{
"cells": [
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"using Plots, ComplexPhasePortrait, ApproxFun\n",
"gr();"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# M3M6: Methods of Mathematical Physics\n",
"\n",
"$$\n",
"\\def\\dashint{{\\int\\!\\!\\!\\!\\!\\!-\\,}}\n",
"\\def\\infdashint{\\dashint_{\\!\\!\\!-\\infty}^{\\,\\infty}}\n",
"\\def\\D{\\,{\\rm d}}\n",
"\\def\\dx{\\D x}\n",
"$$\n",
"\n",
"Dr. Sheehan Olver\n",
"<br>\n",
"s.olver@imperial.ac.uk\n",
"\n",
"\n",
"<br>\n",
"Website: https://github.com/dlfivefifty/M3M6LectureNotes\n",
"\n",
"\n",
"\n",
"# Lecture 7: Integrals over the real line\n",
"\n",
"\n",
"This lecture we cover\n",
"\n",
"1. Integrals over real lines\n",
" - Principal value integral\n",
" - Cauchy's integral formula and Residue theorem on the real line\n",
" - Jordan's lemma\n",
" - Application: Calculating Fourier tranforms of weakly decaying functions\n",
"2. Functions with branch cuts\n",
" - Logarithmic function"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"## Integrals over the real line\n",
"\n",
"Integrals on the real line are always to viewed as improper integrals:\n",
"\n",
"$$ \\int_{-\\infty}^\\infty f(x) \\dx = \\int_0^\\infty f(x) \\dx +\\int_{-\\infty}^0 f(x)\\dx = \\lim_{b \\rightarrow \\infty} \\int_0^b f(x) \\dx + \\lim_{a \\rightarrow -\\infty} \\int_a^0 f(x) \\dx.$$\n",
"\n",
"**Definition (Principal value integral on the real line)** The _(Cauchy) principal value integral on the real line_ is defined as\n",
"$$\n",
"\\infdashint f(x) \\dx := \\lim_{M\\rightarrow \\infty} \\int_{-M}^M f(x) \\dx\n",
"$$\n",
"\n",
"**Proposition (Integability $\\Rightarrow$ Prinipal value integrability)** If \n",
"$ \\int_{-\\infty}^\\infty f(x) \\dx < \\infty$\n",
"then \n",
"$$ \\infdashint f(x) \\dx = \\int_{-\\infty}^\\infty f(x) \\dx.$$\n",
"\n",
"\n",
"### Residue theorem on the real line\n",
"\n",
"The real line doesn't have an _inside_ and _outside_, rather an _above_ and _below_, or _left_ and _right_. Thus we get the following two versions of the Residue theorem:\n",
"\n",
"**Definition (Upper/lower half plane)** Denote the upper/lower half plane by\n",
"$$\n",
"{\\mathbb H}^+ = \\{z : \\Re z > 0 \\} \\\\\n",
"{\\mathbb H}^- = \\{z : \\Re z < 0 \\} \n",
"$$\n",
"The closure is denoted\n",
"$$\n",
"\\bar{\\mathbb H}^+ = {\\mathbb H}^+ \\cup {\\mathbb R} \\cup \\{\\infty\\} \\\\\n",
"\\bar{\\mathbb H}^- = {\\mathbb H}^- \\cup {\\mathbb R} \\cup \\{\\infty\\}\n",
"$$\n",
"\n",
"**Theorem (Residue theorem on the real line)** Suppose $f : \\bar {\\mathbb H}^+ \\backslash \\{z_1,\\ldots,z_r \\} \\rightarrow {\\mathbb C}$ is holomorphic in ${\\mathbb H}^+ \\backslash \\{z_1,\\ldots,z_r \\}$, where $\\Re z_k > 0$, and $\\lim_{\\epsilon \\rightarrow 0} f(x + i \\epsilon) = f(x)$ converges uniformly. If \n",
"$$ \\lim_{z \\rightarrow \\infty} z f(z) = 0\n",
"$$ uniformly for $z \\in \\bar {\\mathbb H}^+$, then\n",
"$$\n",
"\\infdashint f(x) \\dx = 2 \\pi i \\sum_{k=1}^r {\\underset{z = z_k}{\\rm Res}} \\, f(z)\n",
"$$\n",
"Similarly, if the equivalent conditions hold in the lower half plane for $f : \\bar{\\mathbb H}^- \\backslash \\{z_1,\\ldots,z_r \\} \\rightarrow {\\mathbb C}$ then\n",
"$$\n",
"\\infdashint f(x) \\dx = -2 \\pi i \\sum_{k=1}^r {\\underset{z = z_k}{\\rm Res}} \\, f(z)\n",
"$$\n",
"\n",
"\n",
"Examples:"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"scrolled": false
},
"outputs": [],
"source": [
"f = x -> x^2/(x^4+1)\n",
"phaseplot(-3..3, -2..2, f)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This function has poles in the upper plane, but has sufficient decay that we can apply Residue theorem:"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(2.221441469079183 + 3.487868498008632e-16im, 2.221441469084968)"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"z₁,z₂,z₃,z₄ = exp(im*π/4), exp(3im*π/4), exp(5im*π/4), exp(7im*π/4)\n",
"\n",
"res₁ = z₁^2 / ((z₁ - z₂)*(z₁ - z₃)*(z₁ - z₄) )\n",
"res₂ = z₂^2 / ((z₂ - z₁)*(z₂ - z₃)*(z₂ - z₄) )\n",
"\n",
"2π*im*(res₁ + res₂), sum(Fun(f, Line()))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can also apply Resiude theorem in the lower-half plane, and we get the same result:"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(2.221441469079183 + 5.231802747012948e-16im, 2.221441469084968)"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"res₃ = z₃^2 / ((z₃ - z₁)*(z₃ - z₂)*(z₃ - z₄) )\n",
"res₄ = z₄^2 / ((z₄ - z₁)*(z₄ - z₃)*(z₄ - z₂) )\n",
"\n",
"-2π*im*(res₃ + res₄), sum(Fun(f, Line()))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Cauchy's integral formula on the real line\n",
"\n",
"An immediate consequence of the Residue theorem is Cauchy's integral formula on the real line:\n",
"\n",
"**Theorem (Cauchy's integral formula on the real line)** Suppose $f : \\bar {\\mathbb H}^+ \\rightarrow {\\mathbb C}$ is holomorphic in ${\\mathbb H}^+ $, and $\\lim_{\\epsilon \\rightarrow 0} f(x + i \\epsilon) = f(x)$ converges uniformly. If \n",
"$$ \n",
"\\lim_{z \\rightarrow \\infty} f(z) = 0\n",
"$$ \n",
"uniformly for $z \\in \\bar {\\mathbb H}^+$, then\n",
"$$\n",
"f(z) = {1 \\over 2 \\pi i} \\infdashint {f(x) \\over x - z} dx\n",
"$$\n",
"for all $z \\in {\\mathbb H}^+$.\n",
"\n",
"\n",
"_Examples_ Here is a simple example of $f(x) = {x^2 \\over (x+ i)^3}$, which is analytic in the upper half plane:"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2.69638478211931e-13 - 3.032296636007459e-13im"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f = x -> x^2/(x+im)^3\n",
"z = 2.0+2.0im\n",
"sum(Fun(x-> f(x)/(x - z), Line()))/(2π*im) - f(z)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Evaluating in lower half plane doesn't work b ecause it has a pole there:"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(2.2331966377567175e-13 + 2.0218605967538834e-14im, 0.7040000000000001 - 0.128im)"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f = x -> x^2/(x+im)^3\n",
"z = 2.0-2.0im\n",
"sum(Fun(x-> f(x)/(x - z), Line()))/(2π*im) , f(z)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"But does for a function analytic in the lower half plane (with a minus sign):"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(0.03277196176631425 + 0.16750113791564233im, 0.03277196176604461 + 0.1675011379153391im)"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f = x -> x^2/(x-im)^3\n",
"z = 2.0-2.0im\n",
"-sum(Fun(x-> f(x)/(x - z), Line()))/(2π*im) , f(z)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"It also works for functions with exponential decay in the upper-half plane:"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"4.501316791527543e-9 + 5.93330299732131e-7im"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f = x -> exp(im*x)/(x+im)\n",
"z = 2 + 2im\n",
"sum(Fun(x-> f(x)/(x - z), -500 .. 500))/(2π*im) - f(z)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This is difficult as a real integral as the integrand is very oscillatory:"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
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" 320.225,771.62 320.728,771.358 321.231,771.136 321.733,770.956 322.236,770.819 322.739,770.728 323.242,770.683 323.744,770.685 324.247,770.734 324.75,770.829 \n",
" 325.253,770.969 325.755,771.153 326.258,771.38 326.761,771.647 327.264,771.951 327.767,772.289 328.269,772.659 328.772,773.055 329.275,773.476 329.778,773.915 \n",
" 330.28,774.37 330.783,774.835 331.286,775.306 331.789,775.777 332.292,776.245 332.794,776.705 333.297,777.152 333.8,777.581 334.303,777.989 334.805,778.371 \n",
" 335.308,778.723 335.811,779.042 336.314,779.324 336.816,779.567 337.319,779.768 337.822,779.926 338.325,780.037 338.828,780.103 339.33,780.121 339.833,780.091 \n",
" 340.336,780.014 340.839,779.89 341.341,779.721 341.844,779.508 342.347,779.254 342.85,778.96 343.353,778.63 343.855,778.267 344.358,777.874 344.861,777.456 \n",
" 345.364,777.016 345.866,776.559 346.369,776.09 346.872,775.613 347.375,775.133 347.877,774.655 348.38,774.183 348.883,773.722 349.386,773.278 349.889,772.853 \n",
" 350.391,772.453 350.894,772.082 351.397,771.743 351.9,771.44 352.402,771.176 352.905,770.953 353.408,770.774 353.911,770.641 354.414,770.555 354.916,770.516 \n",
" 355.419,770.526 355.922,770.585 356.425,770.691 356.927,770.844 357.43,771.042 357.933,771.284 358.436,771.567 358.938,771.888 359.441,772.244 359.944,772.631 \n",
" 360.447,773.047 360.95,773.485 361.452,773.943 361.955,774.416 362.458,774.899 362.961,775.387 363.463,775.875 363.966,776.359 364.469,776.833 364.972,777.293 \n",
" 365.475,777.734 365.977,778.152 366.48,778.542 366.983,778.901 367.486,779.225 367.988,779.511 368.491,779.755 368.994,779.956 369.497,780.111 369.999,780.219 \n",
" 370.502,780.279 371.005,780.289 371.508,780.25 372.011,780.162 372.513,780.026 373.016,779.843 373.519,779.615 374.022,779.344 374.524,779.033 375.027,778.686 \n",
" 375.53,778.304 376.033,777.892 376.536,777.455 377.038,776.996 377.541,776.521 378.044,776.033 378.547,775.538 379.049,775.04 379.552,774.546 380.055,774.058 \n",
" 380.558,773.584 381.06,773.126 381.563,772.69 382.066,772.281 382.569,771.901 383.072,771.556 383.574,771.249 384.077,770.982 384.58,770.758 385.083,770.581 \n",
" 385.585,770.451 386.088,770.37 386.591,770.338 387.094,770.357 387.597,770.426 388.099,770.545 388.602,770.711 389.105,770.924 389.608,771.182 390.11,771.482 \n",
