. Thus there is no solution fundamentally\ndistinct from~$\\log x$, except the trivial solution $f(x) = 0$, obtained by taking\n$C = 0$.", "markdown": "It can be shown that there is no solution of the equation (1) which possesses a differential coefficient and is fundamentally distinct from $\\log x$. For when we differentiate the functional equation, first with respect to $x$ and then with respect to $y$, we obtain the two equations yf’(xy) = f’(x),0pt minus 3ptxf’(xy) = f’(y); and so, eliminating $f'(xy)$, $xf'(x) = yf'(y)$. But if this is true for every pair of values of $x$ and $y$, then we must have $xf'(x) = C$, or $f'(x) = C/x$, where $C$ is a constant. Hence f(x) = Cx  dx + C’ = Cx + C’, and it is easy to see that $C' = 0$. Thus there is no solution fundamentally distinct from $\\log x$, except the trivial solution $f(x) = 0$, obtained by taking $C = 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxiii/2", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxiii", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "360", "location": "Exercise LXXXIII, problem 2", "problem_latex": "Show in the same way that there is no solution of the equation\n\\[\nf(x) + f(y) = f\\left(\\frac{x + y}{1 - xy}\\right)\n\\]\nwhich possesses a differential coefficient and is fundamentally distinct from\n$\\arctan x$.", "markdown": "Show in the same way that there is no solution of the equation f(x) + f(y) = f(x + y1 - xy) which possesses a differential coefficient and is fundamentally distinct from $\\arctan x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxiv/1", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxiv", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "362", "location": "Exercise LXXXIV, problem 1", "problem_latex": "Between any two terms $f(x)$,~$F(x)$ of the series\nwe can insert a new term~$\\phi(x)$ such that $\\phi(x)$~tends to~$\\infty$ more slowly than~$f(x)$\nand more rapidly than~$F(x)$. [Thus between $\\sqrt{x}$ and~$\\sqrt[3]{x}$ we could insert~$x^{5/12}$:\nbetween $\\sqrtp{\\log x}$ and~$\\sqrtp[3]{\\log x}$ we could insert $(\\log x)^{5/12}$. And, generally,\n$\\phi(x) = \\sqrtb{f(x) F(x)}$ satisfies the conditions stated.]", "markdown": "Between any two terms $f(x)$, $F(x)$ of the series we can insert a new term $\\phi(x)$ such that $\\phi(x)$ tends to $\\infty$ more slowly than $f(x)$ and more rapidly than $F(x)$. [Thus between $\\sqrt{x}$ and $\\sqrt[3]{x}$ we could insert $x^{5/12}$: between $\\sqrtp{\\log x}$ and $\\sqrtp[3]{\\log x}$ we could insert $(\\log x)^{5/12}$. And, generally, $\\phi(x) = \\sqrtb{f(x) F(x)}$ satisfies the conditions stated.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxiv/2", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxiv", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "362", "location": "Exercise LXXXIV, problem 2", "problem_latex": "Find a function which tends to~$\\infty$ more slowly than~$\\sqrt{x}$, but more\nrapidly than~$x^{\\alpha}$, where $\\alpha$~is any rational number less than~$1/2$. [$\\sqrt{x}/(\\log x)$~is\nsuch a function; or $\\sqrt{x}/(\\log x)^{\\beta}$, where $\\beta$~is any positive rational number.]", "markdown": "Find a function which tends to $\\infty$ more slowly than $\\sqrt{x}$, but more rapidly than $x^{\\alpha}$, where $\\alpha$ is any rational number less than $1/2$. [$\\sqrt{x}/(\\log x)$ is such a function; or $\\sqrt{x}/(\\log x)^{\\beta}$, where $\\beta$ is any positive rational number.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxiv/3", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxiv", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "362", "location": "Exercise LXXXIV, problem 3", "problem_latex": "Find a function which tends to~$\\infty$ more slowly than~$\\sqrt{x}$, but more\nrapidly than~$\\sqrt{x}/(\\log x)^{\\alpha}$, where $\\alpha$~is any rational number. [The function\n$\\sqrt{x}/(\\log\\log x)$ is such a function. It will be gathered from these examples that\n\\emph{incompleteness} is an inherent characteristic of the logarithmic scale of infinity.]", "markdown": "Find a function which tends to $\\infty$ more slowly than $\\sqrt{x}$, but more rapidly than $\\sqrt{x}/(\\log x)^{\\alpha}$, where $\\alpha$ is any rational number. [The function $\\sqrt{x}/(\\log\\log x)$ is such a function. It will be gathered from these examples that *incompleteness* is an inherent characteristic of the logarithmic scale of infinity.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxiv/4", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxiv", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "362", "location": "Exercise LXXXIV, problem 4", "problem_latex": "How does the function\n\\[\nf(x) = \\{x^{\\alpha} (\\log x)^{\\alpha'} (\\log\\log x)^{\\alpha''}\\}/\n \\{x^{\\beta} (\\log x)^{\\beta'} (\\log\\log x)^{\\beta''}\\}\n\\]\nbehave as $x$~tends to~$\\infty$? [If $\\alpha \\neq \\beta$ then the behaviour of\n\\[\nf(x) = x^{\\alpha-\\beta} (\\log x)^{\\alpha'-\\beta'} (\\log\\log x)^{\\alpha''-\\beta''}\n\\]\n\\PageSep{363}\nis dominated by that of~$x^{\\alpha-\\beta}$. If $\\alpha = \\beta$ then the power of~$x$ disappears and\nthe behaviour of~$f(x)$ is dominated by that of $(\\log x)^{\\alpha'-\\beta'}$, unless $\\alpha' = \\beta'$, when\nit is dominated by that of $(\\log\\log x)^{\\alpha''-\\beta''}$. Thus $f(x) \\to \\infty$ if $\\alpha > \\beta$, or\n$\\alpha = \\beta$, $\\alpha' > \\beta'$, or $\\alpha = \\beta$, $\\alpha' = \\beta'$, $\\alpha'' > \\beta''$, and $f(x) \\to 0$ if $\\alpha < \\beta$, or $\\alpha = \\beta$, $\\alpha' < \\beta'$, or\n$\\alpha = \\beta$, $\\alpha' = \\beta'$, $\\alpha'' < \\beta''$.]", "markdown": "How does the function f(x) = x^ (x)^’ (x)^”/ x^ (x)^’ (x)^” behave as $x$ tends to $\\infty$? [If $\\alpha \\neq \\beta$ then the behaviour of f(x) = x^- (x)^’-’ (x)^”-” [pg]363 is dominated by that of $x^{\\alpha-\\beta}$. If $\\alpha = \\beta$ then the power of $x$ disappears and the behaviour of $f(x)$ is dominated by that of $(\\log x)^{\\alpha'-\\beta'}$, unless $\\alpha' = \\beta'$, when it is dominated by that of $(\\log\\log x)^{\\alpha''-\\beta''}$. Thus $f(x) \\to \\infty$ if $\\alpha > \\beta$, or $\\alpha = \\beta$, $\\alpha' > \\beta'$, or $\\alpha = \\beta$, $\\alpha' = \\beta'$, $\\alpha'' > \\beta''$, and $f(x) \\to 0$ if $\\alpha < \\beta$, or $\\alpha = \\beta$, $\\alpha' < \\beta'$, or $\\alpha = \\beta$, $\\alpha' = \\beta'$, $\\alpha'' < \\beta''$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxiv/5", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxiv", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "362", "location": "Exercise LXXXIV, problem 5", "problem_latex": "Arrange the functions $x/\\sqrtp{\\log x}$, $x\\sqrtp{\\log x}/\\log\\log x$, $x\\log\\log x/\\sqrtp{\\log x}$,\n$(x\\log\\log\\log x)/\\sqrtp{\\log\\log x}$ according to the rapidity with which they tend\nto infinity as $x \\to \\infty$.", "markdown": "Arrange the functions $x/\\sqrtp{\\log x}$, $x\\sqrtp{\\log x}/\\log\\log x$, $x\\log\\log x/\\sqrtp{\\log x}$, $(x\\log\\log\\log x)/\\sqrtp{\\log\\log x}$ according to the rapidity with which they tend to infinity as $x \\to \\infty$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxiv/6", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxiv", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "362", "location": "Exercise LXXXIV, problem 6", "problem_latex": "Arrange\n\\[\n\\log\\log x/(x\\log x),\\quad\n(\\log x)/x,\\quad\nx\\log\\log x/\\sqrtp{x^{2} + 1},\\quad\n\\{\\sqrtp{x + 1}\\}/x(\\log x)^{2}\n\\]\naccording to the rapidity with which they tend to zero as $x \\to \\infty$.", "markdown": "Arrange x/(xx),0pt minus 3pt(x)/x,0pt minus 3ptxx/x^2 + 1,0pt minus 3ptx + 1/x(x)^2 according to the rapidity with which they tend to zero as $x \\to \\infty$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxiv/7", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxiv", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "362", "location": "Exercise LXXXIV, problem 7", "problem_latex": "Arrange\n\\[\nx\\log\\log(1/x),\\quad\n\\sqrt{x}/\\{\\log(1/x)\\},\\quad\n\\sqrtb{x\\sin x\\log(1/x)},\\quad\n(1 - \\cos x)\\log(1/x)\n\\]\naccording to the rapidity with which they tend to zero as $x \\to +0$.", "markdown": "Arrange x(1/x),0pt minus 3ptx/(1/x),0pt minus 3ptxx(1/x),0pt minus 3pt(1 - x)(1/x) according to the rapidity with which they tend to zero as $x \\to +0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxiv/8a", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxiv", "number": 8, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "362", "location": "Exercise LXXXIV, problem 8a", "problem_latex": "Show that\n\\[\nD_{x}\\log\\log x = 1/(x\\log x),\\quad\nD_{x}\\log\\log\\log x = 1/(x\\log x\\log\\log x),\n\\]\nand so on.", "markdown": "Show that D_xx = 1/(xx),0pt minus 3ptD_xx = 1/(xxx), and so on.", "answer_latex": [ "1/(x\\log x)" ], "answer_markdown": [ "1/(xx)" ], "checks": [ { "task": "differentiate", "verdict": "PASS", "judge_why": null, "problem_expr": "log(log(x))", "answer_expr": "1/(x*log(x))" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "differentiate: log(log(x))" ], "shape": [ "differentiate: log(log(x))" ], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxiv/8b", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxiv", "number": 8, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "362", "location": "Exercise LXXXIV, problem 8b", "problem_latex": "Show that\n\\[\nD_{x}\\log\\log x = 1/(x\\log x),\\quad\nD_{x}\\log\\log\\log x = 1/(x\\log x\\log\\log x),\n\\]\nand so on.", "markdown": "Show that D_xx = 1/(xx),0pt minus 3ptD_xx = 1/(xxx), and so on.", "answer_latex": [ "1/(x\\log x\\log\\log x)" ], "answer_markdown": [ "1/(xxx)" ], "checks": [ { "task": "differentiate", "verdict": "PASS", "judge_why": null, "problem_expr": "log(log(log(x)))", "answer_expr": "1/(x*log(x)*log(log(x)))" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "differentiate: log(log(log(x)))" ], "shape": [ "differentiate: log(log(log(x)))" ], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxiv/9a", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxiv", "number": 9, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "362", "location": "Exercise LXXXIV, problem 9a", "problem_latex": "Show that\n\\[\nD_{x}(\\log x)^{\\alpha} = \\alpha/\\{x(\\log x)^{1-\\alpha}\\},\\quad\nD_{x}(\\log\\log x)^{\\alpha} = \\alpha/\\{x\\log x(\\log\\log x)^{1-\\alpha}\\},\n\\]\nand so on.", "markdown": "Show that D_x(x)^ = /x(x)^1-,0pt minus 3ptD_x(x)^ = /xx(x)^1-, and so on.", "answer_latex": [ "\\alpha/\\{x(\\log x)^{1-\\alpha}\\}" ], "answer_markdown": [ "/x(x)^1-" ], "checks": [ { "task": "differentiate", "verdict": "PASS", "judge_why": null, "problem_expr": "log(x)**alpha", "answer_expr": "alpha/(x*log(x)**(1 - alpha))" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "differentiate: log(x)**a" ], "shape": [ "differentiate: log(x)**a" ], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxiv/9b", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxiv", "number": 9, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "362", "location": "Exercise LXXXIV, problem 9b", "problem_latex": "Show that\n\\[\nD_{x}(\\log x)^{\\alpha} = \\alpha/\\{x(\\log x)^{1-\\alpha}\\},\\quad\nD_{x}(\\log\\log x)^{\\alpha} = \\alpha/\\{x\\log x(\\log\\log x)^{1-\\alpha}\\},\n\\]\nand so on.", "markdown": "Show that D_x(x)^ = /x(x)^1-,0pt minus 3ptD_x(x)^ = /xx(x)^1-, and so on.", "answer_latex": [ "\\alpha/\\{x\\log x(\\log\\log x)^{1-\\alpha}\\}" ], "answer_markdown": [ "/xx(x)^1-" ], "checks": [ { "task": "differentiate", "verdict": "PASS", "judge_why": null, "problem_expr": "log(log(x))**alpha", "answer_expr": "alpha/(x*log(x)*log(log(x))**(1 - alpha))" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "differentiate: log(log(x))**a" ], "shape": [ "differentiate: log(log(x))**a" ], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxix/1", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxix", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "377", "location": "Exercise LXXXIX, problem 1", "problem_latex": "\\Topic{Euler's limit.} Show that\n\\[\n\\phi(n) = 1 + \\frac{1}{2} + \\frac{1}{3} + \\dots + \\frac{1}{n - 1} - \\log n\n\\]\ntends to a limit~$\\gamma$ as $n \\to \\infty$, and that $0 < \\gamma \\leq 1$. [This follows at once from\n\\SecNo[§]{174}. The value of~$\\gamma$ is in fact~$.577\\dots$, and $\\gamma$~is usually called \\Emph{Euler's\nconstant}.]", "markdown": "**’s limit.** Show that (n) = 1 + 12 + 13 + …+ 1n - 1 - n tends to a limit $\\gamma$ as $n \\to \\infty$, and that $0 < \\gamma \\leq 1$. [This follows at once from [§]174. The value of $\\gamma$ is in fact $.577\\dots$, and $\\gamma$ is usually called **’s constant**.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxix/2", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxix", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "377", "location": "Exercise LXXXIX, problem 2", "problem_latex": "If $a$ and~$b$ are positive then\n\\[\n\\frac{1}{a} + \\frac{1}{a + b} + \\frac{1}{a + 2b} + \\dots\n + \\frac{1}{a + (n - 1) b} - \\frac{1}{b}\\log \\DPtypo{(a + nb}{(a + nb)}\n\\]\ntends to a limit as $n \\to \\infty$.", "markdown": "If $a$ and $b$ are positive then 1a + 1a + b + 1a + 2b + … + 1a + (n - 1) b - 1b(a + nb tends to a limit as $n \\to \\infty$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxix/3", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxix", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "377", "location": "Exercise LXXXIX, problem 3", "problem_latex": "If $0 < s < 1$ then\n\\[\n\\phi(n) = 1 + 2^{-s} + 3^{-s} + \\dots + (n - 1)^{-s} - \\frac{n^{1-s}}{1 - s}\n\\]\ntends to a limit as $n \\to \\infty$.", "markdown": "If $0 < s < 1$ then (n) = 1 + 2^-s + 3^-s + …+ (n - 1)^-s - n^1-s1 - s tends to a limit as $n \\to \\infty$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxix/4", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxix", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "377", "location": "Exercise LXXXIX, problem 4", "problem_latex": "Show that the series\n\\[\n\\frac{1}{1}\n + \\frac{1}{2(1 + \\frac{1}{2})}\n + \\frac{1}{3(1 + \\frac{1}{2} + \\frac{1}{3})} + \\dots\n\\]\nis divergent. [Compare the general term of the series with $1/(n\\log n)$.]\nShow also that the series derived from $\\sum n^{-s}$, in the same way that the above\nseries is derived from~$\\sum (1/n)$, is convergent if $s > 1$ and otherwise divergent.", "markdown": "Show that the series 11 + 12(1 + 12) + 13(1 + 12 + 13) + … is divergent. [Compare the general term of the series with $1/(n\\log n)$.] Show also that the series derived from $\\sum n^{-s}$, in the same way that the above series is derived from $\\sum (1/n)$, is convergent if $s > 1$ and otherwise divergent.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "cas.sum" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxix/5", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxix", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "377", "location": "Exercise LXXXIX, problem 5", "problem_latex": "Prove generally that if $\\sum u_{n}$~is a series of positive terms, and\n\\[\ns_{n} = u_{1} + u_{2} + \\dots + u_{n},\n\\]\nthen $\\sum (u_{n}/s_{n-1})$~is convergent or divergent according as $\\sum u_{n}$~is convergent or\n\\PageSep{378}\ndivergent. [If $\\sum u_{n}$~is convergent then $s_{n-1}$~tends to a positive limit~$l$, and so\n$\\sum (u_{n}/s_{n-1})$~is convergent. If $\\sum u_{n}$~is divergent then $s_{n-1} \\to \\infty$, and\n\\[\nu_{n}/s_{n-1} > \\log\\{1 + (u_{n}/s_{n-1})\\} = \\log (s_{n}/s_{n-1})\n\\]\n(\\Ex{lxxxii}.~1); and it is evident that\n\\[\n\\log(s_{2}/s_{1}) + \\log(s_{3}/s_{2}) + \\dots + \\log(s_{n}/s_{n-1})\n = \\log(s_{n}/s_{1})\n\\]\ntends to~$\\infty$ as $n \\to \\infty$.]", "markdown": "Prove generally that if $\\sum u_{n}$ is a series of positive terms, and s_n = u_1 + u_2 + …+ u_n, then $\\sum (u_{n}/s_{n-1})$ is convergent or divergent according as $\\sum u_{n}$ is convergent or [pg]378 divergent. [If $\\sum u_{n}$ is convergent then $s_{n-1}$ tends to a positive limit $l$, and so $\\sum (u_{n}/s_{n-1})$ is convergent. If $\\sum u_{n}$ is divergent then $s_{n-1} \\to \\infty$, and u_n/s_n-1 > 1 + (u_n/s_n-1) = (s_n/s_n-1) (% [examples:lxxxii]Ex. lxxxii%. 1); and it is evident that (s_2/s_1) + (s_3/s_2) + …+ (s_n/s_n-1) = (s_n/s_1) tends to $\\infty$ as $n \\to \\infty$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "cas.sum" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxix/6", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxix", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "377", "location": "Exercise LXXXIX, problem 6", "problem_latex": "Prove that the same result holds for the series $\\sum (u_{n}/s_{n})$. [The proof\nis the same in the case of convergence. If $\\sum u_{n}$~is divergent, and $u_{n} < s_{n-1}$\nfrom a certain value of~$n$ onwards, then $s_{n} < 2s_{n-1}$, and the divergence of\n$\\sum (u_{n}/s_{n})$ follows from that of $\\sum (u_{n}/s_{n-1})$. If on the other hand $u_{n} \\geq s_{n-1}$ for\nan infinity of values of~$n$, as might happen with a rapidly divergent series,\nthen $u_{n}/s_{n} \\geq \\frac{1}{2}$ for all these values of~$n$.]", "markdown": "Prove that the same result holds for the series $\\sum (u_{n}/s_{n})$. [The proof is the same in the case of convergence. If $\\sum u_{n}$ is divergent, and $u_{n} < s_{n-1}$ from a certain value of $n$ onwards, then $s_{n} < 2s_{n-1}$, and the divergence of $\\sum (u_{n}/s_{n})$ follows from that of $\\sum (u_{n}/s_{n-1})$. If on the other hand $u_{n} \\geq s_{n-1}$ for an infinity of values of $n$, as might happen with a rapidly divergent series, then $u_{n}/s_{n} \\geq \\frac{1}{2}$ for all these values of $n$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "cas.sum" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxix/7", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxix", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "377", "location": "Exercise LXXXIX, problem 7", "problem_latex": "Sum the series $1 - \\frac{1}{2} + \\frac{1}{3} - \\dots$. [We have\n\\[\n1 + \\frac{1}{2} + \\dots + \\frac{1}{2n} = \\log(2n + 1) + \\gamma + \\epsilon_{n},\n\\quad\n2\\left(\\frac{1}{2} + \\frac{1}{4} + \\dots + \\frac{1}{2n}\\right)\n = \\log(n + 1) + \\gamma + \\epsilon_{n}',\n\\]\nby Ex.~1, $\\gamma$~denoting Euler's constant, and $\\epsilon_{n}$,~$\\epsilon_{n}'$ being numbers which tend\nto zero as $n \\to \\infty$. Subtracting and making $n \\to \\infty$ we see that the sum of the\ngiven series is~$\\log 2$. See also~\\SecNo[§]{213}.]", "markdown": "Sum the series $1 - \\frac{1}{2} + \\frac{1}{3} - \\dots$. [We have 1 + 12 + …+ 12n = (2n + 1) + + _n, 0pt minus 3pt2(12 + 14 + …+ 12n) = (n + 1) + + _n’, by Ex. 1, $\\gamma$ denoting Euler’s constant, and $\\epsilon_{n}$, $\\epsilon_{n}'$ being numbers which tend to zero as $n \\to \\infty$. Subtracting and making $n \\to \\infty$ we see that the sum of the given series is $\\log 2$. See also [§]213.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "cas.sum", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxix/8", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxix", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "377", "location": "Exercise LXXXIX, problem 8", "problem_latex": "Prove that the series\n\\[\n\\sum_{0}^{\\infty} (-1)^{n}\\left(1 + \\frac{1}{2} + \\dots + \\frac{1}{n + 1} - \\log n - C\\right)\n\\]\noscillates finitely except when $C = \\gamma$, when it converges.", "markdown": "Prove that the series _0^ (-1)^n(1 + 12 + …+ 1n + 1 - n - C) oscillates finitely except when $C = \\gamma$, when it converges.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "cas.sum" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxv/1", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxv", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "366", "location": "Exercise LXXXV, problem 1", "problem_latex": "If $dx/dy = ax$ then $x = Ke^{ay}$, where $K$~is a\nconstant.", "markdown": "If $dx/dy = ax$ then $x = Ke^{ay}$, where $K$ is a constant.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.ode" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxv/2", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxv", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "366", "location": "Exercise LXXXV, problem 2", "problem_latex": "There is no solution of the equation $f(y + z) = f(y)f(z)$ fundamentally\ndistinct from the exponential function. [We assume that $f(y)$~has a differential\ncoefficient. Differentiating the equation with respect to $y$~and~$z$ in turn, we\nobtain\n\\[\nf'(y + z) = f'(y)f(z),\\quad\nf'(y + z) = f(y)f'(z)\n\\]\nand so $f'(y)/f(y) = f'(z)/f(z)$, and therefore each is constant. Thus if $x = f(y)$\nthen $dx/dy = ax$, where $a$~is a constant, so that $x = Ke^{ay}$ (Ex.~1).]", "markdown": "There is no solution of the equation $f(y + z) = f(y)f(z)$ fundamentally distinct from the exponential function. [We assume that $f(y)$ has a differential coefficient. Differentiating the equation with respect to $y$ and $z$ in turn, we obtain f’(y + z) = f’(y)f(z),0pt minus 3ptf’(y + z) = f(y)f’(z) and so $f'(y)/f(y) = f'(z)/f(z)$, and therefore each is constant. Thus if $x = f(y)$ then $dx/dy = ax$, where $a$ is a constant, so that $x = Ke^{ay}$ (Ex. 1).]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.ode" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxv/3", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxv", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "366", "location": "Exercise LXXXV, problem 3", "problem_latex": "Prove that $(e^{ay} - 1)/y \\to a$ as $y \\to 0$. [Applying the Mean Value\nTheorem, we obtain $e^{ay} - 1 = aye^{a\\eta}$, where $0 < |\\eta| < |y|$.]", "markdown": "Prove that $(e^{ay} - 1)/y \\to a$ as $y \\to 0$. [Applying the Mean Value Theorem, we obtain $e^{ay} - 1 = aye^{a\\eta}$, where $0 < |\\eta| < |y|$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxvi/1", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxvi", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "369", "location": "Exercise LXXXVI, problem 1", "problem_latex": "Prove, by taking $y = 1$ and $n = 6$ in the inequalities~\\Eq{(4)}\nof \\SecNo[§]{208}, that $2.5 < e < 2.9$.", "markdown": "Prove, by taking $y = 1$ and $n = 6$ in the inequalities (4) of [§]208, that $2.5 < e < 2.9$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.const" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxvi/2", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxvi", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "369", "location": "Exercise LXXXVI, problem 2", "problem_latex": "Prove that if $t > 1$ then $(t^{1/n} - t^{-1/n})/(t - t^{-1}) < 1/n$, and so that if\n$x > 1$ then\n\\[\n\\int_{1}^{x} \\frac{dt}{t^{1-(1/n)}} - \\int_{1}^{x} \\frac{dt}{t^{1+(1/n)}}\n < \\frac{1}{n} \\int_{1}^{x} \\left(t - \\frac{1}{t}\\right) \\frac{dt}{t}\n = \\frac{1}{n} \\left(x + \\frac{1}{x} - 2\\right).\n\\]\nHence deduce the results of \\SecNo[§]{209}.", "markdown": "Prove that if $t > 1$ then $(t^{1/n} - t^{-1/n})/(t - t^{-1}) < 1/n$, and so that if $x > 1$ then _1^x dtt^1-(1/n) - _1^x dtt^1+(1/n) < 1n _1^x (t - 1t) dtt = 1n (x + 1x - 2). Hence deduce the results of [§]209.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.defint", "cas.integrate" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxvi/3", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxvi", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "369", "location": "Exercise LXXXVI, problem 3", "problem_latex": "If $\\xi_{n}$~is a function of~$n$ such that $n\\xi_{n} \\to l$ as $n \\to \\infty$, then $(1 + \\xi_{n})^{n} \\to e^{l}$.\n[Writing $n\\log(1 + \\xi_{n})$ in the form\n\\[\nl \\left(\\frac{n\\xi_{n}}{l}\\right) \\frac{\\log(1 + \\xi_{n})}{\\xi_{n}},\n\\]\nand using \\Ex{lxxxii}.~4, we see that $n\\log(1 + \\xi_{n})\\to l$.]", "markdown": "If $\\xi_{n}$ is a function of $n$ such that $n\\xi_{n} \\to l$ as $n \\to \\infty$, then $(1 + \\xi_{n})^{n} \\to e^{l}$. [Writing $n\\log(1 + \\xi_{n})$ in the form l (n_nl) (1 + _n)_n, and using % [examples:lxxxii]Ex. lxxxii%. 4, we see that $n\\log(1 + \\xi_{n})\\to l$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxvi/4", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxvi", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "369", "location": "Exercise LXXXVI, problem 4", "problem_latex": "If $n\\xi_{n} \\to \\infty$, then $(1 + \\xi_{n})^{n} \\to \\infty$; and if $1 + \\xi_{n} > 0$ and $n\\xi_{n} \\to -\\infty$, then\n\\[\n(1 + \\xi_{n})^{n} \\to 0.\n\\]", "markdown": "If $n\\xi_{n} \\to \\infty$, then $(1 + \\xi_{n})^{n} \\to \\infty$; and if $1 + \\xi_{n} > 0$ and $n\\xi_{n} \\to -\\infty$, then (1 + _n)^n 0.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxvi/5", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxvi", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "369", "location": "Exercise LXXXVI, problem 5", "problem_latex": "Deduce from~\\Eq{(1)} of \\SecNo[§]{208} the theorem that $e^{y}$~tends to infinity more\nrapidly than any power of~$y$.", "markdown": "Deduce from (1) of [§]208 the theorem that $e^{y}$ tends to infinity more rapidly than any power of $y$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxviii/1", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxviii", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "375", "location": "Exercise LXXXVIII, problem 1", "problem_latex": "The series\n\\[\n\\sum \\frac{1}{n(\\log n)^{2}},\\quad\n\\sum \\frac{(\\log n)^{100}}{n^{101/100}},\\quad\n\\sum \\frac{n^{2} - 1}{n^{2} + 1}\\, \\frac{1}{n(\\log n)^{7/6}}\n\\]\nare convergent. [The convergence of the first series is a direct consequence\nof the theorem of the preceding section. That of the second follows from\nthe fact that $(\\log n)^{100}$~is less than~$n^{\\beta}$ for sufficiently large values of~$n$, however\nsmall $\\beta$ may be, provided that it is positive. And so, taking $\\beta = 1/200$,\n$(\\log n)^{100} n^{-101/100}$ is less than~$n^{-201/200}$ for sufficiently large values of~$n$. The\nconvergence of the third series follows from the comparison test at the end of\nthe last section.]", "markdown": "The series 1n(n)^2,0pt minus 3pt(n)^100n^101/100,0pt minus 3ptn^2 - 1n^2 + 1  1n(n)^7/6 are convergent. [The convergence of the first series is a direct consequence of the theorem of the preceding section. That of the second follows from the fact that $(\\log n)^{100}$ is less than $n^{\\beta}$ for sufficiently large values of $n$, however small $\\beta$ may be, provided that it is positive. And so, taking $\\beta = 1/200$, $(\\log n)^{100} n^{-101/100}$ is less than $n^{-201/200}$ for sufficiently large values of $n$. The convergence of the third series follows from the comparison test at the end of the last section.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:series_convergence_test" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxviii/2", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxviii", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "375", "location": "Exercise LXXXVIII, problem 2", "problem_latex": "The series\n\\[\n\\sum \\frac{1}{n(\\log n)^{6/7}},\\quad\n\\sum \\frac{1}{n^{100/101}(\\log n)^{100}},\\quad\n\\sum \\frac{n\\log n}{(n\\log n)^{2} + 1}\n\\]\nare divergent.", "markdown": "The series 1n(n)^6/7,0pt minus 3pt1n^100/101(n)^100,0pt minus 3ptnn(nn)^2 + 1 are divergent.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:series_convergence_test" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxviii/3", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxviii", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "375", "location": "Exercise LXXXVIII, problem 3", "problem_latex": "The series\n\\[\n\\sum \\frac{(\\log n)^{p}}{n^{1+s}},\\quad\n\\sum \\frac{(\\log n)^{p} (\\log\\log n)^{q}}{n^{1+s}},\\quad\n\\sum \\frac{(\\log\\log n)^{p}}{n(\\log n)^{1+s}},\n\\]\nwhere $s > 0$, are convergent for all values of $p$~and~$q$; similarly the series\n\\[\n\\sum \\frac{1}{n^{1-s}(\\log n)^{p}},\\quad\n\\sum \\frac{1}{n^{1-s}(\\log n)^{p}(\\log\\log n)^{q}},\\quad\n\\sum \\frac{1}{n(\\log n)^{1-s}(\\log\\log n)^{p}}\n\\]\nare divergent.", "markdown": "The series (n)^pn^1+s,0pt minus 3pt(n)^p (n)^qn^1+s,0pt minus 3pt(n)^pn(n)^1+s, where $s > 0$, are convergent for all values of $p$ and $q$; similarly the series 1n^1-s(n)^p,0pt minus 3pt1n^1-s(n)^p(n)^q,0pt minus 3pt1n(n)^1-s(n)^p are divergent.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:series_convergence_test" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxviii/4", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxviii", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "375", "location": "Exercise LXXXVIII, problem 4", "problem_latex": "The question of the convergence or divergence of such series as\n\\[\n\\sum \\frac{1}{n\\log n\\log\\log n},\\quad\n\\sum \\frac{\\log\\log\\log n}{n\\log n\\sqrtp{\\log\\log n}}\n\\]\ncannot be settled by the theorem of \\PageRef{p.}{375}, since in each case the function\nunder the sign of summation tends to zero more rapidly than $1/(n\\log n)$ yet\nless rapidly than $n^{-1}(\\log n)^{-1-\\alpha}$, where $\\alpha$~is any positive number however\nsmall. For such series we need a still more delicate test. The reader should\nbe able, starting from the equations\n\\begin{align*}\nD_{x}(\\log_{k}x)^{1-s}\n &= \\frac{1 - s}{x \\log x \\log_{2}x \\dots \\log_{k-1} x (\\log_{k}x)^{s}},\\\\\nD_{x}\\log_{k+1}x\n &= \\frac{1}{x \\log x \\log_{2}x \\dots \\log_{k-1}x \\log_{k}x},\n\\end{align*}\nwhere $\\log_{2}x = \\log\\log x$, $\\log_{3} x = \\log\\log\\log x$,~\\dots, to prove the following\ntheorem: \\emph{the series and integral\n\\[\n\\sum_{n_{0}}^{\\infty} \\frac{1}{n \\log n \\log_{2}n \\dots \\log_{k-1}n (\\log_{k}n)^{s}},\\quad\n\\int_{a}^{\\infty} \\frac{dx}{x \\log x \\log_{2}x \\dots \\log_{k-1}x (\\log_{k}x)^{s}}\n\\]\nare convergent if $s > 1$ and divergent if $s \\leq 1$}, {\\Loosen$n_{0}$~and~$a$ being any numbers\nsufficiently great to ensure that $\\log_{k}n$ and $\\log_{k}x$ are positive when $n \\geq n_{0}$\nor $x \\geq a$. These values of $n_{0}$ and~$a$ increase very rapidly as $k$~increases:\nthus $\\log x > 0$ requires $x > 1$, $\\log_{2}x > 0$ requires $x > e$, $\\DPtypo{\\log\\log x}{\\log_{3}x} > 0$ requires\n$x > e^{e}$, and so on; and it is easy to see that $e^{e} > 10$, $e^{e^{e}} > e^{10} > 20,000$,\n$e^{e^{e^{e}}} > e^{20,000} > 10^{8000}$.}", "markdown": "The question of the convergence or divergence of such series as 1nnn,0pt minus 3ptnnnn cannot be settled by the theorem of p.375, since in each case the function under the sign of summation tends to zero more rapidly than $1/(n\\log n)$ yet less rapidly than $n^{-1}(\\log n)^{-1-\\alpha}$, where $\\alpha$ is any positive number however small. For such series we need a still more delicate test. The reader should be able, starting from the equations align* D_x(_kx)^1-s &= 1 - sx x _2x …_k-1 x (_kx)^s, D_x_k+1x &= 1x x _2x …_k-1x _kx, align* where $\\log_{2}x = \\log\\log x$, $\\log_{3} x = \\log\\log\\log x$, …, to prove the following theorem: *the series and integral _n_0^ 1n n _2n …_k-1n (_kn)^s,0pt minus 3pt_a^ dxx x _2x …_k-1x (_kx)^s are convergent if $s > 1$ and divergent if $s \\leq 1$*, 0.375em plus 0.75em minus 0.25em$n_{0}$ and $a$ being any numbers sufficiently great to ensure that $\\log_{k}n$ and $\\log_{k}x$ are positive when $n \\geq n_{0}$ or $x \\geq a$. These values of $n_{0}$ and $a$ increase very rapidly as $k$ increases: thus $\\log x > 0$ requires $x > 1$, $\\log_{2}x > 0$ requires $x > e$, $\\DPtypo{\\log\\log x}{\\log_{3}x} > 0$ requires $x > e^{e}$, and so on; and it is easy to see that $e^{e} > 10$, $e^{e^{e}} > e^{10} > 20,000$, $e^{e^{e^{e}}} > e^{20,000} > 10^{8000}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:integral_test_theorem" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxviii/5", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxviii", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "375", "location": "Exercise LXXXVIII, problem 5", "problem_latex": "Prove that the integral $\\ds\\int_{0}^{a} \\frac{1}{x} \\left\\{\\log \\left(\\frac{1}{x}\\right)\\right\\}^{s} dx$, where $0 < a < 1$, is convergent\nif $s < -1$, divergent if $s \\geq -1$. [Consider the behaviour of\n\\[\n\\int_{\\epsilon}^{a} \\frac{1}{x} \\left\\{\\log \\left(\\frac{1}{x}\\right)\\right\\}^{s} dx\n\\]\nas $\\epsilon \\to +0$. This result also may be refined upon by the introduction of\nhigher logarithmic factors.]", "markdown": "Prove that the integral $\\ds\\int_{0}^{a} \\frac{1}{x} \\left\\{\\log \\left(\\frac{1}{x}\\right)\\right\\}^{s} dx$, where $0 < a < 1$, is convergent if $s < -1$, divergent if $s \\geq -1$. [Consider the behaviour of _^a 1x (1x)^s dx as $\\epsilon \\to +0$. This result also may be refined upon by the introduction of higher logarithmic factors.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:improper_integral_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxviii/6", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxviii", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "375", "location": "Exercise LXXXVIII, problem 6", "problem_latex": "Prove that $\\ds\\int_{0}^{1} \\frac{1}{x} \\left\\{\\log \\left(\\frac{1}{x}\\right)\\right\\}^{s} dx$ has no meaning for any value of~$s$.\n[The last example shows that $s < -1$ is a necessary condition for convergence\nat the lower limit: but $\\{\\log(1/x)\\}^{s}$ tends to~$\\infty$ like $(1 - x)^{s}$, as $x \\to 1 - 0$, if $s$~is\nnegative, and so the integral diverges at the upper limit when $s < -1$.]", "markdown": "Prove that $\\ds\\int_{0}^{1} \\frac{1}{x} \\left\\{\\log \\left(\\frac{1}{x}\\right)\\right\\}^{s} dx$ has no meaning for any value of $s$. [The last example shows that $s < -1$ is a necessary condition for convergence at the lower limit: but $\\{\\log(1/x)\\}^{s}$ tends to $\\infty$ like $(1 - x)^{s}$, as $x \\to 1 - 0$, if $s$ is negative, and so the integral diverges at the upper limit when $s < -1$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:improper_integral_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-lxxxviii/7", "set": "hardy-course-of-pure-mathematics-1921/ex-lxxxviii", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "375", "location": "Exercise LXXXVIII, problem 7", "problem_latex": "{\\Loosen The necessary and sufficient conditions for the convergence of\n$\\ds\\int_{0}^{1} x^{a-1} \\left\\{\\log \\left(\\frac{1}{x}\\right)\\right\\}^{s} dx$ are $a > 0$, $s > -1$.}", "markdown": "0.375em plus 0.75em minus 0.25emThe necessary and sufficient conditions for the convergence of $\\ds\\int_{0}^{1} x^{a-1} \\left\\{\\log \\left(\\frac{1}{x}\\right)\\right\\}^{s} dx$ are $a > 0$, $s > -1$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:improper_integral_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ii/1", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ii", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "65", "location": "Exercise Misc-II, problem 1", "problem_latex": "Show that if $y = f(x) = (ax + b)/(cx - a)$ then $x = f(y)$.", "markdown": "Show that if $y = f(x) = (ax + b)/(cx - a)$ then $x = f(y)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ii/10", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ii", "number": 10, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "65", "location": "Exercise Misc-II, problem 10", "problem_latex": "Discuss the graphical solution of the equation\n\\[\nx^{m} + ax^{2} + bx + c = 0\n\\]\nby means of the curves $y = x^{m}$, $y = -ax^{2} - bx - c$. Draw up a table of the\nvarious possible numbers of roots.", "markdown": "Discuss the graphical solution of the equation x^m + ax^2 + bx + c = 0 by means of the curves $y = x^{m}$, $y = -ax^{2} - bx - c$. Draw up a table of the various possible numbers of roots.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ii/11a", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ii", "number": 11, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "65", "location": "Exercise Misc-II, problem 11a", "problem_latex": "Solve the equation $\\sec\\theta + \\cosec\\theta = 2\\sqrt{2}$; and show that the equation\n$\\sec\\theta + \\cosec\\theta = c$ has two roots between $0$~and~$2\\pi$ if $c^{2} < 8$ and four if $c^{2} > 8$.\n\\PageSep{66}", "markdown": "Solve the equation $\\sec\\theta + \\cosec\\theta = 2\\sqrt{2}$; and show that the equation $\\sec\\theta + \\cosec\\theta = c$ has two roots between $0$ and $2\\pi$ if $c^{2} < 8$ and four if $c^{2} > 8$. [pg]66", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "solve", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "Eq(sec(theta) + csc(theta), 2*sqrt(2))", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "solve: Eq(csc(x) + sec(x), 2*sqrt(2))" ], "shape": [ "solve: Eq(csc(x) + sec(x), N*N**N)" ], "same_problem_in": [], "needs": [ "core.solve.num", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ii/11b", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ii", "number": 11, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "65", "location": "Exercise Misc-II, problem 11b", "problem_latex": "Solve the equation $\\sec\\theta + \\cosec\\theta = 2\\sqrt{2}$; and show that the equation\n$\\sec\\theta + \\cosec\\theta = c$ has two roots between $0$~and~$2\\pi$ if $c^{2} < 8$ and four if $c^{2} > 8$.\n\\PageSep{66}", "markdown": "Solve the equation $\\sec\\theta + \\cosec\\theta = 2\\sqrt{2}$; and show that the equation $\\sec\\theta + \\cosec\\theta = c$ has two roots between $0$ and $2\\pi$ if $c^{2} < 8$ and four if $c^{2} > 8$. [pg]66", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ii/12", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ii", "number": 12, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "65", "location": "Exercise Misc-II, problem 12", "problem_latex": "Show that the equation\n\\[\n2x = (2n + 1)\\pi(1 - \\cos x),\n\\]\nwhere $n$~is a positive integer, has $2n + 3$ roots and no more, indicating\ntheir localities roughly. \\MathTrip{1896.}", "markdown": "Show that the equation 2x = (2n + 1)(1 - x), where $n$ is a positive integer, has $2n + 3$ roots and no more, indicating their localities roughly. % [0]% (*Math. Trip.* 1896.)% [1]%", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ii/13", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ii", "number": 13, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "65", "location": "Exercise Misc-II, problem 13", "problem_latex": "Show that the equation $\\frac{2}{3}x\\sin x = 1$ has four roots between $-\\pi$~and~$\\pi$.", "markdown": "Show that the equation $\\frac{2}{3}x\\sin x = 1$ has four roots between $-\\pi$ and $\\pi$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ii/141", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ii", "number": 14, "part": "(1)", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "65", "location": "Exercise Misc-II, problem 14(1)", "problem_latex": "Discuss the number and values of the roots of the equations\n\n%[** TN: Items in multiple columns in the original]\n\\SubItem{(1)} $\\cot x + x - \\frac{3}{2}\\pi = 0$,\n\n\\SubItem{(2)} $x^{2} + \\sin^{2} x = 1$,\n\n\\SubItem{(3)} $\\tan x = 2x/(1 + x^{2})$,\n\n\\SubItem{(4)} $\\sin x - x + \\frac{1}{6}x^{3} = 0$,\n\n\\SubItem{(5)} $(1 - \\cos x)\\tan\\alpha - x + \\sin x = 0$.", "markdown": "Discuss the number and values of the roots of the equations %[** TN: Items in multiple columns in the original] 0pt minus 3pt% [2.25em][l](1)% [2.25em][l](1)% % $\\cot x + x - \\frac{3}{2}\\pi = 0$, 0pt minus 3pt% [2.25em][l](2)% [2.25em][l](2)% % $x^{2} + \\sin^{2} x = 1$, 0pt minus 3pt% [2.25em][l](3)% [2.25em][l](3)% % $\\tan x = 2x/(1 + x^{2})$, 0pt minus 3pt% [2.25em][l](4)% [2.25em][l](4)% % $\\sin x - x + \\frac{1}{6}x^{3} = 0$, 0pt minus 3pt% [2.25em][l](5)% [2.25em][l](5)% % $(1 - \\cos x)\\tan\\alpha - x + \\sin x = 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ii/142", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ii", "number": 14, "part": "(2)", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "65", "location": "Exercise Misc-II, problem 14(2)", "problem_latex": "Discuss the number and values of the roots of the equations\n\n%[** TN: Items in multiple columns in the original]\n\\SubItem{(1)} $\\cot x + x - \\frac{3}{2}\\pi = 0$,\n\n\\SubItem{(2)} $x^{2} + \\sin^{2} x = 1$,\n\n\\SubItem{(3)} $\\tan x = 2x/(1 + x^{2})$,\n\n\\SubItem{(4)} $\\sin x - x + \\frac{1}{6}x^{3} = 0$,\n\n\\SubItem{(5)} $(1 - \\cos x)\\tan\\alpha - x + \\sin x = 0$.", "markdown": "Discuss the number and values of the roots of the equations %[** TN: Items in multiple columns in the original] 0pt minus 3pt% [2.25em][l](1)% [2.25em][l](1)% % $\\cot x + x - \\frac{3}{2}\\pi = 0$, 0pt minus 3pt% [2.25em][l](2)% [2.25em][l](2)% % $x^{2} + \\sin^{2} x = 1$, 0pt minus 3pt% [2.25em][l](3)% [2.25em][l](3)% % $\\tan x = 2x/(1 + x^{2})$, 0pt minus 3pt% [2.25em][l](4)% [2.25em][l](4)% % $\\sin x - x + \\frac{1}{6}x^{3} = 0$, 0pt minus 3pt% [2.25em][l](5)% [2.25em][l](5)% % $(1 - \\cos x)\\tan\\alpha - x + \\sin x = 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ii/143", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ii", "number": 14, "part": "(3)", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "65", "location": "Exercise Misc-II, problem 14(3)", "problem_latex": "Discuss the number and values of the roots of the equations\n\n%[** TN: Items in multiple columns in the original]\n\\SubItem{(1)} $\\cot x + x - \\frac{3}{2}\\pi = 0$,\n\n\\SubItem{(2)} $x^{2} + \\sin^{2} x = 1$,\n\n\\SubItem{(3)} $\\tan x = 2x/(1 + x^{2})$,\n\n\\SubItem{(4)} $\\sin x - x + \\frac{1}{6}x^{3} = 0$,\n\n\\SubItem{(5)} $(1 - \\cos x)\\tan\\alpha - x + \\sin x = 0$.", "markdown": "Discuss the number and values of the roots of the equations %[** TN: Items in multiple columns in the original] 0pt minus 3pt% [2.25em][l](1)% [2.25em][l](1)% % $\\cot x + x - \\frac{3}{2}\\pi = 0$, 0pt minus 3pt% [2.25em][l](2)% [2.25em][l](2)% % $x^{2} + \\sin^{2} x = 1$, 0pt minus 3pt% [2.25em][l](3)% [2.25em][l](3)% % $\\tan x = 2x/(1 + x^{2})$, 0pt minus 3pt% [2.25em][l](4)% [2.25em][l](4)% % $\\sin x - x + \\frac{1}{6}x^{3} = 0$, 0pt minus 3pt% [2.25em][l](5)% [2.25em][l](5)% % $(1 - \\cos x)\\tan\\alpha - x + \\sin x = 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ii/144", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ii", "number": 14, "part": "(4)", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "65", "location": "Exercise Misc-II, problem 14(4)", "problem_latex": "Discuss the number and values of the roots of the equations\n\n%[** TN: Items in multiple columns in the original]\n\\SubItem{(1)} $\\cot x + x - \\frac{3}{2}\\pi = 0$,\n\n\\SubItem{(2)} $x^{2} + \\sin^{2} x = 1$,\n\n\\SubItem{(3)} $\\tan x = 2x/(1 + x^{2})$,\n\n\\SubItem{(4)} $\\sin x - x + \\frac{1}{6}x^{3} = 0$,\n\n\\SubItem{(5)} $(1 - \\cos x)\\tan\\alpha - x + \\sin x = 0$.", "markdown": "Discuss the number and values of the roots of the equations %[** TN: Items in multiple columns in the original] 0pt minus 3pt% [2.25em][l](1)% [2.25em][l](1)% % $\\cot x + x - \\frac{3}{2}\\pi = 0$, 0pt minus 3pt% [2.25em][l](2)% [2.25em][l](2)% % $x^{2} + \\sin^{2} x = 1$, 0pt minus 3pt% [2.25em][l](3)% [2.25em][l](3)% % $\\tan x = 2x/(1 + x^{2})$, 0pt minus 3pt% [2.25em][l](4)% [2.25em][l](4)% % $\\sin x - x + \\frac{1}{6}x^{3} = 0$, 0pt minus 3pt% [2.25em][l](5)% [2.25em][l](5)% % $(1 - \\cos x)\\tan\\alpha - x + \\sin x = 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ii/145", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ii", "number": 14, "part": "(5)", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "65", "location": "Exercise Misc-II, problem 14(5)", "problem_latex": "Discuss the number and values of the roots of the equations\n\n%[** TN: Items in multiple columns in the original]\n\\SubItem{(1)} $\\cot x + x - \\frac{3}{2}\\pi = 0$,\n\n\\SubItem{(2)} $x^{2} + \\sin^{2} x = 1$,\n\n\\SubItem{(3)} $\\tan x = 2x/(1 + x^{2})$,\n\n\\SubItem{(4)} $\\sin x - x + \\frac{1}{6}x^{3} = 0$,\n\n\\SubItem{(5)} $(1 - \\cos x)\\tan\\alpha - x + \\sin x = 0$.", "markdown": "Discuss the number and values of the roots of the equations %[** TN: Items in multiple columns in the original] 0pt minus 3pt% [2.25em][l](1)% [2.25em][l](1)% % $\\cot x + x - \\frac{3}{2}\\pi = 0$, 0pt minus 3pt% [2.25em][l](2)% [2.25em][l](2)% % $x^{2} + \\sin^{2} x = 1$, 0pt minus 3pt% [2.25em][l](3)% [2.25em][l](3)% % $\\tan x = 2x/(1 + x^{2})$, 0pt minus 3pt% [2.25em][l](4)% [2.25em][l](4)% % $\\sin x - x + \\frac{1}{6}x^{3} = 0$, 0pt minus 3pt% [2.25em][l](5)% [2.25em][l](5)% % $(1 - \\cos x)\\tan\\alpha - x + \\sin x = 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ii/15", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ii", "number": 15, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "65", "location": "Exercise Misc-II, problem 15", "problem_latex": "The polynomial of the second degree which assumes, when $x = a$, $b$,~$c$\nthe values $\\alpha$,~$\\beta$,~$\\gamma$ is\n\\[\n\\alpha\\frac{(x - b)(x - c)}{(a - b)(a - c)} +\n\\beta \\frac{(x - c)(x - a)}{(b - c)(b - a)} +\n\\gamma\\frac{(x - a)(x - b)}{(c - a)(c - b)}.\n\\]\nGive a similar formula for the polynomial of the $(n - 1)$th~degree which\nassumes, when $x = a_{1}$, $a_{2}$, \\dots~$a_{n}$, the values $\\alpha_{1}$, $\\alpha_{2}$, \\dots~$\\alpha_{n}$.", "markdown": "The polynomial of the second degree which assumes, when $x = a$, $b$, $c$ the values $\\alpha$, $\\beta$, $\\gamma$ is (x - b)(x - c)(a - b)(a - c) + (x - c)(x - a)(b - c)(b - a) + (x - a)(x - b)(c - a)(c - b). Give a similar formula for the polynomial of the $(n - 1)$th degree which assumes, when $x = a_{1}$, $a_{2}$, … $a_{n}$, the values $\\alpha_{1}$, $\\alpha_{2}$, … $\\alpha_{n}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ii/16", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ii", "number": 16, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "65", "location": "Exercise Misc-II, problem 16", "problem_latex": "Find a polynomial in~$x$ of the second degree which for the values\n$0$,~$1$,~$2$ of~$x$ takes the values $1/c$, $1/(c + 1)$, $1/(c + 2)$; and show that when\n$x = c + 2$ its value is~$1/(c + 1)$. \\MathTrip{1911.}", "markdown": "Find a polynomial in $x$ of the second degree which for the values $0$, $1$, $2$ of $x$ takes the values $1/c$, $1/(c + 1)$, $1/(c + 2)$; and show that when $x = c + 2$ its value is $1/(c + 1)$. % [0]% (*Math. Trip.* 1911.)% [1]%", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ii/17", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ii", "number": 17, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "65", "location": "Exercise Misc-II, problem 17", "problem_latex": "Show that if $x$~is a rational function of~$y$, and $y$~is a rational function\nof~$x$, then $Axy + Bx + Cy + D = 0$.", "markdown": "Show that if $x$ is a rational function of $y$, and $y$ is a rational function of $x$, then $Axy + Bx + Cy + D = 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ii/18", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ii", "number": 18, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "65", "location": "Exercise Misc-II, problem 18", "problem_latex": "If $y$~is an algebraical function of~$x$, then $x$~is an algebraical function\nof~$y$.", "markdown": "If $y$ is an algebraical function of $x$, then $x$ is an algebraical function of $y$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ii/19", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ii", "number": 19, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "65", "location": "Exercise Misc-II, problem 19", "problem_latex": "Verify that the equation\n\\[\n\\cos\\tfrac{1}{2}\\pi x\n = 1 - \\frac{x^{2}}{x + (x - 1)\\bigsqrtp{\\dfrac{2 - x}{3}}}\n\\]\nis approximately true for all values of~$x$ between $0$~and~$1$. [Take $x = 0$, $\\frac{1}{6}$, $\\frac{1}{3}$,\n$\\tfrac{1}{2}$, $\\frac{2}{3}$, $\\frac{5}{6}$,~$1$, and use tables. For which of these values is the formula exact?]", "markdown": "Verify that the equation 12x = 1 - x^2x + (x - 1)2 - x3 is approximately true for all values of $x$ between $0$ and $1$. [Take $x = 0$, $\\frac{1}{6}$, $\\frac{1}{3}$, $\\tfrac{1}{2}$, $\\frac{2}{3}$, $\\frac{5}{6}$, $1$, and use tables. For which of these values is the formula exact?]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.table", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ii/2", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ii", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "65", "location": "Exercise Misc-II, problem 2", "problem_latex": "If $f(x) = f(-x)$ for all values of~$x$, $f(x)$~is called an \\emph{even} function.\nIf $f(x) = -f(-x)$, it is called an \\emph{odd} function. Show that any function of~$x$,\ndefined for all values of~$x$, is the sum of an even and an odd function of~$x$.\n\n[Use the identity $f(x) = \\frac{1}{2}\\{f(x) + f(-x)\\} + \\frac{1}{2}\\{f(x) - f(-x)\\}$.]", "markdown": "If $f(x) = f(-x)$ for all values of $x$, $f(x)$ is called an *even* function. If $f(x) = -f(-x)$, it is called an *odd* function. Show that any function of $x$, defined for all values of $x$, is the sum of an even and an odd function of $x$. [Use the identity $f(x) = \\frac{1}{2}\\{f(x) + f(-x)\\} + \\frac{1}{2}\\{f(x) - f(-x)\\}$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ii/20", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ii", "number": 20, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "65", "location": "Exercise Misc-II, problem 20", "problem_latex": "What is the form of the graph of the functions\n\\[\nz = [x] + [y],\\quad\nz = x + y - [x] - [y]?\n\\]", "markdown": "What is the form of the graph of the functions z = [x] + [y],0pt minus 3ptz = x + y - [x] - [y]?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ii/21", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ii", "number": 21, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "65", "location": "Exercise Misc-II, problem 21", "problem_latex": "What is the form of the graph of the functions $z = \\sin x + \\sin y$,\n$z = \\sin x\\sin y$, $z = \\sin xy$, $z = \\sin(x^{2} + y^{2})$?", "markdown": "What is the form of the graph of the functions $z = \\sin x + \\sin y$, $z = \\sin x\\sin y$, $z = \\sin xy$, $z = \\sin(x^{2} + y^{2})$?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ii/22", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ii", "number": 22, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "65", "location": "Exercise Misc-II, problem 22", "problem_latex": "\\Topic{Geometrical constructions for irrational numbers.} In \\okrickRef{Chapter}{I}\nwe indicated one or two simple geometrical constructions for a length equal to~$\\sqrt{2}$,\nstarting from a given unit length. We also showed how to construct\nthe roots of any quadratic equation $ax^{2} + 2bx + c = 0$, it being supposed that\nwe can construct lines whose lengths are equal to any of the ratios of the\ncoefficients $a$,~$b$,~$c$, as is certainly the case if $a$,~$b$,~$c$ are rational. All these constructions\nwere what may be called Euclidean constructions; they depended\non the ruler and compasses only.\n\\PageSep{67}\n\nIt is fairly obvious that we can construct by these methods the length\nmeasured by any irrational number which is defined by any combination of\nsquare roots, however complicated. Thus\n\\[\n\\bigsqrtb[4]{\\bigsqrtp{\\frac{17 + 3\\sqrt{11}}{17 - 3\\sqrt{11}}}\n - \\bigsqrtp{\\frac{17 - 3\\sqrt{11}}{17 + 3\\sqrt{11}}}}\n\\]\nis a case in point. This expression contains a fourth root, but this is of\ncourse the square root of a square root. We should begin by constructing~$\\sqrt{11}$,\n\\eg\\ as the mean between $1$~and~$11$: then $17 + 3\\sqrt{11}$ and $17 - 3\\sqrt{11}$, and\nso on. Or these two mixed surds might be constructed directly as the roots of\n$x^{2} - 34x + 190 = 0$.\n\nConversely, \\emph{only} irrationals of this kind can be constructed by Euclidean\nmethods. Starting from a unit length we can construct any \\emph{rational} length.\nAnd hence we can construct the line $Ax + By + C = 0$, provided that the ratios\nof $A$,~$B$,~$C$ are rational, and the circle\n\\[\n(x - \\alpha)^{2} + (y - \\beta)^{2} = \\rho ^{2}\n\\]\n(or $x^{2} + y^{2} + 2gx + 2fy + c = 0$), provided that $\\alpha$,~$\\beta$,~$\\rho$ are rational, a condition\nwhich implies that $g$,~$f$,~$c$ are rational.\n\nNow in any Euclidean construction each new point introduced into the\nfigure is determined as the intersection of two lines or circles, or a line and\na circle. But if the coefficients are rational, such a pair of equations as\n\\[\nAx + By + C = 0,\\quad\nx^{2} + y^{2} + 2gx + 2fy + c = 0\n\\]\ngive, on solution, values of $x$~and~$y$ of the form $m + n\\sqrt{p}$, where $m$,~$n$,~$p$ are\nrational: for if we substitute for~$x$ in terms of~$y$ in the second equation we\nobtain a quadratic in~$y$ with rational coefficients. Hence the coordinates of\nall points obtained by means of lines and circles with rational coefficients\nare expressible by rational numbers and quadratic surds. And so the same\nis true of the distance $\\sqrtb{(x_{1} - x_{2})^{2} + (y_{1} - y_{2})^{2}}$ between any two points so\nobtained.\n\nWith the irrational distances thus constructed we may proceed to construct\na number of lines and circles whose coefficients may now themselves involve\nquadratic surds. It is evident, however, that all the lengths which we can\nconstruct by the use of such lines and circles are still expressible by square\nroots only, though our surd expressions may now be of a more complicated\nform. And this remains true however often our constructions are repeated.\nHence \\emph{Euclidean methods will construct any surd expression involving square\nroots only, and no others}.\n\nOne of the famous problems of antiquity was that of the duplication of\nthe cube, that is to say of the construction by Euclidean methods of a\nlength measured by~$\\sqrt[3]{2}$. It can be shown that $\\sqrt[3]{2}$~cannot be expressed by\nmeans of any finite combination of rational numbers and square roots, and so\nthat the problem is an impossible one. See Hobson, \\textit{Squaring the Circle},\npp.~47~\\textit{et~seq.}; the first stage of the proof, viz.\\ the proof that $\\sqrt[3]{2}$~cannot be a\nroot of a quadratic equation $ax^{2} + 2bx + c = 0$ with rational coefficients, was\ngiven in \\okrickRef{Ch.}{I} (\\MiscExs{I}~24).\n\\PageSep{68}", "markdown": "**constructions for irrational numbers.** In ChapterI we indicated one or two simple geometrical constructions for a length equal to $\\sqrt{2}$, starting from a given unit length. We also showed how to construct the roots of any quadratic equation $ax^{2} + 2bx + c = 0$, it being supposed that we can construct lines whose lengths are equal to any of the ratios of the coefficients $a$, $b$, $c$, as is certainly the case if $a$, $b$, $c$ are rational. All these constructions were what may be called Euclidean constructions; they depended on the ruler and compasses only. [pg]67 It is fairly obvious that we can construct by these methods the length measured by any irrational number which is defined by any combination of square roots, however complicated. Thus [4]17 + 31117 - 311 - 17 - 31117 + 311 is a case in point. This expression contains a fourth root, but this is of course the square root of a square root. We should begin by constructing $\\sqrt{11}$, *e.g.* as the mean between $1$ and $11$: then $17 + 3\\sqrt{11}$ and $17 - 3\\sqrt{11}$, and so on. Or these two mixed surds might be constructed directly as the roots of $x^{2} - 34x + 190 = 0$. Conversely, *only* irrationals of this kind can be constructed by Euclidean methods. Starting from a unit length we can construct any *rational* length. And hence we can construct the line $Ax + By + C = 0$, provided that the ratios of $A$, $B$, $C$ are rational, and the circle (x - )^2 + (y - )^2 = ^2 (or $x^{2} + y^{2} + 2gx + 2fy + c = 0$), provided that $\\alpha$, $\\beta$, $\\rho$ are rational, a condition which implies that $g$, $f$, $c$ are rational. Now in any Euclidean construction each new point introduced into the figure is determined as the intersection of two lines or circles, or a line and a circle. But if the coefficients are rational, such a pair of equations as Ax + By + C = 0,0pt minus 3ptx^2 + y^2 + 2gx + 2fy + c = 0 give, on solution, values of $x$ and $y$ of the form $m + n\\sqrt{p}$, where $m$, $n$, $p$ are rational: for if we substitute for $x$ in terms of $y$ in the second equation we obtain a quadratic in $y$ with rational coefficients. Hence the coordinates of all points obtained by means of lines and circles with rational coefficients are expressible by rational numbers and quadratic surds. And so the same is true of the distance $\\sqrtb{(x_{1} - x_{2})^{2} + (y_{1} - y_{2})^{2}}$ between any two points so obtained. With the irrational distances thus constructed we may proceed to construct a number of lines and circles whose coefficients may now themselves involve quadratic surds. It is evident, however, that all the lengths which we can construct by the use of such lines and circles are still expressible by square roots only, though our surd expressions may now be of a more complicated form. And this remains true however often our constructions are repeated. Hence *Euclidean methods will construct any surd expression involving square roots only, and no others*. One of the famous problems of antiquity was that of the duplication of the cube, that is to say of the construction by Euclidean methods of a length measured by $\\sqrt[3]{2}$. It can be shown that $\\sqrt[3]{2}$ cannot be expressed by means of any finite combination of rational numbers and square roots, and so that the problem is an impossible one. See Hobson, *Squaring the Circle*, pp. 47 *et seq.*; the first stage of the proof, viz. the proof that $\\sqrt[3]{2}$ cannot be a root of a quadratic equation $ax^{2} + 2bx + c = 0$ with rational coefficients, was given in Ch.I ([misc:I]Misc. Exs. 24). [pg]68", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ii/23", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ii", "number": 23, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "65", "location": "Exercise Misc-II, problem 23", "problem_latex": "\\Topic{Approximate quadrature of the circle.} Let $O$~be the centre of\na circle of radius~$R$. On the tangent at~$A$ take $AP = \\frac{11}{5}R$ and $AQ = \\frac{13}{5}R$,\nin the same direction. On~$AO$ take $AN = OP$ and draw~$NM$ parallel to~$OQ$\nand cutting~$AP$ in~$M$. Show that\n\\[\nAM/R = \\tfrac{13}{25}\\sqrt{146},\n\\]\nand that to take~$AM$ as being equal to the circumference of the circle would\nlead to a value of~$\\pi$ correct to five places of decimals. If $R$~is the earth's\nradius, the error in supposing $AM$ to be its circumference is less than $11$~yards.", "markdown": "**quadrature of the circle.** Let $O$ be the centre of a circle of radius $R$. On the tangent at $A$ take $AP = \\frac{11}{5}R$ and $AQ = \\frac{13}{5}R$, in the same direction. On $AO$ take $AN = OP$ and draw $NM$ parallel to $OQ$ and cutting $AP$ in $M$. Show that AM/R = 1325146, and that to take $AM$ as being equal to the circumference of the circle would lead to a value of $\\pi$ correct to five places of decimals. If $R$ is the earth’s radius, the error in supposing $AM$ to be its circumference is less than $11$ yards.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.const", "core.units" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ii/24", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ii", "number": 24, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "65", "location": "Exercise Misc-II, problem 24", "problem_latex": "Show that the only lengths which can be constructed with the ruler\nonly, starting from a given unit length, are rational lengths.", "markdown": "Show that the only lengths which can be constructed with the ruler only, starting from a given unit length, are rational lengths.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ii/25", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ii", "number": 25, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "65", "location": "Exercise Misc-II, problem 25", "problem_latex": "\\Topic{Constructions for $\\sqrt[3]{2}$.} $O$~is the vertex and $S$~the focus of the\nparabola $y^{2} = 4x$, and $P$~is one of its points of intersection with the parabola\n$x^{2} = 2y$. Show that $OP$~meets the latus rectum of the first parabola in a point~$Q$\nsuch that $SQ = \\sqrt[3]{2}$.", "markdown": "**for $\\sqrt[3]{2}$.** $O$ is the vertex and $S$ the focus of the parabola $y^{2} = 4x$, and $P$ is one of its points of intersection with the parabola $x^{2} = 2y$. Show that $OP$ meets the latus rectum of the first parabola in a point $Q$ such that $SQ = \\sqrt[3]{2}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.const" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ii/26", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ii", "number": 26, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "65", "location": "Exercise Misc-II, problem 26", "problem_latex": "Take a circle of unit diameter, a diameter~$OA$ and the tangent at~$A$.\nDraw a chord~$OBC$ cutting the circle at~$B$ and the tangent at~$C$. On this\nline take $OM = BC$. Taking $O$~as origin and $OA$~as axis of~$x$, show that the\nlocus of~$M$ is the curve\n\\[\n(x^{2} + y^{2})x - y^{2} = 0\n\\]\n(the \\emph{Cissoid of Diocles}). Sketch the curve. Take along the axis of~$y$ a length\n$OD = 2$. Let $AD$~cut the curve in~$P$ and $OP$~cut the tangent to the circle\nat~$A$ in~$Q$. Show that $AQ = \\sqrt[3]{2}$.", "markdown": "Take a circle of unit diameter, a diameter $OA$ and the tangent at $A$. Draw a chord $OBC$ cutting the circle at $B$ and the tangent at $C$. On this line take $OM = BC$. Taking $O$ as origin and $OA$ as axis of $x$, show that the locus of $M$ is the curve (x^2 + y^2)x - y^2 = 0 (the *Cissoid of Diocles*). Sketch the curve. Take along the axis of $y$ a length $OD = 2$. Let $AD$ cut the curve in $P$ and $OP$ cut the tangent to the circle at $A$ in $Q$. Show that $AQ = \\sqrt[3]{2}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ii/3", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ii", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "65", "location": "Exercise Misc-II, problem 3", "problem_latex": "Draw the graphs of the functions\n\\[\n3\\sin x + 4\\cos x,\\quad\n\\sin\\left(\\frac{\\pi}{\\sqrt{2}} \\sin x\\right).\n\\]\n\\MathTrip{1896.}", "markdown": "Draw the graphs of the functions 3x + 4x,0pt minus 3pt(2 x). % [0]% (*Math. Trip.* 1896.)% [1]%", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ii/4", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ii", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "65", "location": "Exercise Misc-II, problem 4", "problem_latex": "Draw the graphs of the functions\n\\[\n\\sin x(a\\cos^{2} x + b\\sin^{2} x),\\quad\n\\frac{\\sin x}{x}(a\\cos^{2} x + b\\sin^{2} x),\\quad\n\\left(\\frac{\\sin x}{x}\\right)^{2}.\n\\]", "markdown": "Draw the graphs of the functions x(a^2 x + b^2 x),0pt minus 3ptxx(a^2 x + b^2 x),0pt minus 3pt(xx)^2.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ii/5", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ii", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "65", "location": "Exercise Misc-II, problem 5", "problem_latex": "Draw the graphs of the functions $x[1/x]$, $[x]/x$.", "markdown": "Draw the graphs of the functions $x[1/x]$, $[x]/x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ii/6i", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ii", "number": 6, "part": "(i)", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "65", "location": "Exercise Misc-II, problem 6(i)", "problem_latex": "Draw the graphs of the functions\n\\begin{align*}\n\\Itemp{(i)} & \\arccos(2x^{2} - 1) - 2 \\arccos{x}, \\\\\n\\Itemp{(ii)} & \\arctan \\frac{a + x}{1 - ax} - \\arctan{a} - \\arctan{x},\n\\end{align*}\nwhere the symbols $\\arccos a$, $\\arctan a$ denote, for any value of~$a$, the least\npositive (or zero) angle, whose cosine or tangent is~$a$.", "markdown": "Draw the graphs of the functions align* % [2.25em][l](i)% [2.25em][l](i)% % & (2x^2 - 1) - 2 x, % [2.25em][l](ii)% [2.25em][l](ii)% % & a + x1 - ax - a - x, align* where the symbols $\\arccos a$, $\\arctan a$ denote, for any value of $a$, the least positive (or zero) angle, whose cosine or tangent is $a$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ii/6ii", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ii", "number": 6, "part": "(ii)", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "65", "location": "Exercise Misc-II, problem 6(ii)", "problem_latex": "Draw the graphs of the functions\n\\begin{align*}\n\\Itemp{(i)} & \\arccos(2x^{2} - 1) - 2 \\arccos{x}, \\\\\n\\Itemp{(ii)} & \\arctan \\frac{a + x}{1 - ax} - \\arctan{a} - \\arctan{x},\n\\end{align*}\nwhere the symbols $\\arccos a$, $\\arctan a$ denote, for any value of~$a$, the least\npositive (or zero) angle, whose cosine or tangent is~$a$.", "markdown": "Draw the graphs of the functions align* % [2.25em][l](i)% [2.25em][l](i)% % & (2x^2 - 1) - 2 x, % [2.25em][l](ii)% [2.25em][l](ii)% % & a + x1 - ax - a - x, align* where the symbols $\\arccos a$, $\\arctan a$ denote, for any value of $a$, the least positive (or zero) angle, whose cosine or tangent is $a$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ii/7", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ii", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "65", "location": "Exercise Misc-II, problem 7", "problem_latex": "Verify the following method of constructing the graph of $f\\{\\phi(x)\\}$ by\nmeans of the line $y = x$ and the graphs of $f(x)$~and~$\\phi(x)$: take $OA = x$ along~$OX$,\ndraw $AB$ parallel to~$OY$ to meet $y = \\phi(x)$ in~$B$, $BC$~parallel to~$OX$ to\nmeet $y = x$ in~$C$, $CD$~parallel to~$OY$ to meet $y = f(x)$ in~$D$, and $DP$~parallel to~$OX$\nto meet~$AB$ in~$P$; then $P$~is a point on the graph required.", "markdown": "Verify the following method of constructing the graph of $f\\{\\phi(x)\\}$ by means of the line $y = x$ and the graphs of $f(x)$ and $\\phi(x)$: take $OA = x$ along $OX$, draw $AB$ parallel to $OY$ to meet $y = \\phi(x)$ in $B$, $BC$ parallel to $OX$ to meet $y = x$ in $C$, $CD$ parallel to $OY$ to meet $y = f(x)$ in $D$, and $DP$ parallel to $OX$ to meet $AB$ in $P$; then $P$ is a point on the graph required.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ii/8", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ii", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "65", "location": "Exercise Misc-II, problem 8", "problem_latex": "Show that the roots of $x^{3} + px + q = 0$ are the abscissae of the points of\nintersection (other than the origin) of the parabola $y = x^{2}$ and the circle\n\\[\nx^{2} + y^{2} + (p - 1)y + qx = 0.\n\\]", "markdown": "Show that the roots of $x^{3} + px + q = 0$ are the abscissae of the points of intersection (other than the origin) of the parabola $y = x^{2}$ and the circle x^2 + y^2 + (p - 1)y + qx = 0.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ii/9", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ii", "number": 9, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "65", "location": "Exercise Misc-II, problem 9", "problem_latex": "The roots of $x^{4} + nx^{3} + px^{2} + qx + r = 0$ are the abscissae of the points of\nintersection of the parabola $x^{2} = y - \\frac{1}{2}nx$ and the circle\n\\[\nx^{2} + y^{2}\n + (\\tfrac{1}{8}n^{2} - \\tfrac{1}{2}pn + \\tfrac{1}{2}n + q)x\n + (p - 1 - \\tfrac{1}{4}n^{2})y + r = 0.\n\\]", "markdown": "The roots of $x^{4} + nx^{3} + px^{2} + qx + r = 0$ are the abscissae of the points of intersection of the parabola $x^{2} = y - \\frac{1}{2}nx$ and the circle x^2 + y^2 + (18n^2 - 12pn + 12n + q)x + (p - 1 - 14n^2)y + r = 0.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-iii/1", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-iii", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "101", "location": "Exercise Misc-III, problem 1", "problem_latex": "The condition that a triangle~$(xyz)$ should be equilateral is that\n\\[\nx^{2} + y^{2} + z^{2} - yz - zx - xy = 0.\n\\]", "markdown": "The condition that a triangle $(xyz)$ should be equilateral is that x^2 + y^2 + z^2 - yz - zx - xy = 0.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-iii/10", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-iii", "number": 10, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "101", "location": "Exercise Misc-III, problem 10", "problem_latex": "Show that the necessary and sufficient conditions that both the roots\nof the equation $z^{2} + az + b = 0$ should be of unit modulus are\n\\[\n|a| \\leq 2,\\quad\n|b| = 1,\\quad\n\\am b = 2\\am a.\n\\]", "markdown": "Show that the necessary and sufficient conditions that both the roots of the equation $z^{2} + az + b = 0$ should be of unit modulus are |a| 2,0pt minus 3pt|b| = 1,0pt minus 3ptb = 2a.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-iii/11", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-iii", "number": 11, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "101", "location": "Exercise Misc-III, problem 11", "problem_latex": "If $x^{4} + 4a_{1}x^{3} + 6a_{2}x^{2} + 4a_{3}x + a_{4} = 0$ is an equation with real coefficients\nand has two real and two complex roots, concyclic in the Argand diagram, then\n\\[\na_{3}^{2} + a_{1}^{2}a_{4} + a_{2}^{3} - a_{2}a_{4} - 2a_{1}a_{2}a_{3} = 0.\n\\]", "markdown": "If $x^{4} + 4a_{1}x^{3} + 6a_{2}x^{2} + 4a_{3}x + a_{4} = 0$ is an equation with real coefficients and has two real and two complex roots, concyclic in the Argand diagram, then a_3^2 + a_1^2a_4 + a_2^3 - a_2a_4 - 2a_1a_2a_3 = 0.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-iii/12", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-iii", "number": 12, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "101", "location": "Exercise Misc-III, problem 12", "problem_latex": "The four roots of $a_{0}x^{4} + 4a_{1}x^{3} + 6a_{2}x^{2} + 4a_{3}x + a_{4} = 0$ will be harmonically\nrelated if\n\\[\na_{0}a_{3}^{2} + a_{1}^{2}a_{4} + a_{2}^{3}\n - a_{0}a_{2}a_{4} - 2a_{1}a_{2}a_{3} = 0.\n\\]", "markdown": "The four roots of $a_{0}x^{4} + 4a_{1}x^{3} + 6a_{2}x^{2} + 4a_{3}x + a_{4} = 0$ will be harmonically related if a_0a_3^2 + a_1^2a_4 + a_2^3 - a_0a_2a_4 - 2a_1a_2a_3 = 0.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-iii/13", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-iii", "number": 13, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "101", "location": "Exercise Misc-III, problem 13", "problem_latex": "\\Topic{Imaginary points and straight lines.} Let $ax + by + c = 0$ be\nan equation with complex coefficients (which of course may be real in special\ncases).\n\nIf we give $x$ any particular real or complex value, we can find the corresponding\nvalue of~$y$. The aggregate of pairs of real or complex values of $x$~and~$y$\nwhich satisfy the equation is called an \\emph{imaginary straight line}; the\npairs of values are called \\emph{imaginary points}, and are said \\emph{to lie on the line}.\nThe values of $x$~and~$y$ are called the \\emph{coordinates} of the point $(x, y)$. When\n$x$~and~$y$ are real, the point is called a \\emph{real point}: when $a$,~$b$,~$c$ are all real (or\ncan be made all real by division by a common factor), the line is called a \\emph{real\nline}. The points $x = \\alpha + \\beta i$, $y = \\gamma + \\delta i$ and $x = \\alpha - \\beta i$, $y = \\gamma - \\delta i$ are said to be\n\\emph{conjugate}; and so are the lines\n\\[\n(A + A'i)x + (B + B'i)y + C + C'i = 0,\\quad\n(A - A'i)x + (B - B'i)y + C - C'i = 0.\n\\]\n\nVerify the following assertions:---every real line contains infinitely many\npairs of conjugate imaginary points; an imaginary line in general contains\none and only one real point; an imaginary line cannot contain a pair of\nconjugate imaginary points:---and find the conditions (\\ia)~that the line\njoining two given imaginary points should be real, and (\\ib)~that the point\nof intersection of two imaginary lines should be real.", "markdown": "**points and straight lines.** Let $ax + by + c = 0$ be an equation with complex coefficients (which of course may be real in special cases). If we give $x$ any particular real or complex value, we can find the corresponding value of $y$. The aggregate of pairs of real or complex values of $x$ and $y$ which satisfy the equation is called an *imaginary straight line*; the pairs of values are called *imaginary points*, and are said *to lie on the line*. The values of $x$ and $y$ are called the *coordinates* of the point $(x, y)$. When $x$ and $y$ are real, the point is called a *real point*: when $a$, $b$, $c$ are all real (or can be made all real by division by a common factor), the line is called a *real line*. The points $x = \\alpha + \\beta i$, $y = \\gamma + \\delta i$ and $x = \\alpha - \\beta i$, $y = \\gamma - \\delta i$ are said to be *conjugate*; and so are the lines (A + A’i)x + (B + B’i)y + C + C’i = 0,0pt minus 3pt(A - A’i)x + (B - B’i)y + C - C’i = 0. Verify the following assertions:---every real line contains infinitely many pairs of conjugate imaginary points; an imaginary line in general contains one and only one real point; an imaginary line cannot contain a pair of conjugate imaginary points:---and find the conditions (*a*) that the line joining two given imaginary points should be real, and (*b*) that the point of intersection of two imaginary lines should be real.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-iii/14", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-iii", "number": 14, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "101", "location": "Exercise Misc-III, problem 14", "problem_latex": "Prove the identities\n\\begin{gather*}\n(x + y + z) (x + y\\omega_{3} + z\\omega_{3}^{2})\n (x + y\\omega_{3}^{2} + z\\omega_{3})\n = x^{3} + y^{3} + z^{3} - 3xyz,\\\\\n(x + y + z) (x + y\\omega_{5} + z\\omega_{5}^{4})\n (x + y\\omega_{5}^{2} + z\\omega_{5}^{3})\n (x + y\\omega_{5}^{3} + z\\omega_{5}^{2})\n (x + y\\omega_{5}^{4} + z\\omega_{5})\\\\\n= x^{5} + y^{5} + z^{5} - 5x^{3}yz + 5xy^{2}z^{2}.\n\\end{gather*}", "markdown": "Prove the identities gather* (x + y + z) (x + y_3 + z_3^2) (x + y_3^2 + z_3) = x^3 + y^3 + z^3 - 3xyz, (x + y + z) (x + y_5 + z_5^4) (x + y_5^2 + z_5^3) (x + y_5^3 + z_5^2) (x + y_5^4 + z_5) = x^5 + y^5 + z^5 - 5x^3yz + 5xy^2z^2. gather*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-iii/15a", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-iii", "number": 15, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "101", "location": "Exercise Misc-III, problem 15a", "problem_latex": "Solve the equations\n\\[\nx^{3} - 3ax + (a^{3} + 1) = 0,\\quad\nx^{5} - 5ax^{3} + 5a^{2}x + (a^{5} + 1) = 0.\n\\]", "markdown": "Solve the equations x^3 - 3ax + (a^3 + 1) = 0,0pt minus 3ptx^5 - 5ax^3 + 5a^2x + (a^5 + 1) = 0.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "solve", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "Eq(x**3 - 3*a*x + (a**3 + 1), 0)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "solve: Eq(a**3 - 3*a*x + x**3 + 1, 0)" ], "shape": [ "solve: Eq(N*a*x + a**N + x**N + 1, 0)" ], "same_problem_in": [], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-iii/15b", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-iii", "number": 15, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "101", "location": "Exercise Misc-III, problem 15b", "problem_latex": "Solve the equations\n\\[\nx^{3} - 3ax + (a^{3} + 1) = 0,\\quad\nx^{5} - 5ax^{3} + 5a^{2}x + (a^{5} + 1) = 0.\n\\]", "markdown": "Solve the equations x^3 - 3ax + (a^3 + 1) = 0,0pt minus 3ptx^5 - 5ax^3 + 5a^2x + (a^5 + 1) = 0.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "solve", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "Eq(x**5 - 5*a*x**3 + 5*a**2*x + (a**5 + 1), 0)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "solve: Eq(a**5 + 5*a**2*x - 5*a*x**3 + x**5 + 1, 0)" ], "shape": [ "solve: Eq(N*a*x**N + N*a**N*x + a**N + x**N + 1, 0)" ], "same_problem_in": [], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-iii/16", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-iii", "number": 16, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "101", "location": "Exercise Misc-III, problem 16", "problem_latex": "If $f(x) = a_{0} + a_{1}x + \\dots + a_{k}x^{k}$, then\n\\[\n\\{f(x) + f(\\omega x) + \\dots + f(\\omega^{n-1}x)\\}/n\n = a_{0} + a_{n}x^{n} + a_{2n}x^{2n} + \\dots + a_{\\lambda n}x^{\\lambda n},\n\\]\n$\\omega$~being any root of $x^{n} = 1$ (except $x = 1$), and $\\lambda n$~the greatest multiple of~$n$\ncontained in~$k$. Find a similar formula for $a_{\\mu} + a_{\\mu+n}x^{n} + a_{\\mu+2n}x^{2n} + \\dots$.", "markdown": "If $f(x) = a_{0} + a_{1}x + \\dots + a_{k}x^{k}$, then f(x) + f(x) + …+ f(^n-1x)/n = a_0 + a_nx^n + a_2nx^2n + …+ a_nx^n, $\\omega$ being any root of $x^{n} = 1$ (except $x = 1$), and $\\lambda n$ the greatest multiple of $n$ contained in $k$. Find a similar formula for $a_{\\mu} + a_{\\mu+n}x^{n} + a_{\\mu+2n}x^{2n} + \\dots$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-iii/17", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-iii", "number": 17, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "101", "location": "Exercise Misc-III, problem 17", "problem_latex": "If\n\\[\n(1 + x)^{n} = p_{0} + p_{1}x + p_{2}x^{2} + \\dots,\n\\]\n$n$~being a positive integer, then\n\\[\np_{0} - p_{2} + p_{4} - \\dots = 2^{\\frac{1}{2} n} \\cos\\tfrac{1}{4}n\\pi,\\quad\np_{1} - p_{3} + p_{5} - \\dots = 2^{\\frac{1}{2} n} \\sin\\tfrac{1}{4}n\\pi.\n\\]", "markdown": "If (1 + x)^n = p_0 + p_1x + p_2x^2 + …, $n$ being a positive integer, then p_0 - p_2 + p_4 - …= 2^12 n 14n,0pt minus 3ptp_1 - p_3 + p_5 - …= 2^12 n 14n.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-iii/18", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-iii", "number": 18, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "101", "location": "Exercise Misc-III, problem 18", "problem_latex": "Sum the series\n\\[\n\\frac{x}{2! \\DPchg{n - 2!}{(n - 2)!}}\n + \\frac{x^{2}}{5! \\DPchg{n - 5!}{(n - 5)!}}\n + \\frac{x^{3}}{8! \\DPchg{n - 8!}{(n - 8)!}} + \\dots\n + \\frac{x^{n/3}}{\\DPchg{n - 1!}{(n - 1)!}},\n\\]\n$n$~being a multiple of~$3$. \\MathTrip{1899.}", "markdown": "Sum the series x2! n - 2!(n - 2)! + x^25! n - 5!(n - 5)! + x^38! n - 8!(n - 8)! + … + x^n/3n - 1!(n - 1)!, $n$ being a multiple of $3$. % [0]% (*Math. Trip.* 1899.)% [1]%", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.sum" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-iii/19", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-iii", "number": 19, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "101", "location": "Exercise Misc-III, problem 19", "problem_latex": "{\\Loosen If $t$~is a complex number such that $|t| = 1$, then the point\n$x = (at + b)/(t - c)$ describes a circle as $t$~varies, unless $|c| = 1$, when it\ndescribes a straight line.}", "markdown": "0.375em plus 0.75em minus 0.25emIf $t$ is a complex number such that $|t| = 1$, then the point $x = (at + b)/(t - c)$ describes a circle as $t$ varies, unless $|c| = 1$, when it describes a straight line.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-iii/2", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-iii", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "101", "location": "Exercise Misc-III, problem 2", "problem_latex": "If $XYZ$, $X'Y'Z'$ are two triangles, and\n\\[\n\\Seg{YZ} · \\Seg{Y'Z'} = \\Seg{ZX} · \\Seg{Z'X'} = \\Seg{XY} · \\Seg{X'Y'},\n\\]\nthen both triangles are equilateral.", "markdown": "If $XYZ$, $X'Y'Z'$ are two triangles, and YZP’ · Y’Z’P’ = ZXP’ · Z’X’P’ = XYP’ · X’Y’P’, then both triangles are equilateral.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-iii/20", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-iii", "number": 20, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "101", "location": "Exercise Misc-III, problem 20", "problem_latex": "If~$t$ varies as in the last example then the point $x = \\frac{1}{2}\\{at + (b/t)\\}$ in\ngeneral describes an ellipse whose foci are given by $x^{2} = ab$, and whose axes\nare $|a| + |b|$ and $|a| - |b|$. But if $|a| = |b|$ then $x$~describes the finite straight\nline joining the points $-\\sqrtp{ab}$, $\\sqrtp{ab}$.", "markdown": "If $t$ varies as in the last example then the point $x = \\frac{1}{2}\\{at + (b/t)\\}$ in general describes an ellipse whose foci are given by $x^{2} = ab$, and whose axes are $|a| + |b|$ and $|a| - |b|$. But if $|a| = |b|$ then $x$ describes the finite straight line joining the points $-\\sqrtp{ab}$, $\\sqrtp{ab}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-iii/21", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-iii", "number": 21, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "101", "location": "Exercise Misc-III, problem 21", "problem_latex": "Prove that if $t$~is real and $z = t^{2} - 1 + \\sqrtp{t^{4} - t^{2}}$, then, when $t^{2} < 1$, $z$~is\nrepresented by a point which lies on the circle $x^{2} + y^{2} + x = 0$. Assuming that,\nwhen $t^{2} > 1$, $\\sqrtp{t^{4} - t^{2}}$ denotes the positive square root of $t^{4} - t^{2}$, discuss the\nmotion of the point which represents~$z$, as $t$~diminishes from a large positive\nvalue to a large negative value. \\MathTrip{1912.}", "markdown": "Prove that if $t$ is real and $z = t^{2} - 1 + \\sqrtp{t^{4} - t^{2}}$, then, when $t^{2} < 1$, $z$ is represented by a point which lies on the circle $x^{2} + y^{2} + x = 0$. Assuming that, when $t^{2} > 1$, $\\sqrtp{t^{4} - t^{2}}$ denotes the positive square root of $t^{4} - t^{2}$, discuss the motion of the point which represents $z$, as $t$ diminishes from a large positive value to a large negative value. % [0]% (*Math. Trip.* 1912.)% [1]%", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-iii/22", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-iii", "number": 22, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "101", "location": "Exercise Misc-III, problem 22", "problem_latex": "The coefficients of the transformation $z = (aZ + b)/(cZ + d)$ are subject\nto the condition $ad - bc = 1$. Show that, if $c \\neq 0$, there are two \\emph{fixed points}\n$\\alpha$,~$\\beta$, \\ie\\ points unaltered by the transformation, except when $(a + d)^{2} = 4$, when\nthere is only one fixed point~$\\alpha$; and that in these two cases the transformation\nmay be expressed in the forms\n\\[\n\\frac{z - \\alpha}{z - \\beta} = K\\frac{Z - \\alpha}{Z - \\beta},\\quad\n\\frac{1}{z - \\alpha} = \\frac{1}{Z - \\alpha} + K.\n\\]\n\nShow further that, if $c = 0$, there will be one fixed point~$\\alpha$ unless $a = d$,\nand that in these two cases the transformation may be expressed in the\nforms\n\\[\nz - \\alpha = K(Z - \\alpha),\\quad\nz = Z + K.\n\\]\n\nFinally, if $a$,~$b$,~$c$,~$d$ are further restricted to positive integral values (including\nzero), show that the only transformations with less than two fixed\npoints are of the forms $(1/z) = (1/Z) + K$, $z = Z + K$. \\MathTrip{1911.}", "markdown": "The coefficients of the transformation $z = (aZ + b)/(cZ + d)$ are subject to the condition $ad - bc = 1$. Show that, if $c \\neq 0$, there are two *fixed points* $\\alpha$, $\\beta$, *i.e.* points unaltered by the transformation, except when $(a + d)^{2} = 4$, when there is only one fixed point $\\alpha$; and that in these two cases the transformation may be expressed in the forms z - z - = KZ - Z - ,0pt minus 3pt1z - = 1Z - + K. Show further that, if $c = 0$, there will be one fixed point $\\alpha$ unless $a = d$, and that in these two cases the transformation may be expressed in the forms z - = K(Z - ),0pt minus 3ptz = Z + K. Finally, if $a$, $b$, $c$, $d$ are further restricted to positive integral values (including zero), show that the only transformations with less than two fixed points are of the forms $(1/z) = (1/Z) + K$, $z = Z + K$. % [0]% (*Math. Trip.* 1911.)% [1]%", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-iii/23", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-iii", "number": 23, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "101", "location": "Exercise Misc-III, problem 23", "problem_latex": "Prove that the relation $z = (1 + Zi)/(Z + i)$ transforms the part of the\naxis of~$x$ between the points $z = 1$ and $z = -1$ into a semicircle passing\nthrough the points $Z = 1$ and $Z = -1$. Find all the figures that can be obtained\nfrom the originally selected part of the axis of~$x$ by successive applications of\nthe transformation. \\MathTrip{1912.}", "markdown": "Prove that the relation $z = (1 + Zi)/(Z + i)$ transforms the part of the axis of $x$ between the points $z = 1$ and $z = -1$ into a semicircle passing through the points $Z = 1$ and $Z = -1$. Find all the figures that can be obtained from the originally selected part of the axis of $x$ by successive applications of the transformation. % [0]% (*Math. Trip.* 1912.)% [1]%", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-iii/24", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-iii", "number": 24, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "101", "location": "Exercise Misc-III, problem 24", "problem_latex": "If $z = 2Z + Z^{2}$ then the circle $|Z| = 1$ corresponds to a cardioid in the\nplane of~$z$.", "markdown": "If $z = 2Z + Z^{2}$ then the circle $|Z| = 1$ corresponds to a cardioid in the plane of $z$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-iii/25", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-iii", "number": 25, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "101", "location": "Exercise Misc-III, problem 25", "problem_latex": "Discuss the transformation $z = \\frac{1}{2}\\{Z + (1/Z)\\}$, showing in particular\nthat to the circles $X^{2} + Y^{2} = \\alpha^{2}$ correspond the confocal ellipses\n\\[\n\\frac{x^{2}}{\\left\\{\\dfrac{1}{2}\\left(\\alpha + \\dfrac{1}{\\alpha}\\right)\\right\\}^{2}}\n + \\frac{y^{2}}{\\left\\{\\dfrac{1}{2}\\left(\\alpha - \\dfrac{1}{\\alpha}\\right)\\right\\}^{2}}\n = 1.\n\\]", "markdown": "Discuss the transformation $z = \\frac{1}{2}\\{Z + (1/Z)\\}$, showing in particular that to the circles $X^{2} + Y^{2} = \\alpha^{2}$ correspond the confocal ellipses x^212(+ 1)^2 + y^212(- 1)^2 = 1.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-iii/26", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-iii", "number": 26, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "101", "location": "Exercise Misc-III, problem 26", "problem_latex": "If $(z + 1)^{2} = 4/Z$ then the unit circle in the $z$-plane corresponds to the\nparabola $R\\cos^{2} \\frac{1}{2}\\Theta = 1$ in the $Z$-plane, and the inside of the circle to the\noutside of the parabola.", "markdown": "If $(z + 1)^{2} = 4/Z$ then the unit circle in the $z$-plane corresponds to the parabola $R\\cos^{2} \\frac{1}{2}\\Theta = 1$ in the $Z$-plane, and the inside of the circle to the outside of the parabola.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-iii/27", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-iii", "number": 27, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "101", "location": "Exercise Misc-III, problem 27", "problem_latex": "Show that, by means of the transformation $z = \\{(Z - ci)/(Z + ci)\\}^{2}$,\nthe upper half of the $z$-plane may be made to correspond to the interior of\na certain semicircle in the $Z$-plane.", "markdown": "Show that, by means of the transformation $z = \\{(Z - ci)/(Z + ci)\\}^{2}$, the upper half of the $z$-plane may be made to correspond to the interior of a certain semicircle in the $Z$-plane.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-iii/28", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-iii", "number": 28, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "101", "location": "Exercise Misc-III, problem 28", "problem_latex": "If $z = Z^{2} - 1$, then as $z$~describes the circle $|z| = \\kappa$, the two corresponding\npositions of~$Z$ each describe the Cassinian oval $\\rho_{1}\\rho_{2} = \\kappa$, where\n$\\rho_{1}$,~$\\rho_{2}$ are the distances of~$Z$ from the points $-1$,~$1$. Trace the ovals for\ndifferent values of~$\\kappa$.", "markdown": "If $z = Z^{2} - 1$, then as $z$ describes the circle $|z| = \\kappa$, the two corresponding positions of $Z$ each describe the Cassinian oval $\\rho_{1}\\rho_{2} = \\kappa$, where $\\rho_{1}$, $\\rho_{2}$ are the distances of $Z$ from the points $-1$, $1$. Trace the ovals for different values of $\\kappa$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-iii/29", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-iii", "number": 29, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "101", "location": "Exercise Misc-III, problem 29", "problem_latex": "Consider the relation $az^{2} + 2hzZ + bZ^{2} + 2gz + 2fZ + c = 0$. Show that\nthere are two values of~$Z$ for which the corresponding values of~$z$ are equal,\nand \\textit{vice versa}. We call these the \\emph{branch points} in the $Z$ and $z$-planes respectively.\nShow that, if $z$~describes an ellipse whose foci are the branch\npoints, then so does~$Z$.", "markdown": "Consider the relation $az^{2} + 2hzZ + bZ^{2} + 2gz + 2fZ + c = 0$. Show that there are two values of $Z$ for which the corresponding values of $z$ are equal, and *vice versa*. We call these the *branch points* in the $Z$ and $z$-planes respectively. Show that, if $z$ describes an ellipse whose foci are the branch points, then so does $Z$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-iii/3", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-iii", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "101", "location": "Exercise Misc-III, problem 3", "problem_latex": "Similar triangles $BCX$, $CAY$, $ABZ$ are described on the sides of a\ntriangle~$ABC$. Show that the centres of gravity of $ABC$,~$XYZ$ are coincident.", "markdown": "Similar triangles $BCX$, $CAY$, $ABZ$ are described on the sides of a triangle $ABC$. Show that the centres of gravity of $ABC$, $XYZ$ are coincident.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-iii/30", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-iii", "number": 30, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "101", "location": "Exercise Misc-III, problem 30", "problem_latex": "If $z = aZ^{m} + bZ^{n}$, where $m$,~$n$ are positive integers and $a$,~$b$ real, then\nas $Z$~describes the unit circle, $z$~describes a hypo- or epi-cycloid.", "markdown": "If $z = aZ^{m} + bZ^{n}$, where $m$, $n$ are positive integers and $a$, $b$ real, then as $Z$ describes the unit circle, $z$ describes a hypo- or epi-cycloid.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-iii/31", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-iii", "number": 31, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "101", "location": "Exercise Misc-III, problem 31", "problem_latex": "Show that the transformation\n\\[\nz = \\frac{(a + di)Z_{0} + b}{cZ_{0} - (a - di)},\n\\]\nwhere $a$,~$b$,~$c$,~$d$ are real and $a^{2} + d^{2} + bc > 0$, and $Z_{0}$~denotes the conjugate of~$Z$,\nis equivalent to an inversion with respect to the circle\n\\[\nc(x^{2} + y^{2}) - 2ax - 2dy - b = 0.\n\\]\nWhat is the geometrical interpretation of the transformation when\n\\[\na^{2} + d^{2} + bc < 0?\n\\]", "markdown": "Show that the transformation z = (a + di)Z_0 + bcZ_0 - (a - di), where $a$, $b$, $c$, $d$ are real and $a^{2} + d^{2} + bc > 0$, and $Z_{0}$ denotes the conjugate of $Z$, is equivalent to an inversion with respect to the circle c(x^2 + y^2) - 2ax - 2dy - b = 0. What is the geometrical interpretation of the transformation when a^2 + d^2 + bc < 0?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-iii/32", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-iii", "number": 32, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "101", "location": "Exercise Misc-III, problem 32", "problem_latex": "The transformation\n\\[\n\\frac{1 - z}{1 + z} = \\left(\\frac{1 - Z}{1 + Z}\\right)^{c},\n\\]\nwhere $c$~is rational and $0 < c < 1$, transforms the circle $|z| = 1$ into the boundary\nof a circular lune of angle~$\\pi/c$.", "markdown": "The transformation 1 - z1 + z = (1 - Z1 + Z)^c, where $c$ is rational and $0 < c < 1$, transforms the circle $|z| = 1$ into the boundary of a circular lune of angle $\\pi/c$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-iii/4", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-iii", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "101", "location": "Exercise Misc-III, problem 4", "problem_latex": "If $X$,~$Y$,~$Z$ are points on the sides of the triangle $ABC$, such that\n\\[\nBX/XC = CY/YA = AZ/ZB = r,\n\\]\nand if $ABC$, $XYZ$ are similar, then either $r = 1$ or both triangles are\nequilateral.", "markdown": "If $X$, $Y$, $Z$ are points on the sides of the triangle $ABC$, such that BX/XC = CY/YA = AZ/ZB = r, and if $ABC$, $XYZ$ are similar, then either $r = 1$ or both triangles are equilateral.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-iii/5", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-iii", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "101", "location": "Exercise Misc-III, problem 5", "problem_latex": "If $A$,~$B$,~$C$,~$D$ are four points in a plane, then\n\\[\nAD · BC \\leq BD · CA + CD · AB.\n\\]", "markdown": "If $A$, $B$, $C$, $D$ are four points in a plane, then AD · BC BD · CA + CD · AB.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-iii/6", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-iii", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "101", "location": "Exercise Misc-III, problem 6", "problem_latex": "Deduce Ptolemy's Theorem concerning cyclic quadrilaterals from the\nfact that the cross ratios of four concyclic points are real.", "markdown": "Deduce Ptolemy’s Theorem concerning cyclic quadrilaterals from the fact that the cross ratios of four concyclic points are real.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-iii/7", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-iii", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "101", "location": "Exercise Misc-III, problem 7", "problem_latex": "If $z^{2} + z'^{2} = 1$, then the points $z$,~$z'$ are ends of conjugate diameters of an\nellipse whose foci are the points $1$,~$-1$.", "markdown": "If $z^{2} + z'^{2} = 1$, then the points $z$, $z'$ are ends of conjugate diameters of an ellipse whose foci are the points $1$, $-1$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-iii/8", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-iii", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "101", "location": "Exercise Misc-III, problem 8", "problem_latex": "Prove that $|a + b|^{2} + |a - b|^{2} = 2\\{|a|^{2} + |b|^{2}\\}$.", "markdown": "Prove that $|a + b|^{2} + |a - b|^{2} = 2\\{|a|^{2} + |b|^{2}\\}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-iii/9", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-iii", "number": 9, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "101", "location": "Exercise Misc-III, problem 9", "problem_latex": "Deduce from Ex.~8 that\n\\[\n|a + \\sqrtp{a^{2} - b^{2}}| + |a - \\sqrtp{a^{2} - b^{2}}| = |a + b| + |a - b|.\n\\]", "markdown": "Deduce from Ex. 8 that |a + a^2 - b^2| + |a - a^2 - b^2| = |a + b| + |a - b|.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-iv/1", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-iv", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "157", "location": "Exercise Misc-IV, problem 1", "problem_latex": "The function~$\\phi(n)$ takes the values $1$, $0$, $0$, $0$, $1$, $0$, $0$, $0$, $1$,~\\dots\\ when\n$n = 0$, $1$, $2$,~\\dots. Express $\\phi(n)$ in terms of~$n$ by a formula which does not\ninvolve trigonometrical functions.", "markdown": "The function $\\phi(n)$ takes the values $1$, $0$, $0$, $0$, $1$, $0$, $0$, $0$, $1$, … when $n = 0$, $1$, $2$, …. Express $\\phi(n)$ in terms of $n$ by a formula which does not involve trigonometrical functions.", "answer_latex": [ "$\\phi(n) = \\frac{1}{4}\\{1 + (-1)^{n} + i^{n} + (-i)^{n}\\}$." ], "answer_markdown": [ "$\\phi(n) = \\frac{1}{4}\\{1 + (-1)^{n} + i^{n} + (-i)^{n}\\}$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-iv/2", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-iv", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "157", "location": "Exercise Misc-IV, problem 2", "problem_latex": "If $\\phi(n)$~steadily increases, and $\\psi(n)$~steadily decreases, as $n$~tends to~$\\infty$,\nand if $\\psi(n) > \\phi(n)$ for all values of~$n$, then both $\\phi(n)$~and~$\\psi(n)$ tend to\nlimits, and $\\lim\\phi(n) \\leq \\lim\\psi(n)$.", "markdown": "If $\\phi(n)$ steadily increases, and $\\psi(n)$ steadily decreases, as $n$ tends to $\\infty$, and if $\\psi(n) > \\phi(n)$ for all values of $n$, then both $\\phi(n)$ and $\\psi(n)$ tend to limits, and $\\lim\\phi(n) \\leq \\lim\\psi(n)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-iv/3", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-iv", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "157", "location": "Exercise Misc-IV, problem 3", "problem_latex": "Prove that, if\n\\[\n\\phi(n) = \\left(1 + \\frac{1}{n}\\right)^{n},\\quad\n\\psi(n) = \\left(1 - \\frac{1}{n}\\right)^{-n},\n\\]\nthen $\\phi(n + 1) > \\phi(n)$ and $\\psi(n + 1) < \\psi(n)$.", "markdown": "Prove that, if (n) = (1 + 1n)^n,0pt minus 3pt(n) = (1 - 1n)^-n, then $\\phi(n + 1) > \\phi(n)$ and $\\psi(n + 1) < \\psi(n)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/1", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 1", "problem_latex": "Given that $\\log_{10} e = .4343$ and that $2^{10}$ and $3^{21}$ are nearly equal to powers\nof~$10$, calculate $\\log_{10}2$ and $\\log_{10}3$ to four places of decimals. \\MathTrip{1905.}", "markdown": "Given that $\\log_{10} e = .4343$ and that $2^{10}$ and $3^{21}$ are nearly equal to powers of $10$, calculate $\\log_{10}2$ and $\\log_{10}3$ to four places of decimals. % [0]% (*Math. Trip.* 1905.)% [1]%", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/10", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 10, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 10", "problem_latex": "Show that $\\dfrac{1}{\\log(1 + x)} - \\dfrac{1}{x} \\to \\dfrac{1}{2}$ as $x \\to 0$.", "markdown": "Show that $\\dfrac{1}{\\log(1 + x)} - \\dfrac{1}{x} \\to \\dfrac{1}{2}$ as $x \\to 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/11", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 11, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 11", "problem_latex": "Show that $\\dfrac{1}{\\log(1 + x)} - \\dfrac{1}{x}$ decreases steadily from $1$ to~$0$ as $x$~increases\nfrom $-1$ towards~$\\infty$.", "markdown": "Show that $\\dfrac{1}{\\log(1 + x)} - \\dfrac{1}{x}$ decreases steadily from $1$ to $0$ as $x$ increases from $-1$ towards $\\infty$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/12", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 12, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 12", "problem_latex": "Show that the function $(\\log \\xi - \\log x)/(\\xi - x)$, where $\\xi$~is positive,\ndecreases steadily as $x$~increases from $0$ to~$\\xi$, and find its limit as $x \\to \\xi$.", "markdown": "Show that the function $(\\log \\xi - \\log x)/(\\xi - x)$, where $\\xi$ is positive, decreases steadily as $x$ increases from $0$ to $\\xi$, and find its limit as $x \\to \\xi$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/13", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 13, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 13", "problem_latex": "Show that $e^{x} > Mx^{N}$, where $M$~and~$N$ are large positive numbers, \\DPtypo{f}{if}\n$x$~is greater than the greater of $2\\log M$ and~$16N^{2}$.", "markdown": "Show that $e^{x} > Mx^{N}$, where $M$ and $N$ are large positive numbers, f $x$ is greater than the greater of $2\\log M$ and $16N^{2}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/14", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 14, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 14", "problem_latex": "If $f(x)$ and $\\phi(x)$ tend to infinity as $x \\to \\infty$, and $f'(x)/\\phi'(x) \\to \\infty$,\nthen $f(x)/\\phi(x) \\to \\infty$. [Use the result of \\okrickRef{Ch.