{
"cells": [
{
"cell_type": "code",
"execution_count": 43,
"metadata": {},
"outputs": [],
"source": [
"using Plots, ComplexPhasePortrait, ApproxFun, SingularIntegralEquations, OscillatoryIntegrals, QuadGK\n",
"gr();"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# M3M6: Methods of Mathematical Physics 2018\n",
"\n",
"$$\n",
"\\def\\dashint{{\\int\\!\\!\\!\\!\\!\\!-\\,}}\n",
"\\def\\infdashint{\\dashint_{\\!\\!\\!-\\infty}^{\\,\\infty}}\n",
"\\def\\D{\\,{\\rm d}}\n",
"\\def\\E{{\\rm e}}\n",
"\\def\\dx{\\D x}\n",
"\\def\\dt{\\D t}\n",
"\\def\\dz{\\D z}\n",
"\\def\\C{{\\mathbb C}}\n",
"\\def\\R{{\\mathbb R}}\n",
"\\def\\rR{{\\rm R}}\n",
"\\def\\rL{{\\rm L}}\n",
"\\def\\CC{{\\cal C}}\n",
"\\def\\HH{{\\cal H}}\n",
"\\def\\I{{\\rm i}}\n",
"\\def\\qqqquad{\\qquad\\qquad}\n",
"\\def\\qqfor{\\qquad\\hbox{for}\\qquad}\n",
"\\def\\qqwhere{\\qquad\\hbox{where}\\qquad}\n",
"\\def\\Res_#1{\\underset{#1}{\\rm Res}}\\,\n",
"\\def\\sech{{\\rm sech}\\,}\n",
"\\def\\vc#1{{\\mathbf #1}}\n",
"\\def\\Ei{{\\rm Ei}\\,}\n",
"\\def\\pr(#1){\\left({#1}\\right)}\n",
"\\def\\br[#1]{\\left[{#1}\\right]}\n",
"\\def\\set#1{\\left\\{{#1}\\right\\}}\n",
"\\def\\ip<#1>{\\left\\langle{#1}\\right\\rangle}\n",
"\\def\\iip<#1>{\\left\\langle\\!\\langle{#1}\\right\\rangle\\!\\rangle}\n",
"$$\n",
"\n",
"Dr Sheehan Olver\n",
"<br>\n",
"s.olver@imperial.ac.uk\n",
"\n",
"<br>\n",
"Website: https://github.com/dlfivefifty/M3M6LectureNotes\n",
"\n",
"# Solution Sheet 4\n",
"\n",
"\n",
"\n",
"## Problem 1.1\n",
"\n",
"We want to solve\n",
"$$\n",
"\\int_{-1}^1 \\log |x-t| u(t) \\dt = {1 \\over x^2 + 1}\n",
"$$\n",
"differentiating and multiplying though by $1/\\pi$ we have\n",
"$$\n",
"{1 \\over \\pi} \\dashint_{-1}^1 {u(t) \\over t-x} u(t) = {2 x \\over \\pi (1+x^2)^2}\n",
"$$\n",
"\n",
"This is an inverse Hilbert problem so we know\n",
"$$\n",
"u(x) = -{1 \\over \\sqrt{1-x^2} } H[\\sqrt{1-t^2} f(t)] + {C \\over \\sqrt{1-x^2}}\n",
"$$\n",
"for \n",
"$$\n",
"f(x) = {2 x \\over \\pi (1+x^2)^2}.\n",
"$$\n",
"\n",
"We determine the Cauchy transform of $\\sqrt{1-x^2} f(x)$ using the usual methods: start with the ansatz\n",
"$$\n",
"\\phi(z) = {\\sqrt{z-1} \\sqrt{z+1} \\over 2 \\I} {2 z \\over \\pi (1+z^2)^2}\n",
"$$\n",
"This decays at infinity so we just need to remove the poleas at $\\pm \\I$. Here we determine:\n",
"$$\n",
"\\phi(z) = {\\sqrt{\\I -1 } \\sqrt{\\I + 1} \\over \\pi} \\left(-{1 \\over 4 (z-\\I)^2} + {\\I \\over 8 (z-\\I)} + O(1) \\right)\n",
"$$\n",
"and\n",
"$$\n",
"\\phi(z) = {\\sqrt{-\\I -1 } \\sqrt{1-\\I } \\over \\pi} \\left({1 \\over 4 (z+\\I)^2} + {\\I \\over 8 (z+\\I)} + O(1) \\right)\n",
"$$\n",
"Telling us that\n",
"$$\n",
"C[\\sqrt{1-t^2} f(t)](z) = {\\sqrt{z-1} \\sqrt{z+1} \\over 2 \\I} {2 z \\over \\pi (1+z^2)^2} - {\\sqrt{\\I -1 } \\sqrt{\\I + 1} \\over \\pi} \\left(-{1 \\over 4 (z-\\I)^2} + {\\I \\over 8 (z-\\I)} \\right) - {\\sqrt{-\\I -1 } \\sqrt{1-\\I } \\over \\pi} \\left({1 \\over 4 (z+\\I)^2} + {\\I \\over 8 (z+\\I)} \\right)\n",
