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"using Plots, ComplexPhasePortrait, ApproxFun, SingularIntegralEquations\n",
"gr();"
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"# M3M6: Methods of Mathematical Physics\n",
"\n",
"$$\n",
"\\def\\dashint{{\\int\\!\\!\\!\\!\\!\\!-\\,}}\n",
"\\def\\infdashint{\\dashint_{\\!\\!\\!-\\infty}^{\\,\\infty}}\n",
"\\def\\D{\\,{\\rm d}}\n",
"\\def\\dx{\\D x}\n",
"\\def\\dt{\\D t}\n",
"\\def\\C{{\\mathbb C}}\n",
"\\def\\CC{{\\cal C}}\n",
"\\def\\HH{{\\cal H}}\n",
"\\def\\I{{\\rm i}}\n",
"\\def\\qqfor{\\qquad\\hbox{for}\\qquad}\n",
"$$\n",
"\n",
"Dr. Sheehan Olver\n",
"<br>\n",
"s.olver@imperial.ac.uk\n",
"\n",
"Office Hours: 3-4pm Mondays, Huxley 6M40\n",
"<br>\n",
"Website: https://github.com/dlfivefifty/M3M6LectureNotes\n",
"\n",
"# Lecture 19: Logarithmic singular integrals\n",
"\n",
"1. Evaluating logarithmic singular integrals\n",
"2. Logarithmic singular integrals and orthogonal polynomials\n",
"2. Solving a logarithmic singular integral equation\n",
"2. Application: electrostatic potentials in 2D\n",
" - Potential arising from a point charge and a single plate\n",
" \n",
"The motivation behind this lecture is to calculate electrostatic potentials. An example is the Faraday cage: imagine a series of metal plates connected together so that they have the same charge. If configured to surround a region, this configuration will shield the interior from an external charge:\n",
"\n",
"\n",
"\n",
"Here, the coloured lines are equipotential lines, and there is a point source at $x = 2$, which corresponds to a forcing of \n",
"$$\\log\\| (x,y) - (2,0) \\| = \\log|z - 2|$$\n",
"where $z = x + \\I y$. "
]
},
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"source": [
"## Logarithmic singular integrals\n",
"\n",
"From the Green's function of the Laplacian, it is natural to consider logarithmic singular integrals, and we focus yet again on $[-1,1]$:\n",
"\n",
"$$v(z) = {1 \\over \\pi }\\int_{-1}^1 f(t) \\log | z - t| \\dt$$\n",
"\n",
"Note that off $[-1,1]$, $v(z)$ solves Laplace's equation, and is continuous on $[-1,1]$:"
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" 1917.23,439.721 1916.32,438.778 1912.96,435.343 1912.67,435.059 1909.54,431.909 1908.12,430.492 1906.08,428.474 1903.56,426.014 1902.56,425.04 1899.01,421.622 \n",
" 1898.99,421.605 1895.36,418.171 1894.45,417.314 1891.68,414.736 1889.89,413.086 1887.94,411.302 1885.34,408.936 1884.15,407.867 1880.78,404.86 1880.3,404.433 \n",
" 1876.39,400.998 1876.22,400.856 1872.41,397.564 1871.67,396.923 1868.38,394.129 1867.11,393.058 1864.29,390.695 1862.55,389.259 1860.13,387.26 1858,385.524 \n",
" 1855.9,383.826 1853.44,381.85 1851.61,380.391 1848.89,378.238 1847.25,376.957 1844.33,374.684 1842.82,373.522 1839.77,371.187 1838.32,370.088 1835.22,367.747 \n",
" 1833.75,366.653 1830.66,364.36 1829.11,363.219 1826.1,361.027 1824.39,359.784 1821.55,357.745 1819.59,356.35 1816.99,354.514 1814.71,352.915 1812.43,351.333 \n",
" 1809.75,349.481 1807.88,348.2 1804.7,346.046 1803.32,345.114 1799.57,342.612 1798.77,342.074 1794.36,339.177 1794.21,339.08 1789.65,336.13 1789.05,335.743 \n",
" 1785.1,333.223 1783.65,332.308 1780.54,330.359 1778.15,328.874 1775.98,327.537 1772.55,325.439 1771.43,324.757 1766.87,322.017 1766.85,322.005 1762.31,319.315 \n",
" 1761.04,318.57 1757.76,316.652 1755.13,315.136 1753.2,314.028 1749.1,311.701 1748.65,311.442 1744.09,308.892 1742.96,308.267 1739.53,306.379 1736.69,304.832 \n",
" 1734.98,303.901 1730.42,301.46 1730.3,301.398 1725.86,299.051 1723.78,297.963 1721.31,296.677 1717.12,294.529 1716.75,294.338 1712.19,292.03 1710.32,291.094 \n",
" 1707.64,289.756 1703.38,287.66 1703.08,287.515 1698.53,285.303 1696.27,284.225 1693.97,283.124 1689.41,280.977 1689.01,280.791 1684.86,278.858 1681.58,277.356 \n",
" 1680.3,276.771 1675.74,274.713 1673.97,273.922 1671.19,272.685 1666.63,270.686 1666.17,270.487 1662.07,268.714 1658.17,267.052 1657.52,266.773 1652.96,264.858 \n",
" 1649.97,263.618 1648.41,262.972 1643.85,261.112 1641.54,260.183 1639.29,259.28 1634.74,257.475 1632.88,256.749 1630.18,255.696 1625.62,253.943 1623.96,253.314 \n",
" 1621.07,252.216 1616.51,250.516 1614.78,249.88 1611.95,248.84 1607.4,247.19 1605.31,246.445 1602.84,245.565 1598.29,243.965 1595.53,243.011 1593.73,242.389 \n",
" 1589.17,240.837 1585.41,239.576 1584.62,239.311 1580.06,237.807 1575.5,236.328 1574.92,236.142 1570.95,234.871 1566.39,233.438 1564.03,232.707 1561.83,232.029 \n",
" 1557.28,230.642 1552.72,229.279 1552.7,229.273 1548.17,227.936 1543.61,226.618 1540.87,225.838 1539.05,225.322 1534.5,224.046 1529.94,222.795 1528.49,222.404 \n",
" 1525.38,221.563 1520.83,220.354 1516.27,219.168 1515.49,218.969 1511.71,218.001 1507.16,216.856 1502.6,215.733 1501.78,215.535 1498.04,214.63 1493.49,213.547 \n",
" 1488.93,212.487 1487.24,212.1 1484.38,211.446 1479.82,210.426 1475.26,209.427 1471.72,208.666 1470.71,208.448 1466.15,207.488 1461.59,206.549 1457.04,205.631 \n",
" 1455.01,205.231 1452.48,204.732 1447.92,203.851 1443.37,202.992 1438.81,202.153 1436.83,201.797 1434.26,201.332 1429.7,200.53 1425.14,199.748 1420.59,198.986 \n",
" 1416.76,198.362 1416.03,198.244 1411.47,197.518 1406.92,196.812 1402.36,196.126 1397.8,195.459 1394.06,194.928 1393.25,194.812 1388.69,194.181 1384.14,193.569 \n",
" 1379.58,192.976 1375.02,192.403 1370.47,191.848 1367.45,191.493 1365.91,191.311 1361.35,190.792 1356.8,190.291 1352.24,189.809 1347.68,189.345 1343.13,188.9 \n",
" 1338.57,188.474 1334.02,188.065 1333.94,188.059 1329.46,187.673 1324.9,187.3 1320.35,186.944 1315.79,186.607 1311.23,186.287 1306.68,185.986 1302.12,185.703 \n",
" 1297.56,185.437 1293.01,185.19 1288.45,184.96 1283.9,184.749 1280.98,184.624 1279.34,184.554 1274.78,184.377 1270.23,184.217 1265.67,184.076 1261.11,183.951 \n",
" 1256.56,183.845 1252,183.756 1247.44,183.685 1242.89,183.632 1238.33,183.596 1233.78,183.578 1229.22,183.577 1224.66,183.594 1220.11,183.629 1215.55,183.681 \n",
" 1210.99,183.751 1206.44,183.838 1201.88,183.943 1197.32,184.066 1192.77,184.206 1188.21,184.364 1183.66,184.54 1181.67,184.624 1179.1,184.733 1174.54,184.943 \n",
" 1169.99,185.17 1165.43,185.416 1160.87,185.679 1156.32,185.96 1151.76,186.259 1147.2,186.575 1142.65,186.91 1138.09,187.262 1133.54,187.633 1128.98,188.021 \n",
" 1128.56,188.059 1124.42,188.426 1119.87,188.849 1115.31,189.29 1110.75,189.749 1106.2,190.227 1101.64,190.723 1097.08,191.237 1094.89,191.493 1092.53,191.768 \n",
" 1087.97,192.317 1083.42,192.885 1078.86,193.471 1074.3,194.076 1069.75,194.7 1068.13,194.928 1065.19,195.341 1060.63,196 1056.08,196.677 1051.52,197.375 \n",
" 1046.96,198.091 1045.28,198.362 1042.41,198.825 1037.85,199.577 1033.3,200.348 1028.74,201.139 1025.04,201.797 1024.18,201.95 1019.63,202.777 1015.07,203.624 \n",
" 1010.51,204.492 1006.72,205.231 1005.96,205.379 1001.4,206.283 996.845,207.207 992.288,208.153 989.865,208.666 987.732,209.117 983.176,210.099 978.619,211.103 \n",
" 974.185,212.1 974.063,212.128 969.507,213.169 964.95,214.233 960.394,215.317 959.496,215.535 955.837,216.42 951.281,217.544 946.725,218.69 945.632,218.969 \n",
" 942.168,219.854 937.612,221.04 933.056,222.248 932.478,222.404 928.499,223.475 923.943,224.724 919.95,225.838 919.386,225.996 914.83,227.286 910.274,228.6 \n",
" 907.977,229.273 905.717,229.935 901.161,231.291 896.605,232.671 896.487,232.707 892.048,234.07 887.492,235.494 885.45,236.142 882.936,236.94 878.379,238.407 \n",
" 874.809,239.576 873.823,239.899 869.266,241.412 864.71,242.951 864.535,243.011 860.154,244.51 855.597,246.095 854.603,246.445 851.041,247.702 846.485,249.334 \n",
" 844.981,249.88 841.928,250.989 837.372,252.67 835.648,253.314 832.816,254.374 828.259,256.104 826.583,256.749 823.703,257.858 819.146,259.639 817.77,260.183 \n",
" 814.59,261.443 810.034,263.275 809.192,263.618 805.477,265.132 800.921,267.016 800.835,267.052 796.365,268.925 792.691,270.487 791.808,270.863 787.252,272.826 \n",
" 784.744,273.922 782.696,274.818 778.139,276.837 776.982,277.356 773.583,278.884 769.399,280.791 769.026,280.961 764.47,283.064 761.988,284.225 759.914,285.198 \n",
" 755.357,287.361 754.736,287.66 750.801,289.553 747.641,291.094 746.245,291.777 741.688,294.03 740.692,294.529 737.132,296.314 733.886,297.963 732.575,298.631 \n",
" 728.019,300.979 727.215,301.398 723.463,303.358 720.676,304.832 718.906,305.771 714.35,308.218 714.261,308.267 709.794,310.697 707.97,311.701 705.237,313.211 \n",
" 701.795,315.136 700.681,315.761 696.125,318.345 695.732,318.57 691.568,320.965 689.78,322.005 687.012,323.621 683.933,325.439 682.455,326.316 678.188,328.874 \n",
" 677.899,329.048 673.343,331.817 672.544,332.308 668.786,334.626 666.996,335.743 664.23,337.475 661.541,339.177 659.674,340.365 656.177,342.612 655.117,343.296 \n",
" 650.901,346.046 650.561,346.269 646.005,349.286 645.712,349.481 641.448,352.345 640.608,352.915 636.892,355.45 635.585,356.35 632.335,358.601 630.642,359.784 \n",
" 627.779,361.798 625.777,363.219 623.223,365.043 620.988,366.653 618.666,368.337 616.274,370.088 614.11,371.681 611.632,373.522 609.554,375.077 607.061,376.957 \n",
" 604.997,378.525 602.561,380.391 600.441,382.027 598.128,383.826 595.884,385.584 593.762,387.26 591.328,389.198 589.462,390.695 586.772,392.87 585.226,394.129 \n",
" 582.215,396.602 581.052,397.564 577.659,400.395 576.941,400.998 573.103,404.252 572.891,404.433 568.9,407.867 568.546,408.174 564.968,411.302 563.99,412.164 \n",
" 561.093,414.736 559.434,416.224 557.275,418.171 554.877,420.355 553.513,421.605 550.321,424.561 549.806,425.04 546.153,428.474 545.764,428.843 542.553,431.909 \n",
" 541.208,433.206 539.005,435.343 536.652,437.651 535.509,438.778 532.095,442.182 532.065,442.212 528.669,445.647 527.539,446.803 525.324,449.081 522.983,451.517 \n",
" 522.028,452.516 518.78,455.95 518.426,456.328 515.578,459.385 513.87,461.241 512.425,462.819 509.318,466.254 509.314,466.259 506.256,469.688 504.757,471.391 \n",
" 503.24,473.123 500.269,476.557 500.201,476.637 497.341,479.992 495.644,482.009 494.458,483.426 491.618,486.861 491.088,487.51 488.82,490.295 486.532,493.148 \n",
" 486.066,493.73 483.353,497.164 481.975,498.932 480.681,500.599 478.052,504.033 477.419,504.87 475.461,507.468 472.914,510.902 472.863,510.972 470.403,514.337 \n",
" 468.306,517.254 467.935,517.771 465.504,521.206 463.75,523.724 463.113,524.64 460.76,528.075 459.194,530.399 458.446,531.509 456.169,534.944 454.637,537.294 \n",
" 453.932,538.378 451.73,541.813 450.081,544.43 449.567,545.247 447.438,548.682 445.524,551.829 445.35,552.117 443.294,555.551 441.277,558.986 440.968,559.52 \n",
" 439.294,562.42 437.348,565.855 436.412,567.536 435.436,569.289 433.56,572.724 431.855,575.907 431.721,576.158 429.912,579.593 428.142,583.027 427.299,584.693 \n",
" 426.404,586.462 424.7,589.896 423.033,593.331 422.743,593.94 421.396,596.765 419.794,600.2 418.228,603.634 418.186,603.728 416.691,607.069 415.189,610.503 \n",
" 413.723,613.938 413.63,614.16 412.285,617.372 410.882,620.807 409.513,624.241 409.073,625.37 408.174,627.676 406.867,631.11 405.595,634.545 404.517,637.532 \n",
" 404.355,637.979 403.144,641.414 401.966,644.848 400.822,648.283 399.961,650.941 399.709,651.717 398.625,655.152 397.573,658.586 396.554,662.021 395.568,665.455 \n",
" 395.404,666.044 394.61,668.89 393.683,672.324 392.788,675.759 391.925,679.193 391.094,682.628 390.848,683.682 390.291,686.062 389.518,689.497 388.777,692.931 \n",
" 388.068,696.366 387.389,699.8 386.742,703.235 386.292,705.741 386.124,706.669 385.535,710.104 384.977,713.538 384.449,716.973 383.952,720.407 383.486,723.842 \n",
" 383.051,727.276 382.645,730.711 382.27,734.145 381.926,737.58 381.735,739.66 381.611,741.014 381.324,744.449 381.069,747.883 380.843,751.318 380.647,754.752 \n",
" 380.482,758.187 380.347,761.621 380.242,765.056 380.167,768.49 380.122,771.925 380.107,775.359 380.122,778.794 380.167,782.228 380.242,785.663 380.347,789.097 \n",
" 380.482,792.532 380.647,795.966 380.843,799.401 381.069,802.835 381.324,806.27 381.611,809.704 381.735,811.059 381.926,813.139 382.27,816.573 382.645,820.008 \n",
" 383.051,823.442 383.486,826.877 383.952,830.312 384.449,833.746 384.977,837.181 385.535,840.615 386.124,844.05 386.292,844.978 386.742,847.484 387.389,850.919 \n",
" 388.068,854.353 388.777,857.788 389.518,861.222 390.291,864.657 390.848,867.037 391.094,868.091 391.925,871.526 392.788,874.96 393.683,878.395 394.61,881.829 \n",
" 395.404,884.675 395.568,885.264 396.554,888.698 397.573,892.133 398.625,895.567 399.709,899.002 399.961,899.777 400.822,902.436 401.966,905.871 403.144,909.305 \n",
" 404.355,912.74 404.517,913.187 405.595,916.174 406.867,919.609 408.174,923.043 409.073,925.349 409.513,926.478 410.882,929.912 412.285,933.347 413.63,936.558 \n",
" 413.723,936.781 415.189,940.216 416.691,943.65 418.186,946.991 418.228,947.085 419.794,950.519 421.396,953.954 422.743,956.779 423.033,957.388 424.7,960.823 \n",
" 426.404,964.257 427.299,966.026 428.142,967.692 429.912,971.126 431.721,974.561 431.855,974.812 433.56,977.995 435.436,981.43 436.412,983.183 437.348,984.864 \n",
" 439.294,988.299 440.968,991.198 441.277,991.733 443.294,995.168 445.35,998.602 445.524,998.89 447.438,1002.04 449.567,1005.47 450.081,1006.29 451.73,1008.91 \n",
" 453.932,1012.34 454.637,1013.42 456.169,1015.77 458.446,1019.21 459.194,1020.32 460.76,1022.64 463.113,1026.08 463.75,1026.99 465.504,1029.51 467.935,1032.95 \n",
" 468.306,1033.47 470.403,1036.38 472.863,1039.75 472.914,1039.82 475.461,1043.25 477.419,1045.85 478.052,1046.69 480.681,1050.12 481.975,1051.79 483.353,1053.55 \n",
" 486.066,1056.99 486.532,1057.57 488.82,1060.42 491.088,1063.21 491.618,1063.86 494.458,1067.29 495.644,1068.71 497.341,1070.73 500.201,1074.08 500.269,1074.16 \n",
" 503.24,1077.6 504.757,1079.33 506.256,1081.03 509.314,1084.46 509.318,1084.46 512.425,1087.9 513.87,1089.48 515.578,1091.33 518.426,1094.39 518.78,1094.77 \n",
" 522.028,1098.2 522.983,1099.2 525.324,1101.64 527.539,1103.92 528.669,1105.07 532.065,1108.51 532.095,1108.54 535.509,1111.94 536.652,1113.07 539.005,1115.38 \n",
" 541.208,1117.51 542.553,1118.81 545.764,1121.88 546.153,1122.24 549.806,1125.68 550.321,1126.16 553.513,1129.11 554.877,1130.36 557.275,1132.55 559.434,1134.5 \n",
" 561.093,1135.98 563.99,1138.55 564.968,1139.42 568.546,1142.54 568.9,1142.85 572.891,1146.29 573.103,1146.47 576.941,1149.72 577.659,1150.32 581.052,1153.16 \n",
" 582.215,1154.12 585.226,1156.59 586.772,1157.85 589.462,1160.02 591.328,1161.52 593.762,1163.46 595.884,1165.14 598.128,1166.89 600.441,1168.69 602.561,1170.33 \n",
" 604.997,1172.19 607.061,1173.76 609.554,1175.64 611.632,1177.2 614.11,1179.04 616.274,1180.63 618.666,1182.38 620.988,1184.07 623.223,1185.68 625.777,1187.5 \n",
" 627.779,1188.92 630.642,1190.93 632.335,1192.12 635.585,1194.37 636.892,1195.27 640.608,1197.8 641.448,1198.37 645.712,1201.24 646.005,1201.43 650.561,1204.45 \n",
" 650.901,1204.67 655.117,1207.42 656.177,1208.11 659.674,1210.35 661.541,1211.54 664.23,1213.24 666.996,1214.98 668.786,1216.09 672.544,1218.41 673.343,1218.9 \n",
" 677.899,1221.67 678.188,1221.85 682.455,1224.4 683.933,1225.28 687.012,1227.1 689.78,1228.71 691.568,1229.75 695.732,1232.15 696.125,1232.37 700.681,1234.96 \n",
" 701.795,1235.58 705.237,1237.51 707.97,1239.02 709.794,1240.02 714.261,1242.45 714.35,1242.5 718.906,1244.95 720.676,1245.89 723.463,1247.36 727.215,1249.32 \n",
" 728.019,1249.74 732.575,1252.09 733.886,1252.76 737.132,1254.4 740.692,1256.19 741.688,1256.69 746.245,1258.94 747.641,1259.62 750.801,1261.17 754.736,1263.06 \n",
" 755.357,1263.36 759.914,1265.52 761.988,1266.49 764.47,1267.65 769.026,1269.76 769.399,1269.93 773.583,1271.83 776.982,1273.36 778.139,1273.88 782.696,1275.9 \n",
" 784.744,1276.8 787.252,1277.89 791.808,1279.86 792.691,1280.23 796.365,1281.79 800.835,1283.67 800.921,1283.7 805.477,1285.59 809.192,1287.1 810.034,1287.44 \n",
" 814.59,1289.28 817.77,1290.54 819.146,1291.08 823.703,1292.86 826.583,1293.97 828.259,1294.61 832.816,1296.34 835.648,1297.4 837.372,1298.05 841.928,1299.73 \n",
" 844.981,1300.84 846.485,1301.38 851.041,1303.02 854.603,1304.27 855.597,1304.62 860.154,1306.21 864.535,1307.71 864.71,1307.77 869.266,1309.31 873.823,1310.82 \n",
" 874.809,1311.14 878.379,1312.31 882.936,1313.78 885.45,1314.58 887.492,1315.22 892.048,1316.65 896.487,1318.01 896.605,1318.05 901.161,1319.43 905.717,1320.78 \n",
" 907.977,1321.45 910.274,1322.12 914.83,1323.43 919.386,1324.72 919.95,1324.88 923.943,1325.99 928.499,1327.24 932.478,1328.31 933.056,1328.47 937.612,1329.68 \n",
" 942.168,1330.86 945.632,1331.75 946.725,1332.03 951.281,1333.17 955.837,1334.3 959.496,1335.18 960.394,1335.4 964.95,1336.49 969.507,1337.55 974.063,1338.59 \n",
" 974.185,1338.62 978.619,1339.62 983.176,1340.62 987.732,1341.6 989.865,1342.05 992.288,1342.57 996.845,1343.51 1001.4,1344.44 1005.96,1345.34 1006.72,1345.49 \n",
" 1010.51,1346.23 1015.07,1347.09 1019.63,1347.94 1024.18,1348.77 1025.04,1348.92 1028.74,1349.58 1033.3,1350.37 1037.85,1351.14 1042.41,1351.89 1045.28,1352.36 \n",
" 1046.96,1352.63 1051.52,1353.34 1056.08,1354.04 1060.63,1354.72 1065.19,1355.38 1068.13,1355.79 1069.75,1356.02 1074.3,1356.64 1078.86,1357.25 1083.42,1357.83 \n",
" 1087.97,1358.4 1092.53,1358.95 1094.89,1359.23 1097.08,1359.48 1101.64,1360 1106.2,1360.49 1110.75,1360.97 1115.31,1361.43 1119.87,1361.87 1124.42,1362.29 \n",
" 1128.56,1362.66 1128.98,1362.7 1133.54,1363.09 1138.09,1363.46 1142.65,1363.81 1147.2,1364.14 1151.76,1364.46 1156.32,1364.76 1160.87,1365.04 1165.43,1365.3 \n",
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" 1452.48,1345.99 1455.01,1345.49 1457.04,1345.09 1461.59,1344.17 1466.15,1343.23 1470.71,1342.27 1471.72,1342.05 1475.26,1341.29 1479.82,1340.29 1484.38,1339.27 \n",
" 1487.24,1338.62 1488.93,1338.23 1493.49,1337.17 1498.04,1336.09 1501.78,1335.18 1502.6,1334.99 1507.16,1333.86 1511.71,1332.72 1515.49,1331.75 1516.27,1331.55 \n",
" 1520.83,1330.36 1525.38,1329.16 1528.49,1328.31 1529.94,1327.92 1534.5,1326.67 1539.05,1325.4 1540.87,1324.88 1543.61,1324.1 1548.17,1322.78 1552.7,1321.45 \n",
" 1552.72,1321.44 1557.28,1320.08 1561.83,1318.69 1564.03,1318.01 1566.39,1317.28 1570.95,1315.85 1574.92,1314.58 1575.5,1314.39 1580.06,1312.91 1584.62,1311.41 \n",
" 1585.41,1311.14 1589.17,1309.88 1593.73,1308.33 1595.53,1307.71 1598.29,1306.75 1602.84,1305.15 1605.31,1304.27 1607.4,1303.53 1611.95,1301.88 1614.78,1300.84 \n",
" 1616.51,1300.2 1621.07,1298.5 1623.96,1297.4 1625.62,1296.78 1630.18,1295.02 1632.88,1293.97 1634.74,1293.24 1639.29,1291.44 1641.54,1290.54 1643.85,1289.61 \n",
" 1648.41,1287.75 1649.97,1287.1 1652.96,1285.86 1657.52,1283.95 1658.17,1283.67 1662.07,1282 1666.17,1280.23 1666.63,1280.03 1671.19,1278.03 1673.97,1276.8 \n",
" 1675.74,1276.01 1680.3,1273.95 1681.58,1273.36 1684.86,1271.86 1689.01,1269.93 1689.41,1269.74 1693.97,1267.59 1696.27,1266.49 1698.53,1265.42 1703.08,1263.2 \n",
" 1703.38,1263.06 1707.64,1260.96 1710.32,1259.62 1712.19,1258.69 1716.75,1256.38 1717.12,1256.19 1721.31,1254.04 1723.78,1252.76 1725.86,1251.67 1730.3,1249.32 \n",
" 1730.42,1249.26 1734.98,1246.82 1736.69,1245.89 1739.53,1244.34 1742.96,1242.45 1744.09,1241.83 1748.65,1239.28 1749.1,1239.02 1753.2,1236.69 1755.13,1235.58 \n",
" 1757.76,1234.07 1761.04,1232.15 1762.31,1231.4 1766.85,1228.71 1766.87,1228.7 1771.43,1225.96 1772.55,1225.28 1775.98,1223.18 1778.15,1221.85 1780.54,1220.36 \n",
" 1783.65,1218.41 1785.1,1217.5 1789.05,1214.98 1789.65,1214.59 1794.21,1211.64 1794.36,1211.54 1798.77,1208.64 1799.57,1208.11 1803.32,1205.61 1804.7,1204.67 \n",
" 1807.88,1202.52 1809.75,1201.24 1812.43,1199.39 1814.71,1197.8 1816.99,1196.2 1819.59,1194.37 1821.55,1192.97 1824.39,1190.93 1826.1,1189.69 1829.11,1187.5 \n",
" 1830.66,1186.36 1833.75,1184.07 1835.22,1182.97 1838.32,1180.63 1839.77,1179.53 1842.82,1177.2 1844.33,1176.03 1847.25,1173.76 1848.89,1172.48 1851.61,1170.33 \n",
" 1853.44,1168.87 1855.9,1166.89 1858,1165.2 1860.13,1163.46 1862.55,1161.46 1864.29,1160.02 1867.11,1157.66 1868.38,1156.59 1871.67,1153.8 1872.41,1153.16 \n",
" 1876.22,1149.86 1876.39,1149.72 1880.3,1146.29 1880.78,1145.86 1884.15,1142.85 1885.34,1141.78 1887.94,1139.42 1889.89,1137.63 1891.68,1135.98 1894.45,1133.4 \n",
" 1895.36,1132.55 1898.99,1129.11 1899.01,1129.1 1902.56,1125.68 1903.56,1124.7 1906.08,1122.24 1908.12,1120.23 1909.54,1118.81 1912.67,1115.66 1912.96,1115.38 \n",
" 1916.32,1111.94 1917.23,1111 1919.63,1108.51 1921.79,1106.24 1922.89,1105.07 1926.11,1101.64 1926.34,1101.38 1929.27,1098.2 1930.9,1096.41 1932.39,1094.77 \n",
" 1935.46,1091.34 1935.46,1091.33 1938.48,1087.9 1940.01,1086.14 1941.46,1084.46 1944.4,1081.03 1944.57,1080.83 1947.29,1077.6 1949.13,1075.38 1950.13,1074.16 \n",
" 1952.93,1070.73 1953.68,1069.8 1955.69,1067.29 1958.24,1064.07 1958.41,1063.86 1961.08,1060.42 1962.79,1058.19 1963.71,1056.99 1966.3,1053.55 1967.35,1052.15 \n",
" 1968.85,1050.12 1971.36,1046.69 1971.91,1045.93 1973.83,1043.25 1976.26,1039.82 1976.46,1039.53 1978.66,1036.38 1981.01,1032.95 1981.02,1032.93 1983.32,1029.51 \n",
" 1985.58,1026.11 1985.6,1026.08 1987.84,1022.64 1990.04,1019.21 1990.13,1019.06 1992.21,1015.77 1994.33,1012.34 1994.69,1011.75 1996.42,1008.91 1998.48,1005.47 \n",
" 1999.25,1004.17 2000.5,1002.04 2002.48,998.602 2003.8,996.278 2004.43,995.168 2006.35,991.733 2008.22,988.299 2008.36,988.047 2010.07,984.864 2011.88,981.43 \n",