}{VI}, \\MiscEx{VI}~33.] By taking\n$f(x) = x^{\\alpha}$, $\\phi(x) = \\log x$, prove that $(\\log x)/x^{\\alpha} \\to 0$ for all positive values of~$\\alpha$.", "markdown": "If $f(x)$ and $\\phi(x)$ tend to infinity as $x \\to \\infty$, and $f'(x)/\\phi'(x) \\to \\infty$, then $f(x)/\\phi(x) \\to \\infty$. [Use the result of Ch.VI, [misc:VI]Misc. Ex. 33.] By taking $f(x) = x^{\\alpha}$, $\\phi(x) = \\log x$, prove that $(\\log x)/x^{\\alpha} \\to 0$ for all positive values of $\\alpha$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/15", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 15, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 15", "problem_latex": "If $p$ and~$q$ are positive integers then\n\\[\n\\frac{1}{pn + 1} + \\frac{1}{pn + 2} + \\dots + \\frac{1}{qn}\n \\to \\log\\left(\\frac{q}{p}\\right)\n\\]\nas $n \\to \\infty$.", "markdown": "If $p$ and $q$ are positive integers then 1pn + 1 + 1pn + 2 + …+ 1qn (qp) as $n \\to \\infty$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.log", "core.prog" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/16", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 16, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 16", "problem_latex": "Prove that if $x$~is positive then $n\\log\\{\\frac{1}{2}(1 + x^{1/n})\\} \\to -\\frac{1}{2}\\log x$ as\n$n \\to \\infty$.", "markdown": "Prove that if $x$ is positive then $n\\log\\{\\frac{1}{2}(1 + x^{1/n})\\} \\to -\\frac{1}{2}\\log x$ as $n \\to \\infty$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/17", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 17, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 17", "problem_latex": "Prove that if $a$ and~$b$ are positive then\n\\[\n\\{\\tfrac{1}{2}(a^{1/n} + b^{1/n})\\}^{n} \\to \\sqrtp{ab}.\n\\]", "markdown": "Prove that if $a$ and $b$ are positive then 12(a^1/n + b^1/n)^n ab.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/18", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 18, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 18", "problem_latex": "Show that\n\\[\n1 + \\frac{1}{3} + \\frac{1}{5} + \\dots + \\frac{1}{2n - 1}\n = \\tfrac{1}{2}\\log n + \\log 2 + \\tfrac{1}{2} \\gamma + \\epsilon_{n},\n\\]\nwhere $\\gamma$~is Euler's constant (\\Ex{lxxxix}.~1) and $\\epsilon_{n} \\to 0$ as $n \\to \\infty$.", "markdown": "Show that 1 + 13 + 15 + …+ 12n - 1 = 12n + 2 + 12 + _n, where $\\gamma$ is Euler’s constant (% [examples:lxxxix]Ex. lxxxix%. 1) and $\\epsilon_{n} \\to 0$ as $n \\to \\infty$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.const", "core.log", "core.prog" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/19", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 19, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 19", "problem_latex": "Show that\n\\[\n1 + \\tfrac{1}{3} - \\tfrac{1}{2} + \\tfrac{1}{5}\n + \\tfrac{1}{7} - \\tfrac{1}{4} + \\tfrac{1}{9} + \\dots\n = \\tfrac{3}{2} \\log 2,\n\\]\nthe series being formed from the series $1 - \\frac{1}{2} + \\frac{1}{3} - \\dots$ by taking alternately two\npositive terms and then one negative.", "markdown": "Show that 1 + 13 - 12 + 15 + 17 - 14 + 19 + … = 32 2, the series being formed from the series $1 - \\frac{1}{2} + \\frac{1}{3} - \\dots$ by taking alternately two positive terms and then one negative.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.log", "core.prog" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/2", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 2", "problem_latex": "Determine which of $(\\frac{1}{2}e)^{\\sqrt{3}}$ and $(\\sqrt{2})^{\\frac{1}{2}\\pi}$ is the greater.", "markdown": "Determine which of $(\\frac{1}{2}e)^{\\sqrt{3}}$ and $(\\sqrt{2})^{\\frac{1}{2}\\pi}$ is the greater.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.const", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/20", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 20, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 20", "problem_latex": "Show that $1 - \\frac{1}{2} - \\frac{1}{4} + \\frac{1}{3} - \\frac{1}{6} - \\frac{1}{8} + \\frac{1}{5} - \\frac{1}{10} - \\dots = \\frac{1}{2}\\log 2$.", "markdown": "Show that $1 - \\frac{1}{2} - \\frac{1}{4} + \\frac{1}{3} - \\frac{1}{6} - \\frac{1}{8} + \\frac{1}{5} - \\frac{1}{10} - \\dots = \\frac{1}{2}\\log 2$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.log", "core.prog" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/21", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 21, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 21", "problem_latex": "Prove that\n\\[\n\\sum_{1}^{n} \\frac{1}{\\nu(36\\nu^{2} - 1)}\n = -3 + 3\\Sigma_{3n+1} - \\Sigma_{n} - S_{n}\n\\]\nwhere $S_{n} = 1 + \\dfrac{1}{2} + \\dots + \\dfrac{1}{n}$, $\\Sigma_{n} = 1 + \\dfrac{1}{3} + \\dots + \\dfrac{1}{2n - 1}$. Hence prove that the sum\nof the series when continued to infinity is\n\\[\n-3 + \\tfrac{3}{2}\\log 3 + 2\\log 2.\n\\]\n\\MathTrip{1905.}", "markdown": "Prove that _1^n 1(36^2 - 1) = -3 + 3_3n+1 - _n - S_n where $S_{n} = 1 + \\dfrac{1}{2} + \\dots + \\dfrac{1}{n}$, $\\Sigma_{n} = 1 + \\dfrac{1}{3} + \\dots + \\dfrac{1}{2n - 1}$. Hence prove that the sum of the series when continued to infinity is -3 + 323 + 22. % [0]% (*Math. Trip.* 1905.)% [1]%", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.log", "core.prog" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/22", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 22, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 22", "problem_latex": "Show that\n\\[\n\\sum_{1}^{\\infty} \\frac{1}{n(4n^{2} - 1)} = 2\\log 2 - 1, \\quad\n\\sum_{1}^{\\infty} \\frac{1}{n(9n^{2} - 1)} = \\tfrac{3}{2}(\\log 3 - 1).\n\\]", "markdown": "Show that _1^ 1n(4n^2 - 1) = 22 - 1, 0pt minus 3pt_1^ 1n(9n^2 - 1) = 32(3 - 1).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.log", "core.prog" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/23", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 23, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 23", "problem_latex": "Prove that the sums of the four series\n\\[\n\\sum_{1}^{\\infty} \\frac{1}{4n^{2} - 1},\\quad\n\\sum_{1}^{\\infty} \\frac{(-1)^{n-1}}{4n^{2} - 1},\\quad\n\\sum_{1}^{\\infty} \\frac{1}{(2n + 1)^{2} - 1},\\quad\n\\sum_{1}^{\\infty} \\frac{(-1)^{n-1}}{(2n + 1)^{2} - 1}\n\\]\nare $\\frac{1}{2}$, $\\frac{1}{4}\\pi - \\frac{1}{2}$, $\\frac{1}{4}$, $\\frac{1}{2}\\log 2 - \\frac{1}{4}$ respectively.", "markdown": "Prove that the sums of the four series _1^ 14n^2 - 1,0pt minus 3pt_1^ (-1)^n-14n^2 - 1,0pt minus 3pt_1^ 1(2n + 1)^2 - 1,0pt minus 3pt_1^ (-1)^n-1(2n + 1)^2 - 1 are $\\frac{1}{2}$, $\\frac{1}{4}\\pi - \\frac{1}{2}$, $\\frac{1}{4}$, $\\frac{1}{2}\\log 2 - \\frac{1}{4}$ respectively.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.const", "core.log", "core.prog" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/24", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 24, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 24", "problem_latex": "Prove that $n!\\, (a/n)^{n}$ tends to~$0$ or to~$\\infty$ according as $a < e$ or $a > e$.", "markdown": "Prove that $n!\\, (a/n)^{n}$ tends to $0$ or to $\\infty$ according as $a < e$ or $a > e$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.const" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/25", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 25, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 25", "problem_latex": "Find the limit as $x \\to \\infty$ of\n\\[\n\\left(\\frac{a_{0} + a_{1} x + \\dots + a_{r} x^{r}}\n {b_{0} + b_{1} x + \\dots + b_{r} x^{r}}\\right)^{\\lambda_{0}+\\lambda_{1}x},\n\\]\ndistinguishing the different cases which may arise. \\MathTrip{1886.}", "markdown": "Find the limit as $x \\to \\infty$ of (a_0 + a_1 x + …+ a_r x^r b_0 + b_1 x + …+ b_r x^r)^_0+_1x, distinguishing the different cases which may arise. % [0]% (*Math. Trip.* 1886.)% [1]%", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/26", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 26, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 26", "problem_latex": "Prove that\n\\[\n\\sum \\log \\left(1 + \\frac{x}{n}\\right)\\quad (x > 0)\n\\]\ndiverges to~$\\infty$. [Compare with $\\sum (x/n)$.] Deduce that if $x$~is positive then\n\\[\n(1 + x)(2 + x) \\dots (n + x)/n! \\to \\infty\n\\]\nas $n \\to \\infty$. [The logarithm of the function is $\\sum\\limits_{1}^{n} \\log \\left(1 + \\dfrac{x}{\\nu}\\right)$.]", "markdown": "Prove that (1 + xn)0pt minus 3pt(x > 0) diverges to $\\infty$. [Compare with $\\sum (x/n)$.] Deduce that if $x$ is positive then (1 + x)(2 + x) …(n + x)/n! as $n \\to \\infty$. [The logarithm of the function is $\\sum\\limits_{1}^{n} \\log \\left(1 + \\dfrac{x}{\\nu}\\right)$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/27", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 27, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 27", "problem_latex": "Prove that if $x > -1$ then\n\\begin{multline*}\n\\frac{1}{(x + 1)^{2}}\n = \\frac{1}{(x + 1) (x + 2)}\n + \\frac{1!}{(x + 1) (x + 2) (x + 3)}\\\\\n + \\frac{2!}{(x + 1) (x + 2) (x + 3) (x + 4)} + \\dots.\n\\end{multline*}\n\\MathTrip{1908.}", "markdown": "Prove that if $x > -1$ then multline* 1(x + 1)^2 = 1(x + 1) (x + 2) + 1!(x + 1) (x + 2) (x + 3) + 2!(x + 1) (x + 2) (x + 3) (x + 4) + …. multline* % [0]% (*Math. Trip.* 1908.)% [1]%", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.sum", "core.prog" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/28", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 28, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 28", "problem_latex": "No equation of the type\n\\[\nAe^{\\alpha x} + Be^{\\beta x} + \\dots = 0,\n\\]\nwhere $A$, $B$,~\\dots\\ are polynomials and $\\alpha$, $\\beta$,~\\dots\\ different real numbers, can hold\nfor all values of~$x$.", "markdown": "No equation of the type Ae^x + Be^x + …= 0, where $A$, $B$, … are polynomials and $\\alpha$, $\\beta$, … different real numbers, can hold for all values of $x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/29", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 29, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 29", "problem_latex": "Show that the sequence\n\\[\na_{1} = e,\\quad\na_{2} = e^{e^{2}},\\quad\na_{3} = e^{e^{e^{3}}},\\ \\dots\n\\]\ntends to infinity more rapidly than any member of the exponential scale.", "markdown": "Show that the sequence a_1 = e,0pt minus 3pta_2 = e^e^2,0pt minus 3pta_3 = e^e^e^3, … tends to infinity more rapidly than any member of the exponential scale.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/3", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 3", "problem_latex": "Show that $\\log_{10}n$ cannot be a rational number if $n$~is any positive\ninteger not a power of~$10$.", "markdown": "Show that $\\log_{10}n$ cannot be a rational number if $n$ is any positive integer not a power of $10$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/30", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 30, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 30", "problem_latex": "Prove that\n\\[\n\\frac{d}{dx} \\{\\phi(x)\\}^{\\psi(x)}\n = \\frac{d}{dx} \\{\\phi(x)\\}^{\\alpha} + \\frac{d}{dx} \\{\\beta^{\\psi(x)}\\}\n\\]\nwhere $\\alpha$~is to be put equal to~$\\psi(x)$ and $\\beta$ to~$\\phi(x)$ after differentiation.\nEstablish a similar rule for the differentiation of $\\phi(x)^{[\\{\\psi(x)\\}^{\\chi(x)}]}$.", "markdown": "Prove that ddx (x)^(x) = ddx (x)^ + ddx ^(x) where $\\alpha$ is to be put equal to $\\psi(x)$ and $\\beta$ to $\\phi(x)$ after differentiation. Establish a similar rule for the differentiation of $\\phi(x)^{[\\{\\psi(x)\\}^{\\chi(x)}]}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/31", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 31, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 31", "problem_latex": "Prove that if $D_{x}^{n} e^{-x^{2}} = e^{-x^{2}} \\phi_{n}(x)$ then (i)~$\\phi_{n}(x)$ is a polynomial of\ndegree~$n$, (ii)~$\\phi_{n+1} = -2x\\phi_{n} + \\phi_{n}'$, and (iii)~all the roots of $\\phi_{n} = 0$ are real and\ndistinct, and separated by those of $\\phi_{n-1} = 0$. [To prove~(iii) assume the truth\n% [** TN: Typo in original; fixed while swapping roles of n and \\kappa]\nof the result for $\\DPtypo{n}{\\kappa} = 1$, $2$,~\\dots\\Add{,} $\\DPtypo{\\kappa}{n}$, and consider the signs of~$\\DPtypo{\\phi_{\\kappa+1}}{\\phi_{n+1}}$ for the $n$~values\nof~$x$ for which $\\DPtypo{\\phi_{\\kappa}}{\\phi_{n}} = 0$ and for large (positive or negative) values of~$x$.]", "markdown": "Prove that if $D_{x}^{n} e^{-x^{2}} = e^{-x^{2}} \\phi_{n}(x)$ then (i) $\\phi_{n}(x)$ is a polynomial of degree $n$, (ii) $\\phi_{n+1} = -2x\\phi_{n} + \\phi_{n}'$, and (iii) all the roots of $\\phi_{n} = 0$ are real and distinct, and separated by those of $\\phi_{n-1} = 0$. [To prove (iii) assume the truth % [** TN: Typo in original; fixed while swapping roles of n and ] of the result for $\\DPtypo{n}{\\kappa} = 1$, $2$, … $\\DPtypo{\\kappa}{n}$, and consider the signs of $\\DPtypo{\\phi_{\\kappa+1}}{\\phi_{n+1}}$ for the $n$ values of $x$ for which $\\DPtypo{\\phi_{\\kappa}}{\\phi_{n}} = 0$ and for large (positive or negative) values of $x$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/32", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 32, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 32", "problem_latex": "The general solution of $f(xy) = f(x)f(y)$, where $f$~is a differentiable\nfunction, is~$x^{a}$, where $a$~is a constant: and that of\n\\[\nf(x + y) + f(x - y) = 2f(x)f(y)\n\\]\nis $\\cosh ax$ or $\\cos ax$, according as $f''(0)$~is positive or negative.", "markdown": "The general solution of $f(xy) = f(x)f(y)$, where $f$ is a differentiable function, is $x^{a}$, where $a$ is a constant: and that of f(x + y) + f(x - y) = 2f(x)f(y) is $\\cosh ax$ or $\\cos ax$, according as $f''(0)$ is positive or negative.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.hyp" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/33", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 33, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 33", "problem_latex": "How do the functions $x^{\\sin(1/x)}$, $x^{\\sin^{2}(1/x)}$, $x^{\\cosec(1/x)}$ behave as $x \\to +0$?", "markdown": "How do the functions $x^{\\sin(1/x)}$, $x^{\\sin^{2}(1/x)}$, $x^{\\cosec(1/x)}$ behave as $x \\to +0$?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/34", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 34, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 34", "problem_latex": "Trace the curves $y = \\tan x e^{\\tan x}$, $y = \\sin x \\log \\tan \\frac{1}{2}x$.", "markdown": "Trace the curves $y = \\tan x e^{\\tan x}$, $y = \\sin x \\log \\tan \\frac{1}{2}x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/35", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 35, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 35", "problem_latex": "The equation $e^{x} = ax + b$ has one real root if $a < 0$ or $a = 0$, $b > 0$. If\n$a > 0$ then it has two real roots or none, according as $a\\log a > b - a$ or\n$a\\log a < b - a$.", "markdown": "The equation $e^{x} = ax + b$ has one real root if $a < 0$ or $a = 0$, $b > 0$. If $a > 0$ then it has two real roots or none, according as $a\\log a > b - a$ or $a\\log a < b - a$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.log", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/36", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 36, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 36", "problem_latex": "Show by graphical considerations that the equation $e^{x} = ax^{2} + 2bx + c$\nhas one, two, or three real roots if $a > 0$, none, one, or two if $a < 0$; and show\nhow to distinguish between the different cases.", "markdown": "Show by graphical considerations that the equation $e^{x} = ax^{2} + 2bx + c$ has one, two, or three real roots if $a > 0$, none, one, or two if $a < 0$; and show how to distinguish between the different cases.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/37", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 37, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 37", "problem_latex": "Trace the curve $y = \\dfrac{1}{x} \\log\\left(\\dfrac{e^{x} - 1}{x}\\right)$, showing that the point $(0, \\frac{1}{2})$ is\na centre of symmetry, and that as $x$~increases through all real values, $y$~steadily\nincreases from $0$ to~$1$. Deduce that the equation\n\\[\n\\frac{1}{x} \\log\\left(\\frac{e^{x} - 1}{x}\\right) = \\alpha\n\\]\nhas no real root unless $0 < \\alpha < 1$, and then one, whose sign is the same as\nthat of $\\alpha - \\frac{1}{2}$.", "markdown": "Trace the curve $y = \\dfrac{1}{x} \\log\\left(\\dfrac{e^{x} - 1}{x}\\right)$, showing that the point $(0, \\frac{1}{2})$ is a centre of symmetry, and that as $x$ increases through all real values, $y$ steadily increases from $0$ to $1$. Deduce that the equation 1x (e^x - 1x) = has no real root unless $0 < \\alpha < 1$, and then one, whose sign is the same as that of $\\alpha - \\frac{1}{2}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.graph", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/38", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 38, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 38", "problem_latex": "Trace the curve $y = e^{1/x} \\sqrtp{x^{2} + 2x}$, and show that the equation\n\\[\ne^{1/x} \\sqrtp{x^{2} + 2x} = \\alpha\n\\]\nhas no real roots if $\\alpha$~is negative, one negative root if\n\\[\n0 < \\alpha < a = e^{1/\\sqrt{2}} \\sqrtp{2 + 2\\sqrt{2}},\n\\]\nand two positive roots and one negative if $\\alpha > a$.", "markdown": "Trace the curve $y = e^{1/x} \\sqrtp{x^{2} + 2x}$, and show that the equation e^1/x x^2 + 2x = has no real roots if $\\alpha$ is negative, one negative root if 0 < < a = e^1/2 2 + 22, and two positive roots and one negative if $\\alpha > a$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "FLAG-EXTRACTION", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/39", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 39, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 39", "problem_latex": "Show that the equation $f_{n}(x) = 1 + x + \\dfrac{x^{2}}{2!} + \\dots + \\dfrac{x^{n}}{n!} = 0$ has one real\nroot if $n$~is odd and none if $n$~is even.", "markdown": "Show that the equation $f_{n}(x) = 1 + x + \\dfrac{x^{2}}{2!} + \\dots + \\dfrac{x^{n}}{n!} = 0$ has one real root if $n$ is odd and none if $n$ is even.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/4", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 4", "problem_latex": "For what values of~$x$ are the functions $\\log x$, $\\log\\log x$, $\\log\\log\\log x$,~\\dots\\\n(\\ia)~equal to~$0$ (\\ib)~equal to~$1$ (\\ic)~not defined? Consider also the same question\nfor the functions $lx$, $llx$, $lllx$,~\\dots, where $lx = \\log |x|$.", "markdown": "For what values of $x$ are the functions $\\log x$, $\\log\\log x$, $\\log\\log\\log x$, … (*a*) equal to $0$ (*b*) equal to $1$ (*c*) not defined? Consider also the same question for the functions $lx$, $llx$, $lllx$, …, where $lx = \\log |x|$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/40", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 40, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 40", "problem_latex": "Prove that if $a$~and~$b$ are positive and nearly equal then\n\\[\n\\log \\frac{a}{b} = \\frac{1}{2}(a - b) \\left(\\frac{1}{a} + \\frac{1}{b}\\right),\n\\]\napproximately, the error being about $\\frac{1}{6}\\{(a - b)/a\\}^{3}$.", "markdown": "Prove that if $a$ and $b$ are positive and nearly equal then ab = 12(a - b) (1a + 1b), approximately, the error being about $\\frac{1}{6}\\{(a - b)/a\\}^{3}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/41a", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 41, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 41a", "problem_latex": "Prove by multiplication of series that if $-1 < x < 1$ then\n\\begin{align*}\n\\tfrac{1}{2}\\{\\log(1 + x)\\}^{2}\n &= \\tfrac{1}{2} x^{2}\n - \\tfrac{1}{3}(1 + \\tfrac{1}{2})x^{3}\n + \\tfrac{1}{4}(1 + \\tfrac{1}{2} + \\tfrac{1}{3})x^{4} - \\dots,\\\\\n\\tfrac{1}{2}(\\arctan x)^{2}\n &= \\tfrac{1}{2} x^{2}\n - \\tfrac{1}{4}(1 + \\tfrac{1}{3})x^{4}\n + \\tfrac{1}{6}(1 + \\tfrac{1}{3} + \\tfrac{1}{5})x^{6} - \\dots.\n\\end{align*}", "markdown": "Prove by multiplication of series that if $-1 < x < 1$ then align* 12(1 + x)^2 &= 12 x^2 - 13(1 + 12)x^3 + 14(1 + 12 + 13)x^4 - …, 12(x)^2 &= 12 x^2 - 14(1 + 13)x^4 + 16(1 + 13 + 15)x^6 - …. align*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.log", "core.prog", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/42", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 42, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 42", "problem_latex": "Prove that\n\\[\n(1 + \\alpha x)^{1/x}\n = e^{\\alpha}\\{1 - \\tfrac{1}{2} a^{2}x\n + \\tfrac{1}{24}(8 + 3a)a^{3}x^{2}(1 + \\epsilon_{x})\\},\n\\]\nwhere $\\epsilon_{x} \\to 0$ with~$x$.", "markdown": "Prove that (1 + x)^1/x = e^1 - 12 a^2x + 124(8 + 3a)a^3x^2(1 + _x), where $\\epsilon_{x} \\to 0$ with $x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/43", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 43, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 43", "problem_latex": "The first $n + 2$ terms in the expansion of $\\log\\left(1 + x + \\dfrac{x^{2}}{2!} + \\dots + \\dfrac{x^{n}}{n!}\\right)$ in\npowers of~$x$ are\n\\[\nx - \\frac{x^{n+1}}{n!}\n \\left\\{\\frac{1}{n + 1}\n - \\frac{x}{1!\\, (n + 2)}\n + \\frac{x^{2}}{2!\\, (n + 3)} - \\dots\n + (-1)^{n} \\frac{x^{n}}{n!\\, (2n + 1)}\n \\right\\}.\n\\]\n\\MathTrip{1899.}", "markdown": "The first $n + 2$ terms in the expansion of $\\log\\left(1 + x + \\dfrac{x^{2}}{2!} + \\dots + \\dfrac{x^{n}}{n!}\\right)$ in powers of $x$ are x - x^n+1n! 1n + 1 - x1!  (n + 2) + x^22!  (n + 3) - … + (-1)^n x^nn!  (2n + 1) . % [0]% (*Math. Trip.* 1899.)% [1]%", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.log", "core.prog" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/44", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 44, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 44", "problem_latex": "Show that the expansion of\n\\[\n\\exp \\left(-x - \\frac{x^{2}}{2} - \\dots - \\frac{x^{n}}{n}\\right)\n\\]\nin powers of~$x$ begins with the terms\n\\[\n1 - x + \\frac{x^{n+1}}{n + 1}\n - \\sum_{s=1}^{n} \\frac{x^{n+s+1}}{(n + s)(n + s + 1)}.\n\\]\n\\MathTrip{1909.}", "markdown": "Show that the expansion of (-x - x^22 - …- x^nn) in powers of $x$ begins with the terms 1 - x + x^n+1n + 1 - _s=1^n x^n+s+1(n + s)(n + s + 1). % [0]% (*Math. Trip.* 1909.)% [1]%", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.prog" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/45", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 45, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 45", "problem_latex": "Show that if $-1 < x < 1$ then\n\\begin{align*}\n\\frac{1}{3}x + \\frac{1·4}{3·6}2^{2}x^{2} + \\frac{1·4·7}{3·6·9}3^{2}x^{3} + \\dots\n &= \\frac{x(x + 3)}{9(1 - x)^{7/3}},\\\\\n\\frac{1}{3}x + \\frac{1·4}{3·6}2^{3}x^{2} + \\frac{1·4·7}{3·6·9}3^{3}x^{3} + \\dots\n &= \\frac{x(x^{2} + 18x + 9)}{27(1 - x)^{10/3}}.\n\\end{align*}", "markdown": "Show that if $-1 < x < 1$ then align* 13x + 1·43·62^2x^2 + 1·4·73·6·93^2x^3 + … &= x(x + 3)9(1 - x)^7/3, 13x + 1·43·62^3x^2 + 1·4·73·6·93^3x^3 + … &= x(x^2 + 18x + 9)27(1 - x)^10/3. align*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.prog" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/46", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 46, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 46", "problem_latex": "Prove that\n\\begin{align*}\n\\int_{0}^{\\infty} \\frac{dx}{(x + a)(x + b)}\n &= \\frac{1}{a - b} \\log\\left(\\frac{a}{b}\\right), \\\\\n\\int_{0}^{\\infty} \\frac{dx}{(x + a)(x + b)^{2}}\n &= \\frac{1}{(a - b)^{2}b}\\left\\{a - b - b\\log\\left(\\frac{a}{b}\\right)\\right\\},\\\\\n\\int_{0}^{\\infty} \\frac{x\\, dx}{(x + a)(x + b)^{2}}\n &= \\frac{1}{(a - b)^{2}} \\left\\{a\\log\\left(\\frac{a}{b}\\right) - a + b\\right\\},\\\\\n\\int_{0}^{\\infty} \\frac{dx}{(x + a)(x^{2} + b^{2})}\n &= \\frac{1}{(a^{2} + b^{2})b} \\left\\{\\tfrac{1}{2}\\pi a - b\\log\\left(\\frac{a}{b}\\right)\\right\\},\\\\\n\\int_{0}^{\\infty} \\frac{x\\, dx}{(x + a)(x^{2} + b^{2})}\n &= \\frac{1}{a^{2} + b^{2}} \\left\\{\\tfrac{1}{2}\\pi b + a\\log\\left(\\frac{a}{b}\\right)\\right\\},\n\\end{align*}\nprovided that $a$~and~$b$ are positive. Deduce, and verify independently, that\neach of the functions\n\\[\na - 1 - \\log a,\\quad\na\\log a - a + 1,\\quad\n\\tfrac{1}{2}\\pi a - \\log a,\\quad\n\\tfrac{1}{2}\\pi + a\\log a\n\\]\nis positive for all positive values of~$a$.", "markdown": "Prove that align* _0^ dx(x + a)(x + b) &= 1a - b (ab), _0^ dx(x + a)(x + b)^2 &= 1(a - b)^2ba - b - b(ab), _0^ x  dx(x + a)(x + b)^2 &= 1(a - b)^2 a(ab) - a + b, _0^ dx(x + a)(x^2 + b^2) &= 1(a^2 + b^2)b 12a - b(ab), _0^ x  dx(x + a)(x^2 + b^2) &= 1a^2 + b^2 12b + a(ab), align* provided that $a$ and $b$ are positive. Deduce, and verify independently, that each of the functions a - 1 - a,0pt minus 3ptaa - a + 1,0pt minus 3pt12a - a,0pt minus 3pt12+ aa is positive for all positive values of $a$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.defint", "cas.integrate", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/47", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 47, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 47", "problem_latex": "Prove that if $\\alpha$,~$\\beta$,~$\\gamma$ are all positive, and $\\beta^{2} > \\alpha\\gamma$, then\n\\[\n\\int_{0}^{\\infty} \\frac{dx}{\\alpha x^{2} + 2\\beta x + \\gamma}\n = \\frac{1}{\\sqrtp{\\beta^{2} - \\alpha\\gamma}}\n \\log \\left\\{\\frac{\\beta + \\sqrtp{\\beta^{2} - \\alpha\\gamma}}\n {\\sqrtp{\\alpha\\gamma}}\n \\right\\};\n\\]\nwhile if $\\alpha$~is positive and $\\alpha\\gamma > \\beta^{2}$ the value of the integral is\n\\[\n\\frac{1}{\\sqrtp{\\alpha\\gamma - \\beta^{2}}}\n \\arctan \\left\\{\\frac{\\sqrtp{\\alpha\\gamma - \\beta^{2}}}{\\beta}\\right\\},\n\\]\nthat value of the inverse tangent being chosen which lies between $0$ and~$\\pi$.\nAre there any other really different cases in which the integral is convergent?", "markdown": "Prove that if $\\alpha$, $\\beta$, $\\gamma$ are all positive, and $\\beta^{2} > \\alpha\\gamma$, then _0^ dxx^2 + 2x + = 1^2 - + ^2 - ; while if $\\alpha$ is positive and $\\alpha\\gamma > \\beta^{2}$ the value of the integral is 1- ^2 - ^2, that value of the inverse tangent being chosen which lies between $0$ and $\\pi$. Are there any other really different cases in which the integral is convergent?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.defint", "cas.integrate", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/48", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 48, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 48", "problem_latex": "Prove that if $a > -1$ then\n\\[\n\\int_{1}^{\\infty} \\frac{dx}{(x + a)\\sqrtp{x^{2} - 1}}\n = \\int_{0}^{\\infty} \\frac{dt}{\\cosh t + a}\n = 2\\int_{1}^{\\infty}\\frac{du}{u^{2} + 2au + 1};\n\\]\nand deduce that the value of the integral is\n\\[\n\\frac{2}{\\sqrtp{1 - a^{2}}} \\arctan \\bigsqrtp{\\frac{1 - a}{1 + a}}\n\\]\nif $-1 < a < 1$, and\n\\[\n\\frac{1}{\\sqrtp{a^{2} - 1}}\n \\log\\frac{\\sqrtp{a + 1} + \\sqrtp{a - 1}}\n {\\sqrtp{a + 1} - \\sqrtp{a - 1}}\n = \\frac{2}{\\sqrtp{a^{2} - 1}} \\argtanh \\bigsqrtp{\\frac{a - 1}{a + 1}}\n\\]\nif $a > 1$. Discuss the case in which $a = 1$.", "markdown": "Prove that if $a > -1$ then _1^ dx(x + a)x^2 - 1 = _0^ dtt + a = 2_1^duu^2 + 2au + 1; and deduce that the value of the integral is 21 - a^2 1 - a1 + a if $-1 < a < 1$, and 1a^2 - 1 a + 1 + a - 1 a + 1 - a - 1 = 2a^2 - 1 a - 1a + 1 if $a > 1$. Discuss the case in which $a = 1$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "FLAG-EXTRACTION", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate", "core.hyp", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/49", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 49, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 49", "problem_latex": "Transform the integral $\\ds\\int_{0}^{\\infty} \\frac{dx}{(x + a) \\sqrtp{x^{2} + 1}}$, where $a > 0$, in the same\nways, showing that its value is\n\\[\n\\frac{1}{\\sqrtp{a^{2} + 1}}\n \\log\\frac{a + 1 + \\sqrtp{a^{2} + 1}}{a + 1 - \\sqrtp{a^{2} + 1}}\n = \\frac{2}{\\sqrtp{a^{2} + 1}} \\argtanh \\frac{\\sqrtp{a^{2} + 1}}{a + 1}\\Add{.}\n\\]", "markdown": "Transform the integral $\\ds\\int_{0}^{\\infty} \\frac{dx}{(x + a) \\sqrtp{x^{2} + 1}}$, where $a > 0$, in the same ways, showing that its value is 1a^2 + 1 a + 1 + a^2 + 1a + 1 - a^2 + 1 = 2a^2 + 1 a^2 + 1a + 1", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.defint", "cas.integrate", "core.hyp", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/5", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 5", "problem_latex": "Show that\n\\[\n\\log x - \\binom{n}{1} \\log(x + 1) + \\binom{n}{2} \\log(x + 2) - \\dots\n + (-1)^{n} \\log(x + n)\n\\]\nis negative and increases steadily towards $0$ as $x$~increases from $0$ towards~$\\infty$.", "markdown": "Show that x - n1 (x + 1) + n2 (x + 2) - … + (-1)^n (x + n) is negative and increases steadily towards $0$ as $x$ increases from $0$ towards $\\infty$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.log", "core.prog" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/50", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 50, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 50", "problem_latex": "Prove that\n\\[\n\\int_{0}^{1} \\arctan x\\, dx = \\tfrac{1}{4}\\pi - \\tfrac{1}{2}\\log 2.\n\\]", "markdown": "Prove that _0^1 x  dx = 14- 122.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.defint", "core.const", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/51", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 51, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 51", "problem_latex": "If $0 < \\alpha < 1$, $0 < \\beta < 1$, then\n\\[\n\\int_{-1}^{1} \\frac{dx}{\\sqrtb{(1 - 2\\alpha x + \\alpha^{2})(1 - 2\\beta x + \\beta^{2})}}\n = \\frac{1}{\\sqrtp{\\alpha\\beta}}\n \\log \\frac{1 + \\sqrtp{\\alpha\\beta}}{1 - \\sqrtp{\\alpha\\beta}}.\n\\]", "markdown": "If $0 < \\alpha < 1$, $0 < \\beta < 1$, then _-1^1 dx(1 - 2x + ^2)(1 - 2x + ^2) = 1 1 + 1 - .", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.defint", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/52", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 52, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 52", "problem_latex": "Prove that if $a > b > 0$ then\n\\[\n\\int_{-\\infty}^{\\infty} \\frac{d\\theta}{a\\cosh \\theta + b\\sinh \\theta}\n = \\frac{\\pi}{\\sqrtp{a^{2} - b^{2}}}\\Add{.}\n\\]", "markdown": "Prove that if $a > b > 0$ then _-^ da+ b = a^2 - b^2", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.defint", "core.hyp" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/53", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 53, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 53", "problem_latex": "Prove that\n\\[\n\\int_{0}^{1} \\frac{\\log x}{1 + x^{2}}\\, dx\n = -\\int_{1}^{\\infty} \\frac{\\log x}{1 + x^{2}}\\, dx,\\quad\n\\int_{0}^{\\infty} \\frac{\\log x}{1 + x^{2}}\\, dx = 0\\Add{,}\n\\]\nand deduce that if $a > 0$ then\n\\[\n\\int_{0}^{\\infty} \\frac{\\log x}{a^{2} + x^{2}}\\, dx = \\frac{\\pi}{2a}\\log a.\n\\]", "markdown": "Prove that _0^1 x1 + x^2  dx = -_1^ x1 + x^2  dx,0pt minus 3pt_0^ x1 + x^2  dx = 0 and deduce that if $a > 0$ then _0^ xa^2 + x^2  dx = 2aa.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.defint", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/54", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 54, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 54", "problem_latex": "Prove that\n\\[\n%[** TN: In-line in the original]\n\\int_{0}^{\\infty} \\log \\left(1 + \\frac{a^{2}}{x^{2}}\\right) dx = \\pi a\n\\]\nif $a > 0$.", "markdown": "Prove that %[** TN: In-line in the original] _0^ (1 + a^2x^2) dx = a if $a > 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.defint", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/6", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 6", "problem_latex": "Prove that\n\\[\n\\left(\\frac{d}{dx}\\right)^{n} \\frac{\\log x}{x}\n = \\frac{(-1)^{n} n!}{x^{n+1}} \\left(\\log x - 1 - \\frac{1}{2} - \\dots - \\frac{1}{n}\\right).\n\\]\n\\MathTrip{1909.}", "markdown": "Prove that (ddx)^n xx = (-1)^n n!x^n+1 (x - 1 - 12 - …- 1n). % [0]% (*Math. Trip.* 1909.)% [1]%", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/7", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 7", "problem_latex": "If $x > -1$ then $x^{2} > (1 + x) \\{\\log(1 + x)\\}^{2}$. \\MathTrip{1906.}", "markdown": "If $x > -1$ then $x^{2} > (1 + x) \\{\\log(1 + x)\\}^{2}$. % [0]% (*Math. Trip.* 1906.)% [1]%", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.hyp", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/8", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 8", "problem_latex": "Show that $\\{\\log(1 + x)\\}/x$ and $x/\\{(1 + x)\\log(1 + x)\\}$ both decrease steadily\nas $x$~increases from $0$ towards~$\\infty$.", "markdown": "Show that $\\{\\log(1 + x)\\}/x$ and $x/\\{(1 + x)\\log(1 + x)\\}$ both decrease steadily as $x$ increases from $0$ towards $\\infty$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-ix/9", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-ix", "number": 9, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "387", "location": "Exercise Misc-IX, problem 9", "problem_latex": "Show that, as $x$~increases from $-1$ towards~$\\infty$, the function\n$(1 + x)^{-1/x}$ assumes once and only once every value between $0$ and~$1$. \\MathTrip{1910.}", "markdown": "Show that, as $x$ increases from $-1$ towards $\\infty$, the function $(1 + x)^{-1/x}$ assumes once and only once every value between $0$ and $1$. % [0]% (*Math. Trip.* 1910.)% [1]%", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-v/1", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-v", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "194", "location": "Exercise Misc-V, problem 1", "problem_latex": "Show that, if neither $a$~nor~$b$ is zero, then\n\\[\nax^{n} + bx^{n-1} + \\dots + k = ax^{n} (1 + \\epsilon_{x}),\n\\]\nwhere $\\epsilon_{x}$~is of the first order of smallness when $x$~is large.", "markdown": "Show that, if neither $a$ nor $b$ is zero, then ax^n + bx^n-1 + …+ k = ax^n (1 + _x), where $\\epsilon_{x}$ is of the first order of smallness when $x$ is large.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-v/10", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-v", "number": 10, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "194", "location": "Exercise Misc-V, problem 10", "problem_latex": "Prove that $\\phi(x) = 1 - \\cos(1 - \\cos x)$ is of the fourth order of smallness\nwhen $x$~is small; and find the limit of $\\phi(x)/x^{4}$ as $x \\to 0$.", "markdown": "Prove that $\\phi(x) = 1 - \\cos(1 - \\cos x)$ is of the fourth order of smallness when $x$ is small; and find the limit of $\\phi(x)/x^{4}$ as $x \\to 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "cas.series", "cas.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-v/11", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-v", "number": 11, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "194", "location": "Exercise Misc-V, problem 11", "problem_latex": "Prove that $\\phi(x) = x\\sin(\\sin x) - \\sin^{2}x$ is of the sixth order of smallness\nwhen $x$~is small; and find the limit of $\\phi(x)/x^{6}$ as $x \\to 0$.", "markdown": "Prove that $\\phi(x) = x\\sin(\\sin x) - \\sin^{2}x$ is of the sixth order of smallness when $x$ is small; and find the limit of $\\phi(x)/x^{6}$ as $x \\to 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "cas.series", "cas.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-v/12", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-v", "number": 12, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "194", "location": "Exercise Misc-V, problem 12", "problem_latex": "From a point~$P$ on a radius~$OA$ of a circle, produced beyond the circle,\na tangent~$PT$ is drawn to the circle, touching it in~$T$, and $TN$~is drawn perpendicular\nto~$OA$. Show that $NA/AP \\to 1$ as $P$~moves up to~$A$.", "markdown": "From a point $P$ on a radius $OA$ of a circle, produced beyond the circle, a tangent $PT$ is drawn to the circle, touching it in $T$, and $TN$ is drawn perpendicular to $OA$. Show that $NA/AP \\to 1$ as $P$ moves up to $A$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-v/13", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-v", "number": 13, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "194", "location": "Exercise Misc-V, problem 13", "problem_latex": "Tangents are drawn to a circular arc at its middle point and its\nextremities; $\\Delta$~is the area of the triangle formed by the chord of the arc and\nthe two tangents at the extremities, and $\\Delta'$~the area of that formed by the\nthree tangents. Show that $\\Delta/\\Delta' \\to 4$ as the length of the arc tends to zero.", "markdown": "Tangents are drawn to a circular arc at its middle point and its extremities; $\\Delta$ is the area of the triangle formed by the chord of the arc and the two tangents at the extremities, and $\\Delta'$ the area of that formed by the three tangents. Show that $\\Delta/\\Delta' \\to 4$ as the length of the arc tends to zero.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-v/14", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-v", "number": 14, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "194", "location": "Exercise Misc-V, problem 14", "problem_latex": "For what values of~$a$ does $\\{a + \\sin(1/x)\\}/x$ tend to (1)~$\\infty$, (2)~$-\\infty$,\nas $x \\to 0$? [To~$\\infty$ if~$a > 1$, to~$-\\infty$ if~$a < -1$: the function oscillates if\n$-1 \\leq a \\leq 1$.]", "markdown": "For what values of $a$ does $\\{a + \\sin(1/x)\\}/x$ tend to (1) $\\infty$, (2) $-\\infty$, as $x \\to 0$? [To $\\infty$ if $a > 1$, to $-\\infty$ if $a < -1$: the function oscillates if $-1 \\leq a \\leq 1$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-v/15", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-v", "number": 15, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "194", "location": "Exercise Misc-V, problem 15", "problem_latex": "If $\\phi(x) = 1/q$ when $x = p/q$, and $\\phi(x) = 0$ when $x$~is irrational, then\n$\\phi(x)$~is continuous for all irrational and discontinuous for all rational values\nof~$x$.", "markdown": "If $\\phi(x) = 1/q$ when $x = p/q$, and $\\phi(x) = 0$ when $x$ is irrational, then $\\phi(x)$ is continuous for all irrational and discontinuous for all rational values of $x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-v/16", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-v", "number": 16, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "194", "location": "Exercise Misc-V, problem 16", "problem_latex": "Show that the function whose graph is drawn in \\Fig{32} may be represented\nby either of the formulae\n\\[\n1 - x + [x] - [1 - x],\\quad\n1 - x - \\lim_{n\\to\\infty} (\\cos^{2n+1}\\pi x).\n\\]", "markdown": "Show that the function whose graph is drawn in [fig:32]Fig. 32 may be represented by either of the formulae 1 - x + [x] - [1 - x],0pt minus 3pt1 - x - _n (^2n+1x).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-v/17", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-v", "number": 17, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "194", "location": "Exercise Misc-V, problem 17", "problem_latex": "Show that the function~$\\phi(x)$ which is equal to~$0$ when $x = 0$, to~$\\frac{1}{2} - x$\nwhen $0 < x < \\frac{1}{2}$, to~$\\frac{1}{2}$ when $x = \\frac{1}{2}$, to~$\\frac{3}{2} - x$\nwhen $\\frac{1}{2}< x < 1$, and to~$1$ when\n$x = 1$, assumes every value between $0$~and~$1$ once and once only as $x$~increases\nfrom $0$~to~$1$, but is discontinuous for $x = 0$, $x = \\frac{1}{2}$, and $x = 1$. Show also that\nthe function may be represented by the formula\n\\[\n\\tfrac{1}{2} - x - \\tfrac{1}{2}[2x] - \\tfrac{1}{2}[1 - 2x].\n\\]", "markdown": "Show that the function $\\phi(x)$ which is equal to $0$ when $x = 0$, to $\\frac{1}{2} - x$ when $0 < x < \\frac{1}{2}$, to $\\frac{1}{2}$ when $x = \\frac{1}{2}$, to $\\frac{3}{2} - x$ when $\\frac{1}{2}< x < 1$, and to $1$ when $x = 1$, assumes every value between $0$ and $1$ once and once only as $x$ increases from $0$ to $1$, but is discontinuous for $x = 0$, $x = \\frac{1}{2}$, and $x = 1$. Show also that the function may be represented by the formula 12 - x - 12[2x] - 12[1 - 2x].", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-v/18", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-v", "number": 18, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "194", "location": "Exercise Misc-V, problem 18", "problem_latex": "Let $\\phi(x) = x$ when $x$~is rational and $\\phi(x) = 1 - x$ when $x$~is irrational.\nShow that $\\phi(x)$~assumes every value between $0$ and~$1$ once and once only as $x$~increases\nfrom $0$ to~$1$, but is discontinuous for every value of~$x$ except $x = \\frac{1}{2}$.", "markdown": "Let $\\phi(x) = x$ when $x$ is rational and $\\phi(x) = 1 - x$ when $x$ is irrational. Show that $\\phi(x)$ assumes every value between $0$ and $1$ once and once only as $x$ increases from $0$ to $1$, but is discontinuous for every value of $x$ except $x = \\frac{1}{2}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-v/19", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-v", "number": 19, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "194", "location": "Exercise Misc-V, problem 19", "problem_latex": "As $x$~increases from~$-\\frac{1}{2}\\pi$ to~$\\frac{1}{2}\\pi$, $y = \\sin x$ is continuous and steadily\nincreases, in the stricter sense, from~$-1$ to~$1$. Deduce the existence of a\nfunction $x = \\arcsin y$ which is a continuous and steadily increasing function\nof~$y$ from $y = -1$ to~$y = 1$.", "markdown": "As $x$ increases from $-\\frac{1}{2}\\pi$ to $\\frac{1}{2}\\pi$, $y = \\sin x$ is continuous and steadily increases, in the stricter sense, from $-1$ to $1$. Deduce the existence of a function $x = \\arcsin y$ which is a continuous and steadily increasing function of $y$ from $y = -1$ to $y = 1$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-v/2", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-v", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "194", "location": "Exercise Misc-V, problem 2", "problem_latex": "If $P(x) = ax^{n} + bx^{n-1} + \\dots + k$, and $a$~is not zero, then as $x$~increases\n$P(x)$~has ultimately the sign of~$a$; and so has $P(x + \\lambda) - P(x)$, where $\\lambda$~is\nany constant.", "markdown": "If $P(x) = ax^{n} + bx^{n-1} + \\dots + k$, and $a$ is not zero, then as $x$ increases $P(x)$ has ultimately the sign of $a$; and so has $P(x + \\lambda) - P(x)$, where $\\lambda$ is any constant.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-v/20", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-v", "number": 20, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "194", "location": "Exercise Misc-V, problem 20", "problem_latex": "Show that the numerically least value of~$\\arctan y$ is continuous for\nall values of~$y$ and increases steadily from $-\\frac{1}{2}\\pi$ to~$\\frac{1}{2}\\pi$ as $y$~varies through all\nreal values.", "markdown": "Show that the numerically least value of $\\arctan y$ is continuous for all values of $y$ and increases steadily from $-\\frac{1}{2}\\pi$ to $\\frac{1}{2}\\pi$ as $y$ varies through all real values.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-v/21", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-v", "number": 21, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "194", "location": "Exercise Misc-V, problem 21", "problem_latex": "Discuss, on the lines of \\SecNo[§§]{108}--\\SecNo{109}, the solution of the equations\n\\[\ny^{2} - y - x = 0,\\quad\ny^{4} - y^{2} - x^{2} = 0,\\quad\ny^{4} - y^{2} + x^{2} = 0\n\\]\nin the neighbourhood of $x = 0$, $y = 0$.", "markdown": "Discuss, on the lines of [§§]108--109, the solution of the equations y^2 - y - x = 0,0pt minus 3pty^4 - y^2 - x^2 = 0,0pt minus 3pty^4 - y^2 + x^2 = 0 in the neighbourhood of $x = 0$, $y = 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.solve.nonpoly" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-v/22", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-v", "number": 22, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "194", "location": "Exercise Misc-V, problem 22", "problem_latex": "If $ax^{2} + 2bxy + cy^{2} + 2dx + 2ey = 0$ and $\\Delta = 2bde - ae^{2} - cd^{2}$, then one\nvalue of~$y$ is given by $y = \\alpha x + \\beta x^{2} + (\\gamma + \\epsilon_{x}) x^{3}$, where\n\\[\n\\alpha = -d/e,\\quad\n\\beta = \\Delta/2e^{3},\\quad\n\\gamma = (cd - be) \\Delta/2e^{5},\n\\]\nand $\\DPtypo{e_{x}}{\\epsilon_{x}}$~is of the first order of smallness when $x$~is small.\n\n[If $y - \\alpha x = \\eta $ then\n\\[\n-2e\\eta\n = ax^{2} + 2bx(\\eta + \\alpha x) + c(\\eta + \\alpha x)^{2}\n = Ax^{2} + 2Bx \\eta + C\\eta^{2},\n\\]\nsay. It is evident that $\\eta$~is of the second order of smallness, $x\\eta$~of the third,\nand $\\eta^{2}$~of the fourth; and $-2e\\eta = Ax^{2} - (AB/e) x^{3}$, the error being of the fourth\norder.]", "markdown": "If $ax^{2} + 2bxy + cy^{2} + 2dx + 2ey = 0$ and $\\Delta = 2bde - ae^{2} - cd^{2}$, then one value of $y$ is given by $y = \\alpha x + \\beta x^{2} + (\\gamma + \\epsilon_{x}) x^{3}$, where = -d/e,0pt minus 3pt = /2e^3,0pt minus 3pt= (cd - be) /2e^5, and $\\DPtypo{e_{x}}{\\epsilon_{x}}$ is of the first order of smallness when $x$ is small. [If $y - \\alpha x = \\eta $ then -2e = ax^2 + 2bx(+ x) + c(+ x)^2 = Ax^2 + 2Bx + C^2, say. It is evident that $\\eta$ is of the second order of smallness, $x\\eta$ of the third, and $\\eta^{2}$ of the fourth; and $-2e\\eta = Ax^{2} - (AB/e) x^{3}$, the error being of the fourth order.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "cas.series" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-v/23", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-v", "number": 23, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "194", "location": "Exercise Misc-V, problem 23", "problem_latex": "If $x = ay + by^{2} + cy^{3}$ then one value of~$y$ is given by\n\\[\ny = \\alpha x + \\beta x^{2} + (\\gamma + \\epsilon_{x}) x^{3},\n\\]\nwhere $\\alpha = 1/a$, $\\beta = -b/a^{3}$, $\\gamma = (2b^{2} - ac)/a^{5}$, and $\\epsilon_{x}$~is of the first order of smallness\nwhen $x$~is small.", "markdown": "If $x = ay + by^{2} + cy^{3}$ then one value of $y$ is given by y = x + x^2 + (+ _x) x^3, where $\\alpha = 1/a$, $\\beta = -b/a^{3}$, $\\gamma = (2b^{2} - ac)/a^{5}$, and $\\epsilon_{x}$ is of the first order of smallness when $x$ is small.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "cas.series" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-v/24", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-v", "number": 24, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "194", "location": "Exercise Misc-V, problem 24", "problem_latex": "If $x = ay + by^{n}$, where $n$~is an integer greater than unity, then one\nvalue of~$y$ is given by $y = \\alpha x + \\beta x^{n} + (\\gamma + \\epsilon_{x}) x^{2n-1}$, where $\\alpha = 1/a$, $\\beta = -b/a^{n+1}$,\n$\\gamma = nb^{2}/a^{2n+1}$, and $\\epsilon_{x}$~is of the $(n - 1)$th~order of smallness when $x$~is small.", "markdown": "If $x = ay + by^{n}$, where $n$ is an integer greater than unity, then one value of $y$ is given by $y = \\alpha x + \\beta x^{n} + (\\gamma + \\epsilon_{x}) x^{2n-1}$, where $\\alpha = 1/a$, $\\beta = -b/a^{n+1}$, $\\gamma = nb^{2}/a^{2n+1}$, and $\\epsilon_{x}$ is of the $(n - 1)$th order of smallness when $x$ is small.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.series" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-v/25", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-v", "number": 25, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "194", "location": "Exercise Misc-V, problem 25", "problem_latex": "Show that the least positive root of the equation $xy = \\sin x$ is a continuous\nfunction of~$y$ throughout the interval $\\DPmod{(0, 1)}{[0, 1]}$, and decreases steadily\nfrom $\\pi$ to~$0$ as $y$~increases from $0$ to~$1$. [The function is the inverse of\n$(\\sin x)/x$: apply~\\SecNo[§]{109}.]", "markdown": "Show that the least positive root of the equation $xy = \\sin x$ is a continuous function of $y$ throughout the interval $\\DPmod{(0, 1)}{[0, 1]}$, and decreases steadily from $\\pi$ to $0$ as $y$ increases from $0$ to $1$. [The function is the inverse of $(\\sin x)/x$: apply [§]109.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-v/26", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-v", "number": 26, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "194", "location": "Exercise Misc-V, problem 26", "problem_latex": "The least positive root of $xy = \\tan x$ is a continuous function of~$y$\nthroughout the interval $\\DPmod{(1, \\infty)}{[1, \\infty)}$, and increases steadily from $0$ to~$\\frac{1}{2}\\pi$ as $y$~increases\nfrom $1$ towards~$\\infty$.", "markdown": "The least positive root of $xy = \\tan x$ is a continuous function of $y$ throughout the interval $\\DPmod{(1, \\infty)}{[1, \\infty)}$, and increases steadily from $0$ to $\\frac{1}{2}\\pi$ as $y$ increases from $1$ towards $\\infty$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-v/3", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-v", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "194", "location": "Exercise Misc-V, problem 3", "problem_latex": "Show that in general\n\\[\n(ax^{n} + bx^{n-1} + \\dots + k)/(Ax^{n} + Bx^{n-1} + \\dots + K)\n = \\alpha + (\\beta/x) (1 + \\epsilon_{x}),\n\\]\nwhere $\\alpha = a/A$, $\\beta = (bA - aB)/A^{2}$, and $\\epsilon_{x}$~is of the first order of smallness when\n$x$~is large. Indicate any exceptional cases.", "markdown": "Show that in general (ax^n + bx^n-1 + …+ k)/(Ax^n + Bx^n-1 + …+ K) = + (/x) (1 + _x), where $\\alpha = a/A$, $\\beta = (bA - aB)/A^{2}$, and $\\epsilon_{x}$ is of the first order of smallness when $x$ is large. Indicate any exceptional cases.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-v/4", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-v", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "194", "location": "Exercise Misc-V, problem 4", "problem_latex": "Express\n\\[\n(ax^{2} + bx + c)/(Ax^{2} + Bx + C)\n\\]\nin the form\n\\[\n\\alpha + (\\beta/x) + (\\gamma/x^{2})(1 + \\epsilon_{x}),\n\\]\nwhere $\\epsilon_{x}$~is of the first order of smallness when $x$~is large.", "markdown": "Express (ax^2 + bx + c)/(Ax^2 + Bx + C) in the form + (/x) + (/x^2)(1 + _x), where $\\epsilon_{x}$ is of the first order of smallness when $x$ is large.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-v/5", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-v", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "194", "location": "Exercise Misc-V, problem 5", "problem_latex": "Show that\n\\[\n\\lim_{x\\to\\infty}\\sqrt{x}\\{\\sqrtp{x + a} - \\sqrt{x}\\} = \\tfrac{1}{2} a.\n\\]\n\n[Use the formula $\\sqrtp{x + a} - \\sqrt{x} = a/\\{\\sqrtp{x + a} + \\sqrt{x}\\}$.]", "markdown": "Show that _xxx + a - x = 12 a. [Use the formula $\\sqrtp{x + a} - \\sqrt{x} = a/\\{\\sqrtp{x + a} + \\sqrt{x}\\}$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-v/6", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-v", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "194", "location": "Exercise Misc-V, problem 6", "problem_latex": "Show that $\\sqrtp{x + a} = \\sqrt{x} + \\frac{1}{2}(a/\\sqrt{x}) (1 + \\epsilon_{x})$, where $\\epsilon_{x}$~is of the first order\nof smallness when $x$~is large.", "markdown": "Show that $\\sqrtp{x + a} = \\sqrt{x} + \\frac{1}{2}(a/\\sqrt{x}) (1 + \\epsilon_{x})$, where $\\epsilon_{x}$ is of the first order of smallness when $x$ is large.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.series" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-v/7", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-v", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "194", "location": "Exercise Misc-V, problem 7", "problem_latex": "Find values of $\\alpha$~and~$\\beta$ such that $\\sqrtp{a x^{2} + 2bx + c} - \\alpha x - \\beta$ has the limit\nzero as $x \\to \\infty$; and prove that $\\lim x\\{\\sqrtp{ax^{2} + 2bx + c} - \\alpha x - \\beta\\} = (ac - b^{2})/2a$.", "markdown": "Find values of $\\alpha$ and $\\beta$ such that $\\sqrtp{a x^{2} + 2bx + c} - \\alpha x - \\beta$ has the limit zero as $x \\to \\infty$; and prove that $\\lim x\\{\\sqrtp{ax^{2} + 2bx + c} - \\alpha x - \\beta\\} = (ac - b^{2})/2a$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-v/8", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-v", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "194", "location": "Exercise Misc-V, problem 8", "problem_latex": "Evaluate\n\\[\n\\lim_{x \\to\\infty} x\\left\\{\\sqrtbr{x^{2} + \\sqrtp{x^{4} + 1}} - x\\sqrt{2}\\right\\}.\n\\]", "markdown": "Evaluate _x xx^2 + x^4 + 1 - x2.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-v/9", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-v", "number": 9, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "194", "location": "Exercise Misc-V, problem 9", "problem_latex": "Prove that $(\\sec x - \\tan x) \\to 0$ as $x \\to \\frac{1}{2}\\pi$.", "markdown": "Prove that $(\\sec x - \\tan x) \\to 0$ as $x \\to \\frac{1}{2}\\pi$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/1", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 1", "problem_latex": "A function~$f(x)$ is defined as being equal to $1 + x$ when $x \\leq 0$, to~$x$ when $0 < x < 1$, to $2 - x$ when $1 \\leq x \\leq 2$, and to $3x - x^{2}$ when $x > 2$. Discuss the continuity of~$f(x)$ and the existence and continuity of~$f'(x)$ for $x = 0$, $x = 1$, and $x = 2$.", "markdown": "A function $f(x)$ is defined as being equal to $1 + x$ when $x \\leq 0$, to $x$ when $0 < x < 1$, to $2 - x$ when $1 \\leq x \\leq 2$, and to $3x - x^{2}$ when $x > 2$. Discuss the continuity of $f(x)$ and the existence and continuity of $f'(x)$ for $x = 0$, $x = 1$, and $x = 2$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/10", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 10, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 10", "problem_latex": "If $ax + by + c = 0$ then $y_{2} = 0$ (suffixes denoting differentiations with respect to~$x$). We may express this by saying that \\emph{the general differential equation of all straight lines is $y_{2} = 0$}. Find the general differential equations of (i)~all circles with their centres on the axis of~$x$, (ii)~all parabolas with their axes along the axis of~$x$, (iii)~all parabolas with their axes parallel to the axis of~$y$, (iv)~all circles, (v)~all parabolas, (vi)~all conics.", "markdown": "If $ax + by + c = 0$ then $y_{2} = 0$ (suffixes denoting differentiations with respect to $x$). We may express this by saying that *the general differential equation of all straight lines is $y_{2} = 0$*. Find the general differential equations of (i) all circles with their centres on the axis of $x$, (ii) all parabolas with their axes along the axis of $x$, (iii) all parabolas with their axes parallel to the axis of $y$, (iv) all circles, (v) all parabolas, (vi) all conics.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/11", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 11, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 11", "problem_latex": "Show that the general differential equations of all parabolas and of all conics are respectively \\[ D_{x}^{2} (y_{2}^{-2/3}) = 0,\\quad D_{x}^{3} (y_{2}^{-2/3}) = 0. \\]", "markdown": "Show that the general differential equations of all parabolas and of all conics are respectively D_x^2 (y_2^-2/3) = 0,0pt minus 3ptD_x^3 (y_2^-2/3) = 0.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/12", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 12, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 12", "problem_latex": "Denoting $\\dfrac{dy}{dx}$, $\\dfrac{1}{2!}\\, \\dfrac{d^{2}y}{dx^{2}}$, $\\dfrac{1}{3!}\\, \\dfrac{d^{3}y}{dx^{3}}$, $\\dfrac{1}{4!}\\, \\dfrac{d^{4}y}{dx^{4}}$,~\\dots\\ by $t$, $a$, $b$, $c$,~\\dots\\ and $\\dfrac{dx}{dy}$, $\\dfrac{1}{2!}\\, \\dfrac{d^{2}x}{dy^{2}}$, $\\dfrac{1}{3!}\\, \\dfrac{d^{3}x}{dy^{3}}$, $\\dfrac{1}{4!}\\, \\dfrac{d^{4}x}{dy^{4}}$,~\\dots\\ by $\\tau$, $\\alpha$, $\\beta$, $\\gamma$,~\\dots, show that \\[ 4ac - 5b^{2} = (4\\alpha\\gamma - 5\\beta^{2})/\\tau^{8},\\quad bt - a^{2} = - (\\beta\\tau - \\alpha^{2})/\\tau^{6}. \\] Establish similar formulae for the functions $a^{2}d - 3abc - 2b^{3}$, $(1 + t^{2})b - 2a^{2}t$, $2ct - 5ab$.", "markdown": "Denoting $\\dfrac{dy}{dx}$, $\\dfrac{1}{2!}\\, \\dfrac{d^{2}y}{dx^{2}}$, $\\dfrac{1}{3!}\\, \\dfrac{d^{3}y}{dx^{3}}$, $\\dfrac{1}{4!}\\, \\dfrac{d^{4}y}{dx^{4}}$, … by $t$, $a$, $b$, $c$, … and $\\dfrac{dx}{dy}$, $\\dfrac{1}{2!}\\, \\dfrac{d^{2}x}{dy^{2}}$, $\\dfrac{1}{3!}\\, \\dfrac{d^{3}x}{dy^{3}}$, $\\dfrac{1}{4!}\\, \\dfrac{d^{4}x}{dy^{4}}$, … by $\\tau$, $\\alpha$, $\\beta$, $\\gamma$, …, show that 4ac - 5b^2 = (4- 5^2)/^8,0pt minus 3ptbt - a^2 = - (- ^2)/^6. Establish similar formulae for the functions $a^{2}d - 3abc - 2b^{3}$, $(1 + t^{2})b - 2a^{2}t$, $2ct - 5ab$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/13", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 13, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 13", "problem_latex": "Prove that, if $y_{k}$~is the $k$th~derivative of $y = \\sin(n\\arcsin x)$, then \\[ (1 - x^{2})y_{k+2} - (2k + 1)xy_{k+1} + (n^{2} - k^{2})y_{k} = 0. \\]", "markdown": "Prove that, if $y_{k}$ is the $k$th derivative of $y = \\sin(n\\arcsin x)$, then (1 - x^2)y_k+2 - (2k + 1)xy_k+1 + (n^2 - k^2)y_k = 0.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/14", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 14, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 14", "problem_latex": "Prove the formula \\[ vD_{x}^{n}u = D_{x}^{n}(uv) - nD_{x}^{n-1}(uD_{x}v) + \\frac{n(n - 1)}{1·2} D_{x}^{n-2}(uD_{x}^{2}v) - \\dots \\] where $n$~is any positive integer.", "markdown": "Prove the formula vD_x^nu = D_x^n(uv) - nD_x^n-1(uD_xv) + n(n - 1)1·2 D_x^n-2(uD_x^2v) - … where $n$ is any positive integer.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/15", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 15, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 15", "problem_latex": "A curve is given by \\[ x = a(2\\cos t + \\cos 2t),\\quad y = a(2\\sin t - \\sin 2t). \\] Prove (i)~that the equations of the tangent and normal, at the point~$P$ whose parameter is~$t$, are \\[ x\\sin \\tfrac{1}{2} t + y\\cos \\tfrac{1}{2} t = a\\sin \\tfrac{3}{2} t,\\quad x\\cos \\tfrac{1}{2} t - y\\sin \\tfrac{1}{2} t = 3a\\cos \\tfrac{3}{2} t; \\] (ii)~that the tangent at~$P$ meets the curve in the points $Q$,~$R$ whose parameters are $-\\frac{1}{2} t$ and $\\pi - \\frac{1}{2} t$; (iii)~that $QR = 4a$; (iv)~that the tangents at $Q$ and~$R$ are at right angles and intersect on the circle $x^{2} + y^{2} = a^{2}$; (v)~that the normals at $P$,~$Q$, and~$R$ are concurrent and intersect on the circle $x^{2} + y^{2} = 9a^{2}$; (vi)~that the equation of the curve is \\[ (x^{2} + y^{2} + 12ax + 9a^{2})^{2} = 4a(2x + 3a)^{3}. \\] Sketch the form of the curve.", "markdown": "A curve is given by x = a(2t + 2t),0pt minus 3pty = a(2t - 2t). Prove (i) that the equations of the tangent and normal, at the point $P$ whose parameter is $t$, are x12 t + y12 t = a32 t,0pt minus 3ptx12 t - y12 t = 3a32 t; (ii) that the tangent at $P$ meets the curve in the points $Q$, $R$ whose parameters are $-\\frac{1}{2} t$ and $\\pi - \\frac{1}{2} t$; (iii) that $QR = 4a$; (iv) that the tangents at $Q$ and $R$ are at right angles and intersect on the circle $x^{2} + y^{2} = a^{2}$; (v) that the normals at $P$, $Q$, and $R$ are concurrent and intersect on the circle $x^{2} + y^{2} = 9a^{2}$; (vi) that the equation of the curve is (x^2 + y^2 + 12ax + 9a^2)^2 = 4a(2x + 3a)^3. Sketch the form of the curve.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/16", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 16, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 16", "problem_latex": "Show that the equations which define the curve of Ex.~15 may be replaced by $\\xi/a = 2u + (1/u^{2})$, $\\eta/a = (2/u) + u^{2}$, where $\\xi = x + yi$, $\\eta = x - yi$, $u = \\Cis t$. Show that the tangent and normal, at the point defined by~$u$, are \\[ u^{2}\\xi - u\\eta = a(u^{3} - 1),\\quad u^{2}\\xi + u\\eta = 3a(u^{3} + 1), \\] and deduce the properties (ii)--(v) of Ex.~15.", "markdown": "Show that the equations which define the curve of Ex. 15 may be replaced by $\\xi/a = 2u + (1/u^{2})$, $\\eta/a = (2/u) + u^{2}$, where $\\xi = x + yi$, $\\eta = x - yi$, $u = \\Cis t$. Show that the tangent and normal, at the point defined by $u$, are u^2- u= a(u^3 - 1),0pt minus 3ptu^2+ u= 3a(u^3 + 1), and deduce the properties (ii)--(v) of Ex. 15.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/17", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 17, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 17", "problem_latex": "Show that the condition that $x^{4} + 4px^{3} - 4qx - 1 = 0$ should have equal roots may be expressed in the form $(p + q)^{2/3} - (p - q)^{2/3} = 1$.", "markdown": "Show that the condition that $x^{4} + 4px^{3} - 4qx - 1 = 0$ should have equal roots may be expressed in the form $(p + q)^{2/3} - (p - q)^{2/3} = 1$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/18", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 18, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 18", "problem_latex": "The roots of a cubic $f(x) = 0$ are $\\alpha$,~$\\beta$,~$\\gamma$ in ascending order of magnitude. Show that if $\\DPmod{(\\alpha, \\beta)}{[\\alpha, \\beta]}$ and~$\\DPmod{(\\beta, \\gamma)}{[\\beta, \\gamma]}$ are each divided into six equal sub-intervals, then a root of $f'(x) = 0$ will fall in the fourth interval from~$\\beta$ on each side. What will be the nature of the cubic in the two cases when a root of $f'(x) = 0$ falls at a point of division?", "markdown": "The roots of a cubic $f(x) = 0$ are $\\alpha$, $\\beta$, $\\gamma$ in ascending order of magnitude. Show that if $\\DPmod{(\\alpha, \\beta)}{[\\alpha, \\beta]}$ and $\\DPmod{(\\beta, \\gamma)}{[\\beta, \\gamma]}$ are each divided into six equal sub-intervals, then a root of $f'(x) = 0$ will fall in the fourth interval from $\\beta$ on each side. What will be the nature of the cubic in the two cases when a root of $f'(x) = 0$ falls at a point of division?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/19", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 19, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 19", "problem_latex": "Investigate the maxima and minima of~$f(x)$, and the real roots of $f(x) = 0$, $f(x)$~being either of the functions \\[ x - \\sin x - \\tan\\alpha (1 - \\cos x),\\quad x - \\sin x - (\\alpha - \\sin\\alpha) - \\tan \\tfrac{1}{2}\\alpha (\\cos\\alpha - \\cos x), \\] and $\\alpha$~an angle between $0$~and~$\\pi$. Show that in the first case the condition for a double root is that $\\tan\\alpha - \\alpha$ should be a multiple of~$\\pi$.", "markdown": "Investigate the maxima and minima of $f(x)$, and the real roots of $f(x) = 0$, $f(x)$ being either of the functions x - x - (1 - x),0pt minus 3ptx - x - (- ) - 12(- x), and $\\alpha$ an angle between $0$ and $\\pi$. Show that in the first case the condition for a double root is that $\\tan\\alpha - \\alpha$ should be a multiple of $\\pi$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "FLAG-EXTRACTION", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/2", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 2", "problem_latex": "Denoting $a$, $ax + b$, $ax^{2} + 2bx + c$,~\\dots\\ by $u_{0}$,~$u_{1}$, $u_{2}$,~\\dots, show that $u_{0}^{2} u_{3} - 3u_{0} u_{1} u_{2} + 2u_{1}^{3}$ and $u_{0} u_{4} - 4u_{1} u_{3} + 3u_{2}^{2}$ are independent of~$x$.", "markdown": "Denoting $a$, $ax + b$, $ax^{2} + 2bx + c$, … by $u_{0}$, $u_{1}$, $u_{2}$, …, show that $u_{0}^{2} u_{3} - 3u_{0} u_{1} u_{2} + 2u_{1}^{3}$ and $u_{0} u_{4} - 4u_{1} u_{3} + 3u_{2}^{2}$ are independent of $x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/20", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 20, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 20", "problem_latex": "Show that by choice of the ratio~$\\lambda : \\mu$ we can make the roots of $\\lambda(ax^{2} + bx + c) + \\mu(a'x^{2} + b'x + c') = 0$ real and having a difference of any magnitude, unless the roots of the two quadratics are all real and interlace; and that in the excepted case the roots are always real, but there is a lower limit for the magnitude of their difference.", "markdown": "Show that by choice of the ratio $\\lambda : \\mu$ we can make the roots of $\\lambda(ax^{2} + bx + c) + \\mu(a'x^{2} + b'x + c') = 0$ real and having a difference of any magnitude, unless the roots of the two quadratics are all real and interlace; and that in the excepted case the roots are always real, but there is a lower limit for the magnitude of their difference.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/21", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 21, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 21", "problem_latex": "Prove that \\[ \\pi < \\frac{\\sin \\pi x}{x(1 - x)} \\leq 4 \\] when $0 < x < 1$, and draw the graph of the function.", "markdown": "Prove that < xx(1 - x) 4 when $0 < x < 1$, and draw the graph of the function.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/22", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 22, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 22", "problem_latex": "Draw the graph of the function \\[ \\pi \\cot\\pi x - \\frac{1}{x} - \\frac{1}{x - 1}. \\]", "markdown": "Draw the graph of the function x - 1x - 1x - 1.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/23", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 23, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 23", "problem_latex": "Sketch the general form of the graph of~$y$, given that \\[ \\frac{dy}{dx} = \\frac{(6x^{2} + x - 1) (x - 1)^{2} (x + 1)^{3}}{x^{2}}. \\]", "markdown": "Sketch the general form of the graph of $y$, given that dydx = (6x^2 + x - 1) (x - 1)^2 (x + 1)^3x^2.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/24", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 24, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 24", "problem_latex": "A sheet of paper is folded over so that one corner just reaches the opposite side. Show how the paper must be folded to make the length of the crease a maximum.", "markdown": "A sheet of paper is folded over so that one corner just reaches the opposite side. Show how the paper must be folded to make the length of the crease a maximum.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/25", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 25, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 25", "problem_latex": "The greatest acute angle at which the ellipse $(x^{2}/a^{2}) + (y^{2}/b^{2}) = 1$ can be cut by a concentric circle is $\\arctan\\{(a^{2} - b^{2})/2ab\\}$.", "markdown": "The greatest acute angle at which the ellipse $(x^{2}/a^{2}) + (y^{2}/b^{2}) = 1$ can be cut by a concentric circle is $\\arctan\\{(a^{2} - b^{2})/2ab\\}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/26", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 26, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 26", "problem_latex": "In a triangle the area~$\\Delta$ and the semi-perimeter~$s$ are fixed. Show that any maximum or minimum of one of the sides is a root of the equation $s(x - s) x^{2} + 4\\Delta^{2} = 0$. Discuss the reality of the roots of this equation, and whether they correspond to maxima or minima.", "markdown": "In a triangle the area $\\Delta$ and the semi-perimeter $s$ are fixed. Show that any maximum or minimum of one of the sides is a root of the equation $s(x - s) x^{2} + 4\\Delta^{2} = 0$. Discuss the reality of the roots of this equation, and whether they correspond to maxima or minima.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/27", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 27, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 27", "problem_latex": "The area of the greatest equilateral triangle which can be drawn with its sides passing through three given points $A$,~$B$,~$C$ is \\[ 2\\Delta + \\frac{a^{2} + b^{2} + c^{2}}{2\\sqrt{3}}, \\] $a$,~$b$,~$c$ being the sides and $\\Delta$~the area of~$ABC$.", "markdown": "The area of the greatest equilateral triangle which can be drawn with its sides passing through three given points $A$, $B$, $C$ is 2+ a^2 + b^2 + c^223, $a$, $b$, $c$ being the sides and $\\Delta$ the area of $ABC$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/28", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 28, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 28", "problem_latex": "If $\\Delta$,~$\\Delta'$ are the areas of the two maximum isosceles triangles which can be described with their vertices at the origin and their base angles on the cardioid $r = a(1 + \\cos\\theta)$, then $256\\Delta\\Delta' = 25a^{4}\\sqrt{5}$.", "markdown": "If $\\Delta$, $\\Delta'$ are the areas of the two maximum isosceles triangles which can be described with their vertices at the origin and their base angles on the cardioid $r = a(1 + \\cos\\theta)$, then $256\\Delta\\Delta' = 25a^{4}\\sqrt{5}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/29", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 29, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 29", "problem_latex": "Find the limiting values which $(x^{2} - 4y + 8)/(y^{2} - 6x + 3)$ approaches as the point~$(x, y)$ on the curve $x^{2}y - 4x^{2} - 4xy + y^{2} + 16x - 2y - 7 = 0$ approaches the position~$(2, 3)$.", "markdown": "Find the limiting values which $(x^{2} - 4y + 8)/(y^{2} - 6x + 3)$ approaches as the point $(x, y)$ on the curve $x^{2}y - 4x^{2} - 4xy + y^{2} + 16x - 2y - 7 = 0$ approaches the position $(2, 3)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/3", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 3", "problem_latex": "If $a_{0}$, $a_{1}$,~\\dots, $a_{2n}$ are constants and $U_{r} = (a_{0}, a_{1}, \\dots, a_{r} \\btw x, 1)^{r}$, then \\[ U_{0}U_{2n} - 2nU_{1}U_{2n-1} + \\frac{2n(2n - 1)}{1·2} U_{2}U_{2n-2} - \\dots + U_{2n}U_{0} \\] is independent of~$x$.", "markdown": "If $a_{0}$, $a_{1}$, …, $a_{2n}$ are constants and $U_{r} = (a_{0}, a_{1}, \\dots, a_{r} \\btw x, 1)^{r}$, then U_0U_2n - 2nU_1U_2n-1 + 2n(2n - 1)1·2 U_2U_2n-2 - …+ U_2nU_0 is independent of $x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/30", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 30, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 30", "problem_latex": "If $f(x) = \\dfrac{1}{\\sin x - \\sin a} - \\dfrac{1}{(x - a)\\cos a}$, then \\[ \\frac{d}{da}\\{\\lim_{x \\to a} f(x)\\} - \\lim_{x \\to a}f'(x) = \\tfrac{3}{4} \\sec^{3} a - \\tfrac{5}{12} \\sec a. \\]", "markdown": "If $f(x) = \\dfrac{1}{\\sin x - \\sin a} - \\dfrac{1}{(x - a)\\cos a}$, then dda_x a f(x) - _x af’(x) = 34 ^3 a - 512 a.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "FLAG-EXTRACTION", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/31", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 31, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 31", "problem_latex": "Show that if $\\phi(x) = 1/(1 + x^{2})$ then $\\phi^{n} (x) = Q_{n}(x)/(1 + x^{2})^{n+1}$, where $Q_{n}(x)$~is a polynomial of degree~$n$. Show also that (i) $Q_{n+1} = (1 + x^{2}) Q_{n}' - 2(n + 1) x Q_{n}$, (ii) $Q_{n+2} + 2(n + 2) x Q_{n+1} + (n + 2)(n + 1)(1 + x^{2})Q_{n} = 0$, (iii) $(1 + x^{2}) Q_{n}'' - 2nx Q_{n}' + n(n + 1)Q_{n} = 0$, (iv) $Q_{n} = (-1)^{n} n!\\left\\{(n + 1)x^{n} - \\dfrac{(n + 1)n(n - 1)}{3!} x^{n-2} + \\dots\\right\\}$, (v) all the roots of $Q_{n} = 0$ are real and separated by those of $Q_{n-1} = 0$.", "markdown": "Show that if $\\phi(x) = 1/(1 + x^{2})$ then $\\phi^{n} (x) = Q_{n}(x)/(1 + x^{2})^{n+1}$, where $Q_{n}(x)$ is a polynomial of degree $n$. Show also that (i) $Q_{n+1} = (1 + x^{2}) Q_{n}' - 2(n + 1) x Q_{n}$, (ii) $Q_{n+2} + 2(n + 2) x Q_{n+1} + (n + 2)(n + 1)(1 + x^{2})Q_{n} = 0$, (iii) $(1 + x^{2}) Q_{n}'' - 2nx Q_{n}' + n(n + 1)Q_{n} = 0$, (iv) $Q_{n} = (-1)^{n} n!\\left\\{(n + 1)x^{n} - \\dfrac{(n + 1)n(n - 1)}{3!} x^{n-2} + \\dots\\right\\}$, (v) all the roots of $Q_{n} = 0$ are real and separated by those of $Q_{n-1} = 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "FLAG-EXTRACTION", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/32", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 32, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 32", "problem_latex": "If $f(x)$, $\\phi(x)$, $\\psi(x)$ have derivatives when $a \\leq x \\leq b$, then there is a value of~$\\xi$ lying between $a$~and~$b$ and such that \\[ \\begin{vmatrix} f(a) & \\phi(a) & \\psi(a)\\\\ f(b) & \\phi(b) & \\psi(b)\\\\ f'(\\xi)& \\phi'(\\xi)& \\psi'(\\xi) \\end{vmatrix} =0. \\]", "markdown": "If $f(x)$, $\\phi(x)$, $\\psi(x)$ have derivatives when $a \\leq x \\leq b$, then there is a value of $\\xi$ lying between $a$ and $b$ and such that vmatrix f(a) & (a) & (a) f(b) & (b) & (b) f’()& ’()& ’() vmatrix =0.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/33", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 33, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 33", "problem_latex": "Deduce from Ex.~32 the formula \\[ \\frac{f(b) - f(a)}{\\phi(b) - \\phi(a)} = \\frac{f'(\\xi)}{\\phi'(\\xi)}\\Add{.} \\]", "markdown": "Deduce from Ex. 32 the formula f(b) - f(a)(b) - (a) = f’()’()", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/34", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 34, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 34", "problem_latex": "If $\\phi'(x) \\to a$ as $x \\to \\infty$, then $\\phi(x)/x \\to a$. If $\\phi'(x) \\to \\infty$ then $\\phi(x) \\to \\infty$.", "markdown": "If $\\phi'(x) \\to a$ as $x \\to \\infty$, then $\\phi(x)/x \\to a$. If $\\phi'(x) \\to \\infty$ then $\\phi(x) \\to \\infty$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/35", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 35, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 35", "problem_latex": "If $\\phi(x) \\to a$ as $x \\to \\infty$, then $\\phi'(x)$~cannot tend to any limit other than zero.", "markdown": "If $\\phi(x) \\to a$ as $x \\to \\infty$, then $\\phi'(x)$ cannot tend to any limit other than zero.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/36", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 36, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 36", "problem_latex": "If $\\phi(x) + \\phi'(x) \\to a$ as $x \\to \\infty$, then $\\phi(x) \\to a$ and $\\phi'(x) \\to 0$.", "markdown": "If $\\phi(x) + \\phi'(x) \\to a$ as $x \\to \\infty$, then $\\phi(x) \\to a$ and $\\phi'(x) \\to 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/37", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 37, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 37", "problem_latex": "Show how to reduce $\\ds\\int R\\left\\{x, \\bigsqrtp{\\frac{ax + b}{mx + n}}, \\bigsqrtp{\\frac{cx + d}{mx + n}}\\right\\} dx$ to the integral of a rational function.", "markdown": "Show how to reduce $\\ds\\int R\\left\\{x, \\bigsqrtp{\\frac{ax + b}{mx + n}}, \\bigsqrtp{\\frac{cx + d}{mx + n}}\\right\\} dx$ to the integral of a rational function.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/38", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 38, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 38", "problem_latex": "\\int \\frac{dx}{(1 + x^{2})^{3}},\\quad \\int \\bigsqrtp{\\frac{x - 1}{x + 1}}\\, \\frac{dx}{x},\\quad \\int \\frac{x\\, dx}{\\sqrtp{1 + x} - \\sqrtp[3]{1 + x}}, \\int \\bigsqrtb{a^{2} + \\bigsqrtp{b^{2} + \\frac{c}{x}}}\\, dx,\\quad \\int \\cosec^{3}x\\, dx,\\quad \\int \\frac{5\\cos x + 6}{2\\cos x + \\sin x + 3}\\, dx, \\int \\frac{dx}{(2 - \\sin^{2}x) (2 + \\sin x - \\sin^{2} x)},\\quad \\int \\frac{\\cos x\\sin x \\, dx}{\\cos^{4}x + \\sin^{4}x},\\quad \\int \\cosec x \\sqrtp{\\sec 2x}\\, dx, \\int \\frac{dx}{\\sqrtb{(1 + \\sin x) (2 + \\sin x)}},\\quad \\int \\frac{x + \\sin x}{1 + \\cos x}\\, dx,\\quad \\int \\arcsec x\\, dx,\\quad \\int (\\arcsin x)^{2}\\, dx, \\int x\\arcsin x\\, dx,\\quad \\int \\frac{x\\arcsin x}{\\sqrtp{1 - x^{2}}}\\, dx,\\quad \\int \\frac{\\arcsin x}{x^{3}}\\, dx,\\quad \\int \\frac{\\arcsin x}{(1 + x)^{2}}\\, dx, \\int \\frac{\\arctan x}{x^{2}}\\, dx,\\quad \\int \\frac{\\arctan x}{(1 + x^{2})^{3/2}}\\, dx,\\quad \\int \\frac{\\log(\\alpha^{2} + \\beta^{2}x^{2})}{x^{2}}\\, dx,\\quad \\int \\frac{\\log(\\alpha + \\beta x)}{(a + bx)^{2}}\\, dx.", "markdown": "dx(1 + x^2)^3,0pt minus 3ptx - 1x + 1  dxx,0pt minus 3ptx  dx1 + x - [3]1 + x, a^2 + b^2 + cx  dx,0pt minus 3pt^3x  dx,0pt minus 3pt5x + 62x + x + 3  dx, dx(2 - ^2x) (2 + x - ^2 x),0pt minus 3ptxx   dx^4x + ^4x,0pt minus 3ptx 2x  dx, dx(1 + x) (2 + x),0pt minus 3ptx + x1 + x  dx,0pt minus 3ptx  dx,0pt minus 3pt(x)^2  dx, xx  dx,0pt minus 3ptxx1 - x^2  dx,0pt minus 3ptxx^3  dx,0pt minus 3ptx(1 + x)^2  dx, xx^2  dx,0pt minus 3ptx(1 + x^2)^3/2  dx,0pt minus 3pt(^2 + ^2x^2)x^2  dx,0pt minus 3pt(+ x)(a + bx)^2  dx.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "FLAG-EXTRACTION", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/39", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 39, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 39", "problem_latex": "\\Topic{Formulae of reduction.} (i) Show that $2(n - 1)(q - \\tfrac{1}{4}p^{2}) \\int \\frac{dx}{(x^{2} + px + q)^{n}} = \\frac{x + \\frac{1}{2}p}{(x^{2} + px + q)^{n-1}} + (2n - 3) \\int \\frac{dx}{(x^{2} + px + q)^{n-1}}.$ (ii) Show that if $I_{p, q} = \\ds\\int x^{p}(1 + x)^{q}\\, dx$ then $(p + 1) I_{p, q} = x^{p+1}(1 + x)^{q} - qI_{p+1, q-1},$ and obtain a similar formula connecting $I_{p, q}$ with~$I_{p-1, q+1}$. Show also, by means of the substitution $x = -y/(1 + y)$, that $I_{p, q} = (-1)^{p+1} \\int y^{p} (1 + y)^{-p-q-2}\\, dy.$ (iii) Show that if $X = a + bx$ then $\\int xX^{-1/3}\\, dx = -3(3a - 2bx) X^{2/3}/10b^{2}$, $\\int x^{2}X^{-1/3}\\, dx = 3(9a^{2} - 6abx + 5b^{2}x^{2}) X^{2/3}/40b^{3}\\DPchg{.}{,}$, $\\int xX^{-1/4}\\, dx = -4(4a - 3bx) X^{3/4}/21b^{2}$, $\\int x^{2}X^{-1/4}\\, dx = 4(32a^{2} - 24abx + 21b^{2}x^{2}) X^{3/4}/231b^{3}$. (iv) If $I_{m, n} = \\ds\\int \\frac{x^{m}\\, dx}{(1 + x^{2})^{n}}$ then $2(n - 1)I_{m, n} = -x^{m-1} (1 + x^{2})^{-(n-1)} + (m - 1)I_{m-2, n-1}.$ (v) If $I_{n} = \\ds\\int x^{n} \\cos\\beta x\\, dx$ and $J_{n} = \\ds\\int x^{n} \\sin\\beta x\\, dx$ then $\\beta I_{n} = x^{n} \\sin\\beta x - nJ_{n-1}, \\beta J_{n} = -x^{n} \\cos\\beta x + nI_{n-1}.$ (vi) If $I_{n} = \\ds\\int \\cos^{n} x\\, dx$ and $J_{n} = \\ds\\int \\sin^{n} x\\, dx$ then $nI_{n} = \\sin x\\cos^{n-1} x + (n - 1) I_{n-2}, nJ_{n} = -\\cos x\\sin^{n-1} x + (n - 1) J_{n-2}.$ (vii) If $I_{n} = \\ds\\int \\tan^{n}x\\, dx$ then $(n - 1)(I_{n} + I_{n-2}) = \\tan^{n-1}x$. (viii) If $I_{m, n} = \\ds\\int \\cos^{m}x \\sin^{n}x\\, dx$ then $(m+n)I_{m, n} = -\\cos^{m+1}x \\sin^{n-1}x + (n - 1) I_{m, n-2} = \\cos^{m-1}x \\sin^{n+1}x + (m - 1) I_{m-2, n}.$ (ix) Connect $I_{m, n} = \\ds\\int \\sin^{m}x \\sin nx\\, dx$ with~$I_{m-2, n}$. (x) If $I_{m, n} = \\ds\\int x^{m} \\cosec^{n}x\\, dx$ then $(n - 1)(n - 2)I_{m, n} = (n - 2)^{2}I_{m, n-2} + m(m - 1)I_{m-2, n-2} -x^{m-1} \\cosec^{n-1}x \\{m\\sin x + (n - 2) x\\cos x\\}.$ (xi) If $I_{n} = \\ds\\int (a + b\\cos x)^{-n}\\, dx$ then $(n - 1)(a^{2} - b^{2}) I_{n} = -b\\sin x (a + b\\cos x)^{-(n-1)} + (2n - 3)aI_{n-1} - (n - 2)I_{n-2}.$ (xii) If $I_{n} = \\ds\\int (a\\cos^{2} x + 2h\\cos x\\sin x + b\\sin^{2}x)^{-n}\\, dx$ then $4n(n + 1)(ab - h^{2})I_{n+2} - 2n(2n + 1)(a + b)I_{n+1} + 4n^{2}I_{n} = -\\frac{d^{2} I_{n}}{dx^{2}}.$ (xiii) If $I_{m, n} = \\ds\\int x^{m}(\\log x)^{n}\\, dx$ then $(m + 1)I_{m, n} = x^{m+1}(\\log x)^{n} - nI_{m, n-1}.$", "markdown": "**of reduction.** (i) Show that $2(n - 1)(q - \\tfrac{1}{4}p^{2}) \\int \\frac{dx}{(x^{2} + px + q)^{n}} = \\frac{x + \\frac{1}{2}p}{(x^{2} + px + q)^{n-1}} + (2n - 3) \\int \\frac{dx}{(x^{2} + px + q)^{n-1}}.$ (ii) Show that if $I_{p, q} = \\ds\\int x^{p}(1 + x)^{q}\\, dx$ then $(p + 1) I_{p, q} = x^{p+1}(1 + x)^{q} - qI_{p+1, q-1},$ and obtain a similar formula connecting $I_{p, q}$ with $I_{p-1, q+1}$. Show also, by means of the substitution $x = -y/(1 + y)$, that $I_{p, q} = (-1)^{p+1} \\int y^{p} (1 + y)^{-p-q-2}\\, dy.$ (iii) Show that if $X = a + bx$ then $\\int xX^{-1/3}\\, dx = -3(3a - 2bx) X^{2/3}/10b^{2}$, $\\int x^{2}X^{-1/3}\\, dx = 3(9a^{2} - 6abx + 5b^{2}x^{2}) X^{2/3}/40b^{3}\\DPchg{.}{,}$, $\\int xX^{-1/4}\\, dx = -4(4a - 3bx) X^{3/4}/21b^{2}$, $\\int x^{2}X^{-1/4}\\, dx = 4(32a^{2} - 24abx + 21b^{2}x^{2}) X^{3/4}/231b^{3}$. (iv) If $I_{m, n} = \\ds\\int \\frac{x^{m}\\, dx}{(1 + x^{2})^{n}}$ then $2(n - 1)I_{m, n} = -x^{m-1} (1 + x^{2})^{-(n-1)} + (m - 1)I_{m-2, n-1}.$ (v) If $I_{n} = \\ds\\int x^{n} \\cos\\beta x\\, dx$ and $J_{n} = \\ds\\int x^{n} \\sin\\beta x\\, dx$ then $\\beta I_{n} = x^{n} \\sin\\beta x - nJ_{n-1}, \\beta J_{n} = -x^{n} \\cos\\beta x + nI_{n-1}.$ (vi) If $I_{n} = \\ds\\int \\cos^{n} x\\, dx$ and $J_{n} = \\ds\\int \\sin^{n} x\\, dx$ then $nI_{n} = \\sin x\\cos^{n-1} x + (n - 1) I_{n-2}, nJ_{n} = -\\cos x\\sin^{n-1} x + (n - 1) J_{n-2}.$ (vii) If $I_{n} = \\ds\\int \\tan^{n}x\\, dx$ then $(n - 1)(I_{n} + I_{n-2}) = \\tan^{n-1}x$. (viii) If $I_{m, n} = \\ds\\int \\cos^{m}x \\sin^{n}x\\, dx$ then $(m+n)I_{m, n} = -\\cos^{m+1}x \\sin^{n-1}x + (n - 1) I_{m, n-2} = \\cos^{m-1}x \\sin^{n+1}x + (m - 1) I_{m-2, n}.$ (ix) Connect $I_{m, n} = \\ds\\int \\sin^{m}x \\sin nx\\, dx$ with $I_{m-2, n}$. (x) If $I_{m, n} = \\ds\\int x^{m} \\cosec^{n}x\\, dx$ then $(n - 1)(n - 2)I_{m, n} = (n - 2)^{2}I_{m, n-2} + m(m - 1)I_{m-2, n-2} -x^{m-1} \\cosec^{n-1}x \\{m\\sin x + (n - 2) x\\cos x\\}.$ (xi) If $I_{n} = \\ds\\int (a + b\\cos x)^{-n}\\, dx$ then $(n - 1)(a^{2} - b^{2}) I_{n} = -b\\sin x (a + b\\cos x)^{-(n-1)} + (2n - 3)aI_{n-1} - (n - 2)I_{n-2}.$ (xii) If $I_{n} = \\ds\\int (a\\cos^{2} x + 2h\\cos x\\sin x + b\\sin^{2}x)^{-n}\\, dx$ then $4n(n + 1)(ab - h^{2})I_{n+2} - 2n(2n + 1)(a + b)I_{n+1} + 4n^{2}I_{n} = -\\frac{d^{2} I_{n}}{dx^{2}}.$ (xiii) If $I_{m, n} = \\ds\\int x^{m}(\\log x)^{n}\\, dx$ then $(m + 1)I_{m, n} = x^{m+1}(\\log x)^{n} - nI_{m, n-1}.$", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "FLAG-EXTRACTION", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/4", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 4", "problem_latex": "The first three derivatives of the function $\\arcsin(\\mu\\sin x) - x$, where $\\mu > 1$, are positive when $0 \\leq x \\leq \\frac{1}{2} \\pi$.", "markdown": "The first three derivatives of the function $\\arcsin(\\mu\\sin x) - x$, where $\\mu > 1$, are positive when $0 \\leq x \\leq \\frac{1}{2} \\pi$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/40", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 40, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 40", "problem_latex": "If $n$~is a positive integer then the value of $\\ds\\int x^{m}(\\log x)^{n}\\, dx$ is \\[ x^{m+1} \\left\\{\\frac{(\\log x)^{n}}{m + 1} - \\frac{n(\\log x)^{n-1}}{(m + 1)^{2}} + \\frac{n(n - 1)(\\log x)^{n-2}}{(m + 1)^{3}} - \\dots + \\frac{(-1)^{n}n!}{(m + 1)^{n+1}}\\right\\}. \\]", "markdown": "If $n$ is a positive integer then the value of $\\ds\\int x^{m}(\\log x)^{n}\\, dx$ is x^m+1 (x)^nm + 1 - n(x)^n-1(m + 1)^2 + n(n - 1)(x)^n-2(m + 1)^3 - …+ (-1)^nn!(m + 1)^n+1.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/41", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 41, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 41", "problem_latex": "Show that the most general function~$\\phi(x)$, such that $\\phi'' + a^{2}\\phi = 0$ for all values of~$x$, may be expressed in either of the forms $A\\cos ax + B\\sin ax$, $\\rho\\cos(ax + \\epsilon)$, where $A$,~$B$, $\\rho$,~$\\epsilon$ are constants.", "markdown": "Show that the most general function $\\phi(x)$, such that $\\phi'' + a^{2}\\phi = 0$ for all values of $x$, may be expressed in either of the forms $A\\cos ax + B\\sin ax$, $\\rho\\cos(ax + \\epsilon)$, where $A$, $B$, $\\rho$, $\\epsilon$ are constants.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/42", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 42, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 42", "problem_latex": "Determine the most general functions $y$~and~$z$ such that $y' + \\omega z = 0$, and $z' - \\omega y = 0$, where $\\omega$~is a constant and dashes denote differentiation with respect to~$x$.", "markdown": "Determine the most general functions $y$ and $z$ such that $y' + \\omega z = 0$, and $z' - \\omega y = 0$, where $\\omega$ is a constant and dashes denote differentiation with respect to $x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/43", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 43, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 43", "problem_latex": "The area of the curve given by \\[ x = \\cos\\phi + \\frac{\\sin\\alpha \\sin\\phi}{1 - \\cos^{2}\\alpha \\sin^{2}\\phi},\\quad y = \\sin\\phi - \\frac{\\sin\\alpha \\cos\\phi}{1 - \\cos^{2}\\alpha \\sin^{2}\\phi}, \\] where $\\alpha$~is a positive acute angle, is $\\frac{1}{2}\\pi(1 + \\sin\\alpha)^{2}/\\sin\\alpha$.", "markdown": "The area of the curve given by x = + 1 - ^2^2,0pt minus 3pty = - 1 - ^2^2, where $\\alpha$ is a positive acute angle, is $\\frac{1}{2}\\pi(1 + \\sin\\alpha)^{2}/\\sin\\alpha$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/44", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 44, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 44", "problem_latex": "The projection of a chord of a circle of radius~$a$ on a diameter is of constant length~$2a\\cos\\beta$; show that the locus of the middle point of the chord consists of two loops, and that the area of either is $a^{2}(\\beta - \\cos\\beta\\sin\\beta)$.", "markdown": "The projection of a chord of a circle of radius $a$ on a diameter is of constant length $2a\\cos\\beta$; show that the locus of the middle point of the chord consists of two loops, and that the area of either is $a^{2}(\\beta - \\cos\\beta\\sin\\beta)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/45", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 45, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 45", "problem_latex": "Show that the length of a quadrant of the curve $(x/a)^{2/3} + (y/b)^{2/3} = 1$ is $(a^{2} + ab + b^{2})/(a + b)$.", "markdown": "Show that the length of a quadrant of the curve $(x/a)^{2/3} + (y/b)^{2/3} = 1$ is $(a^{2} + ab + b^{2})/(a + b)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/46", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 46, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 46", "problem_latex": "A point $A$~is inside a circle of radius~$a$, at a distance~$b$ from the centre. Show that the locus of the foot of the perpendicular drawn from $A$ to a tangent to the circle encloses an area $\\pi(a^{2} + \\frac{1}{2}b^{2})$.", "markdown": "A point $A$ is inside a circle of radius $a$, at a distance $b$ from the centre. Show that the locus of the foot of the perpendicular drawn from $A$ to a tangent to the circle encloses an area $\\pi(a^{2} + \\frac{1}{2}b^{2})$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/47", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 47, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 47", "problem_latex": "Prove that if $(a, b, c, f, g, h \\btw x, y, 1)^{2} = 0$ is the equation of a conic, then \\[ \\int \\frac{dx}{(lx + my + n)(hx + by + f)} = \\alpha\\log \\frac{PT}{PT'} + \\beta, \\] where $PT$,~$PT'$ are the perpendiculars from a point~$P$ of the conic on the tangents at the ends of the chord $lx + my + n = 0$, and $\\alpha$,~$\\beta$ are constants.", "markdown": "Prove that if $(a, b, c, f, g, h \\btw x, y, 1)^{2} = 0$ is the equation of a conic, then dx(lx + my + n)(hx + by + f) = PTPT’ + , where $PT$, $PT'$ are the perpendiculars from a point $P$ of the conic on the tangents at the ends of the chord $lx + my + n = 0$, and $\\alpha$, $\\beta$ are constants.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/48", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 48, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 48", "problem_latex": "Show that \\[ \\int \\frac{ax^{2} + 2bx + c}{(Ax^{2} + 2Bx + C)^{2}}\\, dx \\] will be a rational function of~$x$ if and only if one or other of $AC - B^{2}$ and $aC + cA - 2bB$ is zero.", "markdown": "Show that ax^2 + 2bx + c(Ax^2 + 2Bx + C)^2  dx will be a rational function of $x$ if and only if one or other of $AC - B^{2}$ and $aC + cA - 2bB$ is zero.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/49", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 49, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 49", "problem_latex": "Show that the necessary and sufficient condition that \\[ \\int \\frac{f(x)}{\\{F(x)\\}^{2}}\\, dx, \\] where $f$~and~$F$ are polynomials of which the latter has no repeated factor, should be a rational function of~$x$, is that $f'F' - fF''$ should be divisible by~$F$.", "markdown": "Show that the necessary and sufficient condition that f(x)F(x)^2  dx, where $f$ and $F$ are polynomials of which the latter has no repeated factor, should be a rational function of $x$, is that $f'F' - fF''$ should be divisible by $F$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/5", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 5", "problem_latex": "The constituents of a determinant are functions of~$x$. Show that its differential coefficient is the sum of the determinants formed by differentiating the constituents of one row only, leaving the rest unaltered.", "markdown": "The constituents of a determinant are functions of $x$. Show that its differential coefficient is the sum of the determinants formed by differentiating the constituents of one row only, leaving the rest unaltered.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/50", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 50, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 50", "problem_latex": "Show that \\[ \\int \\frac{\\alpha\\cos x + \\beta\\sin x + \\gamma}{(1 - e\\cos x)^{2}}\\, dx \\] is a rational function of $\\cos x$ and~$\\sin x$ if and only if $\\alpha e + \\gamma = 0$; and determine the integral when this condition is satisfied.", "markdown": "Show that x + x + (1 - ex)^2  dx is a rational function of $\\cos x$ and $\\sin x$ if and only if $\\alpha e + \\gamma = 0$; and determine the integral when this condition is satisfied.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/6", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 6", "problem_latex": "If $f_{1}$, $f_{2}$, $f_{3}$, $f_{4}$ are polynomials of degree not greater than~$4$, then \\[ \\begin{vmatrix} f_{1}& f_{2}& f_{3}& f_{4}\\\\ f_{1}'& f_{2}'& f_{3}'& f_{4}'\\\\ f_{1}''& f_{2}''& f_{3}''& f_{4}''\\\\ f_{1}'''& f_{2}'''& f_{3}'''& f_{4}''' \\end{vmatrix} \\] is also a polynomial of degree not greater than~$4$.", "markdown": "If $f_{1}$, $f_{2}$, $f_{3}$, $f_{4}$ are polynomials of degree not greater than $4$, then vmatrix f_1& f_2& f_3& f_4 f_1’& f_2’& f_3’& f_4’ f_1”& f_2”& f_3”& f_4” f_1”’& f_2”’& f_3”’& f_4”’ vmatrix is also a polynomial of degree not greater than $4$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/7", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 7", "problem_latex": "If $y^{3} + 3yx + 2x^{3} = 0$ then $x^{2}(1 + x^{3})y'' - \\frac{3}{2}xy' + y = 0$.", "markdown": "If $y^{3} + 3yx + 2x^{3} = 0$ then $x^{2}(1 + x^{3})y'' - \\frac{3}{2}xy' + y = 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/8", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 8", "problem_latex": "Verify that the differential equation $y = \\phi\\{\\psi(y_{1})\\} + \\phi\\{x - \\psi(y_{1})\\}$, where $y_{1}$~is the derivative of~$y$, and $\\psi$~is the function inverse to~$\\phi'$, is satisfied by $y = \\phi(c) + \\phi(x - c)$ or by $y = 2\\phi(\\frac{1}{2}x)$.", "markdown": "Verify that the differential equation $y = \\phi\\{\\psi(y_{1})\\} + \\phi\\{x - \\psi(y_{1})\\}$, where $y_{1}$ is the derivative of $y$, and $\\psi$ is the function inverse to $\\phi'$, is satisfied by $y = \\phi(c) + \\phi(x - c)$ or by $y = 2\\phi(\\frac{1}{2}x)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vi/9", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vi", "number": 9, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "253", "location": "Exercise Misc-VI, problem 9", "problem_latex": "Verify that the differential equation $y = \\{x/\\psi(y_{1})\\} \\phi\\{\\psi(y_{1})\\}$, where the notation is the same as that of Ex.~8, is satisfied by $y = c\\phi(x/c)$ or by $y = \\beta x$, where $\\beta = \\phi(\\alpha)/\\alpha$ and $\\alpha$~is any root of the equation $\\phi(\\alpha) - \\alpha\\phi'(\\alpha) = 0$.", "markdown": "Verify that the differential equation $y = \\{x/\\psi(y_{1})\\} \\phi\\{\\psi(y_{1})\\}$, where the notation is the same as that of Ex. 8, is satisfied by $y = c\\phi(x/c)$ or by $y = \\beta x$, where $\\beta = \\phi(\\alpha)/\\alpha$ and $\\alpha$ is any root of the equation $\\phi(\\alpha) - \\alpha\\phi'(\\alpha) = 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-vii/none", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-vii", "number": null, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "300", "location": "Exercise Misc-VII, problem None", "problem_latex": null, "markdown": "", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "FLAG-EXTRACTION", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/1", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 1", "problem_latex": "Show that the real part of $i^{\\log(1+i)}$ is\n\\[\ne^{(4k+1)\\pi^{2}/8 } \\cos \\{\\tfrac{1}{4}(4k + 1)\\pi\\log 2\\},\n\\]\nwhere $k$~is any integer.", "markdown": "Show that the real part of $i^{\\log(1+i)}$ is e^(4k+1)^2/8 14(4k + 1)2, where $k$ is any integer.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.const", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/10", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 10, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 10", "problem_latex": "The equation $\\tan z = a\\tanh cz$, where $a$ and~$c$ are real, has an infinity\nof real and of purely imaginary roots, but no complex roots.", "markdown": "The equation $\\tan z = a\\tanh cz$, where $a$ and $c$ are real, has an infinity of real and of purely imaginary roots, but no complex roots.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.hyp", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/11", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 11, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 11", "problem_latex": "Show that if $x$~is real then\n\\[\ne^{ax} \\cos bx = \\sum_{0}^{\\infty} \\frac{x^{n}}{n!} \\left\\{\n a^{n} - \\binom{n}{2} a^{n-2} b^{2} + \\binom{n}{4} a^{n-4} b^{4} - \\dots\n\\right\\},\n\\]\nwhere there are $\\frac{1}{2}(n + 1)$ or~$\\frac{1}{2}(n + 2)$ terms inside the large brackets. Find\na similar series for~$e^{ax} \\sin bx$.", "markdown": "Show that if $x$ is real then e^ax bx = _0^ x^nn! a^n - n2 a^n-2 b^2 + n4 a^n-4 b^4 - …, where there are $\\frac{1}{2}(n + 1)$ or $\\frac{1}{2}(n + 2)$ terms inside the large brackets. Find a similar series for $e^{ax} \\sin bx$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.hyp", "core.prog", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/12", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 12, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 12", "problem_latex": "If $n\\phi(z, n) \\to z$ as $n \\to \\infty$, then $\\{1 + \\phi(z, n)\\}^{n} \\to \\exp z$.", "markdown": "If $n\\phi(z, n) \\to z$ as $n \\to \\infty$, then $\\{1 + \\phi(z, n)\\}^{n} \\to \\exp z$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/13", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 13, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 13", "problem_latex": "If $\\phi(t)$~is a complex function of the real variable~$t$, then\n\\[\n\\frac{d}{dt} \\log \\phi(t) = \\frac{\\phi'(t)}{\\phi(t)}.\n\\]\n\n%[** TN: Paragraph break added]\n[Use the formulae\n\\[\n\\phi = \\psi + i\\chi,\\quad\n\\log \\phi = \\tfrac{1}{2}\\log(\\psi^{2} + \\chi^{2}) + i\\arctan(\\chi/\\psi).]\n\\]", "markdown": "If $\\phi(t)$ is a complex function of the real variable $t$, then ddt (t) = ’(t)(t). %[** TN: Paragraph break added] [Use the formulae = + i,0pt minus 3pt= 12(^2 + ^2) + i(/).]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.complex", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/14", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 14, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 14", "problem_latex": "\\Topic{Transformations.} In \\okrickRef{Ch.}{III} (\\Exs{xxi}.\\ 21~\\textit{et~seq.}, and \\MiscExs{III}\\\n22~\\textit{et seq.})\\ we considered some simple examples of the geometrical relations\nbetween figures in the planes of two variables $z$,~$Z$ connected by a relation\n$z = f(Z)$. We shall now consider some cases in which the relation involves\nlogarithmic, exponential, or circular functions.\n\nSuppose firstly that\n\\[\nz = \\exp(\\pi Z/a),\\quad\nZ = (a/\\pi) \\Log z\n\\]\nwhere $a$~is positive. To one value of~$Z$ corresponds one of~$z$, but to one of~$z$\ninfinitely many of~$Z$. If $x$,~$y$, $r$,~$\\theta$ are the coordinates of~$z$ and $X$,~$Y$, $R$,~$\\Theta$\nthose of~$Z$, we have the relations\n\\begin{alignat*}{2}\nx &= e^{\\pi X/a} \\cos(\\pi Y/a),\\qquad & y &= e^{\\pi X/a} \\sin(\\pi Y/a),\\\\\nX &= (a/\\pi) \\log r, & Y &= (a\\theta/\\pi) + 2ka,\n\\end{alignat*}\nwhere $k$~is any integer. If we suppose that $-\\pi < \\theta \\leq \\pi$, and that $\\Log z$~has its\nprincipal value~$\\log z$, then $k = 0$, and $Z$~is confined to a strip of its plane parallel\nto the axis~$OX$ and extending to a distance~$a$ from it on each side, one point\n\\PageSep{427}\nof this strip corresponding to one of the whole $z$-plane, and conversely. By\ntaking a value of~$\\Log z$ other than the principal value we obtain a similar\nrelation between the $z$-plane and another strip of breadth~$2a$ in the $Z$-plane.\n\nTo the lines in the $Z$-plane for which $X$~and~$Y$ are constant correspond the\ncircles and radii vectores in the $z$-plane for which $r$~and~$\\theta$ are constant. To\none of the latter lines corresponds the whole of a parallel to~$OX$, but to a\ncircle for which $r$~is constant corresponds only a part, of length~$2a$, of a\nparallel to~$OY$. To make $Z$~describe the whole of the latter line we must\nmake $z$ move continually round and round the circle.", "markdown": "**.** In Ch.III (xxi. 21 *et seq.*, and [misc:III]Misc. Exs. 22 *et seq.*) we considered some simple examples of the geometrical relations between figures in the planes of two variables $z$, $Z$ connected by a relation $z = f(Z)$. We shall now consider some cases in which the relation involves logarithmic, exponential, or circular functions. Suppose firstly that z = (Z/a),0pt minus 3ptZ = (a/) z where $a$ is positive. To one value of $Z$ corresponds one of $z$, but to one of $z$ infinitely many of $Z$. If $x$, $y$, $r$, $\\theta$ are the coordinates of $z$ and $X$, $Y$, $R$, $\\Theta$ those of $Z$, we have the relations alignat*2 x &= e^X/a (Y/a), & y &= e^X/a (Y/a), X &= (a/) r, & Y &= (a/) + 2ka, alignat* where $k$ is any integer. If we suppose that $-\\pi < \\theta \\leq \\pi$, and that $\\Log z$ has its principal value $\\log z$, then $k = 0$, and $Z$ is confined to a strip of its plane parallel to the axis $OX$ and extending to a distance $a$ from it on each side, one point [pg]427 of this strip corresponding to one of the whole $z$-plane, and conversely. By taking a value of $\\Log z$ other than the principal value we obtain a similar relation between the $z$-plane and another strip of breadth $2a$ in the $Z$-plane. To the lines in the $Z$-plane for which $X$ and $Y$ are constant correspond the circles and radii vectores in the $z$-plane for which $r$ and $\\theta$ are constant. To one of the latter lines corresponds the whole of a parallel to $OX$, but to a circle for which $r$ is constant corresponds only a part, of length $2a$, of a parallel to $OY$. To make $Z$ describe the whole of the latter line we must make $z$ move continually round and round the circle.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.graph", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/15", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 15, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 15", "problem_latex": "Show that to a straight line in the $Z$-plane corresponds an equiangular\nspiral in the $z$-plane.", "markdown": "Show that to a straight line in the $Z$-plane corresponds an equiangular spiral in the $z$-plane.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/16", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 16, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 16", "problem_latex": "Discuss similarly the transformation $z = c\\cosh(\\pi Z/a)$, showing in\nparticular that the whole $z$-plane corresponds to any one of an infinite\nnumber of strips in the $Z$-plane, each parallel to the axis $OX$ and of\nbreadth~$2a$. Show also that to the line $X = X_{0}$ corresponds the ellipse\n\\[\n\\left\\{\\frac{x}{c\\cosh(\\pi X_{0}/a)}\\right\\}^{2} +\n\\left\\{\\frac{y}{c\\sinh(\\pi X_{0}/a)}\\right\\}^{2} = 1,\n\\]\nand that for different values of~$X_{0}$ these ellipses form a confocal system; and\nthat the lines $Y = Y_{0}$ correspond to the associated system of confocal hyperbolas.