" 390.613,771.821 391.116,772.196 391.619,772.603 392.121,773.038 392.624,773.496 393.127,773.974 393.63,774.466 394.133,774.968 394.635,775.474 395.138,775.98 \n",
" 395.641,776.48 396.144,776.969 396.646,777.443 397.149,777.897 397.652,778.325 398.155,778.725 398.658,779.091 399.16,779.421 399.663,779.71 400.166,779.957 \n",
" 400.669,780.157 401.171,780.31 401.674,780.413 402.177,780.466 402.68,780.468 403.182,780.419 403.685,780.32 404.188,780.17 404.691,779.972 405.194,779.728 \n",
" 405.696,779.44 406.199,779.111 406.702,778.744 407.205,778.343 407.707,777.911 408.21,777.454 408.713,776.975 409.216,776.479 409.719,775.971 410.221,775.457 \n",
" 410.724,774.941 411.227,774.428 411.73,773.925 412.232,773.435 412.735,772.963 413.238,772.516 413.741,772.096 414.243,771.708 414.746,771.356 415.249,771.044 \n",
" 415.752,770.774 416.255,770.55 416.757,770.374 417.26,770.248 417.763,770.172 418.266,770.149 418.768,770.177 419.271,770.257 419.774,770.389 420.277,770.57 \n",
" 420.78,770.799 421.282,771.074 421.785,771.392 422.288,771.751 422.791,772.146 423.293,772.573 423.796,773.029 424.299,773.509 424.802,774.008 425.304,774.52 \n",
" 425.807,775.042 426.31,775.568 426.813,776.092 427.316,776.61 427.818,777.116 428.321,777.605 428.824,778.071 429.327,778.512 429.829,778.921 430.332,779.295 \n",
" 430.835,779.631 431.338,779.924 431.841,780.172 432.343,780.372 432.846,780.522 433.349,780.621 433.852,780.667 434.354,780.66 434.857,780.6 435.36,780.488 \n",
" 435.863,780.324 436.365,780.111 436.868,779.849 437.371,779.543 437.874,779.194 438.377,778.807 438.879,778.384 439.382,777.931 439.885,777.451 440.388,776.95 \n",
" 440.89,776.433 441.393,775.904 441.896,775.369 442.399,774.833 442.902,774.302 443.404,773.781 443.907,773.275 444.41,772.789 444.913,772.328 445.415,771.897 \n",
" 445.918,771.5 446.421,771.141 446.924,770.824 447.426,770.552 447.929,770.328 448.432,770.153 448.935,770.031 449.438,769.961 449.94,769.946 450.443,769.984 \n",
" 450.946,770.076 451.449,770.222 451.951,770.419 452.454,770.665 452.957,770.959 453.46,771.297 453.962,771.676 454.465,772.092 454.968,772.542 455.471,773.02 \n",
" 455.974,773.523 456.476,774.044 456.979,774.579 457.482,775.123 457.985,775.67 458.487,776.214 458.99,776.75 459.493,777.273 459.996,777.778 460.499,778.259 \n",
" 461.001,778.712 461.504,779.132 462.007,779.515 462.51,779.856 463.012,780.153 463.515,780.403 464.018,780.602 464.521,780.749 465.023,780.843 465.526,780.882 \n",
" 466.029,780.865 466.532,780.794 467.035,780.668 467.537,780.489 468.04,780.258 468.543,779.978 469.046,779.652 469.548,779.282 470.051,778.873 470.554,778.428 \n",
" 471.057,777.951 471.56,777.448 472.062,776.924 472.565,776.383 473.068,775.831 473.571,775.274 474.073,774.717 474.576,774.166 475.079,773.626 475.582,773.103 \n",
" 476.084,772.601 476.587,772.126 477.09,771.684 477.593,771.277 478.096,770.91 478.598,770.588 479.101,770.313 479.604,770.088 480.107,769.915 480.609,769.797 \n",
" 481.112,769.734 481.615,769.727 482.118,769.777 482.621,769.882 483.123,770.043 483.626,770.256 484.129,770.522 484.632,770.835 485.134,771.195 485.637,771.596 \n",
" 486.14,772.036 486.643,772.509 487.145,773.012 487.648,773.539 488.151,774.084 488.654,774.643 489.157,775.21 489.659,775.78 490.162,776.345 490.665,776.902 \n",
" 491.168,777.444 491.67,777.966 492.173,778.462 492.676,778.928 493.179,779.359 493.682,779.751 494.184,780.099 494.687,780.4 495.19,780.652 495.693,780.85 \n",
" 496.195,780.994 496.698,781.082 497.201,781.113 497.704,781.086 498.206,781.002 498.709,780.862 499.212,780.666 499.715,780.417 500.218,780.117 500.72,779.769 \n",
" 501.223,779.376 501.726,778.943 502.229,778.474 502.731,777.972 503.234,777.444 503.737,776.894 504.24,776.329 504.743,775.752 505.245,775.171 505.748,774.591 \n",
" 506.251,774.019 506.754,773.458 507.256,772.916 507.759,772.398 508.262,771.908 508.765,771.453 509.267,771.036 509.77,770.661 510.273,770.333 510.776,770.055 \n",
" 511.279,769.83 511.781,769.659 512.284,769.545 512.787,769.489 513.29,769.492 513.792,769.554 514.295,769.674 514.798,769.85 515.301,770.082 515.804,770.367 \n",
" 516.306,770.703 516.809,771.085 517.312,771.511 517.815,771.976 518.317,772.475 518.82,773.004 519.323,773.557 519.826,774.128 520.328,774.713 520.831,775.305 \n",
" 521.334,775.899 521.837,776.488 522.34,777.066 522.842,777.628 523.345,778.169 523.848,778.682 524.351,779.162 524.853,779.605 525.356,780.006 525.859,780.362 \n",
" 526.362,780.667 526.865,780.92 527.367,781.118 527.87,781.259 528.373,781.34 528.876,781.363 529.378,781.325 529.881,781.227 530.384,781.07 530.887,780.857 \n",
" 531.389,780.588 531.892,780.266 532.395,779.895 532.898,779.478 533.401,779.019 533.903,778.523 534.406,777.994 534.909,777.439 535.412,776.862 535.914,776.269 \n",
" 536.417,775.666 536.92,775.059 537.423,774.455 537.926,773.858 538.428,773.276 538.931,772.714 539.434,772.177 539.937,771.672 540.439,771.203 540.942,770.774 \n",
" 541.445,770.391 541.948,770.057 542.45,769.776 542.953,769.55 543.456,769.381 543.959,769.273 544.462,769.225 544.964,769.238 545.467,769.313 545.97,769.448 \n",
" 546.473,769.642 546.975,769.894 547.478,770.201 547.981,770.56 548.484,770.967 548.987,771.419 549.489,771.911 549.992,772.438 550.495,772.995 550.998,773.577 \n",
" 551.5,774.177 552.003,774.79 552.506,775.409 553.009,776.029 553.511,776.643 554.014,777.245 554.517,777.829 555.02,778.389 555.523,778.92 556.025,779.416 \n",
" 556.528,779.872 557.031,780.284 557.534,780.646 558.036,780.957 558.539,781.212 559.042,781.409 559.545,781.546 560.048,781.621 560.55,781.633 561.053,781.583 \n",
" 561.556,781.47 562.059,781.297 562.561,781.063 563.064,780.772 563.567,780.427 564.07,780.03 564.572,779.587 565.075,779.1 565.578,778.575 566.081,778.017 \n",
" 566.584,777.432 567.086,776.825 567.589,776.203 568.092,775.571 568.595,774.937 569.097,774.305 569.6,773.684 570.103,773.078 570.606,772.494 571.109,771.937 \n",
" 571.611,771.415 572.114,770.931 572.617,770.49 573.12,770.097 573.622,769.757 574.125,769.472 574.628,769.246 575.131,769.08 575.633,768.977 576.136,768.937 \n",
" 576.639,768.962 577.142,769.051 577.645,769.203 578.147,769.417 578.65,769.69 579.153,770.021 579.656,770.405 580.158,770.84 580.661,771.32 581.164,771.842 \n",
" 581.667,772.399 582.17,772.987 582.672,773.6 583.175,774.231 583.678,774.874 584.181,775.523 584.683,776.171 585.186,776.813 585.689,777.44 586.192,778.048 \n",
" 586.694,778.63 587.197,779.18 587.7,779.693 588.203,780.163 588.706,780.586 589.208,780.957 589.711,781.273 590.214,781.53 590.717,781.726 591.219,781.858 \n",
" 591.722,781.925 592.225,781.928 592.728,781.864 593.231,781.735 593.733,781.542 594.236,781.287 594.739,780.973 595.242,780.602 595.744,780.178 596.247,779.705 \n",
" 596.75,779.188 597.253,778.631 597.755,778.041 598.258,777.424 598.761,776.785 599.264,776.131 599.767,775.467 600.269,774.802 600.772,774.142 601.275,773.492 \n",
" 601.778,772.86 602.28,772.253 602.783,771.675 603.286,771.133 603.789,770.633 604.291,770.179 604.794,769.777 605.297,769.429 605.8,769.141 606.303,768.914 \n",
" 606.805,768.751 607.308,768.654 607.811,768.624 608.314,768.661 608.816,768.765 609.319,768.936 609.822,769.171 610.325,769.468 610.828,769.824 611.33,770.237 \n",
" 611.833,770.701 612.336,771.213 612.839,771.767 613.341,772.357 613.844,772.979 614.347,773.625 614.85,774.29 615.352,774.967 615.855,775.648 616.358,776.328 \n",
" 616.861,776.999 617.364,777.655 617.866,778.289 618.369,778.894 618.872,779.465 619.375,779.996 619.877,780.482 620.38,780.917 620.883,781.297 621.386,781.618 \n",
" 621.889,781.877 622.391,782.072 622.894,782.2 623.397,782.259 623.9,782.249 624.402,782.171 624.905,782.024 625.408,781.811 625.911,781.533 626.413,781.192 \n",
" 626.916,780.793 627.419,780.338 627.922,779.833 628.425,779.283 628.927,778.692 629.43,778.067 629.933,777.414 630.436,776.74 630.938,776.05 631.441,775.353 \n",
" 631.944,774.654 632.447,773.961 632.95,773.282 633.452,772.622 633.955,771.988 634.458,771.387 634.961,770.825 635.463,770.307 635.966,769.838 636.469,769.425 \n",
" 636.972,769.07 637.474,768.777 637.977,768.55 638.48,768.39 638.983,768.3 639.486,768.28 639.988,768.331 640.491,768.453 640.994,768.643 641.497,768.901 \n",
" 641.999,769.225 642.502,769.61 643.005,770.053 643.508,770.55 644.011,771.096 644.513,771.685 645.016,772.312 645.519,772.971 646.022,773.655 646.524,774.357 \n",
" 647.027,775.07 647.53,775.787 648.033,776.501 648.535,777.205 649.038,777.891 649.541,778.553 650.044,779.185 650.547,779.779 651.049,780.33 651.552,780.832 \n",
" 652.055,781.281 652.558,781.671 653.06,781.998 653.563,782.26 654.066,782.453 654.569,782.575 655.072,782.625 655.574,782.603 656.077,782.509 656.58,782.342 \n",
" 657.083,782.106 657.585,781.801 658.088,781.432 658.591,781.002 659.094,780.514 659.596,779.974 660.099,779.386 660.602,778.758 661.105,778.094 661.608,777.402 \n",
" 662.11,776.689 662.613,775.961 663.116,775.225 663.619,774.49 664.121,773.762 664.624,773.049 665.127,772.358 665.63,771.695 666.133,771.069 666.635,770.484 \n",
" 667.138,769.946 667.641,769.462 668.144,769.036 668.646,768.673 669.149,768.376 669.652,768.148 670.155,767.992 670.657,767.91 671.16,767.901 671.663,767.968 \n",