"$$\n",
"Thus we have\n",
"$$\n",
"H[\\sqrt{1-t^2} f](x) = \\I (C^+ + C^-)[\\sqrt{1-t^2} f](x) = - {2\\I \\sqrt{\\I -1 } \\sqrt{\\I + 1} \\over \\pi} \\left(-{1 \\over 4 (x-\\I)^2} + {2\\I \\over 8 (x-\\I)} \\right) - {2\\I \\sqrt{-\\I -1 } \\sqrt{1-\\I } \\over \\pi} \\left({1 \\over 4 (x+\\I)^2} + {\\I \\over 8 (x+\\I)} \\right)\n",
"$$\n",
"\n",
"In other words, for\n",
"$$\n",
"\\tilde u(x) = {2\\I \\sqrt{\\I -1 } \\sqrt{\\I + 1} \\over \\pi \\sqrt{1-x^2}} \\left(-{1 \\over 4 (x-\\I)^2} + {2\\I \\over 8 (x-\\I)} \\right) - {2\\I \\sqrt{-\\I -1 } \\sqrt{1-\\I } \\over \\pi \\sqrt{1-x^2}} \\left({1 \\over 4 (x+\\I)^2} + {\\I \\over 8 (x+\\I)} \\right)\n",
"$$\n",
"we have\n",
"$$\n",
"u(x) = \\tilde u(x) + {C \\over \\sqrt{1-x^2}}.\n",
"$$\n",
"\n",
"\n",
"To find $C$ we impose the condition that\n",
"$$\n",
"\\int_{-1}^1 \\log|t| u(t) \\dt = 1\n",
"$$\n",
"We thus need to determine $C$. Recall from Lecture 19 that\n",
"$$\n",
"{1 \\over \\pi} \\int_{-1}^1 {\\log(z-x) \\over \\sqrt{1-x^2}} \\dx = 2 \\log(\\sqrt{z-1}+\\sqrt{z+1}) - 2 \\log 2\n",
"$$\n",
"Thus for $x \\in [-1,1]$ we have\n",
"$$\n",
"{1 \\over \\pi} \\int_{-1}^1 {\\log|x-t| \\over \\sqrt{1-t^2}} \\dt = 2 \\Re(\\log(\\I \\sqrt{1-x}+\\sqrt{x+1}) - 2 \\log 2)\n",
"$$\n",
"or in particular \n",
"$$\n",
"{1 \\over \\pi} \\int_{-1}^1 {\\log|x| \\over \\sqrt{1-x^2}} \\dx = \\log(1/2)\n",
"$$\n",
"Thus we want to solve\n",
"$$\n",
"\\int_{-1}^1 \\log|t| \\tilde u(t) \\dt + C {\\log(1/2) \\over \\pi} = 1\n",
"$$\n",
"i.e.\n",
"$$\n",
"C = (1 - \\int_{-1}^1 \\log|t| \\tilde u(t) \\dt) {1 \\over \\pi \\log(1/2)}\n",
"$$\n",
"\n",
"\n",
"\n",
"_Check derivation_\n",
"\n",
"Let's check the derivation. First we can calculate $u$ numerically:"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"-2.220446049250313e-16"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"L = SingularIntegral(0) : JacobiWeight(-0.5,-0.5,Chebyshev())\n",
"\n",
"\n",
"x = Fun()\n",
"g = (1/(x^2+1))\n",
"u = (π*L) \\ g\n",
"\n",
"π*logkernel(u, 0.1) - g(0.1)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Differentiating it satisfies $H u = g'/(-\\pi)$:"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"true"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"hilbert(u,0.1) ≈ g'(0.1)/(-π) ≈ (2*0.1)/(π*(1+0.1^2)^2)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The Cauchy transform also works:"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"true"
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"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
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],
"source": [
"f = g'/(-π)\n",
"\n",
"z = 1+im\n",
"ψ = z -> sqrt(z-1)sqrt(z+1)/(2im) * 2z/(π*(1+z^2)^2) - \n",