" 2012.91,979.427 2013.65,977.995 2015.4,974.561 2017.11,971.126 2017.47,970.378 2018.78,967.692 2020.43,964.257 2022.03,960.834 2022.03,960.823 2023.61,957.388 \n",
" 2025.16,953.954 2026.58,950.707 \n",
" \"/>\n",
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" 2026.58,1192.76 2024.97,1194.37 2022.03,1197.27 2021.48,1197.8 2017.95,1201.24 2017.47,1201.7 2014.37,1204.67 2012.91,1206.06 2010.75,1208.11 2008.36,1210.36 \n",
" 2007.09,1211.54 2003.8,1214.58 2003.37,1214.98 1999.62,1218.41 1999.25,1218.75 1995.81,1221.85 1994.69,1222.85 1991.95,1225.28 1990.13,1226.88 1988.04,1228.71 \n",
" 1985.58,1230.86 1984.09,1232.15 1981.02,1234.78 1980.08,1235.58 1976.46,1238.65 1976.02,1239.02 1971.91,1242.45 1971.91,1242.45 1967.74,1245.89 1967.35,1246.21 \n",
" 1963.52,1249.32 1962.79,1249.91 1959.25,1252.76 1958.24,1253.56 1954.92,1256.19 1953.68,1257.16 1950.53,1259.62 1949.13,1260.71 1946.08,1263.06 1944.57,1264.22 \n",
" 1941.57,1266.49 1940.01,1267.67 1937.01,1269.93 1935.46,1271.08 1932.38,1273.36 1930.9,1274.45 1927.68,1276.8 1926.34,1277.77 1922.93,1280.23 1921.79,1281.05 \n",
" 1918.1,1283.67 1917.23,1284.28 1913.21,1287.1 1912.67,1287.48 1908.25,1290.54 1908.12,1290.63 1903.56,1293.74 1903.22,1293.97 1899.01,1296.81 1898.12,1297.4 \n",
" 1894.45,1299.85 1892.95,1300.84 1889.89,1302.85 1887.7,1304.27 1885.34,1305.8 1882.37,1307.71 1880.78,1308.73 1876.97,1311.14 1876.22,1311.61 1871.67,1314.46 \n",
" 1871.48,1314.58 1867.11,1317.28 1865.91,1318.01 1862.55,1320.06 1860.26,1321.45 1858,1322.81 1854.52,1324.88 1853.44,1325.52 1848.89,1328.2 1848.69,1328.31 \n",
" 1844.33,1330.85 1842.76,1331.75 1839.77,1333.46 1836.74,1335.18 1835.22,1336.05 1830.66,1338.6 1830.63,1338.62 1826.1,1341.12 1824.41,1342.05 1821.55,1343.61 \n",
" 1818.08,1345.49 1816.99,1346.08 1812.43,1348.51 1811.65,1348.92 1807.88,1350.91 1805.11,1352.36 1803.32,1353.29 1798.77,1355.63 1798.45,1355.79 1794.21,1357.95 \n",
" 1791.68,1359.23 1789.65,1360.24 1785.1,1362.5 1784.78,1362.66 1780.54,1364.74 1777.75,1366.09 1775.98,1366.95 1771.43,1369.13 1770.59,1369.53 1766.87,1371.29 \n",
" 1763.29,1372.96 1762.31,1373.42 1757.76,1375.52 1755.84,1376.4 1753.2,1377.6 1748.65,1379.65 1748.25,1379.83 1744.09,1381.68 1740.49,1383.27 1739.53,1383.69 \n",
" 1734.98,1385.67 1732.57,1386.7 1730.42,1387.62 1725.86,1389.56 1724.48,1390.14 1721.31,1391.46 1716.75,1393.35 1716.21,1393.57 1712.19,1395.21 1707.75,1397.01 \n",
" 1707.64,1397.05 1703.08,1398.87 1699.08,1400.44 1698.53,1400.66 1693.97,1402.43 1690.2,1403.87 1689.41,1404.18 1684.86,1405.9 1681.1,1407.31 1680.3,1407.61 \n",
" 1675.74,1409.29 1671.76,1410.74 1671.19,1410.95 1666.63,1412.59 1662.16,1414.18 1662.07,1414.21 1657.52,1415.81 1652.96,1417.38 1652.29,1417.61 1648.41,1418.94 \n",
" 1643.85,1420.47 1642.13,1421.05 1639.29,1421.99 1634.74,1423.48 1631.65,1424.48 1630.18,1424.96 1625.62,1426.41 1621.07,1427.85 1620.84,1427.92 1616.51,1429.26 \n",
" 1611.95,1430.66 1609.66,1431.35 1607.4,1432.03 1602.84,1433.39 1598.29,1434.73 1598.08,1434.78 1593.73,1436.04 1589.17,1437.34 1586.05,1438.22 1584.62,1438.62 \n",
" 1580.06,1439.88 1575.5,1441.13 1573.54,1441.65 1570.95,1442.35 1566.39,1443.56 1561.83,1444.74 1560.49,1445.09 1557.28,1445.91 1552.72,1447.06 1548.17,1448.19 \n",
" 1546.82,1448.52 1543.61,1449.31 1539.05,1450.41 1534.5,1451.48 1532.47,1451.96 1529.94,1452.55 1525.38,1453.59 1520.83,1454.62 1517.32,1455.39 1516.27,1455.62 \n",
" 1511.71,1456.62 1507.16,1457.59 1502.6,1458.54 1501.23,1458.83 1498.04,1459.48 1493.49,1460.41 1488.93,1461.31 1484.38,1462.2 1484.05,1462.26 \n",
" \"/>\n",
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" 1917.61,1462.26 1921.79,1460.01 1923.97,1458.83 1926.34,1457.53 1930.24,1455.39 1930.9,1455.03 1935.46,1452.5 1936.42,1451.96 1940.01,1449.94 1942.52,1448.52 \n",
" 1944.57,1447.36 1948.54,1445.09 1949.13,1444.75 1953.68,1442.12 1954.48,1441.65 1958.24,1439.46 1960.34,1438.22 1962.79,1436.77 1966.12,1434.78 1967.35,1434.05 \n",
" 1971.83,1431.35 1971.91,1431.3 1976.46,1428.53 1977.46,1427.92 1981.02,1425.72 1983.02,1424.48 1985.58,1422.89 1988.52,1421.05 1990.13,1420.03 1993.94,1417.61 \n",
" 1994.69,1417.14 1999.25,1414.21 1999.3,1414.18 2003.8,1411.26 2004.59,1410.74 2008.36,1408.27 2009.82,1407.31 2012.91,1405.26 2014.98,1403.87 2017.47,1402.21 \n",
" 2020.09,1400.44 2022.03,1399.12 2025.13,1397.01 2026.58,1396.01 \n",
" \"/>\n",
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" 1698.53,1144.53 1695.63,1146.29 1693.97,1147.29 1689.87,1149.72 1689.41,1149.99 1684.86,1152.65 1683.98,1153.16 1680.3,1155.27 1677.96,1156.59 1675.74,1157.84 \n",
" 1671.8,1160.02 1671.19,1160.36 1666.63,1162.85 1665.5,1163.46 1662.07,1165.29 1659.04,1166.89 1657.52,1167.69 1652.96,1170.05 1652.43,1170.33 1648.41,1172.38 \n",
" 1645.65,1173.76 1643.85,1174.66 1639.29,1176.91 1638.7,1177.2 1634.74,1179.12 1631.56,1180.63 1630.18,1181.29 1625.62,1183.42 1624.23,1184.07 1621.07,1185.52 \n",
" 1616.7,1187.5 1616.51,1187.58 1611.95,1189.62 1608.94,1190.93 1607.4,1191.61 1602.84,1193.57 1600.96,1194.37 1598.29,1195.5 1593.73,1197.4 1592.73,1197.8 \n",
" 1589.17,1199.26 1584.62,1201.09 1584.25,1201.24 1580.06,1202.89 1575.5,1204.66 1575.47,1204.67 1570.95,1206.4 1566.39,1208.11 1566.39,1208.11 1561.83,1209.79 \n",
" 1557.28,1211.44 1556.98,1211.54 1552.72,1213.06 1548.17,1214.65 1547.2,1214.98 1543.61,1216.21 1539.05,1217.74 1537.03,1218.41 1534.5,1219.25 1529.94,1220.73 \n",
" 1526.42,1221.85 1525.38,1222.18 1520.83,1223.6 1516.27,1225 1515.33,1225.28 1511.71,1226.37 1507.16,1227.71 1503.68,1228.71 1502.6,1229.03 1498.04,1230.32 \n",
" 1493.49,1231.58 1491.41,1232.15 1488.93,1232.82 1484.38,1234.04 1479.82,1235.23 1478.43,1235.58 1475.26,1236.39 1470.71,1237.53 1466.15,1238.65 1464.6,1239.02 \n",
" 1461.59,1239.74 1457.04,1240.81 1452.48,1241.85 1449.77,1242.45 1447.92,1242.87 1443.37,1243.86 1438.81,1244.83 1434.26,1245.78 1433.73,1245.89 1429.7,1246.71 \n",
" 1425.14,1247.61 1420.59,1248.49 1416.13,1249.32 1416.03,1249.34 1411.47,1250.18 1406.92,1250.99 1402.36,1251.77 1397.8,1252.54 1396.47,1252.76 1393.25,1253.28 \n",
" 1388.69,1254.01 1384.14,1254.7 1379.58,1255.38 1375.02,1256.04 1373.91,1256.19 1370.47,1256.67 1365.91,1257.28 1361.35,1257.87 1356.8,1258.44 1352.24,1258.99 \n",
" 1347.68,1259.51 1346.69,1259.62 1343.13,1260.02 1338.57,1260.51 1334.02,1260.97 1329.46,1261.41 1324.9,1261.83 1320.35,1262.23 1315.79,1262.61 1311.23,1262.97 \n",
" 1309.97,1263.06 1306.68,1263.3 1302.12,1263.62 1297.56,1263.92 1293.01,1264.19 1288.45,1264.45 1283.9,1264.68 1279.34,1264.9 1274.78,1265.09 1270.23,1265.26 \n",
" 1265.67,1265.42 1261.11,1265.55 1256.56,1265.66 1252,1265.75 1247.44,1265.82 1242.89,1265.87 1238.33,1265.9 1233.78,1265.91 1229.22,1265.9 1224.66,1265.87 \n",
" 1220.11,1265.82 1215.55,1265.75 1210.99,1265.66 1206.44,1265.55 1201.88,1265.42 1197.32,1265.27 1192.77,1265.1 1188.21,1264.9 1183.66,1264.69 1179.1,1264.46 \n",
" 1174.54,1264.21 1169.99,1263.93 1165.43,1263.64 1160.87,1263.32 1157.3,1263.06 1156.32,1262.99 1151.76,1262.63 1147.2,1262.26 1142.65,1261.86 1138.09,1261.45 \n",
" 1133.54,1261.01 1128.98,1260.55 1124.42,1260.07 1120.35,1259.62 1119.87,1259.57 1115.31,1259.05 1110.75,1258.51 1106.2,1257.95 1101.64,1257.37 1097.08,1256.76 \n",
" 1092.91,1256.19 1092.53,1256.14 1087.97,1255.49 1083.42,1254.83 1078.86,1254.14 1074.3,1253.43 1070.12,1252.76 1069.75,1252.7 1065.19,1251.95 1060.63,1251.17 \n",
" 1056.08,1250.38 1051.52,1249.56 1050.25,1249.32 1046.96,1248.72 1042.41,1247.86 1037.85,1246.97 1033.3,1246.07 1032.42,1245.89 1028.74,1245.14 1024.18,1244.19 \n",
" 1019.63,1243.21 1016.14,1242.45 1015.07,1242.22 1010.51,1241.2 1005.96,1240.16 1001.4,1239.09 1001.08,1239.02 996.845,1238.01 992.288,1236.9 987.732,1235.76 \n",
" 987.034,1235.58 983.176,1234.6 978.619,1233.42 974.063,1232.22 973.817,1232.15 969.507,1230.99 964.95,1229.73 961.324,1228.71 960.394,1228.45 955.837,1227.15 \n",
" 951.281,1225.82 949.452,1225.28 946.725,1224.47 942.168,1223.09 938.123,1221.85 937.612,1221.69 933.056,1220.26 928.499,1218.8 927.288,1218.41 923.943,1217.32 \n",
" 919.386,1215.82 916.89,1214.98 914.83,1214.28 910.274,1212.72 906.886,1211.54 905.717,1211.13 901.161,1209.52 897.244,1208.11 896.605,1207.88 892.048,1206.21 \n",
" 887.934,1204.67 887.492,1204.51 882.936,1202.78 878.93,1201.24 878.379,1201.03 873.823,1199.24 870.21,1197.8 869.266,1197.43 864.71,1195.58 861.754,1194.37 \n",
" 860.154,1193.71 855.597,1191.81 853.543,1190.93 851.041,1189.87 846.485,1187.9 845.56,1187.5 841.928,1185.91 837.795,1184.07 837.372,1183.88 832.816,1181.82 \n",
" 830.237,1180.63 828.259,1179.72 823.703,1177.59 822.869,1177.2 819.146,1175.43 815.685,1173.76 814.59,1173.23 810.034,1171 808.675,1170.33 805.477,1168.74 \n",
" 801.831,1166.89 800.921,1166.43 796.365,1164.09 795.145,1163.46 791.808,1161.72 788.609,1160.02 787.252,1159.3 782.696,1156.85 782.217,1156.59 778.139,1154.36 \n",
" 775.966,1153.16 773.583,1151.83 769.846,1149.72 769.026,1149.26 764.47,1146.64 763.854,1146.29 759.914,1143.99 757.986,1142.85 755.357,1141.29 752.237,1139.42 \n",
" 750.801,1138.55 746.601,1135.98 746.245,1135.76 741.688,1132.93 741.077,1132.55 737.132,1130.05 735.661,1129.11 732.575,1127.13 730.348,1125.68 728.019,1124.15 \n",
" 725.135,1122.24 723.463,1121.13 720.02,1118.81 718.906,1118.05 715,1115.38 714.35,1114.93 710.073,1111.94 709.794,1111.74 705.237,1108.51 705.235,1108.51 \n",
" 700.681,1105.22 700.485,1105.07 696.125,1101.86 695.82,1101.64 691.568,1098.45 691.237,1098.2 687.012,1094.98 686.736,1094.77 682.455,1091.44 682.314,1091.33 \n",
" 677.97,1087.9 677.899,1087.84 673.701,1084.46 673.343,1084.17 669.506,1081.03 668.786,1080.44 665.383,1077.6 664.23,1076.62 661.332,1074.16 659.674,1072.74 \n",
" 657.35,1070.73 655.117,1068.78 653.436,1067.29 650.561,1064.73 649.589,1063.86 646.005,1060.6 645.807,1060.42 642.09,1056.99 641.448,1056.39 638.437,1053.55 \n",
" 636.892,1052.08 634.845,1050.12 632.335,1047.68 631.315,1046.69 627.846,1043.25 627.779,1043.18 624.435,1039.82 623.223,1038.58 621.083,1036.38 618.666,1033.87 \n",
" 617.788,1032.95 614.551,1029.51 614.11,1029.04 611.368,1026.08 609.554,1024.09 608.241,1022.64 605.17,1019.21 604.997,1019.01 602.15,1015.77 600.441,1013.8 \n",
" 599.184,1012.34 596.272,1008.91 595.884,1008.44 593.409,1005.47 591.328,1002.93 590.599,1002.04 587.839,998.602 586.772,997.254 585.129,995.168 582.47,991.733 \n",
" 582.215,991.401 579.857,988.299 577.659,985.354 577.295,984.864 574.779,981.43 573.103,979.099 572.312,977.995 569.891,974.561 568.546,972.618 567.518,971.126 \n",
" 565.19,967.692 563.99,965.888 562.909,964.257 560.673,960.823 559.434,958.882 558.482,957.388 556.336,953.954 554.877,951.57 554.236,950.519 552.178,947.085 \n",
" 550.321,943.914 550.167,943.65 548.195,940.216 546.271,936.781 545.764,935.859 544.387,933.347 542.547,929.912 541.208,927.353 540.751,926.478 538.994,923.043 \n",
" 537.283,919.609 536.652,918.311 535.612,916.174 533.983,912.74 532.398,909.305 532.095,908.633 530.851,905.871 529.347,902.436 527.886,899.002 527.539,898.164 \n",
" 526.463,895.567 525.081,892.133 523.743,888.698 522.983,886.686 522.444,885.264 521.184,881.829 519.966,878.395 518.791,874.96 518.426,873.858 517.653,871.526 \n",
" 516.554,868.091 515.498,864.657 514.483,861.222 513.87,859.061 513.507,857.788 512.569,854.353 511.672,850.919 510.816,847.484 510.001,844.05 509.314,841 \n",
" 509.227,840.615 508.488,837.181 507.789,833.746 507.132,830.312 506.515,826.877 505.939,823.442 505.403,820.008 504.907,816.573 504.757,815.447 504.449,813.139 \n",
" 504.029,809.704 503.65,806.27 503.311,802.835 503.012,799.401 502.753,795.966 502.534,792.532 502.355,789.097 502.216,785.663 502.116,782.228 502.056,778.794 \n",
" 502.037,775.359 502.056,771.925 502.116,768.49 502.216,765.056 502.355,761.621 502.534,758.187 502.753,754.752 503.012,751.318 503.311,747.883 503.65,744.449 \n",
" 504.029,741.014 504.449,737.58 504.757,735.272 504.907,734.145 505.403,730.711 505.939,727.276 506.515,723.842 507.132,720.407 507.789,716.973 508.488,713.538 \n",
" 509.227,710.104 509.314,709.719 510.001,706.669 510.816,703.235 511.672,699.8 512.569,696.366 513.507,692.931 513.87,691.658 514.483,689.497 515.498,686.062 \n",
" 516.554,682.628 517.653,679.193 518.426,676.861 518.791,675.759 519.966,672.324 521.184,668.89 522.444,665.455 522.983,664.033 523.743,662.021 525.081,658.586 \n",
" 526.463,655.152 527.539,652.555 527.886,651.717 529.347,648.283 530.851,644.848 532.095,642.086 532.398,641.414 533.983,637.979 535.612,634.545 536.652,632.408 \n",
" 537.283,631.11 538.994,627.676 540.751,624.241 541.208,623.366 542.547,620.807 544.387,617.372 545.764,614.859 546.271,613.938 548.195,610.503 550.167,607.069 \n",
" 550.321,606.805 552.178,603.634 554.236,600.2 554.877,599.149 556.336,596.765 558.482,593.331 559.434,591.837 560.673,589.896 562.909,586.462 563.99,584.831 \n",
" 565.19,583.027 567.518,579.593 568.546,578.101 569.891,576.158 572.312,572.724 573.103,571.62 574.779,569.289 577.295,565.855 577.659,565.365 579.857,562.42 \n",
" 582.215,559.318 582.47,558.986 585.129,555.551 586.772,553.465 587.839,552.117 590.599,548.682 591.328,547.788 593.409,545.247 595.884,542.276 596.272,541.813 \n",
" 599.184,538.378 600.441,536.919 602.15,534.944 604.997,531.705 605.17,531.509 608.241,528.075 609.554,526.629 611.368,524.64 614.11,521.679 614.551,521.206 \n",
" 617.788,517.771 618.666,516.852 621.083,514.337 623.223,512.139 624.435,510.902 627.779,507.535 627.846,507.468 631.315,504.033 632.335,503.036 634.845,500.599 \n",
" 636.892,498.635 638.437,497.164 641.448,494.33 642.09,493.73 645.807,490.295 646.005,490.115 649.589,486.861 650.561,485.988 653.436,483.426 655.117,481.944 \n",
" 657.35,479.992 659.674,477.98 661.332,476.557 664.23,474.095 665.383,473.123 668.786,470.284 669.506,469.688 673.343,466.545 673.701,466.254 677.899,462.876 \n",
" 677.97,462.819 682.314,459.385 682.455,459.274 686.736,455.95 687.012,455.738 691.237,452.516 691.568,452.266 695.82,449.081 696.125,448.855 700.485,445.647 \n",
" 700.681,445.503 705.235,442.212 705.237,442.21 709.794,438.974 710.073,438.778 714.35,435.792 715,435.343 718.906,432.664 720.02,431.909 723.463,429.589 \n",
" 725.135,428.474 728.019,426.564 730.348,425.04 732.575,423.59 735.661,421.605 737.132,420.664 741.077,418.171 741.688,417.786 746.245,414.955 746.601,414.736 \n",
" 750.801,412.169 752.237,411.302 755.357,409.427 757.986,407.867 759.914,406.729 763.854,404.433 764.47,404.075 769.026,401.462 769.846,400.998 773.583,398.89 \n",
" 775.966,397.564 778.139,396.359 782.217,394.129 782.696,393.869 787.252,391.416 788.609,390.695 791.808,389.001 795.145,387.26 796.365,386.626 800.921,384.287 \n",
" 801.831,383.826 805.477,381.983 808.675,380.391 810.034,379.717 814.59,377.486 815.685,376.957 819.146,375.289 822.869,373.522 823.703,373.127 828.259,370.998 \n",
" 830.237,370.088 832.816,368.903 837.372,366.842 837.795,366.653 841.928,364.811 845.56,363.219 846.485,362.814 851.041,360.847 853.543,359.784 855.597,358.912 \n",
" 860.154,357.009 861.754,356.35 864.71,355.134 869.266,353.292 870.21,352.915 873.823,351.477 878.379,349.693 878.93,349.481 882.936,347.936 887.492,346.211 \n",
" 887.934,346.046 892.048,344.511 896.605,342.842 897.244,342.612 901.161,341.199 905.717,339.585 906.886,339.177 910.274,337.996 914.83,336.436 916.89,335.743 \n",
" 919.386,334.902 923.943,333.395 927.288,332.308 928.499,331.915 933.056,330.459 937.612,329.031 938.123,328.874 942.168,327.626 946.725,326.249 949.452,325.439 \n",
" 951.281,324.896 955.837,323.567 960.394,322.266 961.324,322.005 964.95,320.986 969.507,319.732 973.817,318.57 974.063,318.504 978.619,317.296 983.176,316.115 \n",
" 987.034,315.136 987.732,314.958 992.288,313.822 996.845,312.711 1001.08,311.701 1001.4,311.626 1005.96,310.559 1010.51,309.518 1015.07,308.501 1016.14,308.267 \n",
" 1019.63,307.504 1024.18,306.53 1028.74,305.58 1032.42,304.832 1033.3,304.653 1037.85,303.746 1042.41,302.862 1046.96,302.001 1050.25,301.398 1051.52,301.163 \n",
" 1056.08,300.343 1060.63,299.547 1065.19,298.774 1069.75,298.022 1070.12,297.963 1074.3,297.29 1078.86,296.579 1083.42,295.891 1087.97,295.225 1092.53,294.58 \n",
" 1092.91,294.529 1097.08,293.955 1101.64,293.35 1106.2,292.767 1110.75,292.206 1115.31,291.666 1119.87,291.147 1120.35,291.094 1124.42,290.647 1128.98,290.168 \n",
" 1133.54,289.709 1138.09,289.272 1142.65,288.856 1147.2,288.46 1151.76,288.086 1156.32,287.731 1157.3,287.66 1160.87,287.396 1165.43,287.082 1169.99,286.787 \n",
" 1174.54,286.514 1179.1,286.26 1183.66,286.027 1188.21,285.815 1192.77,285.623 1197.32,285.451 1201.88,285.3 1206.44,285.168 1210.99,285.057 1215.55,284.967 \n",
" 1220.11,284.896 1224.66,284.846 1229.22,284.816 1233.78,284.806 1238.33,284.816 1242.89,284.847 1247.44,284.897 1252,284.968 1256.56,285.06 1261.11,285.171 \n",
" 1265.67,285.303 1270.23,285.456 1274.78,285.629 1279.34,285.822 1283.9,286.036 1288.45,286.27 1293.01,286.525 1297.56,286.801 1302.12,287.097 1306.68,287.415 \n",
" 1309.97,287.66 1311.23,287.753 1315.79,288.11 1320.35,288.488 1324.9,288.888 1329.46,289.308 1334.02,289.75 1338.57,290.213 1343.13,290.698 1346.69,291.094 \n",
" 1347.68,291.204 1352.24,291.729 1356.8,292.276 1361.35,292.845 1365.91,293.436 1370.47,294.049 1373.91,294.529 1375.02,294.683 1379.58,295.337 1384.14,296.014 \n",
" 1388.69,296.714 1393.25,297.436 1396.47,297.963 1397.8,298.18 1402.36,298.944 1406.92,299.732 1411.47,300.543 1416.03,301.378 1416.13,301.398 1420.59,302.232 \n",
" 1425.14,303.11 1429.7,304.013 1433.73,304.832 1434.26,304.939 1438.81,305.886 1443.37,306.857 1447.92,307.853 1449.77,308.267 1452.48,308.872 1457.04,309.913 \n",
" 1461.59,310.981 1464.6,311.701 1466.15,312.072 1470.71,313.186 1475.26,314.326 1478.43,315.136 1479.82,315.492 1484.38,316.68 1488.93,317.895 1491.41,318.57 \n",
" 1493.49,319.135 1498.04,320.4 1502.6,321.692 1503.68,322.005 1507.16,323.008 1511.71,324.351 1515.33,325.439 1516.27,325.722 1520.83,327.117 1525.38,328.542 \n",
" 1526.42,328.874 1529.94,329.992 1534.5,331.47 1537.03,332.308 1539.05,332.976 1543.61,334.509 1547.2,335.743 1548.17,336.072 1552.72,337.662 1556.98,339.177 \n",
" 1557.28,339.283 1561.83,340.93 1566.39,342.611 1566.39,342.612 1570.95,344.317 1575.47,346.046 1575.5,346.058 1580.06,347.825 1584.25,349.481 1584.62,349.627 \n",
" 1589.17,351.458 1592.73,352.915 1593.73,353.322 1598.29,355.217 1600.96,356.35 1602.84,357.146 1607.4,359.108 1608.94,359.784 1611.95,361.103 1616.51,363.134 \n",
" 1616.7,363.219 1621.07,365.197 1624.23,366.653 1625.62,367.297 1630.18,369.432 1631.56,370.088 1634.74,371.603 1638.7,373.522 1639.29,373.812 1643.85,376.057 \n",
" 1645.65,376.957 1648.41,378.341 1652.43,380.391 1652.96,380.664 1657.52,383.025 1659.04,383.826 1662.07,385.427 1665.5,387.26 1666.63,387.87 1671.19,390.355 \n",
" 1671.8,390.695 1675.74,392.881 1677.96,394.129 1680.3,395.452 1683.98,397.564 1684.86,398.067 1689.41,400.727 1689.87,400.998 1693.97,403.432 1695.63,404.433 \n",
" 1698.53,406.184 1701.27,407.867 1703.08,408.985 1706.79,411.302 1707.64,411.836 1712.19,414.736 1712.19,414.737 1716.75,417.689 1717.49,418.171 1721.31,420.694 \n",
" 1722.67,421.605 1725.86,423.753 1727.75,425.04 1730.42,426.868 1732.74,428.474 1734.98,430.04 1737.62,431.909 1739.53,433.272 1742.41,435.343 1744.09,436.563 \n",
" 1747.11,438.778 1748.65,439.917 1751.71,442.212 1753.2,443.335 1756.23,445.647 1757.76,446.819 1760.67,449.081 1762.31,450.371 1765.02,452.516 1766.87,453.993 \n",
" 1769.29,455.95 1771.43,457.689 1773.49,459.385 1775.98,461.46 1777.6,462.819 1780.54,465.309 1781.64,466.254 1785.1,469.239 1785.61,469.688 1789.51,473.123 \n",
" 1789.65,473.253 1793.33,476.557 1794.21,477.355 1797.09,479.992 1798.77,481.549 1800.77,483.426 1803.32,485.837 1804.39,486.861 1807.88,490.225 1807.95,490.295 \n",
" 1811.44,493.73 1812.43,494.718 1814.87,497.164 1816.99,499.32 1818.24,500.599 1821.55,504.033 1821.55,504.035 1824.79,507.468 1826.1,508.874 1827.98,510.902 \n",
" 1830.66,513.838 1831.11,514.337 1834.19,517.771 1835.22,518.939 1837.21,521.206 1839.77,524.181 1840.17,524.64 1843.08,528.075 1844.33,529.577 1845.93,531.509 \n",
" 1848.73,534.944 1848.89,535.134 1851.48,538.378 1853.44,540.867 1854.18,541.813 1856.83,545.247 1858,546.787 1859.43,548.682 1861.98,552.117 1862.55,552.908 \n",
" 1864.48,555.551 1866.93,558.986 1867.11,559.247 1869.33,562.42 1871.67,565.826 1871.69,565.855 1874,569.289 1876.22,572.668 1876.26,572.724 1878.48,576.158 \n",
" 1880.65,579.593 1880.78,579.801 1882.78,583.027 1884.86,586.462 1885.34,587.258 1886.9,589.896 1888.9,593.331 1889.89,595.078 1890.85,596.765 1892.76,600.2 \n",
" 1894.45,603.305 1894.63,603.634 1896.46,607.069 1898.24,610.503 1899.01,612.013 1899.98,613.938 1901.68,617.372 1903.34,620.807 1903.56,621.27 1904.97,624.241 \n",
" 1906.55,627.676 1908.08,631.11 1908.12,631.188 1909.59,634.545 1911.05,637.979 1912.47,641.414 1912.67,641.922 1913.86,644.848 1915.2,648.283 1916.51,651.717 \n",
" \n",
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" 1916.51,651.717 1917.23,653.681 1917.77,655.152 1919.01,658.586 1920.2,662.021 1921.35,665.455 1921.79,666.79 1922.47,668.89 1923.55,672.324 1924.6,675.759 \n",
" 1925.6,679.193 1926.34,681.819 1926.57,682.628 1927.51,686.062 1928.41,689.497 1929.27,692.931 1930.09,696.366 1930.88,699.8 1930.9,699.903 1931.63,703.235 \n",
" 1932.35,706.669 1933.03,710.104 1933.68,713.538 1934.29,716.973 1934.86,720.407 1935.4,723.842 1935.46,724.218 1935.91,727.276 1936.38,730.711 1936.81,734.145 \n",