" 672.166,768.108 672.669,768.321 673.171,768.605 673.674,768.957 674.177,769.374 674.68,769.851 675.182,770.384 675.685,770.968 676.188,771.596 676.691,772.264 \n",
" 677.194,772.963 677.696,773.688 678.199,774.431 678.702,775.184 679.205,775.941 679.707,776.693 680.21,777.432 680.713,778.153 681.216,778.847 681.718,779.507 \n",
" 682.221,780.127 682.724,780.7 683.227,781.221 683.73,781.684 684.232,782.084 684.735,782.418 685.238,782.682 685.741,782.874 686.243,782.99 686.746,783.03 \n",
" 687.249,782.994 687.752,782.881 688.255,782.693 688.757,782.431 689.26,782.098 689.763,781.697 690.266,781.232 690.768,780.707 691.271,780.128 691.774,779.5 \n",
" 692.277,778.83 692.779,778.124 693.282,777.388 693.785,776.631 694.288,775.861 694.791,775.083 695.293,774.307 695.796,773.54 696.299,772.79 696.802,772.065 \n",
" 697.304,771.371 697.807,770.715 698.31,770.105 698.813,769.546 699.316,769.045 699.818,768.606 700.321,768.233 700.824,767.931 701.327,767.703 701.829,767.551 \n",
" 702.332,767.477 702.835,767.481 703.338,767.565 703.84,767.726 704.343,767.964 704.846,768.277 705.349,768.661 705.852,769.112 706.354,769.627 706.857,770.201 \n",
" 707.36,770.826 707.863,771.499 708.365,772.211 708.868,772.956 709.371,773.726 709.874,774.515 710.377,775.313 710.879,776.113 711.382,776.907 711.885,777.687 \n",
" 712.388,778.445 712.89,779.174 713.393,779.865 713.896,780.513 714.399,781.111 714.901,781.653 715.404,782.132 715.907,782.544 716.41,782.886 716.913,783.153 \n",
" 717.415,783.342 717.918,783.452 718.421,783.481 718.924,783.429 719.426,783.296 719.929,783.083 720.432,782.793 720.935,782.428 721.438,781.991 721.94,781.488 \n",
" 722.443,780.922 722.946,780.299 723.449,779.626 723.951,778.909 724.454,778.155 724.957,777.371 725.46,776.566 725.962,775.748 726.465,774.924 726.968,774.102 \n",
" 727.471,773.292 727.974,772.501 728.476,771.737 728.979,771.008 729.482,770.32 729.985,769.682 730.487,769.1 730.99,768.579 731.493,768.125 731.996,767.742 \n",
" 732.499,767.435 733.001,767.206 733.504,767.059 734.007,766.994 734.51,767.013 735.012,767.116 735.515,767.301 736.018,767.567 736.521,767.911 737.023,768.331 \n",
" 737.526,768.822 738.029,769.379 738.532,769.997 739.035,770.67 739.537,771.391 740.04,772.153 740.543,772.949 741.046,773.77 741.548,774.609 742.051,775.458 \n",
" 742.554,776.306 743.057,777.147 743.56,777.972 744.062,778.772 744.565,779.54 745.068,780.267 745.571,780.947 746.073,781.572 746.576,782.137 747.079,782.634 \n",
" 747.582,783.06 748.084,783.41 748.587,783.679 749.09,783.866 749.593,783.969 750.096,783.985 750.598,783.915 751.101,783.76 751.604,783.52 752.107,783.198 \n",
" 752.609,782.797 753.112,782.32 753.615,781.773 754.118,781.161 754.62,780.49 755.123,779.766 755.626,778.996 756.129,778.189 756.632,777.351 757.134,776.492 \n",
" 757.637,775.62 758.14,774.743 758.643,773.871 759.145,773.012 759.648,772.175 760.151,771.368 760.654,770.599 761.157,769.876 761.659,769.207 762.162,768.598 \n",
" 762.665,768.055 763.168,767.584 763.67,767.19 764.173,766.877 764.676,766.648 765.179,766.506 765.681,766.452 766.184,766.487 766.687,766.611 767.19,766.822 \n",
" 767.693,767.12 768.195,767.501 768.698,767.961 769.201,768.496 769.704,769.1 770.206,769.769 770.709,770.495 771.212,771.271 771.715,772.089 772.218,772.942 \n",
" 772.72,773.821 773.223,774.718 773.726,775.622 774.229,776.526 774.731,777.42 775.234,778.295 775.737,779.143 776.24,779.954 776.742,780.722 777.245,781.437 \n",
" 777.748,782.093 778.251,782.683 778.754,783.201 779.256,783.642 779.759,784.001 780.262,784.274 780.765,784.458 781.267,784.552 781.77,784.554 782.273,784.464 \n",
" 782.776,784.283 783.279,784.012 783.781,783.654 784.284,783.212 784.787,782.691 785.29,782.095 785.792,781.431 786.295,780.704 786.798,779.922 787.301,779.093 \n",
" 787.803,778.225 788.306,777.327 788.809,776.407 789.312,775.474 789.815,774.538 790.317,773.608 790.82,772.694 791.323,771.805 791.826,770.949 792.328,770.135 \n",
" 792.831,769.372 793.334,768.667 793.837,768.028 794.34,767.461 794.842,766.971 795.345,766.564 795.848,766.244 796.351,766.015 796.853,765.879 797.356,765.837 \n",
" 797.859,765.891 798.362,766.039 798.864,766.281 799.367,766.614 799.87,767.036 800.373,767.542 800.876,768.127 801.378,768.785 801.881,769.511 802.384,770.298 \n",
" 802.887,771.136 803.389,772.019 803.892,772.937 804.395,773.881 804.898,774.842 805.401,775.811 805.903,776.777 806.406,777.731 806.909,778.663 807.412,779.565 \n",
" 807.914,780.427 808.417,781.239 808.92,781.995 809.423,782.686 809.925,783.306 810.428,783.847 810.931,784.304 811.434,784.673 811.937,784.95 812.439,785.132 \n",
" 812.942,785.216 813.445,785.201 813.948,785.088 814.45,784.878 814.953,784.571 815.456,784.172 815.959,783.684 816.462,783.112 816.964,782.46 817.467,781.736 \n",
" 817.97,780.947 818.473,780.099 818.975,779.203 819.478,778.266 819.981,777.297 820.484,776.307 820.986,775.305 821.489,774.302 821.992,773.307 822.495,772.33 \n",
" 822.998,771.381 823.5,770.469 824.003,769.604 824.506,768.795 825.009,768.05 825.511,767.376 826.014,766.781 826.517,766.27 827.02,765.848 827.523,765.521 \n",
" 828.025,765.292 828.528,765.162 829.031,765.135 829.534,765.209 830.036,765.386 830.539,765.662 831.042,766.037 831.545,766.505 832.047,767.063 832.55,767.706 \n",
" 833.053,768.426 833.556,769.218 834.059,770.073 834.561,770.983 835.064,771.939 835.567,772.932 836.07,773.951 836.572,774.987 837.075,776.029 837.578,777.066 \n",
" 838.081,778.09 838.584,779.088 839.086,780.051 839.589,780.97 840.092,781.835 840.595,782.637 841.097,783.368 841.6,784.021 842.103,784.589 842.606,785.066 \n",
" 843.108,785.447 843.611,785.728 844.114,785.906 844.617,785.978 845.12,785.945 845.622,785.806 846.125,785.561 846.628,785.214 847.131,784.768 847.633,784.226 \n",
" 848.136,783.594 848.639,782.879 849.142,782.086 849.645,781.224 850.147,780.301 850.65,779.327 851.153,778.31 851.656,777.261 852.158,776.191 852.661,775.109 \n",
" 853.164,774.028 853.667,772.957 854.169,771.907 854.672,770.889 855.175,769.914 855.678,768.99 856.181,768.128 856.683,767.336 857.186,766.623 857.689,765.995 \n",
" 858.192,765.459 858.694,765.021 859.197,764.685 859.7,764.455 860.203,764.334 860.706,764.323 861.208,764.422 861.711,764.631 862.214,764.948 862.717,765.369 \n",
" 863.219,765.892 863.722,766.511 864.225,767.22 864.728,768.013 865.23,768.881 865.733,769.816 866.236,770.809 866.739,771.85 867.242,772.929 867.744,774.035 \n",
" 868.247,775.157 868.75,776.284 869.253,777.404 869.755,778.508 870.258,779.582 870.761,780.617 871.264,781.603 871.767,782.528 872.269,783.384 872.772,784.162 \n",
" 873.275,784.854 873.778,785.452 874.28,785.952 874.783,786.346 875.286,786.632 875.789,786.805 876.291,786.865 876.794,786.809 877.297,786.639 877.8,786.356 \n",
" 878.303,785.961 878.805,785.46 879.308,784.856 879.811,784.155 880.314,783.364 880.816,782.491 881.319,781.545 881.822,780.534 882.325,779.468 882.828,778.359 \n",
" 883.33,777.216 883.833,776.052 884.336,774.878 884.839,773.705 885.341,772.546 885.844,771.412 886.347,770.314 886.85,769.263 887.352,768.271 887.855,767.347 \n",
" 888.358,766.501 888.861,765.741 889.364,765.075 889.866,764.51 890.369,764.053 890.872,763.707 891.375,763.477 891.877,763.365 892.38,763.373 892.883,763.501 \n",
" 893.386,763.748 893.889,764.112 894.391,764.59 894.894,765.176 895.397,765.866 895.9,766.654 896.402,767.53 896.905,768.488 897.408,769.516 897.911,770.607 \n",
" 898.413,771.747 898.916,772.928 899.419,774.136 899.922,775.359 900.425,776.586 900.927,777.804 901.43,779.002 901.933,780.166 902.436,781.285 902.938,782.348 \n",
" 903.441,783.345 903.944,784.264 904.447,785.097 904.949,785.834 905.452,786.469 905.955,786.995 906.458,787.405 906.961,787.697 907.463,787.865 907.966,787.909 \n",
" 908.469,787.827 908.972,787.621 909.474,787.291 909.977,786.84 910.48,786.273 910.983,785.596 911.486,784.814 911.988,783.935 912.491,782.967 912.994,781.921 \n",
" 913.497,780.806 913.999,779.633 914.502,778.414 915.005,777.16 915.508,775.885 916.01,774.601 916.513,773.32 917.016,772.057 917.519,770.822 918.022,769.629 \n",
" 918.524,768.49 919.027,767.417 919.53,766.42 920.033,765.509 920.535,764.694 921.038,763.983 921.541,763.385 922.044,762.904 922.547,762.546 923.049,762.316 \n",
" 923.552,762.216 924.055,762.246 924.558,762.409 925.06,762.701 925.563,763.121 926.066,763.665 926.569,764.327 927.071,765.103 927.574,765.983 928.077,766.959 \n",
" 928.58,768.023 929.083,769.164 929.585,770.37 930.088,771.629 930.591,772.93 931.094,774.26 931.596,775.604 932.099,776.95 932.602,778.285 933.105,779.594 \n",
" 933.608,780.865 934.11,782.085 934.613,783.242 935.116,784.323 935.619,785.317 936.121,786.216 936.624,787.008 937.127,787.686 937.63,788.243 938.132,788.673 \n",
" 938.635,788.971 939.138,789.133 939.641,789.158 940.144,789.045 940.646,788.794 941.149,788.409 941.652,787.891 942.155,787.246 942.657,786.48 943.16,785.601 \n",
" 943.663,784.615 944.166,783.534 944.669,782.368 945.171,781.128 945.674,779.826 946.177,778.475 946.68,777.088 947.182,775.68 947.685,774.263 948.188,772.853 \n",
" 948.691,771.464 949.193,770.109 949.696,768.802 950.199,767.557 950.702,766.385 951.205,765.3 951.707,764.311 952.21,763.43 952.713,762.666 953.216,762.026 \n",