" sqrt(im-1)sqrt(im+1)/π * (-1/(4*(z-im)^2) + im/(8(z-im))) - \n",
" sqrt(-im-1)sqrt(1-im)/π * (1/(4*(z+im)^2) + im/(8(z+im)))\n",
"\n",
"cauchy(sqrt(1-x^2)*f, z) ≈ ψ(z)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Therefore the Hilbert transform is given by:"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"true"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"H = x-> - 2im*sqrt(im-1)sqrt(im+1)/π * (-1/(4*(x-im)^2) + im/(8(x-im))) - \n",
" 2im*sqrt(-im-1)sqrt(1-im)/π * (1/(4*(x+im)^2) + im/(8(x+im)))\n",
"\n",
"hilbert(sqrt(1-x^2)*f)(0.1) ≈ H(0.1)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We thus have an expression for $u$, we are just missing the constant:"
]
},
{
"cell_type": "code",
"execution_count": 81,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"true"
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},
"execution_count": 81,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"ũ = x -> -H(x)/sqrt(1-x^2)\n",
"\n",
"C = u(0.0) + H(0.0)\n",
"u(0.1) ≈ ũ(0.1) + C/sqrt(1-0.1^2)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can find $C$ interms of the relevant integral, which we call $D$:"
]
},
{
"cell_type": "code",
"execution_count": 92,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(-0.3247204711377926 + 0.0im, -0.32472047136213555 + 0.0im)"
]
},
"execution_count": 92,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"D = quadgk(x -> x == 0 ? 0.0 : ũ(x)*log(abs(x)),-1,1)[1]\n",
"C, (1-D)/(π*log(1/2))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Problem 1.2\n",
"\n",
"Since $\\int_{-1}^1 x \\dx = 0$, we have no logarithmic growth at infinity. Thus by the formula in Lecture 19 we find for\n",
"$$\n",
"U(x) = \\int_x^1 t \\dt = {1 - x^2 \\over 2}\n",
"$$\n",
"that\n",
"$$\n",
"\\int_{-1}^1 \\log|z-x| x \\dx = \\Re{-2 \\I \\pi C U(z)}\n",
"$$\n",
"\n",
"So we just have to work out the Cauchy transform. Try as an ansatz\n",
"$$\n",
"(1 - z^2 \\over 2) {\\log(z-1) - \\log(z+1) \\over 2 \\pi \\I}\n",
"$$\n",
"We only need to remove the growth at infinity. We do so via:\n",
"$$\n",
"\\log(z-1) - \\log(z+1) = -{2 \\over z} + O(z^{-3})\n",
"$$\n",
"telling us\n",
"$$\n",
"C U(z) = {1 - z^2 \\over 2} {\\log(z-1) - \\log(z+1) \\over 2 \\pi \\I} + {z \\over 2 \\pi \\I}\n",
"$$\n",
"and\n",
"$$\n",
"\\int_{-1}^1 \\log|z-x| x \\dx = \\Re \\left({1 - z^2 \\over 2} (\\log(z-1) - \\log(z+1)) - z \\right)\n",
"$$\n",
"\n",
"We can confirm the formula:"
]
},
{
"cell_type": "code",
"execution_count": 93,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"true"
]
},
"execution_count": 93,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"z = 2+im; π*logkernel(x,z) ≈ real((1-z^2)/2 * (log(z-1)-log(z+1)) - z)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Problem 2\n",
"\n",
"### Problem 2.1\n",
"\n",