" 1937.21,737.58 1937.58,741.014 1937.91,744.449 1938.2,747.883 1938.46,751.318 1938.69,754.752 1938.88,758.187 1939.03,761.621 1939.15,765.056 1939.24,768.49 \n",
" 1939.29,771.925 1939.31,775.359 1939.29,778.794 1939.24,782.228 1939.15,785.663 1939.03,789.097 1938.88,792.532 1938.69,795.966 1938.46,799.401 1938.2,802.835 \n",
" 1937.91,806.27 1937.58,809.704 1937.21,813.139 1936.81,816.573 1936.38,820.008 1935.91,823.442 1935.46,826.501 1935.4,826.877 1934.86,830.312 1934.29,833.746 \n",
" 1933.68,837.181 1933.03,840.615 1932.35,844.05 1931.63,847.484 1930.9,850.816 1930.88,850.919 1930.09,854.353 1929.27,857.788 1928.41,861.222 1927.51,864.657 \n",
" 1926.57,868.091 1926.34,868.9 1925.6,871.526 1924.6,874.96 1923.55,878.395 1922.47,881.829 1921.79,883.929 1921.35,885.264 1920.2,888.698 1919.01,892.133 \n",
" 1917.77,895.567 1917.23,897.038 1916.51,899.002 1915.2,902.436 1913.86,905.871 1912.67,908.796 1912.47,909.305 1911.05,912.74 1909.59,916.174 1908.12,919.53 \n",
" 1908.08,919.609 1906.55,923.043 1904.97,926.478 1903.56,929.449 1903.34,929.912 1901.68,933.347 1899.98,936.781 1899.01,938.705 1898.24,940.216 1896.46,943.65 \n",
" 1894.63,947.085 1894.45,947.414 1892.76,950.519 1890.85,953.954 1889.89,955.641 1888.9,957.388 1886.9,960.823 1885.34,963.46 1884.86,964.257 1882.78,967.692 \n",
" 1880.78,970.918 1880.65,971.126 1878.48,974.561 1876.26,977.995 1876.22,978.05 1874,981.43 1871.69,984.864 1871.67,984.893 1869.33,988.299 1867.11,991.471 \n",
" 1866.93,991.733 1864.48,995.168 1862.55,997.811 1861.98,998.602 1859.43,1002.04 1858,1003.93 1856.83,1005.47 1854.18,1008.91 1853.44,1009.85 1851.48,1012.34 \n",
" 1848.89,1015.59 1848.73,1015.77 1845.93,1019.21 1844.33,1021.14 1843.08,1022.64 1840.17,1026.08 1839.77,1026.54 1837.21,1029.51 1835.22,1031.78 1834.19,1032.95 \n",
" 1831.11,1036.38 1830.66,1036.88 1827.98,1039.82 1826.1,1041.85 1824.79,1043.25 1821.55,1046.68 1821.55,1046.69 1818.24,1050.12 1816.99,1051.4 1814.87,1053.55 \n",
" 1812.43,1056 1811.44,1056.99 1807.95,1060.42 1807.88,1060.49 1804.39,1063.86 1803.32,1064.88 1800.77,1067.29 1798.77,1069.17 1797.09,1070.73 1794.21,1073.36 \n",
" 1793.33,1074.16 1789.65,1077.47 1789.51,1077.6 1785.61,1081.03 1785.1,1081.48 1781.64,1084.46 1780.54,1085.41 1777.6,1087.9 1775.98,1089.26 1773.49,1091.33 \n",
" 1771.43,1093.03 1769.29,1094.77 1766.87,1096.73 1765.02,1098.2 1762.31,1100.35 1760.67,1101.64 1757.76,1103.9 1756.23,1105.07 1753.2,1107.38 1751.71,1108.51 \n",
" 1748.65,1110.8 1747.11,1111.94 1744.09,1114.16 1742.41,1115.38 1739.53,1117.45 1737.62,1118.81 1734.98,1120.68 1732.74,1122.24 1730.42,1123.85 1727.75,1125.68 \n",
" 1725.86,1126.97 1722.67,1129.11 1721.31,1130.03 1717.49,1132.55 1716.75,1133.03 1712.19,1135.98 1712.19,1135.98 1707.64,1138.88 1706.79,1139.42 1703.08,1141.73 \n",
" 1701.27,1142.85 1698.53,1144.53 \n",
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" 1621.07,1085.52 1616.81,1087.9 1616.51,1088.07 1611.95,1090.57 1610.53,1091.33 1607.4,1093.02 1604.08,1094.77 1602.84,1095.42 1598.29,1097.77 1597.44,1098.2 \n",
" 1593.73,1100.08 1590.6,1101.64 1589.17,1102.35 1584.62,1104.57 1583.56,1105.07 1580.06,1106.74 1576.29,1108.51 1575.5,1108.87 1570.95,1110.97 1568.78,1111.94 \n",
" 1566.39,1113.02 1561.83,1115.03 1561.03,1115.38 1557.28,1117 1553,1118.81 1552.72,1118.93 1548.17,1120.82 1544.67,1122.24 1543.61,1122.68 1539.05,1124.5 \n",
" 1536.03,1125.68 1534.5,1126.28 1529.94,1128.02 1527.04,1129.11 1525.38,1129.73 1520.83,1131.41 1517.66,1132.55 1516.27,1133.05 1511.71,1134.66 1507.87,1135.98 \n",
" 1507.16,1136.23 1502.6,1137.77 1498.04,1139.28 1497.62,1139.42 1493.49,1140.76 1488.93,1142.2 1486.82,1142.85 1484.38,1143.61 1479.82,1144.99 1475.45,1146.29 \n",
" 1475.26,1146.34 1470.71,1147.66 1466.15,1148.95 1463.37,1149.72 1461.59,1150.21 1457.04,1151.44 1452.48,1152.64 1450.49,1153.16 1447.92,1153.81 1443.37,1154.96 \n",
" 1438.81,1156.07 1436.64,1156.59 1434.26,1157.16 1429.7,1158.22 1425.14,1159.25 1421.62,1160.02 1420.59,1160.25 1416.03,1161.23 1411.47,1162.18 1406.92,1163.1 \n",
" 1405.1,1163.46 1402.36,1164 1397.8,1164.87 1393.25,1165.71 1388.69,1166.53 1386.58,1166.89 1384.14,1167.32 1379.58,1168.09 1375.02,1168.83 1370.47,1169.54 \n",
" 1365.91,1170.23 1365.21,1170.33 1361.35,1170.89 1356.8,1171.53 1352.24,1172.14 1347.68,1172.73 1343.13,1173.3 1339.19,1173.76 1338.57,1173.84 1334.02,1174.35 \n",
" 1329.46,1174.84 1324.9,1175.31 1320.35,1175.75 1315.79,1176.17 1311.23,1176.57 1306.68,1176.94 1303.23,1177.2 1302.12,1177.28 1297.56,1177.61 1293.01,1177.91 \n",
" 1288.45,1178.18 1283.9,1178.43 1279.34,1178.66 1274.78,1178.87 1270.23,1179.05 1265.67,1179.21 1261.11,1179.34 1256.56,1179.46 1252,1179.54 1247.44,1179.61 \n",
" 1242.89,1179.65 1238.33,1179.67 1233.78,1179.67 1229.22,1179.64 1224.66,1179.59 1220.11,1179.52 1215.55,1179.42 1210.99,1179.3 1206.44,1179.16 1201.88,1178.99 \n",
" 1197.32,1178.8 1192.77,1178.59 1188.21,1178.36 1183.66,1178.1 1179.1,1177.82 1174.54,1177.51 1170.19,1177.2 1169.99,1177.18 1165.43,1176.83 1160.87,1176.46 \n",
" 1156.32,1176.06 1151.76,1175.64 1147.2,1175.2 1142.65,1174.73 1138.09,1174.23 1133.93,1173.76 1133.54,1173.72 1128.98,1173.18 1124.42,1172.62 1119.87,1172.03 \n",
" 1115.31,1171.42 1110.75,1170.79 1107.59,1170.33 1106.2,1170.13 1101.64,1169.44 1097.08,1168.74 1092.53,1168.01 1087.97,1167.25 1085.89,1166.89 1083.42,1166.47 \n",
" 1078.86,1165.67 1074.3,1164.84 1069.75,1163.98 1067.05,1163.46 1065.19,1163.1 1060.63,1162.2 1056.08,1161.27 1051.52,1160.31 1050.2,1160.02 1046.96,1159.33 \n",
" 1042.41,1158.32 1037.85,1157.29 1034.85,1156.59 1033.3,1156.23 1028.74,1155.15 1024.18,1154.03 1020.67,1153.16 1019.63,1152.89 1015.07,1151.73 1010.51,1150.54 \n",
" 1007.46,1149.72 1005.96,1149.32 1001.4,1148.07 996.845,1146.8 995.058,1146.29 992.288,1145.5 987.732,1144.17 983.334,1142.85 983.176,1142.8 978.619,1141.42 \n",
" 974.063,1140 972.22,1139.42 969.507,1138.56 964.95,1137.08 961.624,1135.98 960.394,1135.58 955.837,1134.04 951.492,1132.55 951.281,1132.48 946.725,1130.88 \n",
" 942.168,1129.25 941.785,1129.11 937.612,1127.6 933.056,1125.9 932.458,1125.68 928.499,1124.18 923.943,1122.43 923.475,1122.24 919.386,1120.64 914.83,1118.82 \n",
" 914.809,1118.81 910.274,1116.97 906.439,1115.38 905.717,1115.08 901.161,1113.15 898.341,1111.94 896.605,1111.19 892.048,1109.2 890.492,1108.51 887.492,1107.17 \n",
" 882.936,1105.1 882.877,1105.07 878.379,1103 875.49,1101.64 873.823,1100.85 869.266,1098.67 868.307,1098.2 864.71,1096.45 861.321,1094.77 860.154,1094.19 \n",
" 855.597,1091.89 854.521,1091.33 851.041,1089.54 847.898,1087.9 846.485,1087.16 841.928,1084.73 841.441,1084.46 837.372,1082.26 835.147,1081.03 832.816,1079.74 \n",
" 829.002,1077.6 828.259,1077.18 823.703,1074.57 823.004,1074.16 819.146,1071.91 817.147,1070.73 814.59,1069.2 811.423,1067.29 810.034,1066.45 805.828,1063.86 \n",
" 805.477,1063.64 800.921,1060.78 800.358,1060.42 796.365,1057.87 795.008,1056.99 791.808,1054.9 789.773,1053.55 787.252,1051.88 784.649,1050.12 782.696,1048.79 \n",
" 779.634,1046.69 778.139,1045.65 774.723,1043.25 773.583,1042.44 769.914,1039.82 769.026,1039.18 765.204,1036.38 764.47,1035.84 760.589,1032.95 759.914,1032.44 \n",
" 756.068,1029.51 755.357,1028.97 751.637,1026.08 750.801,1025.42 747.295,1022.64 746.245,1021.8 743.039,1019.21 741.688,1018.11 738.867,1015.77 737.132,1014.33 \n",
" 734.778,1012.34 732.575,1010.46 730.769,1008.91 728.019,1006.51 726.838,1005.47 723.463,1002.47 722.984,1002.04 719.206,998.602 718.906,998.327 715.502,995.168 \n",
" 714.35,994.086 711.87,991.733 709.794,989.738 708.309,988.299 705.237,985.28 704.818,984.864 701.396,981.43 700.681,980.703 698.041,977.995 696.125,976 \n",
" 694.753,974.561 691.568,971.166 691.531,971.126 688.372,967.692 687.012,966.189 685.277,964.257 682.455,961.063 682.245,960.823 679.274,957.388 677.899,955.772 \n",
" 676.364,953.954 673.515,950.519 673.343,950.308 670.724,947.085 668.786,944.652 667.993,943.65 665.32,940.216 664.23,938.79 662.704,936.781 660.146,933.347 \n",
" 659.674,932.701 657.643,929.912 655.199,926.478 655.117,926.362 652.806,923.043 650.561,919.74 650.472,919.609 648.189,916.174 646.005,912.803 645.964,912.74 \n",
" 643.788,909.305 641.671,905.871 641.448,905.503 639.602,902.436 637.59,899.002 636.892,897.782 635.628,895.567 633.72,892.133 632.335,889.571 631.865,888.698 \n",
" 630.06,885.264 628.31,881.829 627.779,880.759 626.609,878.395 624.961,874.96 623.368,871.526 623.223,871.204 621.821,868.091 620.328,864.657 618.891,861.222 \n",
" 618.666,860.667 617.501,857.788 616.164,854.353 614.882,850.919 614.11,848.763 613.652,847.484 612.471,844.05 611.345,840.615 610.275,837.181 609.554,834.742 \n",
" 609.258,833.746 608.291,830.312 607.38,826.877 606.525,823.442 605.728,820.008 604.997,816.619 604.987,816.573 604.297,813.139 603.664,809.704 603.091,806.27 \n",
" 602.576,802.835 602.121,799.401 601.725,795.966 601.389,792.532 601.115,789.097 600.9,785.663 600.747,782.228 600.655,778.794 600.625,775.359 600.655,771.925 \n",
" 600.747,768.49 600.9,765.056 601.115,761.621 601.389,758.187 601.725,754.752 602.121,751.318 602.576,747.883 603.091,744.449 603.664,741.014 604.297,737.58 \n",
" 604.987,734.145 604.997,734.099 605.728,730.711 606.525,727.276 607.38,723.842 608.291,720.407 609.258,716.973 609.554,715.977 610.275,713.538 611.345,710.104 \n",
" 612.471,706.669 613.652,703.235 614.11,701.956 614.882,699.8 616.164,696.366 617.501,692.931 618.666,690.052 618.891,689.497 620.328,686.062 621.821,682.628 \n",
" 623.223,679.515 623.368,679.193 624.961,675.759 626.609,672.324 627.779,669.96 628.31,668.89 630.06,665.455 631.865,662.021 632.335,661.148 633.72,658.586 \n",
" 635.628,655.152 636.892,652.936 637.59,651.717 639.602,648.283 641.448,645.216 641.671,644.848 643.788,641.414 645.964,637.979 646.005,637.916 648.189,634.545 \n",
" 650.472,631.11 650.561,630.979 652.806,627.676 655.117,624.357 655.199,624.241 657.643,620.807 659.674,618.018 660.146,617.372 662.704,613.938 664.23,611.929 \n",
" 665.32,610.503 667.993,607.069 668.786,606.067 670.724,603.634 673.343,600.411 673.515,600.2 676.364,596.765 677.899,594.947 679.274,593.331 682.245,589.896 \n",
" 682.455,589.656 685.277,586.462 687.012,584.529 688.372,583.027 691.531,579.593 691.568,579.552 694.753,576.158 696.125,574.718 698.041,572.724 700.681,570.016 \n",
" 701.396,569.289 704.818,565.855 705.237,565.439 708.309,562.42 709.794,560.98 711.87,558.986 714.35,556.633 715.502,555.551 718.906,552.392 719.206,552.117 \n",
" 722.984,548.682 723.463,548.252 726.838,545.247 728.019,544.208 730.769,541.813 732.575,540.256 734.778,538.378 737.132,536.393 738.867,534.944 741.688,532.614 \n",
" 743.039,531.509 746.245,528.916 747.295,528.075 750.801,525.296 751.637,524.64 755.357,521.751 756.068,521.206 759.914,518.279 760.589,517.771 764.47,514.877 \n",
" 765.204,514.337 769.026,511.543 769.914,510.902 773.583,508.275 774.723,507.468 778.139,505.07 779.634,504.033 782.696,501.926 784.649,500.599 787.252,498.843 \n",
" 789.773,497.164 791.808,495.818 795.008,493.73 796.365,492.85 800.358,490.295 800.921,489.937 805.477,487.078 805.828,486.861 810.034,484.27 811.423,483.426 \n",
" 814.59,481.514 817.147,479.992 819.146,478.808 823.004,476.557 823.703,476.152 828.259,473.542 829.002,473.123 832.816,470.979 835.147,469.688 837.372,468.462 \n",
" 841.441,466.254 841.928,465.991 846.485,463.561 847.898,462.819 851.041,461.176 854.521,459.385 855.597,458.833 860.154,456.531 861.321,455.95 864.71,454.27 \n",
" 868.307,452.516 869.266,452.049 873.823,449.867 875.49,449.081 878.379,447.723 882.877,445.647 882.936,445.62 887.492,443.55 890.492,442.212 892.048,441.52 \n",
" 896.605,439.525 898.341,438.778 901.161,437.566 905.717,435.643 906.439,435.343 910.274,433.753 914.809,431.909 914.83,431.9 919.386,430.077 923.475,428.474 \n",
" 923.943,428.291 928.499,426.535 932.458,425.04 933.056,424.814 937.612,423.123 941.785,421.605 942.168,421.466 946.725,419.837 951.281,418.243 951.492,418.171 \n",
" 955.837,416.676 960.394,415.143 961.624,414.736 964.95,413.637 969.507,412.162 972.22,411.302 974.063,410.717 978.619,409.3 983.176,407.915 983.334,407.867 \n",
" 987.732,406.554 992.288,405.224 995.058,404.433 996.845,403.922 1001.4,402.646 1005.96,401.401 1007.46,400.998 1010.51,400.181 1015.07,398.988 1019.63,397.825 \n",
" 1020.67,397.564 1024.18,396.685 1028.74,395.573 1033.3,394.489 1034.85,394.129 1037.85,393.429 1042.41,392.396 1046.96,391.389 1050.2,390.695 1051.52,390.409 \n",
" 1056.08,389.452 1060.63,388.522 1065.19,387.618 1067.05,387.26 1069.75,386.738 1074.3,385.882 1078.86,385.053 1083.42,384.249 1085.89,383.826 1087.97,383.469 \n",
" 1092.53,382.712 1097.08,381.98 1101.64,381.274 1106.2,380.593 1107.59,380.391 1110.75,379.933 1115.31,379.298 1119.87,378.687 1124.42,378.101 1128.98,377.539 \n",
" 1133.54,377.001 1133.93,376.957 1138.09,376.485 1142.65,375.992 1147.2,375.524 1151.76,375.079 1156.32,374.658 1160.87,374.261 1165.43,373.887 1169.99,373.537 \n",
" 1170.19,373.522 1174.54,373.208 1179.1,372.903 1183.66,372.621 1188.21,372.362 1192.77,372.127 1197.32,371.915 1201.88,371.726 1206.44,371.56 1210.99,371.418 \n",
" 1215.55,371.299 1220.11,371.202 1224.66,371.129 1229.22,371.079 1233.78,371.052 1238.33,371.048 1242.89,371.067 1247.44,371.109 1252,371.175 1256.56,371.263 \n",
" 1261.11,371.375 1265.67,371.51 1270.23,371.668 1274.78,371.85 1279.34,372.056 1283.9,372.285 1288.45,372.537 1293.01,372.813 1297.56,373.113 1302.12,373.437 \n",
" 1303.23,373.522 1306.68,373.784 1311.23,374.153 1315.79,374.547 1320.35,374.965 1324.9,375.408 1329.46,375.875 1334.02,376.367 1338.57,376.883 1339.19,376.957 \n",
" 1343.13,377.422 1347.68,377.985 1352.24,378.574 1356.8,379.188 1361.35,379.828 1365.21,380.391 1365.91,380.493 1370.47,381.18 1375.02,381.894 1379.58,382.634 \n",
" 1384.14,383.4 1386.58,383.826 1388.69,384.192 1393.25,385.007 1397.8,385.85 1402.36,386.721 1405.1,387.26 1406.92,387.617 1411.47,388.538 1416.03,389.488 \n",
" 1420.59,390.466 1421.62,390.695 1425.14,391.468 1429.7,392.499 1434.26,393.559 1436.64,394.129 1438.81,394.646 1443.37,395.759 1447.92,396.904 1450.49,397.564 \n",
" 1452.48,398.076 1457.04,399.276 1461.59,400.508 1463.37,400.998 1466.15,401.767 1470.71,403.056 1475.26,404.378 1475.45,404.433 1479.82,405.726 1484.38,407.108 \n",
" 1486.82,407.867 1488.93,408.52 1493.49,409.963 1497.62,411.302 1498.04,411.441 1502.6,412.947 1507.16,414.489 1507.87,414.736 1511.71,416.061 1516.27,417.669 \n",
" 1517.66,418.171 1520.83,419.309 1525.38,420.986 1527.04,421.605 1529.94,422.695 1534.5,424.442 1536.03,425.04 1539.05,426.223 1543.61,428.042 1544.67,428.474 \n",
" 1548.17,429.897 1552.72,431.792 1553,431.909 1557.28,433.721 1561.03,435.343 1561.83,435.693 1566.39,437.702 1568.78,438.778 1570.95,439.752 1575.5,441.844 \n",
" 1576.29,442.212 1580.06,443.976 1583.56,445.647 1584.62,446.153 1589.17,448.372 1590.6,449.081 1593.73,450.635 1597.44,452.516 1598.29,452.944 1602.84,455.299 \n",
" 1604.08,455.95 1607.4,457.701 1610.53,459.385 1611.95,460.152 1616.51,462.653 1616.81,462.819 1621.07,465.202 1622.91,466.254 1625.62,467.804 1628.86,469.688 \n",
" 1630.18,470.46 1634.65,473.123 1634.74,473.171 1639.29,475.937 1640.3,476.557 1643.85,478.76 1645.8,479.992 1648.41,481.643 1651.17,483.426 1652.96,484.587 \n",
" 1656.41,486.861 1657.52,487.594 1661.53,490.295 1662.07,490.666 1666.52,493.73 1666.63,493.804 1671.19,497.011 1671.4,497.164 1675.74,500.29 1676.17,500.599 \n",
" 1680.3,503.643 1680.82,504.033 1684.86,507.072 1685.37,507.468 1689.41,510.581 1689.82,510.902 1693.97,514.174 1694.17,514.337 1698.43,517.771 1698.53,517.852 \n",
" 1702.59,521.206 1703.08,521.621 1706.65,524.64 1707.64,525.483 1710.63,528.075 1712.19,529.445 1714.52,531.509 1716.75,533.509 1718.33,534.944 1721.31,537.682 \n",
" 1722.06,538.378 1725.7,541.813 1725.86,541.969 1729.26,545.247 1730.42,546.378 1732.75,548.682 1734.98,550.914 1736.16,552.117 1739.5,555.551 1739.53,555.584 \n",
" 1742.77,558.986 1744.09,560.402 1745.96,562.42 1748.65,565.371 1749.08,565.855 1752.14,569.289 1753.2,570.507 1755.12,572.724 1757.76,575.82 1758.04,576.158 \n",
" 1760.9,579.593 1762.31,581.328 1763.69,583.027 1766.42,586.462 1766.87,587.043 1769.08,589.896 1771.43,592.988 1771.69,593.331 1774.23,596.765 1775.98,599.187 \n",
" 1776.71,600.2 1779.14,603.634 1780.54,605.665 1781.5,607.069 1783.81,610.503 1785.1,612.455 1786.07,613.938 1788.26,617.372 1789.65,619.599 1790.4,620.807 \n",
" 1792.49,624.241 1794.21,627.147 1794.52,627.676 1796.5,631.11 1798.43,634.545 1798.77,635.167 1800.3,637.979 1802.12,641.414 1803.32,643.744 1803.89,644.848 \n",
" 1805.61,648.283 1807.28,651.717 1807.88,652.994 1808.9,655.152 1810.46,658.586 1811.98,662.021 1812.43,663.08 1813.45,665.455 1814.87,668.89 1816.25,672.324 \n",
" 1816.99,674.258 1817.57,675.759 1818.85,679.193 1820.08,682.628 1821.26,686.062 1821.55,686.933 1822.4,689.497 1823.49,692.931 1824.53,696.366 1825.53,699.8 \n",
" 1826.1,701.888 1826.48,703.235 1827.39,706.669 1828.25,710.104 1829.06,713.538 1829.83,716.973 1830.55,720.407 1830.66,720.945 1831.24,723.842 1831.87,727.276 \n",
" 1832.47,730.711 1833.01,734.145 1833.52,737.58 1833.97,741.014 1834.39,744.449 1834.76,747.883 1835.08,751.318 1835.22,752.922 1835.37,754.752 1835.61,758.187 \n",
" 1835.81,761.621 1835.96,765.056 1836.07,768.49 1836.13,771.925 1836.15,775.359 1836.13,778.794 1836.07,782.228 1835.96,785.663 1835.81,789.097 1835.61,792.532 \n",
" 1835.37,795.966 1835.22,797.797 1835.08,799.401 1834.76,802.835 1834.39,806.27 1833.97,809.704 1833.52,813.139 1833.01,816.573 1832.47,820.008 1831.87,823.442 \n",
" 1831.24,826.877 1830.66,829.773 1830.55,830.312 1829.83,833.746 1829.06,837.181 1828.25,840.615 1827.39,844.05 1826.48,847.484 1826.1,848.831 1825.53,850.919 \n",
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" 1151.76,449.798 1156.32,449.3 1158.44,449.081 1160.87,448.828 1165.43,448.381 1169.99,447.961 1174.54,447.568 1179.1,447.201 1183.66,446.861 1188.21,446.547 \n",
" 1192.77,446.259 1197.32,445.998 1201.88,445.763 1204.41,445.647 1206.44,445.553 1210.99,445.369 1215.55,445.211 1220.11,445.079 1224.66,444.973 1229.22,444.893 \n",
" 1233.78,444.84 1238.33,444.813 1242.89,444.811 1247.44,444.837 1252,444.888 1256.56,444.966 1261.11,445.07 1265.67,445.2 1270.23,445.357 1274.78,445.541 \n",
" 1277.08,445.647 1279.34,445.75 1283.9,445.986 1288.45,446.248 1293.01,446.536 1297.56,446.852 1302.12,447.195 1306.68,447.566 1311.23,447.963 1315.79,448.389 \n",
" 1320.35,448.842 1322.61,449.081 1324.9,449.322 1329.46,449.828 1334.02,450.362 1338.57,450.925 1343.13,451.517 1347.68,452.138 1350.34,452.516 1352.24,452.786 \n",
" 1356.8,453.461 1361.35,454.166 1365.91,454.9 1370.47,455.666 1372.1,455.95 1375.02,456.458 1379.58,457.279 1384.14,458.132 1388.69,459.017 1390.53,459.385 \n",
" 1393.25,459.929 1397.8,460.872 1402.36,461.847 1406.75,462.819 1406.92,462.856 1411.47,463.89 1416.03,464.96 1420.59,466.064 1421.35,466.254 1425.14,467.196 \n",
" 1429.7,468.363 1434.26,469.565 1434.71,469.688 1438.81,470.797 1443.37,472.065 1447.06,473.123 1447.92,473.37 1452.48,474.705 1457.04,476.079 1458.58,476.557 \n",
" 1461.59,477.487 1466.15,478.932 1469.41,479.992 1470.71,480.415 1475.26,481.933 1479.63,483.426 1479.82,483.493 1484.38,485.086 1488.93,486.724 1489.31,486.861 \n",
" 1493.49,488.396 1498.04,490.114 1498.52,490.295 1502.6,491.869 1507.16,493.67 1507.31,493.73 1511.71,495.51 1515.71,497.164 1516.27,497.397 1520.83,499.325 \n",
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" 1552.72,514.165 1553.06,514.337 1557.28,516.486 1559.75,517.771 1561.83,518.864 1566.22,521.206 1566.39,521.3 1570.95,523.79 1572.47,524.64 1575.5,526.341 \n",
" 1578.53,528.075 1580.06,528.954 1584.41,531.509 1584.62,531.631 1589.17,534.372 1590.11,534.944 1593.73,537.18 1595.64,538.378 1598.29,540.058 1601,541.813 \n",
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" 1742.2,713.538 1743.2,716.973 1744.09,720.245 1744.13,720.407 1745.02,723.842 1745.85,727.276 1746.61,730.711 1747.32,734.145 1747.97,737.58 1748.56,741.014 \n",
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"metadata": {},
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],
"source": [
"t = Fun()\n",
"f = sqrt(1-t^2)*exp(t)\n",
"v = z -> logkernel(f, z) # logkernel(f,z) calculates 1/π * \\int f(t)*log|t-z| dt\n",
"\n",
"xx = yy = -2:0.01:2\n",
"V = v.(xx' .+ im*yy)\n",
"\n",
"contour(xx, yy, V)\n",
"plot!(domain(t); color=:black, label=\"contour\")"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
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"source": [
"surface(xx, yy, V)"
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{
"cell_type": "markdown",
"metadata": {},
"source": [
"Continuity follows since $\\log|x-t|$ is integrable for $-1 \\leq x \\leq 1$ provided $f(t)$ has weaker than pole singularities. \n",
"\n",
"For $z \\notin (-\\infty,1]$ this can also be seen since $v$ is the real part of an analytic function:\n",
"$$\n",
" v(z) = \\Re {1 \\over \\pi} \\int_{-1}^1 f(t) \\log ( z - t) \\dt\n",
"$$\n",
"Note that the integrand avoids the branch cut of $\\log z$. To extend this to $z\\in (-\\infty,-1]$ (or more generally, $z \\notin [-1,\\infty)$), we can use the alternative expression\n",
"$$\n",
" v(z) = \\Re {1 \\over \\pi} \\int_{-1}^1 f(t) \\log (t-z) \\dt\n",
"$$\n",
"which follows from $\\log|z-t| = \\log|t-z|$."