" 953.718,761.517 954.221,761.146 954.724,760.915 955.227,760.828 955.73,760.887 956.232,761.091 956.735,761.438 957.238,761.926 957.741,762.55 958.243,763.304 \n",
" 958.746,764.182 959.249,765.175 959.752,766.273 960.254,767.465 960.757,768.741 961.26,770.088 961.763,771.492 962.266,772.939 962.768,774.416 963.271,775.907 \n",
" 963.774,777.398 964.277,778.873 964.779,780.319 965.282,781.72 965.785,783.062 966.288,784.331 966.791,785.515 967.293,786.602 967.796,787.58 968.299,788.439 \n",
" 968.802,789.169 969.304,789.764 969.807,790.217 970.31,790.523 970.813,790.678 971.315,790.68 971.818,790.528 972.321,790.224 972.824,789.771 973.327,789.171 \n",
" 973.829,788.43 974.332,787.557 974.835,786.557 975.338,785.443 975.84,784.223 976.343,782.91 976.846,781.516 977.349,780.056 977.852,778.543 978.354,776.994 \n",
" 978.857,775.422 979.36,773.843 979.863,772.275 980.365,770.731 980.868,769.228 981.371,767.781 981.874,766.405 982.376,765.113 982.879,763.919 983.382,762.835 \n",
" 983.885,761.873 984.388,761.042 984.89,760.352 985.393,759.809 985.896,759.42 986.399,759.189 986.901,759.12 987.404,759.212 987.907,759.467 988.41,759.882 \n",
" 988.913,760.454 989.415,761.177 989.918,762.045 990.421,763.05 990.924,764.182 991.426,765.43 991.929,766.782 992.432,768.225 992.935,769.746 993.437,771.328 \n",
" 993.94,772.957 994.443,774.617 994.946,776.29 995.449,777.96 995.951,779.611 996.454,781.225 996.957,782.787 997.46,784.281 997.962,785.691 998.465,787.003 \n",
" 998.968,788.204 999.471,789.281 999.974,790.222 1000.48,791.018 1000.98,791.661 1001.48,792.143 1001.98,792.458 1002.49,792.604 1002.99,792.577 1003.49,792.377 \n",
" 1004,792.007 1004.5,791.468 1005,790.765 1005.5,789.905 1006.01,788.896 1006.51,787.748 1007.01,786.47 1007.52,785.076 1008.02,783.579 1008.52,781.994 \n",
" 1009.02,780.335 1009.53,778.62 1010.03,776.865 1010.53,775.088 1011.03,773.307 1011.54,771.538 1012.04,769.801 1012.54,768.112 1013.05,766.488 1013.55,764.947 \n",
" 1014.05,763.504 1014.55,762.174 1015.06,760.97 1015.56,759.905 1016.06,758.99 1016.57,758.236 1017.07,757.651 1017.57,757.24 1018.07,757.009 1018.58,756.961 \n",
" 1019.08,757.097 1019.58,757.417 1020.08,757.918 1020.59,758.596 1021.09,759.445 1021.59,760.457 1022.1,761.623 1022.6,762.931 1023.1,764.37 1023.6,765.925 \n",
" 1024.11,767.581 1024.61,769.323 1025.11,771.132 1025.61,772.992 1026.12,774.884 1026.62,776.789 1027.12,778.688 1027.63,780.562 1028.13,782.392 1028.63,784.16 \n",
" 1029.13,785.848 1029.64,787.437 1030.14,788.913 1030.64,790.26 1031.15,791.463 1031.65,792.51 1032.15,793.39 1032.65,794.094 1033.16,794.612 1033.66,794.94 \n",
" 1034.16,795.073 1034.66,795.009 1035.17,794.748 1035.67,794.292 1036.17,793.643 1036.68,792.808 1037.18,791.795 1037.68,790.612 1038.18,789.27 1038.69,787.783 \n",
" 1039.19,786.164 1039.69,784.43 1040.2,782.596 1040.7,780.681 1041.2,778.704 1041.7,776.685 1042.21,774.642 1042.71,772.597 1043.21,770.57 1043.71,768.581 \n",
" 1044.22,766.651 1044.72,764.798 1045.22,763.043 1045.73,761.403 1046.23,759.895 1046.73,758.535 1047.23,757.337 1047.74,756.314 1048.24,755.477 1048.74,754.835 \n",
" 1049.25,754.396 1049.75,754.165 1050.25,754.145 1050.75,754.338 1051.26,754.743 1051.76,755.356 1052.26,756.174 1052.76,757.187 1053.27,758.388 1053.77,759.765 \n",
" 1054.27,761.306 1054.78,762.995 1055.28,764.816 1055.78,766.753 1056.28,768.785 1056.79,770.894 1057.29,773.058 1057.79,775.257 1058.3,777.467 1058.8,779.668 \n",
" 1059.3,781.837 1059.8,783.952 1060.31,785.992 1060.81,787.935 1061.31,789.763 1061.81,791.456 1062.32,792.996 1062.82,794.367 1063.32,795.554 1063.83,796.545 \n",
" 1064.33,797.329 1064.83,797.897 1065.33,798.241 1065.84,798.358 1066.34,798.245 1066.84,797.901 1067.35,797.33 1067.85,796.535 1068.35,795.523 1068.85,794.304 \n",
" 1069.36,792.888 1069.86,791.288 1070.36,789.52 1070.86,787.6 1071.37,785.547 1071.87,783.38 1072.37,781.121 1072.88,778.792 1073.38,776.415 1073.88,774.014 \n",
" 1074.38,771.614 1074.89,769.238 1075.39,766.91 1075.89,764.654 1076.39,762.493 1076.9,760.448 1077.4,758.542 1077.9,756.794 1078.41,755.222 1078.91,753.843 \n",
" 1079.41,752.672 1079.91,751.722 1080.42,751.004 1080.92,750.527 1081.42,750.296 1081.93,750.315 1082.43,750.585 1082.93,751.106 1083.43,751.873 1083.94,752.881 \n",
" 1084.44,754.12 1084.94,755.58 1085.44,757.247 1085.95,759.105 1086.45,761.138 1086.95,763.326 1087.46,765.648 1087.96,768.081 1088.46,770.601 1088.96,773.185 \n",
" 1089.47,775.807 1089.97,778.439 1090.47,781.057 1090.98,783.634 1091.48,786.143 1091.98,788.56 1092.48,790.859 1092.99,793.016 1093.49,795.01 1093.99,796.819 \n",
" 1094.49,798.423 1095,799.806 1095.5,800.952 1096,801.849 1096.51,802.485 1097.01,802.852 1097.51,802.946 1098.01,802.763 1098.52,802.304 1099.02,801.571 \n",
" 1099.52,800.571 1100.03,799.31 1100.53,797.801 1101.03,796.057 1101.53,794.093 1102.04,791.928 1102.54,789.582 1103.04,787.078 1103.54,784.44 1104.05,781.693 \n",
" 1104.55,778.864 1105.05,775.981 1105.56,773.072 1106.06,770.167 1106.56,767.294 1107.06,764.483 1107.57,761.762 1108.07,759.159 1108.57,756.701 1109.08,754.414 \n",
" 1109.58,752.321 1110.08,750.446 1110.58,748.807 1111.09,747.425 1111.59,746.313 1112.09,745.485 1112.59,744.952 1113.1,744.721 1113.6,744.796 1114.1,745.179 \n",
" 1114.61,745.868 1115.11,746.859 1115.61,748.143 1116.11,749.71 1116.62,751.546 1117.12,753.635 1117.62,755.958 1118.13,758.493 1118.63,761.216 1119.13,764.101 \n",
" 1119.63,767.121 1120.14,770.246 1120.64,773.446 1121.14,776.69 1121.64,779.944 1122.15,783.176 1122.65,786.355 1123.15,789.447 1123.66,792.421 1124.16,795.246 \n",
" 1124.66,797.893 1125.16,800.334 1125.67,802.542 1126.17,804.494 1126.67,806.168 1127.17,807.545 1127.68,808.61 1128.18,809.348 1128.68,809.75 1129.19,809.809 \n",
" 1129.69,809.522 1130.19,808.888 1130.69,807.913 1131.2,806.601 1131.7,804.965 1132.2,803.017 1132.71,800.775 1133.21,798.258 1133.71,795.49 1134.21,792.497 \n",
" 1134.72,789.306 1135.22,785.948 1135.72,782.455 1136.22,778.861 1136.73,775.202 1137.23,771.513 1137.73,767.831 1138.24,764.193 1138.74,760.637 1139.24,757.198 \n",
" 1139.74,753.912 1140.25,750.813 1140.75,747.934 1141.25,745.307 1141.76,742.959 1142.26,740.916 1142.76,739.202 1143.26,737.838 1143.77,736.839 1144.27,736.219 \n",
" 1144.77,735.988 1145.27,736.152 1145.78,736.712 1146.28,737.667 1146.78,739.011 1147.29,740.734 1147.79,742.822 1148.29,745.258 1148.79,748.02 1149.3,751.085 \n",
" 1149.8,754.423 1150.3,758.004 1150.81,761.795 1151.31,765.759 1151.81,769.858 1152.31,774.053 1152.82,778.302 1153.32,782.564 1153.82,786.795 1154.32,790.952 \n",
" 1154.83,794.994 1155.33,798.878 1155.83,802.565 1156.34,806.014 1156.84,809.188 1157.34,812.054 1157.84,814.579 1158.35,816.734 1158.85,818.494 1159.35,819.836 \n",
" 1159.86,820.744 1160.36,821.202 1160.86,821.203 1161.36,820.74 1161.87,819.814 1162.37,818.429 1162.87,816.593 1163.37,814.321 1163.88,811.63 1164.38,808.543 \n",
" 1164.88,805.086 1165.39,801.289 1165.89,797.188 1166.39,792.82 1166.89,788.226 1167.4,783.449 1167.9,778.535 1168.4,773.532 1168.91,768.49 1169.41,763.458 \n",
" 1169.91,758.488 1170.41,753.629 1170.92,748.932 1171.42,744.447 1171.92,740.221 1172.42,736.299 1172.93,732.725 1173.43,729.539 1173.93,726.778 1174.44,724.474 \n",
" 1174.94,722.658 1175.44,721.352 1175.94,720.577 1176.45,720.348 1176.95,720.674 1177.45,721.558 1177.96,722.999 1178.46,724.99 1178.96,727.518 1179.46,730.565 \n",
" 1179.97,734.108 1180.47,738.116 1180.97,742.557 1181.47,747.391 1181.98,752.575 1182.48,758.062 1182.98,763.801 1183.49,769.738 1183.99,775.815 1184.49,781.974 \n",
" 1184.99,788.154 1185.5,794.292 1186,800.327 1186.5,806.196 1187,811.838 1187.51,817.192 1188.01,822.2 1188.51,826.807 1189.02,830.96 1189.52,834.61 \n",
" 1190.02,837.712 1190.52,840.228 1191.03,842.121 1191.53,843.364 1192.03,843.933 1192.54,843.812 1193.04,842.99 1193.54,841.465 1194.04,839.241 1194.55,836.327 \n",
" 1195.05,832.742 1195.55,828.51 1196.05,823.664 1196.56,818.241 1197.06,812.286 1197.56,805.849 1198.07,798.988 1198.57,791.764 1199.07,784.242 1199.57,776.494 \n",
" 1200.08,768.594 1200.58,760.618 1201.08,752.645 1201.59,744.757 1202.09,737.034 1202.59,729.557 1203.09,722.407 1203.6,715.662 1204.1,709.399 1204.6,703.69 \n",
" 1205.1,698.605 1205.61,694.207 1206.11,690.556 1206.61,687.703 1207.12,685.695 1207.62,684.57 1208.12,684.358 1208.62,685.082 1209.13,686.754 1209.63,689.38 \n",
" 1210.13,692.954 1210.64,697.461 1211.14,702.878 1211.64,709.171 1212.14,716.298 1212.65,724.205 1213.15,732.833 1213.65,742.11 1214.15,751.96 1214.66,762.297 \n",
" 1215.16,773.028 1215.66,784.055 1216.17,795.274 1216.67,806.576 1217.17,817.848 1217.67,828.976 1218.18,839.841 1218.68,850.326 1219.18,860.315 1219.69,869.689 \n",
" 1220.19,878.337 1220.69,886.148 1221.19,893.017 1221.7,898.846 1222.2,903.541 1222.7,907.02 1223.2,909.206 1223.71,910.036 1224.21,909.455 1224.71,907.42 \n",
" 1225.22,903.903 1225.72,898.886 1226.22,892.367 1226.72,884.357 1227.23,874.884 1227.73,863.988 1228.23,851.727 1228.74,838.171 1229.24,823.409 1229.74,807.543 \n",
" 1230.24,790.69 1230.75,772.982 1231.25,754.564 1231.75,735.598 1232.25,716.253 1232.76,696.715 1233.26,677.179 1233.76,657.851 1234.27,638.944 1234.77,620.681 \n",