\nTrace the variation of~$z$ as $Z$~describes the whole of a line $X = X_{0}$ or\n$Y = Y_{0}$. How does $Z$~vary as $z$~describes the degenerate ellipse and hyperbola\nformed by the segment between the foci of the confocal system and the\nremaining segments of the axis of~$x$?", "markdown": "Discuss similarly the transformation $z = c\\cosh(\\pi Z/a)$, showing in particular that the whole $z$-plane corresponds to any one of an infinite number of strips in the $Z$-plane, each parallel to the axis $OX$ and of breadth $2a$. Show also that to the line $X = X_{0}$ corresponds the ellipse xc(X_0/a)^2 + yc(X_0/a)^2 = 1, and that for different values of $X_{0}$ these ellipses form a confocal system; and that the lines $Y = Y_{0}$ correspond to the associated system of confocal hyperbolas. Trace the variation of $z$ as $Z$ describes the whole of a line $X = X_{0}$ or $Y = Y_{0}$. How does $Z$ vary as $z$ describes the degenerate ellipse and hyperbola formed by the segment between the foci of the confocal system and the remaining segments of the axis of $x$?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.graph", "core.hyp" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/17", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 17, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 17", "problem_latex": "Verify that the results of Ex.~16 are in agreement with those of Ex.~14\nand those of \\okrickRef{Ch.}{III}, \\MiscEx{III}~25. [The transformation $z = c\\cosh(\\pi Z/a)$\nmay be regarded as compounded from the transformations\n\\[\nz = cz_{1},\\quad\nz_{1} = \\tfrac{1}{2}\\{z_{2} + (1/z_{2})\\},\\quad\nz_{2} = \\exp(\\pi Z/a).]\n\\]", "markdown": "Verify that the results of Ex. 16 are in agreement with those of Ex. 14 and those of Ch.III, [misc:III]Misc. Ex. 25. [The transformation $z = c\\cosh(\\pi Z/a)$ may be regarded as compounded from the transformations z = cz_1,0pt minus 3ptz_1 = 12z_2 + (1/z_2),0pt minus 3ptz_2 = (Z/a).]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/18", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 18, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 18", "problem_latex": "Discuss similarly the transformation $z = c\\tanh(\\pi Z/a)$, showing that\nto the lines $X = X_{0}$ correspond the coaxal circles\n\\[\n\\{x - c\\coth(2\\pi X_{0}/a)\\}^{2} + y^{2} = c^{2}\\cosech^{2}(2\\pi X_{0}/a),\n\\]\nand to the lines $Y = Y_{0}$ the orthogonal system of coaxal circles.", "markdown": "Discuss similarly the transformation $z = c\\tanh(\\pi Z/a)$, showing that to the lines $X = X_{0}$ correspond the coaxal circles x - c(2X_0/a)^2 + y^2 = c^2^2(2X_0/a), and to the lines $Y = Y_{0}$ the orthogonal system of coaxal circles.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.graph", "core.hyp" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/19", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 19, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 19", "problem_latex": "\\Topic{The Stereographic and Mercator's Projections.} The points of a\nunit sphere whose centre is the origin are projected from the south pole (whose\ncoordinates are $0$,~$0$,~$-1$) on to the tangent plane at the north pole. The\ncoordinates of a point on the sphere are $\\xi$,~$\\eta$,~$\\zeta$, and Cartesian axes $OX$,~$OY$\nare taken on the tangent plane, parallel to the axes of $\\xi$ and~$\\eta$. Show that\nthe coordinates of the projection of the point are\n\\[\nx = 2\\xi/(1 + \\zeta),\\quad\ny = 2\\eta/(1 + \\zeta),\n\\]\nand that $x + iy = 2\\tan \\frac{1}{2}\\theta \\Cis\\phi$, where $\\phi$~is the longitude (measured from the\nplane $\\eta = 0$) and $\\theta$~the north polar distance of the point on the sphere.\n\\PageSep{428}\n\nThis projection gives a map of the sphere on the tangent plane, generally\nknown as the \\emph{Stereographic Projection}. If now we introduce a new complex\nvariable\n\\[\nZ = X + iY = -i\\log \\tfrac{1}{2}z = -i\\log \\tfrac{1}{2}(x + iy)\n\\]\nso that $X = \\phi$, $Y = \\log \\cot \\frac{1}{2}\\theta$, we obtain another map in the plane of~$Z$,\nusually called \\emph{Mercator's Projection}. In this map parallels of latitude and\nlongitude are represented by straight lines parallel to the axes of $X$ and $Y$\nrespectively.", "markdown": "**Stereographic and Mercator’s Projections.** The points of a unit sphere whose centre is the origin are projected from the south pole (whose coordinates are $0$, $0$, $-1$) on to the tangent plane at the north pole. The coordinates of a point on the sphere are $\\xi$, $\\eta$, $\\zeta$, and Cartesian axes $OX$, $OY$ are taken on the tangent plane, parallel to the axes of $\\xi$ and $\\eta$. Show that the coordinates of the projection of the point are x = 2/(1 + ),0pt minus 3pty = 2/(1 + ), and that $x + iy = 2\\tan \\frac{1}{2}\\theta \\Cis\\phi$, where $\\phi$ is the longitude (measured from the plane $\\eta = 0$) and $\\theta$ the north polar distance of the point on the sphere. [pg]428 This projection gives a map of the sphere on the tangent plane, generally known as the *Stereographic Projection*. If now we introduce a new complex variable Z = X + iY = -i12z = -i12(x + iy) so that $X = \\phi$, $Y = \\log \\cot \\frac{1}{2}\\theta$, we obtain another map in the plane of $Z$, usually called *Mercator’s Projection*. In this map parallels of latitude and longitude are represented by straight lines parallel to the axes of $X$ and $Y$ respectively.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.graph", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/2", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 2", "problem_latex": "If $a\\cos\\theta + b\\sin\\theta + c = 0$, where $a$,~$b$,~$c$ are real and $c^{2} > a^{2} + b^{2}$, then\n\\[\n\\theta = m\\pi + \\alpha\n ± i\\log \\frac{|c| + \\sqrtp{c^{2} - a^{2} - b^{2}}}{\\sqrtp{a^{2} + b^{2}}},\n\\]\nwhere $m$~is any odd or any even integer, according as $c$~is positive or negative,\nand $\\alpha$~is an angle whose cosine and sine are $a/\\sqrtp{a^{2} + b^{2}}$ and $b/\\sqrtp{a^{2} + b^{2}}$.", "markdown": "If $a\\cos\\theta + b\\sin\\theta + c = 0$, where $a$, $b$, $c$ are real and $c^{2} > a^{2} + b^{2}$, then = m+ ± i|c| + c^2 - a^2 - b^2a^2 + b^2, where $m$ is any odd or any even integer, according as $c$ is positive or negative, and $\\alpha$ is an angle whose cosine and sine are $a/\\sqrtp{a^{2} + b^{2}}$ and $b/\\sqrtp{a^{2} + b^{2}}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.log", "core.solve.num", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/20", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 20, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 20", "problem_latex": "Discuss the transformation given by the equation\n\\[\nz = \\Log \\left(\\frac{Z - a}{Z - b}\\right),\n\\]\nshowing that the straight lines for which $x$~and~$y$ are constant correspond to\ntwo orthogonal systems of coaxal circles in the $Z$-plane.", "markdown": "Discuss the transformation given by the equation z = (Z - aZ - b), showing that the straight lines for which $x$ and $y$ are constant correspond to two orthogonal systems of coaxal circles in the $Z$-plane.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.graph", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/21", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 21, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 21", "problem_latex": "Discuss the transformation\n\\[\nz = \\Log \\left\\{\\frac{\\sqrtp{Z - a} + \\sqrtp{Z - b}}{\\sqrtp{b - a}}\\right\\},\n\\]\nshowing that the straight lines for which $x$~and~$y$ are constant correspond to\nsets of confocal ellipses and hyperbolas whose foci are the points $Z = a$ and\n$Z = b$.\n\n[We have\n\\begin{alignat*}{2}\n\\sqrtp{Z - a} + \\sqrtp{Z - b} &= \\sqrtp{b - a}\\, \\exp(& &x + iy), \\\\\n\\sqrtp{Z - a} - \\sqrtp{Z - b} &= \\sqrtp{b - a}\\, \\exp(&-&x - iy);\n\\end{alignat*}\nand it will be found that\n\\[\n|Z - a| + |Z - b| = |b - a|\\cosh 2x,\\quad\n|Z - a| - |Z - b| = |b - a|\\cos 2y.]\n\\]", "markdown": "Discuss the transformation z = Z - a + Z - bb - a, showing that the straight lines for which $x$ and $y$ are constant correspond to sets of confocal ellipses and hyperbolas whose foci are the points $Z = a$ and $Z = b$. [We have alignat*2 Z - a + Z - b &= b - a  (& &x + iy), Z - a - Z - b &= b - a  (&-&x - iy); alignat* and it will be found that |Z - a| + |Z - b| = |b - a|2x,0pt minus 3pt|Z - a| - |Z - b| = |b - a|2y.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.graph", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/22", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 22, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 22", "problem_latex": "\\Topic{The transformation $z = Z^{i}$.} If $z = Z^{i}$, where the imaginary power\nhas its principal value, we have\n\\[\n\\exp(\\log r + i\\theta) = z = \\exp(i\\log Z) = \\exp(i\\log R - \\Theta),\n\\]\nso that $\\log r = -\\Theta$, $\\theta = \\log R + 2k\\pi$, where $k$~is an integer. As all values of~$k$\ngive the same point~$z$, we shall suppose that $k = 0$, so that\n\\[\n\\log r = -\\Theta,\\quad\n\\theta = \\log R.\n\\Tag{(1)}\n\\]\n\nThe whole plane of~$Z$ is covered when $R$~varies through all positive\nvalues and $\\Theta$~from $-\\pi$ to~$\\pi$: then $r$~has the range $\\exp(-\\pi)$ to~$\\exp\\pi$ and $\\theta$~ranges\nthrough all real values. Thus the $Z$-plane corresponds to the ring\nbounded by the circles $r = \\exp(-\\pi)$, $r = \\exp\\pi$; but this ring is covered\ninfinitely often. If however $\\theta$~is allowed to vary only between $-\\pi$ and~$\\pi$,\nso that the ring is covered only once, then $R$~can vary only from $\\exp(-\\pi)$ to~$\\exp \\pi$,\nso that the variation of~$Z$ is restricted to a ring similar in all respects\nto that within which $z$~varies. Each ring, moreover, must be regarded as\nhaving a barrier along the negative real axis which~$z$ (or~$Z$) must not cross, as\nits amplitude must not transgress the limits $-\\pi$ and~$\\pi$.\n\\PageSep{429}\n\nWe thus obtain a correspondence between two rings, given by the pair of\nequations\n\\[\nz = Z^{i},\\quad\nZ = z^{-i},\n\\]\nwhere each power has its principal value. To circles whose centre is the\norigin in one plane correspond straight lines through the origin in the other.", "markdown": "**transformation $z = Z^{i}$.** If $z = Z^{i}$, where the imaginary power has its principal value, we have (r + i) = z = (iZ) = (iR - ), so that $\\log r = -\\Theta$, $\\theta = \\log R + 2k\\pi$, where $k$ is an integer. As all values of $k$ give the same point $z$, we shall suppose that $k = 0$, so that r = -,0pt minus 3pt= R. (1) The whole plane of $Z$ is covered when $R$ varies through all positive values and $\\Theta$ from $-\\pi$ to $\\pi$: then $r$ has the range $\\exp(-\\pi)$ to $\\exp\\pi$ and $\\theta$ ranges through all real values. Thus the $Z$-plane corresponds to the ring bounded by the circles $r = \\exp(-\\pi)$, $r = \\exp\\pi$; but this ring is covered infinitely often. If however $\\theta$ is allowed to vary only between $-\\pi$ and $\\pi$, so that the ring is covered only once, then $R$ can vary only from $\\exp(-\\pi)$ to $\\exp \\pi$, so that the variation of $Z$ is restricted to a ring similar in all respects to that within which $z$ varies. Each ring, moreover, must be regarded as having a barrier along the negative real axis which $z$ (or $Z$) must not cross, as its amplitude must not transgress the limits $-\\pi$ and $\\pi$. [pg]429 We thus obtain a correspondence between two rings, given by the pair of equations z = Z^i,0pt minus 3ptZ = z^-i, where each power has its principal value. To circles whose centre is the origin in one plane correspond straight lines through the origin in the other.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.graph", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/23", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 23, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 23", "problem_latex": "Trace the variation of~$z$ when $Z$, starting at the point~$\\exp \\pi$, moves\nround the larger circle in the positive direction to the point~$-\\exp \\pi$, along\nthe barrier, round the smaller circle in the negative direction, back along the\nbarrier, and round the remainder of the larger circle to its original position.", "markdown": "Trace the variation of $z$ when $Z$, starting at the point $\\exp \\pi$, moves round the larger circle in the positive direction to the point $-\\exp \\pi$, along the barrier, round the smaller circle in the negative direction, back along the barrier, and round the remainder of the larger circle to its original position.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/24", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 24, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 24", "problem_latex": "Suppose each plane to be divided up into an infinite series of rings\nby circles of radii\n\\[\n\\dots,\\quad e^{-(2n+1)\\pi},\\ \\dots,\\quad\ne^{-\\pi},\\quad e^{\\pi},\\quad e^{3\\pi},\\ \\dots,\\quad\ne^{(2n+1)\\pi},\\ \\dots.\n\\]\nShow how to make any ring in one plane correspond to any ring in the\nother, by taking suitable values of the powers in the equations $z = Z^{i}$, $Z = z^{-i}$.", "markdown": "Suppose each plane to be divided up into an infinite series of rings by circles of radii …,0pt minus 3pte^-(2n+1), …,0pt minus 3pte^-,0pt minus 3pte^,0pt minus 3pte^3, …,0pt minus 3pte^(2n+1), …. Show how to make any ring in one plane correspond to any ring in the other, by taking suitable values of the powers in the equations $z = Z^{i}$, $Z = z^{-i}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.graph", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/25", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 25, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 25", "problem_latex": "If $z = Z^{i}$, any value of the power being taken, and $Z$~moves along an\nequiangular spiral whose pole is the origin in its plane, then $z$~moves along an\nequiangular spiral whose pole is the origin in its plane.", "markdown": "If $z = Z^{i}$, any value of the power being taken, and $Z$ moves along an equiangular spiral whose pole is the origin in its plane, then $z$ moves along an equiangular spiral whose pole is the origin in its plane.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.graph", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/26", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 26, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 26", "problem_latex": "How does $Z = z^{ai}$, where $a$~is real, behave as $z$~approaches the origin\nalong the real axis\\DPtypo{.}{?} [$Z$~moves round and round a circle whose centre is the\norigin (the unit circle if $z^{ai}$~has its principal value), and the real and imaginary\nparts of~$Z$ both oscillate finitely.]", "markdown": "How does $Z = z^{ai}$, where $a$ is real, behave as $z$ approaches the origin along the real axis. [$Z$ moves round and round a circle whose centre is the origin (the unit circle if $z^{ai}$ has its principal value), and the real and imaginary parts of $Z$ both oscillate finitely.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/27", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 27, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 27", "problem_latex": "Discuss the same question for $Z = z^{a+bi}$, where $a$~and~$b$ are any real\nnumbers.", "markdown": "Discuss the same question for $Z = z^{a+bi}$, where $a$ and $b$ are any real numbers.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/28", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 28, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 28", "problem_latex": "Show that the region of convergence of a series of the type $\\sum\\limits_{-\\infty}^{\\infty} a_{n}z^{nai}$,\nwhere $a$~is real, is an angle, \\ie\\ a region bounded by inequalities of the type\n$\\theta_{0} < \\am z < \\theta_{1}$ [The angle may reduce to a line, or cover the whole plane.]", "markdown": "Show that the region of convergence of a series of the type $\\sum\\limits_{-\\infty}^{\\infty} a_{n}z^{nai}$, where $a$ is real, is an angle, *i.e.* a region bounded by inequalities of the type $\\theta_{0} < \\am z < \\theta_{1}$ [The angle may reduce to a line, or cover the whole plane.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/29", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 29, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 29", "problem_latex": "\\Topic{Level Curves.} If $f(z)$~is a function of the complex variable~$z$, we\ncall the curves for which $|f(z)|$~is constant the \\emph{level curves} of~$f(z)$. Sketch\nthe forms of the level curves of\n\\begin{alignat*}{2}\nz - a \\quad& \\text{(\\emph{concentric circles})}, \\qquad&\n(z - a)(z - b) \\quad& \\text{(\\emph{Cartesian ovals})}, \\\\\n(z - a)/(z - b) \\quad& \\text{(\\emph{coaxal circles})}, \\qquad&\n\\exp z \\quad& \\text{(\\emph{straight lines})}.\n\\end{alignat*}", "markdown": "**Curves.** If $f(z)$ is a function of the complex variable $z$, we call the curves for which $|f(z)|$ is constant the *level curves* of $f(z)$. Sketch the forms of the level curves of alignat*2 z - a 0pt minus 3pt& (*concentric circles*), & (z - a)(z - b) 0pt minus 3pt& (*Cartesian ovals*), (z - a)/(z - b) 0pt minus 3pt& (*coaxal circles*), & z 0pt minus 3pt& (*straight lines*). alignat*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/3", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 3", "problem_latex": "Prove that if $\\theta$~is real and $\\sin\\theta \\sin\\phi = 1$ then\n\\[\n\\phi = (k + \\tfrac{1}{2})\\pi ± i\\log \\cot \\tfrac{1}{2}(k\\pi + \\theta),\n\\]\nwhere $k$~is any even or any odd integer, according as $\\sin\\theta$~is positive or\nnegative.", "markdown": "Prove that if $\\theta$ is real and $\\sin\\theta \\sin\\phi = 1$ then = (k + 12)± i12(k+ ), where $k$ is any even or any odd integer, according as $\\sin\\theta$ is positive or negative.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/30", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 30, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 30", "problem_latex": "Sketch the forms of the level curves of $(z - a)(z - b)(z - c)$,\n$(1 + z\\sqrt{3} + z^{2})/z$. [Some of the level curves of the latter function are drawn in\n\\Fig{59}, the curves marked \\textsc{i}--\\textsc{vii} corresponding to the values\n\\[\n.10,\\quad 2 - \\sqrt{3} = .27,\\quad\n.40,\\quad 1.00,\\quad 2.00,\\quad\n2 + \\sqrt{3} = 3.73,\\quad 4.53\n\\]\nof~$|f(z)|$. The reader will probably find but little difficulty in arriving at a\ngeneral idea of the forms of the level curves of any given rational function;\nbut to enter into details would carry us into the general theory of functions\nof a complex variable.]", "markdown": "Sketch the forms of the level curves of $(z - a)(z - b)(z - c)$, $(1 + z\\sqrt{3} + z^{2})/z$. [Some of the level curves of the latter function are drawn in [fig:59]Fig. 59, the curves marked i--vii corresponding to the values .10,0pt minus 3pt2 - 3 = .27,0pt minus 3pt.40,0pt minus 3pt1.00,0pt minus 3pt2.00,0pt minus 3pt2 + 3 = 3.73,0pt minus 3pt4.53 of $|f(z)|$. The reader will probably find but little difficulty in arriving at a general idea of the forms of the level curves of any given rational function; but to enter into details would carry us into the general theory of functions of a complex variable.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/31i", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 31, "part": "i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 31i", "problem_latex": "Sketch the forms of the level curves of (i)~$z\\exp z$, (ii)~$\\sin z$. [See\n\\Fig{60}, which represents the level curves of~$\\sin z$. The curves marked \\textsc{i}--\\textsc{viii}\ncorrespond to $k = .35$, $.50$, $.71$, $1.00$, $1.41$, $2.00$, $2.83$,~$4.00$.]", "markdown": "Sketch the forms of the level curves of (i) $z\\exp z$, (ii) $\\sin z$. [See [fig:60]Fig. 60, which represents the level curves of $\\sin z$. The curves marked i--viii correspond to $k = .35$, $.50$, $.71$, $1.00$, $1.41$, $2.00$, $2.83$, $4.00$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.graph", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/31ii", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 31, "part": "ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 31ii", "problem_latex": "Sketch the forms of the level curves of (i)~$z\\exp z$, (ii)~$\\sin z$. [See\n\\Fig{60}, which represents the level curves of~$\\sin z$. The curves marked \\textsc{i}--\\textsc{viii}\ncorrespond to $k = .35$, $.50$, $.71$, $1.00$, $1.41$, $2.00$, $2.83$,~$4.00$.]", "markdown": "Sketch the forms of the level curves of (i) $z\\exp z$, (ii) $\\sin z$. [See [fig:60]Fig. 60, which represents the level curves of $\\sin z$. The curves marked i--viii correspond to $k = .35$, $.50$, $.71$, $1.00$, $1.41$, $2.00$, $2.83$, $4.00$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.graph", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/32", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 32, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 32", "problem_latex": "Sketch the forms of the level curves of~$\\exp z - c$, where $c$~is a real\nconstant. [\\Fig{61} shows the level curves of $|\\exp z - 1|$, the curves \\textsc{i}--\\textsc{vii}\ncorresponding to the values of~$k$ given by $\\log k = -1.00$, $-.20$, $-.05$, $0.00$,\n$.05$, $.20$,~$1.00$.]", "markdown": "Sketch the forms of the level curves of $\\exp z - c$, where $c$ is a real constant. [[fig:61]Fig. 61 shows the level curves of $|\\exp z - 1|$, the curves i--vii corresponding to the values of $k$ given by $\\log k = -1.00$, $-.20$, $-.05$, $0.00$, $.05$, $.20$, $1.00$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.graph", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/33", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 33, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 33", "problem_latex": "The level curves of~$\\sin z - c$, where $c$~is a positive constant, are\nsketched in Figs.~62,~63. [The nature of the curves differs according as\nto whether $c < 1$ or~$c > 1$. In \\Fig{62} we have taken $c = .5$, and the curves\n\\textsc{i}--\\textsc{viii} correspond to $k = .29$, $.37$, $.50$, $.87$, $1.50$, $2.60$, $4.50$,~$7.79$. In \\Fig{63}\nwe have taken $c = 2$, and the curves \\textsc{i}--\\textsc{vii} correspond to $k = .58$, $1.00$, $1.73$,\n$3.00$, $5.20$, $9.00$,~$15.59$. If $c = 1$ then the curves are the same as those of\n\\Fig{60}, except that the origin and scale are different.]", "markdown": "The level curves of $\\sin z - c$, where $c$ is a positive constant, are sketched in Figs. 62, 63. [The nature of the curves differs according as to whether $c < 1$ or $c > 1$. In [fig:62]Fig. 62 we have taken $c = .5$, and the curves i--viii correspond to $k = .29$, $.37$, $.50$, $.87$, $1.50$, $2.60$, $4.50$, $7.79$. In [fig:63]Fig. 63 we have taken $c = 2$, and the curves i--vii correspond to $k = .58$, $1.00$, $1.73$, $3.00$, $5.20$, $9.00$, $15.59$. If $c = 1$ then the curves are the same as those of [fig:60]Fig. 60, except that the origin and scale are different.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.graph", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/34", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 34, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 34", "problem_latex": "Prove that if $0 < \\theta < \\pi$ then\n\\begin{alignat*}{3}\n\\cos\\theta &+ \\tfrac{1}{3} \\cos 3\\theta &&+ \\tfrac{1}{5} \\cos 5\\theta &&+ \\dots\n = \\tfrac{1}{4} \\log \\cot^{2}\\tfrac{1}{2}\\theta,\\\\\n\\sin\\theta &+ \\tfrac{1}{3} \\sin 3\\theta &&+ \\tfrac{1}{5} \\sin 5\\theta &&+ \\dots\n = \\tfrac{1}{4}\\pi,\n\\end{alignat*}\nand determine the sums of the series for all other values of~$\\theta$ for which they\nare convergent. [Use the equation\n\\[\nz + \\tfrac{1}{3}z^{3} + \\tfrac{1}{5}z^{5} + \\dots\n = \\tfrac{1}{2} \\log \\left(\\frac{1 + z}{1 - z}\\right)\n\\]\nwhere $z = \\cos\\theta + i\\sin\\theta$. When $\\theta$~is increased by~$\\pi$ the sum of each series\nsimply changes its sign. It follows that the first formula holds for all values\nof~$\\theta$ save multiples of~$\\pi$ (for which the series diverges), while the sum of the\nsecond series is~$\\frac{1}{4}\\pi$ if $2k\\pi < \\theta < (2k + 1)\\pi$, $-\\frac{1}{4}\\pi$ if $(2k + 1)\\pi < \\theta < (2k + 2)\\pi$,\nand $0$ if $\\theta$~is a multiple of~$\\pi$.]", "markdown": "Prove that if $0 < \\theta < \\pi$ then alignat*3 &+ 13 3&&+ 15 5&&+ … = 14 ^212, &+ 13 3&&+ 15 5&&+ … = 14, alignat* and determine the sums of the series for all other values of $\\theta$ for which they are convergent. [Use the equation z + 13z^3 + 15z^5 + … = 12 (1 + z1 - z) where $z = \\cos\\theta + i\\sin\\theta$. When $\\theta$ is increased by $\\pi$ the sum of each series simply changes its sign. It follows that the first formula holds for all values of $\\theta$ save multiples of $\\pi$ (for which the series diverges), while the sum of the second series is $\\frac{1}{4}\\pi$ if $2k\\pi < \\theta < (2k + 1)\\pi$, $-\\frac{1}{4}\\pi$ if $(2k + 1)\\pi < \\theta < (2k + 2)\\pi$, and $0$ if $\\theta$ is a multiple of $\\pi$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.const", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/35", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 35, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 35", "problem_latex": "Prove that if $0 < \\theta < \\frac{1}{2}\\pi$ then\n\\begin{alignat*}{3}\n\\cos\\theta &- \\tfrac{1}{3} \\cos 3\\theta &&+ \\tfrac{1}{5} \\cos 5\\theta &&- \\dots\n = \\tfrac{1}{4}\\pi,\\\\\n\\sin\\theta &- \\tfrac{1}{3} \\sin 3\\theta &&+ \\tfrac{1}{5} \\sin 5\\theta &&- \\dots\n = \\tfrac{1}{4} \\log (\\sec\\theta + \\tan\\theta)^{2};\n\\end{alignat*}\nand determine the sums of the series for all other values of~$\\theta$ for which they\nare convergent.", "markdown": "Prove that if $0 < \\theta < \\frac{1}{2}\\pi$ then alignat*3 &- 13 3&&+ 15 5&&- … = 14, &- 13 3&&+ 15 5&&- … = 14 (+ )^2; alignat* and determine the sums of the series for all other values of $\\theta$ for which they are convergent.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.const", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/36", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 36, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 36", "problem_latex": "Prove that\n\\[\n\\cos\\theta \\cos\\alpha\n + \\tfrac{1}{2} \\cos 2\\theta \\cos 2\\alpha\n + \\tfrac{1}{3} \\cos 3\\theta \\cos 3\\alpha + \\dots\n = -\\tfrac{1}{4} \\log \\{4(\\cos\\theta - \\cos\\alpha)^{2}\\},\n\\]\nunless $\\theta - \\alpha$ or $\\theta + \\alpha$ is a multiple of~$2\\pi$.", "markdown": "Prove that + 12 22 + 13 33+ … = -14 4(- )^2, unless $\\theta - \\alpha$ or $\\theta + \\alpha$ is a multiple of $2\\pi$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.const", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/37", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 37, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 37", "problem_latex": "Prove that if neither $a$ nor~$b$ is real then\n\\[\n\\int_{0}^{\\infty} \\frac{dx}{(x - a)(x - b)}\n = -\\frac{\\log(-a) - \\log(-b)}{a - b},\n\\]\neach logarithm having its principal value. Verify the result when $a = ci$,\n$b = -ci$, where $c$~is positive. Discuss also the cases in which $a$ or~$b$ or both\nare real and negative.", "markdown": "Prove that if neither $a$ nor $b$ is real then _0^ dx(x - a)(x - b) = -(-a) - (-b)a - b, each logarithm having its principal value. Verify the result when $a = ci$, $b = -ci$, where $c$ is positive. Discuss also the cases in which $a$ or $b$ or both are real and negative.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.defint", "cas.integrate", "core.complex", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/38", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 38, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 38", "problem_latex": "Prove that if $\\alpha$ and~$\\beta$ are real, and $\\beta > 0$, then\n\\[\n\\int_{0}^{\\infty} \\frac{d}{x^{2} - (\\alpha + i\\beta)^{2}}\n = \\frac{\\pi i}{2(\\alpha + i\\beta)}.\n\\]\nWhat is the value of the integral when $\\beta < 0$?", "markdown": "Prove that if $\\alpha$ and $\\beta$ are real, and $\\beta > 0$, then _0^ dx^2 - (+ i)^2 = i2(+ i). What is the value of the integral when $\\beta < 0$?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.defint", "core.complex", "core.const" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/39", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 39, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 39", "problem_latex": "Prove that, if the roots of $Ax^{2} + 2Bx + C = 0$ have their imaginary\nparts of opposite signs, then\n\\[\n\\int_{-\\infty}^{\\infty} \\frac{dx}{Ax^{2} + 2Bx + C}\n = \\frac{\\pi i}{\\sqrtp{B^{2} - AC}},\n\\]\nthe sign of $\\sqrtp{B^{2} - AC}$ being so chosen that the real part of $\\{\\sqrtp{B^{2} - AC}\\}/Ai$\nis positive.", "markdown": "Prove that, if the roots of $Ax^{2} + 2Bx + C = 0$ have their imaginary parts of opposite signs, then _-^ dxAx^2 + 2Bx + C = iB^2 - AC, the sign of $\\sqrtp{B^{2} - AC}$ being so chosen that the real part of $\\{\\sqrtp{B^{2} - AC}\\}/Ai$ is positive.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.defint", "cas.solve.poly", "core.complex" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/4", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 4", "problem_latex": "Show that if $x$~is real then\n\\begin{gather*}\n\\frac{d}{dx} \\exp\\{(a + ib)x\\} = (a + ib) \\exp\\{(a + ib) x\\}, \\\\\n\\int \\exp \\{(a + ib)x\\}\\, dx = \\frac{\\exp{(a + ib)x}}{a + ib}.\n\\end{gather*}\nDeduce the results of \\Ex{lxxxvii}.~3.", "markdown": "Show that if $x$ is real then gather* ddx (a + ib)x = (a + ib) (a + ib) x, (a + ib)x  dx = (a + ib)xa + ib. gather* Deduce the results of % [examples:lxxxvii]Ex. lxxxvii%. 3.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.integrate", "core.complex" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/5", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 5", "problem_latex": "Show that if $a > 0$ then $\\ds\\int_{0}^{\\infty} \\exp\\{-(a + ib)x\\}\\, dx = \\frac{1}{a + ib}$, and deduce the\nresults of \\Ex{lxxxvii}.~5.", "markdown": "Show that if $a > 0$ then $\\ds\\int_{0}^{\\infty} \\exp\\{-(a + ib)x\\}\\, dx = \\frac{1}{a + ib}$, and deduce the results of % [examples:lxxxvii]Ex. lxxxvii%. 5.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.defint", "cas.integrate", "core.complex" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/6", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 6", "problem_latex": "Show that if $(x/a)^{2} + (y/b)^{2} = 1$ is the equation of an ellipse, and $f(x, y)$\ndenotes the terms of highest degree in the equation of any other algebraic\ncurve, then the sum of the eccentric angles of the points of intersection of the\nellipse and the curve differs by a multiple of~$2\\pi$ from\n\\[\n-i\\{\\log f(a, ib) - \\log f(a, -ib)\\}.\n\\]\n\n[The eccentric angles are given by $f(a\\cos\\alpha, b\\sin\\alpha) + \\dots = 0$ or by\n\\[\nf\\left\\{\\tfrac{1}{2} a \\left(u + \\frac{1}{u}\\right),\\\n -\\tfrac{1}{2} ib \\left(u - \\frac{1}{u}\\right) \\right\\} + \\dots = 0,\n\\]\nwhere $u = \\exp i\\alpha$; and $\\sum\\alpha$~is equal to one of the values of~$-i\\Log P$, where $P$~is\nthe product of the roots of this equation.]", "markdown": "Show that if $(x/a)^{2} + (y/b)^{2} = 1$ is the equation of an ellipse, and $f(x, y)$ denotes the terms of highest degree in the equation of any other algebraic curve, then the sum of the eccentric angles of the points of intersection of the ellipse and the curve differs by a multiple of $2\\pi$ from -if(a, ib) - f(a, -ib). [The eccentric angles are given by $f(a\\cos\\alpha, b\\sin\\alpha) + \\dots = 0$ or by f12 a (u + 1u), -12 ib (u - 1u) + …= 0, where $u = \\exp i\\alpha$; and $\\sum\\alpha$ is equal to one of the values of $-i\\Log P$, where $P$ is the product of the roots of this equation.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/7", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 7", "problem_latex": "Determine the number and approximate positions of the roots of the\nequation $\\tan z = az$, where $a$~is real.\n\n[We know already (\\Ex{xvii}.~4) that the equation has infinitely many real\nroots. Now let $z = x + iy$, and equate real and imaginary parts. We obtain\n\\[\n\\sin 2x/(\\cos 2x + \\cosh 2y) = ax,\\quad\n\\sinh 2y/(\\cos 2x + \\cosh 2y) = ay,\n\\]\nso that, unless $x$ or~$y$ is zero, we have\n\\[\n(\\sin 2x)/2x = (\\sinh 2y)/2y.\n\\]\n\\PageSep{426}\nThis is impossible, the left-hand side being numerically less, and the right-hand\nside numerically greater than unity. Thus $x = 0$ or $y = 0$. If $y = 0$ we\ncome back to the real roots of the equation. If $x = 0$ then $\\tanh y = ay$. It is\neasy to see that this equation has no real root other than zero if $a \\leq 0$ or\n$a \\geq 1$, and two such roots if $0 < a < 1$. Thus there are two purely imaginary\nroots if $0 < a < 1$; otherwise all the roots are real.]", "markdown": "Determine the number and approximate positions of the roots of the equation $\\tan z = az$, where $a$ is real. [We know already (% [examples:xvii]Ex. xvii%. 4) that the equation has infinitely many real roots. Now let $z = x + iy$, and equate real and imaginary parts. We obtain 2x/(2x + 2y) = ax,0pt minus 3pt2y/(2x + 2y) = ay, so that, unless $x$ or $y$ is zero, we have (2x)/2x = (2y)/2y. [pg]426 This is impossible, the left-hand side being numerically less, and the right-hand side numerically greater than unity. Thus $x = 0$ or $y = 0$. If $y = 0$ we come back to the real roots of the equation. If $x = 0$ then $\\tanh y = ay$. It is easy to see that this equation has no real root other than zero if $a \\leq 0$ or $a \\geq 1$, and two such roots if $0 < a < 1$. Thus there are two purely imaginary roots if $0 < a < 1$; otherwise all the roots are real.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.solve.num", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/8", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 8", "problem_latex": "The equation $\\tan z = az + b$, where $a$ and~$b$ are real and $b$~is not equal\nto zero, has no complex roots if $a \\leq 0$. If $a > 0$ then the real parts of all the\ncomplex roots are numerically greater than~$|b/2a|$.", "markdown": "The equation $\\tan z = az + b$, where $a$ and $b$ are real and $b$ is not equal to zero, has no complex roots if $a \\leq 0$. If $a > 0$ then the real parts of all the complex roots are numerically greater than $|b/2a|$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-misc-x/9", "set": "hardy-course-of-pure-mathematics-1921/ex-misc-x", "number": 9, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "425", "location": "Exercise Misc-X, problem 9", "problem_latex": "The equation $\\tan z = a/z$, where $a$~is real, has no complex roots, but\nhas two purely imaginary roots if $a < 0$.", "markdown": "The equation $\\tan z = a/z$, where $a$ is real, has no complex roots, but has two purely imaginary roots if $a < 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-v/1", "set": "hardy-course-of-pure-mathematics-1921/ex-v", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "17", "location": "Exercise V, problem 1", "problem_latex": "Prove that $\\alpha + (-\\alpha) = 0$.", "markdown": "Prove that $\\alpha + (-\\alpha) = 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-v/10", "set": "hardy-course-of-pure-mathematics-1921/ex-v", "number": 10, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "17", "location": "Exercise V, problem 10", "problem_latex": "Prove that\n\\[\n\\big||\\alpha| - |\\beta|\\big| \\leq |\\alpha ± \\beta| \\leq |\\alpha| + |\\beta|.\n\\]", "markdown": "Prove that ||| - ||| |± | || + ||.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-v/2", "set": "hardy-course-of-pure-mathematics-1921/ex-v", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "17", "location": "Exercise V, problem 2", "problem_latex": "Prove that $\\alpha + 0 = 0 + \\alpha = \\alpha$.", "markdown": "Prove that $\\alpha + 0 = 0 + \\alpha = \\alpha$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-v/3", "set": "hardy-course-of-pure-mathematics-1921/ex-v", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "17", "location": "Exercise V, problem 3", "problem_latex": "Prove that $\\alpha + \\beta = \\beta + \\alpha$. [This follows at once from the fact that the\nclasses $(a + b)$~and~$(b + a)$, or $(A + B)$~and~$(B + A)$, are the same, since, \\eg,\n$a + b = b + a$ when $a$~and~$b$ are rational.]", "markdown": "Prove that $\\alpha + \\beta = \\beta + \\alpha$. [This follows at once from the fact that the classes $(a + b)$ and $(b + a)$, or $(A + B)$ and $(B + A)$, are the same, since, *e.g.*, $a + b = b + a$ when $a$ and $b$ are rational.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-v/4", "set": "hardy-course-of-pure-mathematics-1921/ex-v", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "17", "location": "Exercise V, problem 4", "problem_latex": "Prove that $\\alpha + (\\beta + \\gamma) = (\\alpha + \\beta) + \\gamma$.", "markdown": "Prove that $\\alpha + (\\beta + \\gamma) = (\\alpha + \\beta) + \\gamma$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-v/5", "set": "hardy-course-of-pure-mathematics-1921/ex-v", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "17", "location": "Exercise V, problem 5", "problem_latex": "Prove that $\\alpha - \\alpha = 0$.", "markdown": "Prove that $\\alpha - \\alpha = 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-v/6", "set": "hardy-course-of-pure-mathematics-1921/ex-v", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "17", "location": "Exercise V, problem 6", "problem_latex": "Prove that $\\alpha - \\beta = -(\\beta - \\alpha)$.", "markdown": "Prove that $\\alpha - \\beta = -(\\beta - \\alpha)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-v/7", "set": "hardy-course-of-pure-mathematics-1921/ex-v", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "17", "location": "Exercise V, problem 7", "problem_latex": "From the definition of subtraction, and Exs.\\ 4,~1, and~2 above, it\nfollows that\n\\[\n(\\alpha - \\beta) + \\beta\n = \\{\\alpha + (-\\beta)\\} + \\beta\n = \\alpha + \\{(-\\beta) + \\beta\\}\n = \\alpha + 0 = \\alpha.\n\\]\nWe might therefore define the difference $\\alpha - \\beta = \\gamma$ by the equation $\\gamma + \\beta = \\alpha$.", "markdown": "From the definition of subtraction, and Exs. 4, 1, and 2 above, it follows that (- ) + = + (-) + = + (-) + = + 0 = . We might therefore define the difference $\\alpha - \\beta = \\gamma$ by the equation $\\gamma + \\beta = \\alpha$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-v/8", "set": "hardy-course-of-pure-mathematics-1921/ex-v", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "17", "location": "Exercise V, problem 8", "problem_latex": "Prove that $\\alpha - (\\beta - \\gamma) = \\alpha - \\beta + \\gamma$.", "markdown": "Prove that $\\alpha - (\\beta - \\gamma) = \\alpha - \\beta + \\gamma$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-v/9", "set": "hardy-course-of-pure-mathematics-1921/ex-v", "number": 9, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "17", "location": "Exercise V, problem 9", "problem_latex": "Give a definition of subtraction which does not depend upon a previous\ndefinition of addition. [To define $\\gamma = \\alpha - \\beta$, form the classes $(c)$,~$(C)$ for which\n$c = a - B$, $C = A - b$. It is easy to show that this definition is equivalent to\nthat which we adopted in the text.]", "markdown": "Give a definition of subtraction which does not depend upon a previous definition of addition. [To define $\\gamma = \\alpha - \\beta$, form the classes $(c)$, $(C)$ for which $c = a - B$, $C = A - b$. It is easy to show that this definition is equivalent to that which we adopted in the text.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-vi/1", "set": "hardy-course-of-pure-mathematics-1921/ex-vi", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "19", "location": "Exercise VI, problem 1", "problem_latex": "$\\alpha × 0 = 0 × \\alpha = 0$.", "markdown": "$\\alpha × 0 = 0 × \\alpha = 0$.", "answer_latex": [ "$\\alpha × 0 = 0 × \\alpha = 0$." ], "answer_markdown": [ "$\\alpha × 0 = 0 × \\alpha = 0$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-vi/2", "set": "hardy-course-of-pure-mathematics-1921/ex-vi", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "19", "location": "Exercise VI, problem 2", "problem_latex": "$\\alpha × 1 = 1 × \\alpha = \\alpha$.", "markdown": "$\\alpha × 1 = 1 × \\alpha = \\alpha$.", "answer_latex": [ "$\\alpha × 1 = 1 × \\alpha = \\alpha$." ], "answer_markdown": [ "$\\alpha × 1 = 1 × \\alpha = \\alpha$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-vi/3", "set": "hardy-course-of-pure-mathematics-1921/ex-vi", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "19", "location": "Exercise VI, problem 3", "problem_latex": "$\\alpha × (1/\\alpha) = 1$.", "markdown": "$\\alpha × (1/\\alpha) = 1$.", "answer_latex": [ "$\\alpha × (1/\\alpha) = 1$." ], "answer_markdown": [ "$\\alpha × (1/\\alpha) = 1$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-vi/4", "set": "hardy-course-of-pure-mathematics-1921/ex-vi", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "19", "location": "Exercise VI, problem 4", "problem_latex": "$\\alpha\\beta = \\beta\\alpha$.", "markdown": "$\\alpha\\beta = \\beta\\alpha$.", "answer_latex": [ "$\\alpha\\beta = \\beta\\alpha$." ], "answer_markdown": [ "$\\alpha\\beta = \\beta\\alpha$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-vi/5", "set": "hardy-course-of-pure-mathematics-1921/ex-vi", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "19", "location": "Exercise VI, problem 5", "problem_latex": "$\\alpha(\\beta\\gamma) = (\\alpha\\beta)\\gamma$.", "markdown": "$\\alpha(\\beta\\gamma) = (\\alpha\\beta)\\gamma$.", "answer_latex": [ "$\\alpha(\\beta\\gamma) = (\\alpha\\beta)\\gamma$." ], "answer_markdown": [ "$\\alpha(\\beta\\gamma) = (\\alpha\\beta)\\gamma$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-vi/6", "set": "hardy-course-of-pure-mathematics-1921/ex-vi", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "19", "location": "Exercise VI, problem 6", "problem_latex": "$\\alpha(\\beta + \\gamma) = \\alpha\\beta + \\alpha\\gamma$.", "markdown": "$\\alpha(\\beta + \\gamma) = \\alpha\\beta + \\alpha\\gamma$.", "answer_latex": [ "$\\alpha(\\beta + \\gamma) = \\alpha\\beta + \\alpha\\gamma$." ], "answer_markdown": [ "$\\alpha(\\beta + \\gamma) = \\alpha\\beta + \\alpha\\gamma$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-vi/7", "set": "hardy-course-of-pure-mathematics-1921/ex-vi", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "19", "location": "Exercise VI, problem 7", "problem_latex": "$(\\alpha + \\beta)\\gamma = \\alpha\\gamma + \\beta\\gamma$.", "markdown": "$(\\alpha + \\beta)\\gamma = \\alpha\\gamma + \\beta\\gamma$.", "answer_latex": [ "$(\\alpha + \\beta)\\gamma = \\alpha\\gamma + \\beta\\gamma$." ], "answer_markdown": [ "$(\\alpha + \\beta)\\gamma = \\alpha\\gamma + \\beta\\gamma$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-vi/8", "set": "hardy-course-of-pure-mathematics-1921/ex-vi", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "19", "location": "Exercise VI, problem 8", "problem_latex": "$|\\alpha\\beta| = |\\alpha|\\, |\\beta|$.", "markdown": "$|\\alpha\\beta| = |\\alpha|\\, |\\beta|$.", "answer_latex": [ "$|\\alpha\\beta| = |\\alpha|\\, |\\beta|$." ], "answer_markdown": [ "$|\\alpha\\beta| = |\\alpha|\\, |\\beta|$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-vii/1", "set": "hardy-course-of-pure-mathematics-1921/ex-vii", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "20", "location": "Exercise VII, problem 1", "problem_latex": "Give geometrical constructions for\n\\[\n\\sqrt{2},\\quad\n\\sqrtp{2 + \\sqrt{2}},\\quad\n\\sqrtb{2 + \\sqrtp{2 + \\sqrt{2}}}.\n\\]", "markdown": "Give geometrical constructions for 2,0pt minus 3pt2 + 2,0pt minus 3pt2 + 2 + 2.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:geometric_construction" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-vii/2", "set": "hardy-course-of-pure-mathematics-1921/ex-vii", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "20", "location": "Exercise VII, problem 2", "problem_latex": "The quadratic equation $ax^{2} + 2bx + c = 0$ has two real roots\\footnote\n {\\Ie\\ there are two values of~$x$ for which $ax^{2} + 2bx + c = 0$. If $b^{2} - ac < 0$ there\n are no such values of~$x$. The reader will remember that in books on elementary\n algebra the equation is said to have two `complex' roots. The meaning to be\n attached to this statement will be explained in \\okrickRef{Ch.}{III}\\@.\n\n When $b^{2} = ac$ the equation has only one root. For the sake of uniformity\n it is generally said in this case to have `two equal' roots, but this is a mere\n convention.}\nif\n$b^{2} - ac > 0$. Suppose $a$,~$b$,~$c$ rational. Nothing is lost by taking all three\nto be integers, for we can multiply the equation by the least common\nmultiple of their denominators.\n\nThe reader will remember that the roots are $\\{-b ± \\sqrtp{b^{2} - ac}\\}/a$. It is\neasy to construct these lengths geometrically, first constructing $\\sqrtp{b^{2} - ac}$.\nA much more elegant, though less straightforward, construction is the\nfollowing.\n\\PageSep{21}\n\n\\begin{Construction}\nDraw a circle of unit radius, a diameter~$PQ$, and the tangents at the ends\nof the diameters.\n%[Illustration: Fig. 5.]\n\\Figure[0.7\\textwidth]{5}{p021}\n\nTake $PP' = -2a/b$ and $QQ' = -c/2b$, having regard to sign.\\footnote\n {The figure is drawn to suit the case in which $b$~and~$c$ have the same and $a$\n the opposite sign. The reader should draw figures for other cases.}\nJoin $P'Q'$,\ncutting the circle in $M$~and~$N$. Draw $PM$~and~$PN$, cutting~$QQ'$ in $X$~and~$Y$.\nThen $QX$~and~$QY$ are the roots of the equation with their proper signs.\\footnote\n {I have taken this construction from Klein's \\textit{Leçons sur certaines questions de\n géométrie élémentaire} (French translation by J.~Griess, Paris, 1896).}\n\\end{Construction}\n\nThe proof is simple and we leave it as an exercise to the reader.\nAnother, perhaps even simpler, construction is the following. \\begin{Construction}[]Take a line\n$AB$ of unit length. Draw $BC = -2b/a$ perpendicular to~$AB$, and $CD = c/a$\n perpendicular to~$BC$ and in the same direction as~$BA$. On~$AD$ as diameter\ndescribe a circle cutting~$BC$ in $X$~and~$Y$. Then $BX$~and~$BY$ are the roots.\n\\end{Construction}", "markdown": "The quadratic equation $ax^{2} + 2bx + c = 0$ has two real roots *I.e.* there are two values of $x$ for which $ax^{2} + 2bx + c = 0$. If $b^{2} - ac < 0$ there are no such values of $x$. The reader will remember that in books on elementary algebra the equation is said to have two ‘complex’ roots. The meaning to be attached to this statement will be explained in Ch.III. When $b^{2} = ac$ the equation has only one root. For the sake of uniformity it is generally said in this case to have ‘two equal’ roots, but this is a mere convention. if $b^{2} - ac > 0$. Suppose $a$, $b$, $c$ rational. Nothing is lost by taking all three to be integers, for we can multiply the equation by the least common multiple of their denominators. The reader will remember that the roots are $\\{-b ± \\sqrtp{b^{2} - ac}\\}/a$. It is easy to construct these lengths geometrically, first constructing $\\sqrtp{b^{2} - ac}$. A much more elegant, though less straightforward, construction is the following. [pg]21 Construction Draw a circle of unit radius, a diameter $PQ$, and the tangents at the ends of the diameters. %[Illustration: Fig. 5.] [0.7]5p021 Take $PP' = -2a/b$ and $QQ' = -c/2b$, having regard to sign. The figure is drawn to suit the case in which $b$ and $c$ have the same and $a$ the opposite sign. The reader should draw figures for other cases. Join $P'Q'$, cutting the circle in $M$ and $N$. Draw $PM$ and $PN$, cutting $QQ'$ in $X$ and $Y$. Then $QX$ and $QY$ are the roots of the equation with their proper signs. I have taken this construction from Klein’s *Leçons sur certaines questions de géométrie élémentaire* (French translation by J. Griess, Paris, 1896). Construction The proof is simple and we leave it as an exercise to the reader. Another, perhaps even simpler, construction is the following. Construction[]Take a line $AB$ of unit length. Draw $BC = -2b/a$ perpendicular to $AB$, and $CD = c/a$ perpendicular to $BC$ and in the same direction as $BA$. On $AD$ as diameter describe a circle cutting $BC$ in $X$ and $Y$. Then $BX$ and $BY$ are the roots. Construction", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:geometric_construction" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-vii/3", "set": "hardy-course-of-pure-mathematics-1921/ex-vii", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "20", "location": "Exercise VII, problem 3", "problem_latex": "If $ac$ is positive $PP'$~and~$QQ'$ will be drawn in the same direction.\nVerify that $P'Q'$~will not meet the circle if $b^{2} < ac$, while if $b^{2} = ac$ it will be\na tangent. Verify also that if $b^{2} = ac$ the circle in the second construction\nwill touch~$BC$.", "markdown": "If $ac$ is positive $PP'$ and $QQ'$ will be drawn in the same direction. Verify that $P'Q'$ will not meet the circle if $b^{2} < ac$, while if $b^{2} = ac$ it will be a tangent. Verify also that if $b^{2} = ac$ the circle in the second construction will touch $BC$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-vii/4", "set": "hardy-course-of-pure-mathematics-1921/ex-vii", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "20", "location": "Exercise VII, problem 4", "problem_latex": "Prove that\n\\[\n\\sqrtp{pq} = \\sqrt{p} × \\sqrt{q},\\quad\n\\sqrtp{p^{2}q} = p\\sqrt{q}.\n\\]", "markdown": "Prove that pq = p × q,0pt minus 3ptp^2q = pq.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-viii/1", "set": "hardy-course-of-pure-mathematics-1921/ex-viii", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "22", "location": "Exercise VIII, problem 1", "problem_latex": "Prove \\textit{ab initio} that $\\sqrt{2}$~and~$\\sqrt{3}$ are not similar\nsurds.", "markdown": "Prove *ab initio* that $\\sqrt{2}$ and $\\sqrt{3}$ are not similar surds.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-viii/10", "set": "hardy-course-of-pure-mathematics-1921/ex-viii", "number": 10, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "22", "location": "Exercise VIII, problem 10", "problem_latex": "If $p$,~$q$, and $p^{2} - q$ are positive, we can express $\\sqrtp{p + \\sqrt{q}}$ in the form\n$\\sqrt{x} + \\sqrt{y}$, where\n\\[\nx = \\tfrac{1}{2}\\{p + \\sqrtp{p^{2} - q}\\},\\quad\ny = \\tfrac{1}{2}\\{p - \\sqrtp{p^{2} - q}\\}.\n\\]", "markdown": "If $p$, $q$, and $p^{2} - q$ are positive, we can express $\\sqrtp{p + \\sqrt{q}}$ in the form $\\sqrt{x} + \\sqrt{y}$, where x = 12p + p^2 - q,0pt minus 3pty = 12p - p^2 - q.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-viii/11", "set": "hardy-course-of-pure-mathematics-1921/ex-viii", "number": 11, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "22", "location": "Exercise VIII, problem 11", "problem_latex": "Determine the conditions that it may be possible to express $\\sqrtp{p + \\sqrt{q}}$,\nwhere $p$~and~$q$ are rational, in the form $\\sqrt{x} + \\sqrt{y}$, where $x$~and~$y$ are rational.", "markdown": "Determine the conditions that it may be possible to express $\\sqrtp{p + \\sqrt{q}}$, where $p$ and $q$ are rational, in the form $\\sqrt{x} + \\sqrt{y}$, where $x$ and $y$ are rational.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-viii/12", "set": "hardy-course-of-pure-mathematics-1921/ex-viii", "number": 12, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "22", "location": "Exercise VIII, problem 12", "problem_latex": "If $a^{2} - b$ is positive, the necessary and sufficient conditions that\n\\[\n\\sqrtp{a + \\sqrt{b}} + \\sqrtp{a - \\sqrt{b}}\n\\]\nshould be rational are that $a^{2} - b$ and $\\frac{1}{2}\\{a + \\sqrtp{a^{2} - b}\\}$ should both be squares\nof rational numbers.", "markdown": "If $a^{2} - b$ is positive, the necessary and sufficient conditions that a + b + a - b should be rational are that $a^{2} - b$ and $\\frac{1}{2}\\{a + \\sqrtp{a^{2} - b}\\}$ should both be squares of rational numbers.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-viii/2", "set": "hardy-course-of-pure-mathematics-1921/ex-viii", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "22", "location": "Exercise VIII, problem 2", "problem_latex": "Prove that $\\sqrt{a}$~and~$\\sqrtp{1/a}$, where $a$~is rational, are similar surds\n(unless both are rational).", "markdown": "Prove that $\\sqrt{a}$ and $\\sqrtp{1/a}$, where $a$ is rational, are similar surds (unless both are rational).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-viii/3", "set": "hardy-course-of-pure-mathematics-1921/ex-viii", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "22", "location": "Exercise VIII, problem 3", "problem_latex": "If $a$~and~$b$ are rational, then $\\sqrt{a} + \\sqrt{b}$ cannot be rational unless $\\sqrt{a}$~and~$\\sqrt{b}$\nare rational. The same is true of $\\sqrt{a}- \\sqrt{b}$, unless $a = b$.", "markdown": "If $a$ and $b$ are rational, then $\\sqrt{a} + \\sqrt{b}$ cannot be rational unless $\\sqrt{a}$ and $\\sqrt{b}$ are rational. The same is true of $\\sqrt{a}- \\sqrt{b}$, unless $a = b$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-viii/4", "set": "hardy-course-of-pure-mathematics-1921/ex-viii", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "22", "location": "Exercise VIII, problem 4", "problem_latex": "If\n\\[\n\\sqrt{A} + \\sqrt{B} = \\sqrt{C} + \\sqrt{D},\n\\]\nthen either (\\ia)~$A = C$ and~$B = D$, or (\\ib)~$A = D$ and~$B = C$, or (\\ic)~$\\sqrt{A}$, $\\sqrt{B}$, $\\sqrt{C}$,\n$\\sqrt{D}$ are all rational or all similar surds. [Square the given equation and\napply the theorem above.]", "markdown": "If A + B = C + D, then either (*a*) $A = C$ and $B = D$, or (*b*) $A = D$ and $B = C$, or (*c*) $\\sqrt{A}$, $\\sqrt{B}$, $\\sqrt{C}$, $\\sqrt{D}$ are all rational or all similar surds. [Square the given equation and apply the theorem above.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-viii/5", "set": "hardy-course-of-pure-mathematics-1921/ex-viii", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "22", "location": "Exercise VIII, problem 5", "problem_latex": "Neither $(a + \\sqrt{b})^{3}$ nor $(a - \\sqrt{b})^{3}$ can be rational unless $\\sqrt{b}$~is rational.", "markdown": "Neither $(a + \\sqrt{b})^{3}$ nor $(a - \\sqrt{b})^{3}$ can be rational unless $\\sqrt{b}$ is rational.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-viii/6", "set": "hardy-course-of-pure-mathematics-1921/ex-viii", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "22", "location": "Exercise VIII, problem 6", "problem_latex": "Prove that if $x = p + \\sqrt{q}$, where $p$~and~$q$ are rational, then $x^{m}$, where\n$m$~is any integer, can be expressed in the form $P + Q \\sqrt{q}$, where $P$~and~$Q$\nare rational. For example,\n\\[\n(p + \\sqrt{q})^{2} = p^{2} + q + 2p\\sqrt{q},\\quad\n(p + \\sqrt{q})^{3} = p^{3} + 3pq + (3p^{2} + q)\\sqrt{q}.\n\\]\nDeduce that any polynomial in~$x$ with rational coefficients (\\ie~any expression\nof the form\n\\[\na_{0}x^{n} + a_{1}x^{n-1} + \\dots + a_{n},\n\\]\nwhere $a_{0}$,~\\dots\\Add{,} $a_{n}$ are rational numbers) can be expressed in the form $P + Q\\sqrt{q}$.", "markdown": "Prove that if $x = p + \\sqrt{q}$, where $p$ and $q$ are rational, then $x^{m}$, where $m$ is any integer, can be expressed in the form $P + Q \\sqrt{q}$, where $P$ and $Q$ are rational. For example, (p + q)^2 = p^2 + q + 2pq,0pt minus 3pt(p + q)^3 = p^3 + 3pq + (3p^2 + q)q. Deduce that any polynomial in $x$ with rational coefficients (*i.e.* any expression of the form a_0x^n + a_1x^n-1 + …+ a_n, where $a_{0}$, … $a_{n}$ are rational numbers) can be expressed in the form $P + Q\\sqrt{q}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-viii/7", "set": "hardy-course-of-pure-mathematics-1921/ex-viii", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "22", "location": "Exercise VIII, problem 7", "problem_latex": "If $a + \\sqrt{b}$, where $b$~is not a perfect square, is the root of an algebraical\nequation with rational coefficients, then $a - \\sqrt{b}$ is another root of the same\nequation.", "markdown": "If $a + \\sqrt{b}$, where $b$ is not a perfect square, is the root of an algebraical equation with rational coefficients, then $a - \\sqrt{b}$ is another root of the same equation.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-viii/8", "set": "hardy-course-of-pure-mathematics-1921/ex-viii", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "22", "location": "Exercise VIII, problem 8", "problem_latex": "Express $1/(p + \\sqrt{q})$ in the form prescribed in Ex.~6. [Multiply\nnumerator and denominator by~$p - \\sqrt{q}$.]", "markdown": "Express $1/(p + \\sqrt{q})$ in the form prescribed in Ex. 6. [Multiply numerator and denominator by $p - \\sqrt{q}$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "identity", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "1/(p + sqrt(q))", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "identity: 1/(a + sqrt(b))" ], "shape": [ "identity: 1/(a + b**N)" ], "same_problem_in": [], "needs": [ "cas.simplify", "core.arith" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-viii/9", "set": "hardy-course-of-pure-mathematics-1921/ex-viii", "number": 9, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "22", "location": "Exercise VIII, problem 9", "problem_latex": "Deduce from Exs.\\ 6~and~8 that any expression of the form $G(x)/H(x)$,\nwhere $G(x)$~and~$H(x)$ are polynomials in~$x$ with rational coefficients, can be\nexpressed in the form $P + Q\\sqrt{q}$, where $P$~and~$Q$ are rational.", "markdown": "Deduce from Exs. 6 and 8 that any expression of the form $G(x)/H(x)$, where $G(x)$ and $H(x)$ are polynomials in $x$ with rational coefficients, can be expressed in the form $P + Q\\sqrt{q}$, where $P$ and $Q$ are rational.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-x/1", "set": "hardy-course-of-pure-mathematics-1921/ex-x", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "39", "location": "Exercise X, problem 1", "problem_latex": "Let $y = x$ or~$2x$ or~$\\frac{1}{2}x$ or $x^{2} +1$. Nothing further need\nbe said at present about cases such as these.", "markdown": "Let $y = x$ or $2x$ or $\\frac{1}{2}x$ or $x^{2} +1$. Nothing further need be said at present about cases such as these.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-x/2", "set": "hardy-course-of-pure-mathematics-1921/ex-x", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "39", "location": "Exercise X, problem 2", "problem_latex": "Let $y = 0$ whatever be the value of~$x$. Then $y$~is a function of~$x$, for we\ncan give~$x$ any value, and the corresponding value of~$y$ (viz.~$0$) is known. In\nthis case the functional relation makes the same value of~$y$ correspond to all\nvalues of~$x$. The same would be true were $y$~equal to~$1$ or~$-\\frac{1}{2}$ or~$\\sqrt{2}$ instead\nof~$0$. Such a function of~$x$ is called \\emph{a constant}.", "markdown": "Let $y = 0$ whatever be the value of $x$. Then $y$ is a function of $x$, for we can give $x$ any value, and the corresponding value of $y$ (viz. $0$) is known. In this case the functional relation makes the same value of $y$ correspond to all values of $x$. The same would be true were $y$ equal to $1$ or $-\\frac{1}{2}$ or $\\sqrt{2}$ instead of $0$. Such a function of $x$ is called *a constant*.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-x/3", "set": "hardy-course-of-pure-mathematics-1921/ex-x", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "39", "location": "Exercise X, problem 3", "problem_latex": "Let $y^{2} = x$. Then if $x$~is positive this equation defines \\emph{two} values of~$y$\ncorresponding to each value of~$x$, viz.~$±\\sqrt{x}$. If $x = 0$, $y = 0$. Hence to the\nparticular value~$0$ of~$x$ corresponds \\emph{one} and only one value of~$y$. But if $x$~is\nnegative there is \\emph{no} value of~$y$ which satisfies the equation. That is to say,\nthe function~$y$ is not defined for negative values of~$x$. This function therefore\npossesses the characteristic~(3), but neither (1)~nor~(2).", "markdown": "Let $y^{2} = x$. Then if $x$ is positive this equation defines *two* values of $y$ corresponding to each value of $x$, viz. $±\\sqrt{x}$. If $x = 0$, $y = 0$. Hence to the particular value $0$ of $x$ corresponds *one* and only one value of $y$. But if $x$ is negative there is *no* value of $y$ which satisfies the equation. That is to say, the function $y$ is not defined for negative values of $x$. This function therefore possesses the characteristic (3), but neither (1) nor (2).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-x/4", "set": "hardy-course-of-pure-mathematics-1921/ex-x", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "39", "location": "Exercise X, problem 4", "problem_latex": "Consider a volume of gas maintained at a constant temperature and\ncontained in a cylinder closed by a sliding piston.\\footnote\n {I borrow this instructive example from Prof.\\ H.~S. Carslaw's \\textit{Introduction to\n the Calculus.}}\n\nLet $A$ be the area of the cross section of the piston and $W$~its weight.\nThe gas, held in a state of compression by the piston, exerts a certain pressure\n$p_{0}$~per unit of area on the piston, which balances the weight~$W$, so that\n\\[\nW = Ap_{0}.\n\\]\n\nLet $v_{0}$ be the volume of the gas when the system is thus in equilibrium.\nIf additional weight is placed upon the piston the latter is forced downwards.\nThe volume~($v$) of the gas diminishes; the pressure~($p$) which it exerts\nupon unit area of the piston increases. Boyle's experimental law asserts that\nthe product of $p$~and~$v$ is very nearly constant, a correspondence which, if\nexact, would be represented by an equation of the type\n\\[\npv = a,\n\\Tag{(i)}\n\\]\nwhere $a$~is a number which can be determined approximately by experiment.\n\nBoyle's law, however, only gives a reasonable approximation to the facts\nprovided the gas is not compressed too much. When $v$~is decreased and $p$~increased\nbeyond a certain point, the relation between them is no longer\nexpressed with tolerable exactness by the equation~\\Eq{(i)}. It is known that a\n\\PageSep{40}\nmuch better approximation to the true relation can then be found by means\nof what is known as `van~der Waals' law', expressed by the equation\n\\[\n\\left(p + \\frac{\\alpha}{v^{2}}\\right)(v - \\beta) = \\gamma,\n\\Tag{(ii)}\n\\]\nwhere $\\alpha$, $\\beta$, $\\gamma$ are numbers which can also be determined approximately by\nexperiment.\n\nOf course the two equations, even taken together, do not give anything\nlike a complete account of the relation between $p$~and~$v$. This relation is no\ndoubt in reality much more complicated, and its form changes, as $v$~varies,\nfrom a form nearly equivalent to~\\Eq{(i)} to a form nearly equivalent to~\\Eq{(ii)}. But,\nfrom a mathematical point of view, there is nothing to prevent us from contemplating\nan ideal state of things in which, for all values of~$v$ not less than\na certain value~$V$, \\Eq{(i)}~would be exactly true, and \\Eq{(ii)}~exactly true for all\nvalues of~$v$ less than~$V$. And then we might regard the two equations as\ntogether defining~$p$ as a function of~$v$. It is an example of a function which\nfor some values of~$v$ is defined by one formula and for other values of~$v$ is\ndefined by another.\n\nThis function possesses the characteristic~(2)\\DPtypo{.}{;} to any value of~$v$ only one\nvalue of~$p$ corresponds: but it does not possess~(1). For $p$~is not defined as\na function of~$v$ for negative values of~$v$; a `negative volume' means\nnothing, and so negative values of~$v$ do not present themselves for consideration\nat all.", "markdown": "Consider a volume of gas maintained at a constant temperature and contained in a cylinder closed by a sliding piston. I borrow this instructive example from Prof. H. S. Carslaw’s *Introduction to the Calculus.* Let $A$ be the area of the cross section of the piston and $W$ its weight. The gas, held in a state of compression by the piston, exerts a certain pressure $p_{0}$ per unit of area on the piston, which balances the weight $W$, so that W = Ap_0. Let $v_{0}$ be the volume of the gas when the system is thus in equilibrium. If additional weight is placed upon the piston the latter is forced downwards. The volume ($v$) of the gas diminishes; the pressure ($p$) which it exerts upon unit area of the piston increases. Boyle’s experimental law asserts that the product of $p$ and $v$ is very nearly constant, a correspondence which, if exact, would be represented by an equation of the type pv = a, (i) where $a$ is a number which can be determined approximately by experiment. Boyle’s law, however, only gives a reasonable approximation to the facts provided the gas is not compressed too much. When $v$ is decreased and $p$ increased beyond a certain point, the relation between them is no longer expressed with tolerable exactness by the equation (i). It is known that a [pg]40 much better approximation to the true relation can then be found by means of what is known as ‘van der Waals’ law’, expressed by the equation (p + v^2)(v - ) = , (ii) where $\\alpha$, $\\beta$, $\\gamma$ are numbers which can also be determined approximately by experiment. Of course the two equations, even taken together, do not give anything like a complete account of the relation between $p$ and $v$. This relation is no doubt in reality much more complicated, and its form changes, as $v$ varies, from a form nearly equivalent to (i) to a form nearly equivalent to (ii). But, from a mathematical point of view, there is nothing to prevent us from contemplating an ideal state of things in which, for all values of $v$ not less than a certain value $V$, (i) would be exactly true, and (ii) exactly true for all values of $v$ less than $V$. And then we might regard the two equations as together defining $p$ as a function of $v$. It is an example of a function which for some values of $v$ is defined by one formula and for other values of $v$ is defined by another. This function possesses the characteristic (2). to any value of $v$ only one value of $p$ corresponds: but it does not possess (1). For $p$ is not defined as a function of $v$ for negative values of $v$; a ‘negative volume’ means nothing, and so negative values of $v$ do not present themselves for consideration at all.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-x/5", "set": "hardy-course-of-pure-mathematics-1921/ex-x", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "39", "location": "Exercise X, problem 5", "problem_latex": "Suppose that a perfectly elastic ball is dropped (without rotation)\nfrom a height~$\\frac{1}{2}g\\tau^{2}$ on to a fixed horizontal plane, and rebounds continually.\n\nThe ordinary formulae of elementary dynamics, with which the reader is\nprobably familiar, show that $h = \\frac{1}{2}gt^{2}$ if $0 \\leq t \\leq \\tau$, $h = \\frac{1}{2}g(2\\tau - t)^{2}$ if $\\tau \\leq t \\leq 3\\tau$, and\ngenerally\n\\[\nh = \\tfrac{1}{2}g(2n\\tau - t)^{2}\n\\]\nif $(2n - 1)\\tau \\leq t \\leq (2n + 1)\\tau$, $h$~being the depth of the ball, at time~$t$, below its\noriginal position. Obviously $h$~is a function of~$t$ which is only defined for\npositive values of~$t$.", "markdown": "Suppose that a perfectly elastic ball is dropped (without rotation) from a height $\\frac{1}{2}g\\tau^{2}$ on to a fixed horizontal plane, and rebounds continually. The ordinary formulae of elementary dynamics, with which the reader is probably familiar, show that $h = \\frac{1}{2}gt^{2}$ if $0 \\leq t \\leq \\tau$, $h = \\frac{1}{2}g(2\\tau - t)^{2}$ if $\\tau \\leq t \\leq 3\\tau$, and generally h = 12g(2n- t)^2 if $(2n - 1)\\tau \\leq t \\leq (2n + 1)\\tau$, $h$ being the depth of the ball, at time $t$, below its original position. Obviously $h$ is a function of $t$ which is only defined for positive values of $t$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-x/6", "set": "hardy-course-of-pure-mathematics-1921/ex-x", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "39", "location": "Exercise X, problem 6", "problem_latex": "Suppose that $y$~is defined as being \\emph{the largest prime factor of~$x$}. This\nis an instance of a definition which only applies to a particular class of values\nof~$x$, viz.\\ \\emph{integral} values. `The largest prime factor of~$\\frac{11}{3}$ or of~$\\sqrt{2}$ or of~$\\pi$'\nmeans nothing, and so our defining relation fails to define for such values of~$x$\nas these. Thus this function does not possess the characteristic~(1). It does\npossess~(2), but not~(3), as there is no simple formula which expresses~$y$ in\nterms of~$x$.", "markdown": "Suppose that $y$ is defined as being *the largest prime factor of $x$*. This is an instance of a definition which only applies to a particular class of values of $x$, viz. *integral* values. ‘The largest prime factor of $\\frac{11}{3}$ or of $\\sqrt{2}$ or of $\\pi$’ means nothing, and so our defining relation fails to define for such values of $x$ as these. Thus this function does not possess the characteristic (1). It does possess (2), but not (3), as there is no simple formula which expresses $y$ in terms of $x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-x/7", "set": "hardy-course-of-pure-mathematics-1921/ex-x", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "39", "location": "Exercise X, problem 7", "problem_latex": "Let $y$~be defined as \\emph{the denominator of~$x$ when $x$~is expressed in its\nlowest terms}. This is an example of a function which is defined if and only\nif $x$~is \\emph{rational}. Thus $y = 7$ if $x = -11/7$: but $y$~is not defined for $x = \\sqrt{2}$, `the\ndenominator of~$\\sqrt{2}$' being a meaningless form of words.", "markdown": "Let $y$ be defined as *the denominator of $x$ when $x$ is expressed in its lowest terms*. This is an example of a function which is defined if and only if $x$ is *rational*. Thus $y = 7$ if $x = -11/7$: but $y$ is not defined for $x = \\sqrt{2}$, ‘the denominator of $\\sqrt{2}$’ being a meaningless form of words.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-x/8", "set": "hardy-course-of-pure-mathematics-1921/ex-x", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "39", "location": "Exercise X, problem 8", "problem_latex": "Let $y$~be defined as \\emph{the height in inches of policeman~$Cx$, in the\nMetropolitan Police, at {\\upshape5.30}~\\DPchg{p.m.}{\\textsc{p.m.}}\\ on {\\upshape8}~Aug.~{\\upshape1907}}. Then $y$~is defined for a\ncertain number of integral values of~$x$, viz.\\ $1$, $2$, \\dots,~$N$, where $N$~is the total\nnumber of policemen in division~$C$ at that particular moment of time.", "markdown": "Let $y$ be defined as *the height in inches of policeman $Cx$, in the Metropolitan Police, at 5.30 p.m.p.m. on 8 Aug. 1907*. Then $y$ is defined for a certain number of integral values of $x$, viz. $1$, $2$, …, $N$, where $N$ is the total number of policemen in division $C$ at that particular moment of time.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xc/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xc", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "379", "location": "Exercise XC, problem 1", "problem_latex": "Show that\n\\[\n\\cosh x = 1 + \\frac{x^{2}}{2!} + \\frac{x^{4}}{4!} + \\dots,\\quad\n\\sinh x = x + \\frac{x^{3}}{3!} + \\frac{x^{5}}{5!} + \\dots.\n\\]", "markdown": "Show that x = 1 + x^22! + x^44! + …,0pt minus 3ptx = x + x^33! + x^55! + ….", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.series", "core.hyp" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xc/10", "set": "hardy-course-of-pure-mathematics-1921/ex-xc", "number": 10, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "379", "location": "Exercise XC, problem 10", "problem_latex": "Prove that $\\sum\\limits_{1}^{\\infty} \\dfrac{(n - 1)x^{n}}{(n + 2)n!} = \\left\\{(x^{2} - 3x + 3)e^{x} + \\frac{1}{2}x^{2} - 3\\right\\}/x^{2}$.\n\n[Multiply numerator and denominator by~$n + 1$, and proceed as in Ex.~7.]", "markdown": "Prove that $\\sum\\limits_{1}^{\\infty} \\dfrac{(n - 1)x^{n}}{(n + 2)n!} = \\left\\{(x^{2} - 3x + 3)e^{x} + \\frac{1}{2}x^{2} - 3\\right\\}/x^{2}$. [Multiply numerator and denominator by $n + 1$, and proceed as in Ex. 7.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.cancel", "cas.expand", "cas.sum" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xc/11", "set": "hardy-course-of-pure-mathematics-1921/ex-xc", "number": 11, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "379", "location": "Exercise XC, problem 11", "problem_latex": "Determine $a$,~$b$,~$c$ so that $\\{(x + a)e^{x} + (bx + c)\\}/x^{3}$ tends to a limit\nas $x \\to 0$, evaluate the limit, and draw the graph of the function $e^{x} + \\dfrac{bx + c}{x + a}$.", "markdown": "Determine $a$, $b$, $c$ so that $\\{(x + a)e^{x} + (bx + c)\\}/x^{3}$ tends to a limit as $x \\to 0$, evaluate the limit, and draw the graph of the function $e^{x} + \\dfrac{bx + c}{x + a}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "cas.series", "core.eqn", "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xc/12", "set": "hardy-course-of-pure-mathematics-1921/ex-xc", "number": 12, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "379", "location": "Exercise XC, problem 12", "problem_latex": "Draw the graphs of $1 + x$, $1 + x + \\frac{1}{2}x^{2}$, $1 + x + \\frac{1}{2}x^{2} + \\frac{1}{6}x^{3}$, and compare\nthem with that of~$e^{x}$.", "markdown": "Draw the graphs of $1 + x$, $1 + x + \\frac{1}{2}x^{2}$, $1 + x + \\frac{1}{2}x^{2} + \\frac{1}{6}x^{3}$, and compare them with that of $e^{x}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xc/13", "set": "hardy-course-of-pure-mathematics-1921/ex-xc", "number": 13, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "379", "location": "Exercise XC, problem 13", "problem_latex": "Prove that $e^{-x} - 1 + x - \\dfrac{x^{n}}{2!} + \\dots - (-1)^{n}\\dfrac{x^{n}}{n!}$ is positive or negative\naccording as $n$~is odd or even. Deduce the exponential theorem.", "markdown": "Prove that $e^{-x} - 1 + x - \\dfrac{x^{n}}{2!} + \\dots - (-1)^{n}\\dfrac{x^{n}}{n!}$ is positive or negative according as $n$ is odd or even. Deduce the exponential theorem.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.series", "core.arith" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xc/14", "set": "hardy-course-of-pure-mathematics-1921/ex-xc", "number": 14, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "379", "location": "Exercise XC, problem 14", "problem_latex": "If\n\\[\nX_{0} = e^{x},\\quad\nX_{1} = e^{x} - 1,\\quad\nX_{2} = e^{x} - 1 - x,\\quad\nX_{3} = e^{x} - 1 - x - (x^{2}/2!),\\ \\dots,\n\\]\nthen $dX_{\\nu}/dx = X_{\\nu-1}$. Hence prove that if $t > 0$ then\n\\[\nX_{1}(t) = \\int_{0}^{t} X_{0}\\, dx < te^{t},\\quad\nX_{2}(t) = \\int_{0}^{t} X_{1}\\, dx < \\int_{0}^{t} xe^{x}\\, dx\n < e^{t} \\int_{0}^{t} x\\, dx = \\frac{t^{2}}{2!} e^{t},\n\\]\nand generally $X_{\\nu}(t) < \\dfrac{t^{\\nu}}{\\nu!} e^{t}$. Deduce the exponential theorem.", "markdown": "If X_0 = e^x,0pt minus 3ptX_1 = e^x - 1,0pt minus 3ptX_2 = e^x - 1 - x,0pt minus 3ptX_3 = e^x - 1 - x - (x^2/2!), …, then $dX_{\\nu}/dx = X_{\\nu-1}$. Hence prove that if $t > 0$ then X_1(t) = _0^t X_0  dx < te^t,0pt minus 3ptX_2(t) = _0^t X_1  dx < _0^t xe^x  dx < e^t _0^t x  dx = t^22! e^t, and generally $X_{\\nu}(t) < \\dfrac{t^{\\nu}}{\\nu!} e^{t}$. Deduce the exponential theorem.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.integrate", "core.const" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xc/15", "set": "hardy-course-of-pure-mathematics-1921/ex-xc", "number": 15, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "379", "location": "Exercise XC, problem 15", "problem_latex": "Show that the expansion in powers of~$p$ of the positive root of\n$x^{2+p} = a^{2}$ begins with the terms\n\\[\na\\{1 - \\tfrac{1}{2} p\\log a + \\tfrac{1}{8} p^{2}\\log a (2 + \\log a)\\}.\n\\]\n\\MathTrip{1909.}", "markdown": "Show that the expansion in powers of $p$ of the positive root of $x^{2+p} = a^{2}$ begins with the terms a1 - 12 pa + 18 p^2a (2 + a). % [0]% (*Math. Trip.* 1909.)% [1]%", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.series", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xc/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xc", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "379", "location": "Exercise XC, problem 2", "problem_latex": "If $x$~is positive then the greatest term in the exponential series is the\n$([x] + 1)$-th, unless $x$~is an integer, when the preceding term is equal to it.", "markdown": "If $x$ is positive then the greatest term in the exponential series is the $([x] + 1)$-th, unless $x$ is an integer, when the preceding term is equal to it.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xc/3", "set": "hardy-course-of-pure-mathematics-1921/ex-xc", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "379", "location": "Exercise XC, problem 3", "problem_latex": "Show that $n! > (n/e)^{n}$. [For $n^{n}/n!$~is one term in the series for~$e^{n}$.]", "markdown": "Show that $n! > (n/e)^{n}$. [For $n^{n}/n!$ is one term in the series for $e^{n}$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.const" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xc/4", "set": "hardy-course-of-pure-mathematics-1921/ex-xc", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "379", "location": "Exercise XC, problem 4", "problem_latex": "Prove that $e^{n} = (n^{n}/n!)(2 + S_{1} + S_{2})$, where\n\\[\nS_{1} = \\frac{1}{1 + \\nu} + \\frac{1}{(1 + \\nu)(1 + 2\\nu)} + \\dots,\\quad\nS_{2} = (1 - \\nu) + (1 - \\nu)(1 - 2\\nu) + \\dots,\n\\]\nand $\\nu = 1/n$; and deduce that $n!$~lies between $2(n/e)^{n}$ and~$2(n + 1)(n/e)^{n}$.", "markdown": "Prove that $e^{n} = (n^{n}/n!)(2 + S_{1} + S_{2})$, where S_1 = 11 + + 1(1 + )(1 + 2) + …,0pt minus 3ptS_2 = (1 - ) + (1 - )(1 - 2) + …, and $\\nu = 1/n$; and deduce that $n!$ lies between $2(n/e)^{n}$ and $2(n + 1)(n/e)^{n}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.sum", "core.const", "core.frac" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xc/5", "set": "hardy-course-of-pure-mathematics-1921/ex-xc", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "379", "location": "Exercise XC, problem 5", "problem_latex": "Employ the exponential series to prove that $e^{x}$~tends to infinity more\nrapidly than any power of~$x$. [Use the inequality $e^{x} > x^{n}/n!$.]", "markdown": "Employ the exponential series to prove that $e^{x}$ tends to infinity more rapidly than any power of $x$. [Use the inequality $e^{x} > x^{n}/n!$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "cas.sum" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xc/6", "set": "hardy-course-of-pure-mathematics-1921/ex-xc", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "379", "location": "Exercise XC, problem 6", "problem_latex": "Show that $e$~is not a rational number. [If $e = p/q$, where $p$ and~$q$ are\nintegers, we must have\n\\[\n\\frac{p}{q} = \\DPtypo{}{1 + {}} 1 + \\frac{1}{2!}+\\frac{1}{3!} + \\dots + \\frac{1}{q!} + \\dots\n\\]\nor, multiplying up by~$q!$,\n\\[\nq! \\left(\\frac{p}{q} - 1 - 1 - \\frac{1}{2!} - \\dots - \\frac{1}{q!}\\right)\n = \\frac{1}{q + 1} + \\frac{1}{(q + 1)(q + 2)} + \\dots\n\\]\nand this is absurd, since the left-hand side is integral, and the right-hand\nside less than $\\{1/(q + 1)\\} + \\{1/(q + 1)\\}^{2} + \\dots = 1/q$.]", "markdown": "Show that $e$ is not a rational number. [If $e = p/q$, where $p$ and $q$ are integers, we must have pq = 1 + 12!+13! + …+ 1q! + … or, multiplying up by $q!$, q! (pq - 1 - 1 - 12! - …- 1q!) = 1q + 1 + 1(q + 1)(q + 2) + … and this is absurd, since the left-hand side is integral, and the right-hand side less than $\\{1/(q + 1)\\} + \\{1/(q + 1)\\}^{2} + \\dots = 1/q$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.sum", "core.frac" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xc/7", "set": "hardy-course-of-pure-mathematics-1921/ex-xc", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "379", "location": "Exercise XC, problem 7", "problem_latex": "Sum the series $\\sum\\limits_{0}^{\\infty} P_{r}(n)\\dfrac{x^{n}}{n!}$, where $P_{r}(n)$~is a polynomial of degree~$r$\nin~$n$. [We can express $P_{r}(n)$ in the form\n\\[\nA_{0} + A_{1}n + A_{2}n(n - 1) + \\dots + A_{r}n(n - 1) \\dots (n - r + 1),\n\\]\nand\n\\begin{align*}\n\\sum_{0}^{\\infty} P_{r}(n) \\frac{x^{n}}{n!}\n &= A_{0}\\sum_{0}^{\\infty}\\frac{x^{n}}{n!}\n + A_{1}\\sum_{1}^{\\infty}\\frac{x^{n}}{(n - 1)!} + \\dots\n + A_{r}\\sum_{r}^{\\infty}\\frac{x^{n}}{(n - r)!}\\\\\n &= (A_{0} + A_{1}x + A_{2}x^{2} + \\dots + A_{r}x^{r})e^{x}.]\n\\end{align*}", "markdown": "Sum the series $\\sum\\limits_{0}^{\\infty} P_{r}(n)\\dfrac{x^{n}}{n!}$, where $P_{r}(n)$ is a polynomial of degree $r$ in $n$. [We can express $P_{r}(n)$ in the form A_0 + A_1n + A_2n(n - 1) + …+ A_rn(n - 1) …(n - r + 1), and align* _0^ P_r(n) x^nn! &= A_0_0^x^nn! + A_1_1^x^n(n - 1)! + … + A_r_r^x^n(n - r)! &= (A_0 + A_1x + A_2x^2 + …+ A_rx^r)e^x.] align*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "cas.sum" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xc/8", "set": "hardy-course-of-pure-mathematics-1921/ex-xc", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "379", "location": "Exercise XC, problem 8", "problem_latex": "Show that\n\\[\n\\sum_{1}^{\\infty} \\frac{n^{3}}{n!} x^{n} = (x + 3x^{2} + x^{3})e^{x},\\quad\n\\sum_{1}^{\\infty} \\frac{n^{4}}{n!} x^{n} = (x + 7x^{2} + 6x^{3} + x^{4})e^{x};\n\\]\nand that if $S_{n} = 1^{3} + 2^{3} + \\dots + n^{3}$ then\n\\[\n\\sum_{1}^{\\infty} S_{n}\\frac{x^{n}}{n!}\n = \\tfrac{1}{4}(4x + 14x^{2} + 8x^{3} + x^{4})e^{x}.\n\\]\nIn particular the last series is equal to zero when $x = -2$. \\MathTrip{1904.}", "markdown": "Show that _1^ n^3n! x^n = (x + 3x^2 + x^3)e^x,0pt minus 3pt_1^ n^4n! x^n = (x + 7x^2 + 6x^3 + x^4)e^x; and that if $S_{n} = 1^{3} + 2^{3} + \\dots + n^{3}$ then _1^ S_nx^nn! = 14(4x + 14x^2 + 8x^3 + x^4)e^x. In particular the last series is equal to zero when $x = -2$. % [0]% (*Math. Trip.* 1904.)% [1]%", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "cas.subst", "cas.sum" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xc/9", "set": "hardy-course-of-pure-mathematics-1921/ex-xc", "number": 9, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "379", "location": "Exercise XC, problem 9", "problem_latex": "Prove that $\\sum (n/n!) = e$, $\\sum (n^{2}/n!) = 2e$, $\\sum (n^{3}/n!) = 5e$, and that $\\sum (n^{k}/n!)$,\nwhere $k$~is any positive integer, is a positive integral multiple of~$e$.", "markdown": "Prove that $\\sum (n/n!) = e$, $\\sum (n^{2}/n!) = 2e$, $\\sum (n^{3}/n!) = 5e$, and that $\\sum (n^{k}/n!)$, where $k$ is any positive integer, is a positive integral multiple of $e$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.sum", "core.const" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xci/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xci", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "382", "location": "Exercise XCI, problem 1", "problem_latex": "$\\log \\left(\\dfrac{1}{1 - x}\\right) = x + \\frac{1}{2} x^{2} + \\frac{1}{3} x^{3} + \\dots$ if $-1 \\leq x < 1$.", "markdown": "$\\log \\left(\\dfrac{1}{1 - x}\\right) = x + \\frac{1}{2} x^{2} + \\frac{1}{3} x^{3} + \\dots$ if $-1 \\leq x < 1$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.series", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xci/10", "set": "hardy-course-of-pure-mathematics-1921/ex-xci", "number": 10, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "382", "location": "Exercise XCI, problem 10", "problem_latex": "Show that\n\\[\n\\tfrac{1}{4}\\pi = \\arctan(1/2) + \\arctan(1/3) = 4\\arctan(1/5) - \\arctan(1/239),\n\\]\nand calculate~$\\pi$ to $6$~places of decimals.", "markdown": "Show that 14= (1/2) + (1/3) = 4(1/5) - (1/239), and calculate $\\pi$ to $6$ places of decimals.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.const", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xci/11", "set": "hardy-course-of-pure-mathematics-1921/ex-xci", "number": 11, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "382", "location": "Exercise XCI, problem 11", "problem_latex": "Show that the expansion of $(1 + x)^{1+x}$ in powers of~$x$ begins with the\nterms $1 + x + x^{2} + 1/2 x^{3}$.", "markdown": "Show that the expansion of $(1 + x)^{1+x}$ in powers of $x$ begins with the terms $1 + x + x^{2} + 1/2 x^{3}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.series" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xci/12", "set": "hardy-course-of-pure-mathematics-1921/ex-xci", "number": 12, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "382", "location": "Exercise XCI, problem 12", "problem_latex": "Show that\n\\[\n\\log_{10} e - \\sqrtb{x(x + 1)} \\log_{10}\\left(\\frac{1 + x}{x}\\right)\n = \\frac{\\log_{10} e}{24x^{2}},\n\\]\napproximately, for large values of~$x$. Apply the formula, when $x = 10$, to\nobtain an approximate value of~$\\log_{10} e$, and estimate the accuracy of the result.", "markdown": "Show that _10 e - x(x + 1) _10(1 + xx) = _10 e24x^2, approximately, for large values of $x$. Apply the formula, when $x = 10$, to obtain an approximate value of $\\log_{10} e$, and estimate the accuracy of the result.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.const", "core.log", "core.units" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xci/13", "set": "hardy-course-of-pure-mathematics-1921/ex-xci", "number": 13, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "382", "location": "Exercise XCI, problem 13", "problem_latex": "Show that\n\\[\n\\frac{1}{1 - x} \\log\\left(\\frac{1}{1 - x}\\right)\n = x + \\left(1 + \\tfrac{1}{2}\\right)x^{2}\n + \\left(1 + \\tfrac{1}{2} + \\tfrac{1}{3}\\right)x^{3} + \\dots,\n\\]\nif $-1 < x < 1$.", "markdown": "Show that 11 - x (11 - x) = x + (1 + 12)x^2 + (1 + 12 + 13)x^3 + …, if $-1 < x < 1$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "cas.series" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xci/14", "set": "hardy-course-of-pure-mathematics-1921/ex-xci", "number": 14, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "382", "location": "Exercise XCI, problem 14", "problem_latex": "{\\Loosen Using the logarithmic series and the facts that $\\log_{10} 2.3758 = .375\\MS809\\MS9\\dots$\nand $\\log_{10} e = .4343\\dots$, show that an approximate solution of the equation\n$x = 100 \\log_{10}x$ is~$237.581\\MS21$.}", "markdown": "0.375em plus 0.75em minus 0.25emUsing the logarithmic series and the facts that $\\log_{10} 2.3758 = .375\\MS809\\MS9\\dots$ and $\\log_{10} e = .4343\\dots$, show that an approximate solution of the equation $x = 100 \\log_{10}x$ is $237.581\\MS21$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.const", "core.log", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xci/15", "set": "hardy-course-of-pure-mathematics-1921/ex-xci", "number": 15, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "382", "location": "Exercise XCI, problem 15", "problem_latex": "Expand $\\log\\cos x$ and $\\log(\\sin x/x)$ in powers of~$x$ as far as~$x^{4}$, and\nverify that, to this order,\n\\[\n\\log\\sin x\n = \\log x - \\tfrac{1}{45} \\log\\cos x + \\tfrac{64}{45}\\log\\cos \\tfrac{1}{2}x.\n\\]", "markdown": "Expand $\\log\\cos x$ and $\\log(\\sin x/x)$ in powers of $x$ as far as $x^{4}$, and verify that, to this order, x = x - 145 x + 644512x.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.series", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xci/16", "set": "hardy-course-of-pure-mathematics-1921/ex-xci", "number": 16, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "382", "location": "Exercise XCI, problem 16", "problem_latex": "Show that\n\\[\n%[** TN: In-line in the original]\n\\int_{0}^{x} \\frac{dt}{1 + t^{4}} = x - \\tfrac{1}{5}x^{5} + \\tfrac{1}{9}x^{9} - \\dots\n\\]\nif $-1 \\leq x \\leq 1$. Deduce that\n\\[\n1 - \\tfrac{1}{5} + \\tfrac{1}{9} - \\dots\n = \\{\\pi + 2\\log(\\sqrt{2} + 1)\\}/4\\sqrt{2}.\n\\]", "markdown": "Show that %[** TN: In-line in the original] _0^x dt1 + t^4 = x - 15x^5 + 19x^9 - … if $-1 \\leq x \\leq 1$. Deduce that 1 - 15 + 19 - … = + 2(2 + 1)/42.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.series", "core.const", "core.integ.num" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xci/17", "set": "hardy-course-of-pure-mathematics-1921/ex-xci", "number": 17, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "382", "location": "Exercise XCI, problem 17", "problem_latex": "Prove similarly that\n\\[\n\\tfrac{1}{3} - \\tfrac{1}{7} + \\tfrac{1}{11} - \\dots\n = \\int_{0}^{1} \\frac{t^{2}\\, dt}{1 + t^{4}}\n = \\{\\pi - 2\\log(\\sqrt{2} + 1)\\}/4\\sqrt{2}.\n\\]", "markdown": "Prove similarly that 13 - 17 + 111 - … = _0^1 t^2  dt1 + t^4 = - 2(2 + 1)/42.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.series", "core.const", "core.integ.num" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xci/18", "set": "hardy-course-of-pure-mathematics-1921/ex-xci", "number": 18, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "382", "location": "Exercise XCI, problem 18", "problem_latex": "Prove generally that if $a$ and~$b$ are positive integers then\n\\[\n\\frac{1}{a} - \\frac{1}{a + b} + \\frac{1}{a + 2b} - \\dots\n = \\int_{0}^{1} \\frac{t^{a-1}\\, dt}{1 + t^{b}},\n\\]\nand so that the sum of the series can be found. Calculate in this way the\nsums of $1 - \\frac{1}{4} + \\frac{1}{7} - \\dots$ and $\\frac{1}{2} - \\frac{1}{5} + \\frac{1}{8} - \\dots$.", "markdown": "Prove generally that if $a$ and $b$ are positive integers then 1a - 1a + b + 1a + 2b - … = _0^1 t^a-1  dt1 + t^b, and so that the sum of the series can be found. Calculate in this way the sums of $1 - \\frac{1}{4} + \\frac{1}{7} - \\dots$ and $\\frac{1}{2} - \\frac{1}{5} + \\frac{1}{8} - \\dots$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate", "core.arith", "core.integ.num" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xci/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xci", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "382", "location": "Exercise XCI, problem 2", "problem_latex": "$\\argtanh x = \\frac{1}{2} \\log\\left(\\dfrac{1 + x}{1 - x}\\right) = x + \\frac{1}{3} x^{3} + \\frac{1}{5} x^{5} + \\dots$ if $-1 < x < 1$.", "markdown": "$\\argtanh x = \\frac{1}{2} \\log\\left(\\dfrac{1 + x}{1 - x}\\right) = x + \\frac{1}{3} x^{3} + \\frac{1}{5} x^{5} + \\dots$ if $-1 < x < 1$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.series", "core.hyp", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xci/3", "set": "hardy-course-of-pure-mathematics-1921/ex-xci", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "382", "location": "Exercise XCI, problem 3", "problem_latex": "Prove that if $x$~is positive then\n\\[\n\\log(1 + x) = \\frac{x}{1 + x}\n + \\tfrac{1}{2} \\left(\\frac{x}{1 + x}\\right)^{2}\n + \\tfrac{1}{3} \\left(\\frac{x}{1 + x}\\right)^{3} + \\dots.\n\\]", "markdown": "Prove that if $x$ is positive then (1 + x) = x1 + x + 12 (x1 + x)^2 + 13 (x1 + x)^3 + ….", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.series", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xci/4", "set": "hardy-course-of-pure-mathematics-1921/ex-xci", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "382", "location": "Exercise XCI, problem 4", "problem_latex": "Obtain the series for $\\log(1 + x)$ and $\\arctan x$ by means of Taylor's\ntheorem.", "markdown": "Obtain the series for $\\log(1 + x)$ and $\\arctan x$ by means of Taylor’s theorem.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.series" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xci/5", "set": "hardy-course-of-pure-mathematics-1921/ex-xci", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "382", "location": "Exercise XCI, problem 5", "problem_latex": "If $y > 0$ then\n\\[\n\\log y = 2 \\left\\{\\frac{y - 1}{y + 1}\n + \\frac{1}{3} \\left(\\frac{y - 1}{y + 1}\\right)^{3}\n + \\frac{1}{5} \\left(\\frac{y - 1}{y + 1}\\right)^{5} + \\dots\\right\\}.\n\\]", "markdown": "If $y > 0$ then y = 2 y - 1y + 1 + 13 (y - 1y + 1)^3 + 15 (y - 1y + 1)^5 + ….", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.series", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xci/6", "set": "hardy-course-of-pure-mathematics-1921/ex-xci", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "382", "location": "Exercise XCI, problem 6", "problem_latex": "Find $\\log 10$ to $3$~places of decimals from the formula\n\\[\n\\log 10 = 3\\log 2 + \\log(1 + \\tfrac{1}{4}).\n\\]", "markdown": "Find $\\log 10$ to $3$ places of decimals from the formula 10 = 32 + (1 + 14).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xci/7", "set": "hardy-course-of-pure-mathematics-1921/ex-xci", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "382", "location": "Exercise XCI, problem 7", "problem_latex": "Prove that\n\\[\n\\log \\left(\\frac{x + 1}{x}\\right)\n = 2\\left\\{\\frac{1}{2x + 1} + \\frac{1}{3(2x + 1)^{3}} + \\frac{1}{5(2x + 1)^{5}} + \\dots\\right\\}\n\\]\nif $x > 0$, and that\n\\[\n\\log \\frac{(x - 1)^{2}(x + 2)}{(x + 1)^{2}(x - 2)}\n = 2\\left\\{\\frac{2}{x^{3} - 3x}\n + \\frac{1}{3}\\left(\\frac{2}{x^{3} - 3x}\\right)^{3}\n + \\frac{1}{5}\\left(\\frac{2}{x^{3} - 3x}\\right)^{5} + \\dots\\right\\}\n\\]\nif $x > 2$. Given that $\\log 2 = .693\\MS147\\MS1\\dots$ and $\\log 3 = 1.098\\MS612\\MS3\\dots$, show, by\nputting $x = 10$ in the second formula, that $\\log 11 = 2.397\\MS895\\dots$.", "markdown": "Prove that (x + 1x) = 212x + 1 + 13(2x + 1)^3 + 15(2x + 1)^5 + … if $x > 0$, and that (x - 1)^2(x + 2)(x + 1)^2(x - 2) = 22x^3 - 3x + 13(2x^3 - 3x)^3 + 15(2x^3 - 3x)^5 + … if $x > 2$. Given that $\\log 2 = .693\\MS147\\MS1\\dots$ and $\\log 3 = 1.098\\MS612\\MS3\\dots$, show, by putting $x = 10$ in the second formula, that $\\log 11 = 2.397\\MS895\\dots$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.series", "core.arith", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xci/8", "set": "hardy-course-of-pure-mathematics-1921/ex-xci", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "382", "location": "Exercise XCI, problem 8", "problem_latex": "Show that if $\\log 2$, $\\log 5$, and $\\log 11$ are known, then the formula\n\\[\n\\log 13 = 3\\log 11 + \\log 5 - 9\\log 2\n\\]\ngives $\\log 13$ with an error practically equal to~$.000\\MS15$.", "markdown": "Show that if $\\log 2$, $\\log 5$, and $\\log 11$ are known, then the formula 13 = 311 + 5 - 92 gives $\\log 13$ with an error practically equal to $.000\\MS15$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xci/9", "set": "hardy-course-of-pure-mathematics-1921/ex-xci", "number": 9, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "382", "location": "Exercise XCI, problem 9", "problem_latex": "Show that\n\\[\n\\tfrac{1}{2} \\log 2 = 7a + 5b + 3c,\\quad\n\\tfrac{1}{2} \\log 3 = 11a + 8b + 5c,\\quad\n\\tfrac{1}{2} \\log 5 = 16a + 12b + 7c,\n\\]\nwhere $a = \\argtanh(1/31)$, $b = \\argtanh(1/49)$, $c = \\argtanh(1/161)$.", "markdown": "Show that 12 2 = 7a + 5b + 3c,0pt minus 3pt12 3 = 11a + 8b + 5c,0pt minus 3pt12 5 = 16a + 12b + 7c, where $a = \\argtanh(1/31)$, $b = \\argtanh(1/49)$, $c = \\argtanh(1/161)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.hyp", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcii/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xcii", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "385", "location": "Exercise XCII, problem 1", "problem_latex": "Prove that if $-1 < x < 1$ then\n\\[\n\\frac{1}{\\sqrtp{1 + x^{2}}} = 1 - \\frac{1}{2}x^{2} + \\frac{1·3}{2·4}x^{4} - \\dots,\\quad\n\\frac{1}{\\sqrtp{1 - x^{2}}} = 1 + \\frac{1}{2}x^{2} + \\frac{1·3}{2·4}x^{4} + \\dots.\n\\]", "markdown": "Prove that if $-1 < x < 1$ then 11 + x^2 = 1 - 12x^2 + 1·32·4x^4 - …,0pt minus 3pt11 - x^2 = 1 + 12x^2 + 1·32·4x^4 + ….", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcii/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xcii", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "385", "location": "Exercise XCII, problem 2", "problem_latex": "\\Topic{Approximation to quadratic and other surds.} {\\Loosen Let $\\sqrt{M}$ be a\nquadratic surd whose numerical value is required. Let $N^{2}$ be the square\nnearest to~$M$; and let $M = N^{2} + x$ or $M = N^{2} - x$, $x$~being positive. Since $x$~cannot\nbe greater than~$N$, $x/N^{2}$~is comparatively small and the surd\n$\\sqrt{M} = N\\sqrtb{1 ± (x/N^{2})}$ can be expressed in a series}\n\\[\n= N\\left\\{\n 1 ± \\frac{1}{2}\\left(\\frac{x}{N^{2}}\\right)\n - \\frac{1·1}{2·4}\\left(\\frac{x}{N^{2}}\\right)^{2} ± \\dots\n\\right\\},\n\\]\nwhich is at any rate fairly rapidly convergent, and may be very rapidly so.\nThus\n\\[\n\\sqrt{67} = \\sqrtp{64 + 3}\n = 8\\left\\{\n 1 + \\frac{1}{2}\\left(\\frac{3}{64}\\right)\n - \\frac{1·1}{2·4}\\left(\\frac{3}{64}\\right)^{2} + \\dots\n\\right\\}.\n\\]\n\nLet us consider the error committed in taking~$8\\frac{3}{16}$ (the value given by\nthe first two terms) as an approximate value. After the second term the\nterms alternate in sign and decrease. Hence the error is one of excess, and\nis less than~$3^{2}/64^{2}$, which is less than~$.003$.", "markdown": "**to quadratic and other surds.** 0.375em plus 0.75em minus 0.25emLet $\\sqrt{M}$ be a quadratic surd whose numerical value is required. Let $N^{2}$ be the square nearest to $M$; and let $M = N^{2} + x$ or $M = N^{2} - x$, $x$ being positive. Since $x$ cannot be greater than $N$, $x/N^{2}$ is comparatively small and the surd $\\sqrt{M} = N\\sqrtb{1 ± (x/N^{2})}$ can be expressed in a series = N 1 ± 12(xN^2) - 1·12·4(xN^2)^2 ± …, which is at any rate fairly rapidly convergent, and may be very rapidly so. Thus 67 = 64 + 3 = 8 1 + 12(364) - 1·12·4(364)^2 + …. Let us consider the error committed in taking $8\\frac{3}{16}$ (the value given by the first two terms) as an approximate value. After the second term the terms alternate in sign and decrease. Hence the error is one of excess, and is less than $3^{2}/64^{2}$, which is less than $.003$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.frac" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcii/3", "set": "hardy-course-of-pure-mathematics-1921/ex-xcii", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "385", "location": "Exercise XCII, problem 3", "problem_latex": "If $x$~is small compared with~$N^{2}$ then\n\\[\n\\sqrtp{N^{2} + x} = N + \\frac{x}{4N} + \\frac{Nx}{2(2N^{2} + x)},\n\\]\nthe error being of the order~$x^{4}/N^{7}$. Apply the process to~$\\sqrt{907}$.\n\n[Expanding by the binomial theorem, we have\n\\[\n\\sqrtp{N^{2} + x}\n = N + \\frac{x}{2N} - \\frac{x^{2}}{8N^{3}} + \\frac{x^{3}}{16N^{5}},\n\\]\nthe error being less than the numerical value of the next term, viz.\\\n$5x^{4}/128N^{7}$. Also\n\\[\n\\frac{Nx}{2(2N^{2} + x)}\n = \\frac{x}{4N} \\left(1 + \\frac{x}{2N^{2}}\\right)^{-1}\n = \\frac{x}{4N} - \\frac{x^{2}}{8N^{3}} + \\frac{x^{3}}{16N^{5}},\n\\]\nthe error being less than~$x^{4}/32N^{7}$. The result follows. The same method\nmay be applied to surds other than quadratic surds, \\eg\\ to~$\\sqrt[3]{1031}$.]", "markdown": "If $x$ is small compared with $N^{2}$ then N^2 + x = N + x4N + Nx2(2N^2 + x), the error being of the order $x^{4}/N^{7}$. Apply the process to $\\sqrt{907}$. [Expanding by the binomial theorem, we have N^2 + x = N + x2N - x^28N^3 + x^316N^5, the error being less than the numerical value of the next term, viz. $5x^{4}/128N^{7}$. Also Nx2(2N^2 + x) = x4N (1 + x2N^2)^-1 = x4N - x^28N^3 + x^316N^5, the error being less than $x^{4}/32N^{7}$. The result follows. The same method may be applied to surds other than quadratic surds, *e.g.* to $\\sqrt[3]{1031}$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.frac", "core.prog" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcii/4", "set": "hardy-course-of-pure-mathematics-1921/ex-xcii", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "385", "location": "Exercise XCII, problem 4", "problem_latex": "If $M$~differs from~$N^{3}$ by less than $1$~per~cent.\\ of either then $\\sqrt[3]{M}$~differs\nfrom $\\frac{2}{3}N + \\frac{1}{3}(M/N^{2})$ by less than $N/90\\MC000$. \\MathTrip{1882.}", "markdown": "If $M$ differs from $N^{3}$ by less than $1$ per cent. of either then $\\sqrt[3]{M}$ differs from $\\frac{2}{3}N + \\frac{1}{3}(M/N^{2})$ by less than $N/90\\MC000$. % [0]% (*Math. Trip.