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"v(z) = 0.7068168642791222\n",
"sum(f * log(abs(z - t))) / π = 0.7068168642791223\n",
"sum(f * log(z - t)) / π = 0.7068168642791222 + 0.5923597267719309im\n",
"sum(f * log(t - z)) / π = 0.7068168642791222 - 1.1831399624402499im\n"
]
}
],
"source": [
"z = 2.0 +3.0im\n",
"@show v(z)\n",
"@show sum(f*log(abs(z-t)))/π\n",
"@show sum(f*log(z-t))/π\n",
"@show sum(f*log(t-z))/π;"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Evaluating logarithmic singular integrals\n",
"\n",
"How can we actually evaluate \n",
"$$\n",
"Lu(z) = {1 \\over \\pi} \\int_{-1}^1 u(t) \\log (z-t) \\dt\n",
"$$\n",
"for $z$ in the complex plane? We relate this to the indefinite integral of the Cauchy transform, and then use that to reduce it to a Cauchy transform itself.\n",
"\n",
"#### Integrated Cauchy transform\n",
"\n",
"For $z$ away from the branch cut our integrand is \"nice\" (assuming $u$ itself is \"nice\") and we can exchange integration and differentiation to determine\n",
"$$\n",
"{\\D \\over \\D z} Lu(z) = {1 \\over \\pi} \\int_{-1}^1 u(t) {1\\over z -t} \\dt = -2 \\I \\CC u(z)\n",
"$$\n",
"thus if we can calculate $\\int^z Cu(\\zeta) \\D \\zeta$, we are in good shape. \n",
"\n",
"\n",
"_Example_ If $u(x) = {1 \\over \\sqrt{1-x^2}}$, then recall\n",
"$$\n",
"Cu(z) = {-1 \\over 2 \\I \\sqrt{z-1} \\sqrt{z+1}}\n",
"$$\n",
"We therefore have\n",
"$$\n",
"{\\D \\over \\D z} Lu(z) = {1 \\over \\sqrt{z-1} \\sqrt{z+1}} \n",
"$$\n",
"and therefore for some constant of integration $D$ we have\n",
"$$\n",
"Lu(z) + D = \\pi \\int^z {\\D z \\over \\sqrt{z-1} \\sqrt{z+1}} = 2 \\pi \\log\\left(\\sqrt{z-1} + \\sqrt{z+1}\\right)\n",
"$$\n",
"We need the right constant of integration. we can find that via the behaviour as $x \\rightarrow \\infty$:\n",
"$$\n",
"Lu(x) ={1 \\over \\pi} \\int_{-1}^1 u(t) \\log(x-t) \\dt = {\\log x \\over \\pi} \\int_{-1}^1 u(t) \\dt + {1 \\over \\pi} \\int_{-1}^1 u(t) \\log(1-t/x) \\dt = {\\log x \\over \\pi}\\int_{-1}^1 u(t) \\dt + O(x^{-1})\n",
"$$\n",
"Since we have\n",
"$$\n",
"\\int_{-1}^1 {1 \\over \\sqrt{1-x^2}} \\dx = \\pi\n",
"$$\n",
"and\n",
"$$\n",
"2 \\log(\\sqrt{x-1} + \\sqrt{x+1}) = 2 \\log \\sqrt x + 2 \\log(\\sqrt{1-1/x} + \\sqrt{1 + 1/x}) = \n",
"\\pi \\log x + 2\\pi \\log(2) + O(1/x)\n",
"$$\n",
"therefore, \n",
"$$\n",
"Lu(z) = 2 \\log\\left(\\sqrt{z-1} + \\sqrt{z+1}\\right) - 2 \\log 2\n",
"$$\n",
"\n",
"_Demonstration_ We want to calculate the log kernel of $1/\\sqrt{1-x^2}$:"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.3681278813450904 + 0.9045568943023814im"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
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],
"source": [
"x = Fun()\n",
"u = 1/sqrt(1-t^2)\n",
"z = 1+im\n",
"sum(u*log(z-x))/π"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We saw that its derivative is given in terms of the Cauchy transform, here we test it using finite difference approximation:"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(0.3515911336637867 - 0.5688482444998755im, 0.35157758425414287 - 0.5688644810057831im, 0.3515775842541429 - 0.568864481005783im)"
]
},
"execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"h = 0.0001;\n",
"(sum(u*log(z+h-x))/π - sum(u*log(z-x))/π)/h , sum(u/(z-x))/π, -2im*cauchy(u,z)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"But we already know the Cauchy transform in closed form:"
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.3515775842541429 - 0.568864481005783im"
]
},
"execution_count": 22,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Cu = z -> -1/(2im*sqrt(z-1)sqrt(z+1))\n",
"\n",
"-2im*Cu(z)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This we can integrate in closed form to determine:"
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.36812788134509056 + 0.9045568943023812im"
]
},
"execution_count": 24,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Lu = z -> 2*log(sqrt(z-1) + sqrt(z+1)) - 2*log(2)\n",
"Lu(z)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Reduction to a Cauchy transform\n",
"\n",
"The representation is not ideal as it involves indefinite integration in the complex plane. However, we will see that we can actually express $L u (z)$ as a Cauchy transform of $\\int_x^1 u(t) \\dt$. \n",
"\n",
"**Lemma (Log kernel as Cauchy transform)** Suppose $u$ is \"nice\" (as in Plemelj). Then \n",
"$$\n",
"Lu(z) = {\\log(z-a) \\over \\pi} \\int_a^b u(x) \\dx +2 \\I C U(z)\n",
"$$\n",
"where \n",
"$$\n",
"U(x) = \\int_x^b u(t) \\dt\n",
"$$\n",
"\n",
"**Proof**\n",
"\n",
"We use Plemelj to prove this result (of course!) and assume $[a,b] = [-1,1]$. Note that $Lu$ satisfies the following Riemann–Hilbert problem:\n",
"\n",
"1. $Lu(z) \\sim {1 \\over \\pi} \\int_{-1}^1 u(t) \\dt \\log z + o(1)$\n",
"2. $L^+ u(x) - L^- u(x) = 2 \\I \\int_x^1 u(t) \\dt $ for $-1 < x < 1$\n",
"3. $L^+ u(x) - L^- u(x) = 2\\I \\int_{-1}^1 u(t) \\dt $ for $x < -1$\n",
"\n",
"\n",
"The first part follows immediately by noting for $z \\rightarrow + \\infty$ we have \n",
"$$\\log(z - x) = \\log(z (1 - x/z)) = \\log z + \\log(1-x/z) = \\log z + o(1)$$\n",
"\n",
"We now see where the jumps come from. Since\n",
" $Lu'(z) = -2 \\I Cu(z)$ and\n",
" This let's us think of $Lu (z)$ as (2 is arbitrary here)\n",
"$$\n",
"L u(z) = Lu(2) + \\int_2^z C u (\\zeta) \\D\\zeta\n",
"$$\n",
"where the contour is a straight line (just like in the problem sheet for $\\log z$ that I'm sure everyone has done by now), but we can think of it also as an arc that avoids $[-1,1]$.\n",
"\n",
"Consider $x < -1$. Given an ellipse $\\gamma$ surrounding $[-1,1]$ we can write the integral representations and exchange orders to get: \n",
"$$\n",
"L u^+(x) - L u^-(x) = -2 \\I \\oint_\\gamma C u(\\zeta) \\D\\zeta = -2 \\I \\int_{-1}^1 u(t) {1\\over 2 \\pi \\I} \\oint_\\gamma {1 \\over t-z} \\dt \\D \\zeta = 2 \\I \\int_{-1}^1 u(t) \\dt \n",
"$$\n",
"\n",
"For $-1 < x < 1$, note that\n",
"$\n",
"L u(1) \n",
"$\n",
"is well-defined as there is only a logarithmic singularity times an algebraic one. Thus deforming the contour onto the interval we have \n",
"$$\n",
"L^\\pm u(x) = -2 \\I \\int_1^x C^\\pm u(t) \\dt + L u(1),\n",
"$$\n",
"hence\n",
"$$\n",
"L^+ u(x) -L^- u(x) = 2 \\I \\int_x^1 (C^+ u(t) - C^- u(t)) \\dt =2 \\I \\int_x^1 u(t) \\dt\n",
"$$\n",
"⬛️\n",
"\n",
"\n",
"\n",
"_Example_\n",
"For the problem above, we have that $\\int_x^1 {1 \\over \\sqrt{1-t^2}}\\dt = {\\rm arccos}\\, x$, which gives us:"
]
},
{
"cell_type": "code",
"execution_count": 37,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(0.36812788134509017 + 0.9045568943023812im, 0.36812788134509056 + 0.9045568943023812im)"
]
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"metadata": {},
"output_type": "execute_result"
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],
"source": [
"x = Fun()\n",
"\n",
"u = 1/sqrt(1-x^2)\n",
"U = acos(x)\n",
"\n",
"\n",
"Lu = z -> 2*log(sqrt(z-1) + sqrt(z+1)) - 2*log(2)\n",
"\n",
"sum(u) * log(z+1)/π + sum(U/(x-z))/(π), Lu(z)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Log transforms of weighted orthogonal polynomials\n",
"\n",
"\n",
"Now consider ${1 \\over \\pi} \\int_a^b p_k(x) w(x) \\log |z-x| \\dx$, which we write in terms of the real part of\n",
"$$\n",
"L_k(z) = L[p_k w](z) = {1 \\over \\pi } \\int_a^b p_k(x) w(x) \\log (z-x) \\dx\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"For $k > 0$ we have $\\int_a^b p_k(x) w(x) dx = 0$ due to orthogonality, and hence we actually have no branch cut:"
]
},
{
"cell_type": "code",
"execution_count": 40,
"metadata": {},
"outputs": [
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},
"execution_count": 40,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"T₅ = Fun(Chebyshev(), [zeros(2);1])\n",
"w = 1/sqrt(1-x^2)\n",
"L₅ = z->cauchyintegral(w*T₅, z)\n",
"\n",
"phaseplot(-3..3, -3..3, L₅)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\n",
"#### Example: Weighted Chebyshev log transform\n",
"\n",
"For classical orthogonal polynomials we can go a step further and relate the indefinite integrals to other orthogonal polynomials.\n",
"\n",
" For example, recall that\n",
"$$\n",
"{\\D\\over \\dx}[\\sqrt{1-x^2} U_n(x)] = -{n+1 \\over \\sqrt{1-x^2}} T_{n+1}(x)\n",
"$$\n",
"in other words, \n",
"$$\n",
"\\int_x^1 {T_k(t) \\over \\sqrt{1-t^2}} \\dt = -{\\sqrt{1-x^2} U_{k-1}(x) \\over k}\n",
"$$\n",
"Thus for $k=1,2,\\ldots$,\n",
"$$\n",
"L_k(z) = -{1 \\over n+1} \\CC[\\sqrt{1-\\diamond^2} U_{k-1}](z)\n",
"$$\n",
"and\n",
"$$\n",
"{1 \\over \\pi} \\int_{-1}^1 {T_k(x) \\over \\sqrt{1-x^2}}\\log(z-x) \\dx = {2 \\I \\over k} \\CC[\\sqrt{1-\\diamond^2} U_{k-1}](z)\n",
"$$\n"
]
},
{
"cell_type": "code",
"execution_count": 42,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
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"metadata": {},
"output_type": "execute_result"
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"source": [
"T₅ = Fun(Chebyshev(), [zeros(5);1])\n",
"U₄ = Fun(Ultraspherical(1), [zeros(4);1])\n",
"x = Fun()\n",
"L₅ = z->sum(T₅/sqrt(1-x^2) * log(z-x))/π\n",
"\n",
"L₅(z), 2im*cauchy(sqrt(1-x^2)*U₄,z)/5"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"As we saw last lecture, Cauchy transforms of OPs satisfy simple recurrences, and this relationship renders log transforms equally calculable. "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\n",
"### Solving logarithmic singular integral equations\n",
"\n",
"Oddly enough, it is easier to solve a singular integral equation involving the logarithmic kernel than to actually calculate the logarithmic singular integral, so we begin here. Consider the problem of calculating $u(x)$ such that\n",
"$$\n",
"{1 \\over \\pi} \\int_{-1}^1 u(t) \\log | x-t| \\dt = f(x) \\qqfor -1 < x < 1\n",
"$$\n",
"\n",
"\n",
"To tackle this problem we first need to understand the additive jump $Re L^\\pm(x) = {L^+(x) + L^-(x) \\over 2}$. Thinking in terms of integrals of Cauchy transforms we see that\n",
"$$\n",
"L^+u(x) + L^-u(x) = -2\\I (2\\int_2^1 \\CC u(x) \\dx + \\int_1^x [C^+ u(t) + C^-u(t)] \\dt) =\n",
" D -2 \\I \\int_1^x (-\\I H u(t)) \\dt = D + 2\\int_x^1 H u(t) dt\n",
"$$\n",
"That is, it is expressed as an indefinite integral of the Hilbert transform. We therefore have:\n",
"$$\n",
"{\\D \\over \\dx} {1 \\over \\pi} \\int_{-1}^1 u(t) \\log | x-t| \\dt = {\\D \\over \\dx} \\int_x^1 H u(t) dt = - Hu(x)\n",
"$$\n",
"where the minus sign comes from the fact that $x$ is the lower limit. In other words, our original SIE becomes equivalent to inverting the Hilbert transform:\n",
"$$\n",
"\\HH u(x) = -f'(x)\n",
"$$\n",
"recall, we can express the solution as\n",
"$$\n",
" u = {1 \\over \\sqrt{1-x^2}} \\HH[\\sqrt{1-\\diamond^2} f'] + {C\\over \\sqrt{1-x^2}}\n",
"$$\n",
"\n",
"Let's do a numerical example:"
]
},
{
"cell_type": "code",
"execution_count": 43,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"hilbert(u, 0.1) + f(0.1) = -4.440892098500626e-16\n",
"logkernel(u, 0.1) - f(0.1) = -0.18115023187927326\n",
"logkernel(u, 0.2) - f(0.2) = -0.18115023187927304\n"
]
}
],
"source": [
"x = Fun()\n",
"f = exp(x)\n",
"\n",
"C = randn()\n",
"\n",
"u₁ = hilbert(sqrt(1-x^2)*f')/sqrt(1-x^2) \n",
"\n",
"u = u₁ + C/sqrt(1-x^2)\n",
"\n",
"@show hilbert(u, 0.1) + f(0.1)\n",
"@show logkernel(u,0.1) - f(0.1) # didn't work 😩\n",
"@show logkernel(u,0.2) - f(0.2); # but we are only off by a constant"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Remember: for inverting the Hilbert transform we had a degree of freedom: every solution plus $C/\\sqrt{1-x^2}$ was also a solution. But here, since we differentiated, we use that degree of freedom to ensure that we have arrived at the right solution. \n",
"\n",
"To choose $C$, we use the fact that\n",
"$$\n",
"f(0) = {1 \\over \\pi} \\int_{-1}^1 u(t) \\log |t| \\dt\n",
"$$\n",
"If we can calculate the right integral, we can use this to choose $C$:"
]
},
{
"cell_type": "code",
"execution_count": 44,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"hilbert(u, 0.1) + f(0.1) = -4.440892098500626e-16\n",
"logkernel(u, 0.1) - f(0.1) = 0.0\n",
"logkernel(u, 0.2) - f(0.2) = 2.220446049250313e-16\n"
]
}
],
"source": [
"# choose C so that\n",
"# logkernel(u₁, 0) + C*logkernel(1/sqrt(1-x^2), 0) == f(0)\n",
"C = (f(0) - logkernel(u₁, 0))/logkernel(1/sqrt(1-x^2), 0)\n",
"u = u₁ + C/sqrt(1-x^2)\n",
"\n",
"@show hilbert(u, 0.1) + f(0.1)\n",
"@show logkernel(u,0.1) - f(0.1) # Works!\n",
"@show logkernel(u,0.2) - f(0.2); # And at all x!"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"*Example* We now do an example which can be solved by hand. Find $u(x)$ so that: \n",
"$$\n",
"\\int_{-1}^1 u(t) \\log | x-t| \\dt = 1.\n",
"$$\n",
"Differentiating, we know that \n",
"$$\n",
"\\int_{-1}^1 {u(t) \\over x-t} \\dt = 0\n",
"$$\n",
"hence $u(x)$ must be of the form ${C \\over \\sqrt{1-x^2}}$. Which $C$? Make it work for $x = 0$:\n",
"$$\n",
"1 = \\int_{-1}^1 u(t) \\log | 0 - t| \\dt = C \\int_{-1}^1 {\\log | t| \\over \\sqrt{1-t^2}} \\dt\n",
"$$\n",
"To evaluate the integral, we can use trigonmetric variables:\n",
"$$\n",
"\\int_0^1 {\\log t \\over \\sqrt{1-t^2}} \\dt = \\int_0^{\\pi \\over 2} {\\cos \\theta \\log \\sin \\theta \\over \\sqrt{1-\\sin^2 \\theta}} \\D\\theta = \\int_0^{\\pi \\over 2} \\log \\sin \\theta \\D\\theta = - {\\pi \\log 2\\over 2}\n",
"$$\n",
"(The last identity takes some work: I'll leave it as an excercise.)\n",
"\n",
"Thus we have $ C = -{1 \\over \\pi \\log 2}$"
]
},
{
"cell_type": "code",
"execution_count": 49,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1.0"
]
},
"execution_count": 49,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x = Fun()\n",
"C = -1/(log(2))\n",
"u = C/sqrt(1-x^2)\n",
"\n",
"logkernel(u, 0.2) "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Physically, this solution gives us the potential field \n",
"$$\n",
"\\int_{-1}^1 u(t) \\log|z-t| \\dt\n",
"$$\n",
"corresponding to holding a metal plate at constant potential:"
]
},
{
"cell_type": "code",
"execution_count": 52,
"metadata": {},
"outputs": [
{
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" 791.808,1364.67 796.365,1365.76 797.813,1366.09 800.921,1366.82 805.477,1367.87 810.034,1368.91 812.834,1369.53 814.59,1369.92 819.146,1370.92 823.703,1371.9 \n",
" 828.259,1372.87 828.729,1372.96 832.816,1373.82 837.372,1374.75 841.928,1375.66 845.667,1376.4 846.485,1376.56 851.041,1377.44 855.597,1378.31 860.154,1379.16 \n",
" 863.848,1379.83 864.71,1379.99 869.266,1380.81 873.823,1381.61 878.379,1382.4 882.936,1383.16 883.564,1383.27 887.492,1383.92 892.048,1384.66 896.605,1385.38 \n",
" 901.161,1386.08 905.251,1386.7 905.717,1386.77 910.274,1387.45 914.83,1388.11 919.386,1388.75 923.943,1389.38 928.499,1389.99 929.603,1390.14 933.056,1390.59 \n",
" 937.612,1391.17 942.168,1391.74 946.725,1392.29 951.281,1392.83 955.837,1393.35 957.843,1393.57 960.394,1393.85 964.95,1394.35 969.507,1394.82 974.063,1395.28 \n",
" 978.619,1395.73 983.176,1396.16 987.732,1396.58 992.288,1396.98 992.609,1397.01 996.845,1397.37 1001.4,1397.74 1005.96,1398.09 1010.51,1398.44 1015.07,1398.76 \n",
" 1019.63,1399.08 1024.18,1399.38 1028.74,1399.66 1033.3,1399.93 1037.85,1400.18 1042.41,1400.42 1042.8,1400.44 1046.96,1400.65 1051.52,1400.86 1056.08,1401.05 \n",
" 1060.63,1401.23 1065.19,1401.4 1069.75,1401.55 1074.3,1401.69 1078.86,1401.81 1083.42,1401.92 1087.97,1402.02 1092.53,1402.1 1097.08,1402.16 1101.64,1402.21 \n",
" 1106.2,1402.25 1110.75,1402.27 1115.31,1402.28 1119.87,1402.27 1124.42,1402.25 1128.98,1402.21 1133.54,1402.16 1138.09,1402.1 1142.65,1402.02 1147.2,1401.92 \n",
" 1151.76,1401.81 1156.32,1401.69 1160.87,1401.55 1165.43,1401.4 1169.99,1401.23 1174.54,1401.05 1179.1,1400.86 1183.66,1400.65 1187.82,1400.44 1188.21,1400.42 \n",
" 1192.77,1400.18 1197.32,1399.93 1201.88,1399.66 1206.44,1399.38 1210.99,1399.08 1215.55,1398.76 1220.11,1398.44 1224.66,1398.09 1229.22,1397.74 1233.78,1397.37 \n",
" 1238.01,1397.01 1238.33,1396.98 1242.89,1396.58 1247.44,1396.16 1252,1395.73 1256.56,1395.28 1261.11,1394.82 1265.67,1394.35 1270.23,1393.85 1272.78,1393.57 \n",
" 1274.78,1393.35 1279.34,1392.83 1283.9,1392.29 1288.45,1391.74 1293.01,1391.17 1297.56,1390.59 1301.02,1390.14 1302.12,1389.99 1306.68,1389.38 1311.23,1388.75 \n",
" 1315.79,1388.11 1320.35,1387.45 1324.9,1386.77 1325.37,1386.7 1329.46,1386.08 1334.02,1385.38 1338.57,1384.66 1343.13,1383.92 1347.06,1383.27 1347.68,1383.16 \n",
" 1352.24,1382.4 1356.8,1381.61 1361.35,1380.81 1365.91,1379.99 1366.77,1379.83 1370.47,1379.16 1375.02,1378.31 1379.58,1377.44 1384.14,1376.56 1384.95,1376.4 \n",
" 1388.69,1375.66 1393.25,1374.75 1397.8,1373.82 1401.89,1372.96 1402.36,1372.87 1406.92,1371.9 1411.47,1370.92 1416.03,1369.92 1417.79,1369.53 1420.59,1368.91 \n",
" 1425.14,1367.87 1429.7,1366.82 1432.81,1366.09 1434.26,1365.76 1438.81,1364.67 1443.37,1363.57 1447.07,1362.66 1447.92,1362.45 1452.48,1361.31 1457.04,1360.16 \n",
" 1460.66,1359.23 1461.59,1358.99 1466.15,1357.8 1470.71,1356.59 1473.66,1355.79 1475.26,1355.36 1479.82,1354.12 1484.38,1352.85 1486.14,1352.36 1488.93,1351.57 \n",
" 1493.49,1350.27 1498.04,1348.95 1498.14,1348.92 1502.6,1347.61 1507.16,1346.26 1509.7,1345.49 1511.71,1344.88 1516.27,1343.48 1520.83,1342.07 1520.87,1342.05 \n",
" 1525.38,1340.63 1529.94,1339.18 1531.67,1338.62 1534.5,1337.71 1539.05,1336.21 1542.14,1335.18 1543.61,1334.7 1548.17,1333.16 1552.3,1331.75 1552.72,1331.6 \n",
" 1557.28,1330.03 1561.83,1328.43 1562.16,1328.31 1566.39,1326.81 1570.95,1325.17 1571.75,1324.88 1575.5,1323.51 1580.06,1321.83 1581.09,1321.45 1584.62,1320.13 \n",
" 1589.17,1318.4 1590.18,1318.01 1593.73,1316.65 1598.29,1314.88 1599.05,1314.58 1602.84,1313.09 1607.4,1311.27 1607.71,1311.14 1611.95,1309.43 1616.16,1307.71 \n",
" 1616.51,1307.57 1621.07,1305.68 1624.42,1304.27 1625.62,1303.77 1630.18,1301.83 1632.49,1300.84 1634.74,1299.87 1639.29,1297.89 1640.39,1297.4 1643.85,1295.88 \n",
" 1648.12,1293.97 1648.41,1293.84 1652.96,1291.78 1655.69,1290.54 1657.52,1289.7 1662.07,1287.59 1663.11,1287.1 1666.63,1285.45 1670.38,1283.67 1671.19,1283.28 \n",
" 1675.74,1281.09 1677.5,1280.23 1680.3,1278.87 1684.5,1276.8 1684.86,1276.62 1689.41,1274.34 1691.35,1273.36 1693.97,1272.04 1698.09,1269.93 1698.53,1269.7 \n",
" 1703.08,1267.34 1704.69,1266.49 1707.64,1264.95 1711.18,1263.06 1712.19,1262.52 1716.75,1260.06 1717.56,1259.62 1721.31,1257.58 1723.82,1256.19 1725.86,1255.06 \n",
" 1729.98,1252.76 1730.42,1252.51 1734.98,1249.93 1736.03,1249.32 1739.53,1247.31 1741.98,1245.89 1744.09,1244.66 1747.83,1242.45 1748.65,1241.97 1753.2,1239.25 \n",
" 1753.59,1239.02 1757.76,1236.5 1759.25,1235.58 1762.31,1233.7 1764.82,1232.15 1766.87,1230.87 1770.31,1228.71 1771.43,1228.01 1775.7,1225.28 1775.98,1225.1 \n",
" 1780.54,1222.16 1781.02,1221.85 1785.1,1219.17 1786.25,1218.41 1789.65,1216.15 1791.4,1214.98 1794.21,1213.08 1796.47,1211.54 1798.77,1209.97 1801.46,1208.11 \n",
" 1803.32,1206.82 1806.38,1204.67 1807.88,1203.62 1811.23,1201.24 1812.43,1200.38 1816,1197.8 1816.99,1197.09 1820.7,1194.37 1821.55,1193.75 1825.34,1190.93 \n",
" 1826.1,1190.36 1829.9,1187.5 1830.66,1186.93 1834.4,1184.07 1835.22,1183.44 1838.84,1180.63 1839.77,1179.9 1843.21,1177.2 1844.33,1176.31 1847.52,1173.76 \n",
" 1848.89,1172.66 1851.76,1170.33 1853.44,1168.95 1855.95,1166.89 1858,1165.19 1860.07,1163.46 1862.55,1161.37 1864.14,1160.02 1867.11,1157.48 1868.14,1156.59 \n",
" 1871.67,1153.53 1872.1,1153.16 1875.99,1149.72 1876.22,1149.51 1879.83,1146.29 1880.78,1145.43 1883.62,1142.85 1885.34,1141.28 1887.35,1139.42 1889.89,1137.05 \n",
" 1891.03,1135.98 1894.45,1132.75 1894.66,1132.55 1898.23,1129.11 1899.01,1128.37 1901.76,1125.68 1903.56,1123.9 1905.24,1122.24 1908.12,1119.36 1908.66,1118.81 \n",
" 1912.04,1115.38 1912.67,1114.72 1915.37,1111.94 1917.23,1110 1918.65,1108.51 1921.79,1105.18 1921.89,1105.07 1925.08,1101.64 1926.34,1100.26 1928.22,1098.2 \n",
" 1930.9,1095.24 1931.32,1094.77 1934.37,1091.33 1935.46,1090.1 1937.38,1087.9 1940.01,1084.86 1940.35,1084.46 1943.27,1081.03 1944.57,1079.49 1946.15,1077.6 \n",
" 1948.99,1074.16 1949.13,1074 1951.79,1070.73 1953.68,1068.37 1954.54,1067.29 1957.25,1063.86 1958.24,1062.6 1959.93,1060.42 1962.56,1056.99 1962.79,1056.68 \n",
" 1965.15,1053.55 1967.35,1050.6 1967.7,1050.12 1970.22,1046.69 1971.91,1044.34 1972.69,1043.25 1975.13,1039.82 1976.46,1037.91 1977.53,1036.38 1979.89,1032.95 \n",
" 1981.02,1031.28 1982.21,1029.51 1984.5,1026.08 1985.58,1024.43 1986.75,1022.64 1988.96,1019.21 1990.13,1017.36 1991.13,1015.77 1993.27,1012.34 1994.69,1010.03 \n",
" 1995.37,1008.91 1997.44,1005.47 1999.25,1002.42 1999.47,1002.04 2001.47,998.602 2003.43,995.168 2003.8,994.514 2005.36,991.733 2007.26,988.299 2008.36,986.264 \n",
" 2009.12,984.864 2010.94,981.43 2012.73,977.995 2012.91,977.637 2014.49,974.561 2016.21,971.126 2017.47,968.573 2017.9,967.692 2019.56,964.257 2021.19,960.823 \n",
" 2022.03,959.014 2022.78,957.388 2024.34,953.954 2025.87,950.519 2026.58,948.879 \n",
" \"/>\n",
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" 2026.58,1142.33 2026.13,1142.85 2023.11,1146.29 2022.03,1147.51 2020.07,1149.72 2017.47,1152.61 2016.98,1153.16 2013.85,1156.59 2012.91,1157.61 2010.68,1160.02 \n",
" 2008.36,1162.51 2007.47,1163.46 2004.23,1166.89 2003.8,1167.34 2000.94,1170.33 1999.25,1172.08 1997.61,1173.76 1994.69,1176.74 1994.24,1177.2 1990.83,1180.63 \n",
" 1990.13,1181.32 1987.37,1184.07 1985.58,1185.83 1983.87,1187.5 1981.02,1190.27 1980.33,1190.93 1976.75,1194.37 1976.46,1194.64 1973.11,1197.8 1971.91,1198.94 \n",
" 1969.44,1201.24 1967.35,1203.17 1965.72,1204.67 1962.79,1207.34 1961.95,1208.11 1958.24,1211.45 1958.13,1211.54 1954.27,1214.98 1953.68,1215.5 1950.36,1218.41 \n",
" 1949.13,1219.48 1946.4,1221.85 1944.57,1223.42 1942.39,1225.28 1940.01,1227.29 1938.33,1228.71 1935.46,1231.12 1934.22,1232.15 1930.9,1234.89 1930.05,1235.58 \n",
" 1926.34,1238.61 1925.83,1239.02 1921.79,1242.27 1921.56,1242.45 1917.24,1245.89 1917.23,1245.89 1912.86,1249.32 1912.67,1249.46 1908.42,1252.76 1908.12,1252.99 \n",
" 1903.93,1256.19 1903.56,1256.47 1899.38,1259.62 1899.01,1259.9 1894.76,1263.06 1894.45,1263.29 1890.09,1266.49 1889.89,1266.64 1885.36,1269.93 1885.34,1269.94 \n",
" 1880.78,1273.21 1880.56,1273.36 1876.22,1276.43 1875.7,1276.8 1871.67,1279.62 1870.78,1280.23 1867.11,1282.76 1865.79,1283.67 1862.55,1285.87 1860.73,1287.1 \n",
" 1858,1288.94 1855.6,1290.54 1853.44,1291.97 1850.4,1293.97 1848.89,1294.97 1845.13,1297.4 1844.33,1297.93 1839.79,1300.84 1839.77,1300.85 1835.22,1303.74 \n",
" 1834.37,1304.27 1830.66,1306.6 1828.88,1307.71 1826.1,1309.42 1823.3,1311.14 1821.55,1312.22 1817.65,1314.58 1816.99,1314.97 1812.43,1317.7 1811.91,1318.01 \n",
" 1807.88,1320.4 1806.09,1321.45 1803.32,1323.06 1800.18,1324.88 1798.77,1325.7 1794.21,1328.3 1794.19,1328.31 1789.65,1330.88 1788.1,1331.75 1785.1,1333.42 \n",
" 1781.91,1335.18 1780.54,1335.94 1775.98,1338.43 1775.64,1338.62 1771.43,1340.89 1769.26,1342.05 1766.87,1343.33 1762.78,1345.49 1762.31,1345.73 1757.76,1348.11 \n",
" 1756.19,1348.92 1753.2,1350.47 1749.5,1352.36 1748.65,1352.79 1744.09,1355.09 1742.69,1355.79 1739.53,1357.37 1735.77,1359.23 1734.98,1359.62 1730.42,1361.84 \n",
" 1728.73,1362.66 1725.86,1364.04 1721.57,1366.09 1721.31,1366.22 1716.75,1368.37 1714.27,1369.53 1712.19,1370.5 1707.64,1372.6 1706.85,1372.96 1703.08,1374.68 \n",
" 1699.28,1376.4 1698.53,1376.74 1693.97,1378.77 1691.57,1379.83 1689.41,1380.79 1684.86,1382.78 1683.72,1383.27 1680.3,1384.74 1675.74,1386.69 1675.71,1386.7 \n",
" 1671.19,1388.61 1667.53,1390.14 1666.63,1390.51 1662.07,1392.39 1659.19,1393.57 1657.52,1394.25 1652.96,1396.09 1650.66,1397.01 1648.41,1397.91 1643.85,1399.7 \n",
" 1641.95,1400.44 1639.29,1401.48 1634.74,1403.23 1633.05,1403.87 1630.18,1404.97 1625.62,1406.68 1623.94,1407.31 1621.07,1408.38 1616.51,1410.05 1614.61,1410.74 \n",
" 1611.95,1411.71 1607.4,1413.35 1605.06,1414.18 1602.84,1414.96 1598.29,1416.56 1595.26,1417.61 1593.73,1418.14 1589.17,1419.7 1585.2,1421.05 1584.62,1421.24 \n",
" 1580.06,1422.77 1575.5,1424.27 1574.86,1424.48 1570.95,1425.76 1566.39,1427.23 1564.23,1427.92 1561.83,1428.68 1557.28,1430.11 1553.29,1431.35 1552.72,1431.53 \n",
" 1548.17,1432.93 1543.61,1434.3 1542,1434.78 1539.05,1435.67 1534.5,1437.01 1530.35,1438.22 1529.94,1438.34 1525.38,1439.65 1520.83,1440.95 1518.3,1441.65 \n",
" 1516.27,1442.22 1511.71,1443.48 1507.16,1444.72 1505.81,1445.09 1502.6,1445.95 1498.04,1447.16 1493.49,1448.35 1492.84,1448.52 1488.93,1449.53 1484.38,1450.69 \n",
" 1479.82,1451.84 1479.34,1451.96 1475.26,1452.97 1470.71,1454.08 1466.15,1455.18 1465.24,1455.39 1461.59,1456.26 1457.04,1457.32 1452.48,1458.37 1450.47,1458.83 \n",
" 1447.92,1459.4 1443.37,1460.42 1438.81,1461.43 1434.95,1462.26 \n",
" \"/>\n",