" 1235.27,603.291 1235.77,587.009 1236.28,572.073 1236.78,558.725 1237.28,547.207 1237.78,537.763 1238.29,530.632 1238.79,526.053 1239.29,524.258 1239.8,525.472 \n",
" 1240.3,529.914 1240.8,537.791 1241.3,549.297 1241.81,564.611 1242.31,583.895 1242.81,607.289 1243.32,634.908 1243.82,666.837 1244.32,703.124 1244.82,743.77 \n",
" 1245.33,788.722 1245.83,837.854 1246.33,890.954 1246.83,947.691 1247.34,1007.59 1247.84,1069.99 1248.34,1133.97 1248.85,1198.32 1249.35,1261.41 1249.85,1321.14 \n",
" 1250.35,1374.81 1250.86,1419.04 1251.36,1449.75 1251.86,1462.26 1252.37,1451.51 1252.87,1412.71 1253.37,1342.2 1253.87,1238.73 1254.38,1104.63 1254.88,946.598 \n",
" 1255.38,775.359 1255.88,604.12 1256.39,446.085 1256.89,311.987 1257.39,208.514 1257.9,138.009 1258.4,99.2046 1258.9,88.4582 1259.4,100.964 1259.91,131.682 \n",
" 1260.41,175.914 1260.91,229.581 1261.42,289.31 1261.92,352.403 1262.42,416.75 1262.92,480.733 1263.43,543.129 1263.93,603.028 1264.43,659.765 1264.93,712.864 \n",
" 1265.44,761.997 1265.94,806.949 1266.44,847.595 1266.95,883.881 1267.45,915.811 1267.95,943.43 1268.45,966.824 1268.96,986.108 1269.46,1001.42 1269.96,1012.93 \n",
" 1270.47,1020.8 1270.97,1025.25 1271.47,1026.46 1271.97,1024.67 1272.48,1020.09 1272.98,1012.96 1273.48,1003.51 1273.98,991.994 1274.49,978.646 1274.99,963.71 \n",
" 1275.49,947.428 1276,930.038 1276.5,911.775 1277,892.868 1277.5,873.539 1278.01,854.003 1278.51,834.466 1279.01,815.121 1279.52,796.154 1280.02,777.737 \n",
" 1280.52,760.029 1281.02,743.176 1281.53,727.31 1282.03,712.548 1282.53,698.992 1283.03,686.73 1283.54,675.835 1284.04,666.361 1284.54,658.352 1285.05,651.833 \n",
" 1285.55,646.816 1286.05,643.299 1286.55,641.264 1287.06,640.683 1287.56,641.512 1288.06,643.699 1288.56,647.177 1289.07,651.873 1289.57,657.701 1290.07,664.571 \n",
" 1290.58,672.382 1291.08,681.03 1291.58,690.404 1292.08,700.392 1292.59,710.878 1293.09,721.743 1293.59,732.87 1294.1,744.143 1294.6,755.445 1295.1,766.663 \n",
" 1295.6,777.691 1296.11,788.422 1296.61,798.759 1297.11,808.609 1297.61,817.886 1298.12,826.513 1298.62,834.421 1299.12,841.547 1299.63,847.841 1300.13,853.258 \n",
" 1300.63,857.765 1301.13,861.339 1301.64,863.965 1302.14,865.637 1302.64,866.361 1303.15,866.149 1303.65,865.024 1304.15,863.016 1304.65,860.163 1305.16,856.512 \n",
" 1305.66,852.114 1306.16,847.029 1306.66,841.32 1307.17,835.057 1307.67,828.312 1308.17,821.162 1308.68,813.685 1309.18,805.962 1309.68,798.074 1310.18,790.101 \n",
" 1310.69,782.125 1311.19,774.225 1311.69,766.477 1312.2,758.955 1312.7,751.731 1313.2,744.869 1313.7,738.433 1314.21,732.478 1314.71,727.055 1315.21,722.208 \n",
" 1315.71,717.977 1316.22,714.392 1316.72,711.478 1317.22,709.253 1317.73,707.728 1318.23,706.907 1318.73,706.786 1319.23,707.355 1319.74,708.597 1320.24,710.491 \n",
" 1320.74,713.006 1321.25,716.109 1321.75,719.759 1322.25,723.912 1322.75,728.518 1323.26,733.527 1323.76,738.881 1324.26,744.523 1324.76,750.392 1325.27,756.427 \n",
" 1325.77,762.565 1326.27,768.745 1326.78,774.904 1327.28,780.981 1327.78,786.918 1328.28,792.657 1328.79,798.144 1329.29,803.328 1329.79,808.162 1330.3,812.603 \n",
" 1330.8,816.611 1331.3,820.153 1331.8,823.2 1332.31,825.729 1332.81,827.72 1333.31,829.161 1333.81,830.045 1334.32,830.371 1334.82,830.141 1335.32,829.367 \n",
" 1335.83,828.061 1336.33,826.245 1336.83,823.941 1337.33,821.18 1337.84,817.994 1338.34,814.42 1338.84,810.498 1339.35,806.272 1339.85,801.786 1340.35,797.09 \n",
" 1340.85,792.231 1341.36,787.261 1341.86,782.229 1342.36,777.186 1342.86,772.184 1343.37,767.27 1343.87,762.493 1344.37,757.899 1344.88,753.531 1345.38,749.43 \n",
" 1345.88,745.633 1346.38,742.176 1346.89,739.089 1347.39,736.398 1347.89,734.125 1348.39,732.29 1348.9,730.905 1349.4,729.979 1349.9,729.516 1350.41,729.516 \n",
" 1350.91,729.975 1351.41,730.883 1351.91,732.225 1352.42,733.985 1352.92,736.14 1353.42,738.665 1353.93,741.53 1354.43,744.705 1354.93,748.154 1355.43,751.84 \n",
" 1355.94,755.725 1356.44,759.766 1356.94,763.924 1357.44,768.155 1357.95,772.416 1358.45,776.666 1358.95,780.861 1359.46,784.96 1359.96,788.924 1360.46,792.715 \n",
" 1360.96,796.296 1361.47,799.634 1361.97,802.698 1362.47,805.461 1362.98,807.896 1363.48,809.985 1363.98,811.707 1364.48,813.051 1364.99,814.007 1365.49,814.567 \n",
" 1365.99,814.731 1366.49,814.5 1367,813.88 1367.5,812.881 1368,811.517 1368.51,809.803 1369.01,807.76 1369.51,805.412 1370.01,802.785 1370.52,799.906 \n",
" 1371.02,796.807 1371.52,793.521 1372.03,790.082 1372.53,786.526 1373.03,782.888 1373.53,779.206 1374.04,775.517 1374.54,771.858 1375.04,768.264 1375.54,764.771 \n",
" 1376.05,761.413 1376.55,758.222 1377.05,755.228 1377.56,752.461 1378.06,749.944 1378.56,747.702 1379.06,745.754 1379.57,744.118 1380.07,742.806 1380.57,741.83 \n",
" 1381.08,741.197 1381.58,740.91 1382.08,740.969 1382.58,741.371 1383.09,742.109 1383.59,743.173 1384.09,744.551 1384.59,746.225 1385.1,748.177 1385.6,750.385 \n",
" 1386.1,752.826 1386.61,755.472 1387.11,758.298 1387.61,761.272 1388.11,764.364 1388.62,767.542 1389.12,770.775 1389.62,774.029 1390.13,777.272 1390.63,780.472 \n",
" 1391.13,783.598 1391.63,786.618 1392.14,789.503 1392.64,792.226 1393.14,794.761 1393.64,797.084 1394.15,799.173 1394.65,801.009 1395.15,802.576 1395.66,803.86 \n",
" 1396.16,804.851 1396.66,805.54 1397.16,805.923 1397.67,805.998 1398.17,805.767 1398.67,805.234 1399.17,804.406 1399.68,803.294 1400.18,801.912 1400.68,800.273 \n",
" 1401.19,798.398 1401.69,796.305 1402.19,794.018 1402.69,791.56 1403.2,788.957 1403.7,786.236 1404.2,783.424 1404.71,780.552 1405.21,777.646 1405.71,774.738 \n",
" 1406.21,771.855 1406.72,769.026 1407.22,766.279 1407.72,763.64 1408.22,761.136 1408.73,758.791 1409.23,756.626 1409.73,754.662 1410.24,752.918 1410.74,751.409 \n",
" 1411.24,750.148 1411.74,749.147 1412.25,748.415 1412.75,747.956 1413.25,747.773 1413.76,747.867 1414.26,748.234 1414.76,748.87 1415.26,749.767 1415.77,750.913 \n",
" 1416.27,752.296 1416.77,753.9 1417.27,755.709 1417.78,757.702 1418.28,759.86 1418.78,762.159 1419.29,764.575 1419.79,767.085 1420.29,769.662 1420.79,772.279 \n",
" 1421.3,774.912 1421.8,777.534 1422.3,780.117 1422.81,782.638 1423.31,785.071 1423.81,787.393 1424.31,789.581 1424.82,791.614 1425.32,793.472 1425.82,795.139 \n",
" 1426.32,796.599 1426.83,797.838 1427.33,798.845 1427.83,799.613 1428.34,800.133 1428.84,800.404 1429.34,800.423 1429.84,800.192 1430.35,799.714 1430.85,798.996 \n",
" 1431.35,798.047 1431.86,796.876 1432.36,795.497 1432.86,793.925 1433.36,792.177 1433.87,790.27 1434.37,788.226 1434.87,786.065 1435.37,783.808 1435.88,781.481 \n",
" 1436.38,779.105 1436.88,776.704 1437.39,774.304 1437.89,771.927 1438.39,769.598 1438.89,767.339 1439.4,765.172 1439.9,763.119 1440.4,761.199 1440.91,759.431 \n",
" 1441.41,757.831 1441.91,756.415 1442.41,755.196 1442.92,754.184 1443.42,753.389 1443.92,752.817 1444.42,752.474 1444.93,752.361 1445.43,752.477 1445.93,752.822 \n",
" 1446.44,753.39 1446.94,754.173 1447.44,755.165 1447.94,756.352 1448.45,757.723 1448.95,759.263 1449.45,760.956 1449.95,762.783 1450.46,764.727 1450.96,766.767 \n",
" 1451.46,768.882 1451.97,771.051 1452.47,773.251 1452.97,775.462 1453.47,777.66 1453.98,779.825 1454.48,781.934 1454.98,783.966 1455.49,785.903 1455.99,787.724 \n",
" 1456.49,789.413 1456.99,790.954 1457.5,792.331 1458,793.532 1458.5,794.545 1459,795.362 1459.51,795.976 1460.01,796.381 1460.51,796.574 1461.02,796.554 \n",
" 1461.52,796.323 1462.02,795.884 1462.52,795.242 1463.03,794.405 1463.53,793.382 1464.03,792.184 1464.54,790.824 1465.04,789.316 1465.54,787.676 1466.04,785.92 \n",
" 1466.55,784.068 1467.05,782.138 1467.55,780.149 1468.05,778.122 1468.56,776.077 1469.06,774.034 1469.56,772.014 1470.07,770.037 1470.57,768.123 1471.07,766.289 \n",
" 1471.57,764.555 1472.08,762.936 1472.58,761.449 1473.08,760.107 1473.59,758.924 1474.09,757.911 1474.59,757.076 1475.09,756.427 1475.6,755.97 1476.1,755.709 \n",
" 1476.6,755.645 1477.1,755.779 1477.61,756.107 1478.11,756.625 1478.61,757.329 1479.12,758.209 1479.62,759.256 1480.12,760.459 1480.62,761.806 1481.13,763.281 \n",
" 1481.63,764.871 1482.13,766.559 1482.64,768.326 1483.14,770.157 1483.64,772.031 1484.14,773.93 1484.65,775.835 1485.15,777.726 1485.65,779.586 1486.15,781.396 \n",
" 1486.66,783.138 1487.16,784.794 1487.66,786.349 1488.17,787.788 1488.67,789.096 1489.17,790.262 1489.67,791.274 1490.18,792.123 1490.68,792.801 1491.18,793.302 \n",
" 1491.69,793.622 1492.19,793.758 1492.69,793.71 1493.19,793.479 1493.7,793.068 1494.2,792.483 1494.7,791.728 1495.2,790.814 1495.71,789.749 1496.21,788.545 \n",
" 1496.71,787.215 1497.22,785.772 1497.72,784.23 1498.22,782.607 1498.72,780.918 1499.23,779.181 1499.73,777.412 1500.23,775.63 1500.73,773.853 1501.24,772.099 \n",
" 1501.74,770.384 1502.24,768.725 1502.75,767.14 1503.25,765.643 1503.75,764.249 1504.25,762.971 1504.76,761.822 1505.26,760.813 1505.76,759.954 1506.27,759.251 \n",
" 1506.77,758.712 1507.27,758.341 1507.77,758.142 1508.28,758.115 1508.78,758.261 1509.28,758.576 1509.78,759.058 1510.29,759.7 1510.79,760.497 1511.29,761.438 \n",
" 1511.8,762.515 1512.3,763.715 1512.8,765.028 1513.3,766.438 1513.81,767.931 1514.31,769.494 1514.81,771.108 1515.32,772.759 1515.82,774.429 1516.32,776.102 \n",