* 1882.)% [1]%", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcii/5", "set": "hardy-course-of-pure-mathematics-1921/ex-xcii", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "385", "location": "Exercise XCII, problem 5", "problem_latex": "If $M = N^{4} + x$, and $x$~is small compared with~$N$, then a good approximation\nfor~$\\sqrt[4]{M}$ is\n\\[\n\\frac{51}{56} N + \\frac{5}{56}\\, \\frac{M}{N^{3}} + \\frac{27Nx}{14(7M + 5N^{4})}.\n\\]\nShow that when $N = 10$, $x = 1$, this approximation is accurate to $16$~places\nof decimals. \\MathTrip{1886.}", "markdown": "If $M = N^{4} + x$, and $x$ is small compared with $N$, then a good approximation for $\\sqrt[4]{M}$ is 5156 N + 556  MN^3 + 27Nx14(7M + 5N^4). Show that when $N = 10$, $x = 1$, this approximation is accurate to $16$ places of decimals. % [0]% (*Math. Trip.* 1886.)% [1]%", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.frac", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcii/6", "set": "hardy-course-of-pure-mathematics-1921/ex-xcii", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "385", "location": "Exercise XCII, problem 6", "problem_latex": "Show how to sum the series\n\\[\n\\sum_{0}^{\\infty} P_{r}(n) \\binom{m}{n} x^{n},\n\\]\nwhere $P_{r}(n)$~is a polynomial of degree~$r$ in~$n$.\n\n[Express $P_{r}(n)$ in the form $A_{0} + A_{1}n + A_{2}n(n - 1) + \\dots$ as in \\Ex{xc}.~7.]", "markdown": "Show how to sum the series _0^ P_r(n) mn x^n, where $P_{r}(n)$ is a polynomial of degree $r$ in $n$. [Express $P_{r}(n)$ in the form $A_{0} + A_{1}n + A_{2}n(n - 1) + \\dots$ as in % [examples:xc]Ex. xc%. 7.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.collect", "cas.expand" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcii/7a", "set": "hardy-course-of-pure-mathematics-1921/ex-xcii", "number": 7, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "385", "location": "Exercise XCII, problem 7a", "problem_latex": "Sum the series $\\sum\\limits_{0}^{\\infty} n \\dbinom{m}{n} x^{n}$, $\\sum\\limits_{0}^{\\infty} n^{2} \\dbinom{m}{n} x^{n}$ and prove that\n\\[\n\\sum_{0}^{\\infty} n^{3} \\binom{m}{n} x^{n}\n = \\{m^{3}x^{3} + m(3m - 1)x^{2} + mx\\}(1 + x)^{m-3}.\n\\]", "markdown": "Sum the series $\\sum\\limits_{0}^{\\infty} n \\dbinom{m}{n} x^{n}$, $\\sum\\limits_{0}^{\\infty} n^{2} \\dbinom{m}{n} x^{n}$ and prove that _0^ n^3 mn x^n = m^3x^3 + m(3m - 1)x^2 + mx(1 + x)^m-3.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcii/7b", "set": "hardy-course-of-pure-mathematics-1921/ex-xcii", "number": 7, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "385", "location": "Exercise XCII, problem 7b", "problem_latex": "Sum the series $\\sum\\limits_{0}^{\\infty} n \\dbinom{m}{n} x^{n}$, $\\sum\\limits_{0}^{\\infty} n^{2} \\dbinom{m}{n} x^{n}$ and prove that\n\\[\n\\sum_{0}^{\\infty} n^{3} \\binom{m}{n} x^{n}\n = \\{m^{3}x^{3} + m(3m - 1)x^{2} + mx\\}(1 + x)^{m-3}.\n\\]", "markdown": "Sum the series $\\sum\\limits_{0}^{\\infty} n \\dbinom{m}{n} x^{n}$, $\\sum\\limits_{0}^{\\infty} n^{2} \\dbinom{m}{n} x^{n}$ and prove that _0^ n^3 mn x^n = m^3x^3 + m(3m - 1)x^2 + mx(1 + x)^m-3.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcii/7c", "set": "hardy-course-of-pure-mathematics-1921/ex-xcii", "number": 7, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "385", "location": "Exercise XCII, problem 7c", "problem_latex": "Sum the series $\\sum\\limits_{0}^{\\infty} n \\dbinom{m}{n} x^{n}$, $\\sum\\limits_{0}^{\\infty} n^{2} \\dbinom{m}{n} x^{n}$ and prove that\n\\[\n\\sum_{0}^{\\infty} n^{3} \\binom{m}{n} x^{n}\n = \\{m^{3}x^{3} + m(3m - 1)x^{2} + mx\\}(1 + x)^{m-3}.\n\\]", "markdown": "Sum the series $\\sum\\limits_{0}^{\\infty} n \\dbinom{m}{n} x^{n}$, $\\sum\\limits_{0}^{\\infty} n^{2} \\dbinom{m}{n} x^{n}$ and prove that _0^ n^3 mn x^n = m^3x^3 + m(3m - 1)x^2 + mx(1 + x)^m-3.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.collect", "cas.derive", "cas.expand" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xciii/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xciii", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "401", "location": "Exercise XCIII, problem 1", "problem_latex": "We supposed above that $-\\pi < \\theta < \\pi$, and so\nexcluded the case in which $z$~is \\emph{real and negative}. In this case the straight\nline from~$1$ to~$z$ passes through~$0$, and is therefore not admissible as a path of\nintegration. Both $\\pi$ and~$-\\pi$ are values of~$\\am z$, and $\\theta$~is equal to one or\nother of them: also $r = -z$. The values of~$\\Log z$ are still the values\nof~$\\log |z| + i\\am z$, viz.\\\n\\[\n\\log (-z) + (2k + 1)\\pi i,\n\\]\nwhere $k$~is an integer. The values~$\\log (-z) + \\pi i$ and~$\\log (-z) - \\pi i$ correspond\nto paths from~$1$ to~$z$ lying respectively entirely above and entirely below the\nreal axis. Either of them may be taken as the principal value of~$\\Log z$, as\nconvenience dictates. We shall choose the value~$\\log (-z) + \\pi$ i corresponding\nto the first path.", "markdown": "We supposed above that $-\\pi < \\theta < \\pi$, and so excluded the case in which $z$ is *real and negative*. In this case the straight line from $1$ to $z$ passes through $0$, and is therefore not admissible as a path of integration. Both $\\pi$ and $-\\pi$ are values of $\\am z$, and $\\theta$ is equal to one or other of them: also $r = -z$. The values of $\\Log z$ are still the values of $\\log |z| + i\\am z$, viz. (-z) + (2k + 1)i, where $k$ is an integer. The values $\\log (-z) + \\pi i$ and $\\log (-z) - \\pi i$ correspond to paths from $1$ to $z$ lying respectively entirely above and entirely below the real axis. Either of them may be taken as the principal value of $\\Log z$, as convenience dictates. We shall choose the value $\\log (-z) + \\pi$ i corresponding to the first path.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xciii/10", "set": "hardy-course-of-pure-mathematics-1921/ex-xciii", "number": 10, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "401", "location": "Exercise XCIII, problem 10", "problem_latex": "The function~$f(x)$ defined by\n\\[\n\\pi f(x) = p\\pi + (q - p)\\Imag\\{\\log(x - 1)\\} + (r - q)\\Imag(\\log x)\n\\]\nis equal to~$p$ when $x > 1$, to~$q$ when $0 < x < 1$, and to~$r$ when $x < 0$.", "markdown": "The function $f(x)$ defined by f(x) = p+ (q - p)(x - 1) + (r - q)(x) is equal to $p$ when $x > 1$, to $q$ when $0 < x < 1$, and to $r$ when $x < 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xciii/11", "set": "hardy-course-of-pure-mathematics-1921/ex-xciii", "number": 11, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "401", "location": "Exercise XCIII, problem 11", "problem_latex": "For what values of~$z$ is (i)~$\\log z$ (ii)~any value of~$\\Log z$ (\\ia)~real or\n(\\ib)~purely imaginary?", "markdown": "For what values of $z$ is (i) $\\log z$ (ii) any value of $\\Log z$ (*a*) real or (*b*) purely imaginary?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xciii/12", "set": "hardy-course-of-pure-mathematics-1921/ex-xciii", "number": 12, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "401", "location": "Exercise XCIII, problem 12", "problem_latex": "If $z = x + iy$ then $\\Log\\Log z = \\log R + i(\\Theta + 2k'\\pi)$, where\n\\[\nR^{2} = (\\log r)^{2} + (\\theta + 2k\\pi)^{2}\n\\]\nand $\\Theta$~is the least positive angle determined by the equations\n\\[\n\\cos\\Theta : \\sin\\Theta : 1 ::\n\\log r : \\theta + 2k\\pi: \\sqrtb{(\\log r)^{2} + (\\theta + 2k\\pi)^{2}}.\n\\]\nPlot roughly the doubly infinite set of values of $\\Log\\Log(1 + i\\sqrt{3})$, indicating\nwhich of them are values of $\\log\\Log(1 + i \\sqrt{3})$ and which of $\\Log\\log(1 + i\\sqrt{3})$.", "markdown": "If $z = x + iy$ then $\\Log\\Log z = \\log R + i(\\Theta + 2k'\\pi)$, where R^2 = (r)^2 + (+ 2k)^2 and $\\Theta$ is the least positive angle determined by the equations : : 1 :: r : + 2k: (r)^2 + (+ 2k)^2. Plot roughly the doubly infinite set of values of $\\Log\\Log(1 + i\\sqrt{3})$, indicating which of them are values of $\\log\\Log(1 + i \\sqrt{3})$ and which of $\\Log\\log(1 + i\\sqrt{3})$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xciii/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xciii", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "401", "location": "Exercise XCIII, problem 2", "problem_latex": "The real and imaginary parts of any value of~$\\Log z$ are both continuous\nfunctions of $x$~and~$y$, except for $x = 0$, $y = 0$.", "markdown": "The real and imaginary parts of any value of $\\Log z$ are both continuous functions of $x$ and $y$, except for $x = 0$, $y = 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xciii/3", "set": "hardy-course-of-pure-mathematics-1921/ex-xciii", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "401", "location": "Exercise XCIII, problem 3", "problem_latex": "\\Topic{The functional equation satisfied by~$\\Log z$.} The function~$\\Log z$\nsatisfies the equation\n\\[\n\\Log z_{1} z_{2} = \\Log z_{1} + \\Log z_{2},\n\\Tag{(1)}\n\\]\nin the sense that \\emph{every} value of either side of this equation is \\emph{one} of the values\nof the other side. This follows at once by putting\n\\[\nz_{1} = r_{1}(\\cos\\theta_{1} + i\\sin\\theta_{1}),\\quad\nz_{2} = r_{2}(\\cos\\theta_{2} + i\\sin\\theta_{2}),\n\\]\nand applying the formula of \\PageRef{p.}{401}. It is however not true that\n\\[\n\\log z_{1}z_{2} = \\log z_{1} + \\log z_{2}\n\\Tag{(2)}\n\\]\nin all circumstances. If, \\eg,\n\\[\nz_{1} = z_{2} = \\tfrac{1}{2}(-1 + i\\sqrt{3})\n = \\cos \\tfrac{2}{3}\\pi + i \\sin \\tfrac{2}{3}\\pi,\n\\]\nthen $\\log z_{1} = \\log z_{2} = \\frac{2}{3}\\pi i$, and $\\log z_{1} + \\log z_{2} = \\frac{4}{3}\\pi i$, which is one of the values of\n$\\Log z_{1}z_{2}$, but not the principal value. In fact $\\log z_{1}z_{2} = -\\frac{2}{3}\\pi i$.\n\nAn equation such as~\\Eq{(1)}, in which every value of either side is a value\nof the other, we shall call a \\emph{complete} equation, or an equation which is\n\\emph{completely true}.", "markdown": "**functional equation satisfied by $\\Log z$.** The function $\\Log z$ satisfies the equation z_1 z_2 = z_1 + z_2, (1) in the sense that *every* value of either side of this equation is *one* of the values of the other side. This follows at once by putting z_1 = r_1(_1 + i_1),0pt minus 3ptz_2 = r_2(_2 + i_2), and applying the formula of p.401. It is however not true that z_1z_2 = z_1 + z_2 (2) in all circumstances. If, *e.g.*, z_1 = z_2 = 12(-1 + i3) = 23+ i 23, then $\\log z_{1} = \\log z_{2} = \\frac{2}{3}\\pi i$, and $\\log z_{1} + \\log z_{2} = \\frac{4}{3}\\pi i$, which is one of the values of $\\Log z_{1}z_{2}$, but not the principal value. In fact $\\log z_{1}z_{2} = -\\frac{2}{3}\\pi i$. An equation such as (1), in which every value of either side is a value of the other, we shall call a *complete* equation, or an equation which is *completely true*.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xciii/4", "set": "hardy-course-of-pure-mathematics-1921/ex-xciii", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "401", "location": "Exercise XCIII, problem 4", "problem_latex": "The equation $\\Log z^{m} = m\\Log z$, where $m$~is an integer, is not completely\ntrue: every value of the right-hand side is a value of the left-hand side, but\nthe converse is not true.", "markdown": "The equation $\\Log z^{m} = m\\Log z$, where $m$ is an integer, is not completely true: every value of the right-hand side is a value of the left-hand side, but the converse is not true.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xciii/5", "set": "hardy-course-of-pure-mathematics-1921/ex-xciii", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "401", "location": "Exercise XCIII, problem 5", "problem_latex": "The equation $\\Log (1/z) = -\\Log z$ is completely true. It is also true\nthat $\\log (1/z) = -\\log z$, except when $z$~is real and negative.", "markdown": "The equation $\\Log (1/z) = -\\Log z$ is completely true. It is also true that $\\log (1/z) = -\\log z$, except when $z$ is real and negative.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xciii/6", "set": "hardy-course-of-pure-mathematics-1921/ex-xciii", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "401", "location": "Exercise XCIII, problem 6", "problem_latex": "The equation\n\\[\n\\log \\left(\\frac{z - a}{z - b}\\right) = \\log (z - a) - \\log (z - b)\n\\]\nis true if $z$~lies outside the region bounded by the line joining the points $z = a$,\n$z = b$, and lines through these points parallel to~$OX$ and extending to infinity\nin the negative direction.", "markdown": "The equation (z - az - b) = (z - a) - (z - b) is true if $z$ lies outside the region bounded by the line joining the points $z = a$, $z = b$, and lines through these points parallel to $OX$ and extending to infinity in the negative direction.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xciii/7", "set": "hardy-course-of-pure-mathematics-1921/ex-xciii", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "401", "location": "Exercise XCIII, problem 7", "problem_latex": "The equation\n\\[\n\\log \\left(\\frac{a - z}{b - z}\\right)\n = \\log \\left(1 - \\frac{a}{z}\\right) - \\log \\left(1 - \\frac{b}{z}\\right)\n\\]\nis true if $z$~lies outside the triangle formed by the three points $O$,~$a$,~$b$.", "markdown": "The equation (a - zb - z) = (1 - az) - (1 - bz) is true if $z$ lies outside the triangle formed by the three points $O$, $a$, $b$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xciii/8", "set": "hardy-course-of-pure-mathematics-1921/ex-xciii", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "401", "location": "Exercise XCIII, problem 8", "problem_latex": "Draw the graph of the function $\\Imag(\\Log x)$ of the real variable~$x$. [The\ngraph consists of the positive halves of the lines $y = 2k\\pi$ and the negative\nhalves of the lines $y = (2k + 1)\\pi$.]", "markdown": "Draw the graph of the function $\\Imag(\\Log x)$ of the real variable $x$. [The graph consists of the positive halves of the lines $y = 2k\\pi$ and the negative halves of the lines $y = (2k + 1)\\pi$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xciii/9", "set": "hardy-course-of-pure-mathematics-1921/ex-xciii", "number": 9, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "401", "location": "Exercise XCIII, problem 9", "problem_latex": "The function~$f(x)$ of the real variable~$x$, defined by\n\\[\n\\pi f(x) = p\\pi + (q - p)\\Imag(\\log x),\n\\]\nis equal to~$p$ when $x$~is positive and to~$q$ when $x$~is negative.", "markdown": "The function $f(x)$ of the real variable $x$, defined by f(x) = p+ (q - p)(x), is equal to $p$ when $x$ is positive and to $q$ when $x$ is negative.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xciv/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xciv", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "407", "location": "Exercise XCIV, problem 1", "problem_latex": "Find all the values of~$i^{i}$. [By definition\n\\[\ni^{i} = \\exp (i\\Log i).\n\\]\nBut\n\\[\ni = \\cos \\tfrac{1}{2}\\pi + i\\sin \\tfrac{1}{2}\\pi,\\quad\n\\Log i = (2k + \\tfrac{1}{2})\\pi i,\n\\]\nwhere $k$~is any integer. Hence\n\\[\ni^{i} = \\exp\\{-(2k + \\tfrac{1}{2})\\pi\\} = e^{-(2k + \\frac{1}{2})\\pi}.\n\\]\nAll the values of~$i^{i}$ are therefore real and positive.]", "markdown": "Find all the values of $i^{i}$. [By definition i^i = (ii). But i = 12+ i12,0pt minus 3pti = (2k + 12)i, where $k$ is any integer. Hence i^i = -(2k + 12) = e^-(2k + 12). All the values of $i^{i}$ are therefore real and positive.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.const", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xciv/10", "set": "hardy-course-of-pure-mathematics-1921/ex-xciv", "number": 10, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "407", "location": "Exercise XCIV, problem 10", "problem_latex": "For what values of~$\\zeta$ is (\\ia)~any value (\\ib)~the principal value of~$e^{\\zeta}$\n(i)~real (ii)~purely imaginary (iii)~of unit modulus?", "markdown": "For what values of $\\zeta$ is (*a*) any value (*b*) the principal value of $e^{\\zeta}$ (i) real (ii) purely imaginary (iii) of unit modulus?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xciv/11", "set": "hardy-course-of-pure-mathematics-1921/ex-xciv", "number": 11, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "407", "location": "Exercise XCIV, problem 11", "problem_latex": "The necessary and sufficient conditions that all the values of~$a^{\\zeta}$ should\nbe real are that $2\\xi$~and~$\\{\\eta\\log |a| + \\xi\\am a\\}/\\pi$, where $\\am a$~denotes any value of\nthe amplitude, should both be integral. What are the corresponding conditions\nthat all the values should be of unit modulus?", "markdown": "The necessary and sufficient conditions that all the values of $a^{\\zeta}$ should be real are that $2\\xi$ and $\\{\\eta\\log |a| + \\xi\\am a\\}/\\pi$, where $\\am a$ denotes any value of the amplitude, should both be integral. What are the corresponding conditions that all the values should be of unit modulus?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xciv/12", "set": "hardy-course-of-pure-mathematics-1921/ex-xciv", "number": 12, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "407", "location": "Exercise XCIV, problem 12", "problem_latex": "The general value of~$|x^{i} + x^{-i}|$, where $x > 0$, is\n\\[\ne^{-(m-n)\\pi} \\sqrtbr{2\\{\\cosh 2(m + n)\\pi + \\cos(2\\log x)\\}}.\n\\]", "markdown": "The general value of $|x^{i} + x^{-i}|$, where $x > 0$, is e^-(m-n) 22(m + n)+ (2x).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.hyp", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xciv/13", "set": "hardy-course-of-pure-mathematics-1921/ex-xciv", "number": 13, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "407", "location": "Exercise XCIV, problem 13", "problem_latex": "Explain the fallacy in the following argument: since $e^{2m\\pi i} = e^{2n\\pi i} = 1$,\nwhere $m$~and~$n$ are any integers, therefore, raising each side to the power~$i$\nwe obtain $e^{-2m\\pi} = e^{-2n\\pi}$.", "markdown": "Explain the fallacy in the following argument: since $e^{2m\\pi i} = e^{2n\\pi i} = 1$, where $m$ and $n$ are any integers, therefore, raising each side to the power $i$ we obtain $e^{-2m\\pi} = e^{-2n\\pi}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xciv/14", "set": "hardy-course-of-pure-mathematics-1921/ex-xciv", "number": 14, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "407", "location": "Exercise XCIV, problem 14", "problem_latex": "In what circumstances are any of the values of~$x^{x}$, where $x$~is real,\nthemselves real? [If $x > 0$ then\n\\[\nx^{x} = \\exp (x\\Log x) = \\exp (x\\log x) \\Cis 2m\\pi x,\n\\]\nthe first factor being real. The principal value, for which $m = 0$, is always\nreal.\n\nIf $x$~is a rational fraction~$p/(2q + 1)$, or is irrational, then there is no other\nreal value. But if $x$~is of the form~$p/2q$, then there is one other real value,\nviz.\\ $-\\exp (x\\log x)$, given by $m = q$.\n\nIf $x = -\\xi < 0$ then\n\\[\nx^{x} = \\exp \\{-\\xi\\Log (-\\xi)\\}\n = \\exp (-\\xi\\log \\xi) \\Cis\\{-(2m + 1)\\pi\\xi\\}.\n\\]\nThe only case in which any value is real is that in which $\\xi = p/(2q + 1)$, when\n$m = q$ gives the real value\n\\[\n\\exp (-\\xi\\log \\xi) \\Cis (-p\\pi) = (-1)^{p} \\xi^{-\\xi}.\n\\]\nThe cases of reality are illustrated by the examples\n\\[\n(\\tfrac{1}{3})^{1/3} = \\sqrt[3]{\\tfrac{1}{3}},\\quad\n(\\tfrac{1}{2})^{\\frac{1}{2}} = ±\\sqrt{\\tfrac{1}{2}},\\quad\n(-\\tfrac{2}{3})^{-\\frac{2}{3}} = \\sqrt[3]{\\tfrac{9}{4}},\\quad\n(-\\tfrac{1}{3})^{-\\frac{1}{3}} = -\\sqrt[3]{3}.]\n\\]", "markdown": "In what circumstances are any of the values of $x^{x}$, where $x$ is real, themselves real? [If $x > 0$ then x^x = (xx) = (xx) 2mx, the first factor being real. The principal value, for which $m = 0$, is always real. If $x$ is a rational fraction $p/(2q + 1)$, or is irrational, then there is no other real value. But if $x$ is of the form $p/2q$, then there is one other real value, viz. $-\\exp (x\\log x)$, given by $m = q$. If $x = -\\xi < 0$ then x^x = -(-) = (-) -(2m + 1). The only case in which any value is real is that in which $\\xi = p/(2q + 1)$, when $m = q$ gives the real value (-) (-p) = (-1)^p ^-. The cases of reality are illustrated by the examples (13)^1/3 = [3]13,0pt minus 3pt(12)^12 = ±12,0pt minus 3pt(-23)^-23 = [3]94,0pt minus 3pt(-13)^-13 = -[3]3.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xciv/15", "set": "hardy-course-of-pure-mathematics-1921/ex-xciv", "number": 15, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "407", "location": "Exercise XCIV, problem 15", "problem_latex": "\\Topic{Logarithms to any base.} We may define $\\zeta = \\Log_{a} z$ in two different\nways. We may say (i)~that $\\zeta = \\Log_{a} z$ if the \\emph{principal} value of~$a^{\\zeta}$ is equal to~$z$;\nor we may say (ii)~that $\\zeta = \\Log_{a} z$ if \\emph{any} value of~$a^{\\zeta}$ is equal to~$z$.\n\nThus if $a = e$ then $\\zeta = \\Log_{e} z$, according to the first definition, if the\nprincipal value of~$e^{\\zeta}$ is equal to~$z$, or if $\\exp \\zeta = z$; and so $\\Log_{e} z$~is identical\nwith~$\\Log z$. But, according to the second definition, $\\zeta = \\Log_{e} z$ if\n\\[\ne^{\\zeta} = \\exp (\\zeta\\Log e) = z,\\quad\n\\zeta\\Log e = \\Log z,\n\\]\nor $\\zeta = (\\Log z)/(\\Log e)$, any values of the logarithms being taken. Thus\n\\[\n\\zeta = \\Log_{e} z = \\frac{\\log |z| + (\\am z + 2m\\pi)i}{1 + 2n\\pi i},\n\\]\nso that $\\zeta$~is a doubly infinitely many-valued function of~$z$. And generally,\naccording to this definition, $\\Log_{a} z = (\\Log z)/(\\Log a)$.", "markdown": "**to any base.** We may define $\\zeta = \\Log_{a} z$ in two different ways. We may say (i) that $\\zeta = \\Log_{a} z$ if the *principal* value of $a^{\\zeta}$ is equal to $z$; or we may say (ii) that $\\zeta = \\Log_{a} z$ if *any* value of $a^{\\zeta}$ is equal to $z$. Thus if $a = e$ then $\\zeta = \\Log_{e} z$, according to the first definition, if the principal value of $e^{\\zeta}$ is equal to $z$, or if $\\exp \\zeta = z$; and so $\\Log_{e} z$ is identical with $\\Log z$. But, according to the second definition, $\\zeta = \\Log_{e} z$ if e^ = (e) = z,0pt minus 3pte = z, or $\\zeta = (\\Log z)/(\\Log e)$, any values of the logarithms being taken. Thus = _e z = |z| + (z + 2m)i1 + 2ni, so that $\\zeta$ is a doubly infinitely many-valued function of $z$. And generally, according to this definition, $\\Log_{a} z = (\\Log z)/(\\Log a)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xciv/16", "set": "hardy-course-of-pure-mathematics-1921/ex-xciv", "number": 16, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "407", "location": "Exercise XCIV, problem 16", "problem_latex": "$\\Log_{e} 1 = 2m\\pi i/(1 + 2n\\pi i)$, $\\Log_{e}(-1) = (2m + 1)\\pi i/(1 + 2n\\pi i)$, where $m$~and~$n$\nare any integers.", "markdown": "$\\Log_{e} 1 = 2m\\pi i/(1 + 2n\\pi i)$, $\\Log_{e}(-1) = (2m + 1)\\pi i/(1 + 2n\\pi i)$, where $m$ and $n$ are any integers.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xciv/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xciv", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "407", "location": "Exercise XCIV, problem 2", "problem_latex": "Find all the values of $(1 + i)^{i}$, $i^{1+i}$, $(1 + i)^{1+i}$.", "markdown": "Find all the values of $(1 + i)^{i}$, $i^{1+i}$, $(1 + i)^{1+i}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xciv/3", "set": "hardy-course-of-pure-mathematics-1921/ex-xciv", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "407", "location": "Exercise XCIV, problem 3", "problem_latex": "The values of~$a^{\\zeta}$, when plotted in the Argand diagram, are the vertices\nof an equiangular polygon inscribed in an equiangular spiral whose angle is\nindependent of~$a$. \\MathTrip{1899.}\n\n[If $a^{\\zeta} = r(\\cos\\theta + i\\sin\\theta)$ we have\n\\[\nr = e^{\\xi\\log \\sigma - \\eta(\\psi + 2m\\pi)},\\quad\n\\theta = \\eta\\log \\sigma + \\xi(\\psi + 2m\\pi);\n\\]\nand all the points lie on the spiral $r = \\sigma^{(\\xi^{2} + \\eta^{2})/\\xi} e^{-\\eta \\theta/\\xi}$.]", "markdown": "The values of $a^{\\zeta}$, when plotted in the Argand diagram, are the vertices of an equiangular polygon inscribed in an equiangular spiral whose angle is independent of $a$. % [0]% (*Math. Trip.* 1899.)% [1]% [If $a^{\\zeta} = r(\\cos\\theta + i\\sin\\theta)$ we have r = e^- (+ 2m),0pt minus 3pt= + (+ 2m); and all the points lie on the spiral $r = \\sigma^{(\\xi^{2} + \\eta^{2})/\\xi} e^{-\\eta \\theta/\\xi}$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xciv/4", "set": "hardy-course-of-pure-mathematics-1921/ex-xciv", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "407", "location": "Exercise XCIV, problem 4", "problem_latex": "\\Topic{The function~$e^{\\zeta}$.} If we write~$e$ for~$a$ in the general formula, so that\n$\\log \\sigma = 1$, $\\psi = 0$, we obtain\n\\[\ne^{\\zeta} = e^{\\xi-2m\\pi\\eta} \\{\\cos(\\eta + 2m\\pi\\xi) + i\\sin(\\eta + 2m\\pi\\xi)\\}.\n\\]\nThe principal value of~$e^{\\zeta}$ is $e^{\\xi}(\\cos\\eta + i\\sin\\eta)$, which is equal to~$\\exp \\zeta$ (\\SecNo[§]{223}).\nIn particular, if $\\zeta$~is real, so that $\\eta = 0$, we obtain\n\\[\ne^{\\zeta} (\\cos 2m\\pi\\zeta + i\\sin 2m\\pi\\zeta)\n\\]\nas the general and $e^{\\zeta}$~as the principal value, $e^{\\zeta}$~denoting here the positive\nvalue of the exponential defined in \\okrickRef{Ch.}{IX}\\@.", "markdown": "**function $e^{\\zeta}$.** If we write $e$ for $a$ in the general formula, so that $\\log \\sigma = 1$, $\\psi = 0$, we obtain e^ = e^-2m (+ 2m) + i(+ 2m). The principal value of $e^{\\zeta}$ is $e^{\\xi}(\\cos\\eta + i\\sin\\eta)$, which is equal to $\\exp \\zeta$ ([§]223). In particular, if $\\zeta$ is real, so that $\\eta = 0$, we obtain e^ (2m+ i2m) as the general and $e^{\\zeta}$ as the principal value, $e^{\\zeta}$ denoting here the positive value of the exponential defined in Ch.IX.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.const", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xciv/5", "set": "hardy-course-of-pure-mathematics-1921/ex-xciv", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "407", "location": "Exercise XCIV, problem 5", "problem_latex": "Show that $\\Log e^{\\zeta} = (1 + 2m\\pi i)\\zeta + 2n\\pi i$, where $m$~and~$n$ are any integers,\nand that in general $\\Log a^{\\zeta}$~has a double infinity of values.", "markdown": "Show that $\\Log e^{\\zeta} = (1 + 2m\\pi i)\\zeta + 2n\\pi i$, where $m$ and $n$ are any integers, and that in general $\\Log a^{\\zeta}$ has a double infinity of values.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xciv/6", "set": "hardy-course-of-pure-mathematics-1921/ex-xciv", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "407", "location": "Exercise XCIV, problem 6", "problem_latex": "The equation $1/a^{\\zeta} = a^{-\\zeta}$ is completely true (\\Ex{xciii}.~3): it is also true\nof the principal values.", "markdown": "The equation $1/a^{\\zeta} = a^{-\\zeta}$ is completely true (% [examples:xciii]Ex. xciii%. 3): it is also true of the principal values.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xciv/7", "set": "hardy-course-of-pure-mathematics-1921/ex-xciv", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "407", "location": "Exercise XCIV, problem 7", "problem_latex": "The equation $a^{\\zeta} × b^{\\zeta} = (ab)^{\\zeta}$ is completely true but not always true of\nthe principal values.", "markdown": "The equation $a^{\\zeta} × b^{\\zeta} = (ab)^{\\zeta}$ is completely true but not always true of the principal values.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xciv/8", "set": "hardy-course-of-pure-mathematics-1921/ex-xciv", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "407", "location": "Exercise XCIV, problem 8", "problem_latex": "The equation $a^{\\zeta} × a^{\\zeta'} = a^{\\zeta+\\zeta'}$ is not completely true, but is true of the\nprincipal values. [Every value of the right-hand side is a value of the left-hand\nside, but the general value of $a^{\\zeta} × a^{\\zeta'}$, viz.\n\\[\n\\exp \\{\\zeta(\\log a + 2m\\pi i) + \\zeta'(\\log a + 2n\\pi i)\\},\n\\]\nis not as a rule a value of~$a^{\\zeta+\\zeta'}$ unless $m = n$.]", "markdown": "The equation $a^{\\zeta} × a^{\\zeta'} = a^{\\zeta+\\zeta'}$ is not completely true, but is true of the principal values. [Every value of the right-hand side is a value of the left-hand side, but the general value of $a^{\\zeta} × a^{\\zeta'}$, viz. (a + 2mi) + ’(a + 2ni), is not as a rule a value of $a^{\\zeta+\\zeta'}$ unless $m = n$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xciv/9", "set": "hardy-course-of-pure-mathematics-1921/ex-xciv", "number": 9, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "407", "location": "Exercise XCIV, problem 9", "problem_latex": "What are the corresponding results as regards the equations\n\\[\n\\Log a^{\\zeta} = \\zeta\\Log a,\\quad\n(a^{\\zeta})^{\\zeta'} = (a^{\\zeta'})^{\\zeta} = a^{\\zeta\\zeta'}?\n\\]", "markdown": "What are the corresponding results as regards the equations a^ = a,0pt minus 3pt(a^)^’ = (a^’)^ = a^’?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcv/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xcv", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "411", "location": "Exercise XCV, problem 1", "problem_latex": "Determine the values of~$\\zeta$ for which $\\cos\\zeta$ and~$\\sin\\zeta$\nare (i)~real (ii)~purely imaginary. [For example $\\cos\\zeta$~is real when $\\eta = 0$ or\nwhen $\\xi$~is any multiple of~$\\pi$.]", "markdown": "Determine the values of $\\zeta$ for which $\\cos\\zeta$ and $\\sin\\zeta$ are (i) real (ii) purely imaginary. [For example $\\cos\\zeta$ is real when $\\eta = 0$ or when $\\xi$ is any multiple of $\\pi$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcv/10", "set": "hardy-course-of-pure-mathematics-1921/ex-xcv", "number": 10, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "411", "location": "Exercise XCV, problem 10", "problem_latex": "\\Topic{Solution of $\\cos\\zeta = \\alpha + i\\beta$, where $\\beta \\neq 0$.} We may suppose $\\beta > 0$,\nsince the results when $\\beta < 0$ may be deduced by merely changing the sign of~$i$.\nIn this case\n\\[\n\\cos\\xi \\cosh\\eta = \\alpha,\\quad\n\\sin\\xi \\sinh\\eta = -\\beta,\n\\Tag{(1)}\n\\]\nand\n\\[\n(\\alpha/\\cosh\\eta)^{2} + (\\beta/\\sinh\\eta)^{2} = 1.\n\\]\n\nIf we put $\\cosh^{2} \\eta = x$ we find that\n\\[\nx^{2} - (1 + \\alpha^{2} + \\beta^{2})x + \\alpha^{2} = 0\n\\]\nor $x = (A_{1} ± A_{2})^{2}$, where\n\\[\nA_{1} = \\tfrac{1}{2}\\sqrtb{(\\alpha + 1)^{2} + \\beta^{2}},\\quad\nA_{2} = \\tfrac{1}{2}\\sqrtb{(\\alpha - 1)^{2} + \\beta^{2}}.\n\\]\nSuppose $\\alpha > 0$. Then $A_{1} > A_{2} > 0$ and $\\cosh\\eta = A_{1} ± A_{2}$. Also\n\\[\n\\cos\\xi = \\alpha/(\\cosh\\eta) = A_{1} \\mp A_{2},\n\\]\nand since $\\cosh\\eta > \\cos\\xi$ we must take\n\\[\n\\cosh\\eta = A_{1} + A_{2},\\quad\n\\cos\\xi = A_{1} - A_{2}.\n\\]\nThe general solutions of these equations are\n\\[\n\\xi = 2k\\pi ± \\arccos M,\\quad\n\\eta = ±\\log \\{L + \\sqrtp{L^{2} - 1}\\},\n\\Tag{(2)}\n\\]\nwhere $L = A_{1} + A_{2}$, $M = A_{1} - A_{2}$, and $\\arccos M$ lies between $0$ and~$\\frac{1}{2}\\pi$.\n\nThe values of $\\eta$ and~$\\xi$ thus found above include, however, the solutions of\nthe equations\n\\[\n\\cos\\xi \\cosh\\eta = \\alpha,\\quad\n\\sin\\xi \\sinh\\eta = \\beta,\n\\Tag{(3)}\n\\]\nas well as those of the equations~\\Eq{(1)}, since we have only used the second of\nthe latter equations after squaring it. To distinguish the two sets of\nsolutions we observe that the sign of~$\\sin\\xi$ is the same as the ambiguous sign\nin the first of the equations~\\Eq{(2)}, and the sign of~$\\sinh\\eta$ is the same as the\nambiguous sign in the second. Since $\\beta > 0$, these two signs must be different.\nHence the general solution required is\n\\[\n\\zeta = 2k\\pi ± [\\arccos M - i\\log \\{L + \\sqrtp{L^{2} - 1}\\}].\n\\]", "markdown": "**of $\\cos\\zeta = \\alpha + i\\beta$, where $\\beta \\neq 0$.** We may suppose $\\beta > 0$, since the results when $\\beta < 0$ may be deduced by merely changing the sign of $i$. In this case = ,0pt minus 3pt= -, (1) and (/)^2 + (/)^2 = 1. If we put $\\cosh^{2} \\eta = x$ we find that x^2 - (1 + ^2 + ^2)x + ^2 = 0 or $x = (A_{1} ± A_{2})^{2}$, where A_1 = 12(+ 1)^2 + ^2,0pt minus 3ptA_2 = 12(- 1)^2 + ^2. Suppose $\\alpha > 0$. Then $A_{1} > A_{2} > 0$ and $\\cosh\\eta = A_{1} ± A_{2}$. Also = /() = A_1 A_2, and since $\\cosh\\eta > \\cos\\xi$ we must take = A_1 + A_2,0pt minus 3pt= A_1 - A_2. The general solutions of these equations are = 2k± M,0pt minus 3pt= ±L + L^2 - 1, (2) where $L = A_{1} + A_{2}$, $M = A_{1} - A_{2}$, and $\\arccos M$ lies between $0$ and $\\frac{1}{2}\\pi$. The values of $\\eta$ and $\\xi$ thus found above include, however, the solutions of the equations = ,0pt minus 3pt= , (3) as well as those of the equations (1), since we have only used the second of the latter equations after squaring it. To distinguish the two sets of solutions we observe that the sign of $\\sin\\xi$ is the same as the ambiguous sign in the first of the equations (2), and the sign of $\\sinh\\eta$ is the same as the ambiguous sign in the second. Since $\\beta > 0$, these two signs must be different. Hence the general solution required is = 2k± [M - iL + L^2 - 1].", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "solve", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "Eq(cos(zeta), alpha + I*beta)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "solve: Eq(cos(x), a*c + b)" ], "shape": [ "solve: Eq(cos(x), a*c + b)" ], "same_problem_in": [], "needs": [ "cas.solve.complex", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcv/11", "set": "hardy-course-of-pure-mathematics-1921/ex-xcv", "number": 11, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "411", "location": "Exercise XCV, problem 11", "problem_latex": "Work out the cases in which $\\alpha < 0$ and $\\alpha = 0$ in the same way.", "markdown": "Work out the cases in which $\\alpha < 0$ and $\\alpha = 0$ in the same way.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.solve.complex", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcv/12", "set": "hardy-course-of-pure-mathematics-1921/ex-xcv", "number": 12, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "411", "location": "Exercise XCV, problem 12", "problem_latex": "If $\\beta = 0$ then $L = \\frac{1}{2}|\\alpha + 1| + \\frac{1}{2}|\\alpha - 1|$ and $M = \\frac{1}{2}|\\alpha + 1| - \\frac{1}{2}|\\alpha - 1|$.\nVerify that the results thus obtained agree with those of Ex.~8.", "markdown": "If $\\beta = 0$ then $L = \\frac{1}{2}|\\alpha + 1| + \\frac{1}{2}|\\alpha - 1|$ and $M = \\frac{1}{2}|\\alpha + 1| - \\frac{1}{2}|\\alpha - 1|$. Verify that the results thus obtained agree with those of Ex. 8.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.log", "core.trig", "core.units" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcv/13", "set": "hardy-course-of-pure-mathematics-1921/ex-xcv", "number": 13, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "411", "location": "Exercise XCV, problem 13", "problem_latex": "{\\Loosen Show that if $\\alpha$~and~$\\beta$ are positive then the general solution of\n$\\sin\\zeta = \\alpha + i\\beta$ is}\n\\[\n\\zeta = k\\pi +(-1)^{k} [\\arcsin M + i\\log \\{L + \\sqrtp{L^{2} - 1}\\}],\n\\]\nwhere $\\arcsin M$ lies between $0$ and~$\\frac{1}{2}\\pi$. Obtain the solution in the other\npossible cases.", "markdown": "0.375em plus 0.75em minus 0.25emShow that if $\\alpha$ and $\\beta$ are positive then the general solution of $\\sin\\zeta = \\alpha + i\\beta$ is = k+(-1)^k [M + iL + L^2 - 1], where $\\arcsin M$ lies between $0$ and $\\frac{1}{2}\\pi$. Obtain the solution in the other possible cases.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.solve.complex", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcv/14", "set": "hardy-course-of-pure-mathematics-1921/ex-xcv", "number": 14, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "411", "location": "Exercise XCV, problem 14", "problem_latex": "Solve $\\tan\\zeta = \\alpha$, where $\\alpha$~is real. [All the roots are real.]", "markdown": "Solve $\\tan\\zeta = \\alpha$, where $\\alpha$ is real. [All the roots are real.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "solve", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "Eq(tan(zeta), alpha)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "solve: Eq(tan(x), a)" ], "shape": [ "solve: Eq(tan(x), a)" ], "same_problem_in": [], "needs": [ "cas.solve.complex", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcv/15", "set": "hardy-course-of-pure-mathematics-1921/ex-xcv", "number": 15, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "411", "location": "Exercise XCV, problem 15", "problem_latex": "Show that the general solution of $\\tan \\zeta = \\alpha + i\\beta$, where $\\beta \\neq 0$, is\n\\[\n\\zeta = k\\pi + \\tfrac{1}{2}\\theta + \\tfrac{1}{4} i\\log\\left\\{\n \\frac{\\alpha^{2} + (1 + \\beta)^{2}}\n {\\alpha^{2} + (1 - \\beta)^{2}}\n \\right\\},\n\\]\nwhere $\\theta$~is the numerically least angle such that\n\\[\n\\cos \\theta : \\sin \\theta : 1 ::\n1 - \\alpha^{2} - \\beta^{2} : 2\\alpha :\n \\sqrtb{(1 - \\alpha^{2} - \\beta^{2})^{2} + 4\\alpha^{2}}.\n\\]", "markdown": "Show that the general solution of $\\tan \\zeta = \\alpha + i\\beta$, where $\\beta \\neq 0$, is = k+ 12+ 14 i ^2 + (1 + )^2 ^2 + (1 - )^2 , where $\\theta$ is the numerically least angle such that : : 1 :: 1 - ^2 - ^2 : 2: (1 - ^2 - ^2)^2 + 4^2.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.solve.complex", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcv/16", "set": "hardy-course-of-pure-mathematics-1921/ex-xcv", "number": 16, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "411", "location": "Exercise XCV, problem 16", "problem_latex": "If $z = \\xi\\exp(\\frac{1}{4}\\pi i)$, where $\\xi$~is real, and $c$~is also real, then the modulus\nof $\\cos 2\\pi z - \\cos 2\\pi c$ is\n\\[\n\\begin{aligned}[b]\n \\surd[\\tfrac{1}{2}\\{1 + \\cos 4\\pi c + \\cos(2\\pi\\xi\\sqrt{2}) &+ \\cosh(2\\pi\\xi\\sqrt{2}) \\\\\n &- 4\\cos 2\\pi c \\cos(\\pi\\xi\\sqrt{2}) \\cosh(\\pi\\xi\\sqrt{2})\\}]\n\\end{aligned}.\n\\]", "markdown": "If $z = \\xi\\exp(\\frac{1}{4}\\pi i)$, where $\\xi$ is real, and $c$ is also real, then the modulus of $\\cos 2\\pi z - \\cos 2\\pi c$ is aligned[b] [121 + 4c + (22) &+ (22) &- 42c (2) (2)] aligned.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.complex", "core.hyp", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcv/17", "set": "hardy-course-of-pure-mathematics-1921/ex-xcv", "number": 17, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "411", "location": "Exercise XCV, problem 17", "problem_latex": "Prove that\n\\begin{gather*}\n|\\exp \\exp(\\xi + i\\eta)| = \\exp(\\exp\\xi \\cos\\eta), \\\\\n\\begin{aligned}\n\\Real \\{\\cos\\cos(\\xi + i\\eta)\\}\n &= \\cos(\\cos\\xi \\cosh\\eta) \\cosh(\\sin\\xi \\sinh\\eta),\\\\\n\\Imag \\{\\sin\\sin(\\xi + i\\eta)\\}\n &= \\cos(\\sin\\xi \\cosh\\eta) \\sinh(\\cos\\xi \\sinh\\eta).\n\\end{aligned}\n\\end{gather*}", "markdown": "Prove that gather* |(+ i)| = (), aligned (+ i) &= () (), (+ i) &= () (). aligned gather*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.hyp", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcv/18", "set": "hardy-course-of-pure-mathematics-1921/ex-xcv", "number": 18, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "411", "location": "Exercise XCV, problem 18", "problem_latex": "Prove that $|\\exp\\zeta|$~tends to~$\\infty$ if $\\zeta$~moves away towards infinity along\nany straight line through the origin making an angle less than~$\\frac{1}{2}\\pi$ with~$OX$,\nand to~$0$ if $\\zeta$~moves away along a similar line making an angle greater than~$\\frac{1}{2}\\pi$\nwith~$OX$.", "markdown": "Prove that $|\\exp\\zeta|$ tends to $\\infty$ if $\\zeta$ moves away towards infinity along any straight line through the origin making an angle less than $\\frac{1}{2}\\pi$ with $OX$, and to $0$ if $\\zeta$ moves away along a similar line making an angle greater than $\\frac{1}{2}\\pi$ with $OX$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcv/19", "set": "hardy-course-of-pure-mathematics-1921/ex-xcv", "number": 19, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "411", "location": "Exercise XCV, problem 19", "problem_latex": "Prove that $|\\cos\\zeta|$ and $|\\sin\\zeta|$ tend to~$\\infty$ if $\\zeta$~moves away towards\ninfinity along any straight line through the origin other than either half of\nthe real axis.", "markdown": "Prove that $|\\cos\\zeta|$ and $|\\sin\\zeta|$ tend to $\\infty$ if $\\zeta$ moves away towards infinity along any straight line through the origin other than either half of the real axis.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcv/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xcv", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "411", "location": "Exercise XCV, problem 2", "problem_latex": "\\begin{alignat*}{2}\n|\\cos (\\xi + i\\eta)|\n &= \\sqrtp{\\cos^{2} \\xi + \\sinh^{2} \\eta}\n &&= \\sqrtb{\\tfrac{1}{2} (\\cosh 2\\eta + \\cos 2\\xi)}, \\\\\n|\\sin (\\xi + i\\eta)|\n &= \\sqrtp{\\sin^{2} \\xi + \\sinh^{2} \\eta}\n &&= \\sqrtb{\\tfrac{1}{2} (\\cosh 2\\eta - \\cos 2\\xi)}.\n\\end{alignat*}\n\n[Use (\\eg)\\ the equation $|\\cos(\\xi + i\\eta)| = \\sqrtb{\\cos(\\xi + i\\eta) \\cos(\\xi - i\\eta)}$.]", "markdown": "alignat*2 |(+ i)| &= ^2 + ^2 &&= 12 (2+ 2), |(+ i)| &= ^2 + ^2 &&= 12 (2- 2). alignat* [Use (*e.g.*) the equation $|\\cos(\\xi + i\\eta)| = \\sqrtb{\\cos(\\xi + i\\eta) \\cos(\\xi - i\\eta)}$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.hyp", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcv/20", "set": "hardy-course-of-pure-mathematics-1921/ex-xcv", "number": 20, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "411", "location": "Exercise XCV, problem 20", "problem_latex": "Prove that $\\tan\\zeta$ tends to~$-i$ or to~$i$ if $\\zeta$~moves away to infinity\nalong the straight line of Ex.~19, to $-i$~if the line lies above the real axis and\nto~$i$ if it lies below.", "markdown": "Prove that $\\tan\\zeta$ tends to $-i$ or to $i$ if $\\zeta$ moves away to infinity along the straight line of Ex. 19, to $-i$ if the line lies above the real axis and to $i$ if it lies below.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcv/3", "set": "hardy-course-of-pure-mathematics-1921/ex-xcv", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "411", "location": "Exercise XCV, problem 3", "problem_latex": "$\\tan (\\xi + i \\eta)\n = \\dfrac{\\sin 2\\xi + i\\sinh 2\\eta}{\\cosh 2\\eta + \\cos 2\\xi}$,\\quad\n$\\cot (\\xi + i \\eta)\n = \\dfrac{\\sin 2\\xi - i\\sinh 2\\eta}{\\cosh 2\\eta - \\cos 2\\xi}$.\n\n[For example\n\\[\n\\tan (\\xi + i\\eta)\n = \\frac{\\sin (\\xi + i\\eta) \\cos (\\xi - i\\eta)}\n {\\cos (\\xi + i\\eta) \\cos (\\xi - i\\eta)}\n = \\frac{\\sin 2\\xi + \\sin 2i\\eta}{\\cos 2\\xi + \\cos 2i\\eta},\n\\]\nwhich leads at once to the result given.]", "markdown": "$\\tan (\\xi + i \\eta) = \\dfrac{\\sin 2\\xi + i\\sinh 2\\eta}{\\cosh 2\\eta + \\cos 2\\xi}$,0pt minus 3pt$\\cot (\\xi + i \\eta) = \\dfrac{\\sin 2\\xi - i\\sinh 2\\eta}{\\cosh 2\\eta - \\cos 2\\xi}$. [For example (+ i) = (+ i) (- i) (+ i) (- i) = 2+ 2i2+ 2i, which leads at once to the result given.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.hyp", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcv/4", "set": "hardy-course-of-pure-mathematics-1921/ex-xcv", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "411", "location": "Exercise XCV, problem 4", "problem_latex": "\\begin{align*}\n\\sec (\\xi + i \\eta)\n &= \\frac{\\cos\\xi \\cosh\\eta + i\\sin\\xi \\sinh\\eta}\n {\\frac{1}{2} (\\cosh 2\\eta + \\cos 2\\xi)}, \\\\\n\\cosec (\\xi + i \\eta)\n &= \\frac{\\sin\\xi \\cosh\\eta - i\\cos\\xi \\sinh\\eta}\n {\\frac{1}{2} (\\cosh 2\\eta - \\cos 2\\xi)}.\n\\end{align*}", "markdown": "align* (+ i ) &= + i 12 (2+ 2), (+ i ) &= - i 12 (2- 2). align*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.hyp", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcv/5", "set": "hardy-course-of-pure-mathematics-1921/ex-xcv", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "411", "location": "Exercise XCV, problem 5", "problem_latex": "If $|\\cos (\\xi + i\\eta)| = 1$ then $\\sin^{2} \\xi = \\sinh^{2} \\eta$, and if $|\\sin (\\xi + i\\eta)| = 1$ then\n$\\cos^{2} \\xi = \\sinh^{2} \\eta$.", "markdown": "If $|\\cos (\\xi + i\\eta)| = 1$ then $\\sin^{2} \\xi = \\sinh^{2} \\eta$, and if $|\\sin (\\xi + i\\eta)| = 1$ then $\\cos^{2} \\xi = \\sinh^{2} \\eta$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.hyp", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcv/6", "set": "hardy-course-of-pure-mathematics-1921/ex-xcv", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "411", "location": "Exercise XCV, problem 6", "problem_latex": "If $|\\cos (\\xi + i\\eta)| = 1$, then\n\\[\n\\sin \\{\\am \\cos (\\xi + i\\eta)\\} = ±\\sin^{2} \\xi = ±\\sinh^{2} \\eta.\n\\]", "markdown": "If $|\\cos (\\xi + i\\eta)| = 1$, then (+ i) = ±^2 = ±^2 .", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.hyp", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcv/7", "set": "hardy-course-of-pure-mathematics-1921/ex-xcv", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "411", "location": "Exercise XCV, problem 7", "problem_latex": "Prove that $\\Log \\cos (\\xi + i\\eta) = A + iB$, where\n\\[\nA = \\tfrac{1}{2} \\log \\{\\tfrac{1}{2} (\\cosh 2\\eta + \\cos 2\\xi)\\}\n\\]\nand $B$~is any angle such that\n\\[\n\\frac{\\cos B}{\\cos\\xi \\cosh\\eta}\n = -\\frac{\\sin B}{\\sin\\xi \\sinh\\eta}\n = \\frac{1}{\\sqrtb{\\frac{1}{2} (\\cosh 2\\eta + \\cos 2\\xi)}}.\n\\]\nFind a similar formula for $\\Log \\sin (\\xi + i\\eta)$.", "markdown": "Prove that $\\Log \\cos (\\xi + i\\eta) = A + iB$, where A = 12 12 (2+ 2) and $B$ is any angle such that B = -B = 112 (2+ 2). Find a similar formula for $\\Log \\sin (\\xi + i\\eta)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.hyp", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcv/8", "set": "hardy-course-of-pure-mathematics-1921/ex-xcv", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "411", "location": "Exercise XCV, problem 8", "problem_latex": "\\Topic{Solution of the equation $\\cos\\zeta = a$, where $a$~is real.} Putting\n$\\zeta = \\xi + i\\eta$, and equating real and imaginary parts, we obtain\n\\[\n\\cos\\xi \\cosh\\eta = a,\\quad\n\\sin\\xi \\sinh\\eta = 0.\n\\]\nHence either $\\eta = 0$ or $\\xi$~is a multiple of~$\\pi$. If (i)~$\\eta = 0$ then $\\cos\\xi = a$, which is\nimpossible unless $-1 \\leq a \\leq 1$. This hypothesis leads to the solution\n\\[\n\\zeta = 2k\\pi ± \\arccos a,\n\\]\nwhere $\\arccos a$ lies between $0$ and~$\\frac{1}{2}\\pi$. If (ii)~$\\xi = m\\pi$ then $\\cosh\\eta = (-1)^{m}a$, so\nthat either $a \\geq 1$ and $m$~is even, or $a \\leq -1$ and $m$~is odd. If $a = ± 1$ then $\\eta = 0$,\nand we are led back to our first case. If $|a| > 1$ then $\\cosh\\eta = |a|$, and we\nare led to the solutions\n\\begin{alignat*}{4}\n\\zeta &=& 2k &\\pi ± i\\log \\{ &&a + \\sqrt{a^{2} - 1}\\}\\quad &&(a > 1), \\\\\n\\zeta &=&(2k + 1) &\\pi ± i\\log \\{-&&a + \\sqrt{a^{2} - 1}\\}\\quad &&(a < -1).\n\\end{alignat*}\nFor example, the general solution of $\\cos\\zeta = -\\frac{5}{3}$ is $\\zeta = (2k + 1)\\pi ± i\\log 3$.", "markdown": "**of the equation $\\cos\\zeta = a$, where $a$ is real.** Putting $\\zeta = \\xi + i\\eta$, and equating real and imaginary parts, we obtain = a,0pt minus 3pt= 0. Hence either $\\eta = 0$ or $\\xi$ is a multiple of $\\pi$. If (i) $\\eta = 0$ then $\\cos\\xi = a$, which is impossible unless $-1 \\leq a \\leq 1$. This hypothesis leads to the solution = 2k± a, where $\\arccos a$ lies between $0$ and $\\frac{1}{2}\\pi$. If (ii) $\\xi = m\\pi$ then $\\cosh\\eta = (-1)^{m}a$, so that either $a \\geq 1$ and $m$ is even, or $a \\leq -1$ and $m$ is odd. If $a = ± 1$ then $\\eta = 0$, and we are led back to our first case. If $|a| > 1$ then $\\cosh\\eta = |a|$, and we are led to the solutions alignat*4 &=& 2k &± i &&a + a^2 - 10pt minus 3pt&&(a > 1), &=&(2k + 1) &± i-&&a + a^2 - 10pt minus 3pt&&(a < -1). alignat* For example, the general solution of $\\cos\\zeta = -\\frac{5}{3}$ is $\\zeta = (2k + 1)\\pi ± i\\log 3$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "solve", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "Eq(cos(zeta), a)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "solve: Eq(cos(x), a)" ], "shape": [ "solve: Eq(cos(x), a)" ], "same_problem_in": [], "needs": [ "cas.solve.complex", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcv/9", "set": "hardy-course-of-pure-mathematics-1921/ex-xcv", "number": 9, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "411", "location": "Exercise XCV, problem 9", "problem_latex": "Solve $\\sin\\zeta = \\alpha$, where $\\alpha$~is real.", "markdown": "Solve $\\sin\\zeta = \\alpha$, where $\\alpha$ is real.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "solve", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "Eq(sin(zeta), alpha)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "solve: Eq(sin(x), a)" ], "shape": [ "solve: Eq(sin(x), a)" ], "same_problem_in": [], "needs": [ "cas.solve.complex", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcvi/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xcvi", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "416", "location": "Exercise XCVI, problem 1", "problem_latex": "Calculate $\\cos i$ and $\\sin i$ to two places of decimals\nby means of the power series for $\\cos z$ and~$\\sin z$.", "markdown": "Calculate $\\cos i$ and $\\sin i$ to two places of decimals by means of the power series for $\\cos z$ and $\\sin z$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.complex" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcvi/10", "set": "hardy-course-of-pure-mathematics-1921/ex-xcvi", "number": 10, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "416", "location": "Exercise XCVI, problem 10", "problem_latex": "Sum\n\\[\n1 + \\frac{a\\cos z}{1!} + \\frac{a^{2}\\cos 2z}{2!} + \\dots,\\quad\n\\frac{a\\sin z}{1!} + \\frac{a^{2}\\sin 2z}{2!} + \\dots.\n\\]", "markdown": "Sum 1 + az1! + a^22z2! + …,0pt minus 3ptaz1! + a^22z2! + ….", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcvi/11", "set": "hardy-course-of-pure-mathematics-1921/ex-xcvi", "number": 11, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "416", "location": "Exercise XCVI, problem 11", "problem_latex": "Sum\n\\[\n1 - \\frac{\\cos 2z}{2!} + \\frac{\\cos 4z}{4!} - \\dots,\\quad\n\\frac{\\cos z}{1!} - \\frac{\\cos 3z}{3!} + \\dots\n\\]\nand the corresponding series involving sines.", "markdown": "Sum 1 - 2z2! + 4z4! - …,0pt minus 3ptz1! - 3z3! + … and the corresponding series involving sines.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.hyp", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcvi/12", "set": "hardy-course-of-pure-mathematics-1921/ex-xcvi", "number": 12, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "416", "location": "Exercise XCVI, problem 12", "problem_latex": "Show that\n\\[\n1 + \\frac{\\cos 4z}{4!} + \\frac{\\cos 8z}{8!} + \\dots\n = \\tfrac{1}{2}\\{\\cos(\\cos z) \\cosh(\\sin z) + \\cos(\\sin z) \\cosh(\\cos z)\\}.\n\\]", "markdown": "Show that 1 + 4z4! + 8z8! + … = 12(z) (z) + (z) (z).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.hyp", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcvi/13", "set": "hardy-course-of-pure-mathematics-1921/ex-xcvi", "number": 13, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "416", "location": "Exercise XCVI, problem 13", "problem_latex": "Show that the expansions of $\\cos(x + h)$ and $\\sin(x + h)$ in powers of~$h$\n(\\Ex{lvi}.~1) are valid for all values of $x$~and~$h$, real or complex.", "markdown": "Show that the expansions of $\\cos(x + h)$ and $\\sin(x + h)$ in powers of $h$ (% [examples:lvi]Ex. lvi%. 1) are valid for all values of $x$ and $h$, real or complex.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.complex", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcvi/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xcvi", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "416", "location": "Exercise XCVI, problem 2", "problem_latex": "Prove that $|\\cos z| \\leq \\cosh|z|$ and $|\\sin z| \\leq \\sinh|z|$.", "markdown": "Prove that $|\\cos z| \\leq \\cosh|z|$ and $|\\sin z| \\leq \\sinh|z|$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.hyp", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcvi/3", "set": "hardy-course-of-pure-mathematics-1921/ex-xcvi", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "416", "location": "Exercise XCVI, problem 3", "problem_latex": "Prove that if $|z| < 1$ then $|\\cos z| < 2$ and $|\\sin z| < \\frac{6}{5}|z|$.", "markdown": "Prove that if $|z| < 1$ then $|\\cos z| < 2$ and $|\\sin z| < \\frac{6}{5}|z|$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcvi/4", "set": "hardy-course-of-pure-mathematics-1921/ex-xcvi", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "416", "location": "Exercise XCVI, problem 4", "problem_latex": "Since $\\sin 2z = 2\\sin z \\cos z$ we have\n\\[\n(2z) - \\frac{(2z)^{3}}{3!} + \\frac{(2z)^{5}}{5!} - \\dots\n = 2\\left(z - \\frac{z^{3}}{3!} + \\dots\\right)\n \\left(1 - \\frac{z^{2}}{2!} + \\dots\\right).\n\\]\nProve by multiplying the two series on the right-hand side (\\SecNo[§]{195}) and\nequating coefficients (\\SecNo[§]{194}) that\n\\[\n\\binom{2n + 1}{1} + \\binom{2n + 1}{3} + \\dots + \\binom{2n + 1}{2n + 1} = 2^{2n}.\n\\]\nVerify the result by means of the binomial theorem. Derive similar identities\nfrom the equations\n\\[\n\\cos^{2}z + \\sin^{2}z = 1,\\quad\n\\cos2z = 2\\cos^{2}z - 1 = 1 - 2\\sin^{2}z.\n\\]", "markdown": "Since $\\sin 2z = 2\\sin z \\cos z$ we have (2z) - (2z)^33! + (2z)^55! - … = 2(z - z^33! + …) (1 - z^22! + …). Prove by multiplying the two series on the right-hand side ([§]195) and equating coefficients ([§]194) that 2n + 11 + 2n + 13 + …+ 2n + 12n + 1 = 2^2n. Verify the result by means of the binomial theorem. Derive similar identities from the equations ^2z + ^2z = 1,0pt minus 3pt2z = 2^2z - 1 = 1 - 2^2z.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.collect", "cas.expand", "core.arith" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcvi/5", "set": "hardy-course-of-pure-mathematics-1921/ex-xcvi", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "416", "location": "Exercise XCVI, problem 5", "problem_latex": "Show that\n\\[\n\\exp\\{(1 + i)z\\}\n = \\sum_{0}^{\\infty} 2^{\\frac{1}{2}n} \\exp(\\tfrac{1}{4}n\\pi i) \\frac{z^{n}}{n!}.\n\\]", "markdown": "Show that (1 + i)z = _0^ 2^12n (14ni) z^nn!.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.complex" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcvi/6", "set": "hardy-course-of-pure-mathematics-1921/ex-xcvi", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "416", "location": "Exercise XCVI, problem 6", "problem_latex": "Expand $\\\\cos z \\\\cosh z$ in powers of~$z$. [We have\n\\begin{align*}\n\\\\cos z \\\\cosh z + i\\\\sin z \\\\sinh z\n &= \\\\cos\\\\{(1 - i)z\\\\}\n = \\\\tfrac{1}{2} [\\\\exp\\\\{(1 + i)z\\\\} + \\\\exp\\\\{-(1 + i)z\\\\}]\\\\\n &= \\\\tfrac{1}{2} \\\\sum_{0}^{\\\\infty} 2^{\\\\frac{1}{2}n}\n \\\\{1 + (-1)^{n}\\\\} \\\\exp(\\\\tfrac{1}{4}n\\\\pi i) \\\\frac{z^{n}}{n!},\n\\end{align*}\nand similarly\n\\[\n\\\\cos z \\\\cosh z - i\\\\sin z \\\\sinh z = \\\\cos (1 + i)z\n = \\\\tfrac{1}{2} \\\\sum_{0}^{\\\\infty} 2^{\\\\frac{1}{2}n}\n \\\\{1 + (-1)^{n}\\\\} \\\\exp(-\\\\tfrac{1}{4}n\\\\pi i) \\\\frac{z^{n}}{n!}.\n\\]\nHence\n\\[\n\\\\cos z \\\\cosh z\n = \\\\tfrac{1}{2} \\\\sum_{0}^{\\\\infty} 2^{\\\\frac{1}{2}n}\\\\{1 + (-1)^{n}\\\\} \\\\cos \\\\tfrac{1}{4}n\\\\pi \\\\frac{z^{n}}{n!}\n = 1 - \\\\frac{2^{2}z^{4}}{4!} + \\\\frac{2^{4}z^{8}}{8!} - \\\\dots.]\n\\]", "markdown": "Expand $\\\\cos z \\\\cosh z$ in powers of $z$. [We have align* cos z cosh z + isin z sinh z &= cos(1 - i)z = tfrac12 [exp(1 + i)z + exp-(1 + i)z] &= tfrac12 sum_0^infty 2^frac12n 1 + (-1)^n exp(tfrac14npi i) fracz^nn!, align* and similarly cos z cosh z - isin z sinh z = cos (1 + i)z = tfrac12 sum_0^infty 2^frac12n 1 + (-1)^n exp(-tfrac14npi i) fracz^nn!. Hence cos z cosh z = tfrac12 sum_0^infty 2^frac12n1 + (-1)^n cos tfrac14npi fracz^nn! = 1 - frac2^2z^44! + frac2^4z^88! - dots.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "FLAG-EXTRACTION", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "core.complex", "core.hyp", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcvi/7", "set": "hardy-course-of-pure-mathematics-1921/ex-xcvi", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "416", "location": "Exercise XCVI, problem 7", "problem_latex": "Expand $\\sin z \\sinh z$, $\\cos z \\sinh z$, and $\\sin z \\cosh z$ in powers of~$z$.", "markdown": "Expand $\\sin z \\sinh z$, $\\cos z \\sinh z$, and $\\sin z \\cosh z$ in powers of $z$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "core.hyp", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcvi/8", "set": "hardy-course-of-pure-mathematics-1921/ex-xcvi", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "416", "location": "Exercise XCVI, problem 8", "problem_latex": "Expand $\\sin^{2} z$ and $\\sin^{3} z$ in powers of~$z$. [Use the formulae\n\\[\n\\sin^{2} z = \\tfrac{1}{2} (1 - \\cos 2z),\\quad\n\\sin^{3} z = \\tfrac{1}{4} (3\\sin z - \\sin 3z),\\ \\dots.\n\\]\nIt is clear that the same method may be used to expand $\\cos^{n} z$ and~$\\sin^{n} z$,\nwhere $n$~is any integer.]", "markdown": "Expand $\\sin^{2} z$ and $\\sin^{3} z$ in powers of $z$. [Use the formulae ^2 z = 12 (1 - 2z),0pt minus 3pt^3 z = 14 (3z - 3z), …. It is clear that the same method may be used to expand $\\cos^{n} z$ and $\\sin^{n} z$, where $n$ is any integer.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcvi/9", "set": "hardy-course-of-pure-mathematics-1921/ex-xcvi", "number": 9, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "416", "location": "Exercise XCVI, problem 9", "problem_latex": "Sum the series\n\\[\nC = 1 + \\frac{\\cos z}{1!} + \\frac{\\cos 2z}{2!} + \\frac{\\cos 3z}{3!} +\\dots,\\quad\nS = \\frac{\\sin z}{1!} + \\frac{\\sin 2z}{2!} + \\frac{\\sin 3z}{3!} + \\dots.\n\\]\n\n[Here\n\\begin{align*}\nC + iS &= 1 + \\dfrac{\\exp(iz)}{1!} + \\dfrac{\\exp(2iz)}{2!} + \\dots\n = \\exp\\{\\exp(iz)\\} \\\\\n &= \\exp(\\cos z) \\{\\cos(\\sin z) + i\\sin(\\sin z)\\},\n\\end{align*}\nand similarly\n\\[\nC - iS = \\exp\\{\\exp(-iz)\\} = \\exp(\\cos z)\\{\\cos(\\sin z) - i\\sin(\\sin z)\\}.\n\\]\nHence\n\\[\nC = \\exp(\\cos z)\\cos(\\sin z),\\quad\nS = \\exp(\\cos z)\\sin(\\sin z).]\n\\]", "markdown": "Sum the series C = 1 + z1! + 2z2! + 3z3! +…,0pt minus 3ptS = z1! + 2z2! + 3z3! + …. [Here align* C + iS &= 1 + (iz)1! + (2iz)2! + … = (iz) &= (z) (z) + i(z), align* and similarly C - iS = (-iz) = (z)(z) - i(z). Hence C = (z)(z),0pt minus 3ptS = (z)(z).]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.hyp", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcvii/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xcvii", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "420", "location": "Exercise XCVII, problem 1", "problem_latex": "Prove that, in any triangle in which $a > b$,\n\\[\n\\log c = \\log a - \\frac{b}{a} \\cos C - \\frac{b^{2}}{2a^{2}} \\cos 2C - \\dots.\n\\]\n\n[Use the formula $\\log c = \\frac{1}{2} \\log(a^{2} + b^{2} - 2ab\\cos C )$.]", "markdown": "Prove that, in any triangle in which $a > b$, c = a - ba C - b^22a^2 2C - …. [Use the formula $\\log c = \\frac{1}{2} \\log(a^{2} + b^{2} - 2ab\\cos C )$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcvii/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xcvii", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "420", "location": "Exercise XCVII, problem 2", "problem_latex": "Prove that if $-1 < r < 1$ and $-\\frac{1}{2}\\pi < \\theta < \\frac{1}{2}\\pi$ then\n\\[\nr\\sin 2\\theta\n - \\tfrac{1}{2}r^{2} \\sin 4\\theta\n + \\tfrac{1}{3}r^{3} \\sin 6\\theta - \\dots\n = \\theta - \\arctan \\left\\{\\left(\\frac{1 - r}{1 + r}\\right) \\tan\\theta\\right\\},\n\\]\nthe inverse tangent lying between $-\\frac{1}{2}\\pi$ and~$\\frac{1}{2}\\pi$. Determine the sum of the\nseries for all other values of~$\\theta$.", "markdown": "Prove that if $-1 < r < 1$ and $-\\frac{1}{2}\\pi < \\theta < \\frac{1}{2}\\pi$ then r2 - 12r^2 4 + 13r^3 6- … = - (1 - r1 + r) , the inverse tangent lying between $-\\frac{1}{2}\\pi$ and $\\frac{1}{2}\\pi$. Determine the sum of the series for all other values of $\\theta$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcvii/3a", "set": "hardy-course-of-pure-mathematics-1921/ex-xcvii", "number": 3, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "420", "location": "Exercise XCVII, problem 3a", "problem_latex": "Prove, by considering the expansions of $\\log(1 + iz)$ and $\\log(1 - iz)$ in\npowers of~$z$, that if $-1 < r < 1$ then\n\\begin{gather*}\n\\begin{alignedat}{4}\nr\\sin\\theta &+ \\tfrac{1}{2}r^{2} \\cos 2\\theta\n &&- \\tfrac{1}{3}r^{3} \\sin 3\\theta\n &&- \\tfrac{1}{4}r^{4} \\cos 4\\theta + \\dots\n &&= \\tfrac{1}{2} \\log(1 + 2r \\sin\\theta + r^{2}),\\\\\nr\\cos\\theta &+ \\tfrac{1}{2}r^{2} \\sin 2\\theta\n &&- \\tfrac{1}{3}r^{3} \\cos 3\\theta\n &&- \\tfrac{1}{4}r^{4} \\sin 4\\theta + \\dots\n &&= \\arctan \\left(\\frac{r\\cos\\theta}{1 - r\\sin\\theta}\\right),\n\\end{alignedat} \\displaybreak[1] \\\\\n\\begin{alignedat}{2}\nr\\sin\\theta &- \\tfrac{1}{3}r^{3} \\sin 3\\theta + \\dots\n &&= \\tfrac{1}{4} \\log\\left(\\frac{1 + 2r \\sin\\theta + r^{2}}\n {1 - 2r \\sin\\theta + r^{2}}\\right),\\\\\nr\\cos\\theta &- \\tfrac{1}{3}r^{3} \\cos 3\\theta + \\dots\n &&= \\tfrac{1}{2} \\arctan \\left(\\frac{2r\\cos\\theta}{1 - r^{2}}\\right),\n\\end{alignedat}\n\\end{gather*}\nthe inverse tangents lying between $-\\frac{1}{2}\\pi$ and~$\\frac{1}{2}\\pi$.", "markdown": "Prove, by considering the expansions of $\\log(1 + iz)$ and $\\log(1 - iz)$ in powers of $z$, that if $-1 < r < 1$ then gather* alignedat4 r&+ 12r^2 2 &&- 13r^3 3 &&- 14r^4 4+ … &&= 12 (1 + 2r + r^2), r&+ 12r^2 2 &&- 13r^3 3 &&- 14r^4 4+ … &&= (r1 - r), alignedat [1] alignedat2 r&- 13r^3 3+ … &&= 14 (1 + 2r + r^2 1 - 2r + r^2), r&- 13r^3 3+ … &&= 12 (2r1 - r^2), alignedat gather* the inverse tangents lying between $-\\frac{1}{2}\\pi$ and $\\frac{1}{2}\\pi$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcvii/3b", "set": "hardy-course-of-pure-mathematics-1921/ex-xcvii", "number": 3, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "420", "location": "Exercise XCVII, problem 3b", "problem_latex": "Prove, by considering the expansions of $\\log(1 + iz)$ and $\\log(1 - iz)$ in\npowers of~$z$, that if $-1 < r < 1$ then\n\\begin{gather*}\n\\begin{alignedat}{4}\nr\\sin\\theta &+ \\tfrac{1}{2}r^{2} \\cos 2\\theta\n &&- \\tfrac{1}{3}r^{3} \\sin 3\\theta\n &&- \\tfrac{1}{4}r^{4} \\cos 4\\theta + \\dots\n &&= \\tfrac{1}{2} \\log(1 + 2r \\sin\\theta + r^{2}),\\\\\nr\\cos\\theta &+ \\tfrac{1}{2}r^{2} \\sin 2\\theta\n &&- \\tfrac{1}{3}r^{3} \\cos 3\\theta\n &&- \\tfrac{1}{4}r^{4} \\sin 4\\theta + \\dots\n &&= \\arctan \\left(\\frac{r\\cos\\theta}{1 - r\\sin\\theta}\\right),\n\\end{alignedat} \\displaybreak[1] \\\\\n\\begin{alignedat}{2}\nr\\sin\\theta &- \\tfrac{1}{3}r^{3} \\sin 3\\theta + \\dots\n &&= \\tfrac{1}{4} \\log\\left(\\frac{1 + 2r \\sin\\theta + r^{2}}\n {1 - 2r \\sin\\theta + r^{2}}\\right),\\\\\nr\\cos\\theta &- \\tfrac{1}{3}r^{3} \\cos 3\\theta + \\dots\n &&= \\tfrac{1}{2} \\arctan \\left(\\frac{2r\\cos\\theta}{1 - r^{2}}\\right),\n\\end{alignedat}\n\\end{gather*}\nthe inverse tangents lying between $-\\frac{1}{2}\\pi$ and~$\\frac{1}{2}\\pi$.", "markdown": "Prove, by considering the expansions of $\\log(1 + iz)$ and $\\log(1 - iz)$ in powers of $z$, that if $-1 < r < 1$ then gather* alignedat4 r&+ 12r^2 2 &&- 13r^3 3 &&- 14r^4 4+ … &&= 12 (1 + 2r + r^2), r&+ 12r^2 2 &&- 13r^3 3 &&- 14r^4 4+ … &&= (r1 - r), alignedat [1] alignedat2 r&- 13r^3 3+ … &&= 14 (1 + 2r + r^2 1 - 2r + r^2), r&- 13r^3 3+ … &&= 12 (2r1 - r^2), alignedat gather* the inverse tangents lying between $-\\frac{1}{2}\\pi$ and $\\frac{1}{2}\\pi$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcvii/3c", "set": "hardy-course-of-pure-mathematics-1921/ex-xcvii", "number": 3, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "420", "location": "Exercise XCVII, problem 3c", "problem_latex": "Prove, by considering the expansions of $\\log(1 + iz)$ and $\\log(1 - iz)$ in\npowers of~$z$, that if $-1 < r < 1$ then\n\\begin{gather*}\n\\begin{alignedat}{4}\nr\\sin\\theta &+ \\tfrac{1}{2}r^{2} \\cos 2\\theta\n &&- \\tfrac{1}{3}r^{3} \\sin 3\\theta\n &&- \\tfrac{1}{4}r^{4} \\cos 4\\theta + \\dots\n &&= \\tfrac{1}{2} \\log(1 + 2r \\sin\\theta + r^{2}),\\\\\nr\\cos\\theta &+ \\tfrac{1}{2}r^{2} \\sin 2\\theta\n &&- \\tfrac{1}{3}r^{3} \\cos 3\\theta\n &&- \\tfrac{1}{4}r^{4} \\sin 4\\theta + \\dots\n &&= \\arctan \\left(\\frac{r\\cos\\theta}{1 - r\\sin\\theta}\\right),\n\\end{alignedat} \\displaybreak[1] \\\\\n\\begin{alignedat}{2}\nr\\sin\\theta &- \\tfrac{1}{3}r^{3} \\sin 3\\theta + \\dots\n &&= \\tfrac{1}{4} \\log\\left(\\frac{1 + 2r \\sin\\theta + r^{2}}\n {1 - 2r \\sin\\theta + r^{2}}\\right),\\\\\nr\\cos\\theta &- \\tfrac{1}{3}r^{3} \\cos 3\\theta + \\dots\n &&= \\tfrac{1}{2} \\arctan \\left(\\frac{2r\\cos\\theta}{1 - r^{2}}\\right),\n\\end{alignedat}\n\\end{gather*}\nthe inverse tangents lying between $-\\frac{1}{2}\\pi$ and~$\\frac{1}{2}\\pi$.", "markdown": "Prove, by considering the expansions of $\\log(1 + iz)$ and $\\log(1 - iz)$ in powers of $z$, that if $-1 < r < 1$ then gather* alignedat4 r&+ 12r^2 2 &&- 13r^3 3 &&- 14r^4 4+ … &&= 12 (1 + 2r + r^2), r&+ 12r^2 2 &&- 13r^3 3 &&- 14r^4 4+ … &&= (r1 - r), alignedat [1] alignedat2 r&- 13r^3 3+ … &&= 14 (1 + 2r + r^2 1 - 2r + r^2), r&- 13r^3 3+ … &&= 12 (2r1 - r^2), alignedat gather* the inverse tangents lying between $-\\frac{1}{2}\\pi$ and $\\frac{1}{2}\\pi$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcvii/3d", "set": "hardy-course-of-pure-mathematics-1921/ex-xcvii", "number": 3, "part": "d", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "420", "location": "Exercise XCVII, problem 3d", "problem_latex": "Prove, by considering the expansions of $\\log(1 + iz)$ and $\\log(1 - iz)$ in\npowers of~$z$, that if $-1 < r < 1$ then\n\\begin{gather*}\n\\begin{alignedat}{4}\nr\\sin\\theta &+ \\tfrac{1}{2}r^{2} \\cos 2\\theta\n &&- \\tfrac{1}{3}r^{3} \\sin 3\\theta\n &&- \\tfrac{1}{4}r^{4} \\cos 4\\theta + \\dots\n &&= \\tfrac{1}{2} \\log(1 + 2r \\sin\\theta + r^{2}),\\\\\nr\\cos\\theta &+ \\tfrac{1}{2}r^{2} \\sin 2\\theta\n &&- \\tfrac{1}{3}r^{3} \\cos 3\\theta\n &&- \\tfrac{1}{4}r^{4} \\sin 4\\theta + \\dots\n &&= \\arctan \\left(\\frac{r\\cos\\theta}{1 - r\\sin\\theta}\\right),\n\\end{alignedat} \\displaybreak[1] \\\\\n\\begin{alignedat}{2}\nr\\sin\\theta &- \\tfrac{1}{3}r^{3} \\sin 3\\theta + \\dots\n &&= \\tfrac{1}{4} \\log\\left(\\frac{1 + 2r \\sin\\theta + r^{2}}\n {1 - 2r \\sin\\theta + r^{2}}\\right),\\\\\nr\\cos\\theta &- \\tfrac{1}{3}r^{3} \\cos 3\\theta + \\dots\n &&= \\tfrac{1}{2} \\arctan \\left(\\frac{2r\\cos\\theta}{1 - r^{2}}\\right),\n\\end{alignedat}\n\\end{gather*}\nthe inverse tangents lying between $-\\frac{1}{2}\\pi$ and~$\\frac{1}{2}\\pi$.", "markdown": "Prove, by considering the expansions of $\\log(1 + iz)$ and $\\log(1 - iz)$ in powers of $z$, that if $-1 < r < 1$ then gather* alignedat4 r&+ 12r^2 2 &&- 13r^3 3 &&- 14r^4 4+ … &&= 12 (1 + 2r + r^2), r&+ 12r^2 2 &&- 13r^3 3 &&- 14r^4 4+ … &&= (r1 - r), alignedat [1] alignedat2 r&- 13r^3 3+ … &&= 14 (1 + 2r + r^2 1 - 2r + r^2), r&- 13r^3 3+ … &&= 12 (2r1 - r^2), alignedat gather* the inverse tangents lying between $-\\frac{1}{2}\\pi$ and $\\frac{1}{2}\\pi$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcvii/4a", "set": "hardy-course-of-pure-mathematics-1921/ex-xcvii", "number": 4, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "420", "location": "Exercise XCVII, problem 4a", "problem_latex": "Prove that\n\\begin{alignat*}{3}\n\\cos\\theta \\cos\\theta &- \\tfrac{1}{2} \\cos 2\\theta \\cos^{2}\\theta\n &&+ \\tfrac{1}{3} \\cos 3\\theta \\cos^{3} \\theta - \\dots\n &&= \\tfrac{1}{2} \\log(1 + 3\\cos^{2} \\theta),\\\\\n\\sin\\theta \\sin\\theta &- \\tfrac{1}{2} \\sin 2\\theta \\sin^{2}\\theta\n &&+ \\tfrac{1}{3} \\sin 3\\theta \\sin^{3} \\theta - \\dots\n &&= \\arccot (1 + \\cot\\theta + \\cot^{2}\\theta),\n\\end{alignat*}\nthe inverse cotangent lying between $-\\frac{1}{2}\\pi$ and~$\\frac{1}{2}\\pi$; and find similar expressions\nfor the sums of the series\n\\[\n\\cos\\theta \\sin\\theta - \\tfrac{1}{2} \\cos 2\\theta \\sin^{2}\\theta + \\dots,\\quad\n\\sin\\theta \\cos\\theta - \\tfrac{1}{2} \\sin 2\\theta \\cos^{2}\\theta + \\dots.\n\\]", "markdown": "Prove that alignat*3 &- 12 2^2 &&+ 13 3^3 - … &&= 12 (1 + 3^2 ), &- 12 2^2 &&+ 13 3^3 - … &&= (1 + + ^2), alignat* the inverse cotangent lying between $-\\frac{1}{2}\\pi$ and $\\frac{1}{2}\\pi$; and find similar expressions for the sums of the series - 12 2^2+ …,0pt minus 3pt- 12 2^2+ ….", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcvii/4b", "set": "hardy-course-of-pure-mathematics-1921/ex-xcvii", "number": 4, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "420", "location": "Exercise XCVII, problem 4b", "problem_latex": "Prove that\n\\begin{alignat*}{3}\n\\cos\\theta \\cos\\theta &- \\tfrac{1}{2} \\cos 2\\theta \\cos^{2}\\theta\n &&+ \\tfrac{1}{3} \\cos 3\\theta \\cos^{3} \\theta - \\dots\n &&= \\tfrac{1}{2} \\log(1 + 3\\cos^{2} \\theta),\\\\\n\\sin\\theta \\sin\\theta &- \\tfrac{1}{2} \\sin 2\\theta \\sin^{2}\\theta\n &&+ \\tfrac{1}{3} \\sin 3\\theta \\sin^{3} \\theta - \\dots\n &&= \\arccot (1 + \\cot\\theta + \\cot^{2}\\theta),\n\\end{alignat*}\nthe inverse cotangent lying between $-\\frac{1}{2}\\pi$ and~$\\frac{1}{2}\\pi$; and find similar expressions\nfor the sums of the series\n\\[\n\\cos\\theta \\sin\\theta - \\tfrac{1}{2} \\cos 2\\theta \\sin^{2}\\theta + \\dots,\\quad\n\\sin\\theta \\cos\\theta - \\tfrac{1}{2} \\sin 2\\theta \\cos^{2}\\theta + \\dots.\n\\]", "markdown": "Prove that alignat*3 &- 12 2^2 &&+ 13 3^3 - … &&= 12 (1 + 3^2 ), &- 12 2^2 &&+ 13 3^3 - … &&= (1 + + ^2), alignat* the inverse cotangent lying between $-\\frac{1}{2}\\pi$ and $\\frac{1}{2}\\pi$; and find similar expressions for the sums of the series - 12 2^2+ …,0pt minus 3pt- 12 2^2+ ….", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcvii/4c", "set": "hardy-course-of-pure-mathematics-1921/ex-xcvii", "number": 4, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "420", "location": "Exercise XCVII, problem 4c", "problem_latex": "Prove that\n\\begin{alignat*}{3}\n\\cos\\theta \\cos\\theta &- \\tfrac{1}{2} \\cos 2\\theta \\cos^{2}\\theta\n &&+ \\tfrac{1}{3} \\cos 3\\theta \\cos^{3} \\theta - \\dots\n &&= \\tfrac{1}{2} \\log(1 + 3\\cos^{2} \\theta),\\\\\n\\sin\\theta \\sin\\theta &- \\tfrac{1}{2} \\sin 2\\theta \\sin^{2}\\theta\n &&+ \\tfrac{1}{3} \\sin 3\\theta \\sin^{3} \\theta - \\dots\n &&= \\arccot (1 + \\cot\\theta + \\cot^{2}\\theta),\n\\end{alignat*}\nthe inverse cotangent lying between $-\\frac{1}{2}\\pi$ and~$\\frac{1}{2}\\pi$; and find similar expressions\nfor the sums of the series\n\\[\n\\cos\\theta \\sin\\theta - \\tfrac{1}{2} \\cos 2\\theta \\sin^{2}\\theta + \\dots,\\quad\n\\sin\\theta \\cos\\theta - \\tfrac{1}{2} \\sin 2\\theta \\cos^{2}\\theta + \\dots.\n\\]", "markdown": "Prove that alignat*3 &- 12 2^2 &&+ 13 3^3 - … &&= 12 (1 + 3^2 ), &- 12 2^2 &&+ 13 3^3 - … &&= (1 + + ^2), alignat* the inverse cotangent lying between $-\\frac{1}{2}\\pi$ and $\\frac{1}{2}\\pi$; and find similar expressions for the sums of the series - 12 2^2+ …,0pt minus 3pt- 12 2^2+ ….", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcviii/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xcviii", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "424", "location": "Exercise XCVIII, problem 1", "problem_latex": "Suppose $m$~real. Then since\n\\[\n\\log(1 + z) = \\tfrac{1}{2} \\log(1 + 2r\\cos\\theta + r^{2})\n + i\\arctan\\left(\\frac{r\\sin\\theta}{1 + r\\cos\\theta}\\right),\n\\]\nwe obtain\n\\begin{align*}\n\\sum_{0}^{\\infty} \\binom{m}{n} z^{n}\n &= \\exp\\{\\tfrac{1}{2}m \\log(1 + 2r\\cos\\theta + r^{2})\\}\n \\Cis \\left\\{m\\arctan\\left(\\frac{r\\sin\\theta}{1 + r\\cos\\theta}\\right)\\right\\} \\\\\n &= (1 + 2r\\cos\\theta + r^{2})^{\\frac{1}{2}m}\n \\Cis \\left\\{m\\arctan\\left(\\frac{r\\sin\\theta}{1 + r\\cos\\theta}\\right)\\right\\},\n\\end{align*}\nall the inverse tangents lying between $-\\frac{1}{2}\\pi$ and~$\\frac{1}{2}\\pi$. In particular, if we\nsuppose $\\theta = \\frac{1}{2}\\pi$, $z = ir$, and equate the real and imaginary parts, we obtain\n\\begin{align*}\n1 - \\binom{m}{2} r^{2} + \\binom{m}{4} r^{4} - \\dots\n &= (1 + r^{2})^{\\frac{1}{2}m} \\cos(m\\arctan r), \\\\\n\\binom{m}{1} r - \\binom{m}{3} r^{3} + \\binom{m}{5} r^{5} - \\dots\n &= (1 + r^{2})^{\\frac{1}{2}m} \\sin(m\\arctan r).\n\\end{align*}", "markdown": "Suppose $m$ real. Then since (1 + z) = 12 (1 + 2r+ r^2) + i(r1 + r), we obtain align* _0^ mn z^n &= 12m (1 + 2r+ r^2) m(r1 + r) &= (1 + 2r+ r^2)^12m m(r1 + r), align* all the inverse tangents lying between $-\\frac{1}{2}\\pi$ and $\\frac{1}{2}\\pi$. In particular, if we suppose $\\theta = \\frac{1}{2}\\pi$, $z = ir$, and equate the real and imaginary parts, we obtain align* 1 - m2 r^2 + m4 r^4 - … &= (1 + r^2)^12m (mr), m1 r - m3 r^3 + m5 r^5 - … &= (1 + r^2)^12m (mr). align*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcviii/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xcviii", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "424", "location": "Exercise XCVIII, problem 2", "problem_latex": "Verify the formulae of Ex.~1 when $m = 1$, $2$, $3$. [Of course when $m$~is\na positive integer the series is finite.]", "markdown": "Verify the formulae of Ex. 1 when $m = 1$, $2$, $3$. [Of course when $m$ is a positive integer the series is finite.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcviii/3", "set": "hardy-course-of-pure-mathematics-1921/ex-xcviii", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "424", "location": "Exercise XCVIII, problem 3", "problem_latex": "Prove that if $0 \\leq r < 1$ then\n\\begin{align*}\n1 - \\frac{1·3}{2·4} r^{2} + \\frac{1·3·5·7}{2·4·6·8} r^{4} - \\dots\n &= \\bigsqrtb{\\frac{\\sqrtp{1 + r^{2}} + 1}{2(1 + r^{2})}}, \\\\\n\\frac{1}{2} r - \\frac{1·3·5}{2·4·6} r^{3} + \\frac{1·3·5·7·9}{2·4·6·8·10} r^{5} - \\dots\n &= \\bigsqrtb{\\frac{\\sqrtp{1 + r^{2}} - 1}{2(1 + r^{2})}}.\n\\end{align*}\n\n[Take $m = -\\frac{1}{2}$ in the last two formulae of Ex.~1.]", "markdown": "Prove that if $0 \\leq r < 1$ then align* 1 - 1·32·4 r^2 + 1·3·5·72·4·6·8 r^4 - … &= 1 + r^2 + 12(1 + r^2), 12 r - 1·3·52·4·6 r^3 + 1·3·5·7·92·4·6·8·10 r^5 - … &= 1 + r^2 - 12(1 + r^2). align* [Take $m = -\\frac{1}{2}$ in the last two formulae of Ex. 1.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.frac" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcviii/4", "set": "hardy-course-of-pure-mathematics-1921/ex-xcviii", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "424", "location": "Exercise XCVIII, problem 4", "problem_latex": "Prove that if $-\\frac{1}{4}\\pi < \\theta < \\frac{1}{4}\\pi$ then\n\\begin{align*}\n\\cos m\\theta &= \\cos^{m} \\theta \\left\\{1 - \\binom{m}{2} \\tan^{2} \\theta\n + \\binom{m}{4} \\tan^{4} \\theta - \\dots\\right\\}, \\\\\n\\sin m\\theta &= \\cos^{m} \\theta \\left\\{\\binom{m}{1} \\tan\\theta\n - \\binom{m}{3} \\tan^{3} \\theta + \\dots\\right\\},\n\\end{align*}\nfor all real values of~$m$. [These results follow at once from the equations\n\\[\n\\cos m\\theta + i\\sin m\\theta\n = (\\cos\\theta + i\\sin\\theta )^{m}\n = \\cos^{m} \\theta(1 + i\\tan\\theta)^{m}.]", "markdown": "Prove that if $-\\frac{1}{4}\\pi < \\theta < \\frac{1}{4}\\pi$ then align* m&= ^m 1 - m2 ^2 + m4 ^4 - …, m&= ^m m1 - m3 ^3 + …, align* for all real values of $m$. [These results follow at once from the equations m+ im = (+ i)^m = ^m (1 + i)^m.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcviii/5", "set": "hardy-course-of-pure-mathematics-1921/ex-xcviii", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "424", "location": "Exercise XCVIII, problem 5", "problem_latex": "We proved (\\Ex{lxxxi}.~6), by direct multiplication of series, that\n$f(m, z) = \\sum\\dbinom{m}{n} z^{n}$, where $|z| < 1$, satisfies the functional equation\n\\[\nf(m, z) f(m', z) = f(m + m', z).\n\\]\nDeduce, by an argument similar to that of \\SecNo[§]{216}, and without assuming the\ngeneral result of \\PageRef{p.}{423}, that if $m$~is real and rational then\n\\[\nf(m, z) = \\exp\\{m\\log(1 + z)\\}.\n\\]", "markdown": "We proved (% [examples:lxxxi]Ex. lxxxi%. 6), by direct multiplication of series, that $f(m, z) = \\sum\\dbinom{m}{n} z^{n}$, where $|z| < 1$, satisfies the functional equation f(m, z) f(m’, z) = f(m + m’, z). Deduce, by an argument similar to that of [§]216, and without assuming the general result of p.423, that if $m$ is real and rational then f(m, z) = m(1 + z).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xcviii/6", "set": "hardy-course-of-pure-mathematics-1921/ex-xcviii", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "424", "location": "Exercise XCVIII, problem 6", "problem_latex": "If $z$~and~$\\mu$ are real, and $-1 < z < 1$, then\n\\[\n\\sum \\binom{i\\mu}{n} z^{n} = \\cos\\{\\mu\\log(1 + z)\\} + i\\sin\\{\\mu\\log(1 + z)\\}.\n\\]", "markdown": "If $z$ and $\\mu$ are real, and $-1 < z < 1$, then in z^n = (1 + z) + i(1 + z).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xi/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xi", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "46", "location": "Exercise XI, problem 1", "problem_latex": "Trace the curves $y = 7x^{4}$, $y = 3x^{5}$, $y = x^{10}$.", "markdown": "Trace the curves $y = 7x^{4}$, $y = 3x^{5}$, $y = x^{10}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xi/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xi", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "46", "location": "Exercise XI, problem 2", "problem_latex": "Compare the relative magnitudes of $x^{12}$, $1,000,000x^{6}$, $1,000,000,000,000x$\nwhen $x = 1$, $10$, $100$,~etc.", "markdown": "Compare the relative magnitudes of $x^{12}$, $1,000,000x^{6}$, $1,000,000,000,000x$ when $x = 1$, $10$, $100$, etc.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.table" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xi/3", "set": "hardy-course-of-pure-mathematics-1921/ex-xi", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "46", "location": "Exercise XI, problem 3", "problem_latex": "Draw the graph of $ax^{2} + 2bx + c$.", "markdown": "Draw the graph of $ax^{2} + 2bx + c$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.complete_square", "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xi/4", "set": "hardy-course-of-pure-mathematics-1921/ex-xi", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "46", "location": "Exercise XI, problem 4", "problem_latex": "Trace the curves $y = x^{3} - 3x + 1$, $y = x^{2}(x - 1)$, $y = x(x - 1)^{2}$.", "markdown": "Trace the curves $y = x^{3} - 3x + 1$, $y = x^{2}(x - 1)$, $y = x(x - 1)^{2}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.solve.poly", "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xii/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xii", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "48", "location": "Exercise XII, problem 1", "problem_latex": "Draw the graphs of $y = 1/x$, $y = 1/x^{2}$, $y = 1/x^{3}$,~\\dots.", "markdown": "Draw the graphs of $y = 1/x$, $y = 1/x^{2}$, $y = 1/x^{3}$, ….", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xii/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xii", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "48", "location": "Exercise XII, problem 2", "problem_latex": "Trace $y = x + (1/x)$, $x - (1/x)$, $x^{2} + (1/x^{2})$, $x^{2} - (1/x^{2})$ and $ax + (b/x)$\ntaking various values, positive and negative, for $a$~and~$b$.", "markdown": "Trace $y = x + (1/x)$, $x - (1/x)$, $x^{2} + (1/x^{2})$, $x^{2} - (1/x^{2})$ and $ax + (b/x)$ taking various values, positive and negative, for $a$ and $b$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xii/3", "set": "hardy-course-of-pure-mathematics-1921/ex-xii", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "48", "location": "Exercise XII, problem 3", "problem_latex": "Trace\n\\[\ny = \\frac{x + 1}{x - 1},\\quad\n\\left(\\frac{x + 1}{x - 1}\\right)^{2},\\quad\n\\frac{1}{(x - 1)^{2}},\\quad\n\\frac{x^{2} + 1}{x^{2} - 1}.\n\\]", "markdown": "Trace y = x + 1x - 1,0pt minus 3pt(x + 1x - 1)^2,0pt minus 3pt1(x - 1)^2,0pt minus 3ptx^2 + 1x^2 - 1.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xii/4", "set": "hardy-course-of-pure-mathematics-1921/ex-xii", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "48", "location": "Exercise XII, problem 4", "problem_latex": "Trace $y = 1/(x - a)(x - b)$, $1/(x - a)(x - b)(x - c)$, where $a < b < c$.", "markdown": "Trace $y = 1/(x - a)(x - b)$, $1/(x - a)(x - b)(x - c)$, where $a < b < c$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xii/5", "set": "hardy-course-of-pure-mathematics-1921/ex-xii", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "48", "location": "Exercise XII, problem 5", "problem_latex": "Sketch the general form assumed by the curves $y = 1/x^{m}$ as $m$~becomes\nlarger and larger, considering separately the cases in which $m$~is\nodd or even.", "markdown": "Sketch the general form assumed by the curves $y = 1/x^{m}$ as $m$ becomes larger and larger, considering separately the cases in which $m$ is odd or even.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xiii/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xiii", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "50", "location": "Exercise XIII, problem 1", "problem_latex": "$\\sqrtb{(x - a)(b - x)}$, where $a < b$, is defined only for\n$a \\leq x \\leq b$. If $a < x < b$ it has two values: if $x = a$ or $b$ only one, viz.~$0$.", "markdown": "$\\sqrtb{(x - a)(b - x)}$, where $a < b$, is defined only for $a \\leq x \\leq b$. If $a < x < b$ it has two values: if $x = a$ or $b$ only one, viz. $0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xiii/2a", "set": "hardy-course-of-pure-mathematics-1921/ex-xiii", "number": 2, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "50", "location": "Exercise XIII, problem 2a", "problem_latex": "Consider similarly\n\\begin{gather*}\n\\sqrtb{(x - a)(x - b)(x - c)} \\quad (a < b < c), \\\\\n\\sqrtb{x(x^{2} - a^{2})},\\quad\n\\sqrtb[3]{(x - a)^{2}(b - x)}\\quad (a < b), \\\\\n\\frac{\\sqrtp{1 + x} - \\sqrtp{1 - x}}\n {\\sqrtp{1 + x} + \\sqrtp{1 - x}},\\quad\n\\sqrtb{x + \\sqrt{x}}.\n\\end{gather*}", "markdown": "Consider similarly gather* (x - a)(x - b)(x - c) 0pt minus 3pt(a < b < c), x(x^2 - a^2),0pt minus 3pt[3](x - a)^2(b - x)0pt minus 3pt(a < b), 1 + x - 1 - x 1 + x + 1 - x,0pt minus 3ptx + x. gather*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xiii/2b", "set": "hardy-course-of-pure-mathematics-1921/ex-xiii", "number": 2, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "50", "location": "Exercise XIII, problem 2b", "problem_latex": "Consider similarly\n\\begin{gather*}\n\\sqrtb{(x - a)(x - b)(x - c)} \\quad (a < b < c), \\\\\n\\sqrtb{x(x^{2} - a^{2})},\\quad\n\\sqrtb[3]{(x - a)^{2}(b - x)}\\quad (a < b), \\\\\n\\frac{\\sqrtp{1 + x} - \\sqrtp{1 - x}}\n {\\sqrtp{1 + x} + \\sqrtp{1 - x}},\\quad\n\\sqrtb{x + \\sqrt{x}}.\n\\end{gather*}", "markdown": "Consider similarly gather* (x - a)(x - b)(x - c) 0pt minus 3pt(a < b < c), x(x^2 - a^2),0pt minus 3pt[3](x - a)^2(b - x)0pt minus 3pt(a < b), 1 + x - 1 - x 1 + x + 1 - x,0pt minus 3ptx + x. gather*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xiii/2c", "set": "hardy-course-of-pure-mathematics-1921/ex-xiii", "number": 2, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "50", "location": "Exercise XIII, problem 2c", "problem_latex": "Consider similarly\n\\begin{gather*}\n\\sqrtb{(x - a)(x - b)(x - c)} \\quad (a < b < c), \\\\\n\\sqrtb{x(x^{2} - a^{2})},\\quad\n\\sqrtb[3]{(x - a)^{2}(b - x)}\\quad (a < b), \\\\\n\\frac{\\sqrtp{1 + x} - \\sqrtp{1 - x}}\n {\\sqrtp{1 + x} + \\sqrtp{1 - x}},\\quad\n\\sqrtb{x + \\sqrt{x}}.\n\\end{gather*}", "markdown": "Consider similarly gather* (x - a)(x - b)(x - c) 0pt minus 3pt(a < b < c), x(x^2 - a^2),0pt minus 3pt[3](x - a)^2(b - x)0pt minus 3pt(a < b), 1 + x - 1 - x 1 + x + 1 - x,0pt minus 3ptx + x. gather*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xiii/2d", "set": "hardy-course-of-pure-mathematics-1921/ex-xiii", "number": 2, "part": "d", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "50", "location": "Exercise XIII, problem 2d", "problem_latex": "Consider similarly\n\\begin{gather*}\n\\sqrtb{(x - a)(x - b)(x - c)} \\quad (a < b < c), \\\\\n\\sqrtb{x(x^{2} - a^{2})},\\quad\n\\sqrtb[3]{(x - a)^{2}(b - x)}\\quad (a < b), \\\\\n\\frac{\\sqrtp{1 + x} - \\sqrtp{1 - x}}\n {\\sqrtp{1 + x} + \\sqrtp{1 - x}},\\quad\n\\sqrtb{x + \\sqrt{x}}.\n\\end{gather*}", "markdown": "Consider similarly gather* (x - a)(x - b)(x - c) 0pt minus 3pt(a < b < c), x(x^2 - a^2),0pt minus 3pt[3](x - a)^2(b - x)0pt minus 3pt(a < b), 1 + x - 1 - x 1 + x + 1 - x,0pt minus 3ptx + x. gather*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xiii/2e", "set": "hardy-course-of-pure-mathematics-1921/ex-xiii", "number": 2, "part": "e", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "50", "location": "Exercise XIII, problem 2e", "problem_latex": "Consider similarly\n\\begin{gather*}\n\\sqrtb{(x - a)(x - b)(x - c)} \\quad (a < b < c), \\\\\n\\sqrtb{x(x^{2} - a^{2})},\\quad\n\\sqrtb[3]{(x - a)^{2}(b - x)}\\quad (a < b), \\\\\n\\frac{\\sqrtp{1 + x} - \\sqrtp{1 - x}}\n {\\sqrtp{1 + x} + \\sqrtp{1 - x}},\\quad\n\\sqrtb{x + \\sqrt{x}}.\n\\end{gather*}", "markdown": "Consider similarly gather* (x - a)(x - b)(x - c) 0pt minus 3pt(a < b < c), x(x^2 - a^2),0pt minus 3pt[3](x - a)^2(b - x)0pt minus 3pt(a < b), 1 + x - 1 - x 1 + x + 1 - x,0pt minus 3ptx + x. gather*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xiii/3a", "set": "hardy-course-of-pure-mathematics-1921/ex-xiii", "number": 3, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "50", "location": "Exercise XIII, problem 3a", "problem_latex": "Trace the curves $y^{2} = x$, $y^{3} = x$, $y^{2} = x^{3}$.", "markdown": "Trace the curves $y^{2} = x$, $y^{3} = x$, $y^{2} = x^{3}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xiii/3b", "set": "hardy-course-of-pure-mathematics-1921/ex-xiii", "number": 3, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "50", "location": "Exercise XIII, problem 3b", "problem_latex": "Trace the curves $y^{2} = x$, $y^{3} = x$, $y^{2} = x^{3}$.", "markdown": "Trace the curves $y^{2} = x$, $y^{3} = x$, $y^{2} = x^{3}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xiii/3c", "set": "hardy-course-of-pure-mathematics-1921/ex-xiii", "number": 3, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "50", "location": "Exercise XIII, problem 3c", "problem_latex": "Trace the curves $y^{2} = x$, $y^{3} = x$, $y^{2} = x^{3}$.", "markdown": "Trace the curves $y^{2} = x$, $y^{3} = x$, $y^{2} = x^{3}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xiii/4a", "set": "hardy-course-of-pure-mathematics-1921/ex-xiii", "number": 4, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "50", "location": "Exercise XIII, problem 4a", "problem_latex": "Draw the graphs of the functions\n%[** TN: Not displayed in the original]\n\\[\ny = \\sqrtp{a^{2} - x^{2}},\\quad\ny = b\\sqrtb{1 - (x^{2}/a^{2})}.\n\\]", "markdown": "Draw the graphs of the functions %[** TN: Not displayed in the original] y = a^2 - x^2,0pt minus 3pty = b1 - (x^2/a^2).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xiii/4b", "set": "hardy-course-of-pure-mathematics-1921/ex-xiii", "number": 4, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "50", "location": "Exercise XIII, problem 4b", "problem_latex": "Draw the graphs of the functions\n%[** TN: Not displayed in the original]\n\\[\ny = \\sqrtp{a^{2} - x^{2}},\\quad\ny = b\\sqrtb{1 - (x^{2}/a^{2})}.\n\\]", "markdown": "Draw the graphs of the functions %[** TN: Not displayed in the original] y = a^2 - x^2,0pt minus 3pty = b1 - (x^2/a^2).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xiv/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xiv", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "51", "location": "Exercise XIV, problem 1", "problem_latex": "If $m = 1$, $y$~is a rational function.", "markdown": "If $m = 1$, $y$ is a rational function.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xiv/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xiv", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "51", "location": "Exercise XIV, problem 2", "problem_latex": "If $m = 2$, the equation is $y^{2} + R_{1}y + R_{2} = 0$, so that\n\\[\ny = \\tfrac{1}{2}\\{-R_{1} ± \\sqrtp{R_{1}^{2} - 4R_{2}}\\}.\n\\]\nThis function is defined for all values of~$x$ for which $R_{1}^{2} \\geq 4R_{2}$. It has two\nvalues if $R_{1}^{2} > 4R_{2}$ and one if $R_{1}^{2} = 4R_{2}$.\n\nIf $m = 3$ or~$4$, we can use the methods explained in treatises on Algebra for\nthe solution of cubic and biquadratic equations. But as a rule the process is\ncomplicated and the results inconvenient in form, and we can generally study\nthe properties of the function better by means of the original equation.", "markdown": "If $m = 2$, the equation is $y^{2} + R_{1}y + R_{2} = 0$, so that y = 12-R_1 ± R_1^2 - 4R_2. This function is defined for all values of $x$ for which $R_{1}^{2} \\geq 4R_{2}$. It has two values if $R_{1}^{2} > 4R_{2}$ and one if $R_{1}^{2} = 4R_{2}$. If $m = 3$ or $4$, we can use the methods explained in treatises on Algebra for the solution of cubic and biquadratic equations. But as a rule the process is complicated and the results inconvenient in form, and we can generally study the properties of the function better by means of the original equation.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xiv/3a", "set": "hardy-course-of-pure-mathematics-1921/ex-xiv", "number": 3, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "51", "location": "Exercise XIV, problem 3a", "problem_latex": "Consider the functions defined by the equations\n\\[\ny^{2} - 2y - x^{2} = 0,\\quad\ny^{2} - 2y + x^{2} = 0,\\quad\ny^{4} - 2y^{2} + x^{2} = 0,\n\\]\nin each case obtaining~$y$ as an explicit function of~$x$, and stating for what\nvalues of~$x$ it is defined.", "markdown": "Consider the functions defined by the equations y^2 - 2y - x^2 = 0,0pt minus 3pty^2 - 2y + x^2 = 0,0pt minus 3pty^4 - 2y^2 + x^2 = 0, in each case obtaining $y$ as an explicit function of $x$, and stating for what values of $x$ it is defined.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "solve", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "Eq(y**2 - 2*y - x**2, 0)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "solve: Eq(-a**2 + x**2 - 2*x, 0)" ], "shape": [ "solve: Eq(N*x - a**N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xiv/3b", "set": "hardy-course-of-pure-mathematics-1921/ex-xiv", "number": 3, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "51", "location": "Exercise XIV, problem 3b", "problem_latex": "Consider the functions defined by the equations\n\\[\ny^{2} - 2y - x^{2} = 0,\\quad\ny^{2} - 2y + x^{2} = 0,\\quad\ny^{4} - 2y^{2} + x^{2} = 0,\n\\]\nin each case obtaining~$y$ as an explicit function of~$x$, and stating for what\nvalues of~$x$ it is defined.", "markdown": "Consider the functions defined by the equations y^2 - 2y - x^2 = 0,0pt minus 3pty^2 - 2y + x^2 = 0,0pt minus 3pty^4 - 2y^2 + x^2 = 0, in each case obtaining $y$ as an explicit function of $x$, and stating for what values of $x$ it is defined.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "solve", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "Eq(y**2 - 2*y + x**2, 0)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "solve: Eq(a**2 + x**2 - 2*x, 0)" ], "shape": [ "solve: Eq(N*x + a**N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xiv/3c", "set": "hardy-course-of-pure-mathematics-1921/ex-xiv", "number": 3, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "51", "location": "Exercise XIV, problem 3c", "problem_latex": "Consider the functions defined by the equations\n\\[\ny^{2} - 2y - x^{2} = 0,\\quad\ny^{2} - 2y + x^{2} = 0,\\quad\ny^{4} - 2y^{2} + x^{2} = 0,\n\\]\nin each case obtaining~$y$ as an explicit function of~$x$, and stating for what\nvalues of~$x$ it is defined.", "markdown": "Consider the functions defined by the equations y^2 - 2y - x^2 = 0,0pt minus 3pty^2 - 2y + x^2 = 0,0pt minus 3pty^4 - 2y^2 + x^2 = 0, in each case obtaining $y$ as an explicit function of $x$, and stating for what values of $x$ it is defined.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "solve", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "Eq(y**4 - 2*y**2 + x**2, 0)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "solve: Eq(a**2 + x**4 - 2*x**2, 0)" ], "shape": [ "solve: Eq(N*x**N + a**N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xiv/4a", "set": "hardy-course-of-pure-mathematics-1921/ex-xiv", "number": 4, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "51", "location": "Exercise XIV, problem 4a", "problem_latex": "Find algebraical equations, with coefficients rational in~$x$, satisfied by\neach of the functions\n\\[\n\\sqrt{x} + \\sqrtp{1/x},\\quad\n\\sqrt[3]{x} + \\sqrtp[3]{1/x},\\quad\n\\sqrtp{x + \\sqrt{x}},\\quad\n\\sqrt{x + \\sqrtp{x + \\sqrt{x}}}.\n\\]", "markdown": "Find algebraical equations, with coefficients rational in $x$, satisfied by each of the functions x + 1/x,0pt minus 3pt[3]x + [3]1/x,0pt minus 3ptx + x,0pt minus 3ptx + x + x.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "sqrt(x) + sqrt(1/x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:eliminate_radicals" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xiv/4b", "set": "hardy-course-of-pure-mathematics-1921/ex-xiv", "number": 4, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "51", "location": "Exercise XIV, problem 4b", "problem_latex": "Find algebraical equations, with coefficients rational in~$x$, satisfied by\neach of the functions\n\\[\n\\sqrt{x} + \\sqrtp{1/x},\\quad\n\\sqrt[3]{x} + \\sqrtp[3]{1/x},\\quad\n\\sqrtp{x + \\sqrt{x}},\\quad\n\\sqrt{x + \\sqrtp{x + \\sqrt{x}}}.\n\\]", "markdown": "Find algebraical equations, with coefficients rational in $x$, satisfied by each of the functions x + 1/x,0pt minus 3pt[3]x + [3]1/x,0pt minus 3ptx + x,0pt minus 3ptx + x + x.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "x**Rational(1,3) + (1/x)**Rational(1,3)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:eliminate_radicals" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xiv/4c", "set": "hardy-course-of-pure-mathematics-1921/ex-xiv", "number": 4, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "51", "location": "Exercise XIV, problem 4c", "problem_latex": "Find algebraical equations, with coefficients rational in~$x$, satisfied by\neach of the functions\n\\[\n\\sqrt{x} + \\sqrtp{1/x},\\quad\n\\sqrt[3]{x} + \\sqrtp[3]{1/x},\\quad\n\\sqrtp{x + \\sqrt{x}},\\quad\n\\sqrt{x + \\sqrtp{x + \\sqrt{x}}}.\n\\]", "markdown": "Find algebraical equations, with coefficients rational in $x$, satisfied by each of the functions x + 1/x,0pt minus 3pt[3]x + [3]1/x,0pt minus 3ptx + x,0pt minus 3ptx + x + x.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "sqrt(x + sqrt(x))", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:eliminate_radicals" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xiv/4d", "set": "hardy-course-of-pure-mathematics-1921/ex-xiv", "number": 4, "part": "d", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "51", "location": "Exercise XIV, problem 4d", "problem_latex": "Find algebraical equations, with coefficients rational in~$x$, satisfied by\neach of the functions\n\\[\n\\sqrt{x} + \\sqrtp{1/x},\\quad\n\\sqrt[3]{x} + \\sqrtp[3]{1/x},\\quad\n\\sqrtp{x + \\sqrt{x}},\\quad\n\\sqrt{x + \\sqrtp{x + \\sqrt{x}}}.\n\\]", "markdown": "Find algebraical equations, with coefficients rational in $x$, satisfied by each of the functions x + 1/x,0pt minus 3pt[3]x + [3]1/x,0pt minus 3ptx + x,0pt minus 3ptx + x + x.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "sqrt(x + sqrt(x + sqrt(x)))", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:eliminate_radicals" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xiv/5", "set": "hardy-course-of-pure-mathematics-1921/ex-xiv", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "51", "location": "Exercise XIV, problem 5", "problem_latex": "Consider the equation $y^{4} = x^{2}$.\n\n[Here $y^{2} = ±x$. If $x$~is positive, $y = \\sqrt{x}$: if negative, $y = \\sqrtp{-x}$. Thus the\nfunction has two values for all values of~$x$ save $x = 0$.]", "markdown": "Consider the equation $y^{4} = x^{2}$. [Here $y^{2} = ±x$. If $x$ is positive, $y = \\sqrt{x}$: if negative, $y = \\sqrtp{-x}$. Thus the function has two values for all values of $x$ save $x = 0$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xiv/6", "set": "hardy-course-of-pure-mathematics-1921/ex-xiv", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "51", "location": "Exercise XIV, problem 6", "problem_latex": "An algebraical function of an algebraical function of~$x$ is itself an\nalgebraical function of~$x$.\n\n[For we have\n\\begin{alignat*}{4}\ny^{m} &+ R_{1}(z)y^{m-1} &&+ \\dots &&+ R_{m}(z) &&= 0,\n\\intertext{where}\nz^{n} &+ S_{1}(x)z^{n-1} &&+ \\dots &&+ S_{n}(x) &&= 0.\n\\intertext{Eliminating~$z$ we find an equation of the form}\ny^{p} &+ T_{1}(x)y^{p-1} &&+ \\dots &&+ T_{p}(x) &&= 0.\n\\end{alignat*}\nHere all the capital letters denote rational functions.]", "markdown": "An algebraical function of an algebraical function of $x$ is itself an algebraical function of $x$. [For we have alignat*4 y^m &+ R_1(z)y^m-1 &&+ …&&+ R_m(z) &&= 0, where z^n &+ S_1(x)z^n-1 &&+ …&&+ S_n(x) &&= 0. Eliminating $z$ we find an equation of the form y^p &+ T_1(x)y^p-1 &&+ …&&+ T_p(x) &&= 0. alignat* Here all the capital letters denote rational functions.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xiv/7", "set": "hardy-course-of-pure-mathematics-1921/ex-xiv", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "51", "location": "Exercise XIV, problem 7", "problem_latex": "An example should perhaps be given of an algebraical function which\ncannot be expressed in an explicit algebraical form. Such an example is the\nfunction~$y$ defined by the equation\n\\[\ny^{5} - y - x = 0.\n\\]\nBut the proof that we cannot find an explicit algebraical expression for~$y$ in\nterms of~$x$ is difficult, and cannot be attempted here.", "markdown": "An example should perhaps be given of an algebraical function which cannot be expressed in an explicit algebraical form. Such an example is the function $y$ defined by the equation y^5 - y - x = 0. But the proof that we cannot find an explicit algebraical expression for $y$ in terms of $x$ is difficult, and cannot be attempted here.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xix/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xix", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "62", "location": "Exercise XIX, problem 1", "problem_latex": "What is represented by \\emph{three} equations of the type\n$f(x, y, z) = 0$?", "markdown": "What is represented by *three* equations of the type $f(x, y, z) = 0$?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xix/10", "set": "hardy-course-of-pure-mathematics-1921/ex-xix", "number": 10, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "62", "location": "Exercise XIX, problem 10", "problem_latex": "\\Topic{Ruled surfaces.} Cylinders and cones are special cases of \\emph{surfaces\ncomposed of straight lines}. Such surfaces are called \\emph{ruled surfaces}.\n\nThe two equations\n\\[\nx = az + b,\\quad\ny = cz + d,\n\\Tag{(1)}\n\\]\nrepresent the intersection of two planes, \\ie\\ a straight line. Now suppose\nthat $a$, $b$, $c$, $d$ instead of being fixed are \\emph{functions of an auxiliary variable~$t$}.\nFor any particular value of~$t$ the equations~\\Eq{(1)} give a line. As $t$~varies,\nthis line moves and generates a surface, whose equation may be found by\neliminating~$t$ between the two equations~\\Eq{(1)}. For instance, in Ex.~7 the\nequations of the line which generates the cone are\n\\[\nx = z\\tan \\alpha\\cos t,\\quad\ny = z\\tan \\alpha\\sin t,\n\\]\nwhere $t$~is the angle between the plane~$XOZ$ and a plane through the line and\nthe axis of~$z$.\n\nAnother simple example of a ruled surface may be constructed as follows.\nTake two sections of a right circular cylinder perpendicular to the axis and\nat a distance~$l$ apart (\\Fig{18a}). We can imagine the surface of the cylinder\nto be made up of a number of thin parallel rigid rods of length~$l$, such as~$PQ$,\nthe ends of the rods being fastened to two circular rods of radius~$a$.