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" 1772.38,1462.26 1775.98,1460.59 1779.73,1458.83 1780.54,1458.45 1785.1,1456.29 1786.97,1455.39 1789.65,1454.11 1794.1,1451.96 1794.21,1451.9 1798.77,1449.68 \n",
" 1801.11,1448.52 1803.32,1447.43 1807.88,1445.16 1808.02,1445.09 1812.43,1442.87 1814.82,1441.65 1816.99,1440.55 1821.53,1438.22 1821.55,1438.21 1826.1,1435.85 \n",
" 1828.14,1434.78 1830.66,1433.46 1834.65,1431.35 1835.22,1431.05 1839.77,1428.62 1841.07,1427.92 1844.33,1426.16 1847.41,1424.48 1848.89,1423.67 1853.44,1421.16 \n",
" 1853.65,1421.05 1858,1418.63 1859.81,1417.61 1862.55,1416.07 1865.89,1414.18 1867.11,1413.48 1871.67,1410.87 1871.88,1410.74 1876.22,1408.23 1877.8,1407.31 \n",
" 1880.78,1405.56 1883.64,1403.87 1885.34,1402.87 1889.4,1400.44 1889.89,1400.15 1894.45,1397.4 1895.09,1397.01 1899.01,1394.62 1900.71,1393.57 1903.56,1391.81 \n",
" 1906.26,1390.14 1908.12,1388.98 1911.74,1386.7 1912.67,1386.11 1917.15,1383.27 1917.23,1383.21 1921.79,1380.29 1922.49,1379.83 1926.34,1377.33 1927.77,1376.4 \n",
" 1930.9,1374.34 1932.98,1372.96 1935.46,1371.32 1938.13,1369.53 1940.01,1368.27 1943.23,1366.09 1944.57,1365.18 1948.26,1362.66 1949.13,1362.06 1953.23,1359.23 \n",
" 1953.68,1358.91 1958.14,1355.79 1958.24,1355.72 1962.79,1352.5 1962.99,1352.36 1967.35,1349.24 1967.79,1348.92 1971.91,1345.95 1972.54,1345.49 1976.46,1342.62 \n",
" 1977.23,1342.05 1981.02,1339.25 1981.86,1338.62 1985.58,1335.84 1986.45,1335.18 1990.13,1332.39 1990.98,1331.75 1994.69,1328.91 1995.46,1328.31 1999.25,1325.38 \n",
" 1999.89,1324.88 2003.8,1321.82 2004.27,1321.45 2008.36,1318.21 2008.6,1318.01 2012.89,1314.58 2012.91,1314.56 2017.12,1311.14 2017.47,1310.86 2021.31,1307.71 \n",
" 2022.03,1307.12 2025.46,1304.27 2026.58,1303.33 \n",
" \"/>\n",
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" 204.037,247.384 205.162,246.445 208.593,243.599 209.306,243.011 213.15,239.859 213.496,239.576 217.706,236.163 217.733,236.142 222.017,232.707 222.262,232.512 \n",
" 226.349,229.273 226.819,228.903 230.73,225.838 231.375,225.336 235.161,222.404 235.932,221.81 239.641,218.969 240.488,218.325 244.173,215.535 245.044,214.879 \n",
" 248.757,212.1 249.601,211.472 253.393,208.666 254.157,208.103 258.082,205.231 258.713,204.772 262.826,201.797 263.27,201.478 267.626,198.362 267.826,198.22 \n",
" 272.382,194.997 272.481,194.928 276.939,191.81 277.394,191.493 281.495,188.656 282.365,188.059 286.052,185.537 287.395,184.624 290.608,182.451 292.486,181.19 \n",
" 295.164,179.398 297.638,177.755 299.721,176.378 302.853,174.321 304.277,173.389 308.131,170.886 308.833,170.432 313.39,167.506 313.474,167.452 317.946,164.609 \n",
" 318.885,164.017 322.503,161.743 324.363,160.583 327.059,158.907 329.91,157.148 331.615,156.1 335.527,153.714 336.172,153.322 340.728,150.572 341.217,150.279 \n",
" 345.284,147.85 346.982,146.845 349.841,145.156 352.821,143.41 354.397,142.49 358.737,139.976 358.953,139.851 363.51,137.237 364.733,136.541 368.066,134.65 \n",
" 370.81,133.107 372.623,132.09 376.969,129.672 377.179,129.556 381.735,127.046 383.215,126.238 386.292,124.561 389.547,122.803 390.848,122.102 395.404,119.667 \n",
" 395.968,119.369 399.961,117.256 402.482,115.934 404.517,114.87 409.073,112.508 409.089,112.5 413.63,110.168 415.796,109.065 418.186,107.851 422.6,105.631 \n",
" 422.743,105.559 427.299,103.288 429.51,102.196 431.855,101.04 436.412,98.8158 436.523,98.7617 440.968,96.6117 443.649,95.3272 445.524,94.4305 450.081,92.2711 \n",
" 450.886,91.8927 454.637,90.1323 458.24,88.4582 \n",
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" 1891.03,414.736 1894.45,417.972 1894.66,418.171 1898.23,421.605 1899.01,422.353 1901.76,425.04 1903.56,426.815 1905.24,428.474 1908.12,431.361 1908.66,431.909 \n",
" 1912.04,435.343 1912.67,435.995 1915.37,438.778 1917.23,440.72 1918.65,442.212 1921.79,445.539 1921.89,445.647 1925.08,449.081 1926.34,450.459 1928.22,452.516 \n",
" 1930.9,455.482 1931.32,455.95 1934.37,459.385 1935.46,460.615 1937.38,462.819 1940.01,465.862 1940.35,466.254 1943.27,469.688 1944.57,471.23 1946.15,473.123 \n",
" 1948.99,476.557 1949.13,476.722 1951.79,479.992 1953.68,482.351 1954.54,483.426 1957.25,486.861 1958.24,488.121 1959.93,490.295 1962.56,493.73 1962.79,494.041 \n",
" 1965.15,497.164 1967.35,500.121 1967.7,500.599 1970.22,504.033 1971.91,506.374 1972.69,507.468 1975.13,510.902 1976.46,512.81 1977.53,514.337 1979.89,517.771 \n",
" 1981.02,519.442 1982.21,521.206 1984.5,524.64 1985.58,526.287 1986.75,528.075 1988.96,531.509 1990.13,533.364 1991.13,534.944 1993.27,538.378 1994.69,540.691 \n",
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" 2009.12,565.855 2010.94,569.289 2012.73,572.724 2012.91,573.082 2014.49,576.158 2016.21,579.593 2017.47,582.145 2017.9,583.027 2019.56,586.462 2021.19,589.896 \n",
" 2022.03,591.704 2022.78,593.331 2024.34,596.765 2025.87,600.2 2026.58,601.84 \n",
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" 1903.93,294.529 1903.56,294.251 1899.38,291.094 1899.01,290.817 1894.76,287.66 1894.45,287.426 1890.09,284.225 1889.89,284.079 1885.36,280.791 1885.34,280.774 \n",
" 1880.78,277.51 1880.56,277.356 1876.22,274.287 1875.7,273.922 1871.67,271.103 1870.78,270.487 1867.11,267.958 1865.79,267.052 1862.55,264.851 1860.73,263.618 \n",
" 1858,261.782 1855.6,260.183 1853.44,258.749 1850.4,256.749 1848.89,255.753 1845.13,253.314 1844.33,252.793 1839.79,249.88 1839.77,249.868 1835.22,246.976 \n",
" 1834.37,246.445 1830.66,244.119 1828.88,243.011 1826.1,241.294 1823.3,239.576 1821.55,238.503 1817.65,236.142 1816.99,235.744 1812.43,233.017 1811.91,232.707 \n",
" 1807.88,230.321 1806.09,229.273 1803.32,227.655 1800.18,225.838 1798.77,225.021 1794.21,222.416 1794.19,222.404 1789.65,219.84 1788.1,218.969 1785.1,217.294 \n",
" 1781.91,215.535 1780.54,214.777 1775.98,212.288 1775.64,212.1 1771.43,209.826 1769.26,208.666 1766.87,207.392 1762.78,205.231 1762.31,204.986 1757.76,202.606 \n",
" 1756.19,201.797 1753.2,200.253 1749.5,198.362 1748.65,197.927 1744.09,195.626 1742.69,194.928 1739.53,193.35 1735.77,191.493 1734.98,191.101 1730.42,188.875 \n",
" 1728.73,188.059 1725.86,186.675 1721.57,184.624 1721.31,184.5 1716.75,182.348 1714.27,181.19 1712.19,180.22 1707.64,178.117 1706.85,177.755 1703.08,176.036 \n",
" 1699.28,174.321 1698.53,173.979 1693.97,171.944 1691.57,170.886 1689.41,169.932 1684.86,167.944 1683.72,167.452 1680.3,165.976 1675.74,164.032 1675.71,164.017 \n",
" 1671.19,162.108 1667.53,160.583 1666.63,160.207 1662.07,158.327 1659.19,157.148 1657.52,156.468 1652.96,154.631 1650.66,153.714 1648.41,152.814 1643.85,151.018 \n",
" 1641.95,150.279 1639.29,149.242 1634.74,147.487 1633.05,146.845 1630.18,145.751 1625.62,144.037 1623.94,143.41 1621.07,142.341 1616.51,140.666 1614.61,139.976 \n",
" 1611.95,139.009 1607.4,137.373 1605.06,136.541 1602.84,135.755 1598.29,134.156 1595.26,133.107 1593.73,132.577 1589.17,131.016 1585.2,129.672 1584.62,129.475 \n",
" 1580.06,127.95 1575.5,126.446 1574.86,126.238 1570.95,124.958 1566.39,123.49 1564.23,122.803 1561.83,122.039 1557.28,120.606 1553.29,119.369 1552.72,119.192 \n",
" 1548.17,117.793 1543.61,116.414 1542,115.934 1539.05,115.051 1534.5,113.706 1530.35,112.5 1529.94,112.379 1525.38,111.067 1520.83,109.774 1518.3,109.065 \n",
" 1516.27,108.497 1511.71,107.236 1507.16,105.994 1505.81,105.631 1502.6,104.767 1498.04,103.557 1493.49,102.364 1492.84,102.196 1488.93,101.186 1484.38,100.025 \n",
" 1479.82,98.8812 1479.34,98.7617 1475.26,97.7515 1470.71,96.6387 1466.15,95.5431 1465.24,95.3272 1461.59,94.4616 1457.04,93.3962 1452.48,92.3478 1450.47,91.8927 \n",
" 1447.92,91.3141 1443.37,90.2955 1438.81,89.2934 1434.95,88.4582 \n",
" \"/>\n",
"<polyline clip-path=\"url(#clip6603)\" style=\"stroke:#0d0829; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
" 1772.38,88.4582 1775.98,90.1323 1779.73,91.8927 1780.54,92.2711 1785.1,94.4305 1786.97,95.3272 1789.65,96.6117 1794.1,98.7617 1794.21,98.8158 1798.77,101.04 \n",
" 1801.11,102.196 1803.32,103.288 1807.88,105.559 1808.02,105.631 1812.43,107.851 1814.82,109.065 1816.99,110.168 1821.53,112.5 1821.55,112.508 1826.1,114.87 \n",
" 1828.14,115.934 1830.66,117.256 1834.65,119.369 1835.22,119.667 1839.77,122.102 1841.07,122.803 1844.33,124.561 1847.41,126.238 1848.89,127.046 1853.44,129.556 \n",
" 1853.65,129.672 1858,132.09 1859.81,133.107 1862.55,134.65 1865.89,136.541 1867.11,137.237 1871.67,139.851 1871.88,139.976 1876.22,142.49 1877.8,143.41 \n",
" 1880.78,145.156 1883.64,146.845 1885.34,147.85 1889.4,150.279 1889.89,150.572 1894.45,153.322 1895.09,153.714 1899.01,156.1 1900.71,157.148 1903.56,158.907 \n",
" 1906.26,160.583 1908.12,161.743 1911.74,164.017 1912.67,164.609 1917.15,167.452 1917.23,167.506 1921.79,170.432 1922.49,170.886 1926.34,173.389 1927.77,174.321 \n",
" 1930.9,176.378 1932.98,177.755 1935.46,179.398 1938.13,181.19 1940.01,182.451 1943.23,184.624 1944.57,185.537 1948.26,188.059 1949.13,188.656 1953.23,191.493 \n",
" 1953.68,191.81 1958.14,194.928 1958.24,194.997 1962.79,198.22 1962.99,198.362 1967.35,201.478 1967.79,201.797 1971.91,204.772 1972.54,205.231 1976.46,208.103 \n",
" 1977.23,208.666 1981.02,211.472 1981.86,212.1 1985.58,214.879 1986.45,215.535 1990.13,218.325 1990.98,218.969 1994.69,221.81 1995.46,222.404 1999.25,225.336 \n",
" 1999.89,225.838 2003.8,228.903 2004.27,229.273 2008.36,232.512 2008.6,232.707 2012.89,236.142 2012.91,236.163 2017.12,239.576 2017.47,239.859 2021.31,243.011 \n",
" 2022.03,243.599 2025.46,246.445 2026.58,247.384 \n",
" \"/>\n",
"<polyline clip-path=\"url(#clip6603)\" style=\"stroke:#63146e; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
" 1779.8,435.343 1780.54,435.927 1784.1,438.778 1785.1,439.582 1788.32,442.212 1789.65,443.304 1792.48,445.647 1794.21,447.093 1796.57,449.081 1798.77,450.953 \n",
" 1800.59,452.516 1803.32,454.885 1804.54,455.95 1807.88,458.893 1808.43,459.385 1812.26,462.819 1812.43,462.979 1816.02,466.254 1816.99,467.147 1819.73,469.688 \n",
" 1821.55,471.4 1823.37,473.123 1826.1,475.741 1826.95,476.557 1830.47,479.992 1830.66,480.174 1833.94,483.426 1835.22,484.704 1837.35,486.861 1839.77,489.335 \n",
" 1840.71,490.295 1844,493.73 1844.33,494.072 1847.25,497.164 1848.89,498.922 1850.44,500.599 1853.44,503.887 1853.57,504.033 1856.66,507.468 1858,508.979 \n",
" 1859.69,510.902 1862.55,514.2 1862.67,514.337 1865.6,517.771 1867.11,519.563 1868.48,521.206 1871.32,524.64 1871.67,525.073 1874.1,528.075 1876.22,530.742 \n",
" 1876.83,531.509 1879.52,534.944 1880.78,536.581 1882.16,538.378 1884.75,541.813 1885.34,542.602 1887.3,545.247 1889.79,548.682 1889.89,548.819 1892.25,552.117 \n",
" 1894.45,555.251 1894.66,555.551 1897.02,558.986 1899.01,561.917 1899.34,562.42 1901.62,565.855 1903.56,568.838 1903.85,569.289 1906.05,572.724 1908.12,576.04 \n",
" 1908.19,576.158 1910.3,579.593 1912.36,583.027 1912.67,583.559 1914.38,586.462 1916.36,589.896 1917.23,591.432 1918.3,593.331 1920.2,596.765 1921.79,599.7 \n",
" 1922.06,600.2 1923.88,603.634 1925.65,607.069 1926.34,608.433 1927.39,610.503 1929.09,613.938 1930.75,617.372 1930.9,617.695 1932.37,620.807 1933.95,624.241 \n",
" 1935.46,627.596 1935.49,627.676 1937,631.11 1938.47,634.545 1939.89,637.979 1940.01,638.275 1941.29,641.414 1942.64,644.848 1943.96,648.283 1944.57,649.931 \n",
" 1945.23,651.717 1946.48,655.152 1947.68,658.586 1948.85,662.021 1949.13,662.853 1949.98,665.455 1951.08,668.89 1952.14,672.324 1953.16,675.759 1953.68,677.558 \n",
" 1954.15,679.193 1955.11,682.628 1956.02,686.062 1956.9,689.497 1957.75,692.931 1958.24,695.008 1958.56,696.366 1959.34,699.8 1960.08,703.235 1960.78,706.669 \n",
" 1961.45,710.104 1962.09,713.538 1962.68,716.973 1962.79,717.64 1963.25,720.407 1963.78,723.842 1964.28,727.276 1964.74,730.711 1965.17,734.145 1965.56,737.58 \n",
" 1965.92,741.014 1966.25,744.449 1966.54,747.883 1966.79,751.318 1967.01,754.752 1967.2,758.187 1967.35,761.553 1967.35,761.621 1967.47,765.056 1967.56,768.49 \n",
" 1967.61,771.925 1967.63,775.359 1967.61,778.794 1967.56,782.228 1967.47,785.663 1967.35,789.097 1967.35,789.165 1967.2,792.532 1967.01,795.966 1966.79,799.401 \n",
" 1966.54,802.835 1966.25,806.27 1965.92,809.704 1965.56,813.139 1965.17,816.573 1964.74,820.008 1964.28,823.442 1963.78,826.877 1963.25,830.312 1962.79,833.079 \n",
" 1962.68,833.746 1962.09,837.181 1961.45,840.615 1960.78,844.05 1960.08,847.484 1959.34,850.919 1958.56,854.353 1958.24,855.711 1957.75,857.788 1956.9,861.222 \n",
" 1956.02,864.657 1955.11,868.091 1954.15,871.526 1953.68,873.161 1953.16,874.96 1952.14,878.395 1951.08,881.829 1949.98,885.264 1949.13,887.866 1948.85,888.698 \n",
" 1947.68,892.133 1946.48,895.567 1945.23,899.002 1944.57,900.788 1943.96,902.436 1942.64,905.871 1941.29,909.305 1940.01,912.444 1939.89,912.74 1938.47,916.174 \n",
" 1937,919.609 1935.49,923.043 1935.46,923.122 1933.95,926.478 1932.37,929.912 1930.9,933.023 1930.75,933.347 1929.09,936.781 1927.39,940.216 1926.34,942.286 \n",
" 1925.65,943.65 1923.88,947.085 1922.06,950.519 1921.79,951.019 1920.2,953.954 1918.3,957.388 1917.23,959.287 1916.36,960.823 1914.38,964.257 1912.67,967.159 \n",
" 1912.36,967.692 1910.3,971.126 1908.19,974.561 1908.12,974.679 1906.05,977.995 1903.85,981.43 1903.56,981.881 1901.62,984.864 1899.34,988.299 1899.01,988.802 \n",
" 1897.02,991.733 1894.66,995.168 1894.45,995.468 1892.25,998.602 1889.89,1001.9 1889.79,1002.04 1887.3,1005.47 1885.34,1008.12 1884.75,1008.91 1882.16,1012.34 \n",
" 1880.78,1014.14 1879.52,1015.77 1876.83,1019.21 1876.22,1019.98 1874.1,1022.64 1871.67,1025.65 1871.32,1026.08 1868.48,1029.51 1867.11,1031.16 1865.6,1032.95 \n",
" 1862.67,1036.38 1862.55,1036.52 1859.69,1039.82 1858,1041.74 1856.66,1043.25 1853.57,1046.69 1853.44,1046.83 1850.44,1050.12 1848.89,1051.8 1847.25,1053.55 \n",
" 1844.33,1056.65 1844,1056.99 1840.71,1060.42 1839.77,1061.38 1837.35,1063.86 1835.22,1066.01 1833.94,1067.29 1830.66,1070.55 1830.47,1070.73 1826.95,1074.16 \n",
" 1826.1,1074.98 1823.37,1077.6 1821.55,1079.32 1819.73,1081.03 1816.99,1083.57 1816.02,1084.46 1812.43,1087.74 1812.26,1087.9 1808.43,1091.33 1807.88,1091.83 \n",
" 1804.54,1094.77 1803.32,1095.83 1800.59,1098.2 1798.77,1099.77 1796.57,1101.64 1794.21,1103.63 1792.48,1105.07 1789.65,1107.41 1788.32,1108.51 1785.1,1111.14 \n",
" 1784.1,1111.94 1780.54,1114.79 1779.8,1115.38 1775.98,1118.38 1775.44,1118.81 1771.43,1121.92 1771,1122.24 1766.87,1125.39 1766.48,1125.68 1762.31,1128.8 \n",
" 1761.89,1129.11 1757.76,1132.16 1757.22,1132.55 1753.2,1135.46 1752.47,1135.98 1748.65,1138.71 1747.64,1139.42 1744.09,1141.91 1742.72,1142.85 1739.53,1145.05 \n",
" 1737.72,1146.29 1734.98,1148.15 1732.64,1149.72 1730.42,1151.2 1727.46,1153.16 1725.86,1154.2 1722.19,1156.59 1721.31,1157.16 1716.83,1160.02 1716.75,1160.07 \n",
" 1712.19,1162.94 1711.36,1163.46 1707.64,1165.77 1705.8,1166.89 1703.08,1168.55 1700.14,1170.33 1698.53,1171.3 1694.37,1173.76 1693.97,1174 1689.41,1176.66 \n",
" 1688.49,1177.2 1684.86,1179.29 1682.5,1180.63 1680.3,1181.88 1676.4,1184.07 1675.74,1184.43 1671.19,1186.95 1670.17,1187.5 1666.63,1189.43 1663.82,1190.93 \n",
" 1662.07,1191.87 1657.52,1194.28 1657.35,1194.37 1652.96,1196.66 1650.73,1197.8 1648.41,1199 1643.99,1201.24 1643.85,1201.31 1639.29,1203.59 1637.1,1204.67 \n",
" 1634.74,1205.84 1630.18,1208.05 1630.06,1208.11 1625.62,1210.24 1622.86,1211.54 1621.07,1212.39 1616.51,1214.52 1615.51,1214.98 1611.95,1216.61 1607.99,1218.41 \n",
" 1607.4,1218.68 1602.84,1220.72 1600.28,1221.85 1598.29,1222.73 1593.73,1224.71 1592.4,1225.28 1589.17,1226.66 1584.62,1228.59 1584.32,1228.71 1580.06,1230.49 \n",
" 1576.04,1232.15 1575.5,1232.37 1570.95,1234.22 1567.53,1235.58 1566.39,1236.04 1561.83,1237.84 1558.8,1239.02 1557.28,1239.61 1552.72,1241.36 1549.83,1242.45 \n",
" 1548.17,1243.08 1543.61,1244.78 1540.6,1245.89 1539.05,1246.46 1534.5,1248.11 1531.1,1249.32 1529.94,1249.74 1525.38,1251.34 1521.3,1252.76 1520.83,1252.92 \n",
" 1516.27,1254.48 1511.71,1256.02 1511.19,1256.19 1507.16,1257.53 1502.6,1259.02 1500.73,1259.62 1498.04,1260.49 1493.49,1261.94 1489.91,1263.06 1488.93,1263.37 \n",
" 1484.38,1264.77 1479.82,1266.15 1478.68,1266.49 1475.26,1267.52 1470.71,1268.86 1467.02,1269.93 1466.15,1270.18 1461.59,1271.48 1457.04,1272.76 1454.86,1273.36 \n",
" 1452.48,1274.02 1447.92,1275.26 1443.37,1276.48 1442.17,1276.8 1438.81,1277.68 1434.26,1278.86 1429.7,1280.02 1428.87,1280.23 1425.14,1281.17 1420.59,1282.29 \n",
" 1416.03,1283.39 1414.89,1283.67 1411.47,1284.48 1406.92,1285.55 1402.36,1286.59 1400.11,1287.1 1397.8,1287.62 1393.25,1288.63 1388.69,1289.63 1384.43,1290.54 \n",
" 1384.14,1290.6 1379.58,1291.56 1375.02,1292.49 1370.47,1293.41 1367.64,1293.97 1365.91,1294.31 1361.35,1295.2 1356.8,1296.06 1352.24,1296.91 1349.54,1297.4 \n",
" 1347.68,1297.74 1343.13,1298.56 1338.57,1299.36 1334.02,1300.13 1329.79,1300.84 1329.46,1300.89 1324.9,1301.64 1320.35,1302.37 1315.79,1303.08 1311.23,1303.77 \n",
" 1307.86,1304.27 1306.68,1304.45 1302.12,1305.11 1297.56,1305.76 1293.01,1306.38 1288.45,1306.99 1283.9,1307.59 1282.92,1307.71 1279.34,1308.16 1274.78,1308.72 \n",
" 1270.23,1309.27 1265.67,1309.8 1261.11,1310.31 1256.56,1310.8 1253.35,1311.14 1252,1311.28 1247.44,1311.75 1242.89,1312.2 1238.33,1312.63 1233.78,1313.04 \n",
" 1229.22,1313.44 1224.66,1313.83 1220.11,1314.19 1215.55,1314.54 1215.1,1314.58 1210.99,1314.88 1206.44,1315.2 1201.88,1315.51 1197.32,1315.79 1192.77,1316.07 \n",
" 1188.21,1316.32 1183.66,1316.57 1179.1,1316.79 1174.54,1317 1169.99,1317.19 1165.43,1317.37 1160.87,1317.54 1156.32,1317.68 1151.76,1317.82 1147.2,1317.93 \n",
" 1143.6,1318.01 1142.65,1318.03 1138.09,1318.12 1133.54,1318.19 1128.98,1318.24 1124.42,1318.28 1119.87,1318.3 1115.31,1318.31 1110.75,1318.3 1106.2,1318.28 \n",
" 1101.64,1318.24 1097.08,1318.19 1092.53,1318.12 1087.97,1318.03 1087.02,1318.01 1083.42,1317.93 1078.86,1317.82 1074.3,1317.68 1069.75,1317.54 1065.19,1317.37 \n",
" 1060.63,1317.19 1056.08,1317 1051.52,1316.79 1046.96,1316.57 1042.41,1316.32 1037.85,1316.07 1033.3,1315.79 1028.74,1315.51 1024.18,1315.2 1019.63,1314.88 \n",
" 1015.52,1314.58 1015.07,1314.54 1010.51,1314.19 1005.96,1313.83 1001.4,1313.44 996.845,1313.04 992.288,1312.63 987.732,1312.2 983.176,1311.75 978.619,1311.28 \n",
" 977.275,1311.14 974.063,1310.8 969.507,1310.31 964.95,1309.8 960.394,1309.27 955.837,1308.72 951.281,1308.16 947.696,1307.71 946.725,1307.59 942.168,1306.99 \n",
" 937.612,1306.38 933.056,1305.76 928.499,1305.11 923.943,1304.45 922.759,1304.27 919.386,1303.77 914.83,1303.08 910.274,1302.37 905.717,1301.64 901.161,1300.89 \n",
" 900.829,1300.84 896.605,1300.13 892.048,1299.36 887.492,1298.56 882.936,1297.74 881.079,1297.4 878.379,1296.91 873.823,1296.06 869.266,1295.2 864.71,1294.31 \n",
" 862.977,1293.97 860.154,1293.41 855.597,1292.49 851.041,1291.56 846.485,1290.6 846.192,1290.54 841.928,1289.63 837.372,1288.63 832.816,1287.62 830.508,1287.1 \n",
" 828.259,1286.59 823.703,1285.55 819.146,1284.48 815.733,1283.67 814.59,1283.39 810.034,1282.29 805.477,1281.17 801.746,1280.23 800.921,1280.02 796.365,1278.86 \n",
" 791.808,1277.68 788.449,1276.8 787.252,1276.48 782.696,1275.26 778.139,1274.02 775.758,1273.36 773.583,1272.76 769.026,1271.48 764.47,1270.18 763.603,1269.93 \n",
" 759.914,1268.86 755.357,1267.52 751.937,1266.49 750.801,1266.15 746.245,1264.77 741.688,1263.37 740.711,1263.06 737.132,1261.94 732.575,1260.49 729.889,1259.62 \n",
" 728.019,1259.02 723.463,1257.53 719.432,1256.19 718.906,1256.02 714.35,1254.48 709.794,1252.92 709.318,1252.76 705.237,1251.34 700.681,1249.74 699.522,1249.32 \n",
" 696.125,1248.11 691.568,1246.46 690.019,1245.89 687.012,1244.78 682.455,1243.08 680.79,1242.45 677.899,1241.36 673.343,1239.61 671.818,1239.02 668.786,1237.84 \n",
" 664.23,1236.04 663.087,1235.58 659.674,1234.22 655.117,1232.37 654.585,1232.15 650.561,1230.49 646.299,1228.71 646.005,1228.59 641.448,1226.66 638.221,1225.28 \n",
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" 607.756,1211.54 604.997,1210.24 600.56,1208.11 600.441,1208.05 595.884,1205.84 593.524,1204.67 591.328,1203.59 586.772,1201.31 586.631,1201.24 582.215,1199 \n",
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" 1835.25,940.216 1835.22,940.274 1833.1,943.65 1830.89,947.085 1830.66,947.433 1828.63,950.519 1826.32,953.954 1826.1,954.263 1823.95,957.388 1821.55,960.802 \n",
" 1821.53,960.823 1819.06,964.257 1816.99,967.075 1816.54,967.692 1813.96,971.126 1812.43,973.112 1811.32,974.561 1808.63,977.995 1807.88,978.933 1805.88,981.43 \n",
" 1803.32,984.557 1803.07,984.864 1800.2,988.299 1798.77,989.995 1797.28,991.733 1794.3,995.168 1794.21,995.267 1791.25,998.602 1789.65,1000.38 1788.15,1002.04 \n",
" 1785.1,1005.34 1784.98,1005.47 1781.75,1008.91 1780.54,1010.17 1778.45,1012.34 1775.98,1014.87 1775.09,1015.77 1771.67,1019.21 1771.43,1019.45 1768.18,1022.64 \n",
" 1766.87,1023.91 1764.62,1026.08 1762.31,1028.26 1760.99,1029.51 1757.76,1032.51 1757.29,1032.95 1753.52,1036.38 1753.2,1036.66 1749.67,1039.82 1748.65,1040.72 \n",
" 1745.75,1043.25 1744.09,1044.69 1741.76,1046.69 1739.53,1048.57 1737.69,1050.12 1734.98,1052.37 1733.54,1053.55 1730.42,1056.09 1729.31,1056.99 1725.86,1059.74 \n",
" 1725,1060.42 1721.31,1063.31 1720.6,1063.86 1716.75,1066.82 1716.12,1067.29 1712.19,1070.25 1711.56,1070.73 1707.64,1073.62 1706.9,1074.16 1703.08,1076.93 \n",
" 1702.15,1077.6 1698.53,1080.18 1697.31,1081.03 1693.97,1083.36 1692.37,1084.46 1689.41,1086.49 1687.33,1087.9 1684.86,1089.57 1682.2,1091.33 1680.3,1092.59 \n",
" 1676.96,1094.77 1675.74,1095.55 1671.61,1098.2 1671.19,1098.47 1666.63,1101.34 1666.15,1101.64 1662.07,1104.16 1660.58,1105.07 1657.52,1106.93 1654.89,1108.51 \n",
" 1652.96,1109.66 1649.08,1111.94 1648.41,1112.34 1643.85,1114.97 1643.15,1115.38 1639.29,1117.57 1637.08,1118.81 1634.74,1120.12 1630.89,1122.24 1630.18,1122.64 \n",
" 1625.62,1125.11 1624.56,1125.68 1621.07,1127.54 1618.08,1129.11 1616.51,1129.94 1611.95,1132.29 1611.45,1132.55 1607.4,1134.61 1604.67,1135.98 1602.84,1136.9 \n",
" 1598.29,1139.15 1597.73,1139.42 1593.73,1141.36 1590.62,1142.85 1589.17,1143.54 1584.62,1145.69 1583.33,1146.29 1580.06,1147.8 1575.86,1149.72 1575.5,1149.88 \n",
" 1570.95,1151.94 1568.19,1153.16 1566.39,1153.95 1561.83,1155.94 1560.32,1156.59 1557.28,1157.9 1552.72,1159.82 1552.24,1160.02 1548.17,1161.72 1543.93,1163.46 \n",
" 1543.61,1163.59 1539.05,1165.43 1535.38,1166.89 1534.5,1167.24 1529.94,1169.03 1526.57,1170.33 1525.38,1170.78 1520.83,1172.52 1517.49,1173.76 1516.27,1174.22 \n",
" 1511.71,1175.9 1508.12,1177.2 1507.16,1177.54 1502.6,1179.17 1498.44,1180.63 1498.04,1180.77 1493.49,1182.34 1488.93,1183.89 1488.42,1184.07 1484.38,1185.42 \n",
" 1479.82,1186.92 1478.03,1187.5 1475.26,1188.4 1470.71,1189.85 1467.25,1190.93 1466.15,1191.28 1461.59,1192.69 1457.04,1194.07 1456.03,1194.37 1452.48,1195.43 \n",
" 1447.92,1196.77 1444.33,1197.8 1443.37,1198.08 1438.81,1199.38 1434.26,1200.65 1432.1,1201.24 1429.7,1201.9 1425.14,1203.13 1420.59,1204.33 1419.28,1204.67 \n",
" 1416.03,1205.52 1411.47,1206.68 1406.92,1207.83 1405.78,1208.11 1402.36,1208.95 1397.8,1210.05 1393.25,1211.14 1391.5,1211.54 1388.69,1212.2 1384.14,1213.24 \n",
" 1379.58,1214.26 1376.33,1214.98 1375.02,1215.26 1370.47,1216.25 1365.91,1217.21 1361.35,1218.15 1360.08,1218.41 1356.8,1219.08 1352.24,1219.98 1347.68,1220.87 \n",
" 1343.13,1221.74 1342.54,1221.85 1338.57,1222.59 1334.02,1223.42 1329.46,1224.23 1324.9,1225.02 1323.36,1225.28 1320.35,1225.79 1315.79,1226.55 1311.23,1227.29 \n",
" 1306.68,1228.01 1302.12,1228.71 1302.07,1228.71 1297.56,1229.39 1293.01,1230.06 1288.45,1230.71 1283.9,1231.34 1279.34,1231.95 1277.8,1232.15 1274.78,1232.54 \n",
" 1270.23,1233.12 1265.67,1233.68 1261.11,1234.23 1256.56,1234.75 1252,1235.26 1248.98,1235.58 1247.44,1235.75 1242.89,1236.22 1238.33,1236.68 1233.78,1237.12 \n",
" 1229.22,1237.54 1224.66,1237.95 1220.11,1238.34 1215.55,1238.71 1211.58,1239.02 1210.99,1239.06 1206.44,1239.4 1201.88,1239.73 1197.32,1240.03 1192.77,1240.32 \n",
" 1188.21,1240.59 1183.66,1240.85 1179.1,1241.08 1174.54,1241.31 1169.99,1241.51 1165.43,1241.7 1160.87,1241.87 1156.32,1242.03 1151.76,1242.17 1147.2,1242.29 \n",
" 1142.65,1242.4 1139.82,1242.45 1138.09,1242.49 1133.54,1242.56 1128.98,1242.62 1124.42,1242.66 1119.87,1242.68 1115.31,1242.69 1110.75,1242.68 1106.2,1242.66 \n",
" 1101.64,1242.62 1097.08,1242.56 1092.53,1242.49 1090.8,1242.45 1087.97,1242.4 1083.42,1242.29 1078.86,1242.17 1074.3,1242.03 1069.75,1241.87 1065.19,1241.7 \n",
" 1060.63,1241.51 1056.08,1241.31 1051.52,1241.08 1046.96,1240.85 1042.41,1240.59 1037.85,1240.32 1033.3,1240.03 1028.74,1239.73 1024.18,1239.4 1019.63,1239.06 \n",
" 1019.04,1239.02 1015.07,1238.71 1010.51,1238.34 1005.96,1237.95 1001.4,1237.54 996.845,1237.12 992.288,1236.68 987.732,1236.22 983.176,1235.75 981.644,1235.58 \n",
" 978.619,1235.26 974.063,1234.75 969.507,1234.23 964.95,1233.68 960.394,1233.12 955.837,1232.54 952.819,1232.15 951.281,1231.95 946.725,1231.34 942.168,1230.71 \n",