" 1516.82,777.762 1517.33,779.39 1517.83,780.973 1518.33,782.493 1518.83,783.937 1519.34,785.289 1519.84,786.537 1520.34,787.669 1520.85,788.674 1521.35,789.542 \n",
" 1521.85,790.265 1522.35,790.836 1522.86,791.252 1523.36,791.506 1523.86,791.599 1524.37,791.53 1524.87,791.299 1525.37,790.91 1525.87,790.367 1526.38,789.677 \n",
" 1526.88,788.846 1527.38,787.884 1527.88,786.8 1528.39,785.606 1528.89,784.314 1529.39,782.938 1529.9,781.491 1530.4,779.988 1530.9,778.444 1531.4,776.875 \n",
" 1531.91,775.297 1532.41,773.725 1532.91,772.175 1533.42,770.663 1533.92,769.203 1534.42,767.809 1534.92,766.496 1535.43,765.276 1535.93,764.161 1536.43,763.162 \n",
" 1536.93,762.288 1537.44,761.548 1537.94,760.948 1538.44,760.494 1538.95,760.19 1539.45,760.039 1539.95,760.041 1540.45,760.196 1540.96,760.502 1541.46,760.954 \n",
" 1541.96,761.55 1542.47,762.28 1542.97,763.139 1543.47,764.117 1543.97,765.203 1544.48,766.388 1544.98,767.657 1545.48,768.999 1545.98,770.4 1546.49,771.845 \n",
" 1546.99,773.321 1547.49,774.812 1548,776.303 1548.5,777.78 1549,779.227 1549.5,780.631 1550.01,781.978 1550.51,783.254 1551.01,784.446 1551.52,785.544 \n",
" 1552.02,786.537 1552.52,787.415 1553.02,788.169 1553.53,788.793 1554.03,789.281 1554.53,789.628 1555.03,789.832 1555.54,789.89 1556.04,789.804 1556.54,789.573 \n",
" 1557.05,789.202 1557.55,788.693 1558.05,788.053 1558.55,787.289 1559.06,786.408 1559.56,785.419 1560.06,784.334 1560.56,783.162 1561.07,781.917 1561.57,780.61 \n",
" 1562.07,779.255 1562.58,777.865 1563.08,776.455 1563.58,775.039 1564.08,773.631 1564.59,772.244 1565.09,770.893 1565.59,769.591 1566.1,768.351 1566.6,767.184 \n",
" 1567.1,766.103 1567.6,765.118 1568.11,764.239 1568.61,763.473 1569.11,762.828 1569.61,762.31 1570.12,761.924 1570.62,761.674 1571.12,761.561 1571.63,761.586 \n",
" 1572.13,761.748 1572.63,762.046 1573.13,762.476 1573.64,763.033 1574.14,763.711 1574.64,764.503 1575.15,765.401 1575.65,766.396 1576.15,767.477 1576.65,768.634 \n",
" 1577.16,769.853 1577.66,771.125 1578.16,772.434 1578.66,773.768 1579.17,775.115 1579.67,776.459 1580.17,777.788 1580.68,779.089 1581.18,780.349 1581.68,781.555 \n",
" 1582.18,782.696 1582.69,783.76 1583.19,784.736 1583.69,785.616 1584.2,786.391 1584.7,787.054 1585.2,787.598 1585.7,788.018 1586.21,788.31 1586.71,788.472 \n",
" 1587.21,788.503 1587.71,788.403 1588.22,788.172 1588.72,787.815 1589.22,787.334 1589.73,786.735 1590.23,786.025 1590.73,785.21 1591.23,784.299 1591.74,783.302 \n",
" 1592.24,782.228 1592.74,781.089 1593.25,779.897 1593.75,778.662 1594.25,777.398 1594.75,776.118 1595.26,774.834 1595.76,773.559 1596.26,772.305 1596.76,771.086 \n",
" 1597.27,769.913 1597.77,768.798 1598.27,767.751 1598.78,766.784 1599.28,765.905 1599.78,765.123 1600.28,764.445 1600.79,763.879 1601.29,763.428 1601.79,763.098 \n",
" 1602.3,762.891 1602.8,762.81 1603.3,762.854 1603.8,763.022 1604.31,763.313 1604.81,763.724 1605.31,764.249 1605.81,764.884 1606.32,765.622 1606.82,766.455 \n",
" 1607.32,767.374 1607.83,768.37 1608.33,769.434 1608.83,770.553 1609.33,771.717 1609.84,772.914 1610.34,774.133 1610.84,775.36 1611.34,776.583 1611.85,777.791 \n",
" 1612.35,778.971 1612.85,780.112 1613.36,781.202 1613.86,782.231 1614.36,783.189 1614.86,784.065 1615.37,784.852 1615.87,785.543 1616.37,786.129 1616.88,786.607 \n",
" 1617.38,786.971 1617.88,787.218 1618.38,787.346 1618.89,787.354 1619.39,787.242 1619.89,787.012 1620.39,786.666 1620.9,786.208 1621.4,785.644 1621.9,784.978 \n",
" 1622.41,784.218 1622.91,783.372 1623.41,782.448 1623.91,781.455 1624.42,780.405 1624.92,779.307 1625.42,778.173 1625.93,777.014 1626.43,775.841 1626.93,774.667 \n",
" 1627.43,773.502 1627.94,772.36 1628.44,771.25 1628.94,770.185 1629.44,769.174 1629.95,768.227 1630.45,767.355 1630.95,766.564 1631.46,765.863 1631.96,765.259 \n",
" 1632.46,764.758 1632.96,764.363 1633.47,764.08 1633.97,763.909 1634.47,763.854 1634.98,763.913 1635.48,764.087 1635.98,764.373 1636.48,764.767 1636.99,765.266 \n",
" 1637.49,765.865 1637.99,766.557 1638.49,767.335 1639,768.191 1639.5,769.116 1640,770.101 1640.51,771.137 1641.01,772.211 1641.51,773.314 1642.01,774.435 \n",
" 1642.52,775.562 1643.02,776.684 1643.52,777.79 1644.03,778.869 1644.53,779.91 1645.03,780.903 1645.53,781.838 1646.04,782.706 1646.54,783.498 1647.04,784.208 \n",
" 1647.54,784.826 1648.05,785.349 1648.55,785.771 1649.05,786.088 1649.56,786.297 1650.06,786.396 1650.56,786.385 1651.06,786.264 1651.57,786.034 1652.07,785.698 \n",
" 1652.57,785.26 1653.08,784.724 1653.58,784.096 1654.08,783.382 1654.58,782.591 1655.09,781.729 1655.59,780.805 1656.09,779.829 1656.59,778.812 1657.1,777.762 \n",
" 1657.6,776.691 1658.1,775.61 1658.61,774.528 1659.11,773.458 1659.61,772.409 1660.11,771.392 1660.62,770.418 1661.12,769.495 1661.62,768.633 1662.12,767.84 \n",
" 1662.63,767.124 1663.13,766.493 1663.63,765.951 1664.14,765.504 1664.64,765.158 1665.14,764.913 1665.64,764.774 1666.15,764.741 1666.65,764.813 1667.15,764.991 \n",
" 1667.66,765.272 1668.16,765.653 1668.66,766.13 1669.16,766.697 1669.67,767.35 1670.17,768.082 1670.67,768.884 1671.17,769.749 1671.68,770.668 1672.18,771.631 \n",
" 1672.68,772.629 1673.19,773.652 1673.69,774.69 1674.19,775.732 1674.69,776.768 1675.2,777.787 1675.7,778.779 1676.2,779.735 1676.71,780.645 1677.21,781.501 \n",
" 1677.71,782.292 1678.21,783.013 1678.72,783.656 1679.22,784.214 1679.72,784.682 1680.22,785.057 1680.73,785.333 1681.23,785.51 1681.73,785.584 1682.24,785.557 \n",
" 1682.74,785.427 1683.24,785.198 1683.74,784.871 1684.25,784.449 1684.75,783.938 1685.25,783.343 1685.76,782.669 1686.26,781.923 1686.76,781.114 1687.26,780.25 \n",
" 1687.77,779.338 1688.27,778.389 1688.77,777.412 1689.27,776.417 1689.78,775.413 1690.28,774.412 1690.78,773.422 1691.29,772.453 1691.79,771.516 1692.29,770.619 \n",
" 1692.79,769.772 1693.3,768.983 1693.8,768.259 1694.3,767.607 1694.81,767.035 1695.31,766.546 1695.81,766.147 1696.31,765.841 1696.82,765.631 1697.32,765.518 \n",
" 1697.82,765.503 1698.32,765.587 1698.83,765.769 1699.33,766.045 1699.83,766.414 1700.34,766.872 1700.84,767.413 1701.34,768.033 1701.84,768.724 1702.35,769.479 \n",
" 1702.85,770.292 1703.35,771.154 1703.86,772.055 1704.36,772.988 1704.86,773.942 1705.36,774.908 1705.87,775.877 1706.37,776.838 1706.87,777.782 1707.37,778.7 \n",
" 1707.88,779.583 1708.38,780.421 1708.88,781.207 1709.39,781.933 1709.89,782.592 1710.39,783.177 1710.89,783.683 1711.4,784.104 1711.9,784.438 1712.4,784.68 \n",
" 1712.91,784.828 1713.41,784.882 1713.91,784.84 1714.41,784.704 1714.92,784.474 1715.42,784.155 1715.92,783.748 1716.42,783.258 1716.93,782.691 1717.43,782.051 \n",
" 1717.93,781.347 1718.44,780.584 1718.94,779.77 1719.44,778.914 1719.94,778.025 1720.45,777.11 1720.95,776.181 1721.45,775.245 1721.95,774.312 1722.46,773.392 \n",
" 1722.96,772.493 1723.46,771.625 1723.97,770.796 1724.47,770.015 1724.97,769.288 1725.47,768.624 1725.98,768.028 1726.48,767.507 1726.98,767.065 1727.49,766.707 \n",
" 1727.99,766.436 1728.49,766.255 1728.99,766.165 1729.5,766.167 1730,766.261 1730.5,766.445 1731,766.718 1731.51,767.077 1732.01,767.518 1732.51,768.036 \n",
" 1733.02,768.626 1733.52,769.282 1734.02,769.997 1734.52,770.764 1735.03,771.576 1735.53,772.424 1736.03,773.299 1736.54,774.193 1737.04,775.097 1737.54,776.001 \n",
" 1738.04,776.897 1738.55,777.776 1739.05,778.629 1739.55,779.448 1740.05,780.224 1740.56,780.95 1741.06,781.619 1741.56,782.223 1742.07,782.758 1742.57,783.218 \n",
" 1743.07,783.599 1743.57,783.896 1744.08,784.108 1744.58,784.232 1745.08,784.267 1745.59,784.213 1746.09,784.071 1746.59,783.842 1747.09,783.529 1747.6,783.135 \n",
" 1748.1,782.664 1748.6,782.121 1749.1,781.512 1749.61,780.843 1750.11,780.12 1750.61,779.351 1751.12,778.544 1751.62,777.707 1752.12,776.848 1752.62,775.975 \n",
" 1753.13,775.099 1753.63,774.227 1754.13,773.368 1754.64,772.53 1755.14,771.723 1755.64,770.953 1756.14,770.229 1756.65,769.558 1757.15,768.945 1757.65,768.398 \n",
" 1758.15,767.922 1758.66,767.521 1759.16,767.199 1759.66,766.959 1760.17,766.804 1760.67,766.734 1761.17,766.75 1761.67,766.852 1762.18,767.04 1762.68,767.309 \n",
" 1763.18,767.659 1763.69,768.085 1764.19,768.582 1764.69,769.146 1765.19,769.772 1765.7,770.451 1766.2,771.179 1766.7,771.946 1767.2,772.747 1767.71,773.571 \n",
" 1768.21,774.412 1768.71,775.261 1769.22,776.109 1769.72,776.948 1770.22,777.77 1770.72,778.566 1771.23,779.328 1771.73,780.049 1772.23,780.722 1772.73,781.34 \n",
" 1773.24,781.897 1773.74,782.388 1774.24,782.808 1774.75,783.152 1775.25,783.418 1775.75,783.603 1776.25,783.706 1776.76,783.725 1777.26,783.66 1777.76,783.513 \n",
" 1778.27,783.284 1778.77,782.977 1779.27,782.594 1779.77,782.14 1780.28,781.619 1780.78,781.036 1781.28,780.398 1781.78,779.711 1782.29,778.982 1782.79,778.218 \n",
" 1783.29,777.427 1783.8,776.616 1784.3,775.795 1784.8,774.971 1785.3,774.152 1785.81,773.347 1786.31,772.564 1786.81,771.81 1787.32,771.093 1787.82,770.42 \n",
" 1788.32,769.797 1788.82,769.231 1789.33,768.727 1789.83,768.291 1790.33,767.926 1790.83,767.636 1791.34,767.423 1791.84,767.29 1792.34,767.238 1792.85,767.267 \n",
" 1793.35,767.377 1793.85,767.566 1794.35,767.833 1794.86,768.174 1795.36,768.587 1795.86,769.066 1796.37,769.608 1796.87,770.205 1797.37,770.853 1797.87,771.545 \n",
" 1798.38,772.274 1798.88,773.032 1799.38,773.812 1799.88,774.606 1800.39,775.406 1800.89,776.204 1801.39,776.992 1801.9,777.763 1802.4,778.508 1802.9,779.22 \n",