"From Lecture 19 we know that we want to solve\n",
"\n",
"1. $v_{xx} + v_{yy} =0$ for $z \\notin [-1,1] \\cup \\I$\n",
"2. $v(z) \\sim \\log|z|$ as $z \\rightarrow \\infty$\n",
"3. $v(z) \\sim \\log|z-\\I|$ as $z \\rightarrow \\I$\n",
"3. $v(x,0) = D$ for some unknown constant $D$ on $[-1,1]$. \n",
"\n",
"### Problem 2.2\n",
"\n",
"\n",
"We write\n",
"$$\n",
"v(x,y) = \\int_{-1}^1 u(t) \\log|z-t| \\dt + \\log|z-\\I|\n",
"$$\n",
"The behaviour at infinity requires that $\\int_{-1}^1 u(t) \\dt = 0$. We further have the singular integral equation\n",
"$$\n",
" \\int_{-1}^1 u(t) \\log|x-t| \\dt = C - \\log|x-\\I|\n",
"$$\n",
"which follows from $D = v(x,0)$\n",
"\n",
"We can actually solve this numerically:"
]
},
{
"cell_type": "code",
"execution_count": 150,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.18822640645958963"
]
},
"execution_count": 150,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"ũ = SingularIntegral(0) \\ (-log(abs(x-im))/π)\n",
"u = ũ - sum(ũ)/(π*sqrt(1-x^2)) # ensure integrates to zero\n",
"v = z -> π*logkernel(u, z) + log(abs(z-im))\n",
"D = v(0)"
]
},
{
"cell_type": "code",
"execution_count": 151,
"metadata": {},
"outputs": [
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" 1226.87,1427.54 1229.38,1427.35 1231.88,1427.15 1234.39,1426.95 1236.9,1426.75 1239.4,1426.54 1241.55,1426.36 1241.91,1426.33 1244.42,1426.11 1246.92,1425.89 \n",
" 1249.43,1425.66 1251.94,1425.43 1254.45,1425.2 1256.95,1424.96 1259.46,1424.72 1261.97,1424.47 1262.01,1424.47 1264.47,1424.22 1266.98,1423.96 1269.49,1423.7 \n",
" 1271.99,1423.44 1274.5,1423.17 1277.01,1422.9 1279.51,1422.62 1279.9,1422.58 1282.02,1422.34 1284.53,1422.05 1287.04,1421.76 1289.54,1421.47 1292.05,1421.17 \n",
" 1294.56,1420.86 1295.98,1420.69 1297.06,1420.55 1299.57,1420.24 1302.08,1419.93 1304.58,1419.6 1307.09,1419.28 1309.6,1418.95 1310.71,1418.8 1312.11,1418.61 \n",
" 1314.61,1418.27 1317.12,1417.93 1319.63,1417.58 1322.13,1417.23 1324.38,1416.91 1324.64,1416.87 1327.15,1416.51 1329.65,1416.14 1332.16,1415.77 1334.67,1415.4 \n",
" 1337.17,1415.02 1337.17,1415.02 1339.68,1414.63 1342.19,1414.25 1344.7,1413.85 1347.2,1413.45 1349.23,1413.13 1349.71,1413.05 1352.22,1412.65 1354.72,1412.23 \n",
" 1357.23,1411.82 1359.74,1411.4 1360.67,1411.24 1362.24,1410.97 1364.75,1410.54 1367.26,1410.11 1369.76,1409.67 1371.58,1409.35 1372.27,1409.23 1374.78,1408.78 \n",
" 1377.29,1408.33 1379.79,1407.87 1382.02,1407.46 1382.3,1407.41 1384.81,1406.94 1387.31,1406.47 1389.82,1405.99 1392.04,1405.57 1392.33,1405.51 1394.83,1405.03 \n",
" 1397.34,1404.54 1399.85,1404.05 1401.68,1403.68 1402.36,1403.55 1404.86,1403.04 1407.37,1402.53 1409.88,1402.02 1410.99,1401.79 1412.38,1401.5 1414.89,1400.98 \n",
" 1417.4,1400.45 1419.9,1399.92 1419.99,1399.9 1422.41,1399.38 1424.92,1398.84 1427.42,1398.29 1428.71,1398.01 1429.93,1397.74 1432.44,1397.19 1434.95,1396.63 \n",