" 937.612,1230.06 933.056,1229.39 928.546,1228.71 928.499,1228.71 923.943,1228.01 919.386,1227.29 914.83,1226.55 910.274,1225.79 907.259,1225.28 905.717,1225.02 \n",
" 901.161,1224.23 896.605,1223.42 892.048,1222.59 888.082,1221.85 887.492,1221.74 882.936,1220.87 878.379,1219.98 873.823,1219.08 870.541,1218.41 869.266,1218.15 \n",
" 864.71,1217.21 860.154,1216.25 855.597,1215.26 854.294,1214.98 851.041,1214.26 846.485,1213.24 841.928,1212.2 839.119,1211.54 837.372,1211.14 832.816,1210.05 \n",
" 828.259,1208.95 824.842,1208.11 823.703,1207.83 819.146,1206.68 814.59,1205.52 811.341,1204.67 810.034,1204.33 805.477,1203.13 800.921,1201.9 798.518,1201.24 \n",
" 796.365,1200.65 791.808,1199.38 787.252,1198.08 786.287,1197.8 782.696,1196.77 778.139,1195.43 774.59,1194.37 773.583,1194.07 769.026,1192.69 764.47,1191.28 \n",
" 763.374,1190.93 759.914,1189.85 755.357,1188.4 752.593,1187.5 750.801,1186.92 746.245,1185.42 742.204,1184.07 741.688,1183.89 737.132,1182.34 732.575,1180.77 \n",
" 732.183,1180.63 728.019,1179.17 723.463,1177.54 722.502,1177.2 718.906,1175.9 714.35,1174.22 713.132,1173.76 709.794,1172.52 705.237,1170.78 704.052,1170.33 \n",
" 700.681,1169.03 696.125,1167.24 695.243,1166.89 691.568,1165.43 687.012,1163.59 686.69,1163.46 682.455,1161.72 678.379,1160.02 677.899,1159.82 673.343,1157.9 \n",
" 670.297,1156.59 668.786,1155.94 664.23,1153.95 662.427,1153.16 659.674,1151.94 655.117,1149.88 654.759,1149.72 650.561,1147.8 647.289,1146.29 646.005,1145.69 \n",
" 641.448,1143.54 640.002,1142.85 636.892,1141.36 632.889,1139.42 632.335,1139.15 627.779,1136.9 625.949,1135.98 623.223,1134.61 619.166,1132.55 618.666,1132.29 \n",
" 614.11,1129.94 612.542,1129.11 609.554,1127.54 606.065,1125.68 604.997,1125.11 600.441,1122.64 599.73,1122.24 595.884,1120.12 593.537,1118.81 591.328,1117.57 \n",
" 587.473,1115.38 586.772,1114.97 582.215,1112.34 581.54,1111.94 577.659,1109.66 575.732,1108.51 573.103,1106.93 570.044,1105.07 568.546,1104.16 564.471,1101.64 \n",
" 563.99,1101.34 559.434,1098.47 559.013,1098.2 554.877,1095.55 553.665,1094.77 550.321,1092.59 548.424,1091.33 545.764,1089.57 543.287,1087.9 541.208,1086.49 \n",
" 538.25,1084.46 536.652,1083.36 533.312,1081.03 532.095,1080.18 528.47,1077.6 527.539,1076.93 523.722,1074.16 522.983,1073.62 519.065,1070.73 518.426,1070.25 \n",
" 514.497,1067.29 513.87,1066.82 510.017,1063.86 509.314,1063.31 505.623,1060.42 504.757,1059.74 501.311,1056.99 500.201,1056.09 497.082,1053.55 495.644,1052.37 \n",
" 492.933,1050.12 491.088,1048.57 488.862,1046.69 486.532,1044.69 484.868,1043.25 481.975,1040.72 480.949,1039.82 477.419,1036.66 477.105,1036.38 473.334,1032.95 \n",
" 472.863,1032.51 469.634,1029.51 468.306,1028.26 466.004,1026.08 463.75,1023.91 462.444,1022.64 459.194,1019.45 458.952,1019.21 455.526,1015.77 454.637,1014.87 \n",
" 452.166,1012.34 450.081,1010.17 448.872,1008.91 445.642,1005.47 445.524,1005.34 442.474,1002.04 440.968,1000.38 439.368,998.602 436.412,995.267 436.325,995.168 \n",
" 433.34,991.733 431.855,989.995 430.416,988.299 427.552,984.864 427.299,984.557 424.744,981.43 422.743,978.933 421.995,977.995 419.302,974.561 418.186,973.112 \n",
" 416.665,971.126 414.085,967.692 413.63,967.075 411.558,964.257 409.088,960.823 409.073,960.802 406.668,957.388 404.517,954.263 404.305,953.954 401.991,950.519 \n",
" 399.961,947.433 399.733,947.085 397.522,943.65 395.404,940.274 395.368,940.216 393.258,936.781 391.204,933.347 390.848,932.739 389.196,929.912 387.24,926.478 \n",
" 386.292,924.772 385.332,923.043 383.472,919.609 381.735,916.31 381.664,916.174 379.898,912.74 378.183,909.305 377.179,907.236 376.516,905.871 374.893,902.436 \n",
" 373.32,899.002 372.623,897.434 371.792,895.567 370.308,892.133 368.873,888.698 368.066,886.704 367.483,885.264 366.135,881.829 364.835,878.395 363.583,874.96 \n",
" 363.51,874.754 362.368,871.526 361.2,868.091 360.079,864.657 359.003,861.222 358.953,861.058 357.965,857.788 356.972,854.353 356.024,850.919 355.121,847.484 \n",
" 354.397,844.589 354.261,844.05 353.44,840.615 352.663,837.181 351.929,833.746 351.239,830.312 350.592,826.877 349.988,823.442 349.841,822.542 349.423,820.008 \n",
" 348.9,816.573 348.419,813.139 347.981,809.704 347.585,806.27 347.231,802.835 346.92,799.401 346.65,795.966 346.421,792.532 346.235,789.097 346.089,785.663 \n",
" 345.986,782.228 345.924,778.794 345.903,775.359 345.924,771.925 345.986,768.49 346.089,765.056 346.235,761.621 346.421,758.187 346.65,754.752 346.92,751.318 \n",
" 347.231,747.883 347.585,744.449 347.981,741.014 348.419,737.58 348.9,734.145 349.423,730.711 349.841,728.177 349.988,727.276 350.592,723.842 351.239,720.407 \n",
" 351.929,716.973 352.663,713.538 353.44,710.104 354.261,706.669 354.397,706.13 355.121,703.235 356.024,699.8 356.972,696.366 357.965,692.931 358.953,689.661 \n",
" 359.003,689.497 360.079,686.062 361.2,682.628 362.368,679.193 363.51,675.965 363.583,675.759 364.835,672.324 366.135,668.89 367.483,665.455 368.066,664.015 \n",
" 368.873,662.021 370.308,658.586 371.792,655.152 372.623,653.285 373.32,651.717 374.893,648.283 376.516,644.848 377.179,643.482 378.183,641.414 379.898,637.979 \n",
" 381.664,634.545 381.735,634.409 383.472,631.11 385.332,627.676 386.292,625.947 387.24,624.241 389.196,620.807 390.848,617.98 391.204,617.372 393.258,613.938 \n",
" 395.368,610.503 395.404,610.444 397.522,607.069 399.733,603.634 399.961,603.286 401.991,600.2 404.305,596.765 404.517,596.456 406.668,593.331 409.073,589.917 \n",
" 409.088,589.896 411.558,586.462 413.63,583.643 414.085,583.027 416.665,579.593 418.186,577.607 419.302,576.158 421.995,572.724 422.743,571.785 424.744,569.289 \n",
" 427.299,566.162 427.552,565.855 430.416,562.42 431.855,560.724 433.34,558.986 436.325,555.551 436.412,555.452 439.368,552.117 440.968,550.341 442.474,548.682 \n",
" 445.524,545.374 445.642,545.247 448.872,541.813 450.081,540.546 452.166,538.378 454.637,535.847 455.526,534.944 458.952,531.509 459.194,531.27 462.444,528.075 \n",
" 463.75,526.808 466.004,524.64 468.306,522.455 469.634,521.206 472.863,518.205 473.334,517.771 477.105,514.337 477.419,514.054 480.949,510.902 481.975,509.996 \n",
" 484.868,507.468 486.532,506.028 488.862,504.033 491.088,502.146 492.933,500.599 495.644,498.346 497.082,497.164 500.201,494.624 501.311,493.73 504.757,490.979 \n",
" 505.623,490.295 509.314,487.406 510.017,486.861 513.87,483.903 514.497,483.426 518.426,480.467 519.065,479.992 522.983,477.097 523.722,476.557 527.539,473.79 \n",
" 528.47,473.123 532.095,470.543 533.312,469.688 536.652,467.356 538.25,466.254 541.208,464.226 543.287,462.819 545.764,461.152 548.424,459.385 550.321,458.132 \n",
" 553.665,455.95 554.877,455.164 559.013,452.516 559.434,452.248 563.99,449.38 564.471,449.081 568.546,446.561 570.044,445.647 573.103,443.788 575.732,442.212 \n",
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" 687.012,387.128 691.568,385.286 695.243,383.826 696.125,383.475 700.681,381.69 704.052,380.391 705.237,379.934 709.794,378.203 713.132,376.957 714.35,376.501 \n",
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" 1165.43,309.019 1169.99,309.207 1174.54,309.413 1179.1,309.634 1183.66,309.873 1188.21,310.128 1192.77,310.4 1197.32,310.688 1201.88,310.994 1206.44,311.316 \n",
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" 1248.98,315.136 1252,315.461 1256.56,315.968 1261.11,316.493 1265.67,317.036 1270.23,317.597 1274.78,318.175 1277.8,318.57 1279.34,318.771 1283.9,319.382 \n",
" 1288.45,320.012 1293.01,320.66 1297.56,321.327 1302.07,322.005 1302.12,322.012 1306.68,322.712 1311.23,323.431 1315.79,324.169 1320.35,324.926 1323.36,325.439 \n",
" 1324.9,325.701 1329.46,326.492 1334.02,327.303 1338.57,328.134 1342.54,328.874 1343.13,328.983 1347.68,329.849 1352.24,330.735 1356.8,331.641 1360.08,332.308 \n",
" 1361.35,332.566 1365.91,333.509 1370.47,334.472 1375.02,335.456 1376.33,335.743 1379.58,336.457 1384.14,337.479 1388.69,338.521 1391.5,339.177 1393.25,339.584 \n",
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" 1466.15,359.44 1467.25,359.784 1470.71,360.869 1475.26,362.322 1478.03,363.219 1479.82,363.799 1484.38,365.299 1488.42,366.653 1488.93,366.826 1493.49,368.374 \n",
" 1498.04,369.95 1498.44,370.088 1502.6,371.548 1507.16,373.174 1508.12,373.522 1511.71,374.824 1516.27,376.501 1517.49,376.957 1520.83,378.203 1525.38,379.934 \n",
" 1526.57,380.391 1529.94,381.69 1534.5,383.475 1535.38,383.826 1539.05,385.286 1543.61,387.128 1543.93,387.26 1548.17,388.995 1552.24,390.695 1552.72,390.895 \n",
" 1557.28,392.821 1560.32,394.129 1561.83,394.778 1566.39,396.766 1568.19,397.564 1570.95,398.784 1575.5,400.835 1575.86,400.998 1580.06,402.915 1583.33,404.433 \n",
" 1584.62,405.029 1589.17,407.176 1590.62,407.867 1593.73,409.356 1597.73,411.302 1598.29,411.571 1602.84,413.82 1604.67,414.736 1607.4,416.104 1611.45,418.171 \n",
" 1611.95,418.426 1616.51,420.782 1618.08,421.605 1621.07,423.177 1624.56,425.04 1625.62,425.611 1630.18,428.084 1630.89,428.474 1634.74,430.595 1637.08,431.909 \n",
" 1639.29,433.148 1643.15,435.343 1643.85,435.744 1648.41,438.382 1649.08,438.778 1652.96,441.062 1654.89,442.212 1657.52,443.788 1660.58,445.647 1662.07,446.561 \n",
" 1666.15,449.081 1666.63,449.38 1671.19,452.248 1671.61,452.516 1675.74,455.164 1676.96,455.95 1680.3,458.132 1682.2,459.385 1684.86,461.152 1687.33,462.819 \n",
" 1689.41,464.226 1692.37,466.254 1693.97,467.356 1697.31,469.688 1698.53,470.543 1702.15,473.123 1703.08,473.79 1706.9,476.557 1707.64,477.097 1711.56,479.992 \n",
" 1712.19,480.467 1716.12,483.426 \n",
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" 1766.37,919.609 1764,923.043 1762.31,925.437 1761.58,926.478 1759.08,929.912 1757.76,931.694 1756.52,933.347 1753.89,936.781 1753.2,937.668 1751.2,940.216 \n",
" 1748.65,943.391 1748.43,943.65 1745.6,947.085 1744.09,948.883 1742.7,950.519 1739.72,953.954 1739.53,954.172 1736.68,957.388 1734.98,959.27 1733.56,960.823 \n",
" 1730.42,964.195 1730.36,964.257 1727.09,967.692 1725.86,968.96 1723.74,971.126 1721.31,973.576 1720.32,974.561 1716.81,977.995 1716.75,978.054 1713.22,981.43 \n",
" 1712.19,982.402 1709.55,984.864 1707.64,986.629 1705.8,988.299 1703.08,990.741 1701.96,991.733 1698.53,994.745 1698.04,995.168 1694.02,998.602 1693.97,998.647 \n",
" 1689.91,1002.04 1689.41,1002.45 1685.72,1005.47 1684.86,1006.17 1681.42,1008.91 1680.3,1009.79 1677.03,1012.34 1675.74,1013.33 1672.54,1015.77 1671.19,1016.8 \n",
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" 1643.85,1036.06 1643.36,1036.38 1639.29,1039.04 1638.09,1039.82 1634.74,1041.97 1632.71,1043.25 1630.18,1044.84 1627.19,1046.69 1625.62,1047.65 1621.55,1050.12 \n",
" 1621.07,1050.41 1616.51,1053.12 1615.77,1053.55 1611.95,1055.78 1609.84,1056.99 1607.4,1058.39 1603.78,1060.42 1602.84,1060.95 1598.29,1063.46 1597.56,1063.86 \n",
" 1593.73,1065.93 1591.18,1067.29 1589.17,1068.36 1584.64,1070.73 1584.62,1070.74 1580.06,1073.08 1577.93,1074.16 1575.5,1075.38 1571.04,1077.6 1570.95,1077.64 \n",
" 1566.39,1079.87 1563.96,1081.03 1561.83,1082.05 1557.28,1084.19 1556.69,1084.46 1552.72,1086.31 1549.22,1087.9 1548.17,1088.38 1543.61,1090.42 1541.52,1091.33 \n",
" 1539.05,1092.42 1534.5,1094.39 1533.61,1094.77 1529.94,1096.33 1525.45,1098.2 1525.38,1098.23 1520.83,1100.11 1517.03,1101.64 1516.27,1101.94 1511.71,1103.76 \n",
" 1508.34,1105.07 1507.16,1105.53 1502.6,1107.28 1499.36,1108.51 1498.04,1109 1493.49,1110.7 1490.07,1111.94 1488.93,1112.36 1484.38,1113.99 1480.45,1115.38 \n",
" 1479.82,1115.6 1475.26,1117.18 1470.71,1118.73 1470.46,1118.81 1466.15,1120.26 1461.59,1121.75 1460.08,1122.24 1457.04,1123.23 1452.48,1124.68 1449.26,1125.68 \n",
" 1447.92,1126.1 1443.37,1127.5 1438.81,1128.87 1437.98,1129.11 1434.26,1130.22 1429.7,1131.54 1426.17,1132.55 1425.14,1132.84 1420.59,1134.12 1416.03,1135.37 \n",
" 1413.76,1135.98 1411.47,1136.6 1406.92,1137.81 1402.36,1138.99 1400.7,1139.42 1397.8,1140.16 1393.25,1141.3 1388.69,1142.41 1386.87,1142.85 1384.14,1143.51 \n",
" 1379.58,1144.59 1375.02,1145.64 1372.17,1146.29 1370.47,1146.67 1365.91,1147.68 1361.35,1148.67 1356.8,1149.64 1356.42,1149.72 1352.24,1150.59 1347.68,1151.52 \n",
" 1343.13,1152.43 1339.4,1153.16 1338.57,1153.32 1334.02,1154.19 1329.46,1155.03 1324.9,1155.86 1320.8,1156.59 1320.35,1156.67 1315.79,1157.46 1311.23,1158.23 \n",
" 1306.68,1158.98 1302.12,1159.71 1300.14,1160.02 1297.56,1160.43 1293.01,1161.12 1288.45,1161.8 1283.9,1162.46 1279.34,1163.09 1276.64,1163.46 1274.78,1163.71 \n",
" 1270.23,1164.31 1265.67,1164.9 1261.11,1165.46 1256.56,1166.01 1252,1166.54 1248.81,1166.89 1247.44,1167.05 1242.89,1167.54 1238.33,1168.02 1233.78,1168.47 \n",
" 1229.22,1168.91 1224.66,1169.33 1220.11,1169.74 1215.55,1170.12 1213,1170.33 1210.99,1170.49 1206.44,1170.84 1201.88,1171.18 1197.32,1171.49 1192.77,1171.79 \n",
" 1188.21,1172.07 1183.66,1172.34 1179.1,1172.59 1174.54,1172.82 1169.99,1173.03 1165.43,1173.22 1160.87,1173.4 1156.32,1173.56 1151.76,1173.71 1149.83,1173.76 \n",
" 1147.2,1173.84 1142.65,1173.95 1138.09,1174.04 1133.54,1174.12 1128.98,1174.18 1124.42,1174.22 1119.87,1174.24 1115.31,1174.25 1110.75,1174.24 1106.2,1174.22 \n",
" 1101.64,1174.18 1097.08,1174.12 1092.53,1174.04 1087.97,1173.95 1083.42,1173.84 1080.79,1173.76 1078.86,1173.71 1074.3,1173.56 1069.75,1173.4 1065.19,1173.22 \n",
" 1060.63,1173.03 1056.08,1172.82 1051.52,1172.59 1046.96,1172.34 1042.41,1172.07 1037.85,1171.79 1033.3,1171.49 1028.74,1171.18 1024.18,1170.84 1019.63,1170.49 \n",
" 1017.62,1170.33 1015.07,1170.12 1010.51,1169.74 1005.96,1169.33 1001.4,1168.91 996.845,1168.47 992.288,1168.02 987.732,1167.54 983.176,1167.05 981.811,1166.89 \n",
" 978.619,1166.54 974.063,1166.01 969.507,1165.46 964.95,1164.9 960.394,1164.31 955.837,1163.71 953.98,1163.46 951.281,1163.09 946.725,1162.46 942.168,1161.8 \n",
" 937.612,1161.12 933.056,1160.43 930.482,1160.02 928.499,1159.71 923.943,1158.98 919.386,1158.23 914.83,1157.46 910.274,1156.67 909.82,1156.59 905.717,1155.86 \n",
" 901.161,1155.03 896.605,1154.19 892.048,1153.32 891.224,1153.16 887.492,1152.43 882.936,1151.52 878.379,1150.59 874.202,1149.72 873.823,1149.64 869.266,1148.67 \n",
" 864.71,1147.68 860.154,1146.67 858.455,1146.29 855.597,1145.64 851.041,1144.59 846.485,1143.51 843.746,1142.85 841.928,1142.41 837.372,1141.3 832.816,1140.16 \n",
" 829.92,1139.42 828.259,1138.99 823.703,1137.81 819.146,1136.6 816.856,1135.98 814.59,1135.37 810.034,1134.12 805.477,1132.84 804.453,1132.55 800.921,1131.54 \n",
" 796.365,1130.22 792.64,1129.11 791.808,1128.87 787.252,1127.5 782.696,1126.1 781.356,1125.68 778.139,1124.68 773.583,1123.23 770.544,1122.24 769.026,1121.75 \n",
" 764.47,1120.26 760.157,1118.81 759.914,1118.73 755.357,1117.18 750.801,1115.6 750.171,1115.38 746.245,1113.99 741.688,1112.36 740.548,1111.94 737.132,1110.7 \n",
" 732.575,1109 731.258,1108.51 728.019,1107.28 723.463,1105.53 722.279,1105.07 718.906,1103.76 714.35,1101.94 713.59,1101.64 709.794,1100.11 705.237,1098.23 \n",
" 705.171,1098.2 700.681,1096.33 697.014,1094.77 696.125,1094.39 691.568,1092.42 689.096,1091.33 687.012,1090.42 682.455,1088.38 681.403,1087.9 677.899,1086.31 \n",
" 673.926,1084.46 673.343,1084.19 668.786,1082.05 666.658,1081.03 664.23,1079.87 659.674,1077.64 659.579,1077.6 655.117,1075.38 652.694,1074.16 650.561,1073.08 \n",
" 646.005,1070.74 645.979,1070.73 641.448,1068.36 639.443,1067.29 636.892,1065.93 633.064,1063.86 632.335,1063.46 627.779,1060.95 626.845,1060.42 623.223,1058.39 \n",
" 620.778,1056.99 618.666,1055.78 614.854,1053.55 614.11,1053.12 609.554,1050.41 609.072,1050.12 604.997,1047.65 603.428,1046.69 600.441,1044.84 597.914,1043.25 \n",
" 595.884,1041.97 592.526,1039.82 591.328,1039.04 587.261,1036.38 586.772,1036.06 582.215,1033.01 582.117,1032.95 577.659,1029.91 577.09,1029.51 573.103,1026.73 \n",
" 572.176,1026.08 568.546,1023.49 567.372,1022.64 563.99,1020.18 562.675,1019.21 559.434,1016.8 558.082,1015.77 554.877,1013.33 553.591,1012.34 550.321,1009.79 \n",
" 549.199,1008.91 545.764,1006.17 544.905,1005.47 541.208,1002.45 540.705,1002.04 536.652,998.647 536.599,998.602 532.584,995.168 532.095,994.745 528.658,991.733 \n",
" 527.539,990.741 524.819,988.299 522.983,986.629 521.066,984.864 518.426,982.402 517.396,981.43 513.87,978.054 513.809,977.995 510.304,974.561 509.314,973.576 \n",
" 506.877,971.126 504.757,968.96 503.529,967.692 500.259,964.257 500.201,964.195 497.063,960.823 495.644,959.27 493.942,957.388 491.088,954.172 490.896,953.954 \n",
" 487.921,950.519 486.532,948.883 485.017,947.085 482.186,943.65 481.975,943.391 479.421,940.216 477.419,937.668 476.727,936.781 474.099,933.347 472.863,931.694 \n",
" 471.539,929.912 469.045,926.478 468.306,925.437 466.616,923.043 464.253,919.609 463.75,918.862 461.951,916.174 459.715,912.74 459.194,911.919 457.539,909.305 \n",
" 455.427,905.871 454.637,904.551 453.375,902.436 451.385,899.002 450.081,896.682 449.456,895.567 447.583,892.133 445.775,888.698 445.524,888.209 444.02,885.264 \n",
" 442.325,881.829 440.968,878.976 440.692,878.395 439.109,874.96 437.588,871.526 436.412,868.763 436.125,868.091 434.713,864.657 433.36,861.222 432.066,857.788 \n",
" 431.855,857.206 430.821,854.353 429.632,850.919 428.5,847.484 427.426,844.05 427.299,843.625 426.399,840.615 425.426,837.181 424.51,833.746 423.649,830.312 \n",
" 422.842,826.877 422.743,826.423 422.083,823.442 421.378,820.008 420.726,816.573 420.127,813.139 419.582,809.704 419.09,806.27 418.651,802.835 418.263,799.401 \n",
" 418.186,798.61 417.926,795.966 417.639,792.532 417.405,789.097 417.223,785.663 417.093,782.228 417.015,778.794 416.989,775.359 417.015,771.925 417.093,768.49 \n",
" 417.223,765.056 417.405,761.621 417.639,758.187 417.926,754.752 418.186,752.109 418.263,751.318 418.651,747.883 419.09,744.449 419.582,741.014 420.127,737.58 \n",
" 420.726,734.145 421.378,730.711 422.083,727.276 422.743,724.296 422.842,723.842 423.649,720.407 424.51,716.973 425.426,713.538 426.399,710.104 427.299,707.094 \n",
" 427.426,706.669 428.5,703.235 429.632,699.8 430.821,696.366 431.855,693.513 432.066,692.931 433.36,689.497 434.713,686.062 436.125,682.628 436.412,681.955 \n",
" 437.588,679.193 439.109,675.759 440.692,672.324 440.968,671.743 442.325,668.89 444.02,665.455 445.524,662.51 445.775,662.021 447.583,658.586 449.456,655.152 \n",
" 450.081,654.037 451.385,651.717 453.375,648.283 454.637,646.168 455.427,644.848 457.539,641.414 459.194,638.8 459.715,637.979 461.951,634.545 463.75,631.857 \n",
" 464.253,631.11 466.616,627.676 468.306,625.281 469.045,624.241 471.539,620.807 472.863,619.025 474.099,617.372 476.727,613.938 477.419,613.05 479.421,610.503 \n",
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" 1001.4,1049.5 1005.96,1049.92 1008.24,1050.12 1010.51,1050.32 1015.07,1050.71 1019.63,1051.07 1024.18,1051.42 1028.74,1051.75 1033.3,1052.07 1037.85,1052.37 \n",
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" 1293.01,1041.71 1297.56,1041.01 1302.12,1040.3 1305.08,1039.82 1306.68,1039.56 1311.23,1038.8 1315.79,1038.03 1320.35,1037.23 1324.9,1036.41 1325.05,1036.38 \n",
" 1329.46,1035.57 1334.02,1034.71 1338.57,1033.83 1343.04,1032.95 1343.13,1032.93 1347.68,1032.01 1352.24,1031.07 1356.8,1030.1 1359.49,1029.51 1361.35,1029.11 \n",
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" 1434.26,1010.02 1437.85,1008.91 1438.81,1008.61 1443.37,1007.16 1447.92,1005.69 1448.58,1005.47 1452.48,1004.19 1457.04,1002.65 1458.82,1002.04 1461.59,1001.08 \n",
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" 1589.17,610.221 1584.62,607.302 1584.24,607.069 1580.06,604.459 1578.7,603.634 1575.5,601.69 1572.99,600.2 1570.95,598.992 1567.09,596.765 1566.39,596.363 \n",
" 1561.83,593.795 \n",
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],
"source": [
"v = z -> π*logkernel(u, z)\n",
"\n",
"xx = yy = -2:0.01:2\n",
"V = v.(xx' .+ im*yy)\n",
"\n",
"contour(xx, yy, V)\n",
"plot!(domain(x); color=:black)"
]
},
{
"cell_type": "code",
"execution_count": 53,
"metadata": {},
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"## Application: Potential arising from a point charge and a single plate\n",
"\n",
"Now imagine we put a point source at $x = 2$, and a metal plate on $[-1,1]$. We know the potential on the plate must be constant, but we don't know what constant. This is equivalent to the following problem (see e.g. [\\[Chapman, Hewett & Trefethen 2015\\]](https://people.maths.ox.ac.uk/trefethen/chapman_hewett_trefethen.pdf)):\n",
"\\begin{align*}\n",
"v_{xx} + v_{yy} = 0 &\\qqfor \\hbox{off $[-1,1]$ and $2$} \\\\\n",
"v(z) \\sim \\log |z - 2| + O(1) &\\qqfor z \\rightarrow 2 \\\\\n",
"v(z) \\sim \\log|z| + o(1) &\\qqfor z \\rightarrow \\infty \\\\\n",
"v(x) = \\kappa &\\qqfor -1 < x < 1\n",
"\\end{align*}\n",
"where $\\kappa$ is an unknown constant. We write the solution as\n",
"$$\n",
"v(z) = {1 \\over \\pi} \\int_{-1}^1 u(t) \\log|t-z| \\dt + \\log|z-2|\n",
"$$\n",
"for a to-be-determined $u$. On $-1 < x < 1$ this satisfies\n",
"$$\n",
" {1 \\over \\pi} \\int_{-1}^1 u(t) \\log|t-x| \\dt = \\kappa - \\log(2-x)\n",
"$$\n",
"\n",
"We can calculate the solution numerically as follows:"
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" 1101.52,129.179 1104.03,128.316 1104.54,128.142 1106.54,127.459 1109.04,126.608 1110.1,126.252 1111.55,125.763 1114.06,124.923 1115.74,124.362 1116.56,124.09 \n",
" 1119.07,123.262 1121.48,122.473 1121.58,122.44 1124.08,121.623 1126.59,120.812 1127.3,120.583 1129.1,120.006 1131.61,119.207 1133.23,118.693 1134.11,118.413 \n",
" 1136.62,117.624 1139.13,116.841 1139.25,116.803 1141.63,116.064 1144.14,115.292 1145.38,114.914 1146.65,114.526 1149.15,113.765 1151.61,113.024 1151.66,113.01 \n",
" 1154.17,112.26 1156.67,111.516 1157.97,111.134 1159.18,110.777 1161.69,110.044 1164.2,109.317 1164.45,109.245 1166.7,108.594 1169.21,107.877 1171.05,107.355 \n",
" 1171.72,107.166 1174.22,106.459 1176.73,105.759 1177.79,105.465 1179.24,105.063 1181.74,104.373 1184.25,103.689 1184.67,103.576 1186.76,103.009 1189.27,102.335 \n",
" 1191.7,101.686 1191.77,101.667 1194.28,101.003 1196.79,100.345 1198.9,99.7963 1199.29,99.6929 1201.8,99.0451 1204.31,98.4029 1206.26,97.9066 1206.81,97.7661 \n",
" 1209.32,97.134 1211.83,96.5075 1213.81,96.0169 1214.33,95.8864 1216.84,95.27 1219.35,94.6591 1221.55,94.1272 1221.86,94.0537 1224.36,93.4528 1226.87,92.8574 \n",
" 1229.38,92.2675 1229.5,92.2375 1231.88,91.682 1234.39,91.102 1236.9,90.5274 1237.69,90.3479 1239.4,89.9576 1241.91,89.3928 1244.42,88.8335 1246.12,88.4582 \n",
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" 1896.22,108.015 1898.73,108.758 1900.36,109.245 1901.24,109.508 1903.74,110.263 1906.25,111.025 1906.61,111.134 1908.76,111.793 1911.26,112.567 1912.73,113.024 \n",
" 1913.77,113.347 1916.28,114.133 1918.75,114.914 1918.78,114.926 1921.29,115.725 1923.8,116.53 1924.64,116.803 1926.31,117.341 1928.81,118.158 1930.44,118.693 \n",
" 1931.32,118.983 1933.83,119.813 1936.13,120.583 1936.33,120.65 1938.84,121.492 1941.35,122.342 1941.73,122.473 1943.85,123.198 1946.36,124.06 1947.23,124.362 \n",
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" 2016.24,150.818 2016.56,150.951 2019.06,152.014 2020.69,152.708 2021.57,153.085 2024.08,154.163 2025.08,154.597 2026.58,155.248 \n",
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" 1956.39,464.877 1955.87,464.506 1953.88,463.098 1953.2,462.616 1951.38,461.341 1950.49,460.727 1948.87,459.605 1947.75,458.837 1946.36,457.889 1944.97,456.947 \n",
" 1943.85,456.194 1942.16,455.057 1941.35,454.518 1939.3,453.168 1938.84,452.863 1936.41,451.278 1936.33,451.226 1933.83,449.609 1933.48,449.388 1931.32,448.011 \n",
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" 1184.25,439.509 1183.44,439.94 1181.74,440.837 1179.88,441.83 1179.24,442.176 1176.73,443.526 1176.37,443.719 1174.22,444.886 1172.9,445.609 1171.72,446.256 \n",
" 1169.46,447.499 1169.21,447.638 1166.7,449.03 1166.06,449.388 1164.2,450.432 1162.69,451.278 1161.69,451.846 1159.36,453.168 1159.18,453.27 1156.67,454.704 \n",
" 1156.06,455.057 1154.17,456.149 1152.79,456.947 1151.66,457.605 1149.55,458.837 1149.15,459.072 1146.65,460.549 1146.35,460.727 1144.14,462.036 1143.17,462.616 \n",
" 1141.63,463.534 1140.02,464.506 1139.13,465.043 1136.89,466.396 1136.62,466.562 1134.11,468.092 1133.8,468.285 1131.61,469.633 1130.73,470.175 1129.1,471.183 \n",
" 1127.68,472.065 1126.59,472.745 1124.66,473.954 1124.08,474.316 1121.66,475.844 1121.58,475.898 1119.07,477.49 1118.69,477.734 1116.56,479.093 1115.74,479.623 \n",
" 1114.06,480.706 1112.81,481.513 1111.55,482.328 1109.9,483.403 1109.04,483.961 1107.01,485.292 1106.54,485.604 1104.14,487.182 1104.03,487.257 1101.52,488.92 \n",
" 1101.29,489.072 1099.02,490.593 1098.46,490.962 1096.51,492.275 1095.65,492.851 1094,493.967 1092.86,494.741 1091.49,495.669 1090.08,496.631 1088.99,497.38 \n",
" 1087.32,498.52 1086.48,499.1 1084.58,500.41 1083.97,500.83 1081.85,502.3 1081.47,502.569 1079.14,504.189 1078.96,504.317 1076.45,506.074 1076.45,506.079 \n",