" 1803.4,779.892 1803.91,780.518 1804.41,781.091 1804.91,781.606 1805.42,782.058 1805.92,782.442 1806.42,782.754 1806.92,782.993 1807.43,783.154 1807.93,783.237 \n",
" 1808.43,783.242 1808.93,783.168 1809.44,783.016 1809.94,782.788 1810.44,782.486 1810.95,782.113 1811.45,781.674 1811.95,781.172 1812.45,780.614 1812.96,780.004 \n",
" 1813.46,779.348 1813.96,778.654 1814.47,777.928 1814.97,777.178 1815.47,776.412 1815.97,775.636 1816.48,774.858 1816.98,774.087 1817.48,773.331 1817.98,772.595 \n",
" 1818.49,771.889 1818.99,771.218 1819.49,770.591 1820,770.011 1820.5,769.487 1821,769.022 1821.5,768.62 1822.01,768.287 1822.51,768.026 1823.01,767.837 \n",
" 1823.51,767.725 1824.02,767.688 1824.52,767.729 1825.02,767.845 1825.53,768.036 1826.03,768.301 1826.53,768.635 1827.03,769.035 1827.54,769.498 1828.04,770.019 \n",
" 1828.54,770.592 1829.05,771.212 1829.55,771.872 1830.05,772.566 1830.55,773.286 1831.06,774.026 1831.56,774.778 1832.06,775.535 1832.56,776.288 1833.07,777.031 \n",
" 1833.57,777.755 1834.07,778.455 1834.58,779.122 1835.08,779.751 1835.58,780.335 1836.08,780.868 1836.59,781.345 1837.09,781.762 1837.59,782.114 1838.1,782.398 \n",
" 1838.6,782.611 1839.1,782.751 1839.6,782.818 1840.11,782.809 1840.61,782.727 1841.11,782.571 1841.61,782.343 1842.12,782.046 1842.62,781.682 1843.12,781.257 \n",
" 1843.63,780.772 1844.13,780.235 1844.63,779.65 1845.13,779.023 1845.64,778.361 1846.14,777.67 1846.64,776.957 1847.15,776.229 1847.65,775.494 1848.15,774.758 \n",
" 1848.65,774.03 1849.16,773.316 1849.66,772.624 1850.16,771.961 1850.66,771.332 1851.17,770.745 1851.67,770.205 1852.17,769.717 1852.68,769.287 1853.18,768.917 \n",
" 1853.68,768.613 1854.18,768.377 1854.69,768.21 1855.19,768.116 1855.69,768.093 1856.2,768.144 1856.7,768.266 1857.2,768.459 1857.7,768.721 1858.21,769.048 \n",
" 1858.71,769.438 1859.21,769.886 1859.71,770.389 1860.22,770.94 1860.72,771.534 1861.22,772.165 1861.73,772.828 1862.23,773.514 1862.73,774.218 1863.23,774.932 \n",
" 1863.74,775.649 1864.24,776.362 1864.74,777.064 1865.25,777.748 1865.75,778.406 1866.25,779.034 1866.75,779.623 1867.26,780.169 1867.76,780.666 1868.26,781.109 \n",
" 1868.76,781.494 1869.27,781.817 1869.77,782.076 1870.27,782.266 1870.78,782.388 1871.28,782.439 1871.78,782.419 1872.28,782.329 1872.79,782.169 1873.29,781.942 \n",
" 1873.79,781.649 1874.3,781.294 1874.8,780.88 1875.3,780.412 1875.8,779.894 1876.31,779.332 1876.81,778.731 1877.31,778.097 1877.81,777.437 1878.32,776.757 \n",
" 1878.82,776.065 1879.32,775.366 1879.83,774.669 1880.33,773.979 1880.83,773.305 1881.33,772.652 1881.84,772.027 1882.34,771.436 1882.84,770.886 1883.34,770.381 \n",
" 1883.85,769.926 1884.35,769.527 1884.85,769.186 1885.36,768.908 1885.86,768.694 1886.36,768.548 1886.86,768.469 1887.37,768.46 1887.87,768.519 1888.37,768.647 \n",
" 1888.88,768.841 1889.38,769.101 1889.88,769.422 1890.38,769.802 1890.89,770.237 1891.39,770.723 1891.89,771.253 1892.39,771.825 1892.9,772.43 1893.4,773.064 \n",
" 1893.9,773.72 1894.41,774.391 1894.91,775.071 1895.41,775.752 1895.91,776.429 1896.42,777.093 1896.92,777.74 1897.42,778.361 1897.93,778.952 1898.43,779.506 \n",
" 1898.93,780.018 1899.43,780.482 1899.94,780.895 1900.44,781.251 1900.94,781.548 1901.44,781.783 1901.95,781.954 1902.45,782.058 1902.95,782.095 1903.46,782.065 \n",
" 1903.96,781.968 1904.46,781.805 1904.96,781.578 1905.47,781.29 1905.97,780.942 1906.47,780.539 1906.98,780.086 1907.48,779.585 1907.98,779.044 1908.48,778.466 \n",
" 1908.99,777.858 1909.49,777.227 1909.99,776.577 1910.49,775.917 1911,775.251 1911.5,774.588 1912,773.934 1912.51,773.295 1913.01,772.677 1913.51,772.088 \n",
" 1914.01,771.531 1914.52,771.014 1915.02,770.541 1915.52,770.117 1916.03,769.746 1916.53,769.431 1917.03,769.176 1917.53,768.984 1918.04,768.855 1918.54,768.791 \n",
" 1919.04,768.793 1919.54,768.861 1920.05,768.993 1920.55,769.189 1921.05,769.446 1921.56,769.762 1922.06,770.133 1922.56,770.556 1923.06,771.026 1923.57,771.539 \n",
" 1924.07,772.089 1924.57,772.67 1925.08,773.278 1925.58,773.906 1926.08,774.547 1926.58,775.196 1927.09,775.845 1927.59,776.488 1928.09,777.119 1928.59,777.732 \n",
" 1929.1,778.32 1929.6,778.877 1930.1,779.399 1930.61,779.879 1931.11,780.314 1931.61,780.698 1932.11,781.029 1932.62,781.302 1933.12,781.516 1933.62,781.668 \n",
" 1934.12,781.757 1934.63,781.781 1935.13,781.742 1935.63,781.639 1936.14,781.473 1936.64,781.247 1937.14,780.962 1937.64,780.621 1938.15,780.229 1938.65,779.788 \n",
" 1939.15,779.304 1939.66,778.781 1940.16,778.225 1940.66,777.641 1941.16,777.035 1941.67,776.414 1942.17,775.782 1942.67,775.147 1943.17,774.516 1943.68,773.894 \n",
" 1944.18,773.287 1944.68,772.702 1945.19,772.144 1945.69,771.619 1946.19,771.132 1946.69,770.688 1947.2,770.292 1947.7,769.947 1948.2,769.656 1948.71,769.422 \n",
" 1949.21,769.249 1949.71,769.136 1950.21,769.086 1950.72,769.098 1951.22,769.173 1951.72,769.31 1952.22,769.507 1952.73,769.762 1953.23,770.072 1953.73,770.435 \n",
" 1954.24,770.847 1954.74,771.303 1955.24,771.799 1955.74,772.329 1956.25,772.89 1956.75,773.474 1957.25,774.076 1957.76,774.69 1958.26,775.31 1958.76,775.929 \n",
" 1959.26,776.542 1959.77,777.142 1960.27,777.724 1960.77,778.281 1961.27,778.808 1961.78,779.3 1962.28,779.752 1962.78,780.159 1963.29,780.518 1963.79,780.825 \n",
" 1964.29,781.077 1964.79,781.271 1965.3,781.406 1965.8,781.481 1966.3,781.494 1966.81,781.446 1967.31,781.337 1967.81,781.169 1968.31,780.943 1968.82,780.662 \n",
" 1969.32,780.328 1969.82,779.945 1970.32,779.516 1970.83,779.047 1971.33,778.541 1971.83,778.005 1972.34,777.443 1972.84,776.861 1973.34,776.264 1973.84,775.66 \n",
" 1974.35,775.053 1974.85,774.45 1975.35,773.857 1975.86,773.28 1976.36,772.725 1976.86,772.196 1977.36,771.7 1977.87,771.241 1978.37,770.824 1978.87,770.453 \n",
" 1979.37,770.131 1979.88,769.862 1980.38,769.648 1980.88,769.492 1981.39,769.394 1981.89,769.356 1982.39,769.378 1982.89,769.46 1983.4,769.601 1983.9,769.798 \n",
" 1984.4,770.052 1984.9,770.357 1985.41,770.713 1985.91,771.114 1986.41,771.557 1986.92,772.037 1987.42,772.55 1987.92,773.09 1988.42,773.653 1988.93,774.231 \n",
" 1989.43,774.82 1989.93,775.413 1990.44,776.006 1990.94,776.59 1991.44,777.162 1991.94,777.715 1992.45,778.244 1992.95,778.743 1993.45,779.208 1993.95,779.634 \n",
" 1994.46,780.016 1994.96,780.352 1995.46,780.637 1995.97,780.869 1996.47,781.045 1996.97,781.165 1997.47,781.226 1997.98,781.229 1998.48,781.174 1998.98,781.06 \n",
" 1999.49,780.889 1999.99,780.664 2000.49,780.386 2000.99,780.058 2001.5,779.683 2002,779.266 2002.5,778.81 2003,778.321 2003.51,777.803 2004.01,777.261 \n",
" 2004.51,776.7 2005.02,776.127 2005.52,775.548 2006.02,774.967 2006.52,774.39 2007.03,773.825 2007.53,773.275 2008.03,772.747 2008.54,772.245 2009.04,771.776 \n",
" 2009.54,771.342 2010.04,770.95 2010.55,770.602 2011.05,770.302 2011.55,770.053 2012.05,769.857 2012.56,769.717 2013.06,769.633 2013.56,769.606 2014.07,769.637 \n",
" 2014.57,769.725 2015.07,769.869 2015.57,770.067 2016.08,770.319 2016.58,770.62 2017.08,770.968 2017.59,771.36 2018.09,771.791 2018.59,772.257 2019.09,772.753 \n",
" 2019.6,773.275 2020.1,773.817 2020.6,774.374 2021.1,774.939 2021.61,775.509 2022.11,776.075 2022.61,776.635 2023.12,777.18 2023.62,777.707 2024.12,778.21 \n",
" 2024.62,778.683 2025.13,779.123 2025.63,779.524 2026.13,779.884 2026.64,780.197 2027.14,780.462 2027.64,780.676 2028.14,780.836 2028.65,780.942 2029.15,780.991 \n",
" 2029.65,780.985 2030.15,780.922 2030.66,780.803 2031.16,780.631 2031.66,780.406 2032.17,780.131 2032.67,779.808 2033.17,779.442 2033.67,779.035 2034.18,778.592 \n",
" 2034.68,778.118 2035.18,777.616 2035.69,777.093 2036.19,776.553 2036.69,776.002 2037.19,775.445 2037.7,774.887 2038.2,774.336 2038.7,773.795 2039.2,773.271 \n",
" 2039.71,772.768 2040.21,772.291 2040.71,771.846 2041.22,771.437 2041.72,771.067 2042.22,770.74 2042.72,770.461 2043.23,770.23 2043.73,770.051 2044.23,769.925 \n",
" 2044.73,769.854 2045.24,769.837 2045.74,769.876 2046.24,769.969 2046.75,770.117 2047.25,770.316 2047.75,770.566 2048.25,770.863 2048.76,771.204 2049.26,771.587 \n",
" 2049.76,772.007 2050.27,772.459 2050.77,772.941 2051.27,773.445 2051.77,773.969 2052.28,774.505 2052.78,775.049 2053.28,775.596 2053.78,776.14 2054.29,776.675 \n",
" 2054.79,777.196 2055.29,777.698 2055.8,778.177 2056.3,778.626 2056.8,779.043 2057.3,779.422 2057.81,779.76 2058.31,780.054 2058.81,780.3 2059.32,780.497 \n",
" 2059.82,780.642 2060.32,780.735 2060.82,780.773 2061.33,780.758 2061.83,780.688 2062.33,780.566 2062.83,780.391 2063.34,780.167 2063.84,779.895 2064.34,779.577 \n",
" 2064.85,779.219 2065.35,778.822 2065.85,778.391 2066.35,777.93 2066.86,777.444 2067.36,776.938 2067.86,776.417 2068.37,775.886 2068.87,775.35 2069.37,774.815 \n",
" 2069.87,774.286 2070.38,773.768 2070.88,773.267 2071.38,772.788 2071.88,772.335 2072.39,771.912 2072.89,771.525 2073.39,771.176 2073.9,770.87 2074.4,770.608 \n",
" 2074.9,770.395 2075.4,770.231 2075.91,770.119 2076.41,770.059 2076.91,770.052 2077.42,770.098 2077.92,770.197 2078.42,770.347 2078.92,770.547 2079.43,770.795 \n",
" 2079.93,771.088 2080.43,771.423 2080.93,771.798 2081.44,772.207 2081.94,772.647 2082.44,773.114 2082.95,773.603 2083.45,774.109 2083.95,774.626 2084.45,775.151 \n",