\n\nNow let us take a third circular rod of the same radius and place it\nround the surface of the cylinder at a distance~$h$ from one of the first two\nrods (see \\Fig{18a}, where $Pq = h$). Unfasten the end~$Q$ of the rod~$PQ$ and\nturn~$PQ$ about~$P$ until $Q$~can be fastened to the third circular rod in the\nposition~$Q'$. The angle $qOQ' = \\alpha$ in the figure is evidently given by\n\\[\nl^{2} - h^{2} = qQ'^{2} = \\left (2a\\sin\\tfrac{1}{2} \\alpha\\right)^{2}.\n\\]\nLet all the other rods of which the cylinder was composed be treated in the\nsame way. We obtain a ruled surface whose form is indicated in \\Fig{18b}.\nIt is entirely built up of straight lines; but the surface is curved everywhere,\nand is in general shape not unlike certain forms of table-napkin rings (\\Fig{18c}).\n%[Illustration: Fig. 18a.]\n%[Illustration: Fig. 18b.]\n%[Illustration: Fig. 18c.]\n\\begin{figure}[hbt!]\n \\begin{minipage}{0.3\\textwidth}\n \\centering\n \\Graphic{1.5in}{p064a}\n \\caption{Fig.~18a.}\n \\label{fig:18a}\n \\end{minipage}\\hfill\n \\begin{minipage}{0.3\\textwidth}\n \\centering\n \\Graphic{1.5in}{p064b}\n \\caption{Fig.~18b.}\n \\label{fig:18b}\n \\end{minipage}\\hfill\n \\begin{minipage}{0.3\\textwidth}\n \\centering\n \\Graphic{1.5in}{p064c}\n \\caption{Fig.~18c.}\n \\label{fig:18c}\n \\end{minipage}\n\\end{figure}", "markdown": "**surfaces.** Cylinders and cones are special cases of *surfaces composed of straight lines*. Such surfaces are called *ruled surfaces*. The two equations x = az + b,0pt minus 3pty = cz + d, (1) represent the intersection of two planes, *i.e.* a straight line. Now suppose that $a$, $b$, $c$, $d$ instead of being fixed are *functions of an auxiliary variable $t$*. For any particular value of $t$ the equations (1) give a line. As $t$ varies, this line moves and generates a surface, whose equation may be found by eliminating $t$ between the two equations (1). For instance, in Ex. 7 the equations of the line which generates the cone are x = zt,0pt minus 3pty = zt, where $t$ is the angle between the plane $XOZ$ and a plane through the line and the axis of $z$. Another simple example of a ruled surface may be constructed as follows. Take two sections of a right circular cylinder perpendicular to the axis and at a distance $l$ apart ([fig:18a]Fig. 18a). We can imagine the surface of the cylinder to be made up of a number of thin parallel rigid rods of length $l$, such as $PQ$, the ends of the rods being fastened to two circular rods of radius $a$. Now let us take a third circular rod of the same radius and place it round the surface of the cylinder at a distance $h$ from one of the first two rods (see [fig:18a]Fig. 18a, where $Pq = h$). Unfasten the end $Q$ of the rod $PQ$ and turn $PQ$ about $P$ until $Q$ can be fastened to the third circular rod in the position $Q'$. The angle $qOQ' = \\alpha$ in the figure is evidently given by l^2 - h^2 = qQ’^2 = (2a12 )^2. Let all the other rods of which the cylinder was composed be treated in the same way. We obtain a ruled surface whose form is indicated in [fig:18b]Fig. 18b. It is entirely built up of straight lines; but the surface is curved everywhere, and is in general shape not unlike certain forms of table-napkin rings ([fig:18c]Fig. 18c). %[Illustration: Fig. 18a.] %[Illustration: Fig. 18b.] %[Illustration: Fig. 18c.] figure[hbt!] minipage0.3 1.5inp064a Fig. 18a. minipage minipage0.3 1.5inp064b Fig. 18b. minipage minipage0.3 1.5inp064c Fig. 18c. minipage figure", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xix/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xix", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "62", "location": "Exercise XIX, problem 2", "problem_latex": "Three linear equations in general represent a single point. What are\nthe exceptional cases?", "markdown": "Three linear equations in general represent a single point. What are the exceptional cases?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xix/3", "set": "hardy-course-of-pure-mathematics-1921/ex-xix", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "62", "location": "Exercise XIX, problem 3", "problem_latex": "What are the equations of a plane curve $f(x, y) = 0$ in the plane~$XOY$,\nwhen regarded as a curve in space? [$f(x, y) = 0$, $z = 0$.]", "markdown": "What are the equations of a plane curve $f(x, y) = 0$ in the plane $XOY$, when regarded as a curve in space? [$f(x, y) = 0$, $z = 0$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xix/4", "set": "hardy-course-of-pure-mathematics-1921/ex-xix", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "62", "location": "Exercise XIX, problem 4", "problem_latex": "\\Topic{Cylinders.} What is the meaning of a single equation $f(x, y) = 0$,\nconsidered as a locus in space of three dimensions?\n\n[All points on the surface satisfy $f(x, y) = 0$, whatever be the value of~$z$. The\ncurve $f(x, y) = 0$, $z = 0$ is the curve in which the locus cuts the plane~$XOY$.\nThe locus is the surface formed by drawing lines parallel to~$OZ$ through all\npoints of this curve. Such a surface is called a \\emph{cylinder}.]", "markdown": "**.** What is the meaning of a single equation $f(x, y) = 0$, considered as a locus in space of three dimensions? [All points on the surface satisfy $f(x, y) = 0$, whatever be the value of $z$. The curve $f(x, y) = 0$, $z = 0$ is the curve in which the locus cuts the plane $XOY$. The locus is the surface formed by drawing lines parallel to $OZ$ through all points of this curve. Such a surface is called a *cylinder*.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xix/5", "set": "hardy-course-of-pure-mathematics-1921/ex-xix", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "62", "location": "Exercise XIX, problem 5", "problem_latex": "\\Topic{Graphical representation of a surface on a plane. Contour Maps.}\nIt might seem to be impossible to represent a surface adequately by a\ndrawing on a plane; and so indeed it is: but a very fair notion of the\nnature of the surface may often be obtained as follows. Let the equation of\nthe surface be $z = f(x, y)$.\n\nIf we give~$z$ a particular value~$a$, we have an equation $f(x, y) = a$, which\nwe may regard as determining a plane curve on the paper. We trace this\ncurve and mark it~$(a)$. Actually the curve~$(a)$ is the projection on the plane~$XOY$\n\\PageSep{63}\nof the section of the surface by the plane $z = a$. We do this for all\nvalues of~$a$ (practically, of course, for a selection of values of~$a$). We obtain\nsome such figure as is shown in \\Fig{17}. It will at once suggest a contoured\nOrdnance Survey map: and in fact this is the principle on which such maps\nare constructed. The contour line~$1000$ is the projection, on the plane of the\nsea level, of the section of the surface of the land by the plane parallel to the\nplane of the sea level and $1000$~ft.\\ above it.\\footnote\n {We assume that the effects of the earth's curvature may be neglected.}\n%[Illustration: Fig. 17.]\n\\Figure{17}{p063}", "markdown": "**representation of a surface on a plane. Contour Maps.** It might seem to be impossible to represent a surface adequately by a drawing on a plane; and so indeed it is: but a very fair notion of the nature of the surface may often be obtained as follows. Let the equation of the surface be $z = f(x, y)$. If we give $z$ a particular value $a$, we have an equation $f(x, y) = a$, which we may regard as determining a plane curve on the paper. We trace this curve and mark it $(a)$. Actually the curve $(a)$ is the projection on the plane $XOY$ [pg]63 of the section of the surface by the plane $z = a$. We do this for all values of $a$ (practically, of course, for a selection of values of $a$). We obtain some such figure as is shown in [fig:17]Fig. 17. It will at once suggest a contoured Ordnance Survey map: and in fact this is the principle on which such maps are constructed. The contour line $1000$ is the projection, on the plane of the sea level, of the section of the surface of the land by the plane parallel to the plane of the sea level and $1000$ ft. above it. We assume that the effects of the earth’s curvature may be neglected. %[Illustration: Fig. 17.] 17p063", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xix/6", "set": "hardy-course-of-pure-mathematics-1921/ex-xix", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "62", "location": "Exercise XIX, problem 6", "problem_latex": "{\\Loosen Draw a series of contour lines to illustrate the form of the surface\n$2z = 3xy$.}", "markdown": "0.375em plus 0.75em minus 0.25emDraw a series of contour lines to illustrate the form of the surface $2z = 3xy$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xix/7", "set": "hardy-course-of-pure-mathematics-1921/ex-xix", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "62", "location": "Exercise XIX, problem 7", "problem_latex": "\\Topic{Right circular cones.} Take the origin of coordinates at the\nvertex of the cone and the axis of~$z$ along the axis of the cone; and let~$\\alpha$ be\nthe semi-vertical angle of the cone. The equation of the cone (which must\nbe regarded as extending both ways from its vertex) is $x^{2} + y^{2} - z^{2}\\tan^{2} \\alpha = 0$.", "markdown": "**circular cones.** Take the origin of coordinates at the vertex of the cone and the axis of $z$ along the axis of the cone; and let $\\alpha$ be the semi-vertical angle of the cone. The equation of the cone (which must be regarded as extending both ways from its vertex) is $x^{2} + y^{2} - z^{2}\\tan^{2} \\alpha = 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xix/8", "set": "hardy-course-of-pure-mathematics-1921/ex-xix", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "62", "location": "Exercise XIX, problem 8", "problem_latex": "\\Topic{Surfaces of revolution in general.} The cone of Ex.~7 cuts~$ZOX$ in\ntwo lines whose equations may be combined in the equation $x^{2} = z^{2}\\tan^{2}\\alpha$.\nThat is to say, the equation of the surface generated by the revolution of\nthe curve $y = 0$, $x^{2} = z^{2}\\tan^{2}\\alpha$ round the axis of~$z$ is derived from the second of\nthese equations by changing~$x^{2}$ into~$x^{2} + y^{2}$. Show generally that the equation\nof the surface generated by the revolution of the curve $y = 0$, $x = f(z)$, round\nthe axis of~$z$, is\n\\[\n\\sqrtp{x^{2} + y^{2}} = f(z).\n\\]", "markdown": "**of revolution in general.** The cone of Ex. 7 cuts $ZOX$ in two lines whose equations may be combined in the equation $x^{2} = z^{2}\\tan^{2}\\alpha$. That is to say, the equation of the surface generated by the revolution of the curve $y = 0$, $x^{2} = z^{2}\\tan^{2}\\alpha$ round the axis of $z$ is derived from the second of these equations by changing $x^{2}$ into $x^{2} + y^{2}$. Show generally that the equation of the surface generated by the revolution of the curve $y = 0$, $x = f(z)$, round the axis of $z$, is x^2 + y^2 = f(z).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xix/9", "set": "hardy-course-of-pure-mathematics-1921/ex-xix", "number": 9, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "62", "location": "Exercise XIX, problem 9", "problem_latex": "\\Topic{Cones in general.} A surface formed by straight lines passing\nthrough a fixed point is called a \\emph{cone}: the point is called the \\emph{vertex}. A\nparticular case is given by the right circular cone of Ex.~7. Show that the\nequation of a cone whose vertex is~$O$ is of the form $f(z/x, z/y) = 0$, and that any\nequation of this form represents a cone. [If $(x, y, z)$ lies on the cone, so must\n$(\\lambda x, \\lambda y, \\lambda z)$, for any value of~$\\lambda$.]", "markdown": "**in general.** A surface formed by straight lines passing through a fixed point is called a *cone*: the point is called the *vertex*. A particular case is given by the right circular cone of Ex. 7. Show that the equation of a cone whose vertex is $O$ is of the form $f(z/x, z/y) = 0$, and that any equation of this form represents a cone. [If $(x, y, z)$ lies on the cone, so must $(\\lambda x, \\lambda y, \\lambda z)$, for any value of $\\lambda$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xl/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xl", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "206", "location": "Exercise XL, problem 1", "problem_latex": "If $y = y_{1}y_{2}y_{3}$ then\n\\[\n\\frac{dy}{dx}\n = y_{2}y_{3}\\, \\frac{dy_{1}}{dx}\n + y_{3}y_{1}\\, \\frac{dy_{2}}{dx}\n + y_{1}y_{2}\\, \\frac{dy_{3}}{dx},\n\\]\nand if $y = y_{1}y_{2} \\dots y_{n}$ then\n\\[\n\\frac{dy}{dx}\n = \\sum_{r=1}^{n} y_{1}y_{2} \\dots y_{r-1}y_{r+1} \\dots y_{n}\\, \\frac{dy_{r}}{dx}.\n\\]\nIn particular, if $y = z^{n}$, then $dy/dx = nz^{n-1}(dz/dx)$; and if $y = x^{n}$, then\n$dy/dx = nx^{n-1}$, as was proved otherwise in \\Ex{xxxix}.~3.", "markdown": "If $y = y_{1}y_{2}y_{3}$ then dydx = y_2y_3  dy_1dx + y_3y_1  dy_2dx + y_1y_2  dy_3dx, and if $y = y_{1}y_{2} \\dots y_{n}$ then dydx = _r=1^n y_1y_2 …y_r-1y_r+1 …y_n  dy_rdx. In particular, if $y = z^{n}$, then $dy/dx = nz^{n-1}(dz/dx)$; and if $y = x^{n}$, then $dy/dx = nx^{n-1}$, as was proved otherwise in % [examples:xxxix]Ex. xxxix%. 3.", "answer_latex": [ "If $y = y_{1}y_{2}y_{3}$ then\n\\[\n\\frac{dy}{dx}\n = y_{2}y_{3}\\, \\frac{dy_{1}}{dx}\n + y_{3}y_{1}\\, \\frac{dy_{2}}{dx}\n + y_{1}y_{2}\\, \\frac{dy_{3}}{dx},\n\\]\nand if $y = y_{1}y_{2} \\dots y_{n}$ then\n\\[\n\\frac{dy}{dx}\n = \\sum_{r=1}^{n} y_{1}y_{2} \\dots y_{r-1}y_{r+1} \\dots y_{n}\\, \\frac{dy_{r}}{dx}.\n\\]\nIn particular, if $y = z^{n}$, then $dy/dx = nz^{n-1}(dz/dx)$; and if $y = x^{n}$, then\n$dy/dx = nx^{n-1}$, as was proved otherwise in \\Ex{xxxix}.~3." ], "answer_markdown": [ "If $y = y_{1}y_{2}y_{3}$ then dydx = y_2y_3  dy_1dx + y_3y_1  dy_2dx + y_1y_2  dy_3dx, and if $y = y_{1}y_{2} \\dots y_{n}$ then dydx = _r=1^n y_1y_2 …y_r-1y_r+1 …y_n  dy_rdx. In particular, if $y = z^{n}$, then $dy/dx = nz^{n-1}(dz/dx)$; and if $y = x^{n}$, then $dy/dx = nx^{n-1}$, as was proved otherwise in % [examples:xxxix]Ex. xxxix%. 3." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xl/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xl", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "206", "location": "Exercise XL, problem 2", "problem_latex": "If $y = y_{1}y_{2}\\dots y_{n}$ then\n\\[\n\\frac{1}{y}\\, \\frac{dy}{dx}\n = \\frac{1}{y_{1}}\\, \\frac{dy_{1}}{dx}\n + \\frac{1}{y_{2}}\\, \\frac{dy_{2}}{dx} + \\dots\n + \\frac{1}{y_{n}}\\, \\frac{dy_{n}}{dx}.\n\\]\nIn particular, if $y = z^{n}$, then $\\dfrac{1}{y}\\, \\dfrac{dy}{dx} = \\dfrac{n}{z}\\, \\dfrac{dz}{dx}$.", "markdown": "If $y = y_{1}y_{2}\\dots y_{n}$ then 1y  dydx = 1y_1  dy_1dx + 1y_2  dy_2dx + … + 1y_n  dy_ndx. In particular, if $y = z^{n}$, then $\\dfrac{1}{y}\\, \\dfrac{dy}{dx} = \\dfrac{n}{z}\\, \\dfrac{dz}{dx}$.", "answer_latex": [ "If $y = y_{1}y_{2}\\dots y_{n}$ then\n\\[\n\\frac{1}{y}\\, \\frac{dy}{dx}\n = \\frac{1}{y_{1}}\\, \\frac{dy_{1}}{dx}\n + \\frac{1}{y_{2}}\\, \\frac{dy_{2}}{dx} + \\dots\n + \\frac{1}{y_{n}}\\, \\frac{dy_{n}}{dx}.\n\\]\nIn particular, if $y = z^{n}$, then $\\dfrac{1}{y}\\, \\dfrac{dy}{dx} = \\dfrac{n}{z}\\, \\dfrac{dz}{dx}$." ], "answer_markdown": [ "If $y = y_{1}y_{2}\\dots y_{n}$ then 1y  dydx = 1y_1  dy_1dx + 1y_2  dy_2dx + … + 1y_n  dy_ndx. In particular, if $y = z^{n}$, then $\\dfrac{1}{y}\\, \\dfrac{dy}{dx} = \\dfrac{n}{z}\\, \\dfrac{dz}{dx}$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xli/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xli", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "208", "location": "Exercise XLI, problem 1", "problem_latex": "Show that if $\\phi(x)$~is a polynomial then $\\phi'(x)$~is\nthe coefficient of~$h$ in the expansion of~$\\phi(x + h)$ in powers of~$h$.", "markdown": "Show that if $\\phi(x)$ is a polynomial then $\\phi'(x)$ is the coefficient of $h$ in the expansion of $\\phi(x + h)$ in powers of $h$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xli/10", "set": "hardy-course-of-pure-mathematics-1921/ex-xli", "number": 10, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "208", "location": "Exercise XLI, problem 10", "problem_latex": "\\Topic{Rolle's Theorem for polynomials.} If $\\phi(x)$~is any polynomial,\nthen between any pair of roots of $\\phi(x) = 0$ lies a root of $\\phi'(x) = 0$.", "markdown": "**’s Theorem for polynomials.** If $\\phi(x)$ is any polynomial, then between any pair of roots of $\\phi(x) = 0$ lies a root of $\\phi'(x) = 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xli/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xli", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "208", "location": "Exercise XLI, problem 2", "problem_latex": "If $\\phi(x)$~is divisible by~$(x - \\alpha)^{2}$, then $\\phi'(x)$~is divisible by~$x - \\alpha$: and\ngenerally, if $\\phi(x)$~is divisible by~$(x - \\alpha)^{m}$, then $\\phi'(x)$~is divisible by~$(x - \\alpha)^{m-1}$.", "markdown": "If $\\phi(x)$ is divisible by $(x - \\alpha)^{2}$, then $\\phi'(x)$ is divisible by $x - \\alpha$: and generally, if $\\phi(x)$ is divisible by $(x - \\alpha)^{m}$, then $\\phi'(x)$ is divisible by $(x - \\alpha)^{m-1}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.pdiv" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xli/3", "set": "hardy-course-of-pure-mathematics-1921/ex-xli", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "208", "location": "Exercise XLI, problem 3", "problem_latex": "Conversely, if $\\phi(x)$ and~$\\phi'(x)$ are \\emph{both} divisible by~$x - \\alpha$, then $\\phi(x)$~is\ndivisible by~$(x - \\alpha)^{2}$; and if $\\phi(x)$~is divisible by~$x - \\alpha$ and $\\phi'(x)$ by~$(x - \\alpha)^{m-1}$,\nthen $\\phi(x)$~is divisible by~$(x - \\alpha)^{m}$.", "markdown": "Conversely, if $\\phi(x)$ and $\\phi'(x)$ are *both* divisible by $x - \\alpha$, then $\\phi(x)$ is divisible by $(x - \\alpha)^{2}$; and if $\\phi(x)$ is divisible by $x - \\alpha$ and $\\phi'(x)$ by $(x - \\alpha)^{m-1}$, then $\\phi(x)$ is divisible by $(x - \\alpha)^{m}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.pdiv" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xli/4", "set": "hardy-course-of-pure-mathematics-1921/ex-xli", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "208", "location": "Exercise XLI, problem 4", "problem_latex": "Show how to determine as completely as possible the multiple roots\nof $P(x) = 0$, where $P(x)$~is a polynomial, with their degrees of multiplicity,\nby means of the elementary algebraical operations.\n\n[If $H_{1}$~is the highest common factor of $P$~and~$P'$, $H_{2}$~the highest common\nfactor of $H_{1}$ and~$P''$, $H_{3}$ that of $H_{2}$ and~$P'''$, and so on, then the roots of\n$H_{1}H_{3}/H_{2}^{2} = 0$ are the \\emph{double} roots of $P = 0$, the roots of $H_{2}H_{4}/H_{3}^{2} = 0$ the \\emph{treble}\nroots, and so on. But it may not be possible to complete the solution of\n$H_{1}H_{3}/H_{2}^{2} = 0$, $H_{2}H_{4}/H_{3}^{2} = 0$,~\\dots. Thus if $P(x) = (x - 1)^{3}(x^{5} - x - 7)^{2}$ then\n$H_{1}H_{3}/H_{2}^{2} = x^{5} - x - 7$ and $H_{2}H_{4}/H_{3}^{2} = x - 1$; and we cannot solve the first\nequation.]", "markdown": "Show how to determine as completely as possible the multiple roots of $P(x) = 0$, where $P(x)$ is a polynomial, with their degrees of multiplicity, by means of the elementary algebraical operations. [If $H_{1}$ is the highest common factor of $P$ and $P'$, $H_{2}$ the highest common factor of $H_{1}$ and $P''$, $H_{3}$ that of $H_{2}$ and $P'''$, and so on, then the roots of $H_{1}H_{3}/H_{2}^{2} = 0$ are the *double* roots of $P = 0$, the roots of $H_{2}H_{4}/H_{3}^{2} = 0$ the *treble* roots, and so on. But it may not be possible to complete the solution of $H_{1}H_{3}/H_{2}^{2} = 0$, $H_{2}H_{4}/H_{3}^{2} = 0$, …. Thus if $P(x) = (x - 1)^{3}(x^{5} - x - 7)^{2}$ then $H_{1}H_{3}/H_{2}^{2} = x^{5} - x - 7$ and $H_{2}H_{4}/H_{3}^{2} = x - 1$; and we cannot solve the first equation.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.cancel", "cas.derive", "cas.pgcd" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xli/5a", "set": "hardy-course-of-pure-mathematics-1921/ex-xli", "number": 5, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "208", "location": "Exercise XLI, problem 5a", "problem_latex": "Find all the roots, with their degrees of multiplicity, of\n\\[\nx^{4} + 3x^{3} - 3x^{2} - 11x - 6 = 0,\\quad\nx^{6} + 2x^{5} - 8x^{4} - 14x^{3} + 11x^{2} + 28x + 12 = 0.\n\\]", "markdown": "Find all the roots, with their degrees of multiplicity, of x^4 + 3x^3 - 3x^2 - 11x - 6 = 0,0pt minus 3ptx^6 + 2x^5 - 8x^4 - 14x^3 + 11x^2 + 28x + 12 = 0.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "solve", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "Eq(x**4 + 3*x**3 - 3*x**2 - 11*x - 6, 0)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "solve: Eq(x**4 + 3*x**3 - 3*x**2 - 11*x - 6, 0)" ], "shape": [ "solve: Eq(N*x + 2*N*x**N + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xli/5b", "set": "hardy-course-of-pure-mathematics-1921/ex-xli", "number": 5, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "208", "location": "Exercise XLI, problem 5b", "problem_latex": "Find all the roots, with their degrees of multiplicity, of\n\\[\nx^{4} + 3x^{3} - 3x^{2} - 11x - 6 = 0,\\quad\nx^{6} + 2x^{5} - 8x^{4} - 14x^{3} + 11x^{2} + 28x + 12 = 0.\n\\]", "markdown": "Find all the roots, with their degrees of multiplicity, of x^4 + 3x^3 - 3x^2 - 11x - 6 = 0,0pt minus 3ptx^6 + 2x^5 - 8x^4 - 14x^3 + 11x^2 + 28x + 12 = 0.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "solve", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "Eq(x**6 + 2*x**5 - 8*x**4 - 14*x**3 + 11*x**2 + 28*x + 12, 0)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "solve: Eq(x**6 + 2*x**5 - 8*x**4 - 14*x**3 + 11*x**2 + 28*x + 12, 0)" ], "shape": [ "solve: Eq(N*x + 4*N*x**N + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.factor", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xli/6", "set": "hardy-course-of-pure-mathematics-1921/ex-xli", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "208", "location": "Exercise XLI, problem 6", "problem_latex": "If $ax^{2} + 2bx + c$ has a double root, \\ie\\ is of the form $a(x - \\alpha)^{2}$, then\n$2(ax + b)$~must be divisible by~$x - \\alpha$, so that $\\alpha = -b/a$. This value of~$x$ must\nsatisfy $ax^{2} + 2bx + c = 0$. Verify that the condition thus arrived at is\n$ac - b^{2} = 0$.", "markdown": "If $ax^{2} + 2bx + c$ has a double root, *i.e.* is of the form $a(x - \\alpha)^{2}$, then $2(ax + b)$ must be divisible by $x - \\alpha$, so that $\\alpha = -b/a$. This value of $x$ must satisfy $ax^{2} + 2bx + c = 0$. Verify that the condition thus arrived at is $ac - b^{2} = 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.expand", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xli/7", "set": "hardy-course-of-pure-mathematics-1921/ex-xli", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "208", "location": "Exercise XLI, problem 7", "problem_latex": "The equation $1/(x - a) + 1/(x - b) + 1/(x - c) = 0$ can have a pair of\nequal roots only if $a = b = c$. \\MathTrip{1905.}", "markdown": "The equation $1/(x - a) + 1/(x - b) + 1/(x - c) = 0$ can have a pair of equal roots only if $a = b = c$. % [0]% (*Math. Trip.* 1905.)% [1]%", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.cancel", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xli/8", "set": "hardy-course-of-pure-mathematics-1921/ex-xli", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "208", "location": "Exercise XLI, problem 8", "problem_latex": "Show that\n\\[\nax^{3} + 3bx^{2} + 3cx + d = 0\n\\]\nhas a double root if $G^{2} + 4H^{3} = 0$, where $H = ac - b^{2}$, $G = a^{2}d - 3abc + 2b^{3}$.\n\n[Put $ax + b = y$, when the equation reduces to $y^{3} + 3Hy + G = 0$. This\nmust have a root in common with $y^{2} + H = 0$.]", "markdown": "Show that ax^3 + 3bx^2 + 3cx + d = 0 has a double root if $G^{2} + 4H^{3} = 0$, where $H = ac - b^{2}$, $G = a^{2}d - 3abc + 2b^{3}$. [Put $ax + b = y$, when the equation reduces to $y^{3} + 3Hy + G = 0$. This must have a root in common with $y^{2} + H = 0$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "cas.pgcd", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xli/9", "set": "hardy-course-of-pure-mathematics-1921/ex-xli", "number": 9, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "208", "location": "Exercise XLI, problem 9", "problem_latex": "The reader may verify that if $\\alpha$,~$\\beta$, $\\gamma$,~$\\delta$ are the roots of\n\\[\nax^{4} + 4bx^{3} + 6cx^{2} + 4dx + e = 0,\n\\]\nthen the equation whose roots are\n\\[\n\\tfrac{1}{12}a \\{\n (\\alpha - \\beta)(\\gamma - \\delta) - (\\gamma - \\alpha)(\\beta - \\delta)\n\\},\n\\]\nand two similar expressions formed by permuting $\\alpha$,~$\\beta$,~$\\gamma$ cyclically, is\n\\[\n4\\theta^{3} - g_{2}\\theta - g_{3} = 0,\n\\]\nwhere\n\\[\ng_{2} = ae - 4bd + 3c^{2},\\quad\ng_{3} = ace + 2bcd - ad^{2} - eb^{2} - c^{3}.\n\\]\nIt is clear that if two of $\\alpha$,~$\\beta$, $\\gamma$,~$\\delta$ are equal then two of the roots of this cubic\nwill be equal. Using the result of Ex.~8 we deduce that $g_{2}^{3} - 27g_{3}^{2} = 0$.", "markdown": "The reader may verify that if $\\alpha$, $\\beta$, $\\gamma$, $\\delta$ are the roots of ax^4 + 4bx^3 + 6cx^2 + 4dx + e = 0, then the equation whose roots are 112a (- )(- ) - (- )(- ) , and two similar expressions formed by permuting $\\alpha$, $\\beta$, $\\gamma$ cyclically, is 4^3 - g_2- g_3 = 0, where g_2 = ae - 4bd + 3c^2,0pt minus 3ptg_3 = ace + 2bcd - ad^2 - eb^2 - c^3. It is clear that if two of $\\alpha$, $\\beta$, $\\gamma$, $\\delta$ are equal then two of the roots of this cubic will be equal. Using the result of Ex. 8 we deduce that $g_{2}^{3} - 27g_{3}^{2} = 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlii/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xlii", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "210", "location": "Exercise XLII, problem 1", "problem_latex": " Prove that\n\\[\n\\frac{d}{dx}\\left(\\frac{x}{1 + x^{2}}\\right)\n = \\frac{1 - x^{2}}{(1 + x^{2})^{2}},\\quad\n\\frac{d}{dx}\\left(\\frac{1 - x^{2}}{1 + x^{2}}\\right)\n = -\\frac{4x}{(1 + x^{2})^{2}}.\n\\]", "markdown": "Prove that ddx(x1 + x^2) = 1 - x^2(1 + x^2)^2,0pt minus 3ptddx(1 - x^21 + x^2) = -4x(1 + x^2)^2.", "answer_latex": [ " Prove that\n\\[\n\\frac{d}{dx}\\left(\\frac{x}{1 + x^{2}}\\right)\n = \\frac{1 - x^{2}}{(1 + x^{2})^{2}},\\quad\n\\frac{d}{dx}\\left(\\frac{1 - x^{2}}{1 + x^{2}}\\right)\n = -\\frac{4x}{(1 + x^{2})^{2}}.\n\\]" ], "answer_markdown": [ "Prove that ddx(x1 + x^2) = 1 - x^2(1 + x^2)^2,0pt minus 3ptddx(1 - x^21 + x^2) = -4x(1 + x^2)^2." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlii/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xlii", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "210", "location": "Exercise XLII, problem 2", "problem_latex": " Prove that\n\\[\n\\frac{d}{dx}\\left(\\frac{ax^{2} + 2bx + c}{Ax^{2} + 2Bx + C}\\right)\n = \\frac{(ax + b) (Bx + C) - (bx + c) (Ax + B)}{(Ax^{2} + 2Bx + C)^{2}}.\n\\]", "markdown": "Prove that ddx(ax^2 + 2bx + cAx^2 + 2Bx + C) = (ax + b) (Bx + C) - (bx + c) (Ax + B)(Ax^2 + 2Bx + C)^2.", "answer_latex": [ " Prove that\n\\[\n\\frac{d}{dx}\\left(\\frac{ax^{2} + 2bx + c}{Ax^{2} + 2Bx + C}\\right)\n = \\frac{(ax + b) (Bx + C) - (bx + c) (Ax + B)}{(Ax^{2} + 2Bx + C)^{2}}.\n\\]" ], "answer_markdown": [ "Prove that ddx(ax^2 + 2bx + cAx^2 + 2Bx + C) = (ax + b) (Bx + C) - (bx + c) (Ax + B)(Ax^2 + 2Bx + C)^2." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.expand", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlii/3", "set": "hardy-course-of-pure-mathematics-1921/ex-xlii", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "210", "location": "Exercise XLII, problem 3", "problem_latex": " If $Q$~has a factor $(x - \\alpha)^{m}$ then the denominator of~$R'$ (when $R'$~is\nreduced to its lowest terms) is divisible by~$(x - \\alpha)^{m+1}$ but by no higher power\nof~$x - \\alpha$.", "markdown": "If $Q$ has a factor $(x - \\alpha)^{m}$ then the denominator of $R'$ (when $R'$ is reduced to its lowest terms) is divisible by $(x - \\alpha)^{m+1}$ but by no higher power of $x - \\alpha$.", "answer_latex": [ " If $Q$~has a factor $(x - \\alpha)^{m}$ then the denominator of~$R'$ (when $R'$~is\nreduced to its lowest terms) is divisible by~$(x - \\alpha)^{m+1}$ but by no higher power\nof~$x - \\alpha$." ], "answer_markdown": [ "If $Q$ has a factor $(x - \\alpha)^{m}$ then the denominator of $R'$ (when $R'$ is reduced to its lowest terms) is divisible by $(x - \\alpha)^{m+1}$ but by no higher power of $x - \\alpha$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlii/4", "set": "hardy-course-of-pure-mathematics-1921/ex-xlii", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "210", "location": "Exercise XLII, problem 4", "problem_latex": " In no case can the denominator of~$R'$ have a \\emph{simple} factor~$x - \\alpha$.\nHence no rational function (such as~$1/x$) whose denominator contains any\nsimple factor can be the derivative of another rational function.", "markdown": "In no case can the denominator of $R'$ have a *simple* factor $x - \\alpha$. Hence no rational function (such as $1/x$) whose denominator contains any simple factor can be the derivative of another rational function.", "answer_latex": [ " In no case can the denominator of~$R'$ have a \\emph{simple} factor~$x - \\alpha$.\nHence no rational function (such as~$1/x$) whose denominator contains any\nsimple factor can be the derivative of another rational function." ], "answer_markdown": [ "In no case can the denominator of $R'$ have a *simple* factor $x - \\alpha$. Hence no rational function (such as $1/x$) whose denominator contains any simple factor can be the derivative of another rational function." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xliii/1a", "set": "hardy-course-of-pure-mathematics-1921/ex-xliii", "number": 1, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "211", "location": "Exercise XLIII, problem 1a", "problem_latex": " Find the derivatives of\n\\[\n\\bigsqrtp{\\frac{1 + x}{1 - x}},\\quad\n\\bigsqrtp{\\frac{ax + b}{cx + d}},\\quad\n\\bigsqrtp{\\frac{ax^{2} + 2bx + c}{Ax^{2} + 2Bx + C}},\\quad\n(ax + b)^{m} (cx + d)^{n}.\n\\]", "markdown": "Find the derivatives of 1 + x1 - x,0pt minus 3ptax + bcx + d,0pt minus 3ptax^2 + 2bx + cAx^2 + 2Bx + C,0pt minus 3pt(ax + b)^m (cx + d)^n.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "differentiate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "sqrt((1 + x)/(1 - x))", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "differentiate: sqrt((x + 1)/(-x + 1))" ], "shape": [ "differentiate: ((x + 1)/(-x + 1))**N" ], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xliii/1b", "set": "hardy-course-of-pure-mathematics-1921/ex-xliii", "number": 1, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "211", "location": "Exercise XLIII, problem 1b", "problem_latex": " Find the derivatives of\n\\[\n\\bigsqrtp{\\frac{1 + x}{1 - x}},\\quad\n\\bigsqrtp{\\frac{ax + b}{cx + d}},\\quad\n\\bigsqrtp{\\frac{ax^{2} + 2bx + c}{Ax^{2} + 2Bx + C}},\\quad\n(ax + b)^{m} (cx + d)^{n}.\n\\]", "markdown": "Find the derivatives of 1 + x1 - x,0pt minus 3ptax + bcx + d,0pt minus 3ptax^2 + 2bx + cAx^2 + 2Bx + C,0pt minus 3pt(ax + b)^m (cx + d)^n.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "differentiate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "sqrt((a*x + b)/(c*x + d))", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "differentiate: sqrt((a*x + b)/(c*x + d))" ], "shape": [ "differentiate: ((a*x + b)/(c*x + d))**N" ], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xliii/1c", "set": "hardy-course-of-pure-mathematics-1921/ex-xliii", "number": 1, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "211", "location": "Exercise XLIII, problem 1c", "problem_latex": " Find the derivatives of\n\\[\n\\bigsqrtp{\\frac{1 + x}{1 - x}},\\quad\n\\bigsqrtp{\\frac{ax + b}{cx + d}},\\quad\n\\bigsqrtp{\\frac{ax^{2} + 2bx + c}{Ax^{2} + 2Bx + C}},\\quad\n(ax + b)^{m} (cx + d)^{n}.\n\\]", "markdown": "Find the derivatives of 1 + x1 - x,0pt minus 3ptax + bcx + d,0pt minus 3ptax^2 + 2bx + cAx^2 + 2Bx + C,0pt minus 3pt(ax + b)^m (cx + d)^n.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "differentiate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "sqrt((a*x**2 + 2*b*x + c)/(A*x**2 + 2*B*x + C))", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "differentiate: sqrt((d*x**2 + 2*e*x + f)/(a*x**2 + 2*b*x + c))" ], "shape": [ "differentiate: ((N*e*x + d*x**N + f)/(N*b*x + a*x**N + c))**N" ], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xliii/1d", "set": "hardy-course-of-pure-mathematics-1921/ex-xliii", "number": 1, "part": "d", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "211", "location": "Exercise XLIII, problem 1d", "problem_latex": " Find the derivatives of\n\\[\n\\bigsqrtp{\\frac{1 + x}{1 - x}},\\quad\n\\bigsqrtp{\\frac{ax + b}{cx + d}},\\quad\n\\bigsqrtp{\\frac{ax^{2} + 2bx + c}{Ax^{2} + 2Bx + C}},\\quad\n(ax + b)^{m} (cx + d)^{n}.\n\\]", "markdown": "Find the derivatives of 1 + x1 - x,0pt minus 3ptax + bcx + d,0pt minus 3ptax^2 + 2bx + cAx^2 + 2Bx + C,0pt minus 3pt(ax + b)^m (cx + d)^n.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "differentiate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "(a*x + b)**m * (c*x + d)**n", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "differentiate: (a*x + b)**e*(c*x + d)**f" ], "shape": [ "differentiate: (a*x + b)**e*(c*x + d)**f" ], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xliii/2a", "set": "hardy-course-of-pure-mathematics-1921/ex-xliii", "number": 2, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "211", "location": "Exercise XLIII, problem 2a", "problem_latex": " Prove that\n\\[\n\\frac{d}{dx}\\left\\{\\frac{x}{\\sqrtp{a^{2} + x^{2}}}\\right\\}\n = \\frac{a^{2}}{(a^{2} + x^{2})^{(3/2)}},\\quad\n\\frac{d}{dx}\\left\\{\\frac{x}{\\sqrtp{a^{2} - x^{2}}}\\right\\}\n = \\frac{a^{2}}{(a^{2} - x^{2})^{3/2}}.\n\\]", "markdown": "Prove that ddxxa^2 + x^2 = a^2(a^2 + x^2)^(3/2),0pt minus 3ptddxxa^2 - x^2 = a^2(a^2 - x^2)^3/2.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "x/sqrt(a**2 + x**2)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xliii/2b", "set": "hardy-course-of-pure-mathematics-1921/ex-xliii", "number": 2, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "211", "location": "Exercise XLIII, problem 2b", "problem_latex": " Prove that\n\\[\n\\frac{d}{dx}\\left\\{\\frac{x}{\\sqrtp{a^{2} + x^{2}}}\\right\\}\n = \\frac{a^{2}}{(a^{2} + x^{2})^{(3/2)}},\\quad\n\\frac{d}{dx}\\left\\{\\frac{x}{\\sqrtp{a^{2} - x^{2}}}\\right\\}\n = \\frac{a^{2}}{(a^{2} - x^{2})^{3/2}}.\n\\]", "markdown": "Prove that ddxxa^2 + x^2 = a^2(a^2 + x^2)^(3/2),0pt minus 3ptddxxa^2 - x^2 = a^2(a^2 - x^2)^3/2.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "x/sqrt(a**2 - x**2)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xliii/3i", "set": "hardy-course-of-pure-mathematics-1921/ex-xliii", "number": 3, "part": "i", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "211", "location": "Exercise XLIII, problem 3i", "problem_latex": " Find the differential coefficient of $y$ when\n\\[\n\\Itemp{(i)} ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0,\\quad\n\\Itemp{(ii)} x^{5} + y^{5} - 5ax^{2}y^{2} = 0.\n\\]", "markdown": "Find the differential coefficient of $y$ when % [2.25em][l](i)% [2.25em][l](i)% % ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0,0pt minus 3pt% [2.25em][l](ii)% [2.25em][l](ii)% % x^5 + y^5 - 5ax^2y^2 = 0.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "differentiate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "a*x**2 + 2*h*x*y + b*y**2 + 2*g*x + 2*f*y + c", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "differentiate: a*x**2 + b*g**2 + c + 2*d*g + 2*e*x + 2*f*g*x" ], "shape": [ "differentiate: N*d*g + N*e*x + N*f*g*x + a*x**N + b*g**N + c" ], "same_problem_in": [], "needs": [ "cas.implicit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xliii/3ii", "set": "hardy-course-of-pure-mathematics-1921/ex-xliii", "number": 3, "part": "ii", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "211", "location": "Exercise XLIII, problem 3ii", "problem_latex": " Find the differential coefficient of $y$ when\n\\[\n\\Itemp{(i)} ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0,\\quad\n\\Itemp{(ii)} x^{5} + y^{5} - 5ax^{2}y^{2} = 0.\n\\]", "markdown": "Find the differential coefficient of $y$ when % [2.25em][l](i)% [2.25em][l](i)% % ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0,0pt minus 3pt% [2.25em][l](ii)% [2.25em][l](ii)% % x^5 + y^5 - 5ax^2y^2 = 0.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "differentiate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "x**5 + y**5 - 5*a*x**2*y**2", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "differentiate: -5*a*b**2*x**2 + b**5 + x**5" ], "shape": [ "differentiate: N*a*b**N*x**N + b**N + x**N" ], "same_problem_in": [], "needs": [ "cas.implicit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xliv/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xliv", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "212", "location": "Exercise XLIV, problem 1", "problem_latex": "Find the derivatives of\\footnotetext\n {In these examples $m$~is a rational number and $a$, $b$,~\\dots, $\\alpha$, $\\beta$~\\dots\\ have such\n values that the functions which involve them are real.}\n\n\\begin{gather*}\n\\cos^{m} x, \\quad \\sin^{m} x, \\quad\n\\cos x^{m}, \\quad \\sin x^{m}, \\quad\n\\cos (\\sin x), \\quad \\sin (\\cos x),\\\\\n\\sqrtp{a^{2}\\cos^{2} x + b^{2}\\sin^{2} x}, \\quad\n\\frac{\\cos x\\sin x}{\\sqrtp{a^{2}\\cos^{2} x + b^{2}\\sin^{2} x}},\\\\\nx\\arcsin x + \\sqrtp{1 - x^{2}}, \\quad\n(1 + x)\\arctan\\sqrt{x} - \\sqrt{x}.\n\\end{gather*}", "markdown": "Find the derivatives of In these examples $m$ is a rational number and $a$, $b$, …, $\\alpha$, $\\beta$ … have such values that the functions which involve them are real. gather* ^m x, 0pt minus 3pt^m x, 0pt minus 3ptx^m, 0pt minus 3ptx^m, 0pt minus 3pt(x), 0pt minus 3pt(x), a^2^2 x + b^2^2 x, 0pt minus 3ptxxa^2^2 x + b^2^2 x, xx + 1 - x^2, 0pt minus 3pt(1 + x)x - x. gather*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "differentiate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "differentiate: (sin(x)*cos(x)/sqrt(a**2*cos(x)**2 + b**2*sin(x)**2), -sqrt(x) + (x + 1)*atan(sqrt(x)), sqrt(a**2*cos(x)**2 + b**2*sin(x)**2), x*asin(x) + sqrt(-x**2 + 1), sin(x)**c, sin(x**c), sin(cos(x)), cos(x)**c, cos(x**c), cos(sin(x)))" ], "shape": [ "differentiate: ((a**N*cos(x)**N + b**N*sin(x)**N)**N*sin(x)*cos(x), -x**N + (x + 1)*atan(x**N), (a**N*cos(x)**N + b**N*sin(x)**N)**N, x*asin(x) + (-x**N + 1)**N, sin(x)**c, sin(x**c), sin(cos(x)), cos(x)**c, cos(x**c), cos(sin(x)))" ], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xliv/10", "set": "hardy-course-of-pure-mathematics-1921/ex-xliv", "number": 10, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "212", "location": "Exercise XLIV, problem 10", "problem_latex": "Prove that the derivative of $F[f\\{\\phi(x)\\}]$ is $F'[f\\{\\phi(x)\\}]\\, f'\\{\\phi(x)\\}\\phi'(x)$,\nand extend the result to still more complicated cases.", "markdown": "Prove that the derivative of $F[f\\{\\phi(x)\\}]$ is $F'[f\\{\\phi(x)\\}]\\, f'\\{\\phi(x)\\}\\phi'(x)$, and extend the result to still more complicated cases.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xliv/11", "set": "hardy-course-of-pure-mathematics-1921/ex-xliv", "number": 11, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "212", "location": "Exercise XLIV, problem 11", "problem_latex": "If $u$~and~$v$ are functions of~$x$, then\n\\[\nD_{x} \\arctan(u/v) = (vD_{x}u - uD_{x}v)/(u^{2} + v^{2}).\n\\]", "markdown": "If $u$ and $v$ are functions of $x$, then D_x (u/v) = (vD_xu - uD_xv)/(u^2 + v^2).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xliv/12", "set": "hardy-course-of-pure-mathematics-1921/ex-xliv", "number": 12, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "212", "location": "Exercise XLIV, problem 12", "problem_latex": "The derivative of $y = (\\tan x + \\sec x)^{m}$ is $my\\sec x$.", "markdown": "The derivative of $y = (\\tan x + \\sec x)^{m}$ is $my\\sec x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xliv/13", "set": "hardy-course-of-pure-mathematics-1921/ex-xliv", "number": 13, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "212", "location": "Exercise XLIV, problem 13", "problem_latex": "The derivative of $y = \\cos x + i\\sin x$ is~$iy$.", "markdown": "The derivative of $y = \\cos x + i\\sin x$ is $iy$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.complex" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xliv/14", "set": "hardy-course-of-pure-mathematics-1921/ex-xliv", "number": 14, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "212", "location": "Exercise XLIV, problem 14", "problem_latex": "Differentiate $x\\cos x$, $(\\sin x)/x$. Show that the values of~$x$ for which\nthe tangents to the curves $y = x\\cos x$, $y = (\\sin x)/x$ are parallel to the axis of~$x$\nare roots of $\\cot x = x$, $\\tan x = x$ respectively.", "markdown": "Differentiate $x\\cos x$, $(\\sin x)/x$. Show that the values of $x$ for which the tangents to the curves $y = x\\cos x$, $y = (\\sin x)/x$ are parallel to the axis of $x$ are roots of $\\cot x = x$, $\\tan x = x$ respectively.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "differentiate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "differentiate: (sin(x)/x, x*cos(x))" ], "shape": [ "differentiate: (sin(x)/x, x*cos(x))" ], "same_problem_in": [], "needs": [ "cas.derive", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xliv/15", "set": "hardy-course-of-pure-mathematics-1921/ex-xliv", "number": 15, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "212", "location": "Exercise XLIV, problem 15", "problem_latex": "It is easy to see (cf.\\ \\Ex{xvii}.~5) that the equation $\\sin x = ax$, where $a$~is\npositive, has no real roots except $x = 0$ if $a \\geq 1$, and if $a < 1$ a finite number of\nroots which increases as $a$~diminishes. Prove that the values of~$a$ for which\nthe number of roots changes are the values of~$\\cos\\xi$, where $\\xi$~is a positive root\nof the equation $\\tan\\xi = \\xi$. [The values required are the values of~$a$ for which\n$y = ax$ touches $y = \\sin x$.]", "markdown": "It is easy to see (cf. % [examples:xvii]Ex. xvii%. 5) that the equation $\\sin x = ax$, where $a$ is positive, has no real roots except $x = 0$ if $a \\geq 1$, and if $a < 1$ a finite number of roots which increases as $a$ diminishes. Prove that the values of $a$ for which the number of roots changes are the values of $\\cos\\xi$, where $\\xi$ is a positive root of the equation $\\tan\\xi = \\xi$. [The values required are the values of $a$ for which $y = ax$ touches $y = \\sin x$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xliv/16", "set": "hardy-course-of-pure-mathematics-1921/ex-xliv", "number": 16, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "212", "location": "Exercise XLIV, problem 16", "problem_latex": "If $\\phi(x) = x^{2}\\sin(1/x)$ when $x \\neq 0$, and $\\phi(0) = 0$, then\n\\[\n\\phi'(x) = 2x\\sin(1/x) - \\cos(1/x)\n\\]\nwhen $x\\neq 0$, and $\\phi'(0) = 0$. And $\\phi'(x)$~is discontinuous for $x = 0$ (cf.\\ \\SecNo[§]{111},~(2)).", "markdown": "If $\\phi(x) = x^{2}\\sin(1/x)$ when $x \\neq 0$, and $\\phi(0) = 0$, then ’(x) = 2x(1/x) - (1/x) when $x\\neq 0$, and $\\phi'(0) = 0$. And $\\phi'(x)$ is discontinuous for $x = 0$ (cf. [§]111, (2)).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xliv/17", "set": "hardy-course-of-pure-mathematics-1921/ex-xliv", "number": 17, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "212", "location": "Exercise XLIV, problem 17", "problem_latex": "Find the equations of the tangent and normal at the point $(x_{0}, y_{0})$\nof the circle $x^{2} + y^{2} = a^{2}$.\n\n[Here $y = \\sqrtp{a^{2} - x^{2}}$, $dy/dx = -x/\\sqrtp{a^{2} - x^{2}}$, and the tangent is\n\\[\ny - y_{0} = (x - x_{0}) \\left\\{-x_{0}/\\sqrtp{a^{2} - x_{0}^{2}}\\right\\},\n\\]\nwhich may be reduced to the form $xx_{0} + yy_{0} = a^{2}$. The normal is $xy_{0} - yx_{0} = 0$,\nwhich of course passes through the origin.]", "markdown": "Find the equations of the tangent and normal at the point $(x_{0}, y_{0})$ of the circle $x^{2} + y^{2} = a^{2}$. [Here $y = \\sqrtp{a^{2} - x^{2}}$, $dy/dx = -x/\\sqrtp{a^{2} - x^{2}}$, and the tangent is y - y_0 = (x - x_0) -x_0/a^2 - x_0^2, which may be reduced to the form $xx_{0} + yy_{0} = a^{2}$. The normal is $xy_{0} - yx_{0} = 0$, which of course passes through the origin.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xliv/18", "set": "hardy-course-of-pure-mathematics-1921/ex-xliv", "number": 18, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "212", "location": "Exercise XLIV, problem 18", "problem_latex": "Find the equations of the tangent and normal at any point of the\nellipse $(x/a)^{2} + (y/b)^{2} = 1$ and the hyperbola $(x/a)^{2} - (y/b)^{2} = 1$.", "markdown": "Find the equations of the tangent and normal at any point of the ellipse $(x/a)^{2} + (y/b)^{2} = 1$ and the hyperbola $(x/a)^{2} - (y/b)^{2} = 1$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xliv/19", "set": "hardy-course-of-pure-mathematics-1921/ex-xliv", "number": 19, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "212", "location": "Exercise XLIV, problem 19", "problem_latex": "The equations of the tangent and normal to the curve $x = \\phi(t)$,\n$y = \\psi(t)$, at the point whose parameter is~$t$, are\n\\[\n\\frac{x - \\phi(t)}{\\phi'(t)} = \\frac{y - \\psi(t)}{\\psi'(t)},\\quad\n\\{x - \\phi(t)\\} \\phi'(t) + \\{y - \\psi(t)\\} \\psi'(t) = 0.\n\\]", "markdown": "The equations of the tangent and normal to the curve $x = \\phi(t)$, $y = \\psi(t)$, at the point whose parameter is $t$, are x - (t)’(t) = y - (t)’(t),0pt minus 3ptx - (t) ’(t) + y - (t) ’(t) = 0.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xliv/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xliv", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "212", "location": "Exercise XLIV, problem 2", "problem_latex": "Verify by differentiation that $\\arcsin x + \\arccos x$ is constant for all\nvalues of~$x$ between $0$~and~$1$, and $\\arctan x + \\arccot x$ for all positive values\nof~$x$.", "markdown": "Verify by differentiation that $\\arcsin x + \\arccos x$ is constant for all values of $x$ between $0$ and $1$, and $\\arctan x + \\arccot x$ for all positive values of $x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xliv/3", "set": "hardy-course-of-pure-mathematics-1921/ex-xliv", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "212", "location": "Exercise XLIV, problem 3", "problem_latex": "\\arcsin\\sqrtp{1 - x^{2}},\\quad\n\\arcsin\\{2x\\sqrtp{1 - x^{2}}\\},\\quad\n\\arctan \\left(\\frac{a + x}{1 - ax}\\right).\n\\]\nHow do you explain the simplicity of the results?", "markdown": "1 - x^2,0pt minus 3pt2x1 - x^2,0pt minus 3pt(a + x1 - ax). How do you explain the simplicity of the results?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "differentiate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "differentiate: (asin(2*x*sqrt(-x**2 + 1)), asin(sqrt(-x**2 + 1)), atan((a + x)/(-a*x + 1)))" ], "shape": [ "differentiate: (asin(N*x*(-x**N + 1)**N), asin((-x**N + 1)**N), atan((a + x)/(-a*x + 1)))" ], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xliv/4", "set": "hardy-course-of-pure-mathematics-1921/ex-xliv", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "212", "location": "Exercise XLIV, problem 4", "problem_latex": "\\frac{1}{\\sqrtp{ac - b^{2}}} \\arctan \\frac{ax + b}{\\sqrtp{ac - b^{2}}},\\quad\n-\\frac{1}{\\sqrtp{-a}} \\arcsin\\frac{ax + b}{\\sqrtp{b^{2} - ac}}.", "markdown": "1ac - b^2 ax + bac - b^2,0pt minus 3pt-1-a ax + bb^2 - ac.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "differentiate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "differentiate: (-asin((a*x + b)/sqrt(-a*c + b**2))/sqrt(-a), atan((a*x + b)/sqrt(a*c - b**2))/sqrt(a*c - b**2))" ], "shape": [ "differentiate: (-(-a)**N*asin((-a*c + b**N)**N*(a*x + b)), (a*c - b**N)**N*atan((a*c - b**N)**N*(a*x + b)))" ], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xliv/5", "set": "hardy-course-of-pure-mathematics-1921/ex-xliv", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "212", "location": "Exercise XLIV, problem 5", "problem_latex": "Show that each of the functions\n\\[\n2\\arcsin \\bigsqrtp{\\frac{x - \\beta}{\\alpha - \\beta}},\\quad\n2\\arctan \\bigsqrtp{\\frac{x - \\beta}{\\alpha - x}},\\quad\n\\arcsin \\frac{2\\sqrtb{(\\alpha - x)(x - \\beta)}}{\\alpha - \\beta}\n\\]\nhas the derivative\n\\[\n\\frac{1}{\\sqrtb{(\\alpha - x)(x - \\beta)}}.\n\\]", "markdown": "Show that each of the functions 2x - - ,0pt minus 3pt2x - - x,0pt minus 3pt2(- x)(x - )- has the derivative 1(- x)(x - ).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xliv/6", "set": "hardy-course-of-pure-mathematics-1921/ex-xliv", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "212", "location": "Exercise XLIV, problem 6", "problem_latex": "Prove that\n\\[\n\\frac{d}{d\\theta}\\left\\{\n \\arccos \\bigsqrtp{\\frac{\\cos 3\\theta}{\\cos^{3}\\theta}}\n\\right\\}\n = \\bigsqrtp{\\frac{3}{\\cos\\theta \\cos 3\\theta}}.\n\\]\n\\MathTrip{1904.}", "markdown": "Prove that dd 3^3 = 33. % [0]% (*Math. Trip.* 1904.)% [1]%", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xliv/7", "set": "hardy-course-of-pure-mathematics-1921/ex-xliv", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "212", "location": "Exercise XLIV, problem 7", "problem_latex": "Show that\n\\[\n\\frac{1}{\\sqrtp{C(Ac - aC)}}\\, \\frac{d}{dx} \\left[\n \\arccos \\bigsqrtb{\\frac{C(ax^{2} + c)}{c(Ax^{2} + C)}}\n\\right]\n = \\frac{1}{(Ax^{2} + C) \\sqrtp{ax^{2} + c}}.\n\\]", "markdown": "Show that 1C(Ac - aC)  ddx [ C(ax^2 + c)c(Ax^2 + C) ] = 1(Ax^2 + C) ax^2 + c.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xliv/8", "set": "hardy-course-of-pure-mathematics-1921/ex-xliv", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "212", "location": "Exercise XLIV, problem 8", "problem_latex": "Each of the functions\n\\[\n\\frac{1}{\\sqrtp{a^{2} - b^{2}}}\n \\arccos \\left(\\frac{a\\cos x + b}{a + b\\cos x}\\right),\\quad\n\\frac{2}{\\sqrtp{a^{2} - b^{2}}}\n \\arctan \\left\\{\\bigsqrtp{\\frac{a - b}{a + b }} \\tan \\tfrac{1}{2}x\\right\\}\n\\]\nhas the derivative~$1/(a + b\\cos x)$.", "markdown": "Each of the functions 1a^2 - b^2 (ax + ba + bx),0pt minus 3pt2a^2 - b^2 a - ba + b 12x has the derivative $1/(a + b\\cos x)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xliv/9", "set": "hardy-course-of-pure-mathematics-1921/ex-xliv", "number": 9, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "212", "location": "Exercise XLIV, problem 9", "problem_latex": "If $X = a + b\\cos x + c\\sin x$, and\n\\[\ny = \\frac{1}{\\sqrtp{a^{2} - b^{2} -c^{2}}}\n \\arccos \\frac{aX - a^{2} + b^{2} + c^{2}}{X \\sqrtp{b^{2} + c^{2}}},\n\\]\nthen $dy/dx = 1/X$.", "markdown": "If $X = a + b\\cos x + c\\sin x$, and y = 1a^2 - b^2 -c^2 aX - a^2 + b^2 + c^2X b^2 + c^2, then $dy/dx = 1/X$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlix/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xlix", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "240", "location": "Exercise XLIX, problem 1", "problem_latex": "Prove that if $a > 0$ then\n\\begin{align*}\n\\int \\sqrtp{x^{2} + a^{2}}\\, dx\n &= \\tfrac{1}{2}x \\sqrtp{x^{2} + a^{2}}\n + \\tfrac{1}{2}a^{2} \\log \\{x + \\sqrtp{x^{2} + a^{2}}\\},\\\\\n\\int \\sqrtp{x^{2} - a^{2}}\\, dx\n &= \\tfrac{1}{2}x \\sqrtp{x^{2} - a^{2}}\n - \\tfrac{1}{2}a^{2} \\log |x + \\sqrtp{x^{2} - a^{2}}|,\\\\\n\\int \\sqrtp{a^{2} - x^{2}}\\, dx\n &= \\tfrac{1}{2}x \\sqrtp{a^{2} - x^{2}}\n + \\tfrac{1}{2}a^{2} \\arcsin(x/a).\n\\end{align*}", "markdown": "Prove that if $a > 0$ then align* x^2 + a^2  dx &= 12x x^2 + a^2 + 12a^2 x + x^2 + a^2, x^2 - a^2  dx &= 12x x^2 - a^2 - 12a^2 |x + x^2 - a^2|, a^2 - x^2  dx &= 12x a^2 - x^2 + 12a^2 (x/a). align*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.integrate.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlix/10", "set": "hardy-course-of-pure-mathematics-1921/ex-xlix", "number": 10, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "240", "location": "Exercise XLIX, problem 10", "problem_latex": "Prove that\n\\[\n\\int f''(x) F(x)\\, dx = f'(x) F(x) - f(x) F'(x) + \\int f(x) F''(x)\\, dx\n\\]\nand generally\n{\\setlength{\\multlinegap}{\\parindent}%\n\\begin{multline*}\n%[** TN: Set on one line in the original]\n\\int f^{(n)}(x) F(x)\\, dx \\\\\n = f^{(n-1)}(x) F(x) - f^{(n-2)}(x) F'(x) + \\dots\n + (-1)^{n} \\int f(x) F^{(n)}(x)\\, dx.\n\\end{multline*}}", "markdown": "Prove that f”(x) F(x)  dx = f’(x) F(x) - f(x) F’(x) + f(x) F”(x)  dx and generally % multline* %[** TN: Set on one line in the original] f^(n)(x) F(x)  dx = f^(n-1)(x) F(x) - f^(n-2)(x) F’(x) + … + (-1)^n f(x) F^(n)(x)  dx. multline*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.integrate.parts" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlix/11", "set": "hardy-course-of-pure-mathematics-1921/ex-xlix", "number": 11, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "240", "location": "Exercise XLIX, problem 11", "problem_latex": "The integral $\\ds\\int (1 + x)^{p} x^{q}\\, dx$, where $p$~and~$q$ are rational, can be found\nin three cases, viz.\\ (i)~if $p$~is an integer, (ii)~if $q$~is an integer, and (iii)~if $p + q$~is an integer. [In case~(i) put $x = u^{s}$, where $s$~is the denominator of~$q$;\nin case~(ii) put $1 + x = t^{s}$, where $s$~is the denominator of~$p$; and in case~(iii) put\n$1 + x = xt^{s}$, where $s$~is the denominator of~$p$.]", "markdown": "The integral $\\ds\\int (1 + x)^{p} x^{q}\\, dx$, where $p$ and $q$ are rational, can be found in three cases, viz. (i) if $p$ is an integer, (ii) if $q$ is an integer, and (iii) if $p + q$ is an integer. [In case (i) put $x = u^{s}$, where $s$ is the denominator of $q$; in case (ii) put $1 + x = t^{s}$, where $s$ is the denominator of $p$; and in case (iii) put $1 + x = xt^{s}$, where $s$ is the denominator of $p$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlix/12", "set": "hardy-course-of-pure-mathematics-1921/ex-xlix", "number": 12, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "240", "location": "Exercise XLIX, problem 12", "problem_latex": "The integral $\\ds\\int x^{m}(ax^{n} + b)^{q}\\, dx$ can be reduced to the preceding\nintegral by the substitution $ax^{n} = bt$. [In practice it is often most convenient\nto calculate a particular integral of this kind by a `formula of\nreduction' (cf.\\ \\MiscEx{VI}~39).]", "markdown": "The integral $\\ds\\int x^{m}(ax^{n} + b)^{q}\\, dx$ can be reduced to the preceding integral by the substitution $ax^{n} = bt$. [In practice it is often most convenient to calculate a particular integral of this kind by a ‘formula of reduction’ (cf. [misc:VI]Misc. Ex. 39).]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate.parts", "cas.integrate.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlix/13", "set": "hardy-course-of-pure-mathematics-1921/ex-xlix", "number": 13, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "240", "location": "Exercise XLIX, problem 13", "problem_latex": "The integral $\\ds\\int R\\{x, \\sqrtp{ax + b}, \\sqrtp{cx + d}\\}\\, dx$ can be reduced to that of\na rational function by the substitution\n\\[\n4x = -(b/a) \\{t + (1/t)\\}^{2} - (d/c)\\{t - (1/t)\\}^{2}.\n\\]", "markdown": "The integral $\\ds\\int R\\{x, \\sqrtp{ax + b}, \\sqrtp{cx + d}\\}\\, dx$ can be reduced to that of a rational function by the substitution 4x = -(b/a) t + (1/t)^2 - (d/c)t - (1/t)^2.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlix/14", "set": "hardy-course-of-pure-mathematics-1921/ex-xlix", "number": 14, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "240", "location": "Exercise XLIX, problem 14", "problem_latex": "Reduce $\\ds\\int R(x, y)\\, dx$, where $y^{2}(x - y) = x^{2}$, to the integral of a rational\nfunction. [Putting $y = tx$ we obtain $x = 1/\\{t^{2}(1 - t)\\}$, $y = 1/\\{t(1 - t)\\}$.]", "markdown": "Reduce $\\ds\\int R(x, y)\\, dx$, where $y^{2}(x - y) = x^{2}$, to the integral of a rational function. [Putting $y = tx$ we obtain $x = 1/\\{t^{2}(1 - t)\\}$, $y = 1/\\{t(1 - t)\\}$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate.subst", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlix/15a", "set": "hardy-course-of-pure-mathematics-1921/ex-xlix", "number": 15, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "240", "location": "Exercise XLIX, problem 15a", "problem_latex": "{\\Loosen Reduce the integral in the same way when (\\ia)~$y(x - y)^{2} = x$,\n(\\ib)~$(x^{2} + y^{2})^{2} = a^{2}(x^{2} - y^{2})$. [In case~(\\ia) put $x - y = t$: in case~(b) put\n$x^{2} + y^{2} = t(x - y)$, when we obtain}\n\\[\n%[** TN: Set in-line in the original]\nx = a^{2}t(t^{2} + a^{2})/(t^{4} + a^{4}),\\quad\ny = a^{2}t(t^{2} - a^{2})/(t^{4} + a^{4}).]\n\\]", "markdown": "0.375em plus 0.75em minus 0.25emReduce the integral in the same way when (*a*) $y(x - y)^{2} = x$, (*b*) $(x^{2} + y^{2})^{2} = a^{2}(x^{2} - y^{2})$. [In case (*a*) put $x - y = t$: in case (b) put $x^{2} + y^{2} = t(x - y)$, when we obtain %[** TN: Set in-line in the original] x = a^2t(t^2 + a^2)/(t^4 + a^4),0pt minus 3pty = a^2t(t^2 - a^2)/(t^4 + a^4).]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate.subst", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlix/15b", "set": "hardy-course-of-pure-mathematics-1921/ex-xlix", "number": 15, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "240", "location": "Exercise XLIX, problem 15b", "problem_latex": "{\\Loosen Reduce the integral in the same way when (\\ia)~$y(x - y)^{2} = x$,\n(\\ib)~$(x^{2} + y^{2})^{2} = a^{2}(x^{2} - y^{2})$. [In case~(\\ia) put $x - y = t$: in case~(b) put\n$x^{2} + y^{2} = t(x - y)$, when we obtain}\n\\[\n%[** TN: Set in-line in the original]\nx = a^{2}t(t^{2} + a^{2})/(t^{4} + a^{4}),\\quad\ny = a^{2}t(t^{2} - a^{2})/(t^{4} + a^{4}).]\n\\]", "markdown": "0.375em plus 0.75em minus 0.25emReduce the integral in the same way when (*a*) $y(x - y)^{2} = x$, (*b*) $(x^{2} + y^{2})^{2} = a^{2}(x^{2} - y^{2})$. [In case (*a*) put $x - y = t$: in case (b) put $x^{2} + y^{2} = t(x - y)$, when we obtain %[** TN: Set in-line in the original] x = a^2t(t^2 + a^2)/(t^4 + a^4),0pt minus 3pty = a^2t(t^2 - a^2)/(t^4 + a^4).]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate.subst", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlix/16", "set": "hardy-course-of-pure-mathematics-1921/ex-xlix", "number": 16, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "240", "location": "Exercise XLIX, problem 16", "problem_latex": "If $y(x - y)^{2} = x$ then\n\\[\n\\int \\frac{dx}{x - 3y} = \\tfrac{1}{2} \\log\\{(x - y)^{2} - 1\\}.\n\\]", "markdown": "If $y(x - y)^{2} = x$ then dxx - 3y = 12 (x - y)^2 - 1.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.integrate.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlix/17", "set": "hardy-course-of-pure-mathematics-1921/ex-xlix", "number": 17, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "240", "location": "Exercise XLIX, problem 17", "problem_latex": "If $(x^{2} + y^{2})^{2} = 2c^{2}(x^{2} - y^{2})$ then\n\\[\n\\int \\frac{dx}{y(x^{2} + y^{2} + c^{2})}\n = - \\frac{1}{c^{2}}\\log\\left(\\frac{x^{2} + y^{2}}{x - y}\\right).\n\\]", "markdown": "If $(x^{2} + y^{2})^{2} = 2c^{2}(x^{2} - y^{2})$ then dxy(x^2 + y^2 + c^2) = - 1c^2(x^2 + y^2x - y).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.integrate.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlix/2a", "set": "hardy-course-of-pure-mathematics-1921/ex-xlix", "number": 2, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "240", "location": "Exercise XLIX, problem 2a", "problem_latex": "Calculate the integrals $\\ds\\int \\frac{dx}{\\sqrtp{a^{2} - x^{2}}}$, $\\ds\\int \\sqrtp{a^{2} - x^{2}}\\, dx$ by means of the\nsubstitution $x = a\\sin\\theta$, and verify that the results agree with those obtained\nin \\SecNo[§]{135} and Ex.~1.", "markdown": "Calculate the integrals $\\ds\\int \\frac{dx}{\\sqrtp{a^{2} - x^{2}}}$, $\\ds\\int \\sqrtp{a^{2} - x^{2}}\\, dx$ by means of the substitution $x = a\\sin\\theta$, and verify that the results agree with those obtained in [§]135 and Ex. 1.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "1/sqrt(a**2 - x**2)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: 1/sqrt(a**2 - x**2)" ], "shape": [ "integrate: (a**N - x**N)**N" ], "same_problem_in": [], "needs": [ "cas.integrate.subst", "cas.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlix/2b", "set": "hardy-course-of-pure-mathematics-1921/ex-xlix", "number": 2, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "240", "location": "Exercise XLIX, problem 2b", "problem_latex": "Calculate the integrals $\\ds\\int \\frac{dx}{\\sqrtp{a^{2} - x^{2}}}$, $\\ds\\int \\sqrtp{a^{2} - x^{2}}\\, dx$ by means of the\nsubstitution $x = a\\sin\\theta$, and verify that the results agree with those obtained\nin \\SecNo[§]{135} and Ex.~1.", "markdown": "Calculate the integrals $\\ds\\int \\frac{dx}{\\sqrtp{a^{2} - x^{2}}}$, $\\ds\\int \\sqrtp{a^{2} - x^{2}}\\, dx$ by means of the substitution $x = a\\sin\\theta$, and verify that the results agree with those obtained in [§]135 and Ex. 1.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "sqrt(a**2 - x**2)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: sqrt(a**2 - x**2)" ], "shape": [ "integrate: (a**N - x**N)**N" ], "same_problem_in": [ "hardy-course-of-pure-mathematics-1921/ex-lxiii/6b", "thompson-calculus-made-easy-1914/ex-xix/1" ], "needs": [ "cas.integrate.subst", "cas.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlix/3", "set": "hardy-course-of-pure-mathematics-1921/ex-xlix", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "240", "location": "Exercise XLIX, problem 3", "problem_latex": "Calculate $\\ds\\int x(x + a)^{m}\\, dx$, where $m$~is any rational number, in three\nways, viz.\\ (i)~by integration by parts, (ii)~by the substitution $(x + a)^{m} = t$, and\n(iii)~by writing $(x + a) - a$ for~$x$; and verify that the results agree.", "markdown": "Calculate $\\ds\\int x(x + a)^{m}\\, dx$, where $m$ is any rational number, in three ways, viz. (i) by integration by parts, (ii) by the substitution $(x + a)^{m} = t$, and (iii) by writing $(x + a) - a$ for $x$; and verify that the results agree.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "x*(x + a)**m", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: x*(a + x)**b" ], "shape": [ "integrate: x*(a + x)**b" ], "same_problem_in": [], "needs": [ "cas.integrate.parts", "cas.integrate.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlix/4a", "set": "hardy-course-of-pure-mathematics-1921/ex-xlix", "number": 4, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "240", "location": "Exercise XLIX, problem 4a", "problem_latex": "Prove, by means of the substitutions $ax + b = 1/t$ and $x = 1/u$, that (in\nthe notation of \\SecNo[§§]{130}~and~\\SecNo{138})\n\\[\n\\int \\frac{dx}{y^{3}} = \\frac{ax + b}{\\Delta y},\\quad\n\\int \\frac{x\\, dx}{y^{3}} = -\\frac{bx + c}{\\Delta y}.\n\\]", "markdown": "Prove, by means of the substitutions $ax + b = 1/t$ and $x = 1/u$, that (in the notation of [§§]130 and 138) dxy^3 = ax + by,0pt minus 3ptx  dxy^3 = -bx + cy.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.integrate.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlix/4b", "set": "hardy-course-of-pure-mathematics-1921/ex-xlix", "number": 4, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "240", "location": "Exercise XLIX, problem 4b", "problem_latex": "Prove, by means of the substitutions $ax + b = 1/t$ and $x = 1/u$, that (in\nthe notation of \\SecNo[§§]{130}~and~\\SecNo{138})\n\\[\n\\int \\frac{dx}{y^{3}} = \\frac{ax + b}{\\Delta y},\\quad\n\\int \\frac{x\\, dx}{y^{3}} = -\\frac{bx + c}{\\Delta y}.\n\\]", "markdown": "Prove, by means of the substitutions $ax + b = 1/t$ and $x = 1/u$, that (in the notation of [§§]130 and 138) dxy^3 = ax + by,0pt minus 3ptx  dxy^3 = -bx + cy.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.integrate.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlix/5", "set": "hardy-course-of-pure-mathematics-1921/ex-xlix", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "240", "location": "Exercise XLIX, problem 5", "problem_latex": "Calculate $\\ds\\int \\frac{dx}{\\sqrtb{(x - a) (b - x)}}$, where $b > a$, in three ways, viz.\\ (i)~by\nthe methods of the preceding sections, (ii)~by the substitution $(b - x)/(x - a) = t^{2}$,\nand (iii)~by the substitution $x = a\\cos^{2}\\theta + b\\sin^{2}\\theta$; and verify that the results\nagree.", "markdown": "Calculate $\\ds\\int \\frac{dx}{\\sqrtb{(x - a) (b - x)}}$, where $b > a$, in three ways, viz. (i) by the methods of the preceding sections, (ii) by the substitution $(b - x)/(x - a) = t^{2}$, and (iii) by the substitution $x = a\\cos^{2}\\theta + b\\sin^{2}\\theta$; and verify that the results agree.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "1/sqrt((x - a)*(b - x))", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: 1/sqrt((-a + x)*(b - x))" ], "shape": [ "integrate: ((-a + x)*(b - x))**N" ], "same_problem_in": [], "needs": [ "cas.integrate.subst", "cas.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlix/6a", "set": "hardy-course-of-pure-mathematics-1921/ex-xlix", "number": 6, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "240", "location": "Exercise XLIX, problem 6a", "problem_latex": "Integrate $\\sqrtb{(x - a) (b - x)}$ and $\\sqrtb{(b - x)/(x - a)}$.", "markdown": "Integrate $\\sqrtb{(x - a) (b - x)}$ and $\\sqrtb{(b - x)/(x - a)}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "sqrt((x - a)*(b - x))", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: sqrt((-a + x)*(b - x))" ], "shape": [ "integrate: ((-a + x)*(b - x))**N" ], "same_problem_in": [], "needs": [ "cas.integrate.subst", "cas.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlix/6b", "set": "hardy-course-of-pure-mathematics-1921/ex-xlix", "number": 6, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "240", "location": "Exercise XLIX, problem 6b", "problem_latex": "Integrate $\\sqrtb{(x - a) (b - x)}$ and $\\sqrtb{(b - x)/(x - a)}$.", "markdown": "Integrate $\\sqrtb{(x - a) (b - x)}$ and $\\sqrtb{(b - x)/(x - a)}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "sqrt((b - x)/(x - a))", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: sqrt((b - x)/(-a + x))" ], "shape": [ "integrate: ((b - x)/(-a + x))**N" ], "same_problem_in": [], "needs": [ "cas.integrate.subst", "cas.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlix/7", "set": "hardy-course-of-pure-mathematics-1921/ex-xlix", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "240", "location": "Exercise XLIX, problem 7", "problem_latex": "Show, by means of the substitution $2x + a + b = \\frac{1}{2}(a - b) \\{t^{2} + (1/t)^{2}\\}$,\nor by multiplying numerator and denominator by $\\sqrtp{x + a} -\\sqrtp{x + b}$, that if $a > b$ then\n\\[\n\\int \\frac{dx}{\\sqrtp{x + a} + \\sqrtp{x + b}}\n = \\tfrac{1}{2}\\sqrtp{a - b} \\left(t + \\frac{1}{3t^{3}}\\right).\n\\]", "markdown": "Show, by means of the substitution $2x + a + b = \\frac{1}{2}(a - b) \\{t^{2} + (1/t)^{2}\\}$, or by multiplying numerator and denominator by $\\sqrtp{x + a} -\\sqrtp{x + b}$, that if $a > b$ then dxx + a + x + b = 12a - b (t + 13t^3).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate.subst", "cas.simplify" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlix/8", "set": "hardy-course-of-pure-mathematics-1921/ex-xlix", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "240", "location": "Exercise XLIX, problem 8", "problem_latex": "Find a substitution which will reduce $\\ds\\int \\frac{dx}{(x + a)^{3/2} + (x - a)^{3/2}}$ to the\nintegral of a rational function. \\MathTrip{1899.}", "markdown": "Find a substitution which will reduce $\\ds\\int \\frac{dx}{(x + a)^{3/2} + (x - a)^{3/2}}$ to the integral of a rational function. % [0]% (*Math. Trip.* 1899.)% [1]%", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlix/9", "set": "hardy-course-of-pure-mathematics-1921/ex-xlix", "number": 9, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "240", "location": "Exercise XLIX, problem 9", "problem_latex": "{\\Loosen Show that $\\ds\\int R\\{x, \\sqrtp[n]{ax + b}\\}\\, dx$ is reduced, by the substitution\n$ax + b = y^{n}$, to the integral of a rational function.}", "markdown": "0.375em plus 0.75em minus 0.25emShow that $\\ds\\int R\\{x, \\sqrtp[n]{ax + b}\\}\\, dx$ is reduced, by the substitution $ax + b = y^{n}$, to the integral of a rational function.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.integrate.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlv/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xlv", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "215", "location": "Exercise XLV, problem 1", "problem_latex": "If $\\phi(x) = x^{m}$ then\n\\[\n\\phi^{(n)}(x) = m(m - 1) \\dots (m - n + 1)x^{m-n}.\n\\]\nThis result enables us to write down the $n$th~derivative of any polynomial.", "markdown": "If $\\phi(x) = x^{m}$ then ^(n)(x) = m(m - 1) …(m - n + 1)x^m-n. This result enables us to write down the $n$th derivative of any polynomial.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlv/10", "set": "hardy-course-of-pure-mathematics-1921/ex-xlv", "number": 10, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "215", "location": "Exercise XLV, problem 10", "problem_latex": "If $U_{n}$~denotes the $n$th~derivative of $(Lx + M)/(x^{2} - 2Bx + C)$, then\n\\[\n\\frac{x^{2} - 2Bx + C}{(n + 1)(n + 2)} U_{n+2}\n + \\frac{2(x - B)}{n + 1} U_{n+1} + U_{n} = 0.\n\\]\n\\MathTrip{1900.}\n\n[First obtain the equation when $n = 0$; then differentiate $n$~times by\n\\DPchg{Leibnitz'}{Leibniz'} Theorem.]", "markdown": "If $U_{n}$ denotes the $n$th derivative of $(Lx + M)/(x^{2} - 2Bx + C)$, then x^2 - 2Bx + C(n + 1)(n + 2) U_n+2 + 2(x - B)n + 1 U_n+1 + U_n = 0. % [0]% (*Math. Trip.* 1900.)% [1]% [First obtain the equation when $n = 0$; then differentiate $n$ times by Leibnitz’Leibniz’ Theorem.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlv/11", "set": "hardy-course-of-pure-mathematics-1921/ex-xlv", "number": 11, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "215", "location": "Exercise XLV, problem 11", "problem_latex": "\\Topic{The $n$th~derivatives of $a/(a^{2} + x^{2})$ and $x/(a^{2} + x^{2})$.} Since\n\\[\n\\frac{a}{a^{2} + x^{2}}\n = \\frac{1}{2i} \\left(\\frac{1}{x - ai} - \\frac{1}{x + ai}\\right), \\quad\n\\frac{x}{a^{2} + x^{2}}\n = \\frac{1}{2} \\left(\\frac{1}{x - ai} + \\frac{1}{x + ai}\\right),\n\\]\nwe have\n\\[\nD_{x}^{n} \\left(\\frac{a}{a^{2} + x^{2}}\\right)\n = \\frac{(-1)^{n} n!}{2i} \\left\\{\n \\frac{1}{(x - ai)^{n+1}} - \\frac{1}{(x + ai)^{n+1}}\n\\right\\},\n\\]\n{\\Loosen and a similar formula for $D_{x}^{n}\\{x/(a^{2} + x^{2})\\}$. If $\\rho = \\sqrtp{x^{2} + a^{2}}$, and $\\theta$~is the\nnumerically smallest angle whose cosine and sine are $x/\\rho$~and~$a/\\rho$, then\n$x + ai = \\rho\\Cis\\theta$ and $x - ai = \\rho\\Cis(-\\theta )$, and so}\n\\begin{align*}\nD_{x}^{n} \\{a/(a^{2} + x^{2})\\}\n &= \\{(-1)^{n} n!/2i\\} \\rho^{-n-1}\n [\\Cis \\{(n + 1)\\theta\\} - \\Cis \\{-(n + 1)\\theta\\}]\\\\\n &= (-1)^{n} n!\\, (x^{2} + a^{2})^{-(n+1)/2} \\sin \\{(n + 1) \\arctan(a/x)\\}.\n\\end{align*}\nSimilarly\n\\[\nD_{x}^{n} \\{x/(a^{2} + x^{2})\\}\n = (-1)^{n} n!\\, (x^{2} + a^{2})^{-(n+1)/2} \\cos \\{(n + 1) \\arctan (a/x)\\}.\n\\]", "markdown": "**$n$th derivatives of $a/(a^{2} + x^{2})$ and $x/(a^{2} + x^{2})$.** Since aa^2 + x^2 = 12i (1x - ai - 1x + ai), 0pt minus 3ptxa^2 + x^2 = 12 (1x - ai + 1x + ai), we have D_x^n (aa^2 + x^2) = (-1)^n n!2i 1(x - ai)^n+1 - 1(x + ai)^n+1 , 0.375em plus 0.75em minus 0.25emand a similar formula for $D_{x}^{n}\\{x/(a^{2} + x^{2})\\}$. If $\\rho = \\sqrtp{x^{2} + a^{2}}$, and $\\theta$ is the numerically smallest angle whose cosine and sine are $x/\\rho$ and $a/\\rho$, then $x + ai = \\rho\\Cis\\theta$ and $x - ai = \\rho\\Cis(-\\theta )$, and so align* D_x^n a/(a^2 + x^2) &= (-1)^n n!/2i ^-n-1 [(n + 1) - -(n + 1)] &= (-1)^n n!  (x^2 + a^2)^-(n+1)/2 (n + 1) (a/x). align* Similarly D_x^n x/(a^2 + x^2) = (-1)^n n!  (x^2 + a^2)^-(n+1)/2 (n + 1) (a/x).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlv/12", "set": "hardy-course-of-pure-mathematics-1921/ex-xlv", "number": 12, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "215", "location": "Exercise XLV, problem 12", "problem_latex": "Prove that\n\\begin{align*}\nD_{x}^{n} \\{(\\cos x)/x\\}\n &= \\{P_{n} \\cos(x + \\tfrac{1}{2}n\\pi)\n + Q_{n} \\sin(x + \\tfrac{1}{2}n\\pi)\\}/x^{n+1},\\\\\nD_{x}^{n} \\{(\\sin x)/x\\}\n &= \\{P_{n} \\sin(x + \\tfrac{1}{2}n\\pi)\n - Q_{n} \\cos(x + \\tfrac{1}{2}n\\pi)\\}/x^{n+1},\n\\end{align*}\nwhere $P_{n}$ and~$Q_{n}$ are polynomials in~$x$ of degree $n$~and~$n-1$ respectively.", "markdown": "Prove that align* D_x^n (x)/x &= P_n (x + 12n) + Q_n (x + 12n)/x^n+1, D_x^n (x)/x &= P_n (x + 12n) - Q_n (x + 12n)/x^n+1, align* where $P_{n}$ and $Q_{n}$ are polynomials in $x$ of degree $n$ and $n-1$ respectively.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlv/13", "set": "hardy-course-of-pure-mathematics-1921/ex-xlv", "number": 13, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "215", "location": "Exercise XLV, problem 13", "problem_latex": "Establish the formulae\n\\begin{gather*}\n%[** TN: Set on one line in the orignal]\n\\frac{dx}{dy} = 1 \\bigg/\\biggl(\\frac{dy}{dx}\\biggr),\\quad\n\\frac{d^{2} x}{dy^{2}}\n = -\\frac{d^{2} y}{dx^{2}} \\bigg/ \\biggl(\\frac{dy}{dx}\\biggr)^{3},\\\\\n\\frac{d^{3} x}{dy^{3}}\n = -\\biggl\\{\\frac{d^{3} y}{dx^{3}}\\, \\frac{dy}{dx}\n - 3\\biggl(\\frac{d^{2} y}{dx^{2}}\\biggr)\\biggr\\} \\bigg/\n \\biggl(\\frac{dy}{dx}\\biggr)^{5}.\n\\end{gather*}", "markdown": "Establish the formulae gather* %[** TN: Set on one line in the orignal] dxdy = 1 /(dydx),0pt minus 3ptd^2 xdy^2 = -d^2 ydx^2 / (dydx)^3, d^3 xdy^3 = -d^3 ydx^3  dydx - 3(d^2 ydx^2) / (dydx)^5. gather*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlv/14", "set": "hardy-course-of-pure-mathematics-1921/ex-xlv", "number": 14, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "215", "location": "Exercise XLV, problem 14", "problem_latex": "If $yz = 1$ and $y_{r} = (1/r!) D_{x}^{r}y$, $z_{s} = (1/s!) D_{x}^{s}z$, then\n\\[\n\\frac{1}{z^{3}}\n\\begin{vmatrix}\nz & z_{1}& z_{2}\\\\\nz_{1}& z_{2}& z_{3}\\\\\nz_{2}& z_{3}& z_{4}\n\\end{vmatrix}\n= \\frac{1}{y^{2}}\n\\begin{vmatrix}\ny_{2}& y_{3}\\\\\ny_{3}& y_{4}\n\\end{vmatrix}.\n\\]\n\\MathTrip{1905.}", "markdown": "If $yz = 1$ and $y_{r} = (1/r!) D_{x}^{r}y$, $z_{s} = (1/s!) D_{x}^{s}z$, then 1z^3 vmatrix z & z_1& z_2 z_1& z_2& z_3 z_2& z_3& z_4 vmatrix = 1y^2 vmatrix y_2& y_3 y_3& y_4 vmatrix. % [0]% (*Math. Trip.* 1905.)% [1]%", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "other:determinant" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlv/15", "set": "hardy-course-of-pure-mathematics-1921/ex-xlv", "number": 15, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "215", "location": "Exercise XLV, problem 15", "problem_latex": "If\n\\[\nW(y, z, u) =\n\\begin{vmatrix}\ny & z & u\\\\\ny' & z' & u'\\\\\ny''& z''& u''\n\\end{vmatrix},\n\\]\ndashes denoting differentiations with\nrespect to~$x$, then\n\\[\nW(y, z, u) = y^{3}\\, W\\left(1, \\frac{z}{y}, \\frac{u}{y}\\right).\n\\]", "markdown": "If W(y, z, u) = vmatrix y & z & u y’ & z’ & u’ y”& z”& u” vmatrix, dashes denoting differentiations with respect to $x$, then W(y, z, u) = y^3  W(1, zy, uy).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "other:determinant" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlv/16", "set": "hardy-course-of-pure-mathematics-1921/ex-xlv", "number": 16, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "215", "location": "Exercise XLV, problem 16", "problem_latex": "If\n\\[\nax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0,\n\\]\nthen\n\\[\ndy/dx = -(ax + hy + g)/(hx + by + f)\n\\]\nand\n\\[\nd^{2}y/dx^{2} = (abc + 2fgh - af^{2} - bg^{2} - ch^{2})/(hx + by + f)^{3}.\n\\]", "markdown": "If ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0, then dy/dx = -(ax + hy + g)/(hx + by + f) and d^2y/dx^2 = (abc + 2fgh - af^2 - bg^2 - ch^2)/(hx + by + f)^3.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.implicit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlv/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xlv", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "215", "location": "Exercise XLV, problem 2", "problem_latex": "If $\\phi(x) = (ax + b)^{m}$ then\n\\[\n\\phi^{(n)}(x) = m(m - 1) \\dots (m - n + 1)a^{n}(ax + b)^{m-n}.\n\\]\nIn these two examples $m$~may have any rational value. If $m$~is a positive\ninteger, and $n > m$, then $\\phi^{(n)}(x) = 0$.", "markdown": "If $\\phi(x) = (ax + b)^{m}$ then ^(n)(x) = m(m - 1) …(m - n + 1)a^n(ax + b)^m-n. In these two examples $m$ may have any rational value. If $m$ is a positive integer, and $n > m$, then $\\phi^{(n)}(x) = 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlv/3", "set": "hardy-course-of-pure-mathematics-1921/ex-xlv", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "215", "location": "Exercise XLV, problem 3", "problem_latex": "The formula\n\\[\n\\left(\\frac{d}{dx}\\right)^{n} \\frac{A}{(x - \\alpha)^{p}}\n = (-1)^{n} \\frac{p(p + 1) \\dots (p + n - 1)A}{(x - \\alpha)^{p+n}}\n\\]\nenables us to write down the $n$th~derivative of any rational function expressed\nin the standard form as a sum of partial fractions.", "markdown": "The formula (ddx)^n A(x - )^p = (-1)^n p(p + 1) …(p + n - 1)A(x - )^p+n enables us to write down the $n$th derivative of any rational function expressed in the standard form as a sum of partial fractions.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlv/4", "set": "hardy-course-of-pure-mathematics-1921/ex-xlv", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "215", "location": "Exercise XLV, problem 4", "problem_latex": "Prove that the $n$th~derivative of $1/(1 - x^{2})$ is\n\\[\n\\tfrac{1}{2}(n!) \\{(1 - x)^{-n-1} + (-1)^{n}(1 + x)^{-n-1}\\}.\n\\]", "markdown": "Prove that the $n$th derivative of $1/(1 - x^{2})$ is 12(n!) (1 - x)^-n-1 + (-1)^n(1 + x)^-n-1.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.partfrac" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlv/5", "set": "hardy-course-of-pure-mathematics-1921/ex-xlv", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "215", "location": "Exercise XLV, problem 5", "problem_latex": "\\Topic{Leibniz' Theorem.} If $y$~is a product~$uv$, and we can form the\nfirst $n$~derivatives of $u$ and~$v$, then we can form the $n$th~derivative of~$y$ by\nmeans of \\emph{Leibniz' Theorem}, which gives the rule\n\\[\n(uv)_{n} = u_{n}v\n + \\binom{n}{1}u_{n-1}v_{1}\n + \\binom{n}{2}u_{n-2}v_{2} + \\dots\n + \\binom{n}{r}u_{n-r}v_{r} + \\dots + uv_{n},\n\\]\nwhere suffixes indicate differentiations, so that $u_{n}$, for example, denotes the\n$n$th~derivative of~$u$. To prove the theorem we observe that\n\\begin{align*}\n(uv)_{1} &= u_{1}v + uv_{1},\\\\\n(uv)_{2} &= u_{2}v + 2u_{1}v_{1} + uv_{2},\n\\end{align*}\nand so on. It is obvious that by repeating this process we arrive at a\nformula of the type\n\\[\n(uv)_{n} = u_{n}v\n + a_{n, 1} u_{n-1} v_{1}\n + a_{n, 2} u_{n-2} v_{2} + \\dots\n + a_{n, r} u_{n-r} v_{r} + \\dots + uv_{n}.\n\\]\n\nLet us assume that $a_{n, r} = \\dbinom{n}{r}$ for $r = 1$, $2$,~\\dots\\Add{,} $n - 1$, and show that if this\nis so then $a_{n+1, r} = \\dbinom{n + 1}{r}$ for $r = 1$, $2$,~\\dots~$n$. It will then follow by the\nprinciple of mathematical induction that $a_{n, r} = \\dbinom{n}{r}$ for all values of $n$ and~$r$\nin question.\n\nWhen we form $(uv)_{n+1}$ by differentiating $(uv)_{n}$ it is clear that the coefficient\nof~$u_{n+1-r}v_{r}$ is\n\\[\na_{n, r} + a_{n, r-1} = \\binom{n}{r} + \\binom{n}{r - 1} = \\binom{n + 1}{r}.\n\\]\nThis establishes the theorem.", "markdown": "**’ Theorem.** If $y$ is a product $uv$, and we can form the first $n$ derivatives of $u$ and $v$, then we can form the $n$th derivative of $y$ by means of *Leibniz’ Theorem*, which gives the rule (uv)_n = u_nv + n1u_n-1v_1 + n2u_n-2v_2 + … + nru_n-rv_r + …+ uv_n, where suffixes indicate differentiations, so that $u_{n}$, for example, denotes the $n$th derivative of $u$. To prove the theorem we observe that align* (uv)_1 &= u_1v + uv_1, (uv)_2 &= u_2v + 2u_1v_1 + uv_2, align* and so on. It is obvious that by repeating this process we arrive at a formula of the type (uv)_n = u_nv + a_n, 1 u_n-1 v_1 + a_n, 2 u_n-2 v_2 + … + a_n, r u_n-r v_r + …+ uv_n. Let us assume that $a_{n, r} = \\dbinom{n}{r}$ for $r = 1$, $2$, … $n - 1$, and show that if this is so then $a_{n+1, r} = \\dbinom{n + 1}{r}$ for $r = 1$, $2$, … $n$. It will then follow by the principle of mathematical induction that $a_{n, r} = \\dbinom{n}{r}$ for all values of $n$ and $r$ in question. When we form $(uv)_{n+1}$ by differentiating $(uv)_{n}$ it is clear that the coefficient of $u_{n+1-r}v_{r}$ is a_n, r + a_n, r-1 = nr + nr - 1 = n + 1r. This establishes the theorem.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlv/6", "set": "hardy-course-of-pure-mathematics-1921/ex-xlv", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "215", "location": "Exercise XLV, problem 6", "problem_latex": "The $n$th~derivative of~$x^{m}f(x)$ is\n\\begin{multline*}\n\\frac{m!}{(m - n)!} x^{m-n} f(x) + n \\frac{m!}{(m - n + 1)!} x^{m-n+1} f'(x)\\\\\n + \\frac{n(n - 1)}{1·2}\\, \\frac{m!}{(m - n + 2)!} x^{m-n+2} f''(x) + \\dots,\n\\end{multline*}\nthe series being continued for $n + 1$~terms or until it terminates.", "markdown": "The $n$th derivative of $x^{m}f(x)$ is multline* m!(m - n)! x^m-n f(x) + n m!(m - n + 1)! x^m-n+1 f’(x) + n(n - 1)1·2  m!(m - n + 2)! x^m-n+2 f”(x) + …, multline* the series being continued for $n + 1$ terms or until it terminates.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlv/7", "set": "hardy-course-of-pure-mathematics-1921/ex-xlv", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "215", "location": "Exercise XLV, problem 7", "problem_latex": "Prove that $D_{x}^{n}\\cos x = \\cos(x + \\frac{1}{2}n\\pi)$, $D_{x}^{n}\\sin x = \\sin(x + \\frac{1}{2}n\\pi)$\\Add{.}", "markdown": "Prove that $D_{x}^{n}\\cos x = \\cos(x + \\frac{1}{2}n\\pi)$, $D_{x}^{n}\\sin x = \\sin(x + \\frac{1}{2}n\\pi)$", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlv/8", "set": "hardy-course-of-pure-mathematics-1921/ex-xlv", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "215", "location": "Exercise XLV, problem 8", "problem_latex": "If $y = A\\cos mx + B\\sin mx$ then $D_{x}^{2} y + m^{2} y = 0$. And if\n\\[\ny = A\\cos mx + B\\sin mx + P_{n}(x),\n\\]\nwhere $P_{n}(x)$~is a polynomial of degree~$n$, then $D_{x}^{n+3} y + m^{2} D_{x}^{n+1} y = 0$.", "markdown": "If $y = A\\cos mx + B\\sin mx$ then $D_{x}^{2} y + m^{2} y = 0$. And if y = Amx + Bmx + P_n(x), where $P_{n}(x)$ is a polynomial of degree $n$, then $D_{x}^{n+3} y + m^{2} D_{x}^{n+1} y = 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlv/9", "set": "hardy-course-of-pure-mathematics-1921/ex-xlv", "number": 9, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "215", "location": "Exercise XLV, problem 9", "problem_latex": "If $x^{2} D_{x}^{2}y + x D_{x} y + y = 0$ then\n\\[\nx^{2} D_{x}^{n+2} y + (2n + 1)x D_{x}^{n+1} y + (n^{2} + 1) D_{x}^{n} y = 0.\n\\]\n\n[Differentiate $n$~times by \\DPchg{Leibnitz'}{Leibniz'} Theorem.]", "markdown": "If $x^{2} D_{x}^{2}y + x D_{x} y + y = 0$ then x^2 D_x^n+2 y + (2n + 1)x D_x^n+1 y + (n^2 + 1) D_x^n y = 0. [Differentiate $n$ times by Leibnitz’Leibniz’ Theorem.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/10", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 10, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 10", "problem_latex": "Discuss similarly the function $(x - a) (x - b)^{2} (x - c)^{3}$, distinguishing\nthe different forms of the graph which correspond to different hypotheses as\nto the relative magnitudes of $a$,~$b$,~$c$.", "markdown": "Discuss similarly the function $(x - a) (x - b)^{2} (x - c)^{3}$, distinguishing the different forms of the graph which correspond to different hypotheses as to the relative magnitudes of $a$, $b$, $c$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "extremum", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "(x - a)*(x - b)**2*(x - c)**3", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "extremum: (-a + x)*(-b + x)**2*(-c + x)**3" ], "shape": [ "extremum: (-a + x)*(-b + x)**N*(-c + x)**N" ], "same_problem_in": [], "needs": [ "cas.derive", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/11", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 11, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 11", "problem_latex": "Show that $(ax + b)/(cx + d)$ has no maxima or minima, whatever\nvalues $a$,~$b$, $c$,~$d$ may have. Draw a graph of the function.", "markdown": "Show that $(ax + b)/(cx + d)$ has no maxima or minima, whatever values $a$, $b$, $c$, $d$ may have. Draw a graph of the function.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "extremum", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "(a*x + b)/(c*x + d)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "extremum: (a*x + b)/(c*x + d)" ], "shape": [ "extremum: (a*x + b)/(c*x + d)" ], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/12", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 12, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 12", "problem_latex": "Discuss the maxima and minima of the function\n\\[\ny = (ax^{2} + 2bx + c)/(Ax^{2} + 2Bx + \\DPtypo{c}{C}),\n\\]\nwhen the denominator has complex roots.", "markdown": "Discuss the maxima and minima of the function y = (ax^2 + 2bx + c)/(Ax^2 + 2Bx + c), when the denominator has complex roots.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "extremum", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "(a*x**2 + 2*b*x + c)/(A*x**2 + 2*B*x + C)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "extremum: (d*x**2 + 2*e*x + f)/(a*x**2 + 2*b*x + c)" ], "shape": [ "extremum: (N*e*x + d*x**N + f)/(N*b*x + a*x**N + c)" ], "same_problem_in": [], "needs": [ "cas.derive", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/13", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 13, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 13", "problem_latex": "The maximum and minimum values themselves are the values of~$\\lambda$\nfor which $ax^{2} + 2bx + c - \\lambda(Ax^{2} + 2Bx + C)$ is a perfect square.", "markdown": "The maximum and minimum values themselves are the values of $\\lambda$ for which $ax^{2} + 2bx + c - \\lambda(Ax^{2} + 2Bx + C)$ is a perfect square.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/14", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 14, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 14", "problem_latex": "In general the maxima and maxima of $R(x) = P(x)/Q(x)$ are among\nthe values of~$\\lambda$ obtained by expressing the condition that $P(x) - \\lambda Q(x) = 0$\nshould have a pair of equal roots.", "markdown": "In general the maxima and maxima of $R(x) = P(x)/Q(x)$ are among the values of $\\lambda$ obtained by expressing the condition that $P(x) - \\lambda Q(x) = 0$ should have a pair of equal roots.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/15", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 15, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 15", "problem_latex": "If $Ax^{2} + 2Bx + C = 0$ has real roots then it is convenient to proceed as\nfollows. We have\n\\[\ny - (a/A) = (\\lambda x + \\mu)/\\{A(Ax^{2} + 2Bx + C)\\},\n\\]\nwhere $\\lambda = bA - aB$, $\\mu = cA - aC$.", "markdown": "If $Ax^{2} + 2Bx + C = 0$ has real roots then it is convenient to proceed as follows. We have y - (a/A) = (x + )/A(Ax^2 + 2Bx + C), where $\\lambda = bA - aB$, $\\mu = cA - aC$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/16", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 16, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 16", "problem_latex": "Show that $(x - \\alpha)(x - \\beta)/(x - \\gamma)$ assumes all real values as $x$~varies, if\n$\\gamma$~lies between $\\alpha$ and~$\\beta$, and otherwise assumes all values except those included\nin an interval of length $4\\sqrtp{|\\alpha - \\gamma||\\beta - \\gamma|}$.", "markdown": "Show that $(x - \\alpha)(x - \\beta)/(x - \\gamma)$ assumes all real values as $x$ varies, if $\\gamma$ lies between $\\alpha$ and $\\beta$, and otherwise assumes all values except those included in an interval of length $4\\sqrtp{|\\alpha - \\gamma||\\beta - \\gamma|}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(x - alpha)*(x - beta)/(x - gamma)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/17", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 17, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 17", "problem_latex": "Show that\n\\[\ny = \\frac{x^{2} + 2x + c}{x^{2} + 4x + 3c}\n\\]\ncan assume any real value if $0 < c < 1$, and draw a graph of the function in\nthis case. \\MathTrip{1910.}", "markdown": "Show that y = x^2 + 2x + cx^2 + 4x + 3c can assume any real value if $0 < c < 1$, and draw a graph of the function in this case. % [0]% (*Math. Trip.* 1910.)% [1]%", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(x**2 + 2*x + c)/(x**2 + 4*x + 3*c)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/18", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 18, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 18", "problem_latex": "Determine the function of the form $(ax^{2} + 2bx + c)/(Ax^{2} + 2Bx + C)$\nwhich has turning values (\\ie\\ maxima or minima) $2$~and~$3$ when $x = 1$ and\n$x = -1$ respectively, and has the value~$2.5$ when $x = 0$. \\MathTrip{1908.}", "markdown": "Determine the function of the form $(ax^{2} + 2bx + c)/(Ax^{2} + 2Bx + C)$ which has turning values (*i.e.* maxima or minima) $2$ and $3$ when $x = 1$ and $x = -1$ respectively, and has the value $2.5$ when $x = 0$. % [0]% (*Math. Trip.* 1908.)% [1]%", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.solve.num" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/19", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 19, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 19", "problem_latex": "The maximum and minimum of $(x + a) (x + b)/(x - a) (x - b)$, where $a$~and~$b$ are positive, are\n\\[\n-\\left(\\frac{\\sqrt{a} + \\sqrt{b}}{\\sqrt{a} - \\sqrt{b}}\\right)^{2},\\quad\n-\\left(\\frac{\\sqrt{a} - \\sqrt{b}}{\\sqrt{a} + \\sqrt{b}}\\right)^{2}.\n\\]", "markdown": "The maximum and minimum of $(x + a) (x + b)/(x - a) (x - b)$, where $a$ and $b$ are positive, are -(a + ba - b)^2,0pt minus 3pt-(a - ba + b)^2.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "extremum", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "(x + a)*(x + b)/((x - a)*(x - b))", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "extremum: (a + x)*(b + x)/((-a + x)*(-b + x))" ], "shape": [ "extremum: (a + x)*(b + x)/((-a + x)*(-b + x))" ], "same_problem_in": [], "needs": [ "cas.derive", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/1a", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 1, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 1a", "problem_latex": "Verify Theorem~B when $\\phi(x) = (x - a)^{m} (x - b)^{n}$ or\n$\\phi(x) = (x - a)^{m} (x - b)^{n} (x - c)^{p}$, where $m$,~$n$,~$p$ are positive integers and $a < b < c$.", "markdown": "Verify Theorem B when $\\phi(x) = (x - a)^{m} (x - b)^{n}$ or $\\phi(x) = (x - a)^{m} (x - b)^{n} (x - c)^{p}$, where $m$, $n$, $p$ are positive integers and $a < b < c$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/1b", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 1, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 1b", "problem_latex": "Verify Theorem~B when $\\phi(x) = (x - a)^{m} (x - b)^{n}$ or\n$\\phi(x) = (x - a)^{m} (x - b)^{n} (x - c)^{p}$, where $m$,~$n$,~$p$ are positive integers and $a < b < c$.", "markdown": "Verify Theorem B when $\\phi(x) = (x - a)^{m} (x - b)^{n}$ or $\\phi(x) = (x - a)^{m} (x - b)^{n} (x - c)^{p}$, where $m$, $n$, $p$ are positive integers and $a < b < c$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 2", "problem_latex": "Show that the polynomials\n\\[\n2x^{3} + 3x^{2} - 12x + 7,\\quad\n3x^{4} + 8x^{3} - 6x^{2} - 24x + 19\n\\]\nare positive when $x > 1$.", "markdown": "Show that the polynomials 2x^3 + 3x^2 - 12x + 7,0pt minus 3pt3x^4 + 8x^3 - 6x^2 - 24x + 19 are positive when $x > 1$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/20", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 20, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 20", "problem_latex": "The maximum value of $(x - 1)^{2}/(x + 1)^{3}$ is~$\\frac{2}{27}$.", "markdown": "The maximum value of $(x - 1)^{2}/(x + 1)^{3}$ is $\\frac{2}{27}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "extremum", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "(x - 1)**2/(x + 1)**3", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "extremum: (x - 1)**2/(x + 1)**3" ], "shape": [ "extremum: (x - 1)**N*(x + 1)**N" ], "same_problem_in": [], "needs": [ "cas.derive", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/21a", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 21, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 21a", "problem_latex": "Discuss the maxima and minima of\n\\begin{gather*}\nx(x - 1)/(x^{2} + 3x + 3),\\quad x^{4}/(x - 1)(x - 3)^{3},\\\\\n(x - 1)^{2}(3x^{2} - 2x - 37)/(x + 5)^{2}(3x^{2} - 14x - 1).\n\\end{gather*}", "markdown": "Discuss the maxima and minima of gather* x(x - 1)/(x^2 + 3x + 3),0pt minus 3ptx^4/(x - 1)(x - 3)^3, (x - 1)^2(3x^2 - 2x - 37)/(x + 5)^2(3x^2 - 14x - 1). gather*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "extremum", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "x*(x - 1)/(x**2 + 3*x + 3)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "extremum: x*(x - 1)/(x**2 + 3*x + 3)" ], "shape": [ "extremum: x*(x - 1)/(N*x + N + x**N)" ], "same_problem_in": [], "needs": [ "cas.derive", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/21b", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 21, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 21b", "problem_latex": "Discuss the maxima and minima of\n\\begin{gather*}\nx(x - 1)/(x^{2} + 3x + 3),\\quad x^{4}/(x - 1)(x - 3)^{3},\\\\\n(x - 1)^{2}(3x^{2} - 2x - 37)/(x + 5)^{2}(3x^{2} - 14x - 1).\n\\end{gather*}", "markdown": "Discuss the maxima and minima of gather* x(x - 1)/(x^2 + 3x + 3),0pt minus 3ptx^4/(x - 1)(x - 3)^3, (x - 1)^2(3x^2 - 2x - 37)/(x + 5)^2(3x^2 - 14x - 1). gather*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "extremum", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "x**4/((x - 1)*(x - 3)**3)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "extremum: x**4/((x - 3)**3*(x - 1))" ], "shape": [ "extremum: x**N*(N + x)**N/(x - 1)" ], "same_problem_in": [], "needs": [ "cas.derive", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/21c", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 21, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 21c", "problem_latex": "Discuss the maxima and minima of\n\\begin{gather*}\nx(x - 1)/(x^{2} + 3x + 3),\\quad x^{4}/(x - 1)(x - 3)^{3},\\\\\n(x - 1)^{2}(3x^{2} - 2x - 37)/(x + 5)^{2}(3x^{2} - 14x - 1).\n\\end{gather*}", "markdown": "Discuss the maxima and minima of gather* x(x - 1)/(x^2 + 3x + 3),0pt minus 3ptx^4/(x - 1)(x - 3)^3, (x - 1)^2(3x^2 - 2x - 37)/(x + 5)^2(3x^2 - 14x - 1). gather*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "extremum", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "(x - 1)**2*(3*x**2 - 2*x - 37)/((x + 5)**2*(3*x**2 - 14*x - 1))", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "extremum: (x - 1)**2*(3*x**2 - 2*x - 37)/((x + 5)**2*(3*x**2 - 14*x - 1))" ], "shape": [ "extremum: (N + x)**N*(x - 1)**N*(N*x + N*x**N + N)/(N*x + N*x**N - 1)" ], "same_problem_in": [], "needs": [ "cas.derive", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/22", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 22, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 22", "problem_latex": "Find the maxima and minima of $a\\cos x + b\\sin x$. Verify the result\nby expressing the function in the form~$A\\cos(x - a)$.", "markdown": "Find the maxima and minima of $a\\cos x + b\\sin x$. Verify the result by expressing the function in the form $A\\cos(x - a)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "extremum", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "a*cos(x) + b*sin(x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "extremum: a*cos(x) + b*sin(x)" ], "shape": [ "extremum: a*cos(x) + b*sin(x)" ], "same_problem_in": [], "needs": [ "cas.derive", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/23a", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 23, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 23a", "problem_latex": "Find the maxima and minima of\n\\[\na^{2}\\cos^{2} x + b^{2}\\sin^{2} x,\\quad\nA\\cos^{2}x + 2H\\cos x\\sin x + B\\sin^{2} x.\n\\]", "markdown": "Find the maxima and minima of a^2^2 x + b^2^2 x,0pt minus 3ptA^2x + 2Hxx + B^2 x.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "extremum", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "a**2*cos(x)**2 + b**2*sin(x)**2", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "extremum: a**2*cos(x)**2 + b**2*sin(x)**2" ], "shape": [ "extremum: a**N*cos(x)**N + b**N*sin(x)**N" ], "same_problem_in": [], "needs": [ "cas.derive", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/23b", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 23, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 23b", "problem_latex": "Find the maxima and minima of\n\\[\na^{2}\\cos^{2} x + b^{2}\\sin^{2} x,\\quad\nA\\cos^{2}x + 2H\\cos x\\sin x + B\\sin^{2} x.\n\\]", "markdown": "Find the maxima and minima of a^2^2 x + b^2^2 x,0pt minus 3ptA^2x + 2Hxx + B^2 x.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "extremum", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "A*cos(x)**2 + 2*H*cos(x)*sin(x) + B*sin(x)**2", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "extremum: a*cos(x)**2 + b*sin(x)**2 + 2*c*sin(x)*cos(x)" ], "shape": [ "extremum: N*c*sin(x)*cos(x) + a*cos(x)**N + b*sin(x)**N" ], "same_problem_in": [], "needs": [ "cas.derive", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/24", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 24, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 24", "problem_latex": "Show that $\\sin(x + a)/\\sin(x + b)$ has no maxima or minima. Draw\na graph of the function.", "markdown": "Show that $\\sin(x + a)/\\sin(x + b)$ has no maxima or minima. Draw a graph of the function.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "extremum", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "sin(x + a)/sin(x + b)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "extremum: sin(a + x)/sin(b + x)" ], "shape": [ "extremum: sin(a + x)/sin(b + x)" ], "same_problem_in": [], "needs": [ "cas.derive", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/25", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 25, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 25", "problem_latex": "Show that the function\n\\[\n\\frac{\\sin^{2}x}{\\sin(x + a)\\sin(x + b)}\\quad\n(0 < a < b < \\pi)\n\\]\nhas an infinity of minima equal to~$0$ and of maxima equal to\n\\[\n-4\\sin a\\sin b/\\sin^{2}(a - b).\n\\]\n\\MathTrip{1909.}", "markdown": "Show that the function ^2x(x + a)(x + b)0pt minus 3pt(0 < a < b < ) has an infinity of minima equal to $0$ and of maxima equal to -4ab/^2(a - b). % [0]% (*Math. Trip.* 1909.)% [1]%", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "extremum", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "sin(x)**2/(sin(x + a)*sin(x + b))", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "extremum: sin(x)**2/(sin(a + x)*sin(b + x))" ], "shape": [ "extremum: sin(x)**N/(sin(a + x)*sin(b + x))" ], "same_problem_in": [], "needs": [ "cas.derive", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/26", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 26, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 26", "problem_latex": "The least value of $a^{2}\\sec^{2}x + b^{2}\\cosec^{2}x$ is $(a + b)^{2}$.", "markdown": "The least value of $a^{2}\\sec^{2}x + b^{2}\\cosec^{2}x$ is $(a + b)^{2}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "extremum", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "a**2*sec(x)**2 + b**2*csc(x)**2", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "extremum: a**2*sec(x)**2 + b**2*csc(x)**2" ], "shape": [ "extremum: a**N*sec(x)**N + b**N*csc(x)**N" ], "same_problem_in": [], "needs": [ "cas.derive", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/27", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 27, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 27", "problem_latex": "Show that $\\tan 3x \\cot 2x$ cannot lie between $\\frac{1}{9}$~and~$\\frac{3}{2}$.", "markdown": "Show that $\\tan 3x \\cot 2x$ cannot lie between $\\frac{1}{9}$ and $\\frac{3}{2}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "tan(3*x)*cot(2*x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/28", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 28, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 28", "problem_latex": "Show that, if the sum of the lengths of the hypothenuse\\DPnote{** [sic], variant spelling} and another\nside of a right-angled triangle is given, then the area of the triangle is a\nmaximum when the angle between those sides is~$60°$. \\MathTrip{1909.