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" 1023.81,544.808 1022.56,545.762 1021.3,546.727 1020.09,547.652 1018.79,548.653 1017.64,549.542 1016.29,550.583 1015.19,551.431 1013.78,552.519 1012.74,553.321 \n",
" 1011.27,554.46 1010.3,555.211 1008.77,556.406 1007.87,557.101 1006.26,558.358 1005.45,558.99 1003.75,560.313 1003.03,560.88 1001.24,562.274 1000.61,562.77 \n",
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" 974.378,583.556 973.668,584.124 972.021,585.446 971.161,586.135 969.668,587.336 968.654,588.15 967.32,589.225 966.147,590.169 964.976,591.115 963.64,592.192 \n",
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" 951.009,602.453 948.699,604.343 948.599,604.425 946.395,606.232 946.092,606.481 944.096,608.122 943.585,608.542 941.803,610.012 941.078,610.609 939.515,611.901 \n",
" 938.571,612.682 937.234,613.791 936.064,614.761 934.96,615.681 933.557,616.847 932.692,617.571 931.05,618.941 930.431,619.46 928.543,621.042 928.178,621.35 \n",
" 926.036,623.151 925.932,623.24 923.695,625.129 923.529,625.269 921.467,627.019 921.022,627.395 919.247,628.909 918.515,629.531 917.037,630.798 916.008,631.678 \n",
" 914.836,632.688 913.501,633.836 912.645,634.578 910.994,636.006 910.465,636.467 908.488,638.188 908.295,638.357 906.138,640.247 905.981,640.384 903.993,642.136 \n",
" 903.474,642.594 901.861,644.026 900.967,644.819 899.742,645.916 898.46,647.062 897.636,647.806 895.953,649.322 895.543,649.695 893.465,651.585 893.446,651.603 \n",
" 891.405,653.475 890.939,653.901 889.359,655.364 888.432,656.223 887.33,657.254 885.925,658.569 885.317,659.144 883.418,660.941 883.322,661.033 881.346,662.923 \n",
" 880.911,663.339 879.389,664.813 878.404,665.767 877.451,666.702 875.897,668.229 875.532,668.592 873.634,670.482 873.39,670.725 871.759,672.371 870.883,673.258 \n",
" 869.905,674.261 868.376,675.833 868.072,676.151 866.263,678.041 865.87,678.453 864.479,679.93 863.363,681.122 862.718,681.82 860.981,683.71 860.856,683.846 \n",
" 859.272,685.599 858.349,686.628 857.587,687.489 855.929,689.379 855.842,689.478 854.299,691.268 853.335,692.399 852.696,693.158 851.121,695.048 850.828,695.402 \n",
" 849.576,696.937 848.321,698.498 848.06,698.827 846.574,700.717 845.814,701.697 845.118,702.606 843.693,704.496 843.307,705.015 842.3,706.386 840.939,708.276 \n",
" 840.8,708.47 839.61,710.165 838.314,712.055 838.293,712.086 837.052,713.945 835.823,715.834 835.786,715.892 834.628,717.724 833.469,719.614 833.279,719.929 \n",
" 832.344,721.503 831.255,723.393 830.772,724.253 830.201,725.283 829.183,727.172 828.265,728.942 828.204,729.062 827.258,730.952 826.352,732.841 825.758,734.129 \n",
" 825.483,734.731 824.65,736.621 823.857,738.511 823.252,740.027 823.103,740.4 822.385,742.29 821.707,744.18 821.069,746.069 820.745,747.093 820.47,747.959 \n",
" 819.908,749.849 819.387,751.738 818.907,753.628 818.469,755.518 818.238,756.611 818.068,757.407 817.706,759.297 817.385,761.187 817.105,763.076 816.866,764.966 \n",
" 816.668,766.856 816.511,768.746 816.394,770.635 816.319,772.525 816.284,774.415 816.289,776.304 816.335,778.194 816.422,780.084 816.55,781.973 816.718,783.863 \n",
" 816.927,785.753 817.177,787.642 817.469,789.532 817.801,791.422 818.174,793.311 818.238,793.602 818.584,795.201 819.034,797.091 819.525,798.98 820.057,800.87 \n",
" 820.63,802.76 820.745,803.114 821.239,804.65 821.887,806.539 822.577,808.429 823.252,810.179 823.306,810.319 824.069,812.208 824.873,814.098 825.717,815.988 \n",
" 825.758,816.077 826.595,817.877 827.512,819.767 828.265,821.26 828.467,821.657 829.457,823.546 830.485,825.436 830.772,825.951 831.548,827.326 832.647,829.215 \n",
" 833.279,830.274 833.782,831.105 834.951,832.995 835.786,834.311 836.155,834.885 837.393,836.774 838.293,838.117 838.664,838.664 839.969,840.554 840.8,841.733 \n",
" 841.307,842.443 842.677,844.333 843.307,845.189 844.079,846.223 845.512,848.112 845.814,848.507 846.976,850.002 848.321,851.706 848.47,851.892 849.995,853.781 \n",
" 850.828,854.802 851.548,855.671 853.13,857.561 853.335,857.804 854.741,859.45 855.842,860.727 856.379,861.34 858.043,863.23 858.349,863.575 859.736,865.12 \n",
" 860.856,866.358 861.453,867.009 863.195,868.899 863.363,869.08 864.963,870.789 865.87,871.751 866.755,872.678 868.376,874.37 868.569,874.568 870.408,876.458 \n",
" 870.883,876.945 872.269,878.347 873.39,879.479 874.151,880.237 875.897,881.974 876.053,882.127 877.977,884.016 878.404,884.436 879.921,885.906 880.911,886.865 \n",
" 881.884,887.796 883.418,889.263 883.865,889.685 885.863,891.575 885.925,891.634 887.881,893.465 888.432,893.98 889.916,895.355 890.939,896.303 891.966,897.244 \n",
" 893.446,898.602 894.032,899.134 895.953,900.881 896.112,901.024 898.208,902.913 898.46,903.141 900.318,904.803 900.967,905.384 902.442,906.693 903.474,907.61 \n",
" 904.578,908.582 905.981,909.82 906.726,910.472 908.488,912.015 908.886,912.362 910.994,914.197 911.057,914.251 913.241,916.141 913.501,916.367 915.435,918.031 \n",
" 916.008,918.525 917.639,919.92 918.515,920.672 919.852,921.81 921.022,922.809 922.074,923.7 923.529,924.935 924.305,925.59 926.036,927.053 926.544,927.479 \n",
" 928.543,929.161 928.791,929.369 931.046,931.259 931.05,931.262 933.309,933.148 933.557,933.356 935.579,935.038 936.064,935.442 937.856,936.928 938.571,937.522 \n",
" 940.139,938.817 941.078,939.595 942.428,940.707 943.585,941.662 944.723,942.597 946.092,943.723 947.023,944.486 948.599,945.779 949.329,946.376 951.106,947.83 \n",
" 951.641,948.266 953.613,949.875 953.957,950.155 956.119,951.916 956.279,952.045 958.606,953.935 958.626,953.952 960.937,955.825 961.133,955.984 963.273,957.714 \n",
" 963.64,958.011 965.614,959.604 966.147,960.034 967.96,961.494 968.654,962.053 970.309,963.383 971.161,964.068 972.663,965.273 973.668,966.079 975.022,967.163 \n",
" 976.175,968.086 977.385,969.052 978.682,970.089 979.752,970.942 981.189,972.088 982.124,972.832 983.696,974.082 984.501,974.721 986.203,976.073 986.882,976.611 \n",
" 988.71,978.06 989.268,978.501 991.217,980.042 991.658,980.39 993.724,982.02 994.053,982.28 996.231,983.994 996.454,984.17 998.737,985.964 998.859,986.06 \n",
" 1001.24,987.929 1001.27,987.949 1003.69,989.839 1003.75,989.89 1006.11,991.729 1006.26,991.846 1008.54,993.618 1008.77,993.797 1010.97,995.508 1011.27,995.743 \n",
" \n",
" \"/>\n",
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" 1011.27,995.743 1013.41,997.398 1013.78,997.684 1015.85,999.287 1016.29,999.62 1018.31,1001.18 1018.79,1001.55 1020.77,1003.07 1021.3,1003.48 1023.23,1004.96 \n",
" 1023.81,1005.4 1025.71,1006.85 1026.31,1007.31 1028.19,1008.74 1028.82,1009.22 1030.68,1010.63 1031.33,1011.12 1033.17,1012.52 1033.83,1013.01 1035.68,1014.4 \n",
" 1036.34,1014.9 1038.19,1016.29 1038.85,1016.79 1040.72,1018.18 1041.36,1018.66 1043.25,1020.07 1043.86,1020.53 1045.79,1021.96 1046.37,1022.39 1048.34,1023.85 \n",
" 1048.88,1024.25 1050.91,1025.74 1051.38,1026.09 1053.48,1027.63 1053.89,1027.93 1056.06,1029.52 1056.4,1029.77 1058.66,1031.41 1058.9,1031.59 1061.27,1033.3 \n",
" 1061.41,1033.41 1063.89,1035.19 1063.92,1035.21 1066.42,1037.01 1066.52,1037.08 1068.93,1038.81 1069.16,1038.97 1071.44,1040.59 1071.82,1040.86 1073.95,1042.36 \n",
" 1074.49,1042.75 1076.45,1044.13 1077.18,1044.64 1078.96,1045.89 1079.88,1046.53 1081.47,1047.63 1082.6,1048.42 1083.97,1049.37 1085.33,1050.31 1086.48,1051.1 \n",
" 1088.07,1052.2 1088.99,1052.82 1090.84,1054.09 1091.49,1054.53 1093.62,1055.98 1094,1056.24 1096.42,1057.87 1096.51,1057.93 1099.02,1059.61 1099.23,1059.76 \n",
" 1101.52,1061.28 1102.07,1061.65 1104.03,1062.95 1104.92,1063.54 1106.54,1064.6 1107.8,1065.43 1109.04,1066.24 1110.69,1067.32 1111.55,1067.88 1113.6,1069.21 \n",
" 1114.06,1069.5 1116.54,1071.1 1116.56,1071.11 1119.07,1072.71 1119.5,1072.99 1121.58,1074.31 1122.48,1074.87 1124.08,1075.89 1125.48,1076.76 1126.59,1077.46 \n",
" 1128.51,1078.65 1129.1,1079.02 1131.56,1080.54 1131.61,1080.57 1134.11,1082.11 1134.64,1082.43 1136.62,1083.64 1137.74,1084.32 1139.13,1085.16 1140.87,1086.21 \n",
" 1141.63,1086.67 1144.03,1088.1 1144.14,1088.17 1146.65,1089.65 1147.22,1089.99 1149.15,1091.13 1150.43,1091.88 1151.66,1092.6 1153.68,1093.77 1154.17,1094.05 \n",
" 1156.67,1095.5 1156.96,1095.66 1159.18,1096.93 1160.27,1097.55 1161.69,1098.36 1163.61,1099.44 1164.2,1099.77 1166.7,1101.17 1166.98,1101.33 1169.21,1102.57 \n",
" 1170.39,1103.22 1171.72,1103.95 1173.84,1105.11 1174.22,1105.32 1176.73,1106.68 1177.33,1107 1179.24,1108.03 1180.85,1108.89 1181.74,1109.37 1184.25,1110.69 \n",
" 1184.41,1110.78 1186.76,1112.01 1188.01,1112.67 1189.27,1113.32 1191.66,1114.56 1191.77,1114.62 1194.28,1115.9 1195.35,1116.45 1196.79,1117.18 1199.08,1118.34 \n",
" 1199.29,1118.44 1201.8,1119.7 1202.87,1120.23 1204.31,1120.94 1206.7,1122.12 1206.81,1122.17 1209.32,1123.4 1210.58,1124.01 1211.83,1124.61 1214.33,1125.81 \n",
" 1214.51,1125.9 1216.84,1127.01 1218.5,1127.79 1219.35,1128.19 1221.86,1129.36 1222.54,1129.68 1224.36,1130.52 1226.64,1131.57 1226.87,1131.67 1229.38,1132.81 \n",
" 1230.8,1133.46 1231.88,1133.94 1234.39,1135.06 1235.02,1135.34 1236.9,1136.17 1239.31,1137.23 1239.4,1137.27 1241.91,1138.36 1243.67,1139.12 1244.42,1139.44 \n",
" 1246.92,1140.51 1248.1,1141.01 1249.43,1141.58 1251.94,1142.63 1252.61,1142.9 1254.45,1143.67 1256.95,1144.7 1257.19,1144.79 1259.46,1145.72 1261.86,1146.68 \n",
" 1261.97,1146.73 1264.47,1147.73 1266.61,1148.57 1266.98,1148.72 1269.49,1149.7 1271.45,1150.46 1271.99,1150.67 1274.5,1151.64 1276.39,1152.35 1277.01,1152.59 \n",
" 1279.51,1153.53 1281.42,1154.24 1282.02,1154.46 1284.53,1155.39 1286.57,1156.13 1287.04,1156.3 1289.54,1157.21 1291.82,1158.02 1292.05,1158.1 1294.56,1158.99 \n",
" 1297.06,1159.86 1297.2,1159.91 1299.57,1160.73 1302.08,1161.59 1302.7,1161.8 1304.58,1162.44 1307.09,1163.28 1308.34,1163.69 1309.6,1164.11 1312.11,1164.93 \n",
" 1314.12,1165.58 1314.61,1165.74 1317.12,1166.54 1319.63,1167.34 1320.05,1167.47 1322.13,1168.12 1324.64,1168.89 1326.16,1169.36 1327.15,1169.66 1329.65,1170.42 \n",
" 1332.16,1171.16 1332.44,1171.25 1334.67,1171.9 1337.17,1172.63 1338.93,1173.14 1339.68,1173.36 1342.19,1174.07 1344.7,1174.77 1345.63,1175.03 1347.2,1175.47 \n",
" 1349.71,1176.15 1352.22,1176.83 1352.56,1176.92 1354.72,1177.5 1357.23,1178.15 1359.74,1178.8 1359.75,1178.81 1362.24,1179.45 1364.75,1180.08 1367.23,1180.7 \n",
" 1367.26,1180.7 1369.76,1181.32 1372.27,1181.93 1374.78,1182.53 1375.04,1182.59 1377.29,1183.12 1379.79,1183.7 1382.3,1184.27 1383.22,1184.48 1384.81,1184.83 \n",
" 1387.31,1185.39 1389.82,1185.94 1391.82,1186.37 1392.33,1186.48 1394.83,1187.01 1397.34,1187.53 1399.85,1188.04 1400.91,1188.26 1402.36,1188.55 1404.86,1189.04 \n",
" 1407.37,1189.53 1409.88,1190.01 1410.58,1190.15 1412.38,1190.48 1414.89,1190.95 1417.4,1191.4 1419.9,1191.85 1420.96,1192.04 1422.41,1192.29 1424.92,1192.72 \n",
" 1427.42,1193.14 1429.93,1193.56 1432.22,1193.93 1432.44,1193.96 1434.95,1194.36 1437.45,1194.75 1439.96,1195.13 1442.47,1195.5 1444.61,1195.81 1444.97,1195.87 \n",
" 1447.48,1196.22 1449.99,1196.57 1452.49,1196.91 1455,1197.25 1457.51,1197.57 1458.57,1197.7 1460.01,1197.89 1462.52,1198.2 1465.03,1198.5 1467.54,1198.79 \n",
" 1470.04,1199.07 1472.55,1199.35 1474.86,1199.59 1475.06,1199.62 1477.56,1199.88 1480.07,1200.13 1482.58,1200.37 1485.08,1200.61 1487.59,1200.84 1490.1,1201.06 \n",
" 1492.6,1201.27 1495.11,1201.47 1495.27,1201.48 1497.62,1201.67 1500.13,1201.86 1502.63,1202.04 1505.14,1202.21 1507.65,1202.37 1510.15,1202.53 1512.66,1202.68 \n",
" 1515.17,1202.82 1517.67,1202.95 1520.18,1203.07 1522.69,1203.19 1525.2,1203.3 1527.05,1203.37 1527.7,1203.4 1530.21,1203.49 1532.72,1203.58 1535.22,1203.65 \n",
" 1537.73,1203.72 1540.24,1203.78 1542.74,1203.84 1545.25,1203.88 1547.76,1203.92 1550.26,1203.95 1552.77,1203.97 1555.28,1203.98 1557.79,1203.99 1560.29,1203.99 \n",
" 1562.8,1203.98 1565.31,1203.96 1567.81,1203.93 1570.32,1203.9 1572.83,1203.85 1575.33,1203.8 1577.84,1203.74 1580.35,1203.68 1582.85,1203.6 1585.36,1203.52 \n",
" 1587.87,1203.43 1589.3,1203.37 1590.38,1203.33 1592.88,1203.22 1595.39,1203.11 1597.9,1202.99 1600.4,1202.86 1602.91,1202.72 1605.42,1202.57 1607.92,1202.42 \n",
" 1610.43,1202.25 1612.94,1202.08 1615.45,1201.9 1617.95,1201.71 1620.46,1201.52 1620.85,1201.48 1622.97,1201.31 1625.47,1201.1 1627.98,1200.88 1630.49,1200.65 \n",
" 1632.99,1200.41 1635.5,1200.17 1638.01,1199.91 1640.51,1199.65 1641.05,1199.59 1643.02,1199.38 1645.53,1199.1 1648.04,1198.82 1650.54,1198.52 1653.05,1198.22 \n",
" 1655.56,1197.9 1657.12,1197.7 1658.06,1197.58 1660.57,1197.25 1663.08,1196.92 1665.58,1196.57 1668.09,1196.22 1670.6,1195.85 1670.85,1195.81 1673.1,1195.48 \n",
" 1675.61,1195.1 1678.12,1194.71 1680.63,1194.31 1683.02,1193.93 1683.13,1193.91 1685.64,1193.49 1688.15,1193.07 1690.65,1192.64 1693.16,1192.19 1694.04,1192.04 \n",
" 1695.67,1191.74 1698.17,1191.28 1700.68,1190.82 1703.19,1190.34 1704.18,1190.15 1705.69,1189.85 1708.2,1189.36 1710.71,1188.85 1713.22,1188.34 1713.62,1188.26 \n",
" 1715.72,1187.82 1718.23,1187.29 1720.74,1186.75 1722.47,1186.37 1723.24,1186.2 1725.75,1185.64 1728.26,1185.07 1730.76,1184.49 1730.83,1184.48 1733.27,1183.91 \n",
" 1735.78,1183.31 1738.29,1182.7 1738.76,1182.59 1740.79,1182.09 1743.3,1181.46 1745.81,1180.83 1746.32,1180.7 1748.31,1180.19 1750.82,1179.53 1753.33,1178.87 \n",
" 1753.55,1178.81 1755.83,1178.2 1758.34,1177.51 1760.49,1176.92 1760.85,1176.82 1763.35,1176.12 1765.86,1175.4 1767.16,1175.03 1768.37,1174.68 1770.88,1173.95 \n",
" 1773.38,1173.2 1773.6,1173.14 1775.89,1172.45 1778.4,1171.69 1779.82,1171.25 1780.9,1170.91 1783.41,1170.13 1785.84,1169.36 1785.92,1169.33 1788.42,1168.53 \n",
" 1790.93,1167.71 1791.67,1167.47 1793.44,1166.89 1795.94,1166.05 1797.33,1165.58 1798.45,1165.2 1800.96,1164.34 1802.83,1163.69 1803.47,1163.47 1805.97,1162.59 \n",
" 1808.19,1161.8 1808.48,1161.7 1810.99,1160.79 1813.41,1159.91 1813.49,1159.88 1816,1158.95 1818.49,1158.02 1818.51,1158.02 1821.01,1157.07 1823.46,1156.13 \n",
" 1823.52,1156.11 1826.03,1155.14 1828.3,1154.24 1828.54,1154.15 1831.04,1153.16 1833.04,1152.35 1833.55,1152.15 1836.06,1151.13 1837.67,1150.46 1838.56,1150.1 \n",
" 1841.07,1149.05 1842.21,1148.57 1843.58,1148 1846.08,1146.93 1846.65,1146.68 1848.59,1145.84 1851,1144.79 1851.1,1144.75 1853.6,1143.64 1855.26,1142.9 \n",
" 1856.11,1142.52 1858.62,1141.39 1859.44,1141.01 1861.13,1140.24 1863.55,1139.12 1863.63,1139.08 1866.14,1137.91 1867.57,1137.23 1868.65,1136.72 1871.15,1135.52 \n",
" 1871.52,1135.34 1873.66,1134.31 1875.41,1133.46 1876.17,1133.08 1878.67,1131.84 1879.22,1131.57 1881.18,1130.58 1882.97,1129.68 1883.69,1129.31 1886.19,1128.03 \n",
" 1886.66,1127.79 1888.7,1126.73 1890.29,1125.9 1891.21,1125.41 1893.72,1124.08 1893.86,1124.01 1896.22,1122.74 1897.37,1122.12 1898.73,1121.38 1900.82,1120.23 \n",
" 1901.24,1120 1903.74,1118.61 1904.23,1118.34 1906.25,1117.2 1907.58,1116.45 1908.76,1115.78 1910.88,1114.56 1911.26,1114.33 1913.77,1112.88 1914.13,1112.67 \n",
" 1916.28,1111.4 1917.33,1110.78 1918.78,1109.91 1920.49,1108.89 1921.29,1108.4 1923.6,1107 1923.8,1106.88 1926.31,1105.33 1926.67,1105.11 1928.81,1103.77 \n",
" 1929.69,1103.22 1931.32,1102.19 1932.68,1101.33 1933.83,1100.59 1935.62,1099.44 1936.33,1098.98 1938.52,1097.55 1938.84,1097.34 1941.35,1095.68 1941.38,1095.66 \n",
" 1943.85,1094.01 1944.21,1093.77 1946.36,1092.31 1947,1091.88 1948.87,1090.6 1949.75,1089.99 1951.38,1088.86 1952.46,1088.1 1953.88,1087.11 1955.14,1086.21 \n",
" 1956.39,1085.33 1957.79,1084.32 1958.9,1083.53 1960.4,1082.43 1961.4,1081.7 1962.98,1080.54 1963.91,1079.86 1965.53,1078.65 1966.42,1077.99 1968.05,1076.76 \n",
" 1968.92,1076.1 1970.53,1074.87 1971.43,1074.18 1972.98,1072.99 1973.94,1072.24 1975.41,1071.1 1976.44,1070.28 1977.8,1069.21 1978.95,1068.29 1980.17,1067.32 \n",
" 1981.46,1066.28 1982.51,1065.43 1983.97,1064.24 1984.82,1063.54 1986.47,1062.17 1987.1,1061.65 1988.98,1060.07 1989.35,1059.76 1991.49,1057.95 1991.58,1057.87 \n",
" 1993.78,1055.98 1993.99,1055.8 1995.96,1054.09 1996.5,1053.61 1998.11,1052.2 1999.01,1051.4 2000.23,1050.31 2001.51,1049.16 2002.34,1048.42 2004.02,1046.89 \n",
" 2004.41,1046.53 2006.46,1044.64 2006.53,1044.58 2008.49,1042.75 2009.03,1042.24 2010.5,1040.86 2011.54,1039.87 2012.48,1038.97 2014.05,1037.46 2014.44,1037.08 \n",
" 2016.38,1035.19 2016.56,1035.01 2018.29,1033.3 2019.06,1032.53 2020.18,1031.41 2021.57,1030.01 2022.05,1029.52 2023.9,1027.63 2024.08,1027.46 2025.73,1025.74 \n",
" 2026.58,1024.86 \n",
" \"/>\n",
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" 2026.58,1394.96 2024.08,1396.04 2023.89,1396.12 2021.57,1397.12 2019.48,1398.01 2019.06,1398.19 2016.56,1399.25 2015.02,1399.9 2014.05,1400.31 2011.54,1401.36 \n",
" 2010.5,1401.79 2009.03,1402.4 2006.53,1403.43 2005.92,1403.68 2004.02,1404.46 2001.51,1405.48 2001.28,1405.57 1999.01,1406.49 1996.58,1407.46 1996.5,1407.49 \n",
" 1993.99,1408.49 1991.82,1409.35 1991.49,1409.48 1988.98,1410.47 1987,1411.24 1986.47,1411.44 1983.97,1412.41 1982.1,1413.13 1981.46,1413.38 1978.95,1414.33 \n",
" 1977.14,1415.02 1976.44,1415.28 1973.94,1416.22 1972.1,1416.91 1971.43,1417.16 1968.92,1418.09 1966.99,1418.8 1966.42,1419.01 1963.91,1419.92 1961.8,1420.69 \n",
" 1961.4,1420.83 1958.9,1421.73 1956.53,1422.58 1956.39,1422.63 1953.88,1423.52 1951.38,1424.4 1951.18,1424.47 1948.87,1425.27 1946.36,1426.14 1945.74,1426.36 \n",
" 1943.85,1427.01 1941.35,1427.86 1940.21,1428.25 1938.84,1428.71 1936.33,1429.55 1934.59,1430.14 1933.83,1430.39 1931.32,1431.22 1928.87,1432.03 1928.81,1432.04 \n",
" 1926.31,1432.86 1923.8,1433.67 1923.05,1433.92 1921.29,1434.48 1918.78,1435.28 1917.12,1435.81 1916.28,1436.07 1913.77,1436.86 1911.26,1437.64 1911.07,1437.69 \n",
" 1908.76,1438.41 1906.25,1439.18 1904.91,1439.58 1903.74,1439.94 1901.24,1440.7 1898.73,1441.44 1898.63,1441.47 1896.22,1442.19 1893.72,1442.93 1892.22,1443.36 \n",
" 1891.21,1443.66 1888.7,1444.38 1886.19,1445.1 1885.66,1445.25 1883.69,1445.82 1881.18,1446.52 1878.97,1447.14 1878.67,1447.23 1876.17,1447.92 1873.66,1448.61 \n",
" 1872.11,1449.03 1871.15,1449.3 1868.65,1449.97 1866.14,1450.65 1865.1,1450.92 1863.63,1451.31 1861.13,1451.97 1858.62,1452.63 1857.91,1452.81 1856.11,1453.28 \n",
" 1853.6,1453.92 1851.1,1454.56 1850.53,1454.7 1848.59,1455.19 1846.08,1455.82 1843.58,1456.44 1842.96,1456.59 1841.07,1457.05 1838.56,1457.66 1836.06,1458.27 \n",
" 1835.16,1458.48 1833.55,1458.87 1831.04,1459.46 1828.54,1460.05 1827.14,1460.37 1826.03,1460.63 1823.52,1461.2 1821.01,1461.78 1818.86,1462.26 \n",
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" 1607.92,1027.7 1605.42,1027.88 1602.91,1028.05 1600.4,1028.2 1597.9,1028.34 1595.39,1028.47 1592.88,1028.58 1590.38,1028.67 1587.87,1028.75 1585.36,1028.82 \n",
" 1582.85,1028.87 1580.35,1028.91 1577.84,1028.93 1575.33,1028.94 1572.83,1028.94 1570.32,1028.92 1567.81,1028.89 1565.31,1028.84 1562.8,1028.78 1560.29,1028.7 \n",
" 1557.79,1028.61 1555.28,1028.5 1552.77,1028.38 1550.26,1028.25 1547.76,1028.1 1545.25,1027.94 1542.74,1027.76 1541.13,1027.63 1540.24,1027.56 1537.73,1027.36 \n",
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" 1512.66,1024.46 1510.15,1024.09 1508.62,1023.85 1507.65,1023.7 1505.14,1023.3 1502.63,1022.88 1500.13,1022.45 1497.62,1021.99 1497.45,1021.96 1495.11,1021.53 \n",
" 1492.6,1021.05 1490.1,1020.55 1487.8,1020.07 1487.59,1020.03 1485.08,1019.5 1482.58,1018.95 1480.07,1018.39 1479.19,1018.18 1477.56,1017.81 1475.06,1017.21 \n",
" 1472.55,1016.59 1471.36,1016.29 1470.04,1015.96 1467.54,1015.31 1465.03,1014.64 1464.15,1014.4 1462.52,1013.96 1460.01,1013.26 1457.51,1012.53 1457.44,1012.52 \n",
" 1455,1011.8 1452.49,1011.04 1451.16,1010.63 1449.99,1010.26 1447.48,1009.47 1445.23,1008.74 1444.97,1008.65 1442.47,1007.82 1439.96,1006.96 1439.62,1006.85 \n",
" 1437.45,1006.09 1434.95,1005.2 1434.29,1004.96 1432.44,1004.28 1429.93,1003.35 1429.2,1003.07 1427.42,1002.39 1424.92,1001.41 1424.33,1001.18 1422.41,1000.41 \n",
" 1419.9,999.39 1419.66,999.287 1417.4,998.347 1415.17,997.398 1414.89,997.279 1412.38,996.189 1410.85,995.508 1409.88,995.074 1407.37,993.936 1406.68,993.618 \n",
" 1404.86,992.773 1402.66,991.729 1402.36,991.584 1399.85,990.37 1398.77,989.839 1397.34,989.13 1395.01,987.949 1394.83,987.862 1392.33,986.567 1391.36,986.06 \n",
" 1389.82,985.244 1387.83,984.17 1387.31,983.892 1384.81,982.51 1384.4,982.28 1382.3,981.098 1381.06,980.39 1379.79,979.654 1377.83,978.501 1377.29,978.179 \n",
" 1374.78,976.67 1374.68,976.611 1372.27,975.127 1371.62,974.721 1369.76,973.549 1368.64,972.832 1367.26,971.934 1365.75,970.942 1364.75,970.282 1362.92,969.052 \n",
" 1362.24,968.591 1360.17,967.163 1359.74,966.859 1357.49,965.273 1357.23,965.085 1354.88,963.383 1354.72,963.267 1352.34,961.494 1352.22,961.404 1349.85,959.604 \n",
" 1349.71,959.493 1347.43,957.714 1347.2,957.531 1345.07,955.825 1344.7,955.517 1342.77,953.935 1342.19,953.448 1340.52,952.045 1339.68,951.321 1338.33,950.155 \n",
" 1337.17,949.132 1336.2,948.266 1334.67,946.878 1334.12,946.376 1332.16,944.555 1332.09,944.486 1330.11,942.597 1329.65,942.156 1328.18,940.707 1327.15,939.677 \n",
" 1326.29,938.817 1324.64,937.113 1324.46,936.928 1322.67,935.038 1322.13,934.453 1320.93,933.148 1319.63,931.692 1319.24,931.259 1317.59,929.369 1317.12,928.817 \n",
" 1315.98,927.479 1314.61,925.819 1314.42,925.59 1312.9,923.7 1312.11,922.677 1311.43,921.81 1310,919.92 1309.6,919.378 1308.61,918.031 1307.26,916.141 \n",
" 1307.09,915.897 1305.95,914.251 1304.69,912.362 1304.58,912.2 1303.46,910.472 1302.28,908.582 1302.08,908.243 1301.14,906.693 1300.04,904.803 1299.57,903.968 \n",
" 1298.98,902.913 1297.96,901.024 1297.06,899.293 1296.98,899.134 1296.04,897.244 1295.14,895.355 1294.56,894.057 1294.29,893.465 1293.47,891.575 1292.7,889.685 \n",
" 1292.05,888.009 1291.97,887.796 1291.27,885.906 1290.62,884.016 1290.02,882.127 1289.54,880.52 1289.46,880.237 1288.93,878.347 1288.46,876.458 1288.02,874.568 \n",
" 1287.64,872.678 1287.3,870.789 1287.04,869.094 1287,868.899 1286.75,867.009 1286.55,865.12 1286.4,863.23 1286.29,861.34 1286.24,859.45 1286.24,857.561 \n",
" 1286.29,855.671 1286.39,853.781 1286.55,851.892 1286.77,850.002 1287.04,848.171 1287.04,848.112 1287.37,846.223 1287.76,844.333 1288.21,842.443 1288.73,840.554 \n",
" 1289.32,838.664 1289.54,838.028 1289.97,836.774 1290.69,834.885 1291.48,832.995 1292.05,831.775 1292.35,831.105 1293.29,829.215 1294.32,827.326 1294.56,826.93 \n",
" 1295.42,825.436 1296.61,823.546 1297.06,822.89 1297.89,821.657 1299.26,819.767 1299.57,819.369 1300.71,817.877 1302.08,816.231 1302.27,815.988 1303.92,814.098 \n",
" 1304.58,813.395 1305.67,812.208 1307.09,810.778 1307.53,810.319 1309.5,808.429 1309.6,808.344 1311.56,806.539 1312.11,806.079 1313.73,804.65 1314.61,803.938 \n",
" 1316.01,802.76 1317.12,801.902 1318.4,800.87 1319.63,799.956 1320.88,798.98 1322.13,798.084 1323.45,797.091 1324.64,796.272 1326.11,795.201 1327.15,794.506 \n",
" 1328.82,793.311 1329.65,792.771 1331.58,791.422 1332.16,791.049 1334.33,789.532 1334.67,789.322 1337.05,787.642 1337.17,787.563 1339.65,785.753 1339.68,785.735 \n",
" 1342.06,783.863 1342.19,783.773 1344.18,781.973 1344.7,781.522 1345.95,780.084 1347.2,778.272 1347.25,778.194 1348.01,776.304 1348.11,774.415 1347.51,772.525 \n",
" 1347.2,771.95 1346.35,770.635 1344.73,768.746 1344.7,768.707 1342.66,766.856 1342.19,766.414 1340.31,764.966 1339.68,764.447 1337.75,763.076 1337.17,762.626 \n",
" 1335.08,761.187 1334.67,760.879 1332.34,759.297 1332.16,759.162 1329.65,757.447 1329.59,757.407 1327.15,755.707 1326.86,755.518 1324.64,753.938 1324.18,753.628 \n",
" 1322.13,752.125 1321.58,751.738 1319.63,750.252 1319.07,749.849 1317.12,748.307 1316.66,747.959 1314.61,746.273 1314.35,746.069 1312.15,744.18 1312.11,744.133 \n",
" 1310.05,742.29 1309.6,741.852 1308.05,740.4 1307.09,739.421 1306.17,738.511 1304.58,736.817 1304.4,736.621 1302.71,734.731 1302.08,733.965 1301.13,732.841 \n",
" 1299.65,730.952 1299.57,730.841 1298.25,729.062 1297.06,727.324 1296.96,727.172 1295.74,725.283 1294.62,723.393 1294.56,723.281 1293.56,721.503 1292.6,719.614 \n",
" 1292.05,718.428 1291.71,717.724 1290.9,715.834 1290.16,713.945 1289.54,712.191 1289.49,712.055 1288.89,710.165 1288.35,708.276 1287.88,706.386 1287.47,704.496 \n",
" 1287.13,702.606 1287.04,701.997 1286.84,700.717 1286.61,698.827 1286.43,696.937 1286.31,695.048 1286.25,693.158 1286.23,691.268 1286.27,689.379 1286.36,687.489 \n",
" 1286.5,685.599 1286.69,683.71 1286.93,681.82 1287.04,681.131 1287.21,679.93 1287.54,678.041 1287.92,676.151 1288.33,674.261 1288.8,672.371 1289.31,670.482 \n",
" 1289.54,669.691 1289.86,668.592 1290.45,666.702 1291.09,664.813 1291.77,662.923 1292.05,662.201 1292.49,661.033 1293.25,659.144 1294.06,657.254 1294.56,656.148 \n",
" 1294.91,655.364 1295.79,653.475 1296.72,651.585 1297.06,650.915 1297.69,649.695 1298.69,647.806 1299.57,646.233 1299.75,645.916 1300.83,644.026 1301.97,642.136 \n",