" 2084.96,775.676 2085.46,776.198 2085.96,776.711 2086.47,777.21 2086.97,777.69 2087.47,778.146 2087.97,778.573 2088.48,778.968 2088.98,779.326 2089.48,779.645 \n",
" 2089.98,779.92 2090.49,780.149 2090.99,780.33 2091.49,780.462 2092,780.542 2092.5,780.57 2093,780.546 2093.5,780.471 2094.01,780.344 2094.51,780.168 \n",
" 2095.01,779.944 2095.51,779.675 2096.02,779.363 2096.52,779.011 2097.02,778.623 2097.53,778.203 2098.03,777.755 2098.53,777.284 2099.03,776.794 2099.54,776.29 \n",
" 2100.04,775.778 2100.54,775.262 2101.05,774.748 2101.55,774.24 2102.05,773.744 2102.55,773.265 2103.06,772.808 2103.56,772.376 2104.06,771.975 2104.56,771.608 \n",
" 2105.07,771.278 2105.57,770.99 2106.07,770.746 2106.58,770.549 2107.08,770.399 2107.58,770.3 2108.08,770.251 2108.59,770.253 2109.09,770.306 2109.59,770.409 \n",
" 2110.1,770.562 2110.6,770.762 2111.1,771.008 2111.6,771.298 2112.11,771.627 2112.61,771.994 2113.11,772.393 2113.61,772.822 2114.12,773.276 2114.62,773.75 \n",
" 2115.12,774.239 2115.63,774.739 2116.13,775.245 2116.63,775.751 2117.13,776.253 2117.64,776.745 2118.14,777.222 2118.64,777.681 2119.15,778.116 2119.65,778.523 \n",
" 2120.15,778.898 2120.65,779.237 2121.16,779.537 2121.66,779.794 2122.16,780.008 2122.66,780.174 2123.17,780.293 2123.67,780.361 2124.17,780.38 2124.68,780.349 \n",
" 2125.18,780.268 2125.68,780.138 2126.18,779.96 2126.69,779.737 2127.19,779.47 2127.69,779.163 2128.2,778.818 2128.7,778.438 2129.2,778.029 2129.7,777.593 \n",
" 2130.21,777.135 2130.71,776.66 2131.21,776.173 2131.71,775.678 2132.22,775.181 2132.72,774.686 2133.22,774.198 2133.73,773.722 2134.23,773.264 2134.73,772.826 \n",
" 2135.23,772.415 2135.74,772.033 2136.24,771.685 2136.74,771.374 2137.25,771.104 2137.75,770.876 2138.25,770.693 2138.75,770.557 2139.26,770.469 2139.76,770.43 \n",
" 2140.26,770.44 2140.76,770.499 2141.27,770.607 2141.77,770.763 2142.27,770.963 2142.78,771.208 2143.28,771.494 2143.78,771.818 2144.28,772.177 2144.79,772.567 \n",
" 2145.29,772.985 2145.79,773.426 2146.29,773.886 2146.8,774.36 2147.3,774.844 2147.8,775.332 2148.31,775.82 2148.81,776.303 2149.31,776.775 2149.81,777.233 \n",
" 2150.32,777.672 2150.82,778.088 2151.32,778.475 2151.83,778.831 2152.33,779.152 2152.83,779.435 2153.33,779.676 2153.84,779.875 2154.34,780.028 2154.84,780.134 \n",
" 2155.34,780.192 2155.85,780.202 2156.35,780.164 2156.85,780.078 2157.36,779.945 2157.86,779.766 2158.36,779.543 2158.86,779.279 2159.37,778.976 2159.87,778.637 \n",
" 2160.37,778.266 2160.88,777.866 2161.38,777.441 2161.88,776.996 2162.38,776.536 2162.89,776.064 2163.39,775.586 2163.89,775.106 2164.39,774.629 2164.9,774.159 \n",
" 2165.4,773.703 2165.9,773.263 2166.41,772.845 2166.91,772.452 2167.41,772.089 2167.91,771.759 2168.42,771.465 2168.92,771.21 2169.42,770.998 2169.93,770.829 \n",
" 2170.43,770.705 2170.93,770.628 2171.43,770.598 2171.94,770.616 2172.44,770.681 2172.94,770.793 2173.44,770.951 2173.95,771.152 2174.45,771.395 2174.95,771.677 \n",
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},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"xx = -200:0.1:200\n",
"plot(xx,real.(f.(xx)))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"An equivalent result holds in the negative real axis, but be careful:"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"-4.529525111429054e-9 + 5.865568293152012e-7im"
]
},
"execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"z = -2-im\n",
"f = x -> exp(im*x)/(x+im)\n",
"sum(Fun(x-> f(x)/(x - z), -500 .. 500))/(2π*im)"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(0.09196985577264773 - 0.09196926921581833im, 0.9007327639404081 + 0.3351305720620013im)"
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
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],
"source": [
"z = -2-im\n",
"f = x -> exp(im*x)/(x-im)\n",
"sum(Fun(x-> f(x)/(x - z), -500 .. 500))/(2π*im), f(z)"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(-0.045354995402086956 - 0.1219015148055592im, -0.04535499089125899 - 0.12190092372837213im)"
]
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"z = -2-im\n",
"f = x -> exp(-im*x)/(x-im)\n",
"-sum(Fun(x-> f(x)/(x - z), -500 .. 500))/(2π*im), f(z)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Jordan's lemma\n",
"\n",
"The case of calculating\n",
"$$\n",
" \\int_{-\\infty}^\\infty e^{i \\omega x} g(x) dx\n",
"$$ \n",
"is important because it is the Fourier transform of $g(x)$. Provided $g$ is defined in the upper half plane and $\\omega > 0$, $f(z) = e^{i \\omega z} g(z)$ has exponential decay."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can use this to get sharper results than ML inequality:\n",
"\n",
"**Lemma (Jordan)** Assume $\\omega > 0$. If $g(z)$ is continuous in on the half circle $C_R = \\{ R e^{i \\theta} : 0 \\leq \\theta \\leq \\pi \\}$ then\n",
"$$\\left| \\int_{C_R} g(z) e^{i \\omega z} dz \\right| \\leq {\\pi \\over \\omega} M$$\n",
"where $M = \\sup_{z \\in C_R} |g(z)|$. \n",
"\n",
"**Sketch of proof** We have\n",
"$$\\left| \\int_{C_R} g(z) e^{i \\omega z} dz \\right| \\leq R \\int_0^\\pi \\left|g(R e^{i \\theta}) e^{i \\omega R e^{i \\theta}}e^{i \\theta}\\right| d\\theta \n",
"\\leq MR \\int_0^\\pi e^{- \\omega R\\sin \\theta } d\\theta \n",
"= 2MR \\int_0^{\\pi\\over 2} e^{- \\omega R\\sin \\theta } d\\theta \n",
"$$\n",
"But we have $\\sin \\theta \\geq {2 \\theta \\over \\pi}$:"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {},
"outputs": [
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" 1671.41,490.885 1691.74,477.008 1712.06,463.132 1732.39,449.255 1752.72,435.378 1773.04,421.501 1793.37,407.624 1813.7,393.748 1834.02,379.871 1854.35,365.994 \n",
" 1874.68,352.117 1895.01,338.24 1915.33,324.364 1935.66,310.487 1955.99,296.61 1976.31,282.733 1996.64,268.856 2016.97,254.98 2037.29,241.103 2057.62,227.226 \n",
" 2077.95,213.349 2098.27,199.473 2118.6,185.596 2138.93,171.719 2159.25,157.842 2179.58,143.965 2199.91,130.089 2220.24,116.212 2240.56,102.335 2260.89,88.4582 \n",
" \n",
" \"/>\n",
"<polygon clip-path=\"url(#clip7101)\" points=\"\n",
"1891.52,312.204 2249.26,312.204 2249.26,130.764 1891.52,130.764 \n",
" \" fill=\"#ffffff\" fill-opacity=\"1\"/>\n",
"<polyline clip-path=\"url(#clip7101)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
" 1891.52,312.204 2249.26,312.204 2249.26,130.764 1891.52,130.764 1891.52,312.204 \n",
" \"/>\n",
"<polyline clip-path=\"url(#clip7101)\" style=\"stroke:#009af9; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
" 1915.52,191.244 2059.52,191.244 \n",
" \"/>\n",
"<g clip-path=\"url(#clip7101)\">\n",
"<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:start;\" transform=\"rotate(0, 2083.52, 208.744)\" x=\"2083.52\" y=\"208.744\">sin t</text>\n",
"</g>\n",
"<polyline clip-path=\"url(#clip7101)\" style=\"stroke:#e26f46; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
" 1915.52,251.724 2059.52,251.724 \n",
" \"/>\n",
"<g clip-path=\"url(#clip7101)\">\n",
"<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:start;\" transform=\"rotate(0, 2083.52, 269.224)\" x=\"2083.52\" y=\"269.224\">2t / pi</text>\n",
"</g>\n",
"</svg>\n"
]
},
"execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"θ = range(0; stop=π/2, length=100)\n",
"plot(θ, sin.(θ); label=\"sin t\")\n",
"plot!(θ, 2θ/π; label = \"2t / pi\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Hence \n",
"$$\\left| \\int_{C_R} g(z) e^{i \\omega z} dz \\right| \\leq 2MR \\int_0^{\\pi\\over 2} e^{- {2\\omega R\\theta \\over \\pi} } d\\theta = {\\pi \\over \\omega} (1 - e^{-\\omega R}) M \\leq {\\pi M \\over \\omega}.\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Application: Calculating Fourier integrals of weakly decaying functions\n",
"\n",
"Why is this useful? We can use it to apply Residue theorem to We already know $O(z^{-2})$ decay gives us the integral via Residue theorem. And if we only have $z^{-1}$ decay our integral does not converge absolutely:"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"7.600902584542065"
]
},
"execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f = x -> exp(im*x)*x/(x^2+1)\n",
"sum(abs.(Fun(f, 0 .. 2000)))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"However, it does converge conditionally: \n",
"$$\n",
"\\dashint_{\\infty}^\\infty f(x) \\dx := \\lim_{M\\rightarrow \\infty} \\int_{-M}^M f(x) \\dx\n",
"$$\n",
"converges:"
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"-6.776263578034403e-21 + 1.1557671135433842im"
]
},
"execution_count": 21,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"f = x -> exp(im*x)*x/(x^2+1)\n",
"\n",
"sum(Fun(f, -30000 .. 30000))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Thus we can construct a Residue theorem for calculating \n",
"$$\n",
"\\infdashint g(x) e^{i \\omega x} \\dx\n",
"$$\n",
"provided that $g(z) \\rightarrow 0$ and is analytic in the upper-half plane."
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.0 + 1.1557273497909217im"
]
},
"execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"2π*im*exp(-1)*im/(im+im) # 2π*im* residue of g(z)exp(im*z) at z = im"
]
}
],
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"display_name": "Julia 1.0.1",
"language": "julia",
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