}", "markdown": "Show that, if the sum of the lengths of the hypothenuse and another side of a right-angled triangle is given, then the area of the triangle is a maximum when the angle between those sides is $60°$. % [0]% (*Math. Trip.* 1909.)% [1]%", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/29a", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 29, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 29a", "problem_latex": "A line is drawn through a fixed point~$(a, b)$ to meet the axes $OX$,~$OY$\nin $P$~and~$Q$. Show that the minimum values of $PQ$, $OP + OQ$, and $OP·OQ$\nare respectively $(a^{2/3} + b^{2/3})^{3/2}$, $(\\sqrt{a} + \\sqrt{b})^{2}$, and~$4ab$.", "markdown": "A line is drawn through a fixed point $(a, b)$ to meet the axes $OX$, $OY$ in $P$ and $Q$. Show that the minimum values of $PQ$, $OP + OQ$, and $OP·OQ$ are respectively $(a^{2/3} + b^{2/3})^{3/2}$, $(\\sqrt{a} + \\sqrt{b})^{2}$, and $4ab$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "extremum", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.solve.num" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/29b", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 29, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 29b", "problem_latex": "A line is drawn through a fixed point~$(a, b)$ to meet the axes $OX$,~$OY$\nin $P$~and~$Q$. Show that the minimum values of $PQ$, $OP + OQ$, and $OP·OQ$\nare respectively $(a^{2/3} + b^{2/3})^{3/2}$, $(\\sqrt{a} + \\sqrt{b})^{2}$, and~$4ab$.", "markdown": "A line is drawn through a fixed point $(a, b)$ to meet the axes $OX$, $OY$ in $P$ and $Q$. Show that the minimum values of $PQ$, $OP + OQ$, and $OP·OQ$ are respectively $(a^{2/3} + b^{2/3})^{3/2}$, $(\\sqrt{a} + \\sqrt{b})^{2}$, and $4ab$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "extremum", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.solve.num" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/29c", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 29, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 29c", "problem_latex": "A line is drawn through a fixed point~$(a, b)$ to meet the axes $OX$,~$OY$\nin $P$~and~$Q$. Show that the minimum values of $PQ$, $OP + OQ$, and $OP·OQ$\nare respectively $(a^{2/3} + b^{2/3})^{3/2}$, $(\\sqrt{a} + \\sqrt{b})^{2}$, and~$4ab$.", "markdown": "A line is drawn through a fixed point $(a, b)$ to meet the axes $OX$, $OY$ in $P$ and $Q$. Show that the minimum values of $PQ$, $OP + OQ$, and $OP·OQ$ are respectively $(a^{2/3} + b^{2/3})^{3/2}$, $(\\sqrt{a} + \\sqrt{b})^{2}$, and $4ab$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "extremum", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.solve.num" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/30", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 30, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 30", "problem_latex": "A tangent to an ellipse meets the axes in $P$~and~$Q$. Show that the\nleast value of~$PQ$ is equal to the sum of the semiaxes of the ellipse.", "markdown": "A tangent to an ellipse meets the axes in $P$ and $Q$. Show that the least value of $PQ$ is equal to the sum of the semiaxes of the ellipse.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.solve.num" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/31", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 31, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 31", "problem_latex": "Find the lengths and directions of the axes of the conic\n\\[\nax^{2} + 2hxy + by^{2} = 1.\n\\]", "markdown": "Find the lengths and directions of the axes of the conic ax^2 + 2hxy + by^2 = 1.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/32", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 32, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 32", "problem_latex": "The greatest value of~$x^{m}y^{n}$, where $x$~and~$y$ are positive and\n$x + y = k$, is\n\\[\nm^{m} n^{n} k^{m+n}/(m + n)^{m+n}.\n\\]", "markdown": "The greatest value of $x^{m}y^{n}$, where $x$ and $y$ are positive and $x + y = k$, is m^m n^n k^m+n/(m + n)^m+n.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "extremum", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "x**m*(k - x)**n", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "extremum: x**b*(a - x)**c" ], "shape": [ "extremum: x**b*(a - x)**c" ], "same_problem_in": [], "needs": [ "cas.derive", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/33", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 33, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 33", "problem_latex": "The greatest value of $ax + by$, where $x$~and~$y$ are positive and\n$x^{2} + xy + y^{2} = 3\\kappa^{2}$, is\n\\[\n2\\kappa \\sqrtp{a^{2} - ab + b^{2}}.\n\\]", "markdown": "The greatest value of $ax + by$, where $x$ and $y$ are positive and $x^{2} + xy + y^{2} = 3\\kappa^{2}$, is 2a^2 - ab + b^2.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "extremum", "verdict": "FLAG-EXTRACTION", "judge_why": "problem or answer expression is null", "problem_expr": "a*x + b*y", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-EXTRACTION" ] }, "form": [ "extremum: a*x + b*c" ], "shape": [ "extremum: a*x + b*c" ], "same_problem_in": [], "needs": [ "cas.derive", "cas.solve.num" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/34", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 34, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 34", "problem_latex": "If $\\theta$ and~$\\phi$ are acute angles connected by the relation $a \\sec\\theta + b \\sec\\phi = c$,\nwhere $a$,~$b$,~$c$ are positive, then $a\\cos\\theta + b\\cos\\phi$ is a minimum when $\\theta = \\phi$.", "markdown": "If $\\theta$ and $\\phi$ are acute angles connected by the relation $a \\sec\\theta + b \\sec\\phi = c$, where $a$, $b$, $c$ are positive, then $a\\cos\\theta + b\\cos\\phi$ is a minimum when $\\theta = \\phi$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "extremum", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "a*cos(theta) + b*cos(phi)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "extremum: Eq(a*sec(x) + b*sec(d), c)" ], "shape": [ "extremum: Eq(a*sec(x) + b*sec(d), c)" ], "same_problem_in": [], "needs": [ "cas.derive", "cas.solve.num", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/3a", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 3, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 3a", "problem_latex": "Show that $x - \\sin x$ is an increasing function throughout any interval\nof values of~$x$, and that $\\tan x - x$ increases as $x$~increases from $-\\frac{1}{2}\\pi$ to~$\\frac{1}{2}\\pi$.\nFor what values of~$a$ is $ax - \\sin x$ a steadily increasing or decreasing function\nof~$x$?", "markdown": "Show that $x - \\sin x$ is an increasing function throughout any interval of values of $x$, and that $\\tan x - x$ increases as $x$ increases from $-\\frac{1}{2}\\pi$ to $\\frac{1}{2}\\pi$. For what values of $a$ is $ax - \\sin x$ a steadily increasing or decreasing function of $x$?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/3b", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 3, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 3b", "problem_latex": "Show that $x - \\sin x$ is an increasing function throughout any interval\nof values of~$x$, and that $\\tan x - x$ increases as $x$~increases from $-\\frac{1}{2}\\pi$ to~$\\frac{1}{2}\\pi$.\nFor what values of~$a$ is $ax - \\sin x$ a steadily increasing or decreasing function\nof~$x$?", "markdown": "Show that $x - \\sin x$ is an increasing function throughout any interval of values of $x$, and that $\\tan x - x$ increases as $x$ increases from $-\\frac{1}{2}\\pi$ to $\\frac{1}{2}\\pi$. For what values of $a$ is $ax - \\sin x$ a steadily increasing or decreasing function of $x$?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/3c", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 3, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 3c", "problem_latex": "Show that $x - \\sin x$ is an increasing function throughout any interval\nof values of~$x$, and that $\\tan x - x$ increases as $x$~increases from $-\\frac{1}{2}\\pi$ to~$\\frac{1}{2}\\pi$.\nFor what values of~$a$ is $ax - \\sin x$ a steadily increasing or decreasing function\nof~$x$?", "markdown": "Show that $x - \\sin x$ is an increasing function throughout any interval of values of $x$, and that $\\tan x - x$ increases as $x$ increases from $-\\frac{1}{2}\\pi$ to $\\frac{1}{2}\\pi$. For what values of $a$ is $ax - \\sin x$ a steadily increasing or decreasing function of $x$?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "a*x - sin(x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/4", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 4", "problem_latex": "Show that $\\tan x - x$ also increases from $x = \\frac{1}{2}\\pi$ to $x = \\frac{3}{2}\\pi$, from $x = \\frac{3}{2}\\pi$\nto $x = \\frac{5}{2}\\pi$, and so on, and deduce that there is one and only one root of the\nequation $\\tan x = x$ in each of these intervals (cf.\\ \\Ex{xvii}.~4).", "markdown": "Show that $\\tan x - x$ also increases from $x = \\frac{1}{2}\\pi$ to $x = \\frac{3}{2}\\pi$, from $x = \\frac{3}{2}\\pi$ to $x = \\frac{5}{2}\\pi$, and so on, and deduce that there is one and only one root of the equation $\\tan x = x$ in each of these intervals (cf. % [examples:xvii]Ex. xvii%. 4).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/5", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 5", "problem_latex": "{\\Loosen Deduce from Ex.~3 that $\\sin x - x < 0$ if $x > 0$, from this that\n$\\cos x - 1 + \\frac{1}{2}x^{2} > 0$, and from this that $\\sin x - x + \\frac{1}{6} x^{3} > 0$. And, generally,\nprove that if}\n\\begin{align*}\nC_{2m} & = \\cos x - 1 + \\frac{x^{2}}{2!} - \\dots - (-1)^{m} \\frac{x^{2m}}{\\DPchg{2m!}{(2m)!}},\\\\\nS_{2m+1}& = \\sin x - x + \\frac{x^{3}}{3!} - \\dots - (-1)^{m} \\frac{x^{2m+1}}{(2m+1)!},\n\\end{align*}\nand $x> 0$, then $C_{2m}$~and~$S_{2m+1}$ are positive or negative according as $m$~is odd\nor even.", "markdown": "0.375em plus 0.75em minus 0.25emDeduce from Ex. 3 that $\\sin x - x < 0$ if $x > 0$, from this that $\\cos x - 1 + \\frac{1}{2}x^{2} > 0$, and from this that $\\sin x - x + \\frac{1}{6} x^{3} > 0$. And, generally, prove that if align* C_2m & = x - 1 + x^22! - …- (-1)^m x^2m2m!(2m)!, S_2m+1& = x - x + x^33! - …- (-1)^m x^2m+1(2m+1)!, align* and $x> 0$, then $C_{2m}$ and $S_{2m+1}$ are positive or negative according as $m$ is odd or even.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/6", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 6", "problem_latex": "If $f(x)$~and~$f''(x)$ are continuous and have the same sign at every\npoint of an interval~$\\DPmod{(a, b)}{[a, b]}$, then this interval can include at most one root of\neither of the equations $f(x) = 0$, $f'(x) = 0$.", "markdown": "If $f(x)$ and $f''(x)$ are continuous and have the same sign at every point of an interval $\\DPmod{(a, b)}{[a, b]}$, then this interval can include at most one root of either of the equations $f(x) = 0$, $f'(x) = 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/7", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 7", "problem_latex": "The functions $u$,~$v$ and their derivatives $u'$,~$v'$ are continuous\nthroughout a certain interval of values of~$x$, and $uv' - u'v$ never vanishes\nat any point of the interval. Show that between any two roots of $u = 0$\nlies one of $v = 0$, and conversely. Verify the theorem when $u = \\cos x$, $v = \\sin x$.", "markdown": "The functions $u$, $v$ and their derivatives $u'$, $v'$ are continuous throughout a certain interval of values of $x$, and $uv' - u'v$ never vanishes at any point of the interval. Show that between any two roots of $u = 0$ lies one of $v = 0$, and conversely. Verify the theorem when $u = \\cos x$, $v = \\sin x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/8a", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 8, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 8a", "problem_latex": "Determine the maxima and minima (if any) of $(x - 1)^{2} (x + 2)$, $x^{3} - 3x$,\n$2x^{3} - 3x^{2} - 36x + 10$, $4x^{3} - 18x^{2} + 27x - 7$, $3x^{4} - 4x^{3} + 1$, $x^{5} - 15x^{3} + 3$. In each\ncase sketch the form of the graph of the function.", "markdown": "Determine the maxima and minima (if any) of $(x - 1)^{2} (x + 2)$, $x^{3} - 3x$, $2x^{3} - 3x^{2} - 36x + 10$, $4x^{3} - 18x^{2} + 27x - 7$, $3x^{4} - 4x^{3} + 1$, $x^{5} - 15x^{3} + 3$. In each case sketch the form of the graph of the function.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "extremum", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "(x - 1)**2*(x + 2)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "extremum: (x - 1)**2*(x + 2)" ], "shape": [ "extremum: (N + x)*(x - 1)**N" ], "same_problem_in": [], "needs": [ "cas.derive", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/8b", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 8, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 8b", "problem_latex": "Determine the maxima and minima (if any) of $(x - 1)^{2} (x + 2)$, $x^{3} - 3x$,\n$2x^{3} - 3x^{2} - 36x + 10$, $4x^{3} - 18x^{2} + 27x - 7$, $3x^{4} - 4x^{3} + 1$, $x^{5} - 15x^{3} + 3$. In each\ncase sketch the form of the graph of the function.", "markdown": "Determine the maxima and minima (if any) of $(x - 1)^{2} (x + 2)$, $x^{3} - 3x$, $2x^{3} - 3x^{2} - 36x + 10$, $4x^{3} - 18x^{2} + 27x - 7$, $3x^{4} - 4x^{3} + 1$, $x^{5} - 15x^{3} + 3$. In each case sketch the form of the graph of the function.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "extremum", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "x**3 - 3*x", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "extremum: x**3 - 3*x" ], "shape": [ "extremum: N*x + x**N" ], "same_problem_in": [], "needs": [ "cas.derive", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/8c", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 8, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 8c", "problem_latex": "Determine the maxima and minima (if any) of $(x - 1)^{2} (x + 2)$, $x^{3} - 3x$,\n$2x^{3} - 3x^{2} - 36x + 10$, $4x^{3} - 18x^{2} + 27x - 7$, $3x^{4} - 4x^{3} + 1$, $x^{5} - 15x^{3} + 3$. In each\ncase sketch the form of the graph of the function.", "markdown": "Determine the maxima and minima (if any) of $(x - 1)^{2} (x + 2)$, $x^{3} - 3x$, $2x^{3} - 3x^{2} - 36x + 10$, $4x^{3} - 18x^{2} + 27x - 7$, $3x^{4} - 4x^{3} + 1$, $x^{5} - 15x^{3} + 3$. In each case sketch the form of the graph of the function.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "extremum", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "2*x**3 - 3*x**2 - 36*x + 10", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "extremum: 2*x**3 - 3*x**2 - 36*x + 10" ], "shape": [ "extremum: N*x + 2*N*x**N + N" ], "same_problem_in": [], "needs": [ "cas.derive", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/8d", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 8, "part": "d", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 8d", "problem_latex": "Determine the maxima and minima (if any) of $(x - 1)^{2} (x + 2)$, $x^{3} - 3x$,\n$2x^{3} - 3x^{2} - 36x + 10$, $4x^{3} - 18x^{2} + 27x - 7$, $3x^{4} - 4x^{3} + 1$, $x^{5} - 15x^{3} + 3$. In each\ncase sketch the form of the graph of the function.", "markdown": "Determine the maxima and minima (if any) of $(x - 1)^{2} (x + 2)$, $x^{3} - 3x$, $2x^{3} - 3x^{2} - 36x + 10$, $4x^{3} - 18x^{2} + 27x - 7$, $3x^{4} - 4x^{3} + 1$, $x^{5} - 15x^{3} + 3$. In each case sketch the form of the graph of the function.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "extremum", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "4*x**3 - 18*x**2 + 27*x - 7", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "extremum: 4*x**3 - 18*x**2 + 27*x - 7" ], "shape": [ "extremum: N*x + 2*N*x**N + N" ], "same_problem_in": [], "needs": [ "cas.derive", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/8e", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 8, "part": "e", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 8e", "problem_latex": "Determine the maxima and minima (if any) of $(x - 1)^{2} (x + 2)$, $x^{3} - 3x$,\n$2x^{3} - 3x^{2} - 36x + 10$, $4x^{3} - 18x^{2} + 27x - 7$, $3x^{4} - 4x^{3} + 1$, $x^{5} - 15x^{3} + 3$. In each\ncase sketch the form of the graph of the function.", "markdown": "Determine the maxima and minima (if any) of $(x - 1)^{2} (x + 2)$, $x^{3} - 3x$, $2x^{3} - 3x^{2} - 36x + 10$, $4x^{3} - 18x^{2} + 27x - 7$, $3x^{4} - 4x^{3} + 1$, $x^{5} - 15x^{3} + 3$. In each case sketch the form of the graph of the function.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "extremum", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "3*x**4 - 4*x**3 + 1", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "extremum: 3*x**4 - 4*x**3 + 1" ], "shape": [ "extremum: 2*N*x**N + 1" ], "same_problem_in": [], "needs": [ "cas.derive", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/8f", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 8, "part": "f", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 8f", "problem_latex": "Determine the maxima and minima (if any) of $(x - 1)^{2} (x + 2)$, $x^{3} - 3x$,\n$2x^{3} - 3x^{2} - 36x + 10$, $4x^{3} - 18x^{2} + 27x - 7$, $3x^{4} - 4x^{3} + 1$, $x^{5} - 15x^{3} + 3$. In each\ncase sketch the form of the graph of the function.", "markdown": "Determine the maxima and minima (if any) of $(x - 1)^{2} (x + 2)$, $x^{3} - 3x$, $2x^{3} - 3x^{2} - 36x + 10$, $4x^{3} - 18x^{2} + 27x - 7$, $3x^{4} - 4x^{3} + 1$, $x^{5} - 15x^{3} + 3$. In each case sketch the form of the graph of the function.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "extremum", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "x**5 - 15*x**3 + 3", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "extremum: x**5 - 15*x**3 + 3" ], "shape": [ "extremum: N*x**N + N + x**N" ], "same_problem_in": [], "needs": [ "cas.derive", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvi/9", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvi", "number": 9, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "222", "location": "Exercise XLVI, problem 9", "problem_latex": "Discuss the maxima and minima of the function $(x - a)^{m} (x - b)^{n}$, where\n$m$~and~$n$ are any positive integers, considering the different cases which occur\naccording as $m$~and~$n$ are odd or even. Sketch the graph of the function.", "markdown": "Discuss the maxima and minima of the function $(x - a)^{m} (x - b)^{n}$, where $m$ and $n$ are any positive integers, considering the different cases which occur according as $m$ and $n$ are odd or even. Sketch the graph of the function.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "extremum", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "(x - a)**m*(x - b)**n", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "extremum: (-a + x)**c*(-b + x)**d" ], "shape": [ "extremum: (-a + x)**c*(-b + x)**d" ], "same_problem_in": [], "needs": [ "cas.derive", "cas.solve.poly" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvii/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvii", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "227", "location": "Exercise XLVII, problem 1", "problem_latex": "Show that\n\\[\n\\phi(b) - \\phi(x) - \\frac{b - x}{b - a}\\{\\phi(b) - \\phi(a)\\}\n\\]\nis the difference between the ordinates of a point on the curve and the\ncorresponding point on the chord.", "markdown": "Show that (b) - (x) - b - xb - a(b) - (a) is the difference between the ordinates of a point on the curve and the corresponding point on the chord.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.simplify", "other:geometric_interpretation" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvii/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvii", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "227", "location": "Exercise XLVII, problem 2", "problem_latex": "Verify the theorem when $\\phi(x) = x^{2}$ and when $\\phi(x) = x^{3}$.", "markdown": "Verify the theorem when $\\phi(x) = x^{2}$ and when $\\phi(x) = x^{3}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "core.arith", "other:verification" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvii/3", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvii", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "227", "location": "Exercise XLVII, problem 3", "problem_latex": "Establish the theorem stated at the end of \\SecNo[§]{124} by means of the Mean\nValue Theorem.", "markdown": "Establish the theorem stated at the end of [§]124 by means of the Mean Value Theorem.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlvii/4", "set": "hardy-course-of-pure-mathematics-1921/ex-xlvii", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "227", "location": "Exercise XLVII, problem 4", "problem_latex": "Use the Mean Value Theorem to prove Theorem~(6) of \\SecNo[§]{113}, assuming\nthat the derivatives which occur are continuous.", "markdown": "Use the Mean Value Theorem to prove Theorem (6) of [§]113, assuming that the derivatives which occur are continuous.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlviii/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xlviii", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "235", "location": "Exercise XLVIII, problem 1", "problem_latex": "Prove that\n\\[\n\\int \\frac{Ax + B}{ax^{2} + 2bx + c}\\, dx\n = \\frac{A}{2a} \\log |X| + \\frac{D}{2a \\sqrtp{-\\Delta}}\n \\log \\left|\\frac{ax + b - \\sqrtp{-\\Delta}}{ax + b + \\sqrtp{-\\Delta}}\\right|\n\\]\n(where $X = ax^{2} + bx + c$) if $\\Delta < 0$, and\n\\[\n\\int \\frac{Ax + B}{ax^{2} + 2bx + c}\\, dx\n = \\frac{A}{2a} \\log |X| + \\frac{D}{2a \\sqrt{\\Delta}}\n \\arctan \\left(\\frac{ax + b}{\\sqrt{\\Delta}}\\right)\n\\]\nif $\\Delta > 0$, $\\Delta$ and~$D$ having the same meanings as on \\PageRef{p.}{234}.", "markdown": "Prove that Ax + Bax^2 + 2bx + c  dx = A2a |X| + D2a - |ax + b - -ax + b + -| (where $X = ax^{2} + bx + c$) if $\\Delta < 0$, and Ax + Bax^2 + 2bx + c  dx = A2a |X| + D2a (ax + b) if $\\Delta > 0$, $\\Delta$ and $D$ having the same meanings as on p.234.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlviii/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xlviii", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "235", "location": "Exercise XLVIII, problem 2", "problem_latex": "In the particular case in which $ac = b^{2}$ the integral is\n\\[\n-\\frac{D}{a(ax + b)} + \\frac{A}{a} \\log |ax + b|.\n\\]", "markdown": "In the particular case in which $ac = b^{2}$ the integral is -Da(ax + b) + Aa |ax + b|.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlviii/3", "set": "hardy-course-of-pure-mathematics-1921/ex-xlviii", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "235", "location": "Exercise XLVIII, problem 3", "problem_latex": "Show that if the roots of $Q(x) = 0$ are all real and distinct, and $P(x)$~is\nof lower degree than~$Q(x)$, then\n\\[\n\\int R(x)\\, dx = \\tsum \\frac{P(\\alpha)}{Q'(\\alpha)} \\log |x - \\alpha|,\n\\]\nthe summation applying to all the roots~$\\alpha$ of $Q(x) = 0$.\n\n[The form of the fraction corresponding to~$\\alpha$ may be deduced from the\nfacts that\n\\[\n\\frac{Q(x)}{x - \\alpha} \\to Q'(\\alpha),\\quad\n(x - \\alpha) R(x) \\to \\frac{P(\\alpha)}{Q'(\\alpha)},\n\\]\nas $x \\to \\alpha$.]", "markdown": "Show that if the roots of $Q(x) = 0$ are all real and distinct, and $P(x)$ is of lower degree than $Q(x)$, then R(x)  dx = P()Q’() |x - |, the summation applying to all the roots $\\alpha$ of $Q(x) = 0$. [The form of the fraction corresponding to $\\alpha$ may be deduced from the facts that Q(x)x - Q’(),0pt minus 3pt(x - ) R(x) P()Q’(), as $x \\to \\alpha$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.solve.poly", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlviii/4", "set": "hardy-course-of-pure-mathematics-1921/ex-xlviii", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "235", "location": "Exercise XLVIII, problem 4", "problem_latex": "If all the roots of~$Q(x)$ are real and $\\alpha$~is a double root, the other roots\nbeing simple roots, and $P(x)$~is of lower degree than~$Q(x)$, then the integral\nis $A/(x - \\alpha) + A'\\log |x - \\alpha| + \\sum B\\log |x - \\beta|$, where\n\\[\nA = -\\frac{2P(\\alpha)}{Q''(\\alpha)},\\quad\nA' = \\frac{2\\{3P'(\\alpha) Q''(\\alpha) - P(a) Q'''(\\alpha)\\}}\n {3\\{Q''(\\alpha)\\}^{2}},\\quad\nB = \\frac{P(\\beta)}{Q'(\\beta)},\n\\]\nand the summation applies to all roots~$\\beta$ of $Q(x) = 0$ other than~$\\alpha$.", "markdown": "If all the roots of $Q(x)$ are real and $\\alpha$ is a double root, the other roots being simple roots, and $P(x)$ is of lower degree than $Q(x)$, then the integral is $A/(x - \\alpha) + A'\\log |x - \\alpha| + \\sum B\\log |x - \\beta|$, where A = -2P()Q”(),0pt minus 3ptA’ = 23P’() Q”() - P(a) Q”’() 3Q”()^2,0pt minus 3ptB = P()Q’(), and the summation applies to all roots $\\beta$ of $Q(x) = 0$ other than $\\alpha$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.solve.poly", "core.log" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlviii/5", "set": "hardy-course-of-pure-mathematics-1921/ex-xlviii", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "235", "location": "Exercise XLVIII, problem 5", "problem_latex": "Calculate\n\\[\n\\int \\frac{dx}{\\{(x - 1) (x^{2} + 1)\\}^{2}}.\n\\]", "markdown": "Calculate dx(x - 1) (x^2 + 1)^2.", "answer_latex": [ "-\\frac{1}{4(x - 1)} - \\frac{1}{4(x^{2} + 1)}\n - \\tfrac{1}{2} \\log |x - 1|\n + \\tfrac{1}{4} \\log (x^{2} + 1)\n + \\tfrac{1}{4} \\arctan x" ], "answer_markdown": [ "-14(x - 1) - 14(x^2 + 1) - 12 |x - 1| + 14 (x^2 + 1) + 14 x" ], "checks": [ { "task": "integrate", "verdict": "PASS", "judge_why": null, "problem_expr": "1/((x - 1)*(x**2 + 1))**2", "answer_expr": "-1/(4*(x - 1)) - 1/(4*(x**2 + 1)) - log(Abs(x - 1))/2 + log(x**2 + 1)/4 + atan(x)/4" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "integrate: 1/((x - 1)**2*(x**2 + 1)**2)" ], "shape": [ "integrate: (x - 1)**N*(x**N + 1)**N" ], "same_problem_in": [], "needs": [ "cas.integrate", "core.log", "core.ratio.partials", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlviii/6a", "set": "hardy-course-of-pure-mathematics-1921/ex-xlviii", "number": 6, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "235", "location": "Exercise XLVIII, problem 6a", "problem_latex": "Integrate\n\\begin{gather*}\n\\frac{x}{(x - a)(x - b)(x - c)},\\quad\n\\frac{x}{(x - a)^{2}(x - b)},\\quad\n\\frac{x}{(x - a)^{2} (x - b)^{2}},\\quad\n\\frac{x}{(x - a)^{3}},\\\\\n%\n\\frac{x}{(x^{2} + a^{2}) (x^{2} + b^{2})},\\quad\n\\frac{x^{2}}{(x^{2} + a^{2}) (x^{2} + b)^{2}},\\quad\n\\frac{x^{2} - a^{2}}{x^{2}(x^{2} + a^{2})},\\quad\n\\frac{x^{2} - a^{2}}{x(x^{2} + a^{2})^{2}}.\n\\end{gather*}", "markdown": "Integrate gather* x(x - a)(x - b)(x - c),0pt minus 3ptx(x - a)^2(x - b),0pt minus 3ptx(x - a)^2 (x - b)^2,0pt minus 3ptx(x - a)^3, % x(x^2 + a^2) (x^2 + b^2),0pt minus 3ptx^2(x^2 + a^2) (x^2 + b)^2,0pt minus 3ptx^2 - a^2x^2(x^2 + a^2),0pt minus 3ptx^2 - a^2x(x^2 + a^2)^2. gather*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "x/((x - a)*(x - b)*(x - c))", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: x/((-a + x)*(-b + x)*(-c + x))" ], "shape": [ "integrate: x/((-a + x)*(-b + x)*(-c + x))" ], "same_problem_in": [], "needs": [ "cas.integrate", "core.log", "core.ratio.partials" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlviii/6b", "set": "hardy-course-of-pure-mathematics-1921/ex-xlviii", "number": 6, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "235", "location": "Exercise XLVIII, problem 6b", "problem_latex": "Integrate\n\\begin{gather*}\n\\frac{x}{(x - a)(x - b)(x - c)},\\quad\n\\frac{x}{(x - a)^{2}(x - b)},\\quad\n\\frac{x}{(x - a)^{2} (x - b)^{2}},\\quad\n\\frac{x}{(x - a)^{3}},\\\\\n%\n\\frac{x}{(x^{2} + a^{2}) (x^{2} + b^{2})},\\quad\n\\frac{x^{2}}{(x^{2} + a^{2}) (x^{2} + b)^{2}},\\quad\n\\frac{x^{2} - a^{2}}{x^{2}(x^{2} + a^{2})},\\quad\n\\frac{x^{2} - a^{2}}{x(x^{2} + a^{2})^{2}}.\n\\end{gather*}", "markdown": "Integrate gather* x(x - a)(x - b)(x - c),0pt minus 3ptx(x - a)^2(x - b),0pt minus 3ptx(x - a)^2 (x - b)^2,0pt minus 3ptx(x - a)^3, % x(x^2 + a^2) (x^2 + b^2),0pt minus 3ptx^2(x^2 + a^2) (x^2 + b)^2,0pt minus 3ptx^2 - a^2x^2(x^2 + a^2),0pt minus 3ptx^2 - a^2x(x^2 + a^2)^2. gather*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "x/((x - a)**2*(x - b))", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: x/((-a + x)**2*(-b + x))" ], "shape": [ "integrate: x*(-a + x)**N/(-b + x)" ], "same_problem_in": [], "needs": [ "cas.integrate", "core.log", "core.ratio.partials" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlviii/6c", "set": "hardy-course-of-pure-mathematics-1921/ex-xlviii", "number": 6, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "235", "location": "Exercise XLVIII, problem 6c", "problem_latex": "Integrate\n\\begin{gather*}\n\\frac{x}{(x - a)(x - b)(x - c)},\\quad\n\\frac{x}{(x - a)^{2}(x - b)},\\quad\n\\frac{x}{(x - a)^{2} (x - b)^{2}},\\quad\n\\frac{x}{(x - a)^{3}},\\\\\n%\n\\frac{x}{(x^{2} + a^{2}) (x^{2} + b^{2})},\\quad\n\\frac{x^{2}}{(x^{2} + a^{2}) (x^{2} + b)^{2}},\\quad\n\\frac{x^{2} - a^{2}}{x^{2}(x^{2} + a^{2})},\\quad\n\\frac{x^{2} - a^{2}}{x(x^{2} + a^{2})^{2}}.\n\\end{gather*}", "markdown": "Integrate gather* x(x - a)(x - b)(x - c),0pt minus 3ptx(x - a)^2(x - b),0pt minus 3ptx(x - a)^2 (x - b)^2,0pt minus 3ptx(x - a)^3, % x(x^2 + a^2) (x^2 + b^2),0pt minus 3ptx^2(x^2 + a^2) (x^2 + b)^2,0pt minus 3ptx^2 - a^2x^2(x^2 + a^2),0pt minus 3ptx^2 - a^2x(x^2 + a^2)^2. gather*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "x/((x - a)**2*(x - b)**2)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: x/((-a + x)**2*(-b + x)**2)" ], "shape": [ "integrate: x*(-a + x)**N*(-b + x)**N" ], "same_problem_in": [], "needs": [ "cas.integrate", "core.log", "core.ratio.partials" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlviii/6d", "set": "hardy-course-of-pure-mathematics-1921/ex-xlviii", "number": 6, "part": "d", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "235", "location": "Exercise XLVIII, problem 6d", "problem_latex": "Integrate\n\\begin{gather*}\n\\frac{x}{(x - a)(x - b)(x - c)},\\quad\n\\frac{x}{(x - a)^{2}(x - b)},\\quad\n\\frac{x}{(x - a)^{2} (x - b)^{2}},\\quad\n\\frac{x}{(x - a)^{3}},\\\\\n%\n\\frac{x}{(x^{2} + a^{2}) (x^{2} + b^{2})},\\quad\n\\frac{x^{2}}{(x^{2} + a^{2}) (x^{2} + b)^{2}},\\quad\n\\frac{x^{2} - a^{2}}{x^{2}(x^{2} + a^{2})},\\quad\n\\frac{x^{2} - a^{2}}{x(x^{2} + a^{2})^{2}}.\n\\end{gather*}", "markdown": "Integrate gather* x(x - a)(x - b)(x - c),0pt minus 3ptx(x - a)^2(x - b),0pt minus 3ptx(x - a)^2 (x - b)^2,0pt minus 3ptx(x - a)^3, % x(x^2 + a^2) (x^2 + b^2),0pt minus 3ptx^2(x^2 + a^2) (x^2 + b)^2,0pt minus 3ptx^2 - a^2x^2(x^2 + a^2),0pt minus 3ptx^2 - a^2x(x^2 + a^2)^2. gather*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "x/(x - a)**3", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: x/(-a + x)**3" ], "shape": [ "integrate: x*(-a + x)**N" ], "same_problem_in": [], "needs": [ "cas.integrate", "core.ratio.partials" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlviii/6e", "set": "hardy-course-of-pure-mathematics-1921/ex-xlviii", "number": 6, "part": "e", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "235", "location": "Exercise XLVIII, problem 6e", "problem_latex": "Integrate\n\\begin{gather*}\n\\frac{x}{(x - a)(x - b)(x - c)},\\quad\n\\frac{x}{(x - a)^{2}(x - b)},\\quad\n\\frac{x}{(x - a)^{2} (x - b)^{2}},\\quad\n\\frac{x}{(x - a)^{3}},\\\\\n%\n\\frac{x}{(x^{2} + a^{2}) (x^{2} + b^{2})},\\quad\n\\frac{x^{2}}{(x^{2} + a^{2}) (x^{2} + b)^{2}},\\quad\n\\frac{x^{2} - a^{2}}{x^{2}(x^{2} + a^{2})},\\quad\n\\frac{x^{2} - a^{2}}{x(x^{2} + a^{2})^{2}}.\n\\end{gather*}", "markdown": "Integrate gather* x(x - a)(x - b)(x - c),0pt minus 3ptx(x - a)^2(x - b),0pt minus 3ptx(x - a)^2 (x - b)^2,0pt minus 3ptx(x - a)^3, % x(x^2 + a^2) (x^2 + b^2),0pt minus 3ptx^2(x^2 + a^2) (x^2 + b)^2,0pt minus 3ptx^2 - a^2x^2(x^2 + a^2),0pt minus 3ptx^2 - a^2x(x^2 + a^2)^2. gather*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "x/((x**2 + a**2)*(x**2 + b**2))", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: x/((a**2 + x**2)*(b**2 + x**2))" ], "shape": [ "integrate: x/((a**N + x**N)*(b**N + x**N))" ], "same_problem_in": [], "needs": [ "cas.integrate", "core.log", "core.ratio.partials", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlviii/6f", "set": "hardy-course-of-pure-mathematics-1921/ex-xlviii", "number": 6, "part": "f", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "235", "location": "Exercise XLVIII, problem 6f", "problem_latex": "Integrate\n\\begin{gather*}\n\\frac{x}{(x - a)(x - b)(x - c)},\\quad\n\\frac{x}{(x - a)^{2}(x - b)},\\quad\n\\frac{x}{(x - a)^{2} (x - b)^{2}},\\quad\n\\frac{x}{(x - a)^{3}},\\\\\n%\n\\frac{x}{(x^{2} + a^{2}) (x^{2} + b^{2})},\\quad\n\\frac{x^{2}}{(x^{2} + a^{2}) (x^{2} + b)^{2}},\\quad\n\\frac{x^{2} - a^{2}}{x^{2}(x^{2} + a^{2})},\\quad\n\\frac{x^{2} - a^{2}}{x(x^{2} + a^{2})^{2}}.\n\\end{gather*}", "markdown": "Integrate gather* x(x - a)(x - b)(x - c),0pt minus 3ptx(x - a)^2(x - b),0pt minus 3ptx(x - a)^2 (x - b)^2,0pt minus 3ptx(x - a)^3, % x(x^2 + a^2) (x^2 + b^2),0pt minus 3ptx^2(x^2 + a^2) (x^2 + b)^2,0pt minus 3ptx^2 - a^2x^2(x^2 + a^2),0pt minus 3ptx^2 - a^2x(x^2 + a^2)^2. gather*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "x**2/((x**2 + a**2)*(x**2 + b)**2)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: x**2/((a**2 + x**2)*(b + x**2)**2)" ], "shape": [ "integrate: x**N*(b + x**N)**N/(a**N + x**N)" ], "same_problem_in": [], "needs": [ "cas.integrate", "core.log", "core.ratio.partials", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlviii/6g", "set": "hardy-course-of-pure-mathematics-1921/ex-xlviii", "number": 6, "part": "g", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "235", "location": "Exercise XLVIII, problem 6g", "problem_latex": "Integrate\n\\begin{gather*}\n\\frac{x}{(x - a)(x - b)(x - c)},\\quad\n\\frac{x}{(x - a)^{2}(x - b)},\\quad\n\\frac{x}{(x - a)^{2} (x - b)^{2}},\\quad\n\\frac{x}{(x - a)^{3}},\\\\\n%\n\\frac{x}{(x^{2} + a^{2}) (x^{2} + b^{2})},\\quad\n\\frac{x^{2}}{(x^{2} + a^{2}) (x^{2} + b)^{2}},\\quad\n\\frac{x^{2} - a^{2}}{x^{2}(x^{2} + a^{2})},\\quad\n\\frac{x^{2} - a^{2}}{x(x^{2} + a^{2})^{2}}.\n\\end{gather*}", "markdown": "Integrate gather* x(x - a)(x - b)(x - c),0pt minus 3ptx(x - a)^2(x - b),0pt minus 3ptx(x - a)^2 (x - b)^2,0pt minus 3ptx(x - a)^3, % x(x^2 + a^2) (x^2 + b^2),0pt minus 3ptx^2(x^2 + a^2) (x^2 + b)^2,0pt minus 3ptx^2 - a^2x^2(x^2 + a^2),0pt minus 3ptx^2 - a^2x(x^2 + a^2)^2. gather*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "(x**2 - a**2)/(x**2*(x**2 + a**2))", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: (-a**2 + x**2)/(x**2*(a**2 + x**2))" ], "shape": [ "integrate: x**N*(-a**N + x**N)/(a**N + x**N)" ], "same_problem_in": [], "needs": [ "cas.integrate", "core.log", "core.ratio.partials", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlviii/6h", "set": "hardy-course-of-pure-mathematics-1921/ex-xlviii", "number": 6, "part": "h", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "235", "location": "Exercise XLVIII, problem 6h", "problem_latex": "Integrate\n\\begin{gather*}\n\\frac{x}{(x - a)(x - b)(x - c)},\\quad\n\\frac{x}{(x - a)^{2}(x - b)},\\quad\n\\frac{x}{(x - a)^{2} (x - b)^{2}},\\quad\n\\frac{x}{(x - a)^{3}},\\\\\n%\n\\frac{x}{(x^{2} + a^{2}) (x^{2} + b^{2})},\\quad\n\\frac{x^{2}}{(x^{2} + a^{2}) (x^{2} + b)^{2}},\\quad\n\\frac{x^{2} - a^{2}}{x^{2}(x^{2} + a^{2})},\\quad\n\\frac{x^{2} - a^{2}}{x(x^{2} + a^{2})^{2}}.\n\\end{gather*}", "markdown": "Integrate gather* x(x - a)(x - b)(x - c),0pt minus 3ptx(x - a)^2(x - b),0pt minus 3ptx(x - a)^2 (x - b)^2,0pt minus 3ptx(x - a)^3, % x(x^2 + a^2) (x^2 + b^2),0pt minus 3ptx^2(x^2 + a^2) (x^2 + b)^2,0pt minus 3ptx^2 - a^2x^2(x^2 + a^2),0pt minus 3ptx^2 - a^2x(x^2 + a^2)^2. gather*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "(x**2 - a**2)/(x*(x**2 + a**2)**2)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: (-a**2 + x**2)/(x*(a**2 + x**2)**2)" ], "shape": [ "integrate: (-a**N + x**N)*(a**N + x**N)**N/x" ], "same_problem_in": [], "needs": [ "cas.integrate", "core.log", "core.ratio.partials", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlviii/7a", "set": "hardy-course-of-pure-mathematics-1921/ex-xlviii", "number": 7, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "235", "location": "Exercise XLVIII, problem 7a", "problem_latex": "Prove the formulae:\n\\begin{alignat*}{3}\n\\int \\frac{dx}{1 + x^{4}}\n &= \\frac{1}{4\\sqrt{2}} \\biggl\\{%\n &&\\log \\biggl(\\frac{1 + x\\sqrt{2} + x^{2}}{1 - x\\sqrt{2} + x^{2}}\\biggr)\n &&+ 2\\arctan \\biggl(\\frac{x\\sqrt{2}}{1 - x^{2}}\\biggr)\\biggr\\},\\\\\n%\n\\int \\frac{x^{2}\\, dx}{1 + x^{4}}\n &= \\frac{1}{4\\sqrt{2}} \\biggl\\{%\n &-&\\log \\biggl(\\frac{1 + x\\sqrt{2} + x^{2}}{1 - x\\sqrt{2} + x^{2}}\\biggr)\n &&+ 2\\arctan \\biggl(\\frac{x\\sqrt{2}}{1 - x^{2}}\\biggr)\\biggr\\},\\\\\n%\n\\int \\frac{dx}{1 + x^{2} + x^{4}}\n &= \\frac{1}{4\\sqrt{3}}\\biggl\\{%\n &\\sqrt{3}&\\log \\biggl(\\frac{1 + x + x^{2}}{1 - x + x^{2}}\\biggr)\n &&+ 2\\arctan \\biggl(\\frac{x\\sqrt{3}}{1 - x^{2}}\\biggr)\\biggr\\}.\n\\end{alignat*}", "markdown": "Prove the formulae: alignat*3 dx1 + x^4 &= 142 % &&(1 + x2 + x^21 - x2 + x^2) &&+ 2(x21 - x^2), % x^2  dx1 + x^4 &= 142 % &-&(1 + x2 + x^21 - x2 + x^2) &&+ 2(x21 - x^2), % dx1 + x^2 + x^4 &= 143% &3&(1 + x + x^21 - x + x^2) &&+ 2(x31 - x^2). alignat*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "1/(1 + x**4)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: 1/(x**4 + 1)" ], "shape": [ "integrate: 1/(x**N + 1)" ], "same_problem_in": [], "needs": [ "cas.integrate", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlviii/7b", "set": "hardy-course-of-pure-mathematics-1921/ex-xlviii", "number": 7, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "235", "location": "Exercise XLVIII, problem 7b", "problem_latex": "Prove the formulae:\n\\begin{alignat*}{3}\n\\int \\frac{dx}{1 + x^{4}}\n &= \\frac{1}{4\\sqrt{2}} \\biggl\\{%\n &&\\log \\biggl(\\frac{1 + x\\sqrt{2} + x^{2}}{1 - x\\sqrt{2} + x^{2}}\\biggr)\n &&+ 2\\arctan \\biggl(\\frac{x\\sqrt{2}}{1 - x^{2}}\\biggr)\\biggr\\},\\\\\n%\n\\int \\frac{x^{2}\\, dx}{1 + x^{4}}\n &= \\frac{1}{4\\sqrt{2}} \\biggl\\{%\n &-&\\log \\biggl(\\frac{1 + x\\sqrt{2} + x^{2}}{1 - x\\sqrt{2} + x^{2}}\\biggr)\n &&+ 2\\arctan \\biggl(\\frac{x\\sqrt{2}}{1 - x^{2}}\\biggr)\\biggr\\},\\\\\n%\n\\int \\frac{dx}{1 + x^{2} + x^{4}}\n &= \\frac{1}{4\\sqrt{3}}\\biggl\\{%\n &\\sqrt{3}&\\log \\biggl(\\frac{1 + x + x^{2}}{1 - x + x^{2}}\\biggr)\n &&+ 2\\arctan \\biggl(\\frac{x\\sqrt{3}}{1 - x^{2}}\\biggr)\\biggr\\}.\n\\end{alignat*}", "markdown": "Prove the formulae: alignat*3 dx1 + x^4 &= 142 % &&(1 + x2 + x^21 - x2 + x^2) &&+ 2(x21 - x^2), % x^2  dx1 + x^4 &= 142 % &-&(1 + x2 + x^21 - x2 + x^2) &&+ 2(x21 - x^2), % dx1 + x^2 + x^4 &= 143% &3&(1 + x + x^21 - x + x^2) &&+ 2(x31 - x^2). alignat*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "integrate", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "x**2/(1 + x**4)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "integrate: x**2/(x**4 + 1)" ], "shape": [ "integrate: x**N/(x**N + 1)" ], "same_problem_in": [], "needs": [ "cas.integrate", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xlviii/7c", "set": "hardy-course-of-pure-mathematics-1921/ex-xlviii", "number": 7, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "235", "location": "Exercise XLVIII, problem 7c", "problem_latex": "\\int \\frac{dx}{1 + x^{2} + x^{4}}", "markdown": "dx1 + x^2 + x^4", "answer_latex": [ "\\frac{1}{4\\sqrt{3}}\\biggl\\{%\n &\\sqrt{3}&\\log \\biggl(\\frac{1 + x + x^{2}}{1 - x + x^{2}}\\biggr)\n &&+ 2\\arctan \\biggl(\\frac{x\\sqrt{3}}{1 - x^{2}}\\biggr)\\biggr\\}" ], "answer_markdown": [ "143% &3&(1 + x + x^21 - x + x^2) &&+ 2(x31 - x^2)" ], "checks": [ { "task": "integrate", "verdict": "PASS", "judge_why": null, "problem_expr": "1/(1 + x**2 + x**4)", "answer_expr": "1/(4*sqrt(3))*(sqrt(3)*log((1 + x + x**2)/(1 - x + x**2)) + 2*atan(x*sqrt(3)/(1 - x**2)))" } ], "verdict": "verified", "judge": { "commit": "4a88328", "verdicts": [ "PASS" ] }, "form": [ "integrate: 1/(x**4 + x**2 + 1)" ], "shape": [ "integrate: 1/(2*x**N + 1)" ], "same_problem_in": [], "needs": [ "cas.integrate", "core.log", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/10a", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 10, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 10a", "problem_latex": "Draw the graphs of $\\arccos x$ and $\\arcsin x$.", "markdown": "Draw the graphs of $\\arccos x$ and $\\arcsin x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "acos(x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/10b", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 10, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 10b", "problem_latex": "Draw the graphs of $\\arccos x$ and $\\arcsin x$.", "markdown": "Draw the graphs of $\\arccos x$ and $\\arcsin x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "asin(x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/11a", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 11, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 11a", "problem_latex": "Draw the graphs of\n\\[\n\\tan x,\\quad\n\\cot x,\\quad\n\\sec x,\\quad\n\\cosec x,\\quad\n\\tan^{2} x,\\quad\n\\cot^{2} x,\\quad\n\\sec^{2} x,\\quad\n\\cosec^{2} x.\n\\]", "markdown": "Draw the graphs of x,0pt minus 3ptx,0pt minus 3ptx,0pt minus 3ptx,0pt minus 3pt^2 x,0pt minus 3pt^2 x,0pt minus 3pt^2 x,0pt minus 3pt^2 x.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "tan(x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/11b", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 11, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 11b", "problem_latex": "Draw the graphs of\n\\[\n\\tan x,\\quad\n\\cot x,\\quad\n\\sec x,\\quad\n\\cosec x,\\quad\n\\tan^{2} x,\\quad\n\\cot^{2} x,\\quad\n\\sec^{2} x,\\quad\n\\cosec^{2} x.\n\\]", "markdown": "Draw the graphs of x,0pt minus 3ptx,0pt minus 3ptx,0pt minus 3ptx,0pt minus 3pt^2 x,0pt minus 3pt^2 x,0pt minus 3pt^2 x,0pt minus 3pt^2 x.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "cot(x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/11c", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 11, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 11c", "problem_latex": "Draw the graphs of\n\\[\n\\tan x,\\quad\n\\cot x,\\quad\n\\sec x,\\quad\n\\cosec x,\\quad\n\\tan^{2} x,\\quad\n\\cot^{2} x,\\quad\n\\sec^{2} x,\\quad\n\\cosec^{2} x.\n\\]", "markdown": "Draw the graphs of x,0pt minus 3ptx,0pt minus 3ptx,0pt minus 3ptx,0pt minus 3pt^2 x,0pt minus 3pt^2 x,0pt minus 3pt^2 x,0pt minus 3pt^2 x.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "sec(x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/11d", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 11, "part": "d", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 11d", "problem_latex": "Draw the graphs of\n\\[\n\\tan x,\\quad\n\\cot x,\\quad\n\\sec x,\\quad\n\\cosec x,\\quad\n\\tan^{2} x,\\quad\n\\cot^{2} x,\\quad\n\\sec^{2} x,\\quad\n\\cosec^{2} x.\n\\]", "markdown": "Draw the graphs of x,0pt minus 3ptx,0pt minus 3ptx,0pt minus 3ptx,0pt minus 3pt^2 x,0pt minus 3pt^2 x,0pt minus 3pt^2 x,0pt minus 3pt^2 x.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "csc(x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/11e", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 11, "part": "e", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 11e", "problem_latex": "Draw the graphs of\n\\[\n\\tan x,\\quad\n\\cot x,\\quad\n\\sec x,\\quad\n\\cosec x,\\quad\n\\tan^{2} x,\\quad\n\\cot^{2} x,\\quad\n\\sec^{2} x,\\quad\n\\cosec^{2} x.\n\\]", "markdown": "Draw the graphs of x,0pt minus 3ptx,0pt minus 3ptx,0pt minus 3ptx,0pt minus 3pt^2 x,0pt minus 3pt^2 x,0pt minus 3pt^2 x,0pt minus 3pt^2 x.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "tan(x)**2", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/11f", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 11, "part": "f", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 11f", "problem_latex": "Draw the graphs of\n\\[\n\\tan x,\\quad\n\\cot x,\\quad\n\\sec x,\\quad\n\\cosec x,\\quad\n\\tan^{2} x,\\quad\n\\cot^{2} x,\\quad\n\\sec^{2} x,\\quad\n\\cosec^{2} x.\n\\]", "markdown": "Draw the graphs of x,0pt minus 3ptx,0pt minus 3ptx,0pt minus 3ptx,0pt minus 3pt^2 x,0pt minus 3pt^2 x,0pt minus 3pt^2 x,0pt minus 3pt^2 x.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "cot(x)**2", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/11g", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 11, "part": "g", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 11g", "problem_latex": "Draw the graphs of\n\\[\n\\tan x,\\quad\n\\cot x,\\quad\n\\sec x,\\quad\n\\cosec x,\\quad\n\\tan^{2} x,\\quad\n\\cot^{2} x,\\quad\n\\sec^{2} x,\\quad\n\\cosec^{2} x.\n\\]", "markdown": "Draw the graphs of x,0pt minus 3ptx,0pt minus 3ptx,0pt minus 3ptx,0pt minus 3pt^2 x,0pt minus 3pt^2 x,0pt minus 3pt^2 x,0pt minus 3pt^2 x.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "sec(x)**2", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/11h", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 11, "part": "h", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 11h", "problem_latex": "Draw the graphs of\n\\[\n\\tan x,\\quad\n\\cot x,\\quad\n\\sec x,\\quad\n\\cosec x,\\quad\n\\tan^{2} x,\\quad\n\\cot^{2} x,\\quad\n\\sec^{2} x,\\quad\n\\cosec^{2} x.\n\\]", "markdown": "Draw the graphs of x,0pt minus 3ptx,0pt minus 3ptx,0pt minus 3ptx,0pt minus 3pt^2 x,0pt minus 3pt^2 x,0pt minus 3pt^2 x,0pt minus 3pt^2 x.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "csc(x)**2", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/12", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 12, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 12", "problem_latex": "Draw the graphs of $\\arctan x$, $\\arccot x$, $\\arcsec x$, $\\arccosec x$. Give\nformulae (as in Ex.~10) expressing all the values of each of these functions\nin terms of any particular value.", "markdown": "Draw the graphs of $\\arctan x$, $\\arccot x$, $\\arcsec x$, $\\arccosec x$. Give formulae (as in Ex. 10) expressing all the values of each of these functions in terms of any particular value.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph", "other:general_branch_formulae" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/13a", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 13, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 13a", "problem_latex": "Draw the graphs of $\\tan(1/x)$, $\\cot(1/x)$, $\\sec(1/x)$, $\\cosec(1/x)$.", "markdown": "Draw the graphs of $\\tan(1/x)$, $\\cot(1/x)$, $\\sec(1/x)$, $\\cosec(1/x)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "tan(1/x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/13b", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 13, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 13b", "problem_latex": "Draw the graphs of $\\tan(1/x)$, $\\cot(1/x)$, $\\sec(1/x)$, $\\cosec(1/x)$.", "markdown": "Draw the graphs of $\\tan(1/x)$, $\\cot(1/x)$, $\\sec(1/x)$, $\\cosec(1/x)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "cot(1/x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/13c", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 13, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 13c", "problem_latex": "Draw the graphs of $\\tan(1/x)$, $\\cot(1/x)$, $\\sec(1/x)$, $\\cosec(1/x)$.", "markdown": "Draw the graphs of $\\tan(1/x)$, $\\cot(1/x)$, $\\sec(1/x)$, $\\cosec(1/x)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "sec(1/x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/13d", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 13, "part": "d", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 13d", "problem_latex": "Draw the graphs of $\\tan(1/x)$, $\\cot(1/x)$, $\\sec(1/x)$, $\\cosec(1/x)$.", "markdown": "Draw the graphs of $\\tan(1/x)$, $\\cot(1/x)$, $\\sec(1/x)$, $\\cosec(1/x)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "csc(1/x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/14", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 14, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 14", "problem_latex": "Show that $\\cos x$ and $\\sin x$ are not rational functions of~$x$.", "markdown": "Show that $\\cos x$ and $\\sin x$ are not rational functions of $x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/15", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 15, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 15", "problem_latex": "Show, more generally, that no function with a period can be an\nalgebraical function of~$x$.", "markdown": "Show, more generally, that no function with a period can be an algebraical function of $x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/16", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 16, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 16", "problem_latex": "The inverse sine and inverse cosine are not rational or algebraical\nfunctions.", "markdown": "The inverse sine and inverse cosine are not rational or algebraical functions.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/1a", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 1, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 1a", "problem_latex": "Draw the graphs of $\\cos x$, $\\sin x$, and $a\\cos x + b\\sin x$.", "markdown": "Draw the graphs of $\\cos x$, $\\sin x$, and $a\\cos x + b\\sin x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "cos(x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/1b", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 1, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 1b", "problem_latex": "Draw the graphs of $\\cos x$, $\\sin x$, and $a\\cos x + b\\sin x$.", "markdown": "Draw the graphs of $\\cos x$, $\\sin x$, and $a\\cos x + b\\sin x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "sin(x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/1c", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 1, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 1c", "problem_latex": "Draw the graphs of $\\cos x$, $\\sin x$, and $a\\cos x + b\\sin x$.", "markdown": "Draw the graphs of $\\cos x$, $\\sin x$, and $a\\cos x + b\\sin x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "a*cos(x) + b*sin(x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/2a", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 2, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 2a", "problem_latex": "Draw the graphs of $\\cos^{2} x$, $\\sin^{2} x$, $a\\cos^{2} x + b\\sin^{2} x$.", "markdown": "Draw the graphs of $\\cos^{2} x$, $\\sin^{2} x$, $a\\cos^{2} x + b\\sin^{2} x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "cos(x)**2", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/2b", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 2, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 2b", "problem_latex": "Draw the graphs of $\\cos^{2} x$, $\\sin^{2} x$, $a\\cos^{2} x + b\\sin^{2} x$.", "markdown": "Draw the graphs of $\\cos^{2} x$, $\\sin^{2} x$, $a\\cos^{2} x + b\\sin^{2} x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "sin(x)**2", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/2c", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 2, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 2c", "problem_latex": "Draw the graphs of $\\cos^{2} x$, $\\sin^{2} x$, $a\\cos^{2} x + b\\sin^{2} x$.", "markdown": "Draw the graphs of $\\cos^{2} x$, $\\sin^{2} x$, $a\\cos^{2} x + b\\sin^{2} x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "a*cos(x)**2 + b*sin(x)**2", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/3", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 3", "problem_latex": "Suppose the graphs of $f(x)$~and~$F(x)$ drawn. Then the graph of\n\\[\nf(x)\\cos^{2} x + F(x)\\sin^{2} x\n\\]\nis a wavy curve which oscillates between the curves $y = f(x)$, $y = F(x)$. Draw\nthe graph when $f(x) = x$, $F(x) = x^{2}$.", "markdown": "Suppose the graphs of $f(x)$ and $F(x)$ drawn. Then the graph of f(x)^2 x + F(x)^2 x is a wavy curve which oscillates between the curves $y = f(x)$, $y = F(x)$. Draw the graph when $f(x) = x$, $F(x) = x^{2}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "x*cos(x)**2 + x**2*sin(x)**2", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/4", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 4", "problem_latex": "Show that the graph of $\\cos px + \\cos qx$ lies between those of\n$2\\cos\\frac{1}{2}(p - q)x$ and $-2\\cos\\frac{1}{2}(p + q)x$, touching each in turn. Sketch the\ngraph when $(p - q)/(p + q)$ is small. \\MathTrip{1908.}", "markdown": "Show that the graph of $\\cos px + \\cos qx$ lies between those of $2\\cos\\frac{1}{2}(p - q)x$ and $-2\\cos\\frac{1}{2}(p + q)x$, touching each in turn. Sketch the graph when $(p - q)/(p + q)$ is small. % [0]% (*Math. Trip.* 1908.)% [1]%", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "cos(p*x) + cos(q*x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph", "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/5a", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 5, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 5a", "problem_latex": "Draw the graphs of $x + \\sin x$, $(1/x) + \\sin x$, $x\\sin x$, $(\\sin x)/x$.", "markdown": "Draw the graphs of $x + \\sin x$, $(1/x) + \\sin x$, $x\\sin x$, $(\\sin x)/x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "x + sin(x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/5b", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 5, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 5b", "problem_latex": "Draw the graphs of $x + \\sin x$, $(1/x) + \\sin x$, $x\\sin x$, $(\\sin x)/x$.", "markdown": "Draw the graphs of $x + \\sin x$, $(1/x) + \\sin x$, $x\\sin x$, $(\\sin x)/x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "1/x + sin(x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/5c", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 5, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 5c", "problem_latex": "Draw the graphs of $x + \\sin x$, $(1/x) + \\sin x$, $x\\sin x$, $(\\sin x)/x$.", "markdown": "Draw the graphs of $x + \\sin x$, $(1/x) + \\sin x$, $x\\sin x$, $(\\sin x)/x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "x*sin(x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/5d", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 5, "part": "d", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 5d", "problem_latex": "Draw the graphs of $x + \\sin x$, $(1/x) + \\sin x$, $x\\sin x$, $(\\sin x)/x$.", "markdown": "Draw the graphs of $x + \\sin x$, $(1/x) + \\sin x$, $x\\sin x$, $(\\sin x)/x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "sin(x)/x", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/6", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 6", "problem_latex": "Draw the graph of~$\\sin(1/x)$.", "markdown": "Draw the graph of $\\sin(1/x)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "sin(1/x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/7", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 7", "problem_latex": "Draw the graph of $x\\sin(1/x)$.", "markdown": "Draw the graph of $x\\sin(1/x)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "x*sin(1/x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/8a", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 8, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 8a", "problem_latex": "Draw the graphs of $x^{2}\\sin(1/x)$, $(1/x)\\sin(1/x)$, $\\sin^{2}(1/x)$, $\\{x\\sin(1/x)\\}^{2}$,\n$a\\cos^{2}(1/x) + b\\sin^{2}(1/x)$, $\\sin x + \\sin(1/x)$, $\\sin x\\sin(1/x)$.", "markdown": "Draw the graphs of $x^{2}\\sin(1/x)$, $(1/x)\\sin(1/x)$, $\\sin^{2}(1/x)$, $\\{x\\sin(1/x)\\}^{2}$, $a\\cos^{2}(1/x) + b\\sin^{2}(1/x)$, $\\sin x + \\sin(1/x)$, $\\sin x\\sin(1/x)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "x**2*sin(1/x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/8b", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 8, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 8b", "problem_latex": "Draw the graphs of $x^{2}\\sin(1/x)$, $(1/x)\\sin(1/x)$, $\\sin^{2}(1/x)$, $\\{x\\sin(1/x)\\}^{2}$,\n$a\\cos^{2}(1/x) + b\\sin^{2}(1/x)$, $\\sin x + \\sin(1/x)$, $\\sin x\\sin(1/x)$.", "markdown": "Draw the graphs of $x^{2}\\sin(1/x)$, $(1/x)\\sin(1/x)$, $\\sin^{2}(1/x)$, $\\{x\\sin(1/x)\\}^{2}$, $a\\cos^{2}(1/x) + b\\sin^{2}(1/x)$, $\\sin x + \\sin(1/x)$, $\\sin x\\sin(1/x)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(1/x)*sin(1/x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/8c", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 8, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 8c", "problem_latex": "Draw the graphs of $x^{2}\\sin(1/x)$, $(1/x)\\sin(1/x)$, $\\sin^{2}(1/x)$, $\\{x\\sin(1/x)\\}^{2}$,\n$a\\cos^{2}(1/x) + b\\sin^{2}(1/x)$, $\\sin x + \\sin(1/x)$, $\\sin x\\sin(1/x)$.", "markdown": "Draw the graphs of $x^{2}\\sin(1/x)$, $(1/x)\\sin(1/x)$, $\\sin^{2}(1/x)$, $\\{x\\sin(1/x)\\}^{2}$, $a\\cos^{2}(1/x) + b\\sin^{2}(1/x)$, $\\sin x + \\sin(1/x)$, $\\sin x\\sin(1/x)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "sin(1/x)**2", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/8d", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 8, "part": "d", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 8d", "problem_latex": "Draw the graphs of $x^{2}\\sin(1/x)$, $(1/x)\\sin(1/x)$, $\\sin^{2}(1/x)$, $\\{x\\sin(1/x)\\}^{2}$,\n$a\\cos^{2}(1/x) + b\\sin^{2}(1/x)$, $\\sin x + \\sin(1/x)$, $\\sin x\\sin(1/x)$.", "markdown": "Draw the graphs of $x^{2}\\sin(1/x)$, $(1/x)\\sin(1/x)$, $\\sin^{2}(1/x)$, $\\{x\\sin(1/x)\\}^{2}$, $a\\cos^{2}(1/x) + b\\sin^{2}(1/x)$, $\\sin x + \\sin(1/x)$, $\\sin x\\sin(1/x)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(x*sin(1/x))**2", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/8e", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 8, "part": "e", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 8e", "problem_latex": "Draw the graphs of $x^{2}\\sin(1/x)$, $(1/x)\\sin(1/x)$, $\\sin^{2}(1/x)$, $\\{x\\sin(1/x)\\}^{2}$,\n$a\\cos^{2}(1/x) + b\\sin^{2}(1/x)$, $\\sin x + \\sin(1/x)$, $\\sin x\\sin(1/x)$.", "markdown": "Draw the graphs of $x^{2}\\sin(1/x)$, $(1/x)\\sin(1/x)$, $\\sin^{2}(1/x)$, $\\{x\\sin(1/x)\\}^{2}$, $a\\cos^{2}(1/x) + b\\sin^{2}(1/x)$, $\\sin x + \\sin(1/x)$, $\\sin x\\sin(1/x)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "a*cos(1/x)**2 + b*sin(1/x)**2", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/8f", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 8, "part": "f", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 8f", "problem_latex": "Draw the graphs of $x^{2}\\sin(1/x)$, $(1/x)\\sin(1/x)$, $\\sin^{2}(1/x)$, $\\{x\\sin(1/x)\\}^{2}$,\n$a\\cos^{2}(1/x) + b\\sin^{2}(1/x)$, $\\sin x + \\sin(1/x)$, $\\sin x\\sin(1/x)$.", "markdown": "Draw the graphs of $x^{2}\\sin(1/x)$, $(1/x)\\sin(1/x)$, $\\sin^{2}(1/x)$, $\\{x\\sin(1/x)\\}^{2}$, $a\\cos^{2}(1/x) + b\\sin^{2}(1/x)$, $\\sin x + \\sin(1/x)$, $\\sin x\\sin(1/x)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "sin(x) + sin(1/x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/8g", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 8, "part": "g", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 8g", "problem_latex": "Draw the graphs of $x^{2}\\sin(1/x)$, $(1/x)\\sin(1/x)$, $\\sin^{2}(1/x)$, $\\{x\\sin(1/x)\\}^{2}$,\n$a\\cos^{2}(1/x) + b\\sin^{2}(1/x)$, $\\sin x + \\sin(1/x)$, $\\sin x\\sin(1/x)$.", "markdown": "Draw the graphs of $x^{2}\\sin(1/x)$, $(1/x)\\sin(1/x)$, $\\sin^{2}(1/x)$, $\\{x\\sin(1/x)\\}^{2}$, $a\\cos^{2}(1/x) + b\\sin^{2}(1/x)$, $\\sin x + \\sin(1/x)$, $\\sin x\\sin(1/x)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "sin(x)*sin(1/x)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/9a", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 9, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 9a", "problem_latex": "Draw the graphs of $\\cos x^{2}$, $\\sin x^{2}$, $a\\cos x^{2} + b\\sin x^{2}$.", "markdown": "Draw the graphs of $\\cos x^{2}$, $\\sin x^{2}$, $a\\cos x^{2} + b\\sin x^{2}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "cos(x**2)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/9b", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 9, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 9b", "problem_latex": "Draw the graphs of $\\cos x^{2}$, $\\sin x^{2}$, $a\\cos x^{2} + b\\sin x^{2}$.", "markdown": "Draw the graphs of $\\cos x^{2}$, $\\sin x^{2}$, $a\\cos x^{2} + b\\sin x^{2}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "sin(x**2)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xv/9c", "set": "hardy-course-of-pure-mathematics-1921/ex-xv", "number": 9, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "53", "location": "Exercise XV, problem 9c", "problem_latex": "Draw the graphs of $\\cos x^{2}$, $\\sin x^{2}$, $a\\cos x^{2} + b\\sin x^{2}$.", "markdown": "Draw the graphs of $\\cos x^{2}$, $\\sin x^{2}$, $a\\cos x^{2} + b\\sin x^{2}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "a*cos(x**2) + b*sin(x**2)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xvi/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xvi", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "55", "location": "Exercise XVI, problem 1", "problem_latex": "Let $y = [x]$, where $[x]$~denotes the greatest integer\nnot greater than~$x$. The graph is shown in \\Fig{15a}. The left-hand end\npoints of the thick lines, but not the right-hand ones, belong to the graph.", "markdown": "Let $y = [x]$, where $[x]$ denotes the greatest integer not greater than $x$. The graph is shown in [fig:15a]Fig. 15a. The left-hand end points of the thick lines, but not the right-hand ones, belong to the graph.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xvi/10", "set": "hardy-course-of-pure-mathematics-1921/ex-xvi", "number": 10, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "55", "location": "Exercise XVI, problem 10", "problem_latex": "Let $y = 1$ when $x$~is rational, but $y = 0$ when $x$~is irrational. The graph\nconsists of two series of points arranged upon the lines $y = 1$ and $y = 0$. To\nthe eye it is not distinguishable from two continuous straight lines, but in\nreality an infinite number of points are missing from each line.", "markdown": "Let $y = 1$ when $x$ is rational, but $y = 0$ when $x$ is irrational. The graph consists of two series of points arranged upon the lines $y = 1$ and $y = 0$. To the eye it is not distinguishable from two continuous straight lines, but in reality an infinite number of points are missing from each line.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xvi/11", "set": "hardy-course-of-pure-mathematics-1921/ex-xvi", "number": 11, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "55", "location": "Exercise XVI, problem 11", "problem_latex": "Let $y = x$ when $x$~is irrational and $y = \\sqrtb{(1 + p^{2})/(1 + q^{2})}$ when $x$~is a\\PageLabel{57}\nrational fraction~$p/q$.", "markdown": "Let $y = x$ when $x$ is irrational and $y = \\sqrtb{(1 + p^{2})/(1 + q^{2})}$ when $x$ is a57 rational fraction $p/q$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.frac", "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xvi/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xvi", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "55", "location": "Exercise XVI, problem 2", "problem_latex": "$y = x - [x]$. (\\Fig{15b}.)", "markdown": "$y = x - [x]$. ([fig:15b]Fig. 15b.)", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xvi/3", "set": "hardy-course-of-pure-mathematics-1921/ex-xvi", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "55", "location": "Exercise XVI, problem 3", "problem_latex": "$y = \\sqrtb{x - [x]}$. (\\Fig{15c}.)", "markdown": "$y = \\sqrtb{x - [x]}$. ([fig:15c]Fig. 15c.)", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xvi/4", "set": "hardy-course-of-pure-mathematics-1921/ex-xvi", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "55", "location": "Exercise XVI, problem 4", "problem_latex": "$y = [x] + \\sqrtb{x - [x]}$. (\\Fig{15d}.)", "markdown": "$y = [x] + \\sqrtb{x - [x]}$. ([fig:15d]Fig. 15d.)", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xvi/5a", "set": "hardy-course-of-pure-mathematics-1921/ex-xvi", "number": 5, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "55", "location": "Exercise XVI, problem 5a", "problem_latex": "$y = (x - [x])^{2}$, $[x] + (x - [x])^{2}$.", "markdown": "$y = (x - [x])^{2}$, $[x] + (x - [x])^{2}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xvi/5b", "set": "hardy-course-of-pure-mathematics-1921/ex-xvi", "number": 5, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "55", "location": "Exercise XVI, problem 5b", "problem_latex": "$y = (x - [x])^{2}$, $[x] + (x - [x])^{2}$.", "markdown": "$y = (x - [x])^{2}$, $[x] + (x - [x])^{2}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xvi/6a", "set": "hardy-course-of-pure-mathematics-1921/ex-xvi", "number": 6, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "55", "location": "Exercise XVI, problem 6a", "problem_latex": "$y = [\\sqrt{x}]$, $[x^{2}]$, $\\sqrt{x} - [\\sqrt{x}]$, $x^{2} - [x^{2}]$, $[1 - x^{2}]$.", "markdown": "$y = [\\sqrt{x}]$, $[x^{2}]$, $\\sqrt{x} - [\\sqrt{x}]$, $x^{2} - [x^{2}]$, $[1 - x^{2}]$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xvi/6b", "set": "hardy-course-of-pure-mathematics-1921/ex-xvi", "number": 6, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "55", "location": "Exercise XVI, problem 6b", "problem_latex": "$y = [\\sqrt{x}]$, $[x^{2}]$, $\\sqrt{x} - [\\sqrt{x}]$, $x^{2} - [x^{2}]$, $[1 - x^{2}]$.", "markdown": "$y = [\\sqrt{x}]$, $[x^{2}]$, $\\sqrt{x} - [\\sqrt{x}]$, $x^{2} - [x^{2}]$, $[1 - x^{2}]$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xvi/6c", "set": "hardy-course-of-pure-mathematics-1921/ex-xvi", "number": 6, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "55", "location": "Exercise XVI, problem 6c", "problem_latex": "$y = [\\sqrt{x}]$, $[x^{2}]$, $\\sqrt{x} - [\\sqrt{x}]$, $x^{2} - [x^{2}]$, $[1 - x^{2}]$.", "markdown": "$y = [\\sqrt{x}]$, $[x^{2}]$, $\\sqrt{x} - [\\sqrt{x}]$, $x^{2} - [x^{2}]$, $[1 - x^{2}]$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xvi/6d", "set": "hardy-course-of-pure-mathematics-1921/ex-xvi", "number": 6, "part": "d", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "55", "location": "Exercise XVI, problem 6d", "problem_latex": "$y = [\\sqrt{x}]$, $[x^{2}]$, $\\sqrt{x} - [\\sqrt{x}]$, $x^{2} - [x^{2}]$, $[1 - x^{2}]$.", "markdown": "$y = [\\sqrt{x}]$, $[x^{2}]$, $\\sqrt{x} - [\\sqrt{x}]$, $x^{2} - [x^{2}]$, $[1 - x^{2}]$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xvi/6e", "set": "hardy-course-of-pure-mathematics-1921/ex-xvi", "number": 6, "part": "e", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "55", "location": "Exercise XVI, problem 6e", "problem_latex": "$y = [\\sqrt{x}]$, $[x^{2}]$, $\\sqrt{x} - [\\sqrt{x}]$, $x^{2} - [x^{2}]$, $[1 - x^{2}]$.", "markdown": "$y = [\\sqrt{x}]$, $[x^{2}]$, $\\sqrt{x} - [\\sqrt{x}]$, $x^{2} - [x^{2}]$, $[1 - x^{2}]$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xvi/7", "set": "hardy-course-of-pure-mathematics-1921/ex-xvi", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "55", "location": "Exercise XVI, problem 7", "problem_latex": "Let $y$~be defined as \\emph{the largest prime factor of~$x$} (cf.\\ \\Exs{x}.~6).\nThen $y$~is defined only for integral values of~$x$. If\n\\begin{alignat*}{3}\nx &= 1,\\ 2,\\ 3,\\ 4,\\ 5,\\ 6,\\ 7,\\ 8,\\ 9,\\ &10,&\\ 11,\\ &12,&\\ 13,\\ \\dots, \\\\\n\\intertext{then}\ny &= 1,\\ 2,\\ 3,\\ 2,\\ 5,\\ 3,\\ 7,\\ 2,\\ 3,\\ & 5,&\\ 11,\\ & 3,&\\ 13,\\ \\dots.\n\\end{alignat*}\nThe graph consists of a number of isolated points.", "markdown": "Let $y$ be defined as *the largest prime factor of $x$* (cf. x. 6). Then $y$ is defined only for integral values of $x$. If alignat*3 x &= 1, 2, 3, 4, 5, 6, 7, 8, 9, &10,& 11, &12,& 13, …, then y &= 1, 2, 3, 2, 5, 3, 7, 2, 3, & 5,& 11, & 3,& 13, …. alignat* The graph consists of a number of isolated points.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xvi/8", "set": "hardy-course-of-pure-mathematics-1921/ex-xvi", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "55", "location": "Exercise XVI, problem 8", "problem_latex": "Let $y$~be \\emph{the denominator of~$x$} (\\Exs{x}.~7). In this case $y$~is defined\nonly for rational values of~$x$. We can mark off as many points on the graph\nas we please, but the result is not in any ordinary sense of the word a curve,\nand there are no points corresponding to any irrational values of~$x$.\n\nDraw the straight line joining the points $(N - 1, N)$,~$(N, N)$, where $N$~is a\npositive integer. Show that the number of points of the locus which lie on\nthis line is equal to the number of positive integers less than and prime to~$N$.", "markdown": "Let $y$ be *the denominator of $x$* (x. 7). In this case $y$ is defined only for rational values of $x$. We can mark off as many points on the graph as we please, but the result is not in any ordinary sense of the word a curve, and there are no points corresponding to any irrational values of $x$. Draw the straight line joining the points $(N - 1, N)$, $(N, N)$, where $N$ is a positive integer. Show that the number of points of the locus which lie on this line is equal to the number of positive integers less than and prime to $N$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xvi/9", "set": "hardy-course-of-pure-mathematics-1921/ex-xvi", "number": 9, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "55", "location": "Exercise XVI, problem 9", "problem_latex": "Let $y = 0$ when $x$~is an integer, $y = x$ when $x$~is not an integer. The\ngraph is derived from the straight line $y = x$ by taking out the points\n\\[\n\\dots\\ (-1, -1),\\quad (0, 0),\\quad (1, 1),\\quad (2, 2),\\ \\dots\n\\]\nand adding the points $(-1, 0)$, $(0, 0)$, $(1, 0)$,~\\dots\\ on the axis of~$x$.", "markdown": "Let $y = 0$ when $x$ is an integer, $y = x$ when $x$ is not an integer. The graph is derived from the straight line $y = x$ by taking out the points … (-1, -1),0pt minus 3pt(0, 0),0pt minus 3pt(1, 1),0pt minus 3pt(2, 2), … and adding the points $(-1, 0)$, $(0, 0)$, $(1, 0)$, … on the axis of $x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xvii/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xvii", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "58", "location": "Exercise XVII, problem 1", "problem_latex": " \\Topic{The quadratic equation $ax^{2} + 2bx + c = 0$.} This\nmay be solved graphically in a variety of ways. For instance we may draw\nthe graphs of\n\\[\ny = ax + 2b,\\quad\ny = -c/x,\n\\]\nwhose intersections, if any, give the roots. Or we may take\n\\[\ny = x^{2},\\quad\ny = -(2bx + c)/a.\n\\]\nBut the most elementary method is probably to draw the circle\n\\[\na(x^{2} + y^{2}) + 2bx + c = 0,\n\\]\nwhose centre is~$(-b/a, 0)$ and radius $\\{\\sqrtp{b^{2} - ac}\\}/a$. The abscissae of its\nintersections with the axis of~$x$ are the roots of the equation.", "markdown": "**quadratic equation $ax^{2} + 2bx + c = 0$.** This may be solved graphically in a variety of ways. For instance we may draw the graphs of y = ax + 2b,0pt minus 3pty = -c/x, whose intersections, if any, give the roots. Or we may take y = x^2,0pt minus 3pty = -(2bx + c)/a. But the most elementary method is probably to draw the circle a(x^2 + y^2) + 2bx + c = 0, whose centre is $(-b/a, 0)$ and radius $\\{\\sqrtp{b^{2} - ac}\\}/a$. The abscissae of its intersections with the axis of $x$ are the roots of the equation.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xvii/2a", "set": "hardy-course-of-pure-mathematics-1921/ex-xvii", "number": 2, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "58", "location": "Exercise XVII, problem 2a", "problem_latex": " Solve by any of these methods\n\\[\nx^{2} + 2x - 3 = 0,\\quad\nx^{2} - 7x + 4 = 0,\\quad\n3x^{2} + 2x - 2 = 0.\n\\]", "markdown": "Solve by any of these methods x^2 + 2x - 3 = 0,0pt minus 3ptx^2 - 7x + 4 = 0,0pt minus 3pt3x^2 + 2x - 2 = 0.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "solve", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "Eq(x**2 + 2*x - 3, 0)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "solve: Eq(x**2 + 2*x - 3, 0)" ], "shape": [ "solve: Eq(N*x + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.solve.poly", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xvii/2b", "set": "hardy-course-of-pure-mathematics-1921/ex-xvii", "number": 2, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "58", "location": "Exercise XVII, problem 2b", "problem_latex": " Solve by any of these methods\n\\[\nx^{2} + 2x - 3 = 0,\\quad\nx^{2} - 7x + 4 = 0,\\quad\n3x^{2} + 2x - 2 = 0.\n\\]", "markdown": "Solve by any of these methods x^2 + 2x - 3 = 0,0pt minus 3ptx^2 - 7x + 4 = 0,0pt minus 3pt3x^2 + 2x - 2 = 0.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "solve", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "Eq(x**2 - 7*x + 4, 0)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "solve: Eq(x**2 - 7*x + 4, 0)" ], "shape": [ "solve: Eq(N*x + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.solve.poly", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xvii/2c", "set": "hardy-course-of-pure-mathematics-1921/ex-xvii", "number": 2, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "58", "location": "Exercise XVII, problem 2c", "problem_latex": " Solve by any of these methods\n\\[\nx^{2} + 2x - 3 = 0,\\quad\nx^{2} - 7x + 4 = 0,\\quad\n3x^{2} + 2x - 2 = 0.\n\\]", "markdown": "Solve by any of these methods x^2 + 2x - 3 = 0,0pt minus 3ptx^2 - 7x + 4 = 0,0pt minus 3pt3x^2 + 2x - 2 = 0.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "solve", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "Eq(3*x**2 + 2*x - 2, 0)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "solve: Eq(3*x**2 + 2*x - 2, 0)" ], "shape": [ "solve: Eq(N*x + N*x**N + N, 0)" ], "same_problem_in": [], "needs": [ "cas.solve.poly", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xvii/3", "set": "hardy-course-of-pure-mathematics-1921/ex-xvii", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "58", "location": "Exercise XVII, problem 3", "problem_latex": " \\Topic{The equation $x^{m} + ax + b = 0$.} This may be solved by constructing\nthe curves $y = x^{m}$, $y = -ax - b$. Verify the following table for the number of\nroots of\n\\begin{gather*}\nx^{m} + ax + b = 0: \\\\\n\\begin{alignedat}{3}\n&\\Item{(\\ia)} &&m~\\emph{even} &&\\left\\{\n \\begin{aligned}\n &\\text{$b$~positive, \\emph{two or none},}\\\\\n &\\text{$b$~negative, \\emph{two}\\Add{;}}\n \\end{aligned}\n\\right. \\\\\n&\\Item{(\\ib)} &&m~\\emph{odd} &&\\left\\{\n \\begin{aligned}\n &\\text{$a$~positive, \\emph{one},}\\\\\n &\\text{$a$~negative, \\emph{three or one}.\\qquad\\qquad\\qquad\\qquad\\qquad}\n \\end{aligned}\n\\right.\n\\end{alignedat}\n\\end{gather*}\nConstruct numerical examples to illustrate all possible cases.", "markdown": "**equation $x^{m} + ax + b = 0$.** This may be solved by constructing the curves $y = x^{m}$, $y = -ax - b$. Verify the following table for the number of roots of gather* x^m + ax + b = 0: alignedat3 &[1.5em][l](*a*) &&m *even* && aligned &$b$ positive, *two or none*, &$b$ negative, *two* aligned . &[1.5em][l](*b*) &&m *odd* && aligned &$a$ positive, *one*, &$a$ negative, *three or one*. aligned . alignedat gather* Construct numerical examples to illustrate all possible cases.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xvii/4", "set": "hardy-course-of-pure-mathematics-1921/ex-xvii", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "58", "location": "Exercise XVII, problem 4", "problem_latex": " Show that the equation $\\tan x = ax + b$ has always an infinite number\nof roots.", "markdown": "Show that the equation $\\tan x = ax + b$ has always an infinite number of roots.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xvii/5a", "set": "hardy-course-of-pure-mathematics-1921/ex-xvii", "number": 5, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "58", "location": "Exercise XVII, problem 5a", "problem_latex": " Determine the number of roots of\n\\[\n\\sin x = x,\\quad\n\\sin x = \\tfrac{1}{3} x,\\quad\n\\sin x = \\tfrac{1}{8} x,\\quad\n\\sin x = \\tfrac{1}{120} x.\n\\]", "markdown": "Determine the number of roots of x = x,0pt minus 3ptx = 13 x,0pt minus 3ptx = 18 x,0pt minus 3ptx = 1120 x.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xvii/5b", "set": "hardy-course-of-pure-mathematics-1921/ex-xvii", "number": 5, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "58", "location": "Exercise XVII, problem 5b", "problem_latex": " Determine the number of roots of\n\\[\n\\sin x = x,\\quad\n\\sin x = \\tfrac{1}{3} x,\\quad\n\\sin x = \\tfrac{1}{8} x,\\quad\n\\sin x = \\tfrac{1}{120} x.\n\\]", "markdown": "Determine the number of roots of x = x,0pt minus 3ptx = 13 x,0pt minus 3ptx = 18 x,0pt minus 3ptx = 1120 x.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xvii/5c", "set": "hardy-course-of-pure-mathematics-1921/ex-xvii", "number": 5, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "58", "location": "Exercise XVII, problem 5c", "problem_latex": " Determine the number of roots of\n\\[\n\\sin x = x,\\quad\n\\sin x = \\tfrac{1}{3} x,\\quad\n\\sin x = \\tfrac{1}{8} x,\\quad\n\\sin x = \\tfrac{1}{120} x.\n\\]", "markdown": "Determine the number of roots of x = x,0pt minus 3ptx = 13 x,0pt minus 3ptx = 18 x,0pt minus 3ptx = 1120 x.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xvii/5d", "set": "hardy-course-of-pure-mathematics-1921/ex-xvii", "number": 5, "part": "d", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "58", "location": "Exercise XVII, problem 5d", "problem_latex": " Determine the number of roots of\n\\[\n\\sin x = x,\\quad\n\\sin x = \\tfrac{1}{3} x,\\quad\n\\sin x = \\tfrac{1}{8} x,\\quad\n\\sin x = \\tfrac{1}{120} x.\n\\]", "markdown": "Determine the number of roots of x = x,0pt minus 3ptx = 13 x,0pt minus 3ptx = 18 x,0pt minus 3ptx = 1120 x.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xvii/6", "set": "hardy-course-of-pure-mathematics-1921/ex-xvii", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "58", "location": "Exercise XVII, problem 6", "problem_latex": " Show that if $a$~is small and positive (\\eg\\ $a = .01$), the equation\n\\[\nx - a = \\tfrac{1}{2}\\pi\\sin^{2} x\n\\]\nhas three roots. Consider also the case in which $a$~is small and negative.\nExplain how the number of roots varies as $a$~varies.", "markdown": "Show that if $a$ is small and positive (*e.g.* $a = .01$), the equation x - a = 12^2 x has three roots. Consider also the case in which $a$ is small and negative. Explain how the number of roots varies as $a$ varies.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xviii/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xviii", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "61", "location": "Exercise XVIII, problem 1", "problem_latex": "The points of intersection of the two curves whose\nequations are $f(x, y) = 0$, $\\phi(x, y) = 0$, where $f$~and~$\\phi$ are polynomials, can be\ndetermined if these equations can be solved as a pair of simultaneous equations\nin $x$~and~$y$. The solution generally consists of a finite number of pairs of\nvalues of $x$~and~$y$. The two equations therefore generally represent a finite\nnumber of isolated points.", "markdown": "The points of intersection of the two curves whose equations are $f(x, y) = 0$, $\\phi(x, y) = 0$, where $f$ and $\\phi$ are polynomials, can be determined if these equations can be solved as a pair of simultaneous equations in $x$ and $y$. The solution generally consists of a finite number of pairs of values of $x$ and $y$. The two equations therefore generally represent a finite number of isolated points.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xviii/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xviii", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "61", "location": "Exercise XVIII, problem 2", "problem_latex": "Trace the curves $(x + y)^{2} = 1$, $xy = 1$, $x^{2} - y^{2} = 1$.", "markdown": "Trace the curves $(x + y)^{2} = 1$, $xy = 1$, $x^{2} - y^{2} = 1$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xviii/3", "set": "hardy-course-of-pure-mathematics-1921/ex-xviii", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "61", "location": "Exercise XVIII, problem 3", "problem_latex": "The curve $f(x, y) + \\lambda\\phi(x, y) = 0$ represents a curve passing through\nthe points of intersection of $f = 0$ and $\\phi = 0$.", "markdown": "The curve $f(x, y) + \\lambda\\phi(x, y) = 0$ represents a curve passing through the points of intersection of $f = 0$ and $\\phi = 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xviii/4a", "set": "hardy-course-of-pure-mathematics-1921/ex-xviii", "number": 4, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "61", "location": "Exercise XVIII, problem 4a", "problem_latex": "What loci are represented by\n\\[\n\\Item{$(\\alpha)$}\\ x = at + b,\\quad y = ct + d,\\qquad\n\\Item{$(\\beta)$}\\ x/a = 2t/(1 + t^{2}),\\quad y/a = (1 - t^{2})/(1 + t^{2}),\n\\]\nwhen $t$~varies through all real values?", "markdown": "What loci are represented by [1.5em][l]$(\\alpha)$ x = at + b,0pt minus 3pty = ct + d, [1.5em][l]$(\\beta)$ x/a = 2t/(1 + t^2),0pt minus 3pty/a = (1 - t^2)/(1 + t^2), when $t$ varies through all real values?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xviii/4b", "set": "hardy-course-of-pure-mathematics-1921/ex-xviii", "number": 4, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "61", "location": "Exercise XVIII, problem 4b", "problem_latex": "What loci are represented by\n\\[\n\\Item{$(\\alpha)$}\\ x = at + b,\\quad y = ct + d,\\qquad\n\\Item{$(\\beta)$}\\ x/a = 2t/(1 + t^{2}),\\quad y/a = (1 - t^{2})/(1 + t^{2}),\n\\]\nwhen $t$~varies through all real values?", "markdown": "What loci are represented by [1.5em][l]$(\\alpha)$ x = at + b,0pt minus 3pty = ct + d, [1.5em][l]$(\\beta)$ x/a = 2t/(1 + t^2),0pt minus 3pty/a = (1 - t^2)/(1 + t^2), when $t$ varies through all real values?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xx/1a", "set": "hardy-course-of-pure-mathematics-1921/ex-xx", "number": 1, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "73", "location": "Exercise XX, problem 1a", "problem_latex": " Prove that\n\n\\SubItem{(i)} $\\alpha [\\beta x, \\beta y] = \\beta [\\alpha x, \\alpha y] = [\\alpha \\beta x, \\alpha \\beta y]$,\n\n\\SubItem{(ii)} $([x, y] + [x', y']) + [x'', y''] = [x, y] + ([x', y'] + [x'', y''])$,\n\n\\SubItem{(iii)} $[x, y] + [x', y'] = [x', y'] + [x, y]$,\n\n\\SubItem{(iv)} $(\\alpha + \\beta) [x, y] = \\alpha [x, y] + \\beta [x, y]$,\n\n\\SubItem{(v)} $\\alpha \\{[x, y] + [x', y']\\} = \\alpha [x, y] + \\alpha [x', y']$.\n\n[We have already proved~(iii). The remaining equations follow with equal\nease from the definitions. The reader should in each case consider the\ngeometrical significance of the equation, as we did above in the case of~(iii).]", "markdown": "Prove that 0pt minus 3pt% [2.25em][l](i)% [2.25em][l](i)% % $\\alpha [\\beta x, \\beta y] = \\beta [\\alpha x, \\alpha y] = [\\alpha \\beta x, \\alpha \\beta y]$, 0pt minus 3pt% [2.25em][l](ii)% [2.25em][l](ii)% % $([x, y] + [x', y']) + [x'', y''] = [x, y] + ([x', y'] + [x'', y''])$, 0pt minus 3pt% [2.25em][l](iii)% [2.25em][l](iii)% % $[x, y] + [x', y'] = [x', y'] + [x, y]$, 0pt minus 3pt% [2.25em][l](iv)% [2.25em][l](iv)% % $(\\alpha + \\beta) [x, y] = \\alpha [x, y] + \\beta [x, y]$, 0pt minus 3pt% [2.25em][l](v)% [2.25em][l](v)% % $\\alpha \\{[x, y] + [x', y']\\} = \\alpha [x, y] + \\alpha [x', y']$. [We have already proved (iii). The remaining equations follow with equal ease from the definitions. The reader should in each case consider the geometrical significance of the equation, as we did above in the case of (iii).]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:vector_algebra" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xx/1b", "set": "hardy-course-of-pure-mathematics-1921/ex-xx", "number": 1, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "73", "location": "Exercise XX, problem 1b", "problem_latex": " Prove that\n\n\\SubItem{(i)} $\\alpha [\\beta x, \\beta y] = \\beta [\\alpha x, \\alpha y] = [\\alpha \\beta x, \\alpha \\beta y]$,\n\n\\SubItem{(ii)} $([x, y] + [x', y']) + [x'', y''] = [x, y] + ([x', y'] + [x'', y''])$,\n\n\\SubItem{(iii)} $[x, y] + [x', y'] = [x', y'] + [x, y]$,\n\n\\SubItem{(iv)} $(\\alpha + \\beta) [x, y] = \\alpha [x, y] + \\beta [x, y]$,\n\n\\SubItem{(v)} $\\alpha \\{[x, y] + [x', y']\\} = \\alpha [x, y] + \\alpha [x', y']$.\n\n[We have already proved~(iii). The remaining equations follow with equal\nease from the definitions. The reader should in each case consider the\ngeometrical significance of the equation, as we did above in the case of~(iii).]", "markdown": "Prove that 0pt minus 3pt% [2.25em][l](i)% [2.25em][l](i)% % $\\alpha [\\beta x, \\beta y] = \\beta [\\alpha x, \\alpha y] = [\\alpha \\beta x, \\alpha \\beta y]$, 0pt minus 3pt% [2.25em][l](ii)% [2.25em][l](ii)% % $([x, y] + [x', y']) + [x'', y''] = [x, y] + ([x', y'] + [x'', y''])$, 0pt minus 3pt% [2.25em][l](iii)% [2.25em][l](iii)% % $[x, y] + [x', y'] = [x', y'] + [x, y]$, 0pt minus 3pt% [2.25em][l](iv)% [2.25em][l](iv)% % $(\\alpha + \\beta) [x, y] = \\alpha [x, y] + \\beta [x, y]$, 0pt minus 3pt% [2.25em][l](v)% [2.25em][l](v)% % $\\alpha \\{[x, y] + [x', y']\\} = \\alpha [x, y] + \\alpha [x', y']$. [We have already proved (iii). The remaining equations follow with equal ease from the definitions. The reader should in each case consider the geometrical significance of the equation, as we did above in the case of (iii).]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:vector_algebra" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xx/1c", "set": "hardy-course-of-pure-mathematics-1921/ex-xx", "number": 1, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "73", "location": "Exercise XX, problem 1c", "problem_latex": " Prove that\n\n\\SubItem{(i)} $\\alpha [\\beta x, \\beta y] = \\beta [\\alpha x, \\alpha y] = [\\alpha \\beta x, \\alpha \\beta y]$,\n\n\\SubItem{(ii)} $([x, y] + [x', y']) + [x'', y''] = [x, y] + ([x', y'] + [x'', y''])$,\n\n\\SubItem{(iii)} $[x, y] + [x', y'] = [x', y'] + [x, y]$,\n\n\\SubItem{(iv)} $(\\alpha + \\beta) [x, y] = \\alpha [x, y] + \\beta [x, y]$,\n\n\\SubItem{(v)} $\\alpha \\{[x, y] + [x', y']\\} = \\alpha [x, y] + \\alpha [x', y']$.\n\n[We have already proved~(iii). The remaining equations follow with equal\nease from the definitions. The reader should in each case consider the\ngeometrical significance of the equation, as we did above in the case of~(iii).]", "markdown": "Prove that 0pt minus 3pt% [2.25em][l](i)% [2.25em][l](i)% % $\\alpha [\\beta x, \\beta y] = \\beta [\\alpha x, \\alpha y] = [\\alpha \\beta x, \\alpha \\beta y]$, 0pt minus 3pt% [2.25em][l](ii)% [2.25em][l](ii)% % $([x, y] + [x', y']) + [x'', y''] = [x, y] + ([x', y'] + [x'', y''])$, 0pt minus 3pt% [2.25em][l](iii)% [2.25em][l](iii)% % $[x, y] + [x', y'] = [x', y'] + [x, y]$, 0pt minus 3pt% [2.25em][l](iv)% [2.25em][l](iv)% % $(\\alpha + \\beta) [x, y] = \\alpha [x, y] + \\beta [x, y]$, 0pt minus 3pt% [2.25em][l](v)% [2.25em][l](v)% % $\\alpha \\{[x, y] + [x', y']\\} = \\alpha [x, y] + \\alpha [x', y']$. [We have already proved (iii). The remaining equations follow with equal ease from the definitions. The reader should in each case consider the geometrical significance of the equation, as we did above in the case of (iii).]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:vector_algebra" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xx/1d", "set": "hardy-course-of-pure-mathematics-1921/ex-xx", "number": 1, "part": "d", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "73", "location": "Exercise XX, problem 1d", "problem_latex": " Prove that\n\n\\SubItem{(i)} $\\alpha [\\beta x, \\beta y] = \\beta [\\alpha x, \\alpha y] = [\\alpha \\beta x, \\alpha \\beta y]$,\n\n\\SubItem{(ii)} $([x, y] + [x', y']) + [x'', y''] = [x, y] + ([x', y'] + [x'', y''])$,\n\n\\SubItem{(iii)} $[x, y] + [x', y'] = [x', y'] + [x, y]$,\n\n\\SubItem{(iv)} $(\\alpha + \\beta) [x, y] = \\alpha [x, y] + \\beta [x, y]$,\n\n\\SubItem{(v)} $\\alpha \\{[x, y] + [x', y']\\} = \\alpha [x, y] + \\alpha [x', y']$.\n\n[We have already proved~(iii). The remaining equations follow with equal\nease from the definitions. The reader should in each case consider the\ngeometrical significance of the equation, as we did above in the case of~(iii).]", "markdown": "Prove that 0pt minus 3pt% [2.25em][l](i)% [2.25em][l](i)% % $\\alpha [\\beta x, \\beta y] = \\beta [\\alpha x, \\alpha y] = [\\alpha \\beta x, \\alpha \\beta y]$, 0pt minus 3pt% [2.25em][l](ii)% [2.25em][l](ii)% % $([x, y] + [x', y']) + [x'', y''] = [x, y] + ([x', y'] + [x'', y''])$, 0pt minus 3pt% [2.25em][l](iii)% [2.25em][l](iii)% % $[x, y] + [x', y'] = [x', y'] + [x, y]$, 0pt minus 3pt% [2.25em][l](iv)% [2.25em][l](iv)% % $(\\alpha + \\beta) [x, y] = \\alpha [x, y] + \\beta [x, y]$, 0pt minus 3pt% [2.25em][l](v)% [2.25em][l](v)% % $\\alpha \\{[x, y] + [x', y']\\} = \\alpha [x, y] + \\alpha [x', y']$. [We have already proved (iii). The remaining equations follow with equal ease from the definitions. The reader should in each case consider the geometrical significance of the equation, as we did above in the case of (iii).]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:vector_algebra" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xx/1e", "set": "hardy-course-of-pure-mathematics-1921/ex-xx", "number": 1, "part": "e", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "73", "location": "Exercise XX, problem 1e", "problem_latex": " Prove that\n\n\\SubItem{(i)} $\\alpha [\\beta x, \\beta y] = \\beta [\\alpha x, \\alpha y] = [\\alpha \\beta x, \\alpha \\beta y]$,\n\n\\SubItem{(ii)} $([x, y] + [x', y']) + [x'', y''] = [x, y] + ([x', y'] + [x'', y''])$,\n\n\\SubItem{(iii)} $[x, y] + [x', y'] = [x', y'] + [x, y]$,\n\n\\SubItem{(iv)} $(\\alpha + \\beta) [x, y] = \\alpha [x, y] + \\beta [x, y]$,\n\n\\SubItem{(v)} $\\alpha \\{[x, y] + [x', y']\\} = \\alpha [x, y] + \\alpha [x', y']$.\n\n[We have already proved~(iii). The remaining equations follow with equal\nease from the definitions. The reader should in each case consider the\ngeometrical significance of the equation, as we did above in the case of~(iii).]", "markdown": "Prove that 0pt minus 3pt% [2.25em][l](i)% [2.25em][l](i)% % $\\alpha [\\beta x, \\beta y] = \\beta [\\alpha x, \\alpha y] = [\\alpha \\beta x, \\alpha \\beta y]$, 0pt minus 3pt% [2.25em][l](ii)% [2.25em][l](ii)% % $([x, y] + [x', y']) + [x'', y''] = [x, y] + ([x', y'] + [x'', y''])$, 0pt minus 3pt% [2.25em][l](iii)% [2.25em][l](iii)% % $[x, y] + [x', y'] = [x', y'] + [x, y]$, 0pt minus 3pt% [2.25em][l](iv)% [2.25em][l](iv)% % $(\\alpha + \\beta) [x, y] = \\alpha [x, y] + \\beta [x, y]$, 0pt minus 3pt% [2.25em][l](v)% [2.25em][l](v)% % $\\alpha \\{[x, y] + [x', y']\\} = \\alpha [x, y] + \\alpha [x', y']$. [We have already proved (iii). The remaining equations follow with equal ease from the definitions. The reader should in each case consider the geometrical significance of the equation, as we did above in the case of (iii).]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:vector_algebra" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xx/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xx", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "73", "location": "Exercise XX, problem 2", "problem_latex": " If $M$~is the middle point of~$PQ$, then $\\Seg{OM} = \\frac{1}{2}(\\Seg{OP} + \\Seg{OQ})$. More generally,\nif $M$~divides~$PQ$ in the ratio~$\\mu : \\lambda$, then\n\\[\n\\Seg{OM}\n = \\frac{\\lambda}{\\lambda + \\mu}\\, \\Seg{OP}\n + \\frac{\\mu}{\\lambda + \\mu}\\, \\Seg{OQ}.\n\\]", "markdown": "If $M$ is the middle point of $PQ$, then $\\Seg{OM} = \\frac{1}{2}(\\Seg{OP} + \\Seg{OQ})$. More generally, if $M$ divides $PQ$ in the ratio $\\mu : \\lambda$, then OMP’ = +   OPP’ + +   OQP’.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:vector_algebra" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xx/3", "set": "hardy-course-of-pure-mathematics-1921/ex-xx", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "73", "location": "Exercise XX, problem 3", "problem_latex": " If $G$~is the centre of mass of equal particles at $P_{1}$, $P_{2}$, \\dots,~$P_{n}$, then\n\\[\n\\Seg{OG} = (\\Seg{OP_{1}} + \\Seg{OP_{2}} + \\dots + \\Seg{OP_{n}})/n.\n\\]", "markdown": "If $G$ is the centre of mass of equal particles at $P_{1}$, $P_{2}$, …, $P_{n}$, then OGP’ = (OP_1P’ + OP_2P’ + …+ OP_nP’)/n.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:vector_algebra" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xx/4", "set": "hardy-course-of-pure-mathematics-1921/ex-xx", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "73", "location": "Exercise XX, problem 4", "problem_latex": " If $P$,~$Q$,~$R$ are collinear points in the plane, then it is possible to find\nreal numbers $\\alpha$,~$\\beta$,~$\\gamma$, not all zero, and such that\n\\[\n\\alpha · \\Seg{OP} + \\beta · \\Seg{OQ} + \\gamma · \\Seg{OR} = 0;\n\\]\nand conversely. [This is really only another way of stating Ex.~2.]", "markdown": "If $P$, $Q$, $R$ are collinear points in the plane, then it is possible to find real numbers $\\alpha$, $\\beta$, $\\gamma$, not all zero, and such that · OPP’ + · OQP’ + · ORP’ = 0; and conversely. [This is really only another way of stating Ex. 2.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:vector_algebra" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xx/5", "set": "hardy-course-of-pure-mathematics-1921/ex-xx", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "73", "location": "Exercise XX, problem 5", "problem_latex": " If $\\Seg{AB}$~and~$\\Seg{AC}$ are two displacements not in the same straight line,\nand\n\\[\n\\alpha · \\Seg{AB} + \\beta · \\Seg{AC} = \\gamma · \\Seg{AB} + \\delta · \\Seg{AC},\n\\]\nthen $\\alpha = \\gamma$ and $\\beta = \\delta$.\n\n[Take $AB_{1} = \\alpha · AB$, $AC_{1} = \\beta · AC$. Complete the parallelogram $AB_{1}P_{1}C_{1}$.\nThen $\\Seg{AP_{1}} = \\alpha · \\Seg{AB} + \\beta · \\Seg{AC}$. It is evident that $\\Seg{AP_{1}}$~can only be expressed\nin this form in one way, whence the theorem follows.]", "markdown": "If $\\Seg{AB}$ and $\\Seg{AC}$ are two displacements not in the same straight line, and · ABP’ + · ACP’ = · ABP’ + · ACP’, then $\\alpha = \\gamma$ and $\\beta = \\delta$. [Take $AB_{1} = \\alpha · AB$, $AC_{1} = \\beta · AC$. Complete the parallelogram $AB_{1}P_{1}C_{1}$. Then $\\Seg{AP_{1}} = \\alpha · \\Seg{AB} + \\beta · \\Seg{AC}$. It is evident that $\\Seg{AP_{1}}$ can only be expressed in this form in one way, whence the theorem follows.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:vector_algebra" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xx/6", "set": "hardy-course-of-pure-mathematics-1921/ex-xx", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "73", "location": "Exercise XX, problem 6", "problem_latex": " $ABCD$~is a parallelogram. Through~$Q$, a point inside the parallelogram,\n$RQS$~and~$TQU$ are drawn\nparallel to the sides. Show that\n$RU$,~$TS$ intersect on~$AC$.\n%[Illustration: Fig. 21.]\n\\Figure[2.75in]{21}{p074}\n\n[Let the ratios $AT:AB$, $AR:AD$\nbe denoted by $\\alpha$,~$\\beta$. Then\n\\begin{gather*}\n\\Seg{AT} = \\alpha · \\Seg{AB},\\quad\n\\Seg{AR} = \\beta · \\Seg{AD}, \\\\\n\\Seg{AU} = \\alpha · \\Seg{AB} + \\Seg{AD},\\quad\n\\Seg{AS} = \\Seg{AB} + \\beta · \\Seg{AD}.\n\\end{gather*}\n\nLet $RU$~meet $AC$ in~$P$. Then,\nsince $R$,~$U$,~$P$ are collinear,\n\\[\n\\Seg{AP}\n = \\frac{\\lambda}{\\lambda + \\mu}\\, \\Seg{AR}\n + \\frac{\\mu}{\\lambda + \\mu}\\, \\Seg{AU},\n\\]\nwhere $\\mu/\\lambda$ is the ratio in which $P$~divides~$RU$. That is to say\n\\[\n\\Seg{AP}\n = \\frac{\\alpha\\mu}{\\lambda + \\mu}\\, \\Seg{AB}\n + \\frac{\\beta\\lambda + \\mu}{\\lambda + \\mu}\\, \\Seg{AD}.\n\\]\n\nBut since $P$~lies on~$AC$, $\\Seg{AP}$~is a numerical multiple of~$\\Seg{AC}$; say\n\\[\n\\Seg{AP} = k · \\Seg{AC} = k · \\Seg{AB} + k · \\Seg{AD}.\n\\]\nHence (Ex.~5) $\\alpha\\mu = \\beta\\lambda + \\mu = (\\lambda + \\mu)k$, from which we deduce\n\\[\nk = \\frac{\\alpha\\beta}{\\alpha + \\beta - 1}.\n\\]\nThe symmetry of this result shows that a similar argument would also give\n\\[\n\\Seg{AP'} = \\frac{\\alpha\\beta}{\\alpha + \\beta - 1}\\, \\Seg{AC},\n\\]\nif $P'$~is the point where $TS$~meets~$AC$. Hence $P$~and~$P'$ are the same point.]", "markdown": "$ABCD$ is a parallelogram. Through $Q$, a point inside the parallelogram, $RQS$ and $TQU$ are drawn parallel to the sides. Show that $RU$, $TS$ intersect on $AC$. %[Illustration: Fig. 21.] [2.75in]21p074 [Let the ratios $AT:AB$, $AR:AD$ be denoted by $\\alpha$, $\\beta$. Then gather* ATP’ = · ABP’,0pt minus 3ptARP’ = · ADP’, AUP’ = · ABP’ + ADP’,0pt minus 3ptASP’ = ABP’ + · ADP’. gather* Let $RU$ meet $AC$ in $P$. Then, since $R$, $U$, $P$ are collinear, APP’ = +   ARP’ + +   AUP’, where $\\mu/\\lambda$ is the ratio in which $P$ divides $RU$. That is to say APP’ = +   ABP’ + + +   ADP’. But since $P$ lies on $AC$, $\\Seg{AP}$ is a numerical multiple of $\\Seg{AC}$; say APP’ = k · ACP’ = k · ABP’ + k · ADP’. Hence (Ex. 5) $\\alpha\\mu = \\beta\\lambda + \\mu = (\\lambda + \\mu)k$, from which we deduce k = + - 1. The symmetry of this result shows that a similar argument would also give AP’P’ = + - 1  ACP’, if $P'$ is the point where $TS$ meets $AC$. Hence $P$ and $P'$ are the same point.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:vector_algebra" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xx/7", "set": "hardy-course-of-pure-mathematics-1921/ex-xx", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "73", "location": "Exercise XX, problem 7", "problem_latex": " $ABCD$~is a parallelogram, and $M$~the middle point of~$AB$. Show that\n$DM$~trisects and is trisected by~$AC$.\\footnote\n {The two preceding examples are taken from Willard Gibbs' \\textit{Vector Analysis}.}", "markdown": "$ABCD$ is a parallelogram, and $M$ the middle point of $AB$. Show that $DM$ trisects and is trisected by $AC$. The two preceding examples are taken from Willard Gibbs’ *Vector Analysis*.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:vector_algebra" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxii/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xxii", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "99", "location": "Exercise XXII, problem 1", "problem_latex": "The two square roots of~$1$ are $1$,~$-1$; the three\ncube roots are $1$, $\\frac{1}{2}(-1 + i\\sqrt{3})$, $\\frac{1}{2}(-1 - i\\sqrt{3})$; the four fourth roots are $1$,\n$i$, $-1$, $-i$; and the five fifth roots are\n\\begin{alignat*}{4}\n1,\\quad &\\tfrac{1}{4} \\Bigl[ &&\\sqrt{5} - 1 + i\\sqrtb{10 + 2\\sqrt{5}}\\Bigr],\\quad\n && \\tfrac{1}{4} \\Bigl[-&&\\sqrt{5} - 1 + i\\sqrtb{10 - 2\\sqrt{5}}\\Bigr],\\\\\n &\\tfrac{1}{4} \\Bigl[-&&\\sqrt{5} - 1 - i\\sqrtb{10 - 2\\sqrt{5}}\\Bigr],\\quad\n && \\tfrac{1}{4} \\Bigl[ &&\\sqrt{5} - 1 - i\\sqrtb{10 + 2\\sqrt{5}}\\Bigr].\n\\end{alignat*}", "markdown": "The two square roots of $1$ are $1$, $-1$; the three cube roots are $1$, $\\frac{1}{2}(-1 + i\\sqrt{3})$, $\\frac{1}{2}(-1 - i\\sqrt{3})$; the four fourth roots are $1$, $i$, $-1$, $-i$; and the five fifth roots are alignat*4 1,0pt minus 3pt&14 [ &&5 - 1 + i10 + 25],0pt minus 3pt && 14 [-&&5 - 1 + i10 - 25], &14 [-&&5 - 1 - i10 - 25],0pt minus 3pt && 14 [ &&5 - 1 - i10 + 25]. alignat*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxii/10", "set": "hardy-course-of-pure-mathematics-1921/ex-xxii", "number": 10, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "99", "location": "Exercise XXII, problem 10", "problem_latex": "The problem of finding the accurate value of~$\\omega_{n}$ in a numerical form\ninvolving square roots only, as in the formula $\\omega_{3} = \\frac{1}{2}(-1 + i\\sqrt{3})$, is the\nalgebraical equivalent of the geometrical problem of inscribing a regular\npolygon of $n$~sides in a circle of unit radius by Euclidean methods, \\ie\\ by ruler\nand compasses. For this construction will be possible if and only if we can\nconstruct lengths measured by $\\cos(2\\pi/n)$ and $\\sin(2\\pi/n)$; and this is possible\n(\\okrickRef{Ch.}{II}, \\MiscExs{II}~22) if and only if these numbers are expressible in a form\ninvolving square roots only.