" 1302.08,641.959 1303.14,640.247 1304.35,638.357 1304.58,638.005 1305.6,636.467 1306.9,634.578 1307.09,634.307 1308.23,632.688 1309.6,630.823 1309.62,630.798 \n",
" 1311.03,628.909 1312.11,627.526 1312.5,627.019 1314,625.129 1314.61,624.386 1315.55,623.24 1317.12,621.384 1317.15,621.35 1318.78,619.46 1319.63,618.512 \n",
" 1320.47,617.571 1322.13,615.749 1322.2,615.681 1323.97,613.791 1324.64,613.092 1325.79,611.901 1327.15,610.527 1327.66,610.012 1329.58,608.122 1329.65,608.047 \n",
" 1331.54,606.232 1332.16,605.65 1333.56,604.343 1334.67,603.326 1335.63,602.453 1337.17,601.071 1337.75,600.563 1339.68,598.882 1339.92,598.674 1342.15,596.784 \n",
" 1342.19,596.754 1344.44,594.894 1344.7,594.686 1346.78,593.005 1347.2,592.672 1349.19,591.115 1349.71,590.711 1351.65,589.225 1352.22,588.8 1354.18,587.336 \n",
" 1354.72,586.936 1356.77,585.446 1357.23,585.119 1359.43,583.556 1359.74,583.345 1362.17,581.666 1362.24,581.613 1364.75,579.921 1364.97,579.777 1367.26,578.269 \n",
" 1367.85,577.887 1369.76,576.655 1370.8,575.997 1372.27,575.076 1373.84,574.108 1374.78,573.534 1376.96,572.218 1377.29,572.025 1379.79,570.549 1380.17,570.328 \n",
" 1382.3,569.106 1383.48,568.439 1384.81,567.693 1386.88,566.549 1387.31,566.312 1389.82,564.96 1390.39,564.659 1392.33,563.636 1394,562.77 1394.83,562.342 \n",
" 1397.34,561.074 1397.73,560.88 1399.85,559.833 1401.59,558.99 1402.36,558.619 1404.86,557.431 1405.57,557.101 1407.37,556.267 1409.7,555.211 1409.88,555.13 \n",
" 1412.38,554.015 1413.97,553.321 1414.89,552.924 1417.4,551.857 1418.42,551.431 1419.9,550.813 1422.41,549.791 1423.03,549.542 1424.92,548.791 1427.42,547.814 \n",
" 1427.85,547.652 1429.93,546.857 1432.44,545.923 1432.88,545.762 1434.95,545.008 1437.45,544.114 1438.14,543.873 1439.96,543.24 1442.47,542.387 1443.68,541.983 \n",
" 1444.97,541.552 1447.48,540.737 1449.51,540.093 1449.99,539.943 1452.49,539.165 1455,538.408 1455.69,538.204 1457.51,537.668 1460.01,536.947 1462.28,536.314 \n",
" 1462.52,536.245 1465.03,535.559 1467.54,534.892 1469.34,534.424 1470.04,534.242 1472.55,533.609 1475.06,532.994 1476.99,532.535 1477.56,532.397 1480.07,531.814 \n",
" 1482.58,531.25 1485.08,530.703 1485.36,530.645 1487.59,530.171 1490.1,529.656 1492.6,529.157 1494.7,528.755 1495.11,528.676 1497.62,528.208 1500.13,527.757 \n",
" 1502.63,527.322 1505.14,526.904 1505.38,526.866 1507.65,526.5 1510.15,526.111 1512.66,525.739 1515.17,525.382 1517.67,525.041 1518.17,524.976 1520.18,524.714 \n",
" 1522.69,524.401 1525.2,524.104 1527.7,523.823 1530.21,523.557 1532.72,523.305 1535.04,523.086 1535.22,523.069 1537.73,522.846 1540.24,522.638 1542.74,522.445 \n",
" 1545.25,522.267 1547.76,522.103 1550.26,521.955 1552.77,521.82 1555.28,521.7 1557.79,521.595 1560.29,521.504 1562.8,521.428 1565.31,521.366 1567.81,521.318 \n",
" 1570.32,521.285 1572.83,521.266 1575.33,521.261 1577.84,521.271 1580.35,521.294 1582.85,521.333 1585.36,521.385 1587.87,521.452 1590.38,521.533 1592.88,521.628 \n",
" 1595.39,521.738 1597.9,521.862 1600.4,522.001 1602.91,522.154 1605.42,522.322 1607.92,522.504 1610.43,522.701 1612.94,522.912 1614.86,523.086 1615.45,523.138 \n",
" 1617.95,523.378 1620.46,523.632 1622.97,523.901 1625.47,524.186 1627.98,524.485 1630.49,524.8 1631.83,524.976 1632.99,525.128 1635.5,525.471 1638.01,525.829 \n",
" 1640.51,526.203 1643.02,526.592 1644.71,526.866 1645.53,526.996 1648.04,527.415 1650.54,527.849 1653.05,528.299 1655.5,528.755 1655.56,528.765 1658.06,529.245 \n",
" 1660.57,529.741 1663.08,530.254 1664.93,530.645 1665.58,530.782 1668.09,531.325 1670.6,531.885 1673.1,532.462 1673.41,532.535 1675.61,533.054 1678.12,533.662 \n",
" 1680.63,534.289 1681.16,534.424 1683.13,534.93 1685.64,535.588 1688.15,536.265 1688.32,536.314 1690.65,536.956 1693.16,537.666 1695.01,538.204 1695.67,538.394 \n",
" 1698.17,539.138 1700.68,539.901 1701.3,540.093 1703.19,540.68 1705.69,541.479 1707.24,541.983 1708.2,542.295 1710.71,543.129 1712.89,543.873 1713.22,543.984 \n",
" 1715.72,544.855 1718.23,545.747 1718.27,545.762 1720.74,546.656 1723.24,547.587 1723.42,547.652 1725.75,548.535 1728.26,549.505 1728.35,549.542 1730.76,550.494 \n",
" 1733.09,551.431 1733.27,551.505 1735.78,552.534 1737.65,553.321 1738.29,553.587 1740.79,554.659 1742.06,555.211 1743.3,555.754 1745.81,556.871 1746.31,557.101 \n",
" 1748.31,558.009 1750.43,558.99 1750.82,559.171 1753.33,560.355 1754.42,560.88 1755.83,561.563 1758.29,562.77 1758.34,562.796 1760.85,564.051 1762.04,564.659 \n",
" 1763.35,565.331 1765.69,566.549 1765.86,566.637 1768.37,567.968 1769.24,568.439 1770.88,569.325 1772.7,570.328 1773.38,570.709 1775.89,572.121 1776.06,572.218 \n",
" 1778.4,573.559 1779.34,574.108 1780.9,575.025 1782.54,575.997 1783.41,576.521 1785.66,577.887 1785.92,578.047 1788.42,579.603 1788.7,579.777 1790.93,581.189 \n",
" 1791.67,581.666 1793.44,582.807 1794.58,583.556 1795.94,584.458 1797.42,585.446 1798.45,586.142 1800.2,587.336 1800.96,587.861 1802.91,589.225 1803.47,589.615 \n",
" 1805.57,591.115 1805.97,591.405 1808.17,593.005 1808.48,593.232 1810.71,594.894 1810.99,595.098 1813.21,596.784 1813.49,597.004 1815.65,598.674 1816,598.95 \n",
" 1818.04,600.563 1818.51,600.939 1820.38,602.453 1821.01,602.972 1822.67,604.343 1823.52,605.05 1824.92,606.232 1826.03,607.175 1827.13,608.122 1828.54,609.349 \n",
" 1829.29,610.012 1831.04,611.574 1831.41,611.901 1833.48,613.791 1833.55,613.851 1835.52,615.681 1836.06,616.185 1837.52,617.571 1838.56,618.575 1839.48,619.46 \n",
" 1841.07,621.026 1841.4,621.35 1843.28,623.24 1843.58,623.54 1845.13,625.129 1846.08,626.121 1846.94,627.019 1848.59,628.771 1848.72,628.909 1850.46,630.798 \n",
" 1851.1,631.496 1852.17,632.688 1853.6,634.299 1853.85,634.578 1855.5,636.467 1856.11,637.185 1857.11,638.357 1858.62,640.157 1858.69,640.247 1860.25,642.136 \n",
" 1861.13,643.227 1861.77,644.026 1863.26,645.916 1863.63,646.395 1864.72,647.806 1866.14,649.671 1866.16,649.695 1867.56,651.585 1868.65,653.067 1868.94,653.475 \n",
" 1870.29,655.364 1871.15,656.59 1871.62,657.254 1872.91,659.144 1873.66,660.251 1874.19,661.033 1875.43,662.923 1876.17,664.066 1876.65,664.813 1877.84,666.702 \n",
" 1878.67,668.049 1879.01,668.592 1880.15,670.482 1881.18,672.22 1881.27,672.371 1882.37,674.261 1883.44,676.151 1883.69,676.606 1884.48,678.041 1885.51,679.93 \n",
" 1886.19,681.234 1886.5,681.82 1887.48,683.71 1888.43,685.599 1888.7,686.143 1889.37,687.489 1890.27,689.379 1891.16,691.268 1891.21,691.379 1892.02,693.158 \n",
" 1892.87,695.048 1893.68,696.937 1893.72,697.013 1894.48,698.827 1895.26,700.717 1896.02,702.606 1896.22,703.14 1896.75,704.496 1897.46,706.386 1898.16,708.276 \n",
" 1898.73,709.889 1898.83,710.165 1899.48,712.055 1900.11,713.945 1900.72,715.834 1901.24,717.487 1901.31,717.724 1901.88,719.614 1902.43,721.503 1902.96,723.393 \n",
" 1903.47,725.283 1903.74,726.33 1903.96,727.172 1904.43,729.062 1904.89,730.952 1905.32,732.841 1905.73,734.731 1906.12,736.621 1906.25,737.27 1906.5,738.511 \n",
" 1906.85,740.4 1907.19,742.29 1907.5,744.18 1907.8,746.069 1908.08,747.959 1908.34,749.849 1908.58,751.738 1908.76,753.267 1908.8,753.628 1909,755.518 \n",
" 1909.19,757.407 1909.35,759.297 1909.5,761.187 1909.63,763.076 1909.74,764.966 1909.83,766.856 1909.9,768.746 1909.95,770.635 1909.99,772.525 1910,774.415 \n",
" 1910,776.304 1909.98,778.194 1909.94,780.084 1909.88,781.973 1909.81,783.863 1909.71,785.753 1909.6,787.642 1909.46,789.532 1909.31,791.422 1909.14,793.311 \n",
" 1908.95,795.201 1908.76,796.943 1908.74,797.091 1908.51,798.98 1908.27,800.87 1908.01,802.76 1907.72,804.65 1907.42,806.539 1907.1,808.429 1906.76,810.319 \n",
" 1906.4,812.208 1906.25,812.933 1906.02,814.098 1905.62,815.988 1905.2,817.877 1904.77,819.767 1904.31,821.657 1903.83,823.546 1903.74,823.877 1903.34,825.436 \n",
" 1902.82,827.326 1902.28,829.215 1901.73,831.105 1901.24,832.716 1901.15,832.995 1900.56,834.885 1899.94,836.774 1899.3,838.664 1898.73,840.315 1898.65,840.554 \n",
" 1897.97,842.443 1897.27,844.333 1896.55,846.223 1896.22,847.063 1895.81,848.112 1895.05,850.002 1894.27,851.892 1893.72,853.187 1893.46,853.781 1892.64,855.671 \n",
" 1891.79,857.561 1891.21,858.822 1890.92,859.45 1890.03,861.34 1889.11,863.23 1888.7,864.06 1888.18,865.12 1887.22,867.009 1886.23,868.899 1886.19,868.971 \n",
" 1885.23,870.789 1884.2,872.678 1883.69,873.596 1883.15,874.568 1882.07,876.458 1881.18,877.983 1880.97,878.347 1879.84,880.237 1878.69,882.127 1878.67,882.157 \n",
" 1877.52,884.016 1876.32,885.906 1876.17,886.139 1875.09,887.796 1873.84,889.685 1873.66,889.953 1872.56,891.575 1871.26,893.465 1871.15,893.615 1869.93,895.355 \n",
" 1868.65,897.137 1868.57,897.244 1867.18,899.134 1866.14,900.531 1865.77,901.024 1864.33,902.913 1863.63,903.808 1862.86,904.803 1861.36,906.693 1861.13,906.978 \n",
" 1859.83,908.582 1858.62,910.045 1858.26,910.472 1856.67,912.362 1856.11,913.019 1855.05,914.251 1853.6,915.905 1853.4,916.141 1851.71,918.031 1851.1,918.707 \n",
" 1849.99,919.92 1848.59,921.432 1848.24,921.81 1846.45,923.7 1846.08,924.083 1844.63,925.59 1843.58,926.663 1842.77,927.479 1841.07,929.178 1840.88,929.369 \n",
" 1838.95,931.259 1838.56,931.628 1836.98,933.148 1836.06,934.019 1834.97,935.038 1833.55,936.352 1832.92,936.928 1831.04,938.63 1830.83,938.817 1828.7,940.707 \n",
" 1828.54,940.854 1826.53,942.597 1826.03,943.028 1824.31,944.486 1823.52,945.153 1822.05,946.376 1821.01,947.231 1819.75,948.266 1818.51,949.264 1817.39,950.155 \n",
" 1816,951.253 1814.99,952.045 1813.49,953.2 1812.53,953.935 1810.99,955.105 1810.03,955.825 1808.48,956.971 1807.47,957.714 1805.97,958.799 1804.85,959.604 \n",
" 1803.47,960.589 1802.18,961.494 1800.96,962.343 1799.45,963.383 1798.45,964.061 1796.65,965.273 1795.94,965.745 1793.79,967.163 1793.44,967.396 1790.93,969.014 \n",
" 1790.87,969.052 1788.42,970.601 1787.88,970.942 1785.92,972.157 1784.81,972.832 1783.41,973.682 1781.67,974.721 1780.9,975.178 1778.45,976.611 1778.4,976.644 \n",
" 1775.89,978.083 1775.15,978.501 1773.38,979.494 1771.76,980.39 1770.88,980.878 1768.37,982.235 1768.28,982.28 1765.86,983.567 1764.71,984.17 1763.35,984.872 \n",
" \n",
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" 1763.35,984.872 1761.03,986.06 1760.85,986.152 1758.34,987.409 1757.24,987.949 1755.83,988.64 1753.34,989.839 1753.33,989.847 1750.82,991.033 1749.32,991.729 \n",
" 1748.31,992.194 1745.81,993.333 1745.17,993.618 1743.3,994.45 1740.87,995.508 1740.79,995.543 1738.29,996.617 1736.43,997.398 1735.78,997.669 1733.27,998.7 \n",
" 1731.81,999.287 1730.76,999.709 1728.26,1000.7 1727.02,1001.18 1725.75,1001.67 1723.24,1002.62 1722.03,1003.07 1720.74,1003.55 1718.23,1004.46 1716.83,1004.96 \n",
" 1715.72,1005.35 1713.22,1006.22 1711.38,1006.85 1710.71,1007.07 1708.2,1007.91 1705.69,1008.72 1705.66,1008.74 1703.19,1009.52 1700.68,1010.3 1699.62,1010.63 \n",
" 1698.17,1011.07 1695.67,1011.81 1693.23,1012.52 1693.16,1012.54 1690.65,1013.25 1688.15,1013.94 1686.42,1014.4 1685.64,1014.61 1683.13,1015.27 1680.63,1015.92 \n",
" 1679.1,1016.29 1678.12,1016.54 1675.61,1017.15 1673.1,1017.74 1671.18,1018.18 1670.6,1018.32 1668.09,1018.88 1665.58,1019.42 1663.08,1019.95 1662.47,1020.07 \n",
" 1660.57,1020.46 1658.06,1020.96 1655.56,1021.44 1653.05,1021.9 1652.71,1021.96 1650.54,1022.35 1648.04,1022.79 1645.53,1023.21 1643.02,1023.61 1641.46,1023.85 \n",
" 1640.51,1024 1638.01,1024.37 1635.5,1024.73 1632.99,1025.08 1630.49,1025.4 1627.98,1025.72 1627.76,1025.74 1625.47,1026.02 1622.97,1026.3 1620.46,1026.57 \n",
" 1617.95,1026.83 1615.45,1027.07 1612.94,1027.29 1610.43,1027.5 1608.76,1027.63 1607.92,1027.7 \n",
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" 1592.88,925.664 1590.38,925.801 1587.87,925.913 1585.36,926.001 1582.85,926.063 1580.35,926.1 1577.84,926.113 1575.33,926.101 1572.83,926.063 1570.32,926.001 \n",
" 1567.81,925.914 1565.31,925.801 1562.8,925.663 1561.68,925.59 1560.29,925.5 1557.79,925.313 1555.28,925.1 1552.77,924.862 1550.26,924.597 1547.76,924.305 \n",
" 1545.25,923.986 1543.17,923.7 1542.74,923.641 1540.24,923.272 1537.73,922.875 1535.22,922.449 1532.72,921.994 1531.76,921.81 1530.21,921.514 1527.7,921.006 \n",
" 1525.2,920.467 1522.79,919.92 1522.69,919.898 1520.18,919.303 1517.67,918.676 1515.23,918.031 1515.17,918.015 1512.66,917.329 1510.15,916.608 1508.6,916.141 \n",
" 1507.65,915.854 1505.14,915.069 1502.65,914.251 1502.63,914.246 1500.13,913.393 1497.62,912.501 1497.24,912.362 1495.11,911.576 1492.6,910.61 1492.26,910.472 \n",
" 1490.1,909.61 1487.63,908.582 1487.59,908.565 1485.08,907.485 1483.32,906.693 1482.58,906.359 1480.07,905.19 1479.27,904.803 1477.56,903.977 1475.45,902.913 \n",
" 1475.06,902.716 1472.55,901.407 1471.84,901.024 1470.04,900.047 1468.42,899.134 1467.54,898.634 1465.16,897.244 1465.03,897.164 1462.52,895.638 1462.07,895.355 \n",
" 1460.01,894.05 1459.12,893.465 1457.51,892.397 1456.3,891.575 1455,890.675 1453.61,889.685 1452.49,888.881 1451.03,887.796 1449.99,887.008 1448.56,885.906 \n",
" 1447.48,885.052 1446.2,884.016 1444.97,883.006 1443.93,882.127 1442.47,880.863 1441.76,880.237 1439.96,878.613 1439.67,878.347 1437.67,876.458 1437.45,876.248 \n",
" 1435.75,874.568 1434.95,873.754 1433.9,872.678 1432.44,871.118 1432.13,870.789 1430.44,868.899 1429.93,868.319 1428.81,867.009 1427.42,865.338 1427.25,865.12 \n",
" 1425.75,863.23 1424.92,862.142 1424.31,861.34 1422.94,859.45 1422.41,858.697 1421.62,857.561 1420.37,855.671 1419.9,854.951 1419.16,853.781 1418.01,851.892 \n",
" 1417.4,850.833 1416.92,850.002 1415.87,848.112 1414.89,846.242 1414.88,846.223 1413.93,844.333 1413.03,842.443 1412.38,841.001 1412.18,840.554 1411.37,838.664 \n",
" 1410.61,836.774 1409.89,834.885 1409.88,834.833 1409.21,832.995 1408.58,831.105 1407.98,829.215 1407.42,827.326 1407.37,827.128 1406.9,825.436 1406.42,823.546 \n",
" 1405.97,821.657 1405.55,819.767 1405.17,817.877 1404.86,816.235 1404.82,815.988 1404.49,814.098 1404.2,812.208 1403.93,810.319 1403.69,808.429 1403.48,806.539 \n",
" 1403.29,804.65 1403.12,802.76 1402.97,800.87 1402.84,798.98 1402.72,797.091 1402.62,795.201 1402.54,793.311 1402.47,791.422 1402.41,789.532 1402.36,787.642 \n",
" 1402.36,787.332 1402.32,785.753 1402.29,783.863 1402.27,781.973 1402.25,780.084 1402.24,778.194 1402.23,776.304 1402.23,774.415 1402.23,772.525 1402.24,770.635 \n",
" 1402.26,768.746 1402.28,766.856 1402.31,764.966 1402.35,763.076 1402.36,762.865 1402.4,761.187 1402.45,759.297 1402.52,757.407 1402.6,755.518 1402.69,753.628 \n",
" 1402.8,751.738 1402.93,749.849 1403.07,747.959 1403.24,746.069 1403.42,744.18 1403.63,742.29 1403.87,740.4 1404.12,738.511 1404.41,736.621 1404.73,734.731 \n",
" 1404.86,733.976 1405.07,732.841 1405.44,730.952 1405.85,729.062 1406.29,727.172 1406.77,725.283 1407.28,723.393 1407.37,723.077 1407.83,721.503 1408.41,719.614 \n",
" 1409.04,717.724 1409.71,715.834 1409.88,715.375 1410.41,713.945 1411.16,712.055 1411.96,710.165 1412.38,709.204 1412.8,708.276 1413.68,706.386 1414.62,704.496 \n",
" 1414.89,703.965 1415.6,702.606 1416.63,700.717 1417.4,699.37 1417.71,698.827 1418.84,696.937 1419.9,695.25 1420.03,695.048 1421.27,693.158 1422.41,691.505 \n",
" 1422.58,691.268 1423.93,689.379 1424.92,688.061 1425.35,687.489 1426.83,685.599 1427.42,684.867 1428.38,683.71 1429.93,681.884 1429.99,681.82 1431.66,679.93 \n",
" 1432.44,679.086 1433.41,678.041 1434.95,676.449 1435.24,676.151 1437.14,674.261 1437.45,673.955 1439.12,672.371 1439.96,671.59 1441.18,670.482 1442.47,669.341 \n",
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" 1470.04,650.156 1470.89,649.695 1472.55,648.796 1474.44,647.806 1475.06,647.487 1477.56,646.227 1478.2,645.916 1480.07,645.012 1482.19,644.026 1482.58,643.845 \n",
" 1485.08,642.719 1486.43,642.136 1487.59,641.636 1490.1,640.595 1490.97,640.247 1492.6,639.591 1495.11,638.628 1495.84,638.357 1497.62,637.701 1500.13,636.811 \n",
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" 1520.18,630.903 1520.62,630.798 1522.69,630.304 1525.2,629.736 1527.7,629.199 1529.14,628.909 1530.21,628.69 1532.72,628.207 1535.22,627.754 1537.73,627.33 \n",
" 1539.7,627.019 1540.24,626.933 1542.74,626.561 1545.25,626.216 1547.76,625.898 1550.26,625.607 1552.77,625.343 1555.03,625.129 1555.28,625.105 1557.79,624.89 \n",
" 1560.29,624.702 1562.8,624.539 1565.31,624.401 1567.81,624.289 1570.32,624.202 1572.83,624.139 1575.33,624.102 1577.84,624.09 1580.35,624.102 1582.85,624.14 \n",
" 1585.36,624.202 1587.87,624.289 1590.38,624.4 1592.88,624.537 1595.39,624.699 1597.9,624.886 1600.4,625.098 1600.73,625.129 1602.91,625.333 1605.42,625.594 \n",
" 1607.92,625.88 1610.43,626.191 1612.94,626.53 1615.45,626.894 1616.25,627.019 1617.95,627.282 1620.46,627.696 1622.97,628.137 1625.47,628.606 1627,628.909 \n",
" 1627.98,629.101 1630.49,629.621 1632.99,630.169 1635.5,630.747 1635.71,630.798 1638.01,631.349 1640.51,631.981 1643.02,632.643 1643.18,632.688 1645.53,633.33 \n",
" 1648.04,634.049 1649.8,634.578 1650.54,634.798 1653.05,635.576 1655.56,636.387 1655.8,636.467 1658.06,637.225 1660.57,638.099 1661.29,638.357 1663.08,639.002 \n",
" 1665.58,639.94 1666.38,640.247 1668.09,640.911 1670.6,641.918 1671.13,642.136 1673.1,642.958 1675.59,644.026 1675.61,644.036 1678.12,645.148 1679.79,645.916 \n",
" 1680.63,646.301 1683.13,647.491 1683.78,647.806 1685.64,648.721 1687.56,649.695 1688.15,649.993 1690.65,651.307 1691.17,651.585 1693.16,652.665 1694.61,653.475 \n",
" 1695.67,654.068 1697.91,655.364 1698.17,655.518 1700.68,657.016 1701.07,657.254 1703.19,658.564 1704.1,659.144 1705.69,660.165 1707.02,661.033 1708.2,661.821 \n",
" 1709.82,662.923 1710.71,663.534 1712.53,664.813 1713.22,665.306 1715.13,666.702 1715.72,667.141 1717.64,668.592 1718.23,669.042 1720.07,670.482 1720.74,671.012 \n",
" 1722.42,672.371 1723.24,673.056 1724.68,674.261 1725.75,675.177 1726.87,676.151 1728.26,677.382 1728.99,678.041 1730.76,679.675 1731.04,679.93 1733.02,681.82 \n",
" 1733.27,682.064 1734.94,683.71 1735.78,684.556 1736.8,685.599 1738.29,687.158 1738.6,687.489 1740.34,689.379 1740.79,689.884 1742.02,691.268 1743.3,692.743 \n",
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" 1753.33,705.86 1753.69,706.386 1754.93,708.276 1755.83,709.686 1756.14,710.165 1757.3,712.055 1758.34,713.806 1758.42,713.945 1759.51,715.834 1760.55,717.724 \n",
" 1760.85,718.294 1761.55,719.614 1762.51,721.503 1763.35,723.23 1763.43,723.393 1764.32,725.283 1765.17,727.172 1765.86,728.771 1765.99,729.062 1766.77,730.952 \n",
" 1767.51,732.841 1768.22,734.731 1768.37,735.163 1768.89,736.621 1769.53,738.511 1770.13,740.4 1770.7,742.29 1770.88,742.906 1771.24,744.18 1771.74,746.069 \n",
" 1772.22,747.959 1772.65,749.849 1773.06,751.738 1773.38,753.393 1773.43,753.628 1773.77,755.518 1774.08,757.407 1774.36,759.297 1774.61,761.187 1774.82,763.076 \n",
" 1775,764.966 1775.16,766.856 1775.28,768.746 1775.36,770.635 1775.42,772.525 1775.45,774.415 1775.44,776.304 1775.41,778.194 1775.34,780.084 1775.25,781.973 \n",
" 1775.12,783.863 1774.96,785.753 1774.77,787.642 1774.54,789.532 1774.29,791.422 1774,793.311 1773.68,795.201 1773.38,796.809 1773.33,797.091 1772.95,798.98 \n",
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},
"execution_count": 133,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x = Fun()\n",
"u₀ = SingularIntegral(0) \\ (-log(2-x)) # solution with κ = 0\n",
"u = u₀ - sum(u₀)/(π*sqrt(1-x^2)) # ensure correct asymptotics\n",
"v = z -> logkernel(u, z) + log(abs(2-z))\n",
"xx = yy = -4:0.011:4\n",
"V = v.(xx' .+ im*yy)\n",
"\n",
"contour(xx, yy, V)\n",
"plot!(domain(x); color=:black, legend=false)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can solve this equation explicitly. By differentiating we see that $u$ satisfies the following:\n",
"$$\n",
"Hu(x) = -f'(x) = {1 \\over x-2}\n",
"$$\n",
"We now use the inverse Hilbert formula to determine:\n",
"$$\n",
"u(x) = -{1\\over\\sqrt{1-x^2}} H[{1 \\over \\diamond-2} \\sqrt{1-\\diamond^2}](x) + {D \\over \\sqrt{1-x^2}}\n",
"$$\n",
"Using the usual procedure of taking an obvious ansatz and subtracting off the singularities at poles and $\\infty$ we find that:\n",
"$$\n",
"C[{\\sqrt{1-\\diamond^2} \\over x -2}](z) = \\underbrace{\\sqrt{z-1} \\sqrt{z+1} \\over 2 \\I (z-2)}_{\\hbox{ansatz}} - \\underbrace{\\sqrt 3\\over 2 \\I (z-2)}_{\\hbox{remove pole near $z=2$}} - \\underbrace{{1 \\over 2 \\I}}_{\\hbox{remove constant at $\\infty$}}\n",
"$$\n",
"This implies that\n",
"$$\n",
"H[{1 \\over\\diamond-2} \\sqrt{1-\\diamond^2}](x) = \\I(C^+ + C^-)[{1 \\over\\diamond-2} \\sqrt{1-\\diamond^2}](x) = -{\\sqrt 3 \\over x -2} - 1\n",
"$$\n",
"in other words, \n",
"$$\n",
"u = {\\sqrt{3} \\over (x - 2) \\sqrt{1-x^2}} + {D \\over \\sqrt{1-x^2}}\n",
"$$\n",
"We still need to determine $D$. This is chosen so that we tend to $\\log |z|$ near $\\infty$. In particular, we know that\n",
"$$\n",
"{1 \\over \\pi} \\int_{-1}^1 \\log|z-x| u(x) \\dx \\sim {\\int_{-1}^1 u(x) \\dx \\over \\pi} \\log |z|\n",
"$$\n",
"so we need to choose $D$ so that $\\int_{-1}^1 u(x) \\dx = 0$ and only the $\\log|z-2|$ singularity appears near $\\infty$. We first note that \n",
"\\begin{align*}\n",
"C[{1 \\over (\\diamond - 2) \\sqrt{1-\\diamond^2}}](z) &= \\underbrace{{\\I \\over 2 \\sqrt{z-1} \\sqrt{z+1} (z-2)}}_{\\hbox{ansatz}} - \\underbrace{{\\I \\over 2\\sqrt{3} (z-2)}}_{\\hbox{remove pole}} \\\\\n",
"& = -{\\I \\over 2 \\sqrt{3} z} + O(z^{-2})\n",
"\\end{align*}\n",
"near $\\infty$. Since \n",
"$$\n",
"Cw(z) ={ \\int_{-1}^1 w(x) \\dx \\over -2\\pi \\I z} + O(z^{-2}) \n",
"$$\n",
"we can infer the integral from the Cauchy transform and find that\n",
"$$\n",
"\\int_{-1}^1 {1 \\over (x - 2) \\sqrt{1-x^2}} \\dx = -{ \\pi \\over \\sqrt{3}}\n",
"$$\n",
"On the other hand,\n",
"$$\n",
"\\int_{-1}^1 {\\dx \\over \\sqrt{1-x^2}} = \\pi\n",
"$$\n",
"Thus we require $D = 1$ and have the solution:\n",
"$$\n",
"u(x) = {\\sqrt{3} \\over (x - 2) \\sqrt{1-x^2}} - {1 \\over \\sqrt{1-x^2}}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"_Demonstration_ We first check that differentiation reduces to the Hilbert transform:"
]
},
{
"cell_type": "code",
"execution_count": 149,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"true"
]
},
"execution_count": 149,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"t = Fun()\n",
"f = -log(2-t)\n",
"hilbert(u, 0.1) ≈ -f'(0.1) ≈ 1/(0.1-2)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In the inverse Hilbert transform we calculated the following Cauchy transform:"
]
},
{
"cell_type": "code",
"execution_count": 150,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"true"
]
},
"execution_count": 150,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"z = 1+im\n",
"cauchy((-f')*sqrt(1-t^2), z) ≈ sqrt(z-1)sqrt(z+1)/(2im*(z-2)) - sqrt(3)/(2im*(z-2)) - 1/(2im)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Which means that:"
]
},
{
"cell_type": "code",
"execution_count": 153,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(-0.08839431180585401, -0.08839431180585411)"
]
},
"execution_count": 153,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"hilbert((-f')*sqrt(1-t^2), 0.1), -sqrt(3)/(0.1-2) - 1"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Thus we have the inverse Hilbert transform:"
]
},
{
"cell_type": "code",
"execution_count": 140,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"true"
]
},
"execution_count": 140,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"D = randn()\n",
"ũ = sqrt(3)/((x-2)*sqrt(1-x^2)) + D/sqrt(1-x^2)\n",
"hilbert(ũ,0.1) ≈ -f'(0.1)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We still need to determine the constant $D$. What goes wrong if we have the wrong choice is that we don't tend to precisely $\\log z$ near $\\infty$:"
]
},
{
"cell_type": "code",
"execution_count": 141,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(-24.76573192820616, 12.206072645530174)"
]
},
"execution_count": 141,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"z = 200000\n",
"logkernel(ũ,z), log(z)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We determined the integral"
]
},
{
"cell_type": "code",
"execution_count": 142,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"true"
]
},
"execution_count": 142,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sum(1/((x-2)*sqrt(1-x^2))) ≈ -2π/(2*sqrt(3))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Therefore:"
]
},
{
"cell_type": "code",
"execution_count": 160,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1.743934249004316e-16"
]
},
"execution_count": 160,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sum(u)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We thus found $u$:"
]
},
{
"cell_type": "code",
"execution_count": 154,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(0.08883962601869715, 0.08883962601869722)"
]
},
"execution_count": 154,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x = 0.1; u(x), sqrt(3)/((x-2)*sqrt(1-x^2)) + 1/sqrt(1-x^2)"
]
}
],
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