\n\nEuclid gives constructions for $n = 3$, $4$, $5$, $6$, $8$, $10$, $12$, and~$15$. It is\nevident that the construction is possible for any value of~$n$ which can be\nfound from these by multiplication by any power of~$2$. There are other\nspecial values of~$n$ for which such constructions are possible, the most interesting\nbeing~$n = 17$.", "markdown": "The problem of finding the accurate value of $\\omega_{n}$ in a numerical form involving square roots only, as in the formula $\\omega_{3} = \\frac{1}{2}(-1 + i\\sqrt{3})$, is the algebraical equivalent of the geometrical problem of inscribing a regular polygon of $n$ sides in a circle of unit radius by Euclidean methods, *i.e.* by ruler and compasses. For this construction will be possible if and only if we can construct lengths measured by $\\cos(2\\pi/n)$ and $\\sin(2\\pi/n)$; and this is possible (Ch.II, [misc:II]Misc. Exs. 22) if and only if these numbers are expressible in a form involving square roots only. Euclid gives constructions for $n = 3$, $4$, $5$, $6$, $8$, $10$, $12$, and $15$. It is evident that the construction is possible for any value of $n$ which can be found from these by multiplication by any power of $2$. There are other special values of $n$ for which such constructions are possible, the most interesting being $n = 17$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxii/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xxii", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "99", "location": "Exercise XXII, problem 2", "problem_latex": "Prove that\n\\[\n1 + \\omega_{n} + \\omega_{n}^{2} + \\dots + \\omega_{n}^{n-1} = 0.\n\\]", "markdown": "Prove that 1 + _n + _n^2 + …+ _n^n-1 = 0.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxii/3", "set": "hardy-course-of-pure-mathematics-1921/ex-xxii", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "99", "location": "Exercise XXII, problem 3", "problem_latex": "Prove that\n\\[\n(x + y\\omega_{3} + z\\omega_{3}^{2})\n(x + y\\omega_{3}^{2} + z\\omega_{3})\n = x^{2} + y^{2} + z^{2} - yz - zx - xy.\n\\]", "markdown": "Prove that (x + y_3 + z_3^2) (x + y_3^2 + z_3) = x^2 + y^2 + z^2 - yz - zx - xy.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxii/4", "set": "hardy-course-of-pure-mathematics-1921/ex-xxii", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "99", "location": "Exercise XXII, problem 4", "problem_latex": "The $n$th~roots of~$a$ are the products of the $n$th~roots of unity by the\nprincipal value of~$\\sqrt[n]{a}$.", "markdown": "The $n$th roots of $a$ are the products of the $n$th roots of unity by the principal value of $\\sqrt[n]{a}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxii/5", "set": "hardy-course-of-pure-mathematics-1921/ex-xxii", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "99", "location": "Exercise XXII, problem 5", "problem_latex": "It follows from \\Exs{xxi}.~14 that the roots of\n\\[\nz^{2} = \\alpha + \\beta i\n\\]\nare\n\\[\n± \\sqrtbr{\\tfrac{1}{2} \\{\\sqrtp{\\alpha^{2} + \\beta^{2}} + \\alpha\\}}\n± i\\sqrtbr{\\tfrac{1}{2} \\{\\sqrtp{\\alpha^{2} + \\beta^{2}} - \\alpha\\}},\n\\]\nlike or unlike signs being chosen according as $\\beta$~is positive or negative. Show\nthat this result agrees with the result of \\SecNo[§]{48}.", "markdown": "It follows from xxi. 14 that the roots of z^2 = + i are ± 12 ^2 + ^2 + ± i12 ^2 + ^2 - , like or unlike signs being chosen according as $\\beta$ is positive or negative. Show that this result agrees with the result of [§]48.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.complex" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxii/6", "set": "hardy-course-of-pure-mathematics-1921/ex-xxii", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "99", "location": "Exercise XXII, problem 6", "problem_latex": "Show that $(x^{2m} - a^{2m})/(x^{2} - a^{2})$ is equal to\n\\[\n\\Bigl(x^{2} - 2ax\\cos\\frac{\\pi}{m} + a^{2}\\Bigr)\n\\Bigl(x^{2} - 2ax\\cos\\frac{2\\pi}{m} + a^{2}\\Bigr) \\dots\n\\Bigl(x^{2} - 2ax\\cos\\frac{(m - 1)\\pi}{m} + a^{2}\\Bigr).\n\\]\n\n[The factors of $x^{2m} - a^{2m}$ are\n\\[\n(x - a),\\quad\n(x - a\\omega_{2m}),\\quad\n(x - a\\omega_{2m}^{2}),\\ \\dots\\quad\n(x - a\\omega_{2m}^{2m-1}).\n\\]\nThe factor $x - a\\omega_{2m}^{m}$ is $x + a$. The factors $(x - a\\omega_{2m}^{s})$, $(x - a\\omega_{2m}^{2m-s})$ taken together\ngive a factor $x^{2} - 2ax \\cos(s\\pi/m) + a^{2}$.]", "markdown": "Show that $(x^{2m} - a^{2m})/(x^{2} - a^{2})$ is equal to (x^2 - 2axm + a^2) (x^2 - 2ax2m + a^2) …(x^2 - 2ax(m - 1)m + a^2). [The factors of $x^{2m} - a^{2m}$ are (x - a),0pt minus 3pt(x - a_2m),0pt minus 3pt(x - a_2m^2), …0pt minus 3pt(x - a_2m^2m-1). The factor $x - a\\omega_{2m}^{m}$ is $x + a$. The factors $(x - a\\omega_{2m}^{s})$, $(x - a\\omega_{2m}^{2m-s})$ taken together give a factor $x^{2} - 2ax \\cos(s\\pi/m) + a^{2}$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxii/7", "set": "hardy-course-of-pure-mathematics-1921/ex-xxii", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "99", "location": "Exercise XXII, problem 7", "problem_latex": "Resolve $x^{2m+1} - a^{2m+1}$, $x^{2m} + a^{2m}$, and $x^{2m+1} + a^{2m+1}$ into factors in a\nsimilar way.", "markdown": "Resolve $x^{2m+1} - a^{2m+1}$, $x^{2m} + a^{2m}$, and $x^{2m+1} + a^{2m+1}$ into factors in a similar way.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.factor", "core.complex" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxii/8", "set": "hardy-course-of-pure-mathematics-1921/ex-xxii", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "99", "location": "Exercise XXII, problem 8", "problem_latex": "Show that $x^{2n} - 2x^{n}a^{n} \\cos\\theta + a^{2n}$ is equal to\n\\begin{multline*}\n\\left(x^{2} - 2xa\\cos\\frac{\\theta}{n} + a^{2}\\right)\n\\left(x^{2} - 2xa\\cos\\frac{\\theta + 2\\pi}{n} + a^{2}\\right) \\dots \\\\\n\\dots\\left(x^{2} - 2xa\\cos\\frac{\\theta + 2(n - 1)\\pi}{n} + a^{2}\\right).\n\\end{multline*}\n\n[Use the formula\n\\[\nx^{2n} - 2x^{n}a^{n} \\cos\\theta + a^{2n}\n = \\{x^{n} - a^{n}(\\cos\\theta + i\\sin\\theta)\\}\n \\{x^{n} - a^{n}(\\cos\\theta - i\\sin\\theta)\\},\n\\]\nand split up each of the last two expressions into $n$~factors.]", "markdown": "Show that $x^{2n} - 2x^{n}a^{n} \\cos\\theta + a^{2n}$ is equal to multline* (x^2 - 2xan + a^2) (x^2 - 2xa+ 2n + a^2) … …(x^2 - 2xa+ 2(n - 1)n + a^2). multline* [Use the formula x^2n - 2x^na^n + a^2n = x^n - a^n(+ i) x^n - a^n(- i), and split up each of the last two expressions into $n$ factors.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.complex" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxii/9", "set": "hardy-course-of-pure-mathematics-1921/ex-xxii", "number": 9, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "99", "location": "Exercise XXII, problem 9", "problem_latex": "Find all the roots of the equation $x^{6} - 2x^{3} + 2 = 0$. \\MathTrip{1910.}", "markdown": "Find all the roots of the equation $x^{6} - 2x^{3} + 2 = 0$. % [0]% (*Math. Trip.* 1910.)% [1]%", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "solve", "verdict": "FLAG-MISSING", "judge_why": "problem or answer expression is null", "problem_expr": "Eq(x**6 - 2*x**3 + 2, 0)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "FLAG-MISSING" ] }, "form": [ "solve: Eq(x**6 - 2*x**3 + 2, 0)" ], "shape": [ "solve: Eq(N*x**N + N + x**N, 0)" ], "same_problem_in": [], "needs": [ "cas.solve.poly", "core.complex" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiii/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiii", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "120", "location": "Exercise XXIII, problem 1", "problem_latex": "$\\phi(n) = n^{k}$, where $k$~is a positive or negative integer or rational fraction.", "markdown": "$\\phi(n) = n^{k}$, where $k$ is a positive or negative integer or rational fraction.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiii/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiii", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "120", "location": "Exercise XXIII, problem 2", "problem_latex": "$\\phi(n) = p_{n}$, where $p_{n}$~is the $n$th~prime number.", "markdown": "$\\phi(n) = p_{n}$, where $p_{n}$ is the $n$th prime number.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiii/3", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiii", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "120", "location": "Exercise XXIII, problem 3", "problem_latex": "Let $\\phi(n)$~be the number of primes less than~$n$.", "markdown": "Let $\\phi(n)$ be the number of primes less than $n$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiii/4", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiii", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "120", "location": "Exercise XXIII, problem 4", "problem_latex": "$\\phi(n) = [\\alpha n]$, where $\\alpha$~is any positive number.", "markdown": "$\\phi(n) = [\\alpha n]$, where $\\alpha$ is any positive number.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiii/5", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiii", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "120", "location": "Exercise XXIII, problem 5", "problem_latex": "If $\\phi(n) = 1\\MC000\\MC000/n$, then $\\lim\\phi(n) = 0$: and if $\\psi(n) = n/1\\MC000\\MC000$, then\n$\\psi(n) \\to +\\infty$.", "markdown": "If $\\phi(n) = 1\\MC000\\MC000/n$, then $\\lim\\phi(n) = 0$: and if $\\psi(n) = n/1\\MC000\\MC000$, then $\\psi(n) \\to +\\infty$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiii/6", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiii", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "120", "location": "Exercise XXIII, problem 6", "problem_latex": "$\\phi(n) = 1/\\{n - (-1)^{n}\\}$, $n - (-1)^{n}$, $n\\{1 - (-1)^{n}\\}$.", "markdown": "$\\phi(n) = 1/\\{n - (-1)^{n}\\}$, $n - (-1)^{n}$, $n\\{1 - (-1)^{n}\\}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiii/7", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiii", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "120", "location": "Exercise XXIII, problem 7", "problem_latex": "$\\phi(n) = (\\sin n\\theta\\pi)/n$, where $\\theta$~is any real number.", "markdown": "$\\phi(n) = (\\sin n\\theta\\pi)/n$, where $\\theta$ is any real number.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiii/8", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiii", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "120", "location": "Exercise XXIII, problem 8", "problem_latex": "$\\phi(n) = (\\sin n\\theta\\pi)/\\sqrt{n}$, $(a\\cos^{2} n\\theta + b\\sin^{2}n\\theta)/n$, where $a$~and~$b$ are any real\nnumbers.", "markdown": "$\\phi(n) = (\\sin n\\theta\\pi)/\\sqrt{n}$, $(a\\cos^{2} n\\theta + b\\sin^{2}n\\theta)/n$, where $a$ and $b$ are any real numbers.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiii/9", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiii", "number": 9, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "120", "location": "Exercise XXIII, problem 9", "problem_latex": "$\\phi(n) = \\sin n\\theta\\pi$.", "markdown": "$\\phi(n) = \\sin n\\theta\\pi$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiv/10", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiv", "number": 10, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "122", "location": "Exercise XXIV, problem 10", "problem_latex": "$a + bn + (-1)^{n} (c + dn) + e\\cos n\\theta\\pi + f\\sin n\\theta\\pi$.", "markdown": "$a + bn + (-1)^{n} (c + dn) + e\\cos n\\theta\\pi + f\\sin n\\theta\\pi$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "a + b*n + (-1)**n*(c + d*n) + e*cos(n*theta*pi) + f*sin(n*theta*pi)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiv/11", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiv", "number": 11, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "122", "location": "Exercise XXIV, problem 11", "problem_latex": "$n\\sin n\\theta\\pi$. If $\\DPtypo{n}{\\theta}$ is integral, then $\\phi(n) = 0$, $\\phi(n) \\to 0$. If $\\theta$~is rational\nbut not integral, or irrational, then $\\phi(n)$~oscillates infinitely.", "markdown": "$n\\sin n\\theta\\pi$. If $\\DPtypo{n}{\\theta}$ is integral, then $\\phi(n) = 0$, $\\phi(n) \\to 0$. If $\\theta$ is rational but not integral, or irrational, then $\\phi(n)$ oscillates infinitely.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "n*sin(n*theta*pi)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiv/12", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiv", "number": 12, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "122", "location": "Exercise XXIV, problem 12", "problem_latex": "$n(a\\cos^{2} n\\theta\\pi + b\\sin^{2} n\\theta\\pi)$. In this case $\\phi(n)$~tends to~$+\\infty$ if $a$~and~$b$\nare both positive, but to~$-\\infty$ if both are negative. Consider the special\ncases in which $a = 0$, $b > 0$, or $a > 0$, $b = 0$, or $a = 0$, $b = 0$. If $a$~and~$b$ have\nopposite signs $\\phi(n)$~generally oscillates infinitely. Consider any exceptional\ncases.", "markdown": "$n(a\\cos^{2} n\\theta\\pi + b\\sin^{2} n\\theta\\pi)$. In this case $\\phi(n)$ tends to $+\\infty$ if $a$ and $b$ are both positive, but to $-\\infty$ if both are negative. Consider the special cases in which $a = 0$, $b > 0$, or $a > 0$, $b = 0$, or $a = 0$, $b = 0$. If $a$ and $b$ have opposite signs $\\phi(n)$ generally oscillates infinitely. Consider any exceptional cases.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "n*(a*cos(n*theta*pi)**2 + b*sin(n*theta*pi)**2)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiv/13", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiv", "number": 13, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "122", "location": "Exercise XXIV, problem 13", "problem_latex": "$\\sin(n^{2}\\theta\\pi)$. If $\\theta$~is integral, then $\\phi(n) \\to 0$. Otherwise $\\phi(n)$~oscillates\nfinitely, as may be shown by arguments similar to though more complex\nthan those used in \\Exs{xxiii}.~9 and \\Exs[]{xxiv}.~7.\\footnote\n {See Bromwich's \\textit{Infinite Series}, p.~485.}", "markdown": "$\\sin(n^{2}\\theta\\pi)$. If $\\theta$ is integral, then $\\phi(n) \\to 0$. Otherwise $\\phi(n)$ oscillates finitely, as may be shown by arguments similar to though more complex than those used in xxiii. 9 and []xxiv. 7. See Bromwich’s *Infinite Series*, p. 485.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "sin(n**2*theta*pi)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiv/14", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiv", "number": 14, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "122", "location": "Exercise XXIV, problem 14", "problem_latex": "$\\sin(n!\\, \\theta\\pi)$. If $\\theta$~has a rational value~$p/q$, then $n!\\, \\theta$~is certainly\nintegral for all values of $n$ greater than or equal to~$q$. Hence $\\phi(n) \\to 0$. The\ncase in which $\\theta$~is irrational cannot be dealt with without the aid of considerations\nof a much more difficult character.", "markdown": "$\\sin(n!\\, \\theta\\pi)$. If $\\theta$ has a rational value $p/q$, then $n!\\, \\theta$ is certainly integral for all values of $n$ greater than or equal to $q$. Hence $\\phi(n) \\to 0$. The case in which $\\theta$ is irrational cannot be dealt with without the aid of considerations of a much more difficult character.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "sin(factorial(n)*theta*pi)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.arith", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiv/15a", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiv", "number": 15, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "122", "location": "Exercise XXIV, problem 15a", "problem_latex": "$\\cos(n!\\, \\theta\\pi)$, $a\\cos^{2}(n!\\, \\theta\\pi) + b\\sin^{2}(n!\\, \\theta\\pi)$, where $\\theta$~is rational.", "markdown": "$\\cos(n!\\, \\theta\\pi)$, $a\\cos^{2}(n!\\, \\theta\\pi) + b\\sin^{2}(n!\\, \\theta\\pi)$, where $\\theta$ is rational.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "cos(factorial(n)*theta*pi)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.arith", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiv/15b", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiv", "number": 15, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "122", "location": "Exercise XXIV, problem 15b", "problem_latex": "$\\cos(n!\\, \\theta\\pi)$, $a\\cos^{2}(n!\\, \\theta\\pi) + b\\sin^{2}(n!\\, \\theta\\pi)$, where $\\theta$~is rational.", "markdown": "$\\cos(n!\\, \\theta\\pi)$, $a\\cos^{2}(n!\\, \\theta\\pi) + b\\sin^{2}(n!\\, \\theta\\pi)$, where $\\theta$ is rational.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "a*cos(factorial(n)*theta*pi)**2 + b*sin(factorial(n)*theta*pi)**2", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.arith", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiv/16a", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiv", "number": 16, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "122", "location": "Exercise XXIV, problem 16a", "problem_latex": "$an - [bn]$, $(-1)^{n}(an - [bn])$.", "markdown": "$an - [bn]$, $(-1)^{n}(an - [bn])$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "a*n - floor(b*n)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.arith" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiv/16b", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiv", "number": 16, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "122", "location": "Exercise XXIV, problem 16b", "problem_latex": "$an - [bn]$, $(-1)^{n}(an - [bn])$.", "markdown": "$an - [bn]$, $(-1)^{n}(an - [bn])$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(-1)**n*(a*n - floor(b*n))", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.arith" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiv/17a", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiv", "number": 17, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "122", "location": "Exercise XXIV, problem 17a", "problem_latex": "$[\\sqrt{n}]$, $(-1)^{n}[\\sqrt{n}]$, $\\sqrt{n} - [\\sqrt{n}]$.", "markdown": "$[\\sqrt{n}]$, $(-1)^{n}[\\sqrt{n}]$, $\\sqrt{n} - [\\sqrt{n}]$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "floor(sqrt(n))", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.arith" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiv/17b", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiv", "number": 17, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "122", "location": "Exercise XXIV, problem 17b", "problem_latex": "$[\\sqrt{n}]$, $(-1)^{n}[\\sqrt{n}]$, $\\sqrt{n} - [\\sqrt{n}]$.", "markdown": "$[\\sqrt{n}]$, $(-1)^{n}[\\sqrt{n}]$, $\\sqrt{n} - [\\sqrt{n}]$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(-1)**n*floor(sqrt(n))", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.arith" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiv/17c", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiv", "number": 17, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "122", "location": "Exercise XXIV, problem 17c", "problem_latex": "$[\\sqrt{n}]$, $(-1)^{n}[\\sqrt{n}]$, $\\sqrt{n} - [\\sqrt{n}]$.", "markdown": "$[\\sqrt{n}]$, $(-1)^{n}[\\sqrt{n}]$, $\\sqrt{n} - [\\sqrt{n}]$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "sqrt(n) - floor(sqrt(n))", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.arith" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiv/18", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiv", "number": 18, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "122", "location": "Exercise XXIV, problem 18", "problem_latex": "\\emph{The smallest prime factor of~$n$}. When $n$~is a prime, $\\phi(n) = n$. When\n$n$~is even, $\\phi(n) = 2$. Thus $\\phi(n)$ oscillates infinitely.", "markdown": "*The smallest prime factor of $n$*. When $n$ is a prime, $\\phi(n) = n$. When $n$ is even, $\\phi(n) = 2$. Thus $\\phi(n)$ oscillates infinitely.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "other:number_theory" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiv/19", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiv", "number": 19, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "122", "location": "Exercise XXIV, problem 19", "problem_latex": "\\emph{The largest prime factor of~$n$}.", "markdown": "*The largest prime factor of $n$*.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "other:number_theory" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiv/1a", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiv", "number": 1, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "122", "location": "Exercise XXIV, problem 1a", "problem_latex": "$(-1)^{n}$, $5 + 3(-1)^{n}$, $(1\\MC000\\MC000/n) + (-1)^{n}$, $1\\MC000\\MC000(-1)^{n} + (1/n)$.", "markdown": "$(-1)^{n}$, $5 + 3(-1)^{n}$, $(1\\MC000\\MC000/n) + (-1)^{n}$, $1\\MC000\\MC000(-1)^{n} + (1/n)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(-1)**n", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiv/1b", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiv", "number": 1, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "122", "location": "Exercise XXIV, problem 1b", "problem_latex": "$(-1)^{n}$, $5 + 3(-1)^{n}$, $(1\\MC000\\MC000/n) + (-1)^{n}$, $1\\MC000\\MC000(-1)^{n} + (1/n)$.", "markdown": "$(-1)^{n}$, $5 + 3(-1)^{n}$, $(1\\MC000\\MC000/n) + (-1)^{n}$, $1\\MC000\\MC000(-1)^{n} + (1/n)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "5 + 3*(-1)**n", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiv/1c", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiv", "number": 1, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "122", "location": "Exercise XXIV, problem 1c", "problem_latex": "$(-1)^{n}$, $5 + 3(-1)^{n}$, $(1\\MC000\\MC000/n) + (-1)^{n}$, $1\\MC000\\MC000(-1)^{n} + (1/n)$.", "markdown": "$(-1)^{n}$, $5 + 3(-1)^{n}$, $(1\\MC000\\MC000/n) + (-1)^{n}$, $1\\MC000\\MC000(-1)^{n} + (1/n)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "1000000/n + (-1)**n", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiv/1d", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiv", "number": 1, "part": "d", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "122", "location": "Exercise XXIV, problem 1d", "problem_latex": "$(-1)^{n}$, $5 + 3(-1)^{n}$, $(1\\MC000\\MC000/n) + (-1)^{n}$, $1\\MC000\\MC000(-1)^{n} + (1/n)$.", "markdown": "$(-1)^{n}$, $5 + 3(-1)^{n}$, $(1\\MC000\\MC000/n) + (-1)^{n}$, $1\\MC000\\MC000(-1)^{n} + (1/n)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "1000000*(-1)**n + 1/n", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiv/20", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiv", "number": 20, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "122", "location": "Exercise XXIV, problem 20", "problem_latex": "\\emph{The number of days in the year~$n$~\\textsc{a.d.}}", "markdown": "*The number of days in the year $n$ a.d.*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "other:calendar" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiv/2a", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiv", "number": 2, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "122", "location": "Exercise XXIV, problem 2a", "problem_latex": "$(-1)^{n}n$, $1\\MC000\\MC000 + (-1)^{n}n$.", "markdown": "$(-1)^{n}n$, $1\\MC000\\MC000 + (-1)^{n}n$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(-1)**n*n", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiv/2b", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiv", "number": 2, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "122", "location": "Exercise XXIV, problem 2b", "problem_latex": "$(-1)^{n}n$, $1\\MC000\\MC000 + (-1)^{n}n$.", "markdown": "$(-1)^{n}n$, $1\\MC000\\MC000 + (-1)^{n}n$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "1000000 + (-1)**n*n", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiv/3a", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiv", "number": 3, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "122", "location": "Exercise XXIV, problem 3a", "problem_latex": "$1\\MC000\\MC000 - n$, $(-1)^{n}(1\\MC000\\MC000 - n)$.", "markdown": "$1\\MC000\\MC000 - n$, $(-1)^{n}(1\\MC000\\MC000 - n)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "1000000 - n", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiv/3b", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiv", "number": 3, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "122", "location": "Exercise XXIV, problem 3b", "problem_latex": "$1\\MC000\\MC000 - n$, $(-1)^{n}(1\\MC000\\MC000 - n)$.", "markdown": "$1\\MC000\\MC000 - n$, $(-1)^{n}(1\\MC000\\MC000 - n)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(-1)**n*(1000000 - n)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiv/4", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiv", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "122", "location": "Exercise XXIV, problem 4", "problem_latex": "$n\\{1 + (-1)^{n}\\}$. In this case the values of~$\\phi(n)$ are\n\\[\n0,\\quad 4,\\quad 0,\\quad 8,\\quad 0,\\quad 12,\\quad 0,\\quad 16,\\ \\dots.\n\\]\nThe odd terms are all zero and the even terms tend to~$+\\infty$: $\\phi(n)$~oscillates\ninfinitely.", "markdown": "$n\\{1 + (-1)^{n}\\}$. In this case the values of $\\phi(n)$ are 0,0pt minus 3pt4,0pt minus 3pt0,0pt minus 3pt8,0pt minus 3pt0,0pt minus 3pt12,0pt minus 3pt0,0pt minus 3pt16, …. The odd terms are all zero and the even terms tend to $+\\infty$: $\\phi(n)$ oscillates infinitely.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "n*(1 + (-1)**n)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiv/5", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiv", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "122", "location": "Exercise XXIV, problem 5", "problem_latex": "$n^{2} + (-1)^{n}2n$. The second term oscillates infinitely, but the first is\nvery much larger than the second when $n$~is large. In fact $\\phi(n) \\geq n^{2} - 2n$ and\n$n^{2} - 2n = (n - 1)^{2} - 1$ is greater than any assigned value~$\\Delta$ if $n > 1 + \\sqrtp{\\Delta + 1}$.\nThus $\\phi(n) \\to +\\infty$. It should be observed that in this case $\\phi(2k + 1)$~is\nalways less than~$\\phi(2k)$, so that the function progresses to infinity by a continual\nseries of steps forwards and backwards. It does not however `oscillate'\naccording to our definition of the term.", "markdown": "$n^{2} + (-1)^{n}2n$. The second term oscillates infinitely, but the first is very much larger than the second when $n$ is large. In fact $\\phi(n) \\geq n^{2} - 2n$ and $n^{2} - 2n = (n - 1)^{2} - 1$ is greater than any assigned value $\\Delta$ if $n > 1 + \\sqrtp{\\Delta + 1}$. Thus $\\phi(n) \\to +\\infty$. It should be observed that in this case $\\phi(2k + 1)$ is always less than $\\phi(2k)$, so that the function progresses to infinity by a continual series of steps forwards and backwards. It does not however ‘oscillate’ according to our definition of the term.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "n**2 + (-1)**n*2*n", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiv/6a", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiv", "number": 6, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "122", "location": "Exercise XXIV, problem 6a", "problem_latex": "$n^{2}\\{1 + (-1)^{n}\\}$, $(-1)^{n}n^{2} + n$, $n^{3} + (-1)^{n}n^{2}$.", "markdown": "$n^{2}\\{1 + (-1)^{n}\\}$, $(-1)^{n}n^{2} + n$, $n^{3} + (-1)^{n}n^{2}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "n**2*(1 + (-1)**n)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiv/6b", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiv", "number": 6, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "122", "location": "Exercise XXIV, problem 6b", "problem_latex": "$n^{2}\\{1 + (-1)^{n}\\}$, $(-1)^{n}n^{2} + n$, $n^{3} + (-1)^{n}n^{2}$.", "markdown": "$n^{2}\\{1 + (-1)^{n}\\}$, $(-1)^{n}n^{2} + n$, $n^{3} + (-1)^{n}n^{2}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(-1)**n*n**2 + n", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiv/6c", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiv", "number": 6, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "122", "location": "Exercise XXIV, problem 6c", "problem_latex": "$n^{2}\\{1 + (-1)^{n}\\}$, $(-1)^{n}n^{2} + n$, $n^{3} + (-1)^{n}n^{2}$.", "markdown": "$n^{2}\\{1 + (-1)^{n}\\}$, $(-1)^{n}n^{2} + n$, $n^{3} + (-1)^{n}n^{2}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "n**3 + (-1)**n*n**2", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiv/7", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiv", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "122", "location": "Exercise XXIV, problem 7", "problem_latex": "$\\sin n\\theta\\pi$. We have already seen (\\Exs{xxiii}.~9) that $\\phi(n)$~oscillates\nfinitely when $\\theta$~is rational, unless $\\theta$~is an integer, when $\\phi(n)= 0$, $\\phi(n) \\to 0$.\n\nThe case in which $\\theta$~is irrational is a little more difficult. But it is not\ndifficult to see that $\\phi(n)$~still oscillates finitely. We can without loss of\ngenerality suppose $0 < \\theta < 1$. In the first place $|\\phi(n)| < 1$. Hence $\\phi(n)$~must\noscillate finitely or tend to a limit. We shall consider whether the\nsecond alternative is really possible. Let us suppose that\n\\[\n\\lim \\sin n\\theta\\pi = l.\n\\]\n{\\Loosen Then, however small $\\DELTA$ may be, we can choose~$n_{0}$ so that $\\sin n\\theta\\pi$ lies between\n$l - \\DELTA$ and $l + \\DELTA$ for all values of~$n$ greater than or equal to~$n_{0}$. Hence\n$\\sin(n + 1)\\theta\\pi - \\sin n\\theta\\pi$ is numerically less than~$2\\DELTA$ for all such values of~$n$,\nand so $|\\sin \\frac{1}{2}\\theta\\pi \\cos(n + \\frac{1}{2})\\theta\\pi| < \\DELTA$.}\n\nHence\n\\[\n\\cos(n + \\tfrac{1}{2})\\theta\\pi\n = \\cos n\\theta\\pi \\cos\\tfrac{1}{2}\\theta\\pi\n - \\sin n\\theta\\pi \\sin\\tfrac{1}{2}\\theta\\pi\n\\]\nmust be numerically less than~$\\DELTA/|\\sin\\frac{1}{2}\\theta\\pi|$. Similarly\n\\[\n\\cos(n - \\tfrac{1}{2})\\theta\\pi\n = \\cos n\\theta\\pi \\cos\\tfrac{1}{2}\\theta\\pi\n + \\sin n\\theta\\pi \\sin\\tfrac{1}{2}\\theta\\pi\n\\]\nmust be numerically less than~$\\DELTA/|\\sin\\frac{1}{2}\\theta\\pi|$; and so each of $\\cos n\\theta\\pi \\cos\\frac{1}{2}\\theta\\pi$,\n$\\sin n\\theta\\pi \\sin\\frac{1}{2}\\theta\\pi$ must be numerically less than $\\DELTA/|\\sin\\frac{1}{2}\\theta\\pi|$. That is to say,\n$\\cos n\\theta\\pi \\cos\\frac{1}{2}\\theta\\pi$ is very small if $n$~is large, and this can only be the case\nif $\\cos n\\theta\\pi$ is very small. Similarly $\\sin n\\theta\\pi$ must be very small, so that $l$~must\nbe zero. But it is impossible that $\\cos n\\theta\\pi$ and $\\sin n\\theta\\pi$ can \\emph{both} be\nvery small, as the sum of their squares is unity. Thus the hypothesis that\n$\\sin n\\theta\\pi$ tends to a limit~$l$ is impossible, and therefore $\\sin n\\theta\\pi$ oscillates\nas $n$~tends to~$\\infty$.\n\n{\\Loosen The reader should consider with particular care the argument\n`$\\cos n\\theta\\pi \\cos\\frac{1}{2}\\theta\\pi$ is very small, and this can only be the case if $\\cos n\\theta\\pi$\nis very small'. Why, he may ask, should it not be the other factor $\\cos\\frac{1}{2}\\theta\\pi$\nwhich is `very small'? The answer is to be found, of course, in the meaning\nof the phrase `very small' as used in this connection. When we say `$\\phi(n)$~is\nvery small' for large values of~$n$, we mean that we can choose~$n_{0}$ so\nthat $\\phi(n)$~is numerically smaller than \\emph{any} assigned number, if \\DPchg{$n$~is sufficiently\nlarge}{$n \\geq n_{0}$}. Such an assertion is palpably absurd when made of a \\emph{fixed} number\nsuch as~$\\cos\\frac{1}{2}\\theta\\pi$, which is not zero.}\n\nProve similarly that $\\cos n\\theta\\pi$ oscillates finitely, unless $\\theta$~is an even integer.", "markdown": "$\\sin n\\theta\\pi$. We have already seen (xxiii. 9) that $\\phi(n)$ oscillates finitely when $\\theta$ is rational, unless $\\theta$ is an integer, when $\\phi(n)= 0$, $\\phi(n) \\to 0$. The case in which $\\theta$ is irrational is a little more difficult. But it is not difficult to see that $\\phi(n)$ still oscillates finitely. We can without loss of generality suppose $0 < \\theta < 1$. In the first place $|\\phi(n)| < 1$. Hence $\\phi(n)$ must oscillate finitely or tend to a limit. We shall consider whether the second alternative is really possible. Let us suppose that n= l. 0.375em plus 0.75em minus 0.25emThen, however small $\\DELTA$ may be, we can choose $n_{0}$ so that $\\sin n\\theta\\pi$ lies between $l - \\DELTA$ and $l + \\DELTA$ for all values of $n$ greater than or equal to $n_{0}$. Hence $\\sin(n + 1)\\theta\\pi - \\sin n\\theta\\pi$ is numerically less than $2\\DELTA$ for all such values of $n$, and so $|\\sin \\frac{1}{2}\\theta\\pi \\cos(n + \\frac{1}{2})\\theta\\pi| < \\DELTA$. Hence (n + 12) = n12 - n12 must be numerically less than $\\DELTA/|\\sin\\frac{1}{2}\\theta\\pi|$. Similarly (n - 12) = n12 + n12 must be numerically less than $\\DELTA/|\\sin\\frac{1}{2}\\theta\\pi|$; and so each of $\\cos n\\theta\\pi \\cos\\frac{1}{2}\\theta\\pi$, $\\sin n\\theta\\pi \\sin\\frac{1}{2}\\theta\\pi$ must be numerically less than $\\DELTA/|\\sin\\frac{1}{2}\\theta\\pi|$. That is to say, $\\cos n\\theta\\pi \\cos\\frac{1}{2}\\theta\\pi$ is very small if $n$ is large, and this can only be the case if $\\cos n\\theta\\pi$ is very small. Similarly $\\sin n\\theta\\pi$ must be very small, so that $l$ must be zero. But it is impossible that $\\cos n\\theta\\pi$ and $\\sin n\\theta\\pi$ can *both* be very small, as the sum of their squares is unity. Thus the hypothesis that $\\sin n\\theta\\pi$ tends to a limit $l$ is impossible, and therefore $\\sin n\\theta\\pi$ oscillates as $n$ tends to $\\infty$. 0.375em plus 0.75em minus 0.25emThe reader should consider with particular care the argument ‘$\\cos n\\theta\\pi \\cos\\frac{1}{2}\\theta\\pi$ is very small, and this can only be the case if $\\cos n\\theta\\pi$ is very small’. Why, he may ask, should it not be the other factor $\\cos\\frac{1}{2}\\theta\\pi$ which is ‘very small’? The answer is to be found, of course, in the meaning of the phrase ‘very small’ as used in this connection. When we say ‘$\\phi(n)$ is very small’ for large values of $n$, we mean that we can choose $n_{0}$ so that $\\phi(n)$ is numerically smaller than *any* assigned number, if $n$ is sufficiently large$n \\geq n_{0}$. Such an assertion is palpably absurd when made of a *fixed* number such as $\\cos\\frac{1}{2}\\theta\\pi$, which is not zero. Prove similarly that $\\cos n\\theta\\pi$ oscillates finitely, unless $\\theta$ is an even integer.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "sin(n*theta*pi)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiv/8a", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiv", "number": 8, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "122", "location": "Exercise XXIV, problem 8a", "problem_latex": "$\\sin n\\theta\\pi + (1/n)$, $\\sin n\\theta\\pi + 1$, $\\sin n\\theta\\pi + n$, $(-1)^{n} \\sin n\\theta\\pi$.", "markdown": "$\\sin n\\theta\\pi + (1/n)$, $\\sin n\\theta\\pi + 1$, $\\sin n\\theta\\pi + n$, $(-1)^{n} \\sin n\\theta\\pi$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "sin(n*theta*pi) + 1/n", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiv/8b", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiv", "number": 8, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "122", "location": "Exercise XXIV, problem 8b", "problem_latex": "$\\sin n\\theta\\pi + (1/n)$, $\\sin n\\theta\\pi + 1$, $\\sin n\\theta\\pi + n$, $(-1)^{n} \\sin n\\theta\\pi$.", "markdown": "$\\sin n\\theta\\pi + (1/n)$, $\\sin n\\theta\\pi + 1$, $\\sin n\\theta\\pi + n$, $(-1)^{n} \\sin n\\theta\\pi$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "sin(n*theta*pi) + 1", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiv/8c", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiv", "number": 8, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "122", "location": "Exercise XXIV, problem 8c", "problem_latex": "$\\sin n\\theta\\pi + (1/n)$, $\\sin n\\theta\\pi + 1$, $\\sin n\\theta\\pi + n$, $(-1)^{n} \\sin n\\theta\\pi$.", "markdown": "$\\sin n\\theta\\pi + (1/n)$, $\\sin n\\theta\\pi + 1$, $\\sin n\\theta\\pi + n$, $(-1)^{n} \\sin n\\theta\\pi$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "sin(n*theta*pi) + n", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiv/8d", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiv", "number": 8, "part": "d", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "122", "location": "Exercise XXIV, problem 8d", "problem_latex": "$\\sin n\\theta\\pi + (1/n)$, $\\sin n\\theta\\pi + 1$, $\\sin n\\theta\\pi + n$, $(-1)^{n} \\sin n\\theta\\pi$.", "markdown": "$\\sin n\\theta\\pi + (1/n)$, $\\sin n\\theta\\pi + 1$, $\\sin n\\theta\\pi + n$, $(-1)^{n} \\sin n\\theta\\pi$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(-1)**n*sin(n*theta*pi)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiv/9a", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiv", "number": 9, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "122", "location": "Exercise XXIV, problem 9a", "problem_latex": "$a\\cos n\\theta\\pi + b\\sin n\\theta\\pi$, $\\sin^{2}n\\theta\\pi$, $a\\cos^{2}n\\theta\\pi + b\\sin^{2}n\\theta\\pi$.", "markdown": "$a\\cos n\\theta\\pi + b\\sin n\\theta\\pi$, $\\sin^{2}n\\theta\\pi$, $a\\cos^{2}n\\theta\\pi + b\\sin^{2}n\\theta\\pi$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "a*cos(n*theta*pi) + b*sin(n*theta*pi)", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiv/9b", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiv", "number": 9, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "122", "location": "Exercise XXIV, problem 9b", "problem_latex": "$a\\cos n\\theta\\pi + b\\sin n\\theta\\pi$, $\\sin^{2}n\\theta\\pi$, $a\\cos^{2}n\\theta\\pi + b\\sin^{2}n\\theta\\pi$.", "markdown": "$a\\cos n\\theta\\pi + b\\sin n\\theta\\pi$, $\\sin^{2}n\\theta\\pi$, $a\\cos^{2}n\\theta\\pi + b\\sin^{2}n\\theta\\pi$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "sin(n*theta*pi)**2", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxiv/9c", "set": "hardy-course-of-pure-mathematics-1921/ex-xxiv", "number": 9, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "122", "location": "Exercise XXIV, problem 9c", "problem_latex": "$a\\cos n\\theta\\pi + b\\sin n\\theta\\pi$, $\\sin^{2}n\\theta\\pi$, $a\\cos^{2}n\\theta\\pi + b\\sin^{2}n\\theta\\pi$.", "markdown": "$a\\cos n\\theta\\pi + b\\sin n\\theta\\pi$, $\\sin^{2}n\\theta\\pi$, $a\\cos^{2}n\\theta\\pi + b\\sin^{2}n\\theta\\pi$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "a*cos(n*theta*pi)**2 + b*sin(n*theta*pi)**2", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxix/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xxix", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "143", "location": "Exercise XXIX, problem 1", "problem_latex": "\\Topic{Recurring decimals.} The commonest example\nof an infinite geometric series is given by an ordinary recurring decimal.\n\\PageSep{144}\nConsider, for example, the decimal $.217\\DPmod{\\dot{1}\\dot{3}}{\\Repeat{13}}$. This stands, according to the\nordinary rules of arithmetic, for\n\\[\n\\frac{2}{10} + \\frac{1}{10^{2}} + \\frac{7}{10^{3}}\n + \\frac{1}{10^{4}} + \\frac{3}{10^{5}} + \\frac{1}{10^{6}} + \\frac{3}{10^{7}}\n + \\dots\n = \\frac{217}{1000}\n + \\frac{13}{10^{5}} \\bigg/ \\left(1 - \\frac{1}{10^{2}}\\right)\n = \\frac{2687}{12\\MC375}.\n\\]\nThe reader should consider where and how any of the general theorems of\n\\SecNo[§]{77} have been used in this reduction.", "markdown": "**decimals.** The commonest example of an infinite geometric series is given by an ordinary recurring decimal. [pg]144 Consider, for example, the decimal $.217\\DPmod{\\dot{1}\\dot{3}}{\\Repeat{13}}$. This stands, according to the ordinary rules of arithmetic, for 210 + 110^2 + 710^3 + 110^4 + 310^5 + 110^6 + 310^7 + … = 2171000 + 1310^5 / (1 - 110^2) = 268712375. The reader should consider where and how any of the general theorems of [§]77 have been used in this reduction.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.frac" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxix/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xxix", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "143", "location": "Exercise XXIX, problem 2", "problem_latex": "Show that in general\n\\[\n.a_{1}a_{2}\\dots a_{m} \\DPmod{\\dot{\\alpha}_{1}\\alpha_{2}\\dots\\dot{\\alpha}_{n}}\n {\\Repeat{\\alpha_{1}\\alpha_{2}\\dots \\alpha_{n}}}\n = \\frac{a_{1}a_{2}\\dots a_{m}\\alpha_{1}\\dots \\alpha_{n} - a_{1}a_{2}\\dots a_{n}}\n {99\\dots 900\\dots 0},\n\\]\nthe denominator containing~$n$ $9$'s and $m$~$0$'s.", "markdown": "Show that in general .a_1a_2…a_m _1_2…_n _1_2…_n| = a_1a_2…a_m_1…_n - a_1a_2…a_n 99…900…0, the denominator containing $n$ $9$’s and $m$ $0$’s.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxix/3", "set": "hardy-course-of-pure-mathematics-1921/ex-xxix", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "143", "location": "Exercise XXIX, problem 3", "problem_latex": "Show that a pure recurring decimal is always equal to a proper\nfraction whose denominator does not contain $2$~or~$5$ as a factor.", "markdown": "Show that a pure recurring decimal is always equal to a proper fraction whose denominator does not contain $2$ or $5$ as a factor.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxix/4", "set": "hardy-course-of-pure-mathematics-1921/ex-xxix", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "143", "location": "Exercise XXIX, problem 4", "problem_latex": "A decimal with $m$~non-recurring and $n$~recurring decimal figures is\nequal to a proper fraction whose denominator is divisible by $2^{m}$~or~$5^{m}$ but by\nno higher power of either.", "markdown": "A decimal with $m$ non-recurring and $n$ recurring decimal figures is equal to a proper fraction whose denominator is divisible by $2^{m}$ or $5^{m}$ but by no higher power of either.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxix/5", "set": "hardy-course-of-pure-mathematics-1921/ex-xxix", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "143", "location": "Exercise XXIX, problem 5", "problem_latex": "The converses of Exs.~3,~4 are also true. Let $r = p/q$, and suppose first\nthat $q$~is prime to~$10$. If we divide all powers of~$10$ by~$q$ we can obtain at most\n$q$~different remainders. It is therefore possible to find two numbers $n_{1}$~and~$n_{2}$,\nwhere $\\DPtypo{n_{2} > n_{1}}{n_{1} > n_{2}}$, such that $10^{n_{1}}$ and $10^{n_{2}}$ give the same remainder. Hence\n$10^{n_{1}} - 10^{n_{2}} = 10^{n_{2}}(10^{n_{1}-n_{2}} - 1)$ is divisible by~$q$, and so $10^{n} - 1$, where $n = n_{1} - n_{2}$,\nis divisible by~$q$. Hence $r$~may be expressed in the form~$P/(10^{n} - 1)$, or in the\nform\n\\[\n\\frac{P}{10^{n}} + \\frac{P}{10^{2n}} + \\dots,\n\\]\n\\ie\\ as a pure recurring decimal with $n$~figures. If on the other hand $q = 2^{\\alpha}5^{\\beta}Q$,\nwhere $Q$~is prime to~$10$, and $m$~is the greater of $\\alpha$~and~$\\beta$, then $10^{m}r$~has a denominator\nprime to~$10$, and is therefore expressible as the sum of an integer\nand a pure recurring decimal. But this is not true of~$10^{\\mu}r$, for any value of~$\\mu$\nless than~$m$; hence the decimal for~$r$ has exactly~$m$ non-recurring figures.", "markdown": "The converses of Exs. 3, 4 are also true. Let $r = p/q$, and suppose first that $q$ is prime to $10$. If we divide all powers of $10$ by $q$ we can obtain at most $q$ different remainders. It is therefore possible to find two numbers $n_{1}$ and $n_{2}$, where $\\DPtypo{n_{2} > n_{1}}{n_{1} > n_{2}}$, such that $10^{n_{1}}$ and $10^{n_{2}}$ give the same remainder. Hence $10^{n_{1}} - 10^{n_{2}} = 10^{n_{2}}(10^{n_{1}-n_{2}} - 1)$ is divisible by $q$, and so $10^{n} - 1$, where $n = n_{1} - n_{2}$, is divisible by $q$. Hence $r$ may be expressed in the form $P/(10^{n} - 1)$, or in the form P10^n + P10^2n + …, *i.e.* as a pure recurring decimal with $n$ figures. If on the other hand $q = 2^{\\alpha}5^{\\beta}Q$, where $Q$ is prime to $10$, and $m$ is the greater of $\\alpha$ and $\\beta$, then $10^{m}r$ has a denominator prime to $10$, and is therefore expressible as the sum of an integer and a pure recurring decimal. But this is not true of $10^{\\mu}r$, for any value of $\\mu$ less than $m$; hence the decimal for $r$ has exactly $m$ non-recurring figures.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxix/6", "set": "hardy-course-of-pure-mathematics-1921/ex-xxix", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "143", "location": "Exercise XXIX, problem 6", "problem_latex": "To the results of Exs.~2--5 we must add that of \\Ex{i}.~3. Finally, if\nwe observe that\n\\[\n.\\DPmod{\\dot{9}}{\\Repeat{9}}\n = \\frac{9}{10} + \\frac{9}{10^{2}} + \\frac{9}{10^{3}} + \\dots\n = 1,\n\\]\nwe see that every terminating decimal can also be expressed as a mixed\nrecurring decimal whose recurring part is composed entirely of~$9$'s. For\nexample, $.217 = .216\\DPmod{\\dot{9}}{\\Repeat{9}}$. Thus every proper fraction can be expressed as a\nrecurring decimal, and conversely.", "markdown": "To the results of Exs. 2--5 we must add that of % [examples:i]Ex. i%. 3. Finally, if we observe that .99| = 910 + 910^2 + 910^3 + … = 1, we see that every terminating decimal can also be expressed as a mixed recurring decimal whose recurring part is composed entirely of $9$’s. For example, $.217 = .216\\DPmod{\\dot{9}}{\\Repeat{9}}$. Thus every proper fraction can be expressed as a recurring decimal, and conversely.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxix/7", "set": "hardy-course-of-pure-mathematics-1921/ex-xxix", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "143", "location": "Exercise XXIX, problem 7", "problem_latex": "\\Topic{Decimals in general. The expression of irrational numbers as\nnon-recurring decimals.} Any decimal, whether recurring or not, corresponds\nto a definite number between $0$~and~$1$. For the decimal $.a_{1}a_{2}a_{3}a_{4}\\dots$ stands\nfor the series\n\\[\n\\frac{a_{1}}{10} + \\frac{a_{2}}{10^{2}} + \\frac{a_{3}}{10^{3}} + \\dots.\n\\]\n\\PageSep{145}\nSince all the digits~$a_{r}$ are positive, the sum~$s_{n}$ of the first $n$~terms of this\nseries increases with~$n$, and it is certainly not greater than~$.\\DPmod{\\dot{9}}{\\Repeat{9}}$ or~$1$. Hence $s_{n}$~tends to a limit between $0$~and~$1$.\n\nMoreover no two decimals can correspond to the same number (except in\nthe special case noticed in Ex.~6). For suppose that $.a_{1}a_{2}a_{3} \\dots$, $.b_{1}b_{2}b_{3} \\dots$ are\ntwo decimals which agree as far as the figures $a_{r-1}$,~$b_{r-1}$, while $a_{r} > b_{r}$.\nThen $a_{r}\\geq b_{r} + 1 > b_{r}.b_{r+1}b_{r+2} \\dots$ (unless $b_{r+1}$, $b_{r+2}$,~\\dots\\ are all~$9$'s), and so\n\\[\n.a_{1}a_{2} \\dots a_{r}a_{r+1} \\dots > .b_{1}b_{2} \\dots b_{r}b_{r+1} \\dots.\n\\]\nIt follows that the expression of a rational fraction as a recurring decimal\n(Exs.\\ 2--6) is unique. It also follows that every decimal which does not\nrecur represents some \\emph{irrational} number between $0$~and~$1$. Conversely, any\nsuch number can be expressed as such a decimal. For it must lie in one of\nthe intervals\n\\[\n0,\\ 1/10;\\quad 1/10,\\ 2/10;\\ \\dots;\\quad 9/10,\\ 1.\n\\]\nIf it lies between $r/10$ and $(r + 1)/10$, then the first figure is~$r$. By subdividing\nthis interval into $10$~parts we can determine the second figure; and so on.\nBut (Exs.~3,~4) the decimal cannot recur. Thus, for example, the decimal\n$1.414\\dots$, obtained by the ordinary process for the extraction of~$\\sqrt{2}$, cannot\nrecur.", "markdown": "**in general. The expression of irrational numbers as non-recurring decimals.** Any decimal, whether recurring or not, corresponds to a definite number between $0$ and $1$. For the decimal $.a_{1}a_{2}a_{3}a_{4}\\dots$ stands for the series a_110 + a_210^2 + a_310^3 + …. [pg]145 Since all the digits $a_{r}$ are positive, the sum $s_{n}$ of the first $n$ terms of this series increases with $n$, and it is certainly not greater than $.\\DPmod{\\dot{9}}{\\Repeat{9}}$ or $1$. Hence $s_{n}$ tends to a limit between $0$ and $1$. Moreover no two decimals can correspond to the same number (except in the special case noticed in Ex. 6). For suppose that $.a_{1}a_{2}a_{3} \\dots$, $.b_{1}b_{2}b_{3} \\dots$ are two decimals which agree as far as the figures $a_{r-1}$, $b_{r-1}$, while $a_{r} > b_{r}$. Then $a_{r}\\geq b_{r} + 1 > b_{r}.b_{r+1}b_{r+2} \\dots$ (unless $b_{r+1}$, $b_{r+2}$, … are all $9$’s), and so .a_1a_2 …a_ra_r+1 …> .b_1b_2 …b_rb_r+1 …. It follows that the expression of a rational fraction as a recurring decimal (Exs. 2--6) is unique. It also follows that every decimal which does not recur represents some *irrational* number between $0$ and $1$. Conversely, any such number can be expressed as such a decimal. For it must lie in one of the intervals 0, 1/10;0pt minus 3pt1/10, 2/10; …;0pt minus 3pt9/10, 1. If it lies between $r/10$ and $(r + 1)/10$, then the first figure is $r$. By subdividing this interval into $10$ parts we can determine the second figure; and so on. But (Exs. 3, 4) the decimal cannot recur. Thus, for example, the decimal $1.414\\dots$, obtained by the ordinary process for the extraction of $\\sqrt{2}$, cannot recur.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxix/8", "set": "hardy-course-of-pure-mathematics-1921/ex-xxix", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "143", "location": "Exercise XXIX, problem 8", "problem_latex": "The decimals $.101\\MS001\\MS000\\MS100\\MS001\\MS0\\dots$ and $.202\\MS002\\MS000\\MS200\\MS002\\MS0\\dots$, in\nwhich the number of zeros between two~$1$'s or $2$'s increases by one at each\nstage, represent irrational numbers.", "markdown": "The decimals $.101\\MS001\\MS000\\MS100\\MS001\\MS0\\dots$ and $.202\\MS002\\MS000\\MS200\\MS002\\MS0\\dots$, in which the number of zeros between two $1$’s or $2$’s increases by one at each stage, represent irrational numbers.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxix/9", "set": "hardy-course-of-pure-mathematics-1921/ex-xxix", "number": 9, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "143", "location": "Exercise XXIX, problem 9", "problem_latex": "The decimal $.111\\MS010\\MS100\\MS010\\MS10\\dots$, in which the $n$th~figure is~$1$ if $n$~is\nprime, and zero otherwise, represents an irrational number. [Since the\nnumber of primes is infinite the decimal does not terminate. Nor can it\nrecur: for if it did we could determine $m$~and~$p$ so that $m$,~$m + p$, $m + 2p$,\n$m + 3p$,~\\dots\\ are all prime numbers; and this is absurd, since the series includes\n$m + mp$.]\\footnote\n {All the results of \\Exs{xxix} may be extended, with suitable modifications, to\n decimals in any scale of notation. For a fuller discussion see Bromwich, \\textit{Infinite\n Series}, Appendix~I.}", "markdown": "The decimal $.111\\MS010\\MS100\\MS010\\MS10\\dots$, in which the $n$th figure is $1$ if $n$ is prime, and zero otherwise, represents an irrational number. [Since the number of primes is infinite the decimal does not terminate. Nor can it recur: for if it did we could determine $m$ and $p$ so that $m$, $m + p$, $m + 2p$, $m + 3p$, … are all prime numbers; and this is absurd, since the series includes $m + mp$.] All the results of xxix may be extended, with suitable modifications, to decimals in any scale of notation. For a fuller discussion see Bromwich, *Infinite Series*, Appendix I.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxv/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xxv", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "124", "location": "Exercise XXV, problem 1", "problem_latex": "If $\\phi(n) \\to +\\infty$ and $\\psi(n) \\geq \\phi(n)$ for all values of~$n$,\nthen $\\psi(n) \\to +\\infty$.", "markdown": "If $\\phi(n) \\to +\\infty$ and $\\psi(n) \\geq \\phi(n)$ for all values of $n$, then $\\psi(n) \\to +\\infty$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxv/10a", "set": "hardy-course-of-pure-mathematics-1921/ex-xxv", "number": 10, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "124", "location": "Exercise XXV, problem 10a", "problem_latex": "Determine the least value of~$n_{0}$ for which it is true that\n\\[\n\\Item{(\\ia)}\\ n + (-1)^{n} > 1000\\quad (n \\geq n_{0}),\\qquad\n\\Item{(\\ib)}\\ n + (-1)^{n} > 1\\MC000\\MC000\\quad (n \\geq n_{0}).\n\\]", "markdown": "Determine the least value of $n_{0}$ for which it is true that [1.5em][l](*a*) n + (-1)^n > 10000pt minus 3pt(n n_0), [1.5em][l](*b*) n + (-1)^n > 10000000pt minus 3pt(n n_0).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.solve.num", "core.table" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxv/10b", "set": "hardy-course-of-pure-mathematics-1921/ex-xxv", "number": 10, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "124", "location": "Exercise XXV, problem 10b", "problem_latex": "Determine the least value of~$n_{0}$ for which it is true that\n\\[\n\\Item{(\\ia)}\\ n + (-1)^{n} > 1000\\quad (n \\geq n_{0}),\\qquad\n\\Item{(\\ib)}\\ n + (-1)^{n} > 1\\MC000\\MC000\\quad (n \\geq n_{0}).\n\\]", "markdown": "Determine the least value of $n_{0}$ for which it is true that [1.5em][l](*a*) n + (-1)^n > 10000pt minus 3pt(n n_0), [1.5em][l](*b*) n + (-1)^n > 10000000pt minus 3pt(n n_0).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.solve.num", "core.table" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxv/11a", "set": "hardy-course-of-pure-mathematics-1921/ex-xxv", "number": 11, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "124", "location": "Exercise XXV, problem 11a", "problem_latex": "Determine the least value of~$n_{0}$ for which it is true that\n\\[\n\\Item{(\\ia)}\\ n^{2} + 2n > \\Delta\\quad (n \\geq n_{0}),\\qquad\n\\Item{(\\ib)}\\ n + (-1)^{n} > \\Delta\\quad (n \\geq n_{0}),\n\\]\n$\\Delta$~being any positive number.", "markdown": "Determine the least value of $n_{0}$ for which it is true that [1.5em][l](*a*) n^2 + 2n > 0pt minus 3pt(n n_0), [1.5em][l](*b*) n + (-1)^n > 0pt minus 3pt(n n_0), $\\Delta$ being any positive number.", "answer_latex": [ "[(\\ia)~$n_{0} = [\\sqrtp{\\Delta + 1}]$: (\\ib)~$n_{0} = 1 + [\\Delta]$ or $2 + [\\Delta]$, according as $[\\Delta]$~is odd or\neven, \\ie\\ $n_{0} = 1 + [\\Delta] + \\frac{1}{2} \\{1 + (-1)^{[\\Delta]}\\}$.]" ], "answer_markdown": [ "[(*a*) $n_{0} = [\\sqrtp{\\Delta + 1}]$: (*b*) $n_{0} = 1 + [\\Delta]$ or $2 + [\\Delta]$, according as $[\\Delta]$ is odd or even, *i.e.* $n_{0} = 1 + [\\Delta] + \\frac{1}{2} \\{1 + (-1)^{[\\Delta]}\\}$.]" ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "floor(sqrt(Delta + 1))" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxv/11b", "set": "hardy-course-of-pure-mathematics-1921/ex-xxv", "number": 11, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "124", "location": "Exercise XXV, problem 11b", "problem_latex": "Determine the least value of~$n_{0}$ for which it is true that\n\\[\n\\Item{(\\ia)}\\ n^{2} + 2n > \\Delta\\quad (n \\geq n_{0}),\\qquad\n\\Item{(\\ib)}\\ n + (-1)^{n} > \\Delta\\quad (n \\geq n_{0}),\n\\]\n$\\Delta$~being any positive number.", "markdown": "Determine the least value of $n_{0}$ for which it is true that [1.5em][l](*a*) n^2 + 2n > 0pt minus 3pt(n n_0), [1.5em][l](*b*) n + (-1)^n > 0pt minus 3pt(n n_0), $\\Delta$ being any positive number.", "answer_latex": [ "[(\\ia)~$n_{0} = [\\sqrtp{\\Delta + 1}]$: (\\ib)~$n_{0} = 1 + [\\Delta]$ or $2 + [\\Delta]$, according as $[\\Delta]$~is odd or\neven, \\ie\\ $n_{0} = 1 + [\\Delta] + \\frac{1}{2} \\{1 + (-1)^{[\\Delta]}\\}$.]" ], "answer_markdown": [ "[(*a*) $n_{0} = [\\sqrtp{\\Delta + 1}]$: (*b*) $n_{0} = 1 + [\\Delta]$ or $2 + [\\Delta]$, according as $[\\Delta]$ is odd or even, *i.e.* $n_{0} = 1 + [\\Delta] + \\frac{1}{2} \\{1 + (-1)^{[\\Delta]}\\}$.]" ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "1 + floor(Delta) + Rational(1, 2)*(1 + (-1)**floor(Delta))" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.solve.num" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxv/12a", "set": "hardy-course-of-pure-mathematics-1921/ex-xxv", "number": 12, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "124", "location": "Exercise XXV, problem 12a", "problem_latex": "Determine the least value of~$n_{0}$ such that\n\\[\n\\Item{(\\ia)}\\ n/(n^{2} + 1) < .0001,\\qquad\n\\Item{(\\ib)}\\ (1/n) + \\{(-1)^{n}/n^{2}\\} < .000\\MS01,\n\\]\nwhen $n \\geq n_{0}$.", "markdown": "Determine the least value of $n_{0}$ such that [1.5em][l](*a*) n/(n^2 + 1) < .0001, [1.5em][l](*b*) (1/n) + (-1)^n/n^2 < .00001, when $n \\geq n_{0}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.table" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxv/12b", "set": "hardy-course-of-pure-mathematics-1921/ex-xxv", "number": 12, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "124", "location": "Exercise XXV, problem 12b", "problem_latex": "Determine the least value of~$n_{0}$ such that\n\\[\n\\Item{(\\ia)}\\ n/(n^{2} + 1) < .0001,\\qquad\n\\Item{(\\ib)}\\ (1/n) + \\{(-1)^{n}/n^{2}\\} < .000\\MS01,\n\\]\nwhen $n \\geq n_{0}$.", "markdown": "Determine the least value of $n_{0}$ such that [1.5em][l](*a*) n/(n^2 + 1) < .0001, [1.5em][l](*b*) (1/n) + (-1)^n/n^2 < .00001, when $n \\geq n_{0}$.", "answer_latex": [ "[Let us take the latter case. In the first place\n\\[\n(1/n) + \\{(-1)^{n}/n^{2}\\} \\leq (n + 1)/n^{2},\n\\]\nand it is easy to see that the least value of~$n_{0}$, such that $(n + 1)/n^{2} < .000\\MS001$\nwhen $n \\geq n_{0}$, is~$1\\MC000\\MC002$. But the inequality given is satisfied by $n = 1\\MC000\\MC001$,\nand this is the value of~$n_{0}$ required.]" ], "answer_markdown": [ "[Let us take the latter case. In the first place (1/n) + (-1)^n/n^2 (n + 1)/n^2, and it is easy to see that the least value of $n_{0}$, such that $(n + 1)/n^{2} < .000\\MS001$ when $n \\geq n_{0}$, is $1\\MC000\\MC002$. But the inequality given is satisfied by $n = 1\\MC000\\MC001$, and this is the value of $n_{0}$ required.]" ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": "1000001" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.table" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxv/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xxv", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "124", "location": "Exercise XXV, problem 2", "problem_latex": "If $\\phi(n) \\to 0$, and $|\\psi(n)| \\leq |\\phi(n)|$ for all values of~$n$, then $\\psi(n) \\to 0$.", "markdown": "If $\\phi(n) \\to 0$, and $|\\psi(n)| \\leq |\\phi(n)|$ for all values of $n$, then $\\psi(n) \\to 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxv/3", "set": "hardy-course-of-pure-mathematics-1921/ex-xxv", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "124", "location": "Exercise XXV, problem 3", "problem_latex": "If $\\lim |\\phi(n)| = 0$, then $\\lim \\phi(n) = 0$.", "markdown": "If $\\lim |\\phi(n)| = 0$, then $\\lim \\phi(n) = 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxv/4", "set": "hardy-course-of-pure-mathematics-1921/ex-xxv", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "124", "location": "Exercise XXV, problem 4", "problem_latex": "If $\\phi(n)$ tends to a limit or oscillates finitely, and $|\\psi(n)| \\leq |\\phi(n)|$ when\n$n \\geq n_{0}$, then $\\psi(n)$~tends to a limit or oscillates finitely.", "markdown": "If $\\phi(n)$ tends to a limit or oscillates finitely, and $|\\psi(n)| \\leq |\\phi(n)|$ when $n \\geq n_{0}$, then $\\psi(n)$ tends to a limit or oscillates finitely.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxv/5", "set": "hardy-course-of-pure-mathematics-1921/ex-xxv", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "124", "location": "Exercise XXV, problem 5", "problem_latex": "If $\\phi(n)$ tends to~$+\\infty$, or to~$-\\infty$, or oscillates infinitely, and\n\\[\n|\\psi(n)| \\geq |\\phi(n)|\n\\]\nwhen $n \\geq n_{0}$, then $\\psi(n)$~tends to~$+\\infty$ or to~$-\\infty$ or oscillates infinitely.", "markdown": "If $\\phi(n)$ tends to $+\\infty$, or to $-\\infty$, or oscillates infinitely, and |(n)| |(n)| when $n \\geq n_{0}$, then $\\psi(n)$ tends to $+\\infty$ or to $-\\infty$ or oscillates infinitely.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxv/6", "set": "hardy-course-of-pure-mathematics-1921/ex-xxv", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "124", "location": "Exercise XXV, problem 6", "problem_latex": "`If $\\phi(n)$~oscillates and, however great be~$n_{0}$, we can find values of~$n$\ngreater than~$n_{0}$ for\nwhich $\\psi(n) > \\phi(n)$, and values of~$n$ greater than~$n_{0}$ for\nwhich $\\psi(n) < \\phi(n)$, then $\\psi(n)$ oscillates'. Is this true? If not give an\nexample to the contrary.", "markdown": "‘If $\\phi(n)$ oscillates and, however great be $n_{0}$, we can find values of $n$ greater than $n_{0}$ for which $\\psi(n) > \\phi(n)$, and values of $n$ greater than $n_{0}$ for which $\\psi(n) < \\phi(n)$, then $\\psi(n)$ oscillates’. Is this true? If not give an example to the contrary.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxv/7", "set": "hardy-course-of-pure-mathematics-1921/ex-xxv", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "124", "location": "Exercise XXV, problem 7", "problem_latex": "If $\\phi(n) \\to l$ as $n \\to \\infty$, then also $\\phi(n + p) \\to l$, $p$~being any fixed integer.\n[This follows at once from the definition. Similarly we see that if $\\phi(n)$~tends\nto~$+\\infty$ or~$-\\infty$ or oscillates so also does~$\\phi(n + p)$.]", "markdown": "If $\\phi(n) \\to l$ as $n \\to \\infty$, then also $\\phi(n + p) \\to l$, $p$ being any fixed integer. [This follows at once from the definition. Similarly we see that if $\\phi(n)$ tends to $+\\infty$ or $-\\infty$ or oscillates so also does $\\phi(n + p)$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxv/8", "set": "hardy-course-of-pure-mathematics-1921/ex-xxv", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "124", "location": "Exercise XXV, problem 8", "problem_latex": "The same conclusions hold (except in the case of oscillation) if $p$~varies\nwith~$n$ but is always numerically less than a fixed positive integer~$N$; or if $p$~varies\nwith~$n$ in any way, so long as it is always positive.", "markdown": "The same conclusions hold (except in the case of oscillation) if $p$ varies with $n$ but is always numerically less than a fixed positive integer $N$; or if $p$ varies with $n$ in any way, so long as it is always positive.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:proof" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxv/9a", "set": "hardy-course-of-pure-mathematics-1921/ex-xxv", "number": 9, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "124", "location": "Exercise XXV, problem 9a", "problem_latex": "Determine the least value of~$n_{0}$ for which it is true that\n\\[\n\\Item{(\\ia)}\\ n^{2} + 2n > 999\\MC999\\quad (n \\geq n_{0}),\\qquad\n\\Item{(\\ib)}\\ n^{2} + 2n > 1\\MC000\\MC000\\quad (n \\geq n_{0}).\n\\]", "markdown": "Determine the least value of $n_{0}$ for which it is true that [1.5em][l](*a*) n^2 + 2n > 9999990pt minus 3pt(n n_0), [1.5em][l](*b*) n^2 + 2n > 10000000pt minus 3pt(n n_0).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.solve.num", "core.table" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxv/9b", "set": "hardy-course-of-pure-mathematics-1921/ex-xxv", "number": 9, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "124", "location": "Exercise XXV, problem 9b", "problem_latex": "Determine the least value of~$n_{0}$ for which it is true that\n\\[\n\\Item{(\\ia)}\\ n^{2} + 2n > 999\\MC999\\quad (n \\geq n_{0}),\\qquad\n\\Item{(\\ib)}\\ n^{2} + 2n > 1\\MC000\\MC000\\quad (n \\geq n_{0}).\n\\]", "markdown": "Determine the least value of $n_{0}$ for which it is true that [1.5em][l](*a*) n^2 + 2n > 9999990pt minus 3pt(n n_0), [1.5em][l](*b*) n^2 + 2n > 10000000pt minus 3pt(n n_0).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.solve.num", "core.table" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxvi/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xxvi", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "131", "location": "Exercise XXVI, problem 1", "problem_latex": " What is the behaviour of the functions\n\\[\n\\left(\\frac{n - 1}{n + 1}\\right)^{2},\\quad\n(-1)^{n} \\left(\\frac{n - 1}{n + 1}\\right)^{2},\\quad\n\\frac{n^{2} + 1}{n},\\quad\n(-1)^{n} \\frac{n^{2} + 1}{n},\n\\]\nas $n\\to\\infty$?", "markdown": "What is the behaviour of the functions (n - 1n + 1)^2,0pt minus 3pt(-1)^n (n - 1n + 1)^2,0pt minus 3ptn^2 + 1n,0pt minus 3pt(-1)^n n^2 + 1n, as $n\\to\\infty$?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.arith" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxvi/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xxvi", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "131", "location": "Exercise XXVI, problem 2", "problem_latex": " Which (if any) of the functions\n\\begin{gather*}\n1/(\\cos^{2}\\tfrac{1}{2}n\\pi + n\\sin^{2}\\tfrac{1}{2}n\\pi),\\quad\n1/\\{n(\\cos^{2}\\tfrac{1}{2}n\\pi + n\\sin^{2}\\tfrac{1}{2}n\\pi)\\}, \\\\\n (n\\cos^{2}\\tfrac{1}{2}n\\pi + \\sin^{2}\\tfrac{1}{2}n\\pi)/\n\\{n(\\cos^{2}\\tfrac{1}{2}n\\pi + n\\sin^{2}\\tfrac{1}{2}n\\pi)\\}\n\\end{gather*}\ntend to a limit as $n \\to \\infty$?", "markdown": "Which (if any) of the functions gather* 1/(^212n+ n^212n),0pt minus 3pt1/n(^212n+ n^212n), (n^212n+ ^212n)/ n(^212n+ n^212n) gather* tend to a limit as $n \\to \\infty$?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.arith", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxvi/3", "set": "hardy-course-of-pure-mathematics-1921/ex-xxvi", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "131", "location": "Exercise XXVI, problem 3", "problem_latex": " Denoting by~$S(n)$ the general rational function of~$n$ considered above,\nshow that in all cases\n\\[\n\\lim\\frac{S(n + 1)}{S(n)} = 1,\\quad\n\\lim\\frac{S\\{n + (1/n)\\}}{S(n)} = 1.\n\\]", "markdown": "Denoting by $S(n)$ the general rational function of $n$ considered above, show that in all cases S(n + 1)S(n) = 1,0pt minus 3ptSn + (1/n)S(n) = 1.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxvii/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xxvii", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "135", "location": "Exercise XXVII, problem 1", "problem_latex": "If $\\phi(n)$~is positive and $\\phi(n + 1) > K \\phi(n)$, where\n$K > 1$, for all values of~$n$, then $\\phi(n) \\to +\\infty$.", "markdown": "If $\\phi(n)$ is positive and $\\phi(n + 1) > K \\phi(n)$, where $K > 1$, for all values of $n$, then $\\phi(n) \\to +\\infty$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxvii/10", "set": "hardy-course-of-pure-mathematics-1921/ex-xxvii", "number": 10, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "135", "location": "Exercise XXVII, problem 10", "problem_latex": "Prove that if $x$~is positive then $\\sqrt[n]{x} \\to 1$ as $n \\to \\infty$.", "markdown": "Prove that if $x$ is positive then $\\sqrt[n]{x} \\to 1$ as $n \\to \\infty$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxvii/11", "set": "hardy-course-of-pure-mathematics-1921/ex-xxvii", "number": 11, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "135", "location": "Exercise XXVII, problem 11", "problem_latex": "$\\sqrt[n]{n}\\to 1$.", "markdown": "$\\sqrt[n]{n}\\to 1$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxvii/12", "set": "hardy-course-of-pure-mathematics-1921/ex-xxvii", "number": 12, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "135", "location": "Exercise XXVII, problem 12", "problem_latex": "$\\sqrtp[n]{n!} \\to +\\infty$.", "markdown": "$\\sqrtp[n]{n!} \\to +\\infty$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxvii/13", "set": "hardy-course-of-pure-mathematics-1921/ex-xxvii", "number": 13, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "135", "location": "Exercise XXVII, problem 13", "problem_latex": "Show that if $-1 < x < 1$ then\n\\[\nu_{n} = \\frac{m(m - 1) \\dots (m - n + 1)}{n!} x^{n} = \\binom{m}{n} x^{n}\n\\]\ntends to zero as $n \\to \\infty$.", "markdown": "Show that if $-1 < x < 1$ then u_n = m(m - 1) …(m - n + 1)n! x^n = mn x^n tends to zero as $n \\to \\infty$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.frac" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxvii/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xxvii", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "135", "location": "Exercise XXVII, problem 2", "problem_latex": "The same result is true if the conditions above stated are satisfied\nonly when $n \\geq n_{0}$.", "markdown": "The same result is true if the conditions above stated are satisfied only when $n \\geq n_{0}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxvii/3", "set": "hardy-course-of-pure-mathematics-1921/ex-xxvii", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "135", "location": "Exercise XXVII, problem 3", "problem_latex": "If $\\phi(n)$~is positive and $\\phi(n + 1) < K\\phi(n)$, where $0 < K < 1$, then\n$\\lim\\phi(n) = 0$. This result also is true if the conditions are satisfied only when\n$n \\geq n_{0}$.", "markdown": "If $\\phi(n)$ is positive and $\\phi(n + 1) < K\\phi(n)$, where $0 < K < 1$, then $\\lim\\phi(n) = 0$. This result also is true if the conditions are satisfied only when $n \\geq n_{0}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxvii/4", "set": "hardy-course-of-pure-mathematics-1921/ex-xxvii", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "135", "location": "Exercise XXVII, problem 4", "problem_latex": "If $|\\phi(n + 1)| < K|\\phi(n)|$ when $n \\geq n_{0}$, and $0 < K < 1$, then $\\lim\\phi(n) = 0$.", "markdown": "If $|\\phi(n + 1)| < K|\\phi(n)|$ when $n \\geq n_{0}$, and $0 < K < 1$, then $\\lim\\phi(n) = 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxvii/5", "set": "hardy-course-of-pure-mathematics-1921/ex-xxvii", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "135", "location": "Exercise XXVII, problem 5", "problem_latex": "If $\\phi(n)$ is positive and $\\lim\\{\\phi(n + 1)\\}/\\{\\phi(n)\\} = l > 1$, then $\\phi(n) \\to +\\infty$.", "markdown": "If $\\phi(n)$ is positive and $\\lim\\{\\phi(n + 1)\\}/\\{\\phi(n)\\} = l > 1$, then $\\phi(n) \\to +\\infty$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxvii/6", "set": "hardy-course-of-pure-mathematics-1921/ex-xxvii", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "135", "location": "Exercise XXVII, problem 6", "problem_latex": "If $\\lim\\{\\phi(n + 1)\\}/\\{\\phi(n)\\} = l$, where $l$~is numerically less than unity,\nthen $\\lim\\phi(n) = 0$.", "markdown": "If $\\lim\\{\\phi(n + 1)\\}/\\{\\phi(n)\\} = l$, where $l$ is numerically less than unity, then $\\lim\\phi(n) = 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxvii/7", "set": "hardy-course-of-pure-mathematics-1921/ex-xxvii", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "135", "location": "Exercise XXVII, problem 7", "problem_latex": "Determine the behaviour, as $n \\to \\infty$, of $\\phi(n) = n^{r}x^{n}$, where $r$~is any\npositive integer.", "markdown": "Determine the behaviour, as $n \\to \\infty$, of $\\phi(n) = n^{r}x^{n}$, where $r$ is any positive integer.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxvii/8", "set": "hardy-course-of-pure-mathematics-1921/ex-xxvii", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "135", "location": "Exercise XXVII, problem 8", "problem_latex": "Discuss $n^{-r}x^{n}$ in the same way.", "markdown": "Discuss $n^{-r}x^{n}$ in the same way.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxvii/9", "set": "hardy-course-of-pure-mathematics-1921/ex-xxvii", "number": 9, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "135", "location": "Exercise XXVII, problem 9", "problem_latex": "Draw up a table to show how $n^{k}x^{n}$ behaves as $n \\to \\infty$, for all real\nvalues of~$x$, and all positive and negative integral values of~$k$.", "markdown": "Draw up a table to show how $n^{k}x^{n}$ behaves as $n \\to \\infty$, for all real values of $x$, and all positive and negative integral values of $k$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.table" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxviii/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xxviii", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "139", "location": "Exercise XXVIII, problem 1", "problem_latex": "Verify \\Eq{(9)} for $r = 2$,~$3$, and \\Eq{(10)} for $s = \\frac{1}{2}$,~$\\frac{1}{3}$.", "markdown": "Verify (9) for $r = 2$, $3$, and (10) for $s = \\frac{1}{2}$, $\\frac{1}{3}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.arith", "core.frac" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxviii/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xxviii", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "139", "location": "Exercise XXVIII, problem 2", "problem_latex": "Show that \\Eq{(9)}~and~\\Eq{(10)} are also true if $y > x > 0$.", "markdown": "Show that (9) and (10) are also true if $y > x > 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxviii/3", "set": "hardy-course-of-pure-mathematics-1921/ex-xxviii", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "139", "location": "Exercise XXVIII, problem 3", "problem_latex": "Show that \\Eq{(9)}~also holds for $r < 0$. [See Chrystal's \\textit{Algebra}, vol.~ii,\npp.~43--45.]", "markdown": "Show that (9) also holds for $r < 0$. [See Chrystal’s *Algebra*, vol. ii, pp. 43--45.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxviii/4", "set": "hardy-course-of-pure-mathematics-1921/ex-xxviii", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "139", "location": "Exercise XXVIII, problem 4", "problem_latex": "If $\\phi(n) \\to l$, where $l > 0$, as $n \\to \\infty$, then $\\phi^{k} \\to l^{k}$, $k$~being any rational number.", "markdown": "If $\\phi(n) \\to l$, where $l > 0$, as $n \\to \\infty$, then $\\phi^{k} \\to l^{k}$, $k$ being any rational number.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxviii/5", "set": "hardy-course-of-pure-mathematics-1921/ex-xxviii", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "139", "location": "Exercise XXVIII, problem 5", "problem_latex": "Extend the results of \\Exs{xxvii}.\\ 7,~8,~9 to the case in which $r$~or~$k$\nare any rational numbers.", "markdown": "Extend the results of xxvii. 7, 8, 9 to the case in which $r$ or $k$ are any rational numbers.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxx/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xxx", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "145", "location": "Exercise XXX, problem 1", "problem_latex": "{\\Loosen The series $r^{m} + r^{m+1} + \\dots$ is convergent if $-1 < r < 1$,\nand its sum is $1/(1 - r) - 1 - r - \\dots - r^{m-1}$ (\\SecNo[§]{77},~\\Eq{(2)}).}", "markdown": "0.375em plus 0.75em minus 0.25emThe series $r^{m} + r^{m+1} + \\dots$ is convergent if $-1 < r < 1$, and its sum is $1/(1 - r) - 1 - r - \\dots - r^{m-1}$ ([§]77, (2)).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxx/10a", "set": "hardy-course-of-pure-mathematics-1921/ex-xxx", "number": 10, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "145", "location": "Exercise XXX, problem 10a", "problem_latex": "Consider the convergence of the series\n\\begin{align*}\n& (1 + r) + (r^{2} + r^{3}) + \\dots,\n&& (1 + r + r^{2}) + (r^{3} + r^{4} + r^{5}) + \\dots,\\\\\n& 1 - 2r + r^{2} + r^{3} - 2r^{4} + r^{5} + \\dots,\n&& (1 - 2r + r^{2}) + (r^{3} - 2r^{4} + r^{5}) + \\dots,\n\\end{align*}\nand find their sums when they are convergent.", "markdown": "Consider the convergence of the series align* & (1 + r) + (r^2 + r^3) + …, && (1 + r + r^2) + (r^3 + r^4 + r^5) + …, & 1 - 2r + r^2 + r^3 - 2r^4 + r^5 + …, && (1 - 2r + r^2) + (r^3 - 2r^4 + r^5) + …, align* and find their sums when they are convergent.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxx/10b", "set": "hardy-course-of-pure-mathematics-1921/ex-xxx", "number": 10, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "145", "location": "Exercise XXX, problem 10b", "problem_latex": "Consider the convergence of the series\n\\begin{align*}\n& (1 + r) + (r^{2} + r^{3}) + \\dots,\n&& (1 + r + r^{2}) + (r^{3} + r^{4} + r^{5}) + \\dots,\\\\\n& 1 - 2r + r^{2} + r^{3} - 2r^{4} + r^{5} + \\dots,\n&& (1 - 2r + r^{2}) + (r^{3} - 2r^{4} + r^{5}) + \\dots,\n\\end{align*}\nand find their sums when they are convergent.", "markdown": "Consider the convergence of the series align* & (1 + r) + (r^2 + r^3) + …, && (1 + r + r^2) + (r^3 + r^4 + r^5) + …, & 1 - 2r + r^2 + r^3 - 2r^4 + r^5 + …, && (1 - 2r + r^2) + (r^3 - 2r^4 + r^5) + …, align* and find their sums when they are convergent.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxx/10c", "set": "hardy-course-of-pure-mathematics-1921/ex-xxx", "number": 10, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "145", "location": "Exercise XXX, problem 10c", "problem_latex": "Consider the convergence of the series\n\\begin{align*}\n& (1 + r) + (r^{2} + r^{3}) + \\dots,\n&& (1 + r + r^{2}) + (r^{3} + r^{4} + r^{5}) + \\dots,\\\\\n& 1 - 2r + r^{2} + r^{3} - 2r^{4} + r^{5} + \\dots,\n&& (1 - 2r + r^{2}) + (r^{3} - 2r^{4} + r^{5}) + \\dots,\n\\end{align*}\nand find their sums when they are convergent.", "markdown": "Consider the convergence of the series align* & (1 + r) + (r^2 + r^3) + …, && (1 + r + r^2) + (r^3 + r^4 + r^5) + …, & 1 - 2r + r^2 + r^3 - 2r^4 + r^5 + …, && (1 - 2r + r^2) + (r^3 - 2r^4 + r^5) + …, align* and find their sums when they are convergent.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxx/10d", "set": "hardy-course-of-pure-mathematics-1921/ex-xxx", "number": 10, "part": "d", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "145", "location": "Exercise XXX, problem 10d", "problem_latex": "Consider the convergence of the series\n\\begin{align*}\n& (1 + r) + (r^{2} + r^{3}) + \\dots,\n&& (1 + r + r^{2}) + (r^{3} + r^{4} + r^{5}) + \\dots,\\\\\n& 1 - 2r + r^{2} + r^{3} - 2r^{4} + r^{5} + \\dots,\n&& (1 - 2r + r^{2}) + (r^{3} - 2r^{4} + r^{5}) + \\dots,\n\\end{align*}\nand find their sums when they are convergent.", "markdown": "Consider the convergence of the series align* & (1 + r) + (r^2 + r^3) + …, && (1 + r + r^2) + (r^3 + r^4 + r^5) + …, & 1 - 2r + r^2 + r^3 - 2r^4 + r^5 + …, && (1 - 2r + r^2) + (r^3 - 2r^4 + r^5) + …, align* and find their sums when they are convergent.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxx/11", "set": "hardy-course-of-pure-mathematics-1921/ex-xxx", "number": 11, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "145", "location": "Exercise XXX, problem 11", "problem_latex": "If $0 \\leq a_{n} \\leq 1$ then the series $a_{0} + a_{1}r + a_{2}r^{2} + \\dots$ is convergent for\n$0 \\leq r < 1$, and its sum is not greater than~$1/(1 - r)$.", "markdown": "If $0 \\leq a_{n} \\leq 1$ then the series $a_{0} + a_{1}r + a_{2}r^{2} + \\dots$ is convergent for $0 \\leq r < 1$, and its sum is not greater than $1/(1 - r)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxx/12", "set": "hardy-course-of-pure-mathematics-1921/ex-xxx", "number": 12, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "145", "location": "Exercise XXX, problem 12", "problem_latex": "If in addition the series $a_{0} + a_{1} + a_{2} + \\dots$ is convergent, then the series\n$a_{0} + a_{1}r + a_{2}r^{2} + \\dots$ is convergent for $0 \\leq r \\leq 1$, and its sum is not greater than\nthe lesser of $a_{0} + a_{1} + a_{2} + \\dots$ and~$1/(1 - r)$.", "markdown": "If in addition the series $a_{0} + a_{1} + a_{2} + \\dots$ is convergent, then the series $a_{0} + a_{1}r + a_{2}r^{2} + \\dots$ is convergent for $0 \\leq r \\leq 1$, and its sum is not greater than the lesser of $a_{0} + a_{1} + a_{2} + \\dots$ and $1/(1 - r)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxx/13", "set": "hardy-course-of-pure-mathematics-1921/ex-xxx", "number": 13, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "145", "location": "Exercise XXX, problem 13", "problem_latex": "The series\n\\[\n1 + \\frac{1}{1} + \\frac{1}{1·2} + \\frac{1}{1·2·3} + \\dots\n\\]\nis convergent. [For $1/(1·2 \\dots n) \\leq 1/2^{n-1}$.]", "markdown": "The series 1 + 11 + 11·2 + 11·2·3 + … is convergent. [For $1/(1·2 \\dots n) \\leq 1/2^{n-1}$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.frac", "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxx/14a", "set": "hardy-course-of-pure-mathematics-1921/ex-xxx", "number": 14, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "145", "location": "Exercise XXX, problem 14a", "problem_latex": "The series\n\\[\n1 + \\frac{1}{1·2} + \\frac{1}{1·2·3·4} + \\dots,\\quad\n\\frac{1}{1} + \\frac{1}{1·2·3} + \\frac{1}{1·2·3·4·5} + \\dots\n\\]\nare convergent.", "markdown": "The series 1 + 11·2 + 11·2·3·4 + …,0pt minus 3pt11 + 11·2·3 + 11·2·3·4·5 + … are convergent.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.frac", "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxx/14b", "set": "hardy-course-of-pure-mathematics-1921/ex-xxx", "number": 14, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "145", "location": "Exercise XXX, problem 14b", "problem_latex": "The series\n\\[\n1 + \\frac{1}{1·2} + \\frac{1}{1·2·3·4} + \\dots,\\quad\n\\frac{1}{1} + \\frac{1}{1·2·3} + \\frac{1}{1·2·3·4·5} + \\dots\n\\]\nare convergent.", "markdown": "The series 1 + 11·2 + 11·2·3·4 + …,0pt minus 3pt11 + 11·2·3 + 11·2·3·4·5 + … are convergent.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.frac", "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxx/15", "set": "hardy-course-of-pure-mathematics-1921/ex-xxx", "number": 15, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "145", "location": "Exercise XXX, problem 15", "problem_latex": "The general harmonic series\n\\[\n\\frac{1}{a} + \\frac{1}{a + b} + \\frac{1}{a + 2b} + \\dots,\n\\]\nwhere $a$~and~$b$ are positive, diverges to~$+\\infty$.\n\n[For $u_{n} = 1/(a + nb) > 1/\\{n(a + b)\\}$. Now compare with $1 + \\frac{1}{2} + \\frac{1}{3} + \\dots$.]", "markdown": "The general harmonic series 1a + 1a + b + 1a + 2b + …, where $a$ and $b$ are positive, diverges to $+\\infty$. [For $u_{n} = 1/(a + nb) > 1/\\{n(a + b)\\}$. Now compare with $1 + \\frac{1}{2} + \\frac{1}{3} + \\dots$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.frac", "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxx/16", "set": "hardy-course-of-pure-mathematics-1921/ex-xxx", "number": 16, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "145", "location": "Exercise XXX, problem 16", "problem_latex": "Show that the series\n\\[\n(u_{0} - u_{1}) + (u_{1} - u_{2}) + (u_{2} - u_{3}) + \\dots\n\\]\nis convergent if and only if $u_{n}$~tends to a limit as $n \\to \\infty$.", "markdown": "Show that the series (u_0 - u_1) + (u_1 - u_2) + (u_2 - u_3) + … is convergent if and only if $u_{n}$ tends to a limit as $n \\to \\infty$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxx/17", "set": "hardy-course-of-pure-mathematics-1921/ex-xxx", "number": 17, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "145", "location": "Exercise XXX, problem 17", "problem_latex": "If $u_{1} + u_{2} + u_{3} + \\dots$ is divergent then so is any series formed by\ngrouping the terms in brackets in any way to form new single terms.", "markdown": "If $u_{1} + u_{2} + u_{3} + \\dots$ is divergent then so is any series formed by grouping the terms in brackets in any way to form new single terms.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxx/18", "set": "hardy-course-of-pure-mathematics-1921/ex-xxx", "number": 18, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "145", "location": "Exercise XXX, problem 18", "problem_latex": "Any series, formed by taking a selection of the terms of a convergent\nseries of positive terms, is itself convergent.", "markdown": "Any series, formed by taking a selection of the terms of a convergent series of positive terms, is itself convergent.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxx/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xxx", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "145", "location": "Exercise XXX, problem 2", "problem_latex": "The series $r^{m} + r^{m+1} + \\dots$ is convergent if $-1 < r < 1$, and its sum is\n$r^{m}/(1 - r)$ (\\SecNo[§]{77},~\\Eq{(4)}). Verify that the results of Exs.\\ 1~and~2 are in agreement.", "markdown": "The series $r^{m} + r^{m+1} + \\dots$ is convergent if $-1 < r < 1$, and its sum is $r^{m}/(1 - r)$ ([§]77, (4)). Verify that the results of Exs. 1 and 2 are in agreement.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.frac", "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxx/3a", "set": "hardy-course-of-pure-mathematics-1921/ex-xxx", "number": 3, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "145", "location": "Exercise XXX, problem 3a", "problem_latex": "Prove that the series $1 + 2r + 2r^{2} + \\dots$ is convergent, and that its sum\nis~$(1 + r)/(1 - r)$, ($\\alpha$)~by writing it in the form $-1 + 2(1 + r + r^{2} + \\dots)$, ($\\beta$)~by\nwriting it in the form $1 + 2(r + r^{2} + \\dots)$, ($\\gamma$)~by adding the two series\n$1 + r + r^{2} + \\dots$, $r + r^{2} + \\dots$. In each case mention which of the theorems of\n\\SecNo[§]{77} are used in your proof.", "markdown": "Prove that the series $1 + 2r + 2r^{2} + \\dots$ is convergent, and that its sum is $(1 + r)/(1 - r)$, ($\\alpha$) by writing it in the form $-1 + 2(1 + r + r^{2} + \\dots)$, ($\\beta$) by writing it in the form $1 + 2(r + r^{2} + \\dots)$, ($\\gamma$) by adding the two series $1 + r + r^{2} + \\dots$, $r + r^{2} + \\dots$. In each case mention which of the theorems of [§]77 are used in your proof.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxx/3b", "set": "hardy-course-of-pure-mathematics-1921/ex-xxx", "number": 3, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "145", "location": "Exercise XXX, problem 3b", "problem_latex": "Prove that the series $1 + 2r + 2r^{2} + \\dots$ is convergent, and that its sum\nis~$(1 + r)/(1 - r)$, ($\\alpha$)~by writing it in the form $-1 + 2(1 + r + r^{2} + \\dots)$, ($\\beta$)~by\nwriting it in the form $1 + 2(r + r^{2} + \\dots)$, ($\\gamma$)~by adding the two series\n$1 + r + r^{2} + \\dots$, $r + r^{2} + \\dots$. In each case mention which of the theorems of\n\\SecNo[§]{77} are used in your proof.", "markdown": "Prove that the series $1 + 2r + 2r^{2} + \\dots$ is convergent, and that its sum is $(1 + r)/(1 - r)$, ($\\alpha$) by writing it in the form $-1 + 2(1 + r + r^{2} + \\dots)$, ($\\beta$) by writing it in the form $1 + 2(r + r^{2} + \\dots)$, ($\\gamma$) by adding the two series $1 + r + r^{2} + \\dots$, $r + r^{2} + \\dots$. In each case mention which of the theorems of [§]77 are used in your proof.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxx/3c", "set": "hardy-course-of-pure-mathematics-1921/ex-xxx", "number": 3, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "145", "location": "Exercise XXX, problem 3c", "problem_latex": "Prove that the series $1 + 2r + 2r^{2} + \\dots$ is convergent, and that its sum\nis~$(1 + r)/(1 - r)$, ($\\alpha$)~by writing it in the form $-1 + 2(1 + r + r^{2} + \\dots)$, ($\\beta$)~by\nwriting it in the form $1 + 2(r + r^{2} + \\dots)$, ($\\gamma$)~by adding the two series\n$1 + r + r^{2} + \\dots$, $r + r^{2} + \\dots$. In each case mention which of the theorems of\n\\SecNo[§]{77} are used in your proof.", "markdown": "Prove that the series $1 + 2r + 2r^{2} + \\dots$ is convergent, and that its sum is $(1 + r)/(1 - r)$, ($\\alpha$) by writing it in the form $-1 + 2(1 + r + r^{2} + \\dots)$, ($\\beta$) by writing it in the form $1 + 2(r + r^{2} + \\dots)$, ($\\gamma$) by adding the two series $1 + r + r^{2} + \\dots$, $r + r^{2} + \\dots$. In each case mention which of the theorems of [§]77 are used in your proof.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxx/4", "set": "hardy-course-of-pure-mathematics-1921/ex-xxx", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "145", "location": "Exercise XXX, problem 4", "problem_latex": "Prove that the `arithmetic' series\n\\[\na + (a + b) + (a + 2b) + \\dots\n\\]\nis always divergent, unless both $a$~and~$b$ are zero. Show that, if $b$ is not\nzero, the series diverges to~$+\\infty$ or to~$-\\infty$ according to the sign of~$b$, while if\n$b = 0$ it diverges to~$+\\infty$ or~$-\\infty$ according to the sign of~$a$.", "markdown": "Prove that the ‘arithmetic’ series a + (a + b) + (a + 2b) + … is always divergent, unless both $a$ and $b$ are zero. Show that, if $b$ is not zero, the series diverges to $+\\infty$ or to $-\\infty$ according to the sign of $b$, while if $b = 0$ it diverges to $+\\infty$ or $-\\infty$ according to the sign of $a$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxx/5", "set": "hardy-course-of-pure-mathematics-1921/ex-xxx", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "145", "location": "Exercise XXX, problem 5", "problem_latex": "What is the sum of the series\n\\[\n(1 - r) + (r - r^{2}) + (r^{2} - r^{3}) + \\dots\n\\]\nwhen the series is convergent? [The series converges only if $-1 < r \\leq 1$. Its\nsum is~$1$, except when $r = 1$, when its sum is~$0$.]", "markdown": "What is the sum of the series (1 - r) + (r - r^2) + (r^2 - r^3) + … when the series is convergent? [The series converges only if $-1 < r \\leq 1$. Its sum is $1$, except when $r = 1$, when its sum is $0$.]", "answer_latex": [ "[The series converges only if $-1 < r \\leq 1$. Its\nsum is~$1$, except when $r = 1$, when its sum is~$0$.]" ], "answer_markdown": [ "[The series converges only if $-1 < r \\leq 1$. Its sum is $1$, except when $r = 1$, when its sum is $0$.]" ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxx/6", "set": "hardy-course-of-pure-mathematics-1921/ex-xxx", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "145", "location": "Exercise XXX, problem 6", "problem_latex": "Sum the series\n\\[\n%[** TN: In-line equation in the original]\nr^{2} + \\frac{r^{2}}{1 + r^{2}} + \\frac{r^{2}}{(1 + r^{2})^{2}} + \\dots.\n\\]\n[The series is always convergent.\nIts sum is~$1 + r^{2}$, except when $r = 0$, when its sum is~$0$.]", "markdown": "Sum the series %[** TN: In-line equation in the original] r^2 + r^21 + r^2 + r^2(1 + r^2)^2 + …. [The series is always convergent. Its sum is $1 + r^{2}$, except when $r = 0$, when its sum is $0$.]", "answer_latex": [ "[The series is always convergent.\nIts sum is~$1 + r^{2}$, except when $r = 0$, when its sum is~$0$.]" ], "answer_markdown": [ "[The series is always convergent. Its sum is $1 + r^{2}$, except when $r = 0$, when its sum is $0$.]" ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.frac", "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxx/7", "set": "hardy-course-of-pure-mathematics-1921/ex-xxx", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "145", "location": "Exercise XXX, problem 7", "problem_latex": "If we assume that $1 + r + r^{2} + \\dots$ is convergent then we can prove that its\nsum is~$1/(1 - r)$ by means of \\SecNo[§]{77}, \\Eq{(1)}~and~\\Eq{(4)}. For if $1 + r + r^{2} + \\dots = s$ then\n\\[\ns = 1 + r(1 + r^{2} + \\dots) = 1 + rs.\n\\]", "markdown": "If we assume that $1 + r + r^{2} + \\dots$ is convergent then we can prove that its sum is $1/(1 - r)$ by means of [§]77, (1) and (4). For if $1 + r + r^{2} + \\dots = s$ then s = 1 + r(1 + r^2 + …) = 1 + rs.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxx/8", "set": "hardy-course-of-pure-mathematics-1921/ex-xxx", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "145", "location": "Exercise XXX, problem 8", "problem_latex": "Sum the series\n\\[\nr + \\frac{r}{1 + r} + \\frac{r}{(1 + r)^{2}} + \\dots\n\\]\nwhen it is convergent. [The series is convergent if $-1 < 1/(1 + r) < 1$, \\ie\\ if\n$r < -2$ or if $r > 0$, and its sum is~$1 + r$. It is also convergent when $r = 0$, when\nits sum is~$0$.]", "markdown": "Sum the series r + r1 + r + r(1 + r)^2 + … when it is convergent. [The series is convergent if $-1 < 1/(1 + r) < 1$, *i.e.* if $r < -2$ or if $r > 0$, and its sum is $1 + r$. It is also convergent when $r = 0$, when its sum is $0$.]", "answer_latex": [ "[The series is convergent if $-1 < 1/(1 + r) < 1$, \\ie\\ if\n$r < -2$ or if $r > 0$, and its sum is~$1 + r$. It is also convergent when $r = 0$, when\nits sum is~$0$.]" ], "answer_markdown": [ "[The series is convergent if $-1 < 1/(1 + r) < 1$, *i.e.* if $r < -2$ or if $r > 0$, and its sum is $1 + r$. It is also convergent when $r = 0$, when its sum is $0$.]" ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.frac", "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxx/9a", "set": "hardy-course-of-pure-mathematics-1921/ex-xxx", "number": 9, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "145", "location": "Exercise XXX, problem 9a", "problem_latex": "Answer the same question for the series\n\\begin{align*}\n& r - \\frac{r}{1 + r} + \\frac{r}{(1 + r)^{2}} - \\dots,\n&& r + \\frac{r}{1 - r} + \\frac{r}{(1 - r)^{2}} + \\dots,\\\\\n& 1 - \\frac{r}{1 + r} + \\left(\\frac{r}{1 + r}\\right)^{2} - \\dots,\n&& 1 + \\frac{r}{1 - r} + \\left(\\frac{r}{1 - r}\\right)^{2} + \\dots.\n\\end{align*}", "markdown": "Answer the same question for the series align* & r - r1 + r + r(1 + r)^2 - …, && r + r1 - r + r(1 - r)^2 + …, & 1 - r1 + r + (r1 + r)^2 - …, && 1 + r1 - r + (r1 - r)^2 + …. align*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.frac", "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxx/9b", "set": "hardy-course-of-pure-mathematics-1921/ex-xxx", "number": 9, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "145", "location": "Exercise XXX, problem 9b", "problem_latex": "Answer the same question for the series\n\\begin{align*}\n& r - \\frac{r}{1 + r} + \\frac{r}{(1 + r)^{2}} - \\dots,\n&& r + \\frac{r}{1 - r} + \\frac{r}{(1 - r)^{2}} + \\dots,\\\\\n& 1 - \\frac{r}{1 + r} + \\left(\\frac{r}{1 + r}\\right)^{2} - \\dots,\n&& 1 + \\frac{r}{1 - r} + \\left(\\frac{r}{1 - r}\\right)^{2} + \\dots.\n\\end{align*}", "markdown": "Answer the same question for the series align* & r - r1 + r + r(1 + r)^2 - …, && r + r1 - r + r(1 - r)^2 + …, & 1 - r1 + r + (r1 + r)^2 - …, && 1 + r1 - r + (r1 - r)^2 + …. align*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.frac", "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxx/9c", "set": "hardy-course-of-pure-mathematics-1921/ex-xxx", "number": 9, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "145", "location": "Exercise XXX, problem 9c", "problem_latex": "Answer the same question for the series\n\\begin{align*}\n& r - \\frac{r}{1 + r} + \\frac{r}{(1 + r)^{2}} - \\dots,\n&& r + \\frac{r}{1 - r} + \\frac{r}{(1 - r)^{2}} + \\dots,\\\\\n& 1 - \\frac{r}{1 + r} + \\left(\\frac{r}{1 + r}\\right)^{2} - \\dots,\n&& 1 + \\frac{r}{1 - r} + \\left(\\frac{r}{1 - r}\\right)^{2} + \\dots.\n\\end{align*}", "markdown": "Answer the same question for the series align* & r - r1 + r + r(1 + r)^2 - …, && r + r1 - r + r(1 - r)^2 + …, & 1 - r1 + r + (r1 + r)^2 - …, && 1 + r1 - r + (r1 - r)^2 + …. align*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.frac", "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxx/9d", "set": "hardy-course-of-pure-mathematics-1921/ex-xxx", "number": 9, "part": "d", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "145", "location": "Exercise XXX, problem 9d", "problem_latex": "Answer the same question for the series\n\\begin{align*}\n& r - \\frac{r}{1 + r} + \\frac{r}{(1 + r)^{2}} - \\dots,\n&& r + \\frac{r}{1 - r} + \\frac{r}{(1 - r)^{2}} + \\dots,\\\\\n& 1 - \\frac{r}{1 + r} + \\left(\\frac{r}{1 + r}\\right)^{2} - \\dots,\n&& 1 + \\frac{r}{1 - r} + \\left(\\frac{r}{1 - r}\\right)^{2} + \\dots.\n\\end{align*}", "markdown": "Answer the same question for the series align* & r - r1 + r + r(1 + r)^2 - …, && r + r1 - r + r(1 - r)^2 + …, & 1 - r1 + r + (r1 + r)^2 - …, && 1 + r1 - r + (r1 - r)^2 + …. align*", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.frac", "other:series_convergence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxi/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxi", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "148", "location": "Exercise XXXI, problem 1", "problem_latex": "$\\phi_{n}(x) = x$. Here $n$~does not appear at all in the\nexpression of~$\\phi_{n}(x)$, and $\\phi(x) = \\lim\\phi_{n}(x) = x$ for all values of~$x$.", "markdown": "$\\phi_{n}(x) = x$. Here $n$ does not appear at all in the expression of $\\phi_{n}(x)$, and $\\phi(x) = \\lim\\phi_{n}(x) = x$ for all values of $x$.", "answer_latex": [ "$\\phi_{n}(x) = x$. Here $n$~does not appear at all in the\nexpression of~$\\phi_{n}(x)$, and $\\phi(x) = \\lim\\phi_{n}(x) = x$ for all values of~$x$." ], "answer_markdown": [ "$\\phi_{n}(x) = x$. Here $n$ does not appear at all in the expression of $\\phi_{n}(x)$, and $\\phi(x) = \\lim\\phi_{n}(x) = x$ for all values of $x$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxi/10a", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxi", "number": 10, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "148", "location": "Exercise XXXI, problem 10a", "problem_latex": "$\\phi_{n}(x) = (x^{n} - 1)/(x^{n} + 1)$, $(nx^{n} - 1)/(nx^{n} + 1)$, $(x^{n} - n)/(x^{n} + n)$. [In the\nfirst case $\\phi(x) = 1$ when $|x| > 1$, $\\phi(x) = -1$ when $|x| < 1$, $\\phi(x) = 0$ when $x = 1$\nand $\\phi(x)$~is not defined when $x = -1$. The second and third functions differ\nfrom the first in that they are defined both when $x = 1$ and when $x = -1$: the\nsecond has the value~$1$ and the third the value~$-1$ for both these values of~$x$.]", "markdown": "$\\phi_{n}(x) = (x^{n} - 1)/(x^{n} + 1)$, $(nx^{n} - 1)/(nx^{n} + 1)$, $(x^{n} - n)/(x^{n} + n)$. [In the first case $\\phi(x) = 1$ when $|x| > 1$, $\\phi(x) = -1$ when $|x| < 1$, $\\phi(x) = 0$ when $x = 1$ and $\\phi(x)$ is not defined when $x = -1$. The second and third functions differ from the first in that they are defined both when $x = 1$ and when $x = -1$: the second has the value $1$ and the third the value $-1$ for both these values of $x$.]", "answer_latex": [ "$\\phi_{n}(x) = (x^{n} - 1)/(x^{n} + 1)$, $(nx^{n} - 1)/(nx^{n} + 1)$, $(x^{n} - n)/(x^{n} + n)$. [In the\nfirst case $\\phi(x) = 1$ when $|x| > 1$, $\\phi(x) = -1$ when $|x| < 1$, $\\phi(x) = 0$ when $x = 1$\nand $\\phi(x)$~is not defined when $x = -1$. The second and third functions differ\nfrom the first in that they are defined both when $x = 1$ and when $x = -1$: the\nsecond has the value~$1$ and the third the value~$-1$ for both these values of~$x$.]" ], "answer_markdown": [ "$\\phi_{n}(x) = (x^{n} - 1)/(x^{n} + 1)$, $(nx^{n} - 1)/(nx^{n} + 1)$, $(x^{n} - n)/(x^{n} + n)$. [In the first case $\\phi(x) = 1$ when $|x| > 1$, $\\phi(x) = -1$ when $|x| < 1$, $\\phi(x) = 0$ when $x = 1$ and $\\phi(x)$ is not defined when $x = -1$. The second and third functions differ from the first in that they are defined both when $x = 1$ and when $x = -1$: the second has the value $1$ and the third the value $-1$ for both these values of $x$.]" ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxi/10b", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxi", "number": 10, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "148", "location": "Exercise XXXI, problem 10b", "problem_latex": "$\\phi_{n}(x) = (x^{n} - 1)/(x^{n} + 1)$, $(nx^{n} - 1)/(nx^{n} + 1)$, $(x^{n} - n)/(x^{n} + n)$. [In the\nfirst case $\\phi(x) = 1$ when $|x| > 1$, $\\phi(x) = -1$ when $|x| < 1$, $\\phi(x) = 0$ when $x = 1$\nand $\\phi(x)$~is not defined when $x = -1$. The second and third functions differ\nfrom the first in that they are defined both when $x = 1$ and when $x = -1$: the\nsecond has the value~$1$ and the third the value~$-1$ for both these values of~$x$.]", "markdown": "$\\phi_{n}(x) = (x^{n} - 1)/(x^{n} + 1)$, $(nx^{n} - 1)/(nx^{n} + 1)$, $(x^{n} - n)/(x^{n} + n)$. [In the first case $\\phi(x) = 1$ when $|x| > 1$, $\\phi(x) = -1$ when $|x| < 1$, $\\phi(x) = 0$ when $x = 1$ and $\\phi(x)$ is not defined when $x = -1$. The second and third functions differ from the first in that they are defined both when $x = 1$ and when $x = -1$: the second has the value $1$ and the third the value $-1$ for both these values of $x$.]", "answer_latex": [ "$\\phi_{n}(x) = (x^{n} - 1)/(x^{n} + 1)$, $(nx^{n} - 1)/(nx^{n} + 1)$, $(x^{n} - n)/(x^{n} + n)$. [In the\nfirst case $\\phi(x) = 1$ when $|x| > 1$, $\\phi(x) = -1$ when $|x| < 1$, $\\phi(x) = 0$ when $x = 1$\nand $\\phi(x)$~is not defined when $x = -1$. The second and third functions differ\nfrom the first in that they are defined both when $x = 1$ and when $x = -1$: the\nsecond has the value~$1$ and the third the value~$-1$ for both these values of~$x$.]" ], "answer_markdown": [ "$\\phi_{n}(x) = (x^{n} - 1)/(x^{n} + 1)$, $(nx^{n} - 1)/(nx^{n} + 1)$, $(x^{n} - n)/(x^{n} + n)$. [In the first case $\\phi(x) = 1$ when $|x| > 1$, $\\phi(x) = -1$ when $|x| < 1$, $\\phi(x) = 0$ when $x = 1$ and $\\phi(x)$ is not defined when $x = -1$. The second and third functions differ from the first in that they are defined both when $x = 1$ and when $x = -1$: the second has the value $1$ and the third the value $-1$ for both these values of $x$.]" ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxi/10c", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxi", "number": 10, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "148", "location": "Exercise XXXI, problem 10c", "problem_latex": "$\\phi_{n}(x) = (x^{n} - 1)/(x^{n} + 1)$, $(nx^{n} - 1)/(nx^{n} + 1)$, $(x^{n} - n)/(x^{n} + n)$. [In the\nfirst case $\\phi(x) = 1$ when $|x| > 1$, $\\phi(x) = -1$ when $|x| < 1$, $\\phi(x) = 0$ when $x = 1$\nand $\\phi(x)$~is not defined when $x = -1$. The second and third functions differ\nfrom the first in that they are defined both when $x = 1$ and when $x = -1$: the\nsecond has the value~$1$ and the third the value~$-1$ for both these values of~$x$.]", "markdown": "$\\phi_{n}(x) = (x^{n} - 1)/(x^{n} + 1)$, $(nx^{n} - 1)/(nx^{n} + 1)$, $(x^{n} - n)/(x^{n} + n)$. [In the first case $\\phi(x) = 1$ when $|x| > 1$, $\\phi(x) = -1$ when $|x| < 1$, $\\phi(x) = 0$ when $x = 1$ and $\\phi(x)$ is not defined when $x = -1$. The second and third functions differ from the first in that they are defined both when $x = 1$ and when $x = -1$: the second has the value $1$ and the third the value $-1$ for both these values of $x$.]", "answer_latex": [ "$\\phi_{n}(x) = (x^{n} - 1)/(x^{n} + 1)$, $(nx^{n} - 1)/(nx^{n} + 1)$, $(x^{n} - n)/(x^{n} + n)$. [In the\nfirst case $\\phi(x) = 1$ when $|x| > 1$, $\\phi(x) = -1$ when $|x| < 1$, $\\phi(x) = 0$ when $x = 1$\nand $\\phi(x)$~is not defined when $x = -1$. The second and third functions differ\nfrom the first in that they are defined both when $x = 1$ and when $x = -1$: the\nsecond has the value~$1$ and the third the value~$-1$ for both these values of~$x$.]" ], "answer_markdown": [ "$\\phi_{n}(x) = (x^{n} - 1)/(x^{n} + 1)$, $(nx^{n} - 1)/(nx^{n} + 1)$, $(x^{n} - n)/(x^{n} + n)$. [In the first case $\\phi(x) = 1$ when $|x| > 1$, $\\phi(x) = -1$ when $|x| < 1$, $\\phi(x) = 0$ when $x = 1$ and $\\phi(x)$ is not defined when $x = -1$. The second and third functions differ from the first in that they are defined both when $x = 1$ and when $x = -1$: the second has the value $1$ and the third the value $-1$ for both these values of $x$.]" ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxi/11", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxi", "number": 11, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "148", "location": "Exercise XXXI, problem 11", "problem_latex": "Construct an example in which $\\phi(x) = 1$, ($|x| > 1$); $\\phi(x) = -1$,\n($|x| < 1$); and $\\phi(x) = 0$, ($x = 1$ and $x = -1$).", "markdown": "Construct an example in which $\\phi(x) = 1$, ($|x| > 1$); $\\phi(x) = -1$, ($|x| < 1$); and $\\phi(x) = 0$, ($x = 1$ and $x = -1$).", "answer_latex": [ "Construct an example in which $\\phi(x) = 1$, ($|x| > 1$); $\\phi(x) = -1$,\n($|x| < 1$); and $\\phi(x) = 0$, ($x = 1$ and $x = -1$)." ], "answer_markdown": [ "Construct an example in which $\\phi(x) = 1$, ($|x| > 1$); $\\phi(x) = -1$, ($|x| < 1$); and $\\phi(x) = 0$, ($x = 1$ and $x = -1$)." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:construction" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxi/12a", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxi", "number": 12, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "148", "location": "Exercise XXXI, problem 12a", "problem_latex": "$\\phi_{n}(x) = x\\{(x^{2n} - 1)/(x^{2n} + 1)\\}^{2}$, $n/(x^{n} + x^{-n} + n)$.", "markdown": "$\\phi_{n}(x) = x\\{(x^{2n} - 1)/(x^{2n} + 1)\\}^{2}$, $n/(x^{n} + x^{-n} + n)$.", "answer_latex": [ "$\\phi_{n}(x) = x\\{(x^{2n} - 1)/(x^{2n} + 1)\\}^{2}$, $n/(x^{n} + x^{-n} + n)$." ], "answer_markdown": [ "$\\phi_{n}(x) = x\\{(x^{2n} - 1)/(x^{2n} + 1)\\}^{2}$, $n/(x^{n} + x^{-n} + n)$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxi/12b", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxi", "number": 12, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "148", "location": "Exercise XXXI, problem 12b", "problem_latex": "$\\phi_{n}(x) = x\\{(x^{2n} - 1)/(x^{2n} + 1)\\}^{2}$, $n/(x^{n} + x^{-n} + n)$.", "markdown": "$\\phi_{n}(x) = x\\{(x^{2n} - 1)/(x^{2n} + 1)\\}^{2}$, $n/(x^{n} + x^{-n} + n)$.", "answer_latex": [ "$\\phi_{n}(x) = x\\{(x^{2n} - 1)/(x^{2n} + 1)\\}^{2}$, $n/(x^{n} + x^{-n} + n)$." ], "answer_markdown": [ "$\\phi_{n}(x) = x\\{(x^{2n} - 1)/(x^{2n} + 1)\\}^{2}$, $n/(x^{n} + x^{-n} + n)$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxi/13", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxi", "number": 13, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "148", "location": "Exercise XXXI, problem 13", "problem_latex": "$\\phi_{n}(x) = \\{x^{n}f(x) + g(x)\\}/(x^{n} + 1)$. [Here $\\phi(x) = f(x)$, ($|x| > 1$);\n$\\phi(x) = g(x)$, ($|x| < 1$); $\\phi(x) = \\frac{1}{2}\\{f(x) + g(x)\\}$, ($x = 1$); and $\\phi(x)$~is undefined\nwhen $x = -1$.]", "markdown": "$\\phi_{n}(x) = \\{x^{n}f(x) + g(x)\\}/(x^{n} + 1)$. [Here $\\phi(x) = f(x)$, ($|x| > 1$); $\\phi(x) = g(x)$, ($|x| < 1$); $\\phi(x) = \\frac{1}{2}\\{f(x) + g(x)\\}$, ($x = 1$); and $\\phi(x)$ is undefined when $x = -1$.]", "answer_latex": [ "$\\phi_{n}(x) = \\{x^{n}f(x) + g(x)\\}/(x^{n} + 1)$. [Here $\\phi(x) = f(x)$, ($|x| > 1$);\n$\\phi(x) = g(x)$, ($|x| < 1$); $\\phi(x) = \\frac{1}{2}\\{f(x) + g(x)\\}$, ($x = 1$); and $\\phi(x)$~is undefined\nwhen $x = -1$.]" ], "answer_markdown": [ "$\\phi_{n}(x) = \\{x^{n}f(x) + g(x)\\}/(x^{n} + 1)$. [Here $\\phi(x) = f(x)$, ($|x| > 1$); $\\phi(x) = g(x)$, ($|x| < 1$); $\\phi(x) = \\frac{1}{2}\\{f(x) + g(x)\\}$, ($x = 1$); and $\\phi(x)$ is undefined when $x = -1$.]" ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxi/14", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxi", "number": 14, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "148", "location": "Exercise XXXI, problem 14", "problem_latex": "$\\phi_{n}(x) = (2/\\pi) \\arctan(nx)$. [$\\phi(x) = 1$, ($x > 0$); $\\phi(x) = 0$, ($x = 0$);\n$\\phi(x) = -1$, ($x < 0$). This function is important in the Theory of Numbers,\nand is usually denoted by~$\\sgn x$.]", "markdown": "$\\phi_{n}(x) = (2/\\pi) \\arctan(nx)$. [$\\phi(x) = 1$, ($x > 0$); $\\phi(x) = 0$, ($x = 0$); $\\phi(x) = -1$, ($x < 0$). This function is important in the Theory of Numbers, and is usually denoted by $\\sgn x$.]", "answer_latex": [ "$\\phi_{n}(x) = (2/\\pi) \\arctan(nx)$. [$\\phi(x) = 1$, ($x > 0$); $\\phi(x) = 0$, ($x = 0$);\n$\\phi(x) = -1$, ($x < 0$). This function is important in the Theory of Numbers,\nand is usually denoted by~$\\sgn x$.]" ], "answer_markdown": [ "$\\phi_{n}(x) = (2/\\pi) \\arctan(nx)$. [$\\phi(x) = 1$, ($x > 0$); $\\phi(x) = 0$, ($x = 0$); $\\phi(x) = -1$, ($x < 0$). This function is important in the Theory of Numbers, and is usually denoted by $\\sgn x$.]" ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxi/15", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxi", "number": 15, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "148", "location": "Exercise XXXI, problem 15", "problem_latex": "$\\phi_{n}(x) = \\sin nx\\pi$. [$\\phi(x) = 0$ when $x$~is an integer; and $\\phi(x)$~is\notherwise undefined (\\Ex{xxiv}.~7).]", "markdown": "$\\phi_{n}(x) = \\sin nx\\pi$. [$\\phi(x) = 0$ when $x$ is an integer; and $\\phi(x)$ is otherwise undefined (% [examples:xxiv]Ex. xxiv%. 7).]", "answer_latex": [ "$\\phi_{n}(x) = \\sin nx\\pi$. [$\\phi(x) = 0$ when $x$~is an integer; and $\\phi(x)$~is\notherwise undefined (\\Ex{xxiv}.~7).]" ], "answer_markdown": [ "$\\phi_{n}(x) = \\sin nx\\pi$. [$\\phi(x) = 0$ when $x$ is an integer; and $\\phi(x)$ is otherwise undefined (% [examples:xxiv]Ex. xxiv%. 7).]" ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxi/16", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxi", "number": 16, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "148", "location": "Exercise XXXI, problem 16", "problem_latex": "If $\\phi_{n}(x) = \\sin (n!\\, x\\pi)$ then $\\phi(x) = 0$ for all rational values of~$x$ (\\Ex{xxiv}.~14).\n[The consideration of irrational values presents greater difficulties.]", "markdown": "If $\\phi_{n}(x) = \\sin (n!\\, x\\pi)$ then $\\phi(x) = 0$ for all rational values of $x$ (% [examples:xxiv]Ex. xxiv%. 14). [The consideration of irrational values presents greater difficulties.]", "answer_latex": [ "If $\\phi_{n}(x) = \\sin (n!\\, x\\pi)$ then $\\phi(x) = 0$ for all rational values of~$x$ (\\Ex{xxiv}.~14).\n[The consideration of irrational values presents greater difficulties.]" ], "answer_markdown": [ "If $\\phi_{n}(x) = \\sin (n!\\, x\\pi)$ then $\\phi(x) = 0$ for all rational values of $x$ (% [examples:xxiv]Ex. xxiv%. 14). [The consideration of irrational values presents greater difficulties.]" ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxi/17", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxi", "number": 17, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "148", "location": "Exercise XXXI, problem 17", "problem_latex": "$\\phi_{n}(x) = (\\cos^{2} x\\pi)^{n}$. [$\\phi(x) = 0$ except when $x$~is integral, when\n$\\phi(x) = 1$.]", "markdown": "$\\phi_{n}(x) = (\\cos^{2} x\\pi)^{n}$. [$\\phi(x) = 0$ except when $x$ is integral, when $\\phi(x) = 1$.]", "answer_latex": [ "$\\phi_{n}(x) = (\\cos^{2} x\\pi)^{n}$. [$\\phi(x) = 0$ except when $x$~is integral, when\n$\\phi(x) = 1$.]" ], "answer_markdown": [ "$\\phi_{n}(x) = (\\cos^{2} x\\pi)^{n}$. [$\\phi(x) = 0$ except when $x$ is integral, when $\\phi(x) = 1$.]" ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxi/18", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxi", "number": 18, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "148", "location": "Exercise XXXI, problem 18", "problem_latex": "If $N \\geq 1752$ then the number of days in the year $N$~\\textsc{a.d.}\\ is\n\\[\n\\lim \\{365\n + (\\cos^{2} \\tfrac{1}{4} N\\pi)^{n} - (\\cos^{2} \\tfrac{1}{100} N\\pi)^{n}\n + (\\cos^{2} \\tfrac{1}{400} N\\pi)^{n}\\}.\n\\]", "markdown": "If $N \\geq 1752$ then the number of days in the year $N$ a.d. is 365 + (^2 14 N)^n - (^2 1100 N)^n + (^2 1400 N)^n.", "answer_latex": [ "If $N \\geq 1752$ then the number of days in the year $N$~\\textsc{a.d.}\\ is\n\\[\n\\lim \\{365\n + (\\cos^{2} \\tfrac{1}{4} N\\pi)^{n} - (\\cos^{2} \\tfrac{1}{100} N\\pi)^{n}\n + (\\cos^{2} \\tfrac{1}{400} N\\pi)^{n}\\}.\n\\]" ], "answer_markdown": [ "If $N \\geq 1752$ then the number of days in the year $N$ a.d. is 365 + (^2 14 N)^n - (^2 1100 N)^n + (^2 1400 N)^n." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxi/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxi", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "148", "location": "Exercise XXXI, problem 2", "problem_latex": "$\\phi_{n}(x) = x/n$. Here $\\phi(x) = \\lim\\phi_{n}(x) = 0$ for all values of~$x$.", "markdown": "$\\phi_{n}(x) = x/n$. Here $\\phi(x) = \\lim\\phi_{n}(x) = 0$ for all values of $x$.", "answer_latex": [ "$\\phi_{n}(x) = x/n$. Here $\\phi(x) = \\lim\\phi_{n}(x) = 0$ for all values of~$x$." ], "answer_markdown": [ "$\\phi_{n}(x) = x/n$. Here $\\phi(x) = \\lim\\phi_{n}(x) = 0$ for all values of $x$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxi/3", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxi", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "148", "location": "Exercise XXXI, problem 3", "problem_latex": "$\\phi_{n}(x) = nx$. If $x > 0$, $\\phi_{n}(x) \\to +\\infty$; if $x < 0$, $\\phi_{n}(x) \\to -\\infty$: only when\n$x = 0$ has $\\phi_{n}(x)$ a limit (viz.~$0$) as $n \\to \\infty$. Thus $\\phi(x) = 0$ when $x = 0$ and is\nnot defined for any other value of~$x$.", "markdown": "$\\phi_{n}(x) = nx$. If $x > 0$, $\\phi_{n}(x) \\to +\\infty$; if $x < 0$, $\\phi_{n}(x) \\to -\\infty$: only when $x = 0$ has $\\phi_{n}(x)$ a limit (viz. $0$) as $n \\to \\infty$. Thus $\\phi(x) = 0$ when $x = 0$ and is not defined for any other value of $x$.", "answer_latex": [ "$\\phi_{n}(x) = nx$. If $x > 0$, $\\phi_{n}(x) \\to +\\infty$; if $x < 0$, $\\phi_{n}(x) \\to -\\infty$: only when\n$x = 0$ has $\\phi_{n}(x)$ a limit (viz.~$0$) as $n \\to \\infty$. Thus $\\phi(x) = 0$ when $x = 0$ and is\nnot defined for any other value of~$x$." ], "answer_markdown": [ "$\\phi_{n}(x) = nx$. If $x > 0$, $\\phi_{n}(x) \\to +\\infty$; if $x < 0$, $\\phi_{n}(x) \\to -\\infty$: only when $x = 0$ has $\\phi_{n}(x)$ a limit (viz. $0$) as $n \\to \\infty$. Thus $\\phi(x) = 0$ when $x = 0$ and is not defined for any other value of $x$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxi/4a", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxi", "number": 4, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "148", "location": "Exercise XXXI, problem 4a", "problem_latex": "$\\phi_{n}(x) = 1/nx$, $nx/(nx + 1)$.", "markdown": "$\\phi_{n}(x) = 1/nx$, $nx/(nx + 1)$.", "answer_latex": [ "$\\phi_{n}(x) = 1/nx$, $nx/(nx + 1)$." ], "answer_markdown": [ "$\\phi_{n}(x) = 1/nx$, $nx/(nx + 1)$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxi/4b", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxi", "number": 4, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "148", "location": "Exercise XXXI, problem 4b", "problem_latex": "$\\phi_{n}(x) = 1/nx$, $nx/(nx + 1)$.", "markdown": "$\\phi_{n}(x) = 1/nx$, $nx/(nx + 1)$.", "answer_latex": [ "$\\phi_{n}(x) = 1/nx$, $nx/(nx + 1)$." ], "answer_markdown": [ "$\\phi_{n}(x) = 1/nx$, $nx/(nx + 1)$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxi/5", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxi", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "148", "location": "Exercise XXXI, problem 5", "problem_latex": "$\\phi_{n}(x) = x^{n}$. Here $\\phi(x) = 0$, ($-1 < x < 1$); $\\phi(x) = 1$, ($x = 1$); and $\\phi(x)$\nis not defined for any other value of~$x$.", "markdown": "$\\phi_{n}(x) = x^{n}$. Here $\\phi(x) = 0$, ($-1 < x < 1$); $\\phi(x) = 1$, ($x = 1$); and $\\phi(x)$ is not defined for any other value of $x$.", "answer_latex": [ "$\\phi_{n}(x) = x^{n}$. Here $\\phi(x) = 0$, ($-1 < x < 1$); $\\phi(x) = 1$, ($x = 1$); and $\\phi(x)$\nis not defined for any other value of~$x$." ], "answer_markdown": [ "$\\phi_{n}(x) = x^{n}$. Here $\\phi(x) = 0$, ($-1 < x < 1$); $\\phi(x) = 1$, ($x = 1$); and $\\phi(x)$ is not defined for any other value of $x$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxi/6", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxi", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "148", "location": "Exercise XXXI, problem 6", "problem_latex": "$\\phi_{n}(x) = x^{n}(1 - x)$. Here $\\phi(x)$~differs from the $\\phi(x)$ of Ex.~5 in that\nit has the value~$0$ when $x = 1$.", "markdown": "$\\phi_{n}(x) = x^{n}(1 - x)$. Here $\\phi(x)$ differs from the $\\phi(x)$ of Ex. 5 in that it has the value $0$ when $x = 1$.", "answer_latex": [ "$\\phi_{n}(x) = x^{n}(1 - x)$. Here $\\phi(x)$~differs from the $\\phi(x)$ of Ex.~5 in that\nit has the value~$0$ when $x = 1$." ], "answer_markdown": [ "$\\phi_{n}(x) = x^{n}(1 - x)$. Here $\\phi(x)$ differs from the $\\phi(x)$ of Ex. 5 in that it has the value $0$ when $x = 1$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxi/7", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxi", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "148", "location": "Exercise XXXI, problem 7", "problem_latex": "$\\phi_{n}(x) = x^{n}/n$. Here $\\phi(x)$ differs from the $\\phi(x)$ of Ex.~6 in that it has\nthe value~$0$ when $x = -1$ as well as when $x = 1$.", "markdown": "$\\phi_{n}(x) = x^{n}/n$. Here $\\phi(x)$ differs from the $\\phi(x)$ of Ex. 6 in that it has the value $0$ when $x = -1$ as well as when $x = 1$.", "answer_latex": [ "$\\phi_{n}(x) = x^{n}/n$. Here $\\phi(x)$ differs from the $\\phi(x)$ of Ex.~6 in that it has\nthe value~$0$ when $x = -1$ as well as when $x = 1$." ], "answer_markdown": [ "$\\phi_{n}(x) = x^{n}/n$. Here $\\phi(x)$ differs from the $\\phi(x)$ of Ex. 6 in that it has the value $0$ when $x = -1$ as well as when $x = 1$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxi/8", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxi", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "148", "location": "Exercise XXXI, problem 8", "problem_latex": "$\\phi_{n}(x) = x^{n}/(x^{n} + 1)$. [$\\phi(x) = 0$, ($-1 < x < 1$); $\\phi(x) = \\frac{1}{2}$, ($x = 1$); $\\phi(x) = 1$,\n($x < -1$ or $x > 1$); and $\\phi(x)$~is not defined when $x = -1$.]", "markdown": "$\\phi_{n}(x) = x^{n}/(x^{n} + 1)$. [$\\phi(x) = 0$, ($-1 < x < 1$); $\\phi(x) = \\frac{1}{2}$, ($x = 1$); $\\phi(x) = 1$, ($x < -1$ or $x > 1$); and $\\phi(x)$ is not defined when $x = -1$.]", "answer_latex": [ "$\\phi_{n}(x) = x^{n}/(x^{n} + 1)$. [$\\phi(x) = 0$, ($-1 < x < 1$); $\\phi(x) = \\frac{1}{2}$, ($x = 1$); $\\phi(x) = 1$,\n($x < -1$ or $x > 1$); and $\\phi(x)$~is not defined when $x = -1$.]" ], "answer_markdown": [ "$\\phi_{n}(x) = x^{n}/(x^{n} + 1)$. [$\\phi(x) = 0$, ($-1 < x < 1$); $\\phi(x) = \\frac{1}{2}$, ($x = 1$); $\\phi(x) = 1$, ($x < -1$ or $x > 1$); and $\\phi(x)$ is not defined when $x = -1$.]" ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxi/9a", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxi", "number": 9, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "148", "location": "Exercise XXXI, problem 9a", "problem_latex": "$\\phi_{n}(x) = x^{n}/(x^{n} - 1)$, $1/(x^{n} + 1)$, $1/(x^{n} - 1)$, $1/(x^{n} + x^{-n})$, $1/(x^{n} - x^{-n})$.", "markdown": "$\\phi_{n}(x) = x^{n}/(x^{n} - 1)$, $1/(x^{n} + 1)$, $1/(x^{n} - 1)$, $1/(x^{n} + x^{-n})$, $1/(x^{n} - x^{-n})$.", "answer_latex": [ "$\\phi_{n}(x) = x^{n}/(x^{n} - 1)$, $1/(x^{n} + 1)$, $1/(x^{n} - 1)$, $1/(x^{n} + x^{-n})$, $1/(x^{n} - x^{-n})$." ], "answer_markdown": [ "$\\phi_{n}(x) = x^{n}/(x^{n} - 1)$, $1/(x^{n} + 1)$, $1/(x^{n} - 1)$, $1/(x^{n} + x^{-n})$, $1/(x^{n} - x^{-n})$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxi/9b", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxi", "number": 9, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "148", "location": "Exercise XXXI, problem 9b", "problem_latex": "$\\phi_{n}(x) = x^{n}/(x^{n} - 1)$, $1/(x^{n} + 1)$, $1/(x^{n} - 1)$, $1/(x^{n} + x^{-n})$, $1/(x^{n} - x^{-n})$.", "markdown": "$\\phi_{n}(x) = x^{n}/(x^{n} - 1)$, $1/(x^{n} + 1)$, $1/(x^{n} - 1)$, $1/(x^{n} + x^{-n})$, $1/(x^{n} - x^{-n})$.", "answer_latex": [ "$\\phi_{n}(x) = x^{n}/(x^{n} - 1)$, $1/(x^{n} + 1)$, $1/(x^{n} - 1)$, $1/(x^{n} + x^{-n})$, $1/(x^{n} - x^{-n})$." ], "answer_markdown": [ "$\\phi_{n}(x) = x^{n}/(x^{n} - 1)$, $1/(x^{n} + 1)$, $1/(x^{n} - 1)$, $1/(x^{n} + x^{-n})$, $1/(x^{n} - x^{-n})$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxi/9c", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxi", "number": 9, "part": "c", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "148", "location": "Exercise XXXI, problem 9c", "problem_latex": "$\\phi_{n}(x) = x^{n}/(x^{n} - 1)$, $1/(x^{n} + 1)$, $1/(x^{n} - 1)$, $1/(x^{n} + x^{-n})$, $1/(x^{n} - x^{-n})$.", "markdown": "$\\phi_{n}(x) = x^{n}/(x^{n} - 1)$, $1/(x^{n} + 1)$, $1/(x^{n} - 1)$, $1/(x^{n} + x^{-n})$, $1/(x^{n} - x^{-n})$.", "answer_latex": [ "$\\phi_{n}(x) = x^{n}/(x^{n} - 1)$, $1/(x^{n} + 1)$, $1/(x^{n} - 1)$, $1/(x^{n} + x^{-n})$, $1/(x^{n} - x^{-n})$." ], "answer_markdown": [ "$\\phi_{n}(x) = x^{n}/(x^{n} - 1)$, $1/(x^{n} + 1)$, $1/(x^{n} - 1)$, $1/(x^{n} + x^{-n})$, $1/(x^{n} - x^{-n})$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxi/9d", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxi", "number": 9, "part": "d", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "148", "location": "Exercise XXXI, problem 9d", "problem_latex": "$\\phi_{n}(x) = x^{n}/(x^{n} - 1)$, $1/(x^{n} + 1)$, $1/(x^{n} - 1)$, $1/(x^{n} + x^{-n})$, $1/(x^{n} - x^{-n})$.", "markdown": "$\\phi_{n}(x) = x^{n}/(x^{n} - 1)$, $1/(x^{n} + 1)$, $1/(x^{n} - 1)$, $1/(x^{n} + x^{-n})$, $1/(x^{n} - x^{-n})$.", "answer_latex": [ "$\\phi_{n}(x) = x^{n}/(x^{n} - 1)$, $1/(x^{n} + 1)$, $1/(x^{n} - 1)$, $1/(x^{n} + x^{-n})$, $1/(x^{n} - x^{-n})$." ], "answer_markdown": [ "$\\phi_{n}(x) = x^{n}/(x^{n} - 1)$, $1/(x^{n} + 1)$, $1/(x^{n} - 1)$, $1/(x^{n} + x^{-n})$, $1/(x^{n} - x^{-n})$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxi/9e", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxi", "number": 9, "part": "e", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "148", "location": "Exercise XXXI, problem 9e", "problem_latex": "$\\phi_{n}(x) = x^{n}/(x^{n} - 1)$, $1/(x^{n} + 1)$, $1/(x^{n} - 1)$, $1/(x^{n} + x^{-n})$, $1/(x^{n} - x^{-n})$.", "markdown": "$\\phi_{n}(x) = x^{n}/(x^{n} - 1)$, $1/(x^{n} + 1)$, $1/(x^{n} - 1)$, $1/(x^{n} + x^{-n})$, $1/(x^{n} - x^{-n})$.", "answer_latex": [ "$\\phi_{n}(x) = x^{n}/(x^{n} - 1)$, $1/(x^{n} + 1)$, $1/(x^{n} - 1)$, $1/(x^{n} + x^{-n})$, $1/(x^{n} - x^{-n})$." ], "answer_markdown": [ "$\\phi_{n}(x) = x^{n}/(x^{n} - 1)$, $1/(x^{n} + 1)$, $1/(x^{n} - 1)$, $1/(x^{n} + x^{-n})$, $1/(x^{n} - x^{-n})$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxii/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxii", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "151", "location": "Exercise XXXII, problem 1", "problem_latex": "Neither $\\Lambda$~nor~$\\lambda$ is affected by any alteration in\nany finite number of values of~$\\phi(n)$.", "markdown": "Neither $\\Lambda$ nor $\\lambda$ is affected by any alteration in any finite number of values of $\\phi(n)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:limit_points_of_sequence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxii/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxii", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "151", "location": "Exercise XXXII, problem 2", "problem_latex": "If $\\phi(n) = a$ for all values of~$n$, then $m = \\lambda = \\Lambda = M = a$.", "markdown": "If $\\phi(n) = a$ for all values of $n$, then $m = \\lambda = \\Lambda = M = a$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:limit_points_of_sequence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxii/3", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxii", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "151", "location": "Exercise XXXII, problem 3", "problem_latex": "If $\\phi(n) = 1/n$, then $m = \\lambda = \\Lambda = 0$ and $M = 1$.", "markdown": "If $\\phi(n) = 1/n$, then $m = \\lambda = \\Lambda = 0$ and $M = 1$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:limit_points_of_sequence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxii/4", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxii", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "151", "location": "Exercise XXXII, problem 4", "problem_latex": "If $\\phi(n) = (-1)^{n}$, then $m = \\lambda = -1$ and $\\Lambda = M = 1$.", "markdown": "If $\\phi(n) = (-1)^{n}$, then $m = \\lambda = -1$ and $\\Lambda = M = 1$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:limit_points_of_sequence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxii/5", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxii", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "151", "location": "Exercise XXXII, problem 5", "problem_latex": "If $\\phi(n) = (-1)^{n}/n$, then $m = -1$, $\\lambda = \\Lambda = 0$, $M = \\frac{1}{2}$.", "markdown": "If $\\phi(n) = (-1)^{n}/n$, then $m = -1$, $\\lambda = \\Lambda = 0$, $M = \\frac{1}{2}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:limit_points_of_sequence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxii/6", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxii", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "151", "location": "Exercise XXXII, problem 6", "problem_latex": "If $\\phi(n) = (-1)^{n}\\{1 + (1/n)\\}$, then $m = -2$, $\\lambda = -1$, $\\Lambda = 1$, $M = \\frac{3}{2}$.", "markdown": "If $\\phi(n) = (-1)^{n}\\{1 + (1/n)\\}$, then $m = -2$, $\\lambda = -1$, $\\Lambda = 1$, $M = \\frac{3}{2}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:limit_points_of_sequence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxii/7", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxii", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "151", "location": "Exercise XXXII, problem 7", "problem_latex": "Let $\\phi(n) = \\sin n\\theta\\pi$, where $\\theta > 0$. If $\\theta$~is an integer then $m = \\lambda = \\Lambda = M = 0$.\nIf $\\theta$~is rational but not integral a variety of cases arise. Suppose, \\eg, that\n$\\theta = p/q$, $p$~and~$q$ being positive, odd, and prime to one another, and $q > 1$.\nThen $\\phi(n)$~assumes the cyclical sequence of values\n\\[\n\\sin(p\\pi/q),\\quad\n\\sin(2p\\pi/q),\\ \\dots,\\quad\n\\sin\\{(2q - 1)p\\pi/q\\},\\quad\n\\sin(2qp\\pi/q),\\ \\dots.\n\\]\nIt is easily verified that the numerically greatest and least values of~$\\phi(n)$ are\n$\\cos(\\pi/2q)$ and $-\\cos(\\pi/2q)$, so that\n\\[\nm = \\lambda = -\\cos(\\pi/2q),\\quad\n\\Lambda = M = \\cos(\\pi/2q).\n\\]\nThe reader may discuss similarly the cases which arise when $p$~and~$q$ are\nnot both odd.\n\nThe case in which $\\theta$~is irrational is more difficult: it may be shown that\nin this case $m = \\lambda = -1$ and $\\Lambda = M = 1$. It may also be shown that the values\nof~$\\phi(n)$ are scattered all over the interval $\\DPmod{(-1, 1)}{[-1, 1]}$ in such a way that, if $\\xi$~is\n\\PageSep{152}\n\\emph{any} number of the interval, then there is a sequence $n_{1}$, $n_{2}$,~\\dots\\ such that\n$\\phi(n_{k}) \\to \\xi$ as $k \\to \\infty$.\\footnote\n {A number of simple proofs of this result are given by Hardy and Littlewood,\n ``Some Problems of Diophantine Approximation'', \\textit{Acta Mathematica}, vol.~xxxvii.}\n\nThe results are very similar when $\\phi(n)$~is the fractional part of~$n\\theta$.", "markdown": "Let $\\phi(n) = \\sin n\\theta\\pi$, where $\\theta > 0$. If $\\theta$ is an integer then $m = \\lambda = \\Lambda = M = 0$. If $\\theta$ is rational but not integral a variety of cases arise. Suppose, *e.g.*, that $\\theta = p/q$, $p$ and $q$ being positive, odd, and prime to one another, and $q > 1$. Then $\\phi(n)$ assumes the cyclical sequence of values (p/q),0pt minus 3pt(2p/q), …,0pt minus 3pt(2q - 1)p/q,0pt minus 3pt(2qp/q), …. It is easily verified that the numerically greatest and least values of $\\phi(n)$ are $\\cos(\\pi/2q)$ and $-\\cos(\\pi/2q)$, so that m = = -(/2q),0pt minus 3pt= M = (/2q). The reader may discuss similarly the cases which arise when $p$ and $q$ are not both odd. The case in which $\\theta$ is irrational is more difficult: it may be shown that in this case $m = \\lambda = -1$ and $\\Lambda = M = 1$. It may also be shown that the values of $\\phi(n)$ are scattered all over the interval $\\DPmod{(-1, 1)}{[-1, 1]}$ in such a way that, if $\\xi$ is [pg]152 *any* number of the interval, then there is a sequence $n_{1}$, $n_{2}$, … such that $\\phi(n_{k}) \\to \\xi$ as $k \\to \\infty$. A number of simple proofs of this result are given by Hardy and Littlewood, “Some Problems of Diophantine Approximation”, *Acta Mathematica*, vol. xxxvii. The results are very similar when $\\phi(n)$ is the fractional part of $n\\theta$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.const", "core.trig", "other:limit_points_of_sequence" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxiii/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxiii", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "157", "location": "Exercise XXXIII, problem 1", "problem_latex": "Prove directly that $\\phi(n) = r^{n} \\cos n\\theta$ converges\nto~$0$ when $r < 1$ and to~$1$ when $r = 1$ and $\\theta$~is a multiple of~$2\\pi$. Prove further\nthat if $r = 1$ and $\\theta$~is not a multiple of~$2\\pi$, then $\\phi(n)$~oscillates finitely; if\n$r > 1$ and $\\theta$~is a multiple of~$2\\pi$, then $\\phi(n) \\to +\\infty$; and if $r > 1$ and $\\theta$~is not a\nmultiple of~$2\\pi$, then $\\phi(n)$~oscillates infinitely.", "markdown": "Prove directly that $\\phi(n) = r^{n} \\cos n\\theta$ converges to $0$ when $r < 1$ and to $1$ when $r = 1$ and $\\theta$ is a multiple of $2\\pi$. Prove further that if $r = 1$ and $\\theta$ is not a multiple of $2\\pi$, then $\\phi(n)$ oscillates finitely; if $r > 1$ and $\\theta$ is a multiple of $2\\pi$, then $\\phi(n) \\to +\\infty$; and if $r > 1$ and $\\theta$ is not a multiple of $2\\pi$, then $\\phi(n)$ oscillates infinitely.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxiii/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxiii", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "157", "location": "Exercise XXXIII, problem 2", "problem_latex": "Establish a similar series of results for $\\phi(n) = r^{n} \\sin n\\theta$.", "markdown": "Establish a similar series of results for $\\phi(n) = r^{n} \\sin n\\theta$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxiii/3", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxiii", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "157", "location": "Exercise XXXIII, problem 3", "problem_latex": "Prove that\n\\begin{gather*}\nz^{m} + z^{m+1} + \\dots = z^{m}/(1 - z),\\\\\nz^{m} + 2z^{m+1} + 2z^{m+2} + \\dots = z^{m}(1 + z)/(1 - z),\n\\end{gather*}\nif and only if $|z| < 1$. Which of the theorems of \\SecNo[§]{86} do you use?", "markdown": "Prove that gather* z^m + z^m+1 + …= z^m/(1 - z), z^m + 2z^m+1 + 2z^m+2 + …= z^m(1 + z)/(1 - z), gather* if and only if $|z| < 1$. Which of the theorems of [§]86 do you use?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "cas.sum" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxiii/4", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxiii", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "157", "location": "Exercise XXXIII, problem 4", "problem_latex": "Prove that if $-1 < r < 1$ then\n\\[\n1 + 2r\\cos\\theta + 2r^{2}\\cos 2\\theta + \\dots\n = (1 - r^{2})/(1 - 2r\\cos\\theta + r^{2}).\n\\]", "markdown": "Prove that if $-1 < r < 1$ then 1 + 2r+ 2r^22+ … = (1 - r^2)/(1 - 2r+ r^2).", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.sum", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxiii/5", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxiii", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "157", "location": "Exercise XXXIII, problem 5", "problem_latex": "The series\n\\[\n1 + \\frac{z}{1 + z} + \\left(\\frac{z}{1 + z}\\right)^{2} + \\dots\n\\]\nconverges to the sum $1\\bigg/\\left(1 - \\dfrac{z}{1 + z}\\right) = 1 + z$ if $|z/(1 + z) | < 1$. Show that this\ncondition is equivalent to the condition that $z$~has a real part greater than~$-\\frac{1}{2}$.", "markdown": "The series 1 + z1 + z + (z1 + z)^2 + … converges to the sum $1\\bigg/\\left(1 - \\dfrac{z}{1 + z}\\right) = 1 + z$ if $|z/(1 + z) | < 1$. Show that this condition is equivalent to the condition that $z$ has a real part greater than $-\\frac{1}{2}$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.simplify", "core.complex" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxiv/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxiv", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "164", "location": "Exercise XXXIV, problem 1", "problem_latex": "Consider the behaviour of the following functions\nas $x \\to \\infty$: $1/x$, $1 + (1/x)$, $x^{2}$, $x^{k}$, $[x]$, $x - [x]$, $[x] + \\sqrtb{x - [x]}$.\n\nThe first four functions correspond exactly to functions of~$n$ fully discussed\nin \\okrickRef{Ch.}{IV}\\@. The graphs of the last three were constructed in \\okrickRef{Ch.}{II}\n(\\Exs{xvi}.\\ 1,~2,~4), and the reader will see at once that $[x] \\to \\infty$, $x - [x]$ oscillates\nfinitely, and $[x] + \\sqrtb{x - [x]} \\to \\infty$.\n\nOne simple remark may be inserted here. The function $\\phi(x) = x - [x]$\noscillates between $0$~and~$1$, as is obvious from the form of its graph. It is\nequal to zero whenever $x$~is an integer, so that the function~$\\phi(n)$ derived\nfrom it is always zero and so tends to the limit zero. The same is true if\n\\[\n\\phi(x) = \\sin x\\pi,\\quad\n\\phi(n) = \\sin n\\pi = 0.\n\\]\nIt is evident that $\\phi(x) \\to l$ or $\\phi(x) \\to \\infty$ or $\\phi(x) \\to -\\infty$ involves the corresponding\nproperty for~$\\phi(n)$, but that the converse is by no means always\ntrue.", "markdown": "Consider the behaviour of the following functions as $x \\to \\infty$: $1/x$, $1 + (1/x)$, $x^{2}$, $x^{k}$, $[x]$, $x - [x]$, $[x] + \\sqrtb{x - [x]}$. The first four functions correspond exactly to functions of $n$ fully discussed in Ch.IV. The graphs of the last three were constructed in Ch.II (xvi. 1, 2, 4), and the reader will see at once that $[x] \\to \\infty$, $x - [x]$ oscillates finitely, and $[x] + \\sqrtb{x - [x]} \\to \\infty$. One simple remark may be inserted here. The function $\\phi(x) = x - [x]$ oscillates between $0$ and $1$, as is obvious from the form of its graph. It is equal to zero whenever $x$ is an integer, so that the function $\\phi(n)$ derived from it is always zero and so tends to the limit zero. The same is true if (x) = x,0pt minus 3pt(n) = n= 0. It is evident that $\\phi(x) \\to l$ or $\\phi(x) \\to \\infty$ or $\\phi(x) \\to -\\infty$ involves the corresponding property for $\\phi(n)$, but that the converse is by no means always true.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxiv/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxiv", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "164", "location": "Exercise XXXIV, problem 2", "problem_latex": "Consider in the same way the functions:\n\\[\n(\\sin x\\pi)/x,\\quad\nx\\sin x\\pi,\\quad\n(x\\sin x\\pi)^{2},\\quad\n\\tan x\\pi,\\quad\na\\cos^{2} x\\pi + b\\sin^{2} x\\pi,\n\\]\nillustrating your remarks by means of the graphs of the functions.", "markdown": "Consider in the same way the functions: (x)/x,0pt minus 3ptxx,0pt minus 3pt(xx)^2,0pt minus 3ptx,0pt minus 3pta^2 x+ b^2 x, illustrating your remarks by means of the graphs of the functions.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxiv/3", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxiv", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "164", "location": "Exercise XXXIV, problem 3", "problem_latex": "Give a geometrical explanation of Def.~1, analogous to the geometrical\nexplanation of \\okrickRef{Ch.}{IV}, \\SecNo[§]{59}.", "markdown": "Give a geometrical explanation of Def. 1, analogous to the geometrical explanation of Ch.IV, [§]59.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxiv/4", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxiv", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "164", "location": "Exercise XXXIV, problem 4", "problem_latex": "If $\\phi(x) \\to l$, and $l$~is not zero, then $\\phi(x)\\cos x\\pi$ and $\\phi(x)\\sin x\\pi$ oscillate\nfinitely. If $\\phi(x) \\to \\infty$ or $\\phi(x) \\to -\\infty$, then they oscillate infinitely. The\ngraph of either function is a wavy curve oscillating between the curves\n$y = \\phi(x)$ and $y = -\\phi(x)$.", "markdown": "If $\\phi(x) \\to l$, and $l$ is not zero, then $\\phi(x)\\cos x\\pi$ and $\\phi(x)\\sin x\\pi$ oscillate finitely. If $\\phi(x) \\to \\infty$ or $\\phi(x) \\to -\\infty$, then they oscillate infinitely. The graph of either function is a wavy curve oscillating between the curves $y = \\phi(x)$ and $y = -\\phi(x)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxiv/5", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxiv", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "164", "location": "Exercise XXXIV, problem 5", "problem_latex": "Discuss the behaviour, as $x \\to \\infty$, of the function\n\\[\ny = f(x)\\cos^{2} x\\pi + F(x)\\sin^{2} x\\pi,\n\\]\nwhere $f(x)$~and~$F(x)$ are some pair of simple functions (\\eg\\ $x$~and~$x^{2}$). [The\ngraph of~$y$ is a curve oscillating between the curves $y = f(x)$, $y = F(x)$.]", "markdown": "Discuss the behaviour, as $x \\to \\infty$, of the function y = f(x)^2 x+ F(x)^2 x, where $f(x)$ and $F(x)$ are some pair of simple functions (*e.g.* $x$ and $x^{2}$). [The graph of $y$ is a curve oscillating between the curves $y = f(x)$, $y = F(x)$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxix/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxix", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "201", "location": "Exercise XXXIX, problem 1", "problem_latex": "If $\\phi(x)$~is a constant then $\\phi'(x) = 0$. Interpret\nthis result geometrically.", "markdown": "If $\\phi(x)$ is a constant then $\\phi'(x) = 0$. Interpret this result geometrically.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxix/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxix", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "201", "location": "Exercise XXXIX, problem 2", "problem_latex": "If $\\phi(x) = ax + b$ then $\\phi'(x) = a$. Prove this (i)~from the formal definition\nand (ii)~by geometrical considerations.", "markdown": "If $\\phi(x) = ax + b$ then $\\phi'(x) = a$. Prove this (i) from the formal definition and (ii) by geometrical considerations.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxix/3", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxix", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "201", "location": "Exercise XXXIX, problem 3", "problem_latex": "If $\\phi(x) = x^{m}$, where $m$~is a positive integer, then $\\phi'(x) = mx^{m-1}$.\n\n[For\n\\begin{align*}\n\\phi'(x) &= \\lim \\frac{(x + h)^{m} - x^{m}}{h}\\\\\n &= \\lim \\left\\{mx^{m-1} + \\frac{m(m - 1)}{1·2} x^{m-2} h + \\dots + h^{m-1}\\right\\}.\n\\end{align*}\n\nThe reader should observe that this method cannot be applied to~$x^{p/q}$,\nwhere $p/q$~is a rational fraction, as we have no means of expressing $(x + h)^{p/q}$\nas a finite series of powers of~$h$. We shall show later on (\\SecNo[§]{118}) that the result\nof this example holds for all rational values of~$m$. Meanwhile the reader\nwill find it instructive to determine $\\phi'(x)$ when $m$~has some special fractional\nvalue (\\eg~$\\frac{1}{2}$), by means of some special device.]", "markdown": "If $\\phi(x) = x^{m}$, where $m$ is a positive integer, then $\\phi'(x) = mx^{m-1}$. [For align* ’(x) &= (x + h)^m - x^mh &= mx^m-1 + m(m - 1)1·2 x^m-2 h + …+ h^m-1. align* The reader should observe that this method cannot be applied to $x^{p/q}$, where $p/q$ is a rational fraction, as we have no means of expressing $(x + h)^{p/q}$ as a finite series of powers of $h$. We shall show later on ([§]118) that the result of this example holds for all rational values of $m$. Meanwhile the reader will find it instructive to determine $\\phi'(x)$ when $m$ has some special fractional value (*e.g.* $\\frac{1}{2}$), by means of some special device.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.expand", "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxix/4", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxix", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "201", "location": "Exercise XXXIX, problem 4", "problem_latex": "{\\Loosen If $\\phi(x) = \\sin x$, then $\\phi'(x) = \\cos x$; and if $\\phi(x) = \\cos x$, then\n$\\phi'(x) = -\\sin x$.}\n\n[For example, if $\\phi(x) = \\sin x$, we have\n\\[\n\\{\\phi(x + h) - \\phi(x)\\}/h\n = \\{2\\sin \\tfrac{1}{2}h \\cos(x + \\tfrac{1}{2}h)\\}/h,\n\\]\nthe limit of which, when $h \\to 0$, is $\\cos x$, since $\\lim\\cos(x + \\frac{1}{2}h) = \\cos x$ (the cosine\nbeing a continuous function) and $\\lim\\{(\\sin \\frac{1}{2}h)/\\frac{1}{2}h\\} = 1$ (\\Ex{xxxvi}.~13).]", "markdown": "0.375em plus 0.75em minus 0.25emIf $\\phi(x) = \\sin x$, then $\\phi'(x) = \\cos x$; and if $\\phi(x) = \\cos x$, then $\\phi'(x) = -\\sin x$. [For example, if $\\phi(x) = \\sin x$, we have (x + h) - (x)/h = 212h (x + 12h)/h, the limit of which, when $h \\to 0$, is $\\cos x$, since $\\lim\\cos(x + \\frac{1}{2}h) = \\cos x$ (the cosine being a continuous function) and $\\lim\\{(\\sin \\frac{1}{2}h)/\\frac{1}{2}h\\} = 1$ (% [examples:xxxvi]Ex. xxxvi%. 13).]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxix/5", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxix", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "201", "location": "Exercise XXXIX, problem 5", "problem_latex": "\\Topic{Equations of the tangent and normal to a curve $y = \\phi(x)$.} The\ntangent to the curve at the point $(x_{0}, y_{0})$ is the line through $(x_{0}, y_{0})$ which\nmakes with~$OX$ an angle~$\\psi$, where $\\tan\\psi = \\phi'(x_{0})$. Its equation is therefore\n\\[\ny - y_{0} = (x - x_{0}) \\phi'(x_{0});\n\\]\nand the equation of the normal (the perpendicular to the tangent at the\npoint of contact) is\n\\[\n(y - y_{0}) \\phi'(x_{0}) + x - x_{0} = 0.\n\\]\nWe have assumed that the tangent is not parallel to the axis of~$y$. In\nthis special case it is obvious that the tangent and normal are $x = x_{0}$ and\n$y = y_{0}$ respectively.", "markdown": "**of the tangent and normal to a curve $y = \\phi(x)$.** The tangent to the curve at the point $(x_{0}, y_{0})$ is the line through $(x_{0}, y_{0})$ which makes with $OX$ an angle $\\psi$, where $\\tan\\psi = \\phi'(x_{0})$. Its equation is therefore y - y_0 = (x - x_0) ’(x_0); and the equation of the normal (the perpendicular to the tangent at the point of contact) is (y - y_0) ’(x_0) + x - x_0 = 0. We have assumed that the tangent is not parallel to the axis of $y$. In this special case it is obvious that the tangent and normal are $x = x_{0}$ and $y = y_{0}$ respectively.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxix/6", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxix", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "201", "location": "Exercise XXXIX, problem 6", "problem_latex": "Write down the equations of the tangent and normal at any point of\nthe parabola $x^{2} = 4ay$. Show that if $x_{0} = 2a/m$, $y_{0} = a/m^{2}$, then the tangent\nat $(x_{0}, y_{0})$ is $x = my + (a/m)$.", "markdown": "Write down the equations of the tangent and normal at any point of the parabola $x^{2} = 4ay$. Show that if $x_{0} = 2a/m$, $y_{0} = a/m^{2}$, then the tangent at $(x_{0}, y_{0})$ is $x = my + (a/m)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.derive", "cas.simplify", "cas.subst" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxv/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxv", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "168", "location": "Exercise XXXV, problem 1", "problem_latex": "If\n\\[\n\\phi(x) \\to l,\\quad\n\\psi(x) \\to l',\n\\]\nas $x \\to a$, then $\\phi(x) + \\psi(x) \\to l + l'$, $\\phi(x)\\psi(x) \\to ll'$, and $\\phi(x)/\\psi(x) \\to l/l'$,\nunless in the last case $l' = 0$.\n\n[We saw in \\SecNo[§]{91} that the theorems of \\okrickRef{Ch.}{IV}, \\SecNo[§§]{63}~\\textit{et~seq.}\\ hold also for\nfunctions of~$x$ when $x \\to \\infty$ or $x \\to -\\infty$. By putting $x = 1/y$ we may extend\nthem to functions of~$y$, when $y \\to 0$, and by putting $y = z - a$ to functions of~$z$,\nwhen $z \\to a$.\n\\PageSep{169}\n\nThe reader should however try to prove them directly from the formal\ndefinition given above. Thus, in order to obtain a strict direct proof of the\nfirst result he need only take the proof of Theorem~I of \\SecNo[§]{63} and write\nthroughout $x$~for~$n$, $a$~for~$\\infty$ and $0 < |x - a| \\leq \\EPSILON$ for $n \\geq n_{0}$.]", "markdown": "If (x) l,0pt minus 3pt(x) l’, as $x \\to a$, then $\\phi(x) + \\psi(x) \\to l + l'$, $\\phi(x)\\psi(x) \\to ll'$, and $\\phi(x)/\\psi(x) \\to l/l'$, unless in the last case $l' = 0$. [We saw in [§]91 that the theorems of Ch.IV, [§§]63 *et seq.* hold also for functions of $x$ when $x \\to \\infty$ or $x \\to -\\infty$. By putting $x = 1/y$ we may extend them to functions of $y$, when $y \\to 0$, and by putting $y = z - a$ to functions of $z$, when $z \\to a$. [pg]169 The reader should however try to prove them directly from the formal definition given above. Thus, in order to obtain a strict direct proof of the first result he need only take the proof of Theorem I of [§]63 and write throughout $x$ for $n$, $a$ for $\\infty$ and $0 < |x - a| \\leq \\EPSILON$ for $n \\geq n_{0}$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxv/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxv", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "168", "location": "Exercise XXXV, problem 2", "problem_latex": "If $m$~is a positive integer then $x^{m} \\to 0$ as $x \\to 0$.", "markdown": "If $m$ is a positive integer then $x^{m} \\to 0$ as $x \\to 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxv/3", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxv", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "168", "location": "Exercise XXXV, problem 3", "problem_latex": "If $m$~is a negative integer then $x^{m} \\to +\\infty$ as $x \\to +0$, while $x^{m} \\to -\\infty$ or\n$x^{m} \\to +\\infty$ as $x \\to -0$, according as $m$~is odd or even. If $m = 0$ then $x^{m} = 1$\nand $x^{m} \\to 1$.", "markdown": "If $m$ is a negative integer then $x^{m} \\to +\\infty$ as $x \\to +0$, while $x^{m} \\to -\\infty$ or $x^{m} \\to +\\infty$ as $x \\to -0$, according as $m$ is odd or even. If $m = 0$ then $x^{m} = 1$ and $x^{m} \\to 1$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxv/4", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxv", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "168", "location": "Exercise XXXV, problem 4", "problem_latex": "$\\lim\\limits_{x \\to 0} (a + bx + cx^{2} + \\dots + kx^{m}) = a$.", "markdown": "$\\lim\\limits_{x \\to 0} (a + bx + cx^{2} + \\dots + kx^{m}) = a$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxv/5", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxv", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "168", "location": "Exercise XXXV, problem 5", "problem_latex": "$\\lim\\limits_{x \\to 0} \\left\\{(a + bx + \\dots + kx^{m})/(\\alpha + \\beta x + \\dots + \\kappa x^{\\mu})\\right\\} = a/\\alpha$, unless $\\alpha = 0$. If $\\alpha = 0$\nand $a \\neq 0$, $\\beta \\neq 0$, then the function tends to $+\\infty$~or~$-\\infty$, as $x \\to +0$, according\nas $a$~and~$\\beta$ have like or unlike signs; the case is reversed if $x \\to -0$. The\ncase in which both $a$~and~$\\alpha$ vanish is considered in \\Ex{xxxvi}.~5. Discuss the\ncases which arise when $a \\neq 0$ and more than one of the first coefficients in the\ndenominator vanish.", "markdown": "$\\lim\\limits_{x \\to 0} \\left\\{(a + bx + \\dots + kx^{m})/(\\alpha + \\beta x + \\dots + \\kappa x^{\\mu})\\right\\} = a/\\alpha$, unless $\\alpha = 0$. If $\\alpha = 0$ and $a \\neq 0$, $\\beta \\neq 0$, then the function tends to $+\\infty$ or $-\\infty$, as $x \\to +0$, according as $a$ and $\\beta$ have like or unlike signs; the case is reversed if $x \\to -0$. The case in which both $a$ and $\\alpha$ vanish is considered in % [examples:xxxvi]Ex. xxxvi%. 5. Discuss the cases which arise when $a \\neq 0$ and more than one of the first coefficients in the denominator vanish.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxv/6", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxv", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "168", "location": "Exercise XXXV, problem 6", "problem_latex": "$\\lim\\limits_{x \\to a} x^{m} = a^{m}$, if $m$~is any positive or negative integer, except when $a = 0$\nand $m$~is negative. [If $m > 0$, put $x = y + a$ and apply Ex.~4. When $m < 0$,\nthe result follows from Ex.~1 above. It follows at once that $\\lim P(x) = P(a)$,\nif $P(x)$~is any polynomial.]", "markdown": "$\\lim\\limits_{x \\to a} x^{m} = a^{m}$, if $m$ is any positive or negative integer, except when $a = 0$ and $m$ is negative. [If $m > 0$, put $x = y + a$ and apply Ex. 4. When $m < 0$, the result follows from Ex. 1 above. It follows at once that $\\lim P(x) = P(a)$, if $P(x)$ is any polynomial.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxv/7", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxv", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "168", "location": "Exercise XXXV, problem 7", "problem_latex": "$\\lim\\limits_{x \\to a} R(x) = R(a)$, if $R$~denotes any rational function and $a$~is not one\nof the roots of its denominator.", "markdown": "$\\lim\\limits_{x \\to a} R(x) = R(a)$, if $R$ denotes any rational function and $a$ is not one of the roots of its denominator.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxv/8", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxv", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "168", "location": "Exercise XXXV, problem 8", "problem_latex": "Show that $\\lim\\limits_{x \\to a} x^{m} = a^{m}$ for all rational values of~$m$, except when $a = 0$\nand $m$~is negative. [This follows at once, when $a$~is positive, from the inequalities\n\\Eq{(9)}~or~\\Eq{(10)} of \\SecNo[§]{74}. For $|x^{m} - a^{m}| < H|x - a|$, where $H$~is the greater\nof the absolute values of $mx^{m-1}$ and~$ma^{m-1}$ (cf.\\ \\Ex{xxviii}.~4). If $a$~is negative\nwe write $x = -y$ and $a = -b$. Then\n\\[\n\\lim x^{m} = \\lim (-1)^{m}y^{m} = (-1)^{m}b^{m} = a^{m}.]\n\\]", "markdown": "Show that $\\lim\\limits_{x \\to a} x^{m} = a^{m}$ for all rational values of $m$, except when $a = 0$ and $m$ is negative. [This follows at once, when $a$ is positive, from the inequalities (9) or (10) of [§]74. For $|x^{m} - a^{m}| < H|x - a|$, where $H$ is the greater of the absolute values of $mx^{m-1}$ and $ma^{m-1}$ (cf. % [examples:xxviii]Ex. xxviii%. 4). If $a$ is negative we write $x = -y$ and $a = -b$. Then x^m = (-1)^my^m = (-1)^mb^m = a^m.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvi/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvi", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "171", "location": "Exercise XXXVI, problem 1", "problem_latex": "$\\lim\\limits_{x \\to a} (x^{2} - a^{2})/(x - a) = 2a$.", "markdown": "$\\lim\\limits_{x \\to a} (x^{2} - a^{2})/(x - a) = 2a$.", "answer_latex": [ "$\\lim\\limits_{x \\to a} (x^{2} - a^{2})/(x - a) = 2a$." ], "answer_markdown": [ "$\\lim\\limits_{x \\to a} (x^{2} - a^{2})/(x - a) = 2a$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(x**2 - a**2)/(x - a)", "answer_expr": "2*a" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvi/10", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvi", "number": 10, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "171", "location": "Exercise XXXVI, problem 10", "problem_latex": "$\\lim\\{\\sqrtp{1 + x + x^{2}} - 1\\}/x = \\frac{1}{2}$.", "markdown": "$\\lim\\{\\sqrtp{1 + x + x^{2}} - 1\\}/x = \\frac{1}{2}$.", "answer_latex": [ "$\\lim\\{\\sqrtp{1 + x + x^{2}} - 1\\}/x = \\frac{1}{2}$." ], "answer_markdown": [ "$\\lim\\{\\sqrtp{1 + x + x^{2}} - 1\\}/x = \\frac{1}{2}$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(sqrt(1 + x + x**2) - 1)/x", "answer_expr": "1/2" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvi/11", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvi", "number": 11, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "171", "location": "Exercise XXXVI, problem 11", "problem_latex": "$\\lim\\dfrac{\\sqrtp{1 + x} - \\sqrtp{1 + x^{2}}}{\\sqrtp{1 - x^{2}} - \\sqrtp{1 - x}} = 1$.", "markdown": "$\\lim\\dfrac{\\sqrtp{1 + x} - \\sqrtp{1 + x^{2}}}{\\sqrtp{1 - x^{2}} - \\sqrtp{1 - x}} = 1$.", "answer_latex": [ "$\\lim\\dfrac{\\sqrtp{1 + x} - \\sqrtp{1 + x^{2}}}{\\sqrtp{1 - x^{2}} - \\sqrtp{1 - x}} = 1$." ], "answer_markdown": [ "$\\lim\\dfrac{\\sqrtp{1 + x} - \\sqrtp{1 + x^{2}}}{\\sqrtp{1 - x^{2}} - \\sqrtp{1 - x}} = 1$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(sqrt(1 + x) - sqrt(1 + x**2))/(sqrt(1 - x**2) - sqrt(1 - x))", "answer_expr": "1" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvi/12", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvi", "number": 12, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "171", "location": "Exercise XXXVI, problem 12", "problem_latex": "Draw a graph of the function\n\\[\ny = \\biggl\\{\\frac{1}{x - 1}\n + \\frac{1}{x - \\tfrac{1}{2}}\n + \\frac{1}{x - \\tfrac{1}{3}}\n + \\frac{1}{x - \\tfrac{1}{4}}\\biggr\\} \\bigg/\n \\biggl\\{\\frac{1}{x - 1}\n + \\frac{1}{x - \\tfrac{1}{2}}\n + \\frac{1}{x - \\tfrac{1}{3}}\n + \\frac{1}{x - \\tfrac{1}{4}}\\biggr\\}.\n\\]\n\nHas it a limit as $x \\to 0$?", "markdown": "Draw a graph of the function y = 1x - 1 + 1x - 12 + 1x - 13 + 1x - 14 / 1x - 1 + 1x - 12 + 1x - 13 + 1x - 14. Has it a limit as $x \\to 0$?", "answer_latex": [ "Here $y = 1$ except for $x = 1$, $\\frac{1}{2}$,~$\\frac{1}{3}$,~$\\frac{1}{4}$, when $y$~is\nnot defined, and $y \\to 1$ as $x \\to 0$." ], "answer_markdown": [ "Here $y = 1$ except for $x = 1$, $\\frac{1}{2}$, $\\frac{1}{3}$, $\\frac{1}{4}$, when $y$ is not defined, and $y \\to 1$ as $x \\to 0$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.graph" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvi/13", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvi", "number": 13, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "171", "location": "Exercise XXXVI, problem 13", "problem_latex": "$\\lim\\dfrac{\\sin x}{x} = 1$.", "markdown": "$\\lim\\dfrac{\\sin x}{x} = 1$.", "answer_latex": [ "$\\lim\\dfrac{\\sin x}{x} = 1$." ], "answer_markdown": [ "$\\lim\\dfrac{\\sin x}{x} = 1$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "sin(x)/x", "answer_expr": "1" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvi/14", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvi", "number": 14, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "171", "location": "Exercise XXXVI, problem 14", "problem_latex": "$\\lim \\dfrac{1 - \\cos x}{x^{2}} = \\frac{1}{2}$.", "markdown": "$\\lim \\dfrac{1 - \\cos x}{x^{2}} = \\frac{1}{2}$.", "answer_latex": [ "$\\lim \\dfrac{1 - \\cos x}{x^{2}} = \\frac{1}{2}$." ], "answer_markdown": [ "$\\lim \\dfrac{1 - \\cos x}{x^{2}} = \\frac{1}{2}$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(1 - cos(x))/x**2", "answer_expr": "1/2" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvi/15", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvi", "number": 15, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "171", "location": "Exercise XXXVI, problem 15", "problem_latex": "$\\lim \\dfrac{\\sin \\alpha x}{x} = \\alpha$. Is this true if $\\alpha = 0$?", "markdown": "$\\lim \\dfrac{\\sin \\alpha x}{x} = \\alpha$. Is this true if $\\alpha = 0$?", "answer_latex": [ "$\\lim \\dfrac{\\sin \\alpha x}{x} = \\alpha$. Is this true if $\\alpha = 0$?" ], "answer_markdown": [ "$\\lim \\dfrac{\\sin \\alpha x}{x} = \\alpha$. Is this true if $\\alpha = 0$?" ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "sin(alpha*x)/x", "answer_expr": "alpha" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvi/16", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvi", "number": 16, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "171", "location": "Exercise XXXVI, problem 16", "problem_latex": "$\\lim \\dfrac{\\arcsin x}{x} = 1$.", "markdown": "$\\lim \\dfrac{\\arcsin x}{x} = 1$.", "answer_latex": [ "$\\lim \\dfrac{\\arcsin x}{x} = 1$." ], "answer_markdown": [ "$\\lim \\dfrac{\\arcsin x}{x} = 1$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "asin(x)/x", "answer_expr": "1" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvi/17a", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvi", "number": 17, "part": "a", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "171", "location": "Exercise XXXVI, problem 17a", "problem_latex": "$\\lim \\dfrac{\\tan \\alpha x}{x}= \\alpha$,\\quad $\\lim\\dfrac{\\arctan \\alpha x}{x} = \\alpha$.", "markdown": "$\\lim \\dfrac{\\tan \\alpha x}{x}= \\alpha$,0pt minus 3pt$\\lim\\dfrac{\\arctan \\alpha x}{x} = \\alpha$.", "answer_latex": [ "$\\lim \\dfrac{\\tan \\alpha x}{x}= \\alpha$,\\quad $\\lim\\dfrac{\\arctan \\alpha x}{x} = \\alpha$." ], "answer_markdown": [ "$\\lim \\dfrac{\\tan \\alpha x}{x}= \\alpha$,0pt minus 3pt$\\lim\\dfrac{\\arctan \\alpha x}{x} = \\alpha$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "tan(alpha*x)/x", "answer_expr": "alpha" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvi/17b", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvi", "number": 17, "part": "b", "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "171", "location": "Exercise XXXVI, problem 17b", "problem_latex": "$\\lim \\dfrac{\\tan \\alpha x}{x}= \\alpha$,\\quad $\\lim\\dfrac{\\arctan \\alpha x}{x} = \\alpha$.", "markdown": "$\\lim \\dfrac{\\tan \\alpha x}{x}= \\alpha$,0pt minus 3pt$\\lim\\dfrac{\\arctan \\alpha x}{x} = \\alpha$.", "answer_latex": [ "$\\lim \\dfrac{\\tan \\alpha x}{x}= \\alpha$,\\quad $\\lim\\dfrac{\\arctan \\alpha x}{x} = \\alpha$." ], "answer_markdown": [ "$\\lim \\dfrac{\\tan \\alpha x}{x}= \\alpha$,0pt minus 3pt$\\lim\\dfrac{\\arctan \\alpha x}{x} = \\alpha$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "atan(alpha*x)/x", "answer_expr": "alpha" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvi/18", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvi", "number": 18, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "171", "location": "Exercise XXXVI, problem 18", "problem_latex": "$\\lim \\dfrac{\\cosec x - \\cot x}{x} = \\frac{1}{2}$.", "markdown": "$\\lim \\dfrac{\\cosec x - \\cot x}{x} = \\frac{1}{2}$.", "answer_latex": [ "$\\lim \\dfrac{\\cosec x - \\cot x}{x} = \\frac{1}{2}$." ], "answer_markdown": [ "$\\lim \\dfrac{\\cosec x - \\cot x}{x} = \\frac{1}{2}$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(csc(x) - cot(x))/x", "answer_expr": "1/2" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvi/19", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvi", "number": 19, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "171", "location": "Exercise XXXVI, problem 19", "problem_latex": "$\\lim\\limits_{x \\to 1} \\dfrac{1 + \\cos \\pi x}{\\tan^{2}\\pi x} = \\frac{1}{2}$.", "markdown": "$\\lim\\limits_{x \\to 1} \\dfrac{1 + \\cos \\pi x}{\\tan^{2}\\pi x} = \\frac{1}{2}$.", "answer_latex": [ "$\\lim\\limits_{x \\to 1} \\dfrac{1 + \\cos \\pi x}{\\tan^{2}\\pi x} = \\frac{1}{2}$." ], "answer_markdown": [ "$\\lim\\limits_{x \\to 1} \\dfrac{1 + \\cos \\pi x}{\\tan^{2}\\pi x} = \\frac{1}{2}$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(1 + cos(pi*x))/tan(pi*x)**2", "answer_expr": "1/2" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvi/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvi", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "171", "location": "Exercise XXXVI, problem 2", "problem_latex": "$\\lim\\limits_{x \\to a} (x^{m} - a^{m})/(x - a) = ma^{m-1}$, if $m$~is any integer (zero included).", "markdown": "$\\lim\\limits_{x \\to a} (x^{m} - a^{m})/(x - a) = ma^{m-1}$, if $m$ is any integer (zero included).", "answer_latex": [ "$\\lim\\limits_{x \\to a} (x^{m} - a^{m})/(x - a) = ma^{m-1}$, if $m$~is any integer (zero included)." ], "answer_markdown": [ "$\\lim\\limits_{x \\to a} (x^{m} - a^{m})/(x - a) = ma^{m-1}$, if $m$ is any integer (zero included)." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(x**m - a**m)/(x - a)", "answer_expr": "m*a**(m-1)" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvi/20", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvi", "number": 20, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "171", "location": "Exercise XXXVI, problem 20", "problem_latex": "{\\Loosen How do the functions $\\sin(1/x)$, $(1/x)\\sin(1/x)$, $x\\sin(1/x)$ behave\nas $x \\to 0$?", "markdown": "0.375em plus 0.75em minus 0.25emHow do the functions $\\sin(1/x)$, $(1/x)\\sin(1/x)$, $x\\sin(1/x)$ behave as $x \\to 0$?", "answer_latex": [ "[The first oscillates finitely, the second infinitely, the third\ntends to the limit~$0$. None is defined when $x = 0$. See \\Exs{xv}.\\ 6,~7,~8.]" ], "answer_markdown": [ "[The first oscillates finitely, the second infinitely, the third tends to the limit $0$. None is defined when $x = 0$. See xv. 6, 7, 8.]" ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvi/21", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvi", "number": 21, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "171", "location": "Exercise XXXVI, problem 21", "problem_latex": "Does the function\n\\[\ny = \\biggl(\\sin \\frac{1}{x}\\biggr)\\bigg/\\biggr(\\sin \\frac{1}{x}\\biggr)\n\\]\ntend to a limit as $x$~tends to~$0$?", "markdown": "Does the function y = (1x)/(1x) tend to a limit as $x$ tends to $0$?", "answer_latex": [ "\\emph{No}. The function is equal to~$1$ except when\n$\\sin(1/x) = 0$; \\ie\\ when $x = 1/\\pi$, $1/2\\pi$,~\\dots, $-1/\\pi$, $-1/2\\pi$,~\\dots. For these values the\nformula for~$y$ assumes the meaningless form~$0/0$, and $y$~is therefore not defined\nfor an infinity of values of~$x$ near $x = 0$." ], "answer_markdown": [ "*No*. The function is equal to $1$ except when $\\sin(1/x) = 0$; *i.e.* when $x = 1/\\pi$, $1/2\\pi$, …, $-1/\\pi$, $-1/2\\pi$, …. For these values the formula for $y$ assumes the meaningless form $0/0$, and $y$ is therefore not defined for an infinity of values of $x$ near $x = 0$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvi/22", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvi", "number": 22, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "171", "location": "Exercise XXXVI, problem 22", "problem_latex": "Prove that if $m$ is any integer then $[x] \\to m$ and $x - [x] \\to 0$ as\n$x \\to m+0$, and $[x] \\to m - 1$, $x - [x] \\to 1$ as $x \\to m-0$.", "markdown": "Prove that if $m$ is any integer then $[x] \\to m$ and $x - [x] \\to 0$ as $x \\to m+0$, and $[x] \\to m - 1$, $x - [x] \\to 1$ as $x \\to m-0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvi/3", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvi", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "171", "location": "Exercise XXXVI, problem 3", "problem_latex": "Show that the result of Ex.~2 remains true for all rational values\nof~$m$, provided $a$~is positive.", "markdown": "Show that the result of Ex. 2 remains true for all rational values of $m$, provided $a$ is positive.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvi/4", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvi", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "171", "location": "Exercise XXXVI, problem 4", "problem_latex": "$\\lim\\limits_{x \\to 1} (x^{7} - 2x^{5} + 1)/(x^{3} - 3x^{2} + 2) = 1$.", "markdown": "$\\lim\\limits_{x \\to 1} (x^{7} - 2x^{5} + 1)/(x^{3} - 3x^{2} + 2) = 1$.", "answer_latex": [ "$\\lim\\limits_{x \\to 1} (x^{7} - 2x^{5} + 1)/(x^{3} - 3x^{2} + 2) = 1$." ], "answer_markdown": [ "$\\lim\\limits_{x \\to 1} (x^{7} - 2x^{5} + 1)/(x^{3} - 3x^{2} + 2) = 1$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(x**7 - 2*x**5 + 1)/(x**3 - 3*x**2 + 2)", "answer_expr": "1" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.factor", "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvi/5", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvi", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "171", "location": "Exercise XXXVI, problem 5", "problem_latex": "Discuss the behaviour of\n\\[\n\\phi(x) = (a_{0}x^{m} + a_{1}x^{m+1} + \\dots + a_{k}x^{m+k})\n /(b_{0}x^{n} + b_{1}x^{n+1} + \\dots + b_{l}x^{n+l})\n\\]\nas $x$~tends to~$0$ by positive or negative values.", "markdown": "Discuss the behaviour of (x) = (a_0x^m + a_1x^m+1 + …+ a_kx^m+k) /(b_0x^n + b_1x^n+1 + …+ b_lx^n+l) as $x$ tends to $0$ by positive or negative values.", "answer_latex": [ "If $m > n$, $\\lim\\phi(x) = 0$. If $m = n$, $\\lim\\phi(x) = a_{0}/b_{0}$. If $m < n$ and $n - m$ is\neven, $\\phi(x) \\to +\\infty$ or $\\phi(x) \\to -\\infty$ according as $a_{0}/b_{0} > 0$ or $a_{0}/b_{0} < 0$. If $m < n$ and\n$n - m$ is odd, $\\phi(x) \\to +\\infty$ as $x \\to +0$ and $\\phi(x) \\to -\\infty$ as $x \\to -0$, or $\\phi(x) \\to -\\infty$\nas $x \\to +0$ and $\\phi(x) \\to +\\infty$ as $x \\to -0$, according as $a_{0}/b_{0} > 0$ or $a_{0}/b_{0} < 0$." ], "answer_markdown": [ "If $m > n$, $\\lim\\phi(x) = 0$. If $m = n$, $\\lim\\phi(x) = a_{0}/b_{0}$. If $m < n$ and $n - m$ is even, $\\phi(x) \\to +\\infty$ or $\\phi(x) \\to -\\infty$ according as $a_{0}/b_{0} > 0$ or $a_{0}/b_{0} < 0$. If $m < n$ and $n - m$ is odd, $\\phi(x) \\to +\\infty$ as $x \\to +0$ and $\\phi(x) \\to -\\infty$ as $x \\to -0$, or $\\phi(x) \\to -\\infty$ as $x \\to +0$ and $\\phi(x) \\to +\\infty$ as $x \\to -0$, according as $a_{0}/b_{0} > 0$ or $a_{0}/b_{0} < 0$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(a0*x**m + a1*x**(m + 1) + a_k*x**(m + k))/(b0*x**n + b1*x**(n + 1) + b_l*x**(n + l))", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvi/6", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvi", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "171", "location": "Exercise XXXVI, problem 6", "problem_latex": "\\Topic{Orders of smallness}. When $x$~is small $x^{2}$~is very much smaller,\n$x^{3}$~much smaller still, and so on: in other words\n\\[\n\\lim_{x\\to 0} (x^{2}/x) = 0,\\quad\n\\lim_{x\\to 0} (x^{3}/x^{2}) = 0,\\ \\dots.\n\\]\n\nAnother way of stating the matter is to say that, when $x$~tends to~$0$,\n$x^{2}$, $x^{3}$,~\\dots\\ all also tend to~$0$, but $x^{2}$~tends to~$0$ more rapidly than~$x$, $x^{3}$~than~$x^{2}$,\nand so on. It is convenient to have some scale by which to measure\nthe rapidity with which a function, whose limit, as $x$~tends to~$0$, is~$0$,\ndiminishes with~$x$, and it is natural to take the simple functions $x$,~$x^{2}$, $x^{3}$,~\\dots\\\nas the measures of our scale.\n\nWe say, therefore, that \\emph{$\\phi(x)$~is of the first order of smallness} if $\\phi(x)/x$\ntends to a limit other than~$0$ as $x$~tends to~$0$. Thus $2x + 3x^{2} + x^{7}$ is of the\nfirst order of smallness, since $\\lim(2x + 3x^{2} + x^{7})/x = 2$.\n\nSimilarly we define the second, third, fourth,~\\dots\\ orders of smallness. It\nmust not be imagined that this scale of orders of smallness is in any way\ncomplete. If it were complete, then every function~$\\phi(x)$ which tends to zero\nwith~$x$ would be of either the first or second or some higher order of smallness.\nThis is obviously not the case. For example $\\phi(x) = x^{7/5}$ tends to zero more\nrapidly than~$x$ and less rapidly than~$x^{2}$.\n\nThe reader may not unnaturally think that our scale might be made\ncomplete by including in it \\emph{fractional} orders of smallness. Thus we might\nsay that $x^{7/5}$~was of the $\\frac{7}{5}$th~order of smallness. We shall however see later\non that such a scale of orders would still be altogether incomplete. And\nas a matter of fact the \\emph{integral} orders of smallness defined above are so\nmuch more important in applications than any others that it is hardly\nnecessary to attempt to make our definitions more precise.\n\n\\Topic{Orders of greatness.} Similar definitions are at once suggested to\nmeet the case in which $\\phi(x)$~is large (positively or negatively) when $x$~is\nsmall. We shall say that $\\phi(x)$~is of the $k$th~order of greatness when $x$~is small\nif $\\phi(x)/x^{-k} = x^{k}\\phi(x)$ tends to a limit different from~$0$ as $x$~tends to~$0$.\n\nThese definitions have reference to the case in which $x \\to 0$. There are of\ncourse corresponding definitions relating to the cases in which $x \\to \\infty$ or $x \\to a$.\nThus if $x^{k}\\phi(x)$~tends to a limit other than zero, as $x \\to \\infty$, then we say that\n$\\phi(x)$~is of the $k$th~order of smallness when $x$~is large: while if $(x - a)^{k}\\phi(x)$\ntends to a limit other than zero, as $x \\to a$, then we say that $\\phi(x)$~is of the $k$th~order\nof greatness when $x$~is nearly equal to~$a$.", "markdown": "**of smallness**. When $x$ is small $x^{2}$ is very much smaller, $x^{3}$ much smaller still, and so on: in other words _x0 (x^2/x) = 0,0pt minus 3pt_x0 (x^3/x^2) = 0, …. Another way of stating the matter is to say that, when $x$ tends to $0$, $x^{2}$, $x^{3}$, … all also tend to $0$, but $x^{2}$ tends to $0$ more rapidly than $x$, $x^{3}$ than $x^{2}$, and so on. It is convenient to have some scale by which to measure the rapidity with which a function, whose limit, as $x$ tends to $0$, is $0$, diminishes with $x$, and it is natural to take the simple functions $x$, $x^{2}$, $x^{3}$, … as the measures of our scale. We say, therefore, that *$\\phi(x)$ is of the first order of smallness* if $\\phi(x)/x$ tends to a limit other than $0$ as $x$ tends to $0$. Thus $2x + 3x^{2} + x^{7}$ is of the first order of smallness, since $\\lim(2x + 3x^{2} + x^{7})/x = 2$. Similarly we define the second, third, fourth, … orders of smallness. It must not be imagined that this scale of orders of smallness is in any way complete. If it were complete, then every function $\\phi(x)$ which tends to zero with $x$ would be of either the first or second or some higher order of smallness. This is obviously not the case. For example $\\phi(x) = x^{7/5}$ tends to zero more rapidly than $x$ and less rapidly than $x^{2}$. The reader may not unnaturally think that our scale might be made complete by including in it *fractional* orders of smallness. Thus we might say that $x^{7/5}$ was of the $\\frac{7}{5}$th order of smallness. We shall however see later on that such a scale of orders would still be altogether incomplete. And as a matter of fact the *integral* orders of smallness defined above are so much more important in applications than any others that it is hardly necessary to attempt to make our definitions more precise. **of greatness.** Similar definitions are at once suggested to meet the case in which $\\phi(x)$ is large (positively or negatively) when $x$ is small. We shall say that $\\phi(x)$ is of the $k$th order of greatness when $x$ is small if $\\phi(x)/x^{-k} = x^{k}\\phi(x)$ tends to a limit different from $0$ as $x$ tends to $0$. These definitions have reference to the case in which $x \\to 0$. There are of course corresponding definitions relating to the cases in which $x \\to \\infty$ or $x \\to a$. Thus if $x^{k}\\phi(x)$ tends to a limit other than zero, as $x \\to \\infty$, then we say that $\\phi(x)$ is of the $k$th order of smallness when $x$ is large: while if $(x - a)^{k}\\phi(x)$ tends to a limit other than zero, as $x \\to a$, then we say that $\\phi(x)$ is of the $k$th order of greatness when $x$ is nearly equal to $a$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvi/7", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvi", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "171", "location": "Exercise XXXVI, problem 7", "problem_latex": "$\\lim\\sqrtp{1 + x} = \\lim\\sqrtp{1 - x} = 1$.", "markdown": "$\\lim\\sqrtp{1 + x} = \\lim\\sqrtp{1 - x} = 1$.", "answer_latex": [ "$\\lim\\sqrtp{1 + x} = \\lim\\sqrtp{1 - x} = 1$." ], "answer_markdown": [ "$\\lim\\sqrtp{1 + x} = \\lim\\sqrtp{1 - x} = 1$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "sqrt(1 + x)", "answer_expr": "1" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvi/8", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvi", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "171", "location": "Exercise XXXVI, problem 8", "problem_latex": "$\\lim\\{\\sqrtp{1 + x} - \\sqrtp{1 - x}\\}/x = 1$.", "markdown": "$\\lim\\{\\sqrtp{1 + x} - \\sqrtp{1 - x}\\}/x = 1$.", "answer_latex": [ "$\\lim\\{\\sqrtp{1 + x} - \\sqrtp{1 - x}\\}/x = 1$." ], "answer_markdown": [ "$\\lim\\{\\sqrtp{1 + x} - \\sqrtp{1 - x}\\}/x = 1$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(sqrt(1 + x) - sqrt(1 - x))/x", "answer_expr": "1" } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvi/9", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvi", "number": 9, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "171", "location": "Exercise XXXVI, problem 9", "problem_latex": "Consider the behaviour of $\\{\\sqrtp{1 + x^{m}} - \\sqrtp{1 - x^{m}}\\}/x^{n}$ as $x \\to 0$, $m$~and~$n$\nbeing positive integers.", "markdown": "Consider the behaviour of $\\{\\sqrtp{1 + x^{m}} - \\sqrtp{1 - x^{m}}\\}/x^{n}$ as $x \\to 0$, $m$ and $n$ being positive integers.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": "(sqrt(1 + x**m) - sqrt(1 - x**m))/x**n", "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvii/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvii", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "176", "location": "Exercise XXXVII, problem 1", "problem_latex": "The sum or product of two functions continuous\nat a point is continuous at that point. The quotient is also continuous\nunless the denominator vanishes at the point. [This follows at once from\n\\Ex{xxxv}.~1.]", "markdown": "The sum or product of two functions continuous at a point is continuous at that point. The quotient is also continuous unless the denominator vanishes at the point. [This follows at once from % [examples:xxxv]Ex. xxxv%. 1.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:continuity" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvii/10", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvii", "number": 10, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "176", "location": "Exercise XXXVII, problem 10", "problem_latex": "For what values of~$x$ are $\\tan x$, $\\cot x$, $\\sec x$, and $\\cosec x$ continuous\nor discontinuous?", "markdown": "For what values of $x$ are $\\tan x$, $\\cot x$, $\\sec x$, and $\\cosec x$ continuous or discontinuous?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig", "other:continuity" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvii/11", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvii", "number": 11, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "176", "location": "Exercise XXXVII, problem 11", "problem_latex": "If $f(y)$~is continuous for $y = \\eta$, and $\\phi(x)$~is a continuous function of~$x$\nwhich is equal to~$\\eta$ when $x = \\xi$, then $f\\{\\phi(x)\\}$~is continuous for $x = \\xi$.", "markdown": "If $f(y)$ is continuous for $y = \\eta$, and $\\phi(x)$ is a continuous function of $x$ which is equal to $\\eta$ when $x = \\xi$, then $f\\{\\phi(x)\\}$ is continuous for $x = \\xi$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:continuity" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvii/12", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvii", "number": 12, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "176", "location": "Exercise XXXVII, problem 12", "problem_latex": "If $\\phi(x)$~is continuous for any particular value of~$x$, then any polynomial\nin~$\\phi(x)$, such as $a\\{\\phi(x)\\}^{m} + \\dots$, is so too.", "markdown": "If $\\phi(x)$ is continuous for any particular value of $x$, then any polynomial in $\\phi(x)$, such as $a\\{\\phi(x)\\}^{m} + \\dots$, is so too.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:continuity" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvii/13", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvii", "number": 13, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "176", "location": "Exercise XXXVII, problem 13", "problem_latex": "Discuss the continuity of\n\\[\n1/(a\\cos^{2} x + b\\sin^{2} x),\\quad\n\\sqrtp{2 + \\cos x},\\quad\n\\sqrtp{1 + \\sin x},\\quad\n1/\\sqrtp{1 + \\sin x}.\n\\]", "markdown": "Discuss the continuity of 1/(a^2 x + b^2 x),0pt minus 3pt2 + x,0pt minus 3pt1 + x,0pt minus 3pt1/1 + x.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig", "other:continuity" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvii/14", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvii", "number": 14, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "176", "location": "Exercise XXXVII, problem 14", "problem_latex": "$\\sin(1/x)$, $x\\sin(1/x)$, and $x^{2}\\sin(1/x)$ are continuous except for $x = 0$.", "markdown": "$\\sin(1/x)$, $x\\sin(1/x)$, and $x^{2}\\sin(1/x)$ are continuous except for $x = 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig", "other:continuity" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvii/15", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvii", "number": 15, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "176", "location": "Exercise XXXVII, problem 15", "problem_latex": "The function which is equal to $x\\sin(1/x)$ except when $x = 0$, and to\nzero when $x = 0$, is continuous for all values of~$x$.", "markdown": "The function which is equal to $x\\sin(1/x)$ except when $x = 0$, and to zero when $x = 0$, is continuous for all values of $x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig", "other:continuity" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvii/16", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvii", "number": 16, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "176", "location": "Exercise XXXVII, problem 16", "problem_latex": "$[x]$ and $x - [x]$ are discontinuous for all integral values of~$x$.", "markdown": "$[x]$ and $x - [x]$ are discontinuous for all integral values of $x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:continuity" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvii/17", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvii", "number": 17, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "176", "location": "Exercise XXXVII, problem 17", "problem_latex": "For what (if any) values of~$x$ are the following functions discontinuous:\n$[x^{2}]$, $[\\sqrt{x}\\,]$, $\\sqrtp{x - [x]}$, $[x] + \\sqrtp{x - [x]}$, $[2x]$, $[x] + [-x]$?", "markdown": "For what (if any) values of $x$ are the following functions discontinuous: $[x^{2}]$, $[\\sqrt{x}\\,]$, $\\sqrtp{x - [x]}$, $[x] + \\sqrtp{x - [x]}$, $[2x]$, $[x] + [-x]$?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:continuity" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvii/18", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvii", "number": 18, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "176", "location": "Exercise XXXVII, problem 18", "problem_latex": "\\Topic{Classification of discontinuities.} Some of the preceding examples\nsuggest a classification of different types of discontinuity.\n\n\\SubItem{(1)} Suppose that $\\phi(x)$~tends to a limit as $x \\to a$ either by values less\nthan or by values greater than~$a$. Denote these limits, as in \\SecNo[§]{95}, by $\\phi(a - 0)$\nand $\\phi(a + 0)$ respectively. Then, for continuity, it is necessary and sufficient\nthat $\\phi(x)$~should be defined for $x = a$, and that $\\phi(a - 0) = \\phi(a) = \\phi(a + 0)$. Discontinuity\nmay arise in a variety of ways.\n\n\\Item{($\\alpha$)} $\\phi(a - 0)$ may be equal to $\\phi(a + 0)$, but $\\phi(a)$~may not be defined, or\nmay differ from $\\phi(a - 0)$ and~$\\phi(a + 0)$. Thus if $\\phi(x) = x \\sin(1/x)$ and $a = 0$,\n$\\phi(0 - 0) = \\phi(0 + 0) = 0$, but $\\phi(x)$~is not defined for $x = 0$. Or if $\\phi(x) = [1 - x^{2}]$\nand $a = 0$, $\\phi(0 - 0) = \\phi(0 + 0) = 0$, but $\\phi(0) = 1$.\n\n\\Item{($\\beta$)} {\\Loosen$\\phi(a - 0)$ and $\\phi(a + 0)$ may be unequal. In this case $\\phi(a)$~may be\nequal to one or to neither, or be undefined. The first case is illustrated\nby $\\phi(x) = [x]$, for which $\\phi(0 - 0) = -1$, $\\phi(0 + 0) = \\phi(0) = 0$; the second by\n$\\phi(x) = [x] - [-x]$, for which $\\phi(0 - 0) = -1$, $\\phi(0 + 0) = 1$, $\\phi(0) = 0$; and the third\nby $\\phi(x) = [x] + x \\sin(1/x)$, for which $\\phi(0 - 0)= -1$, $\\phi(0 + 0) = 0$, and $\\phi(0)$~is\nundefined.}\n\nIn any of these cases we say that $\\phi(x)$~has a \\Emph{simple discontinuity} at\n$x = a$. And to these cases we may add those in which $\\phi(x)$~is defined only\non one side of $x = a$, and $\\phi(a - 0)$ or~$\\phi(a + 0)$, as the case may be, exists, but\n$\\phi(x)$~is either not defined when $x = a$ or has when $x = a$ a value different from\n$\\phi(a - 0)$ or~$\\phi(a + 0)$.\n\nIt is plain from \\SecNo[§]{95} that \\emph{a function which increases or decreases steadily\nin the neighbourhood of $x = a$ can have at most a simple discontinuity for $x = a$}.\n\n\\SubItem{(2)} It may be the case that only one (or neither) of $\\phi(a - 0)$ and $\\phi(a + 0)$\nexists, but that, supposing for example $\\phi(a + 0)$ not to exist, $\\phi(x) \\to +\\infty$ or\n$\\phi(x) \\to -\\infty$ as $x \\to a+0$, so that $\\phi(x)$~tends to a limit or to~$+\\infty$ or to~$-\\infty$ as\n$x$~approaches~$a$ from either side. Such is the case, for instance, if $\\phi(x) = 1/x$ or\n$\\phi(x) = 1/x^{2}$, and $a = 0$. In such cases we say (cf.\\ Ex.~7) that $x = a$ is an \\Emph{infinity}\nof~$\\phi(x)$. And again we may add to these cases those in which $\\phi(x) \\to +\\infty$\nor $\\phi(x) \\to -\\infty$ as $x \\to a$ from one side, but $\\phi(x)$~is not defined at all on the\nother side of $x = a$.\n\n\\SubItem{(3)} Any point of discontinuity which is not a point of simple discontinuity\nnor an infinity is called a point of \\Emph{oscillatory discontinuity}. Such\nis the point $x = 0$ for the functions $\\sin(1/x)$, $(1/x)\\sin(1/x)$.", "markdown": "**of discontinuities.** Some of the preceding examples suggest a classification of different types of discontinuity. 0pt minus 3pt% [2.25em][l](1)% [2.25em][l](1)% % Suppose that $\\phi(x)$ tends to a limit as $x \\to a$ either by values less than or by values greater than $a$. Denote these limits, as in [§]95, by $\\phi(a - 0)$ and $\\phi(a + 0)$ respectively. Then, for continuity, it is necessary and sufficient that $\\phi(x)$ should be defined for $x = a$, and that $\\phi(a - 0) = \\phi(a) = \\phi(a + 0)$. Discontinuity may arise in a variety of ways. [1.5em][l]($\\alpha$) $\\phi(a - 0)$ may be equal to $\\phi(a + 0)$, but $\\phi(a)$ may not be defined, or may differ from $\\phi(a - 0)$ and $\\phi(a + 0)$. Thus if $\\phi(x) = x \\sin(1/x)$ and $a = 0$, $\\phi(0 - 0) = \\phi(0 + 0) = 0$, but $\\phi(x)$ is not defined for $x = 0$. Or if $\\phi(x) = [1 - x^{2}]$ and $a = 0$, $\\phi(0 - 0) = \\phi(0 + 0) = 0$, but $\\phi(0) = 1$. [1.5em][l]($\\beta$) 0.375em plus 0.75em minus 0.25em$\\phi(a - 0)$ and $\\phi(a + 0)$ may be unequal. In this case $\\phi(a)$ may be equal to one or to neither, or be undefined. The first case is illustrated by $\\phi(x) = [x]$, for which $\\phi(0 - 0) = -1$, $\\phi(0 + 0) = \\phi(0) = 0$; the second by $\\phi(x) = [x] - [-x]$, for which $\\phi(0 - 0) = -1$, $\\phi(0 + 0) = 1$, $\\phi(0) = 0$; and the third by $\\phi(x) = [x] + x \\sin(1/x)$, for which $\\phi(0 - 0)= -1$, $\\phi(0 + 0) = 0$, and $\\phi(0)$ is undefined. In any of these cases we say that $\\phi(x)$ has a **discontinuity** at $x = a$. And to these cases we may add those in which $\\phi(x)$ is defined only on one side of $x = a$, and $\\phi(a - 0)$ or $\\phi(a + 0)$, as the case may be, exists, but $\\phi(x)$ is either not defined when $x = a$ or has when $x = a$ a value different from $\\phi(a - 0)$ or $\\phi(a + 0)$. It is plain from [§]95 that *a function which increases or decreases steadily in the neighbourhood of $x = a$ can have at most a simple discontinuity for $x = a$*. 0pt minus 3pt% [2.25em][l](2)% [2.25em][l](2)% % It may be the case that only one (or neither) of $\\phi(a - 0)$ and $\\phi(a + 0)$ exists, but that, supposing for example $\\phi(a + 0)$ not to exist, $\\phi(x) \\to +\\infty$ or $\\phi(x) \\to -\\infty$ as $x \\to a+0$, so that $\\phi(x)$ tends to a limit or to $+\\infty$ or to $-\\infty$ as $x$ approaches $a$ from either side. Such is the case, for instance, if $\\phi(x) = 1/x$ or $\\phi(x) = 1/x^{2}$, and $a = 0$. In such cases we say (cf. Ex. 7) that $x = a$ is an **** of $\\phi(x)$. And again we may add to these cases those in which $\\phi(x) \\to +\\infty$ or $\\phi(x) \\to -\\infty$ as $x \\to a$ from one side, but $\\phi(x)$ is not defined at all on the other side of $x = a$. 0pt minus 3pt% [2.25em][l](3)% [2.25em][l](3)% % Any point of discontinuity which is not a point of simple discontinuity nor an infinity is called a point of **discontinuity**. Such is the point $x = 0$ for the functions $\\sin(1/x)$, $(1/x)\\sin(1/x)$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig", "other:continuity" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvii/19", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvii", "number": 19, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "176", "location": "Exercise XXXVII, problem 19", "problem_latex": "What is the nature of the discontinuities at $x = 0$ of the functions\n$(\\sin x)/x$, $[x] + [-x]$, $\\cosec x$, $\\sqrtp{1/x}$, $\\sqrtp[3]{1/x}$, $\\cosec(1/x)$, $\\sin(1/x)/\\sin(1/x)$?", "markdown": "What is the nature of the discontinuities at $x = 0$ of the functions $(\\sin x)/x$, $[x] + [-x]$, $\\cosec x$, $\\sqrtp{1/x}$, $\\sqrtp[3]{1/x}$, $\\cosec(1/x)$, $\\sin(1/x)/\\sin(1/x)$?", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig", "other:continuity" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvii/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvii", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "176", "location": "Exercise XXXVII, problem 2", "problem_latex": "Any polynomial is continuous for all values of~$x$. Any rational\nfraction is continuous except for values of~$x$ for which the denominator\nvanishes. [This follows from \\Exs{xxxv}.\\ 6,~7.]", "markdown": "Any polynomial is continuous for all values of $x$. Any rational fraction is continuous except for values of $x$ for which the denominator vanishes. [This follows from xxxv. 6, 7.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:continuity" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvii/20", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvii", "number": 20, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "176", "location": "Exercise XXXVII, problem 20", "problem_latex": "The function which is equal to~$1$ when $x$~is rational and to~$0$ when\n$x$~is irrational (\\okrickRef{Ch.}{II}, \\Ex{xvi}.~10) is discontinuous for all values of~$x$. So too\nis any function which is defined only for rational or for irrational values of~$x$.", "markdown": "The function which is equal to $1$ when $x$ is rational and to $0$ when $x$ is irrational (Ch.II, % [examples:xvi]Ex. xvi%. 10) is discontinuous for all values of $x$. So too is any function which is defined only for rational or for irrational values of $x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:continuity" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvii/21", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvii", "number": 21, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "176", "location": "Exercise XXXVII, problem 21", "problem_latex": "{\\Loosen The function which is equal to~$x$ when $x$~is irrational and to\n$\\sqrtb{(1 + p^{2})/(1 + q^{2})}$ when $x$~is a rational fraction~$p/q$ (\\okrickRef{Ch.}{II}, \\Ex{xvi}.~11) is\ndiscontinuous for all negative and for positive rational values of~$x$, but\ncontinuous for positive irrational values.}", "markdown": "0.375em plus 0.75em minus 0.25emThe function which is equal to $x$ when $x$ is irrational and to $\\sqrtb{(1 + p^{2})/(1 + q^{2})}$ when $x$ is a rational fraction $p/q$ (Ch.II, % [examples:xvi]Ex. xvi%. 11) is discontinuous for all negative and for positive rational values of $x$, but continuous for positive irrational values.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:continuity" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvii/22", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvii", "number": 22, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "176", "location": "Exercise XXXVII, problem 22", "problem_latex": "For what points are the functions considered in \\okrickRef{Ch.}{IV}, \\Exs{xxxi}\ndiscontinuous, and what is the nature of their discontinuities? [Consider,\n\\eg, the function $y = \\lim x^{n}$ (Ex.~5). Here $y$~is only defined when $-1 < x \\leq 1$:\nit is equal to~$0$ when $-1 < x < 1$ and to~$1$ when $x = 1$. The points $x = 1$ and\n$x = -1$ are points of simple discontinuity.]", "markdown": "For what points are the functions considered in Ch.IV, xxxi discontinuous, and what is the nature of their discontinuities? [Consider, *e.g.*, the function $y = \\lim x^{n}$ (Ex. 5). Here $y$ is only defined when $-1 < x \\leq 1$: it is equal to $0$ when $-1 < x < 1$ and to $1$ when $x = 1$. The points $x = 1$ and $x = -1$ are points of simple discontinuity.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:continuity", "other:limits" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvii/3", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvii", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "176", "location": "Exercise XXXVII, problem 3", "problem_latex": "$\\sqrt{x}$~is continuous for all positive values of~$x$ (\\Ex{xxxv}.~8). It is not\ndefined when $x < 0$, but is continuous for $x = 0$ in virtue of the remark made at\nthe end of \\SecNo[§]{98}. The same is true of~$x^{m/n}$, where $m$~and~$n$ are any positive\nintegers of which $n$ is even.", "markdown": "$\\sqrt{x}$ is continuous for all positive values of $x$ (% [examples:xxxv]Ex. xxxv%. 8). It is not defined when $x < 0$, but is continuous for $x = 0$ in virtue of the remark made at the end of [§]98. The same is true of $x^{m/n}$, where $m$ and $n$ are any positive integers of which $n$ is even.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:continuity" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvii/4", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvii", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "176", "location": "Exercise XXXVII, problem 4", "problem_latex": "The function~$x^{m/n}$, where $n$~is odd, is continuous for all values of~$x$.", "markdown": "The function $x^{m/n}$, where $n$ is odd, is continuous for all values of $x$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:continuity" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvii/5", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvii", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "176", "location": "Exercise XXXVII, problem 5", "problem_latex": "$1/x$~is not continuous for $x = 0$. It has no value for $x = 0$, nor does it\ntend to a limit as $x \\to 0$. In fact $1/x \\to +\\infty$ or $1/x \\to -\\infty$ according as $x \\to 0$\nby positive or negative values.", "markdown": "$1/x$ is not continuous for $x = 0$. It has no value for $x = 0$, nor does it tend to a limit as $x \\to 0$. In fact $1/x \\to +\\infty$ or $1/x \\to -\\infty$ according as $x \\to 0$ by positive or negative values.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:continuity" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvii/6", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvii", "number": 6, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "176", "location": "Exercise XXXVII, problem 6", "problem_latex": "Discuss the continuity of~$x^{-m/n}$, where $m$~and~$n$ are positive integers,\nfor $x = 0$.", "markdown": "Discuss the continuity of $x^{-m/n}$, where $m$ and $n$ are positive integers, for $x = 0$.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:continuity" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvii/7", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvii", "number": 7, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "176", "location": "Exercise XXXVII, problem 7", "problem_latex": "The standard rational function $R(x) = P(x)/Q(x)$ is discontinuous for\n$x = a$, where $a$~is any root of $Q(x) = 0$. Thus $(x^{2} + 1)/(x^{2} - 3x + 2)$ is discontinuous\nfor $x = 1$. It will be noticed that in the case of rational functions a\ndiscontinuity is always associated with (\\ia)~a failure of the definition for a\nparticular value of~$x$ and (\\ib)~a tending of the function to~$+\\infty$ or~$-\\infty$ as $x$~approaches\nthis value from either side. Such a particular kind of point of\ndiscontinuity is usually described as an \\Emph{infinity} of the function. An `infinity'\nis the kind of discontinuity of most common occurrence in ordinary work.", "markdown": "The standard rational function $R(x) = P(x)/Q(x)$ is discontinuous for $x = a$, where $a$ is any root of $Q(x) = 0$. Thus $(x^{2} + 1)/(x^{2} - 3x + 2)$ is discontinuous for $x = 1$. It will be noticed that in the case of rational functions a discontinuity is always associated with (*a*) a failure of the definition for a particular value of $x$ and (*b*) a tending of the function to $+\\infty$ or $-\\infty$ as $x$ approaches this value from either side. Such a particular kind of point of discontinuity is usually described as an **** of the function. An ‘infinity’ is the kind of discontinuity of most common occurrence in ordinary work.", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:continuity" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvii/8", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvii", "number": 8, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "176", "location": "Exercise XXXVII, problem 8", "problem_latex": "Discuss the continuity of\n\\[\n\\sqrtb{(x - a)(b - x)},\\quad\n\\sqrtb[3]{(x - a)(b - x)},\\quad\n\\sqrtb{(x - a)/(b - x)},\\quad\n\\sqrtb[3]{(x - a)/(b - x)}\\Add{.}\n\\]", "markdown": "Discuss the continuity of (x - a)(b - x),0pt minus 3pt[3](x - a)(b - x),0pt minus 3pt(x - a)/(b - x),0pt minus 3pt[3](x - a)/(b - x)", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "other:continuity" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxvii/9", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxvii", "number": 9, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "176", "location": "Exercise XXXVII, problem 9", "problem_latex": "$\\sin x$ and $\\cos x$ are continuous for all values of~$x$.\n\n[We have\n\\[\n\\sin(x + h) - \\sin x = 2\\sin \\tfrac{1}{2}h \\cos(x + \\tfrac{1}{2}h),\n\\]\nwhich is numerically less than the numerical value of~$h$.]", "markdown": "$\\sin x$ and $\\cos x$ are continuous for all values of $x$. [We have (x + h) - x = 212h (x + 12h), which is numerically less than the numerical value of $h$.]", "answer_latex": [], "answer_markdown": [], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "core.trig", "other:continuity" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxviii/1", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxviii", "number": 1, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "184", "location": "Exercise XXXVIII, problem 1", "problem_latex": " {\\Loosen If $\\phi(x) = 1/x$ except when $x = 0$, and $\\phi(x) = 0$\nwhen $x = 0$, then $\\phi(x)$~has neither an upper nor a lower bound in any\ninterval which includes $x = 0$ in its interior, as \\eg\\ the interval~$\\DPmod{(-1, +1)}{[-1, +1]}$.}", "markdown": "0.375em plus 0.75em minus 0.25emIf $\\phi(x) = 1/x$ except when $x = 0$, and $\\phi(x) = 0$ when $x = 0$, then $\\phi(x)$ has neither an upper nor a lower bound in any interval which includes $x = 0$ in its interior, as *e.g.* the interval $\\DPmod{(-1, +1)}{[-1, +1]}$.", "answer_latex": [ " {\\Loosen If $\\phi(x) = 1/x$ except when $x = 0$, and $\\phi(x) = 0$\nwhen $x = 0$, then $\\phi(x)$~has neither an upper nor a lower bound in any\ninterval which includes $x = 0$ in its interior, as \\eg\\ the interval~$\\DPmod{(-1, +1)}{[-1, +1]}$.}" ], "answer_markdown": [ "0.375em plus 0.75em minus 0.25emIf $\\phi(x) = 1/x$ except when $x = 0$, and $\\phi(x) = 0$ when $x = 0$, then $\\phi(x)$ has neither an upper nor a lower bound in any interval which includes $x = 0$ in its interior, as *e.g.* the interval $\\DPmod{(-1, +1)}{[-1, +1]}$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxviii/2", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxviii", "number": 2, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "184", "location": "Exercise XXXVIII, problem 2", "problem_latex": " If $\\phi(x) = 1/x^{2}$ except when $x = 0$, and $\\phi(x) = 0$ when $x = 0$, then $\\phi(x)$~has\nthe lower bound~$0$, but no upper bound, in the interval~$\\DPmod{(-1, +1)}{[-1, +1]}$.", "markdown": "If $\\phi(x) = 1/x^{2}$ except when $x = 0$, and $\\phi(x) = 0$ when $x = 0$, then $\\phi(x)$ has the lower bound $0$, but no upper bound, in the interval $\\DPmod{(-1, +1)}{[-1, +1]}$.", "answer_latex": [ " If $\\phi(x) = 1/x^{2}$ except when $x = 0$, and $\\phi(x) = 0$ when $x = 0$, then $\\phi(x)$~has\nthe lower bound~$0$, but no upper bound, in the interval~$\\DPmod{(-1, +1)}{[-1, +1]}$." ], "answer_markdown": [ "If $\\phi(x) = 1/x^{2}$ except when $x = 0$, and $\\phi(x) = 0$ when $x = 0$, then $\\phi(x)$ has the lower bound $0$, but no upper bound, in the interval $\\DPmod{(-1, +1)}{[-1, +1]}$." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxviii/3", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxviii", "number": 3, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "184", "location": "Exercise XXXVIII, problem 3", "problem_latex": " Let $\\phi(x) = \\sin(1/x)$ except when $x = 0$, and $\\phi(x) = 0$ when $x = 0$. Then\n$\\phi(x)$~is discontinuous for $x = 0$. In any interval~$\\DPmod{(-\\DELTA, +\\DELTA)}{[-\\DELTA, +\\DELTA]}$ the lower bound is~$-1$\nand the upper bound~$+1$, and each of these values is assumed by~$\\phi(x)$ an\ninfinity of times.", "markdown": "Let $\\phi(x) = \\sin(1/x)$ except when $x = 0$, and $\\phi(x) = 0$ when $x = 0$. Then $\\phi(x)$ is discontinuous for $x = 0$. In any interval $\\DPmod{(-\\DELTA, +\\DELTA)}{[-\\DELTA, +\\DELTA]}$ the lower bound is $-1$ and the upper bound $+1$, and each of these values is assumed by $\\phi(x)$ an infinity of times.", "answer_latex": [ " Let $\\phi(x) = \\sin(1/x)$ except when $x = 0$, and $\\phi(x) = 0$ when $x = 0$. Then\n$\\phi(x)$~is discontinuous for $x = 0$. In any interval~$\\DPmod{(-\\DELTA, +\\DELTA)}{[-\\DELTA, +\\DELTA]}$ the lower bound is~$-1$\nand the upper bound~$+1$, and each of these values is assumed by~$\\phi(x)$ an\ninfinity of times." ], "answer_markdown": [ "Let $\\phi(x) = \\sin(1/x)$ except when $x = 0$, and $\\phi(x) = 0$ when $x = 0$. Then $\\phi(x)$ is discontinuous for $x = 0$. In any interval $\\DPmod{(-\\DELTA, +\\DELTA)}{[-\\DELTA, +\\DELTA]}$ the lower bound is $-1$ and the upper bound $+1$, and each of these values is assumed by $\\phi(x)$ an infinity of times." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [ "cas.limit", "core.trig" ], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxviii/4", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxviii", "number": 4, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "184", "location": "Exercise XXXVIII, problem 4", "problem_latex": " Let $\\phi(x) = x - [x]$. This function is discontinuous for all integral\nvalues of~$x$. In the interval~$\\DPmod{(0, 1)}{[0, 1]}$ its lower bound is~$0$ and its upper bound~$1$.\nIt is equal to~$0$ when $x = 0$ or $x = 1$, but it is never equal to~$1$. Thus $\\phi(x)$~never\nassumes a value equal to its upper bound.", "markdown": "Let $\\phi(x) = x - [x]$. This function is discontinuous for all integral values of $x$. In the interval $\\DPmod{(0, 1)}{[0, 1]}$ its lower bound is $0$ and its upper bound $1$. It is equal to $0$ when $x = 0$ or $x = 1$, but it is never equal to $1$. Thus $\\phi(x)$ never assumes a value equal to its upper bound.", "answer_latex": [ " Let $\\phi(x) = x - [x]$. This function is discontinuous for all integral\nvalues of~$x$. In the interval~$\\DPmod{(0, 1)}{[0, 1]}$ its lower bound is~$0$ and its upper bound~$1$.\nIt is equal to~$0$ when $x = 0$ or $x = 1$, but it is never equal to~$1$. Thus $\\phi(x)$~never\nassumes a value equal to its upper bound." ], "answer_markdown": [ "Let $\\phi(x) = x - [x]$. This function is discontinuous for all integral values of $x$. In the interval $\\DPmod{(0, 1)}{[0, 1]}$ its lower bound is $0$ and its upper bound $1$. It is equal to $0$ when $x = 0$ or $x = 1$, but it is never equal to $1$. Thus $\\phi(x)$ never assumes a value equal to its upper bound." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] }, { "id": "hardy-course-of-pure-mathematics-1921/ex-xxxviii/5", "set": "hardy-course-of-pure-mathematics-1921/ex-xxxviii", "number": 5, "part": null, "book": "hardy-course-of-pure-mathematics-1921", "edition": "Cambridge University Press, 3rd ed., 1921", "page": "184", "location": "Exercise XXXVIII, problem 5", "problem_latex": " Let $\\phi(x) = 0$ when $x$~is irrational, and $\\phi(x) = q$ when $x$~is a rational\nfraction~$p/q$. Then $\\phi(x)$~has the lower bound~$0$, but no upper bound, in any\ninterval~$\\DPmod{(a, b)}{[a, b]}$. But if $\\phi(x) = (-1)^{p}q$ when $x = p/q$, then $\\phi(x)$~has neither an\nupper nor a lower bound in any interval.", "markdown": "Let $\\phi(x) = 0$ when $x$ is irrational, and $\\phi(x) = q$ when $x$ is a rational fraction $p/q$. Then $\\phi(x)$ has the lower bound $0$, but no upper bound, in any interval $\\DPmod{(a, b)}{[a, b]}$. But if $\\phi(x) = (-1)^{p}q$ when $x = p/q$, then $\\phi(x)$ has neither an upper nor a lower bound in any interval.", "answer_latex": [ " Let $\\phi(x) = 0$ when $x$~is irrational, and $\\phi(x) = q$ when $x$~is a rational\nfraction~$p/q$. Then $\\phi(x)$~has the lower bound~$0$, but no upper bound, in any\ninterval~$\\DPmod{(a, b)}{[a, b]}$. But if $\\phi(x) = (-1)^{p}q$ when $x = p/q$, then $\\phi(x)$~has neither an\nupper nor a lower bound in any interval." ], "answer_markdown": [ "Let $\\phi(x) = 0$ when $x$ is irrational, and $\\phi(x) = q$ when $x$ is a rational fraction $p/q$. Then $\\phi(x)$ has the lower bound $0$, but no upper bound, in any interval $\\DPmod{(a, b)}{[a, b]}$. But if $\\phi(x) = (-1)^{p}q$ when $x = p/q$, then $\\phi(x)$ has neither an upper nor a lower bound in any interval." ], "checks": [ { "task": "other", "verdict": "UNJUDGED-OTHER", "judge_why": "task other: verbatim only", "problem_expr": null, "answer_expr": null } ], "verdict": "unverified", "judge": { "commit": "4a88328", "verdicts": [ "UNJUDGED-OTHER" ] }, "form": [], "shape": [], "same_problem_in": [], "needs": [], "expectation": null, "keys": [] } ], "errata_statuses": [ "transcriber_marked", "candidate", "probable", "confirmed", "dismissed", "note" ], "errata": [] }