" 1437.18,1396.12 1437.45,1396.06 1439.96,1395.49 1442.47,1394.91 1444.97,1394.33 1445.4,1394.23 1447.48,1393.75 1449.99,1393.16 1452.49,1392.56 1453.41,1392.34 \n",
" 1455,1391.96 1457.51,1391.36 1460.01,1390.75 1461.21,1390.45 1462.52,1390.13 1465.03,1389.51 1467.54,1388.89 1468.82,1388.56 1470.04,1388.26 1472.55,1387.62 \n",
" 1475.06,1386.98 1476.26,1386.67 1477.56,1386.34 1480.07,1385.69 1482.58,1385.03 1483.52,1384.78 1485.08,1384.37 1487.59,1383.71 1490.1,1383.04 1490.63,1382.89 \n",
" 1492.6,1382.36 1495.11,1381.68 1497.59,1381 1497.62,1381 1500.13,1380.31 1502.63,1379.61 1504.4,1379.11 1505.14,1378.91 1507.65,1378.2 1510.15,1377.49 \n",
" 1511.09,1377.22 1512.66,1376.77 1515.17,1376.05 1517.64,1375.34 1517.67,1375.33 1520.18,1374.59 1522.69,1373.86 1524.07,1373.45 1525.2,1373.11 1527.7,1372.36 \n",
" 1530.21,1371.61 1530.39,1371.56 1532.72,1370.85 1535.22,1370.09 1536.6,1369.67 1537.73,1369.32 1540.24,1368.54 1542.7,1367.78 1542.74,1367.76 1545.25,1366.98 \n",
" 1547.76,1366.19 1548.7,1365.89 1550.26,1365.39 1552.77,1364.59 1554.6,1364 1555.28,1363.78 1557.79,1362.97 1560.29,1362.15 1560.42,1362.11 1562.8,1361.32 \n",
" 1565.31,1360.49 1566.14,1360.22 1567.81,1359.66 1570.32,1358.82 1571.77,1358.33 1572.83,1357.97 1575.33,1357.12 1577.32,1356.44 1577.84,1356.26 1580.35,1355.4 \n",
" 1582.79,1354.55 1582.85,1354.53 1585.36,1353.65 1587.87,1352.77 1588.19,1352.66 1590.38,1351.89 1592.88,1350.99 1593.51,1350.77 1595.39,1350.1 1597.9,1349.19 \n",
" 1598.76,1348.88 1600.4,1348.28 1602.91,1347.37 1603.93,1346.99 1605.42,1346.44 1607.92,1345.52 1609.04,1345.1 1610.43,1344.58 1612.94,1343.64 1614.09,1343.21 \n",
" 1615.45,1342.7 1617.95,1341.75 1619.07,1341.32 1620.46,1340.79 1622.97,1339.83 1623.99,1339.43 1625.47,1338.86 1627.98,1337.88 1628.85,1337.54 1630.49,1336.9 \n",
" 1632.99,1335.91 1633.65,1335.65 1635.5,1334.92 1638.01,1333.92 1638.4,1333.76 1640.51,1332.91 1643.02,1331.9 1643.09,1331.87 1645.53,1330.88 1647.72,1329.98 \n",
" 1648.04,1329.85 1650.54,1328.82 1652.31,1328.09 1653.05,1327.78 1655.56,1326.74 1656.84,1326.2 1658.06,1325.69 1660.57,1324.63 1661.32,1324.31 1663.08,1323.57 \n",
" 1665.58,1322.5 1665.76,1322.42 1668.09,1321.42 1670.15,1320.53 1670.6,1320.34 1673.1,1319.25 1674.49,1318.64 1675.61,1318.15 1678.12,1317.05 1678.78,1316.75 \n",
" 1680.63,1315.94 1683.04,1314.87 1683.13,1314.82 1685.64,1313.7 1687.24,1312.98 1688.15,1312.57 1690.65,1311.43 1691.41,1311.09 1693.16,1310.29 1695.54,1309.2 \n",
" 1695.67,1309.14 1698.17,1307.98 1699.62,1307.31 1700.68,1306.81 1703.19,1305.64 1703.67,1305.42 1705.69,1304.46 1707.68,1303.53 1708.2,1303.28 1710.71,1302.09 \n",
" 1711.65,1301.64 1713.22,1300.89 1715.58,1299.75 1715.72,1299.68 1718.23,1298.46 1719.48,1297.86 1720.74,1297.24 1723.24,1296.01 1723.34,1295.97 1725.75,1294.78 \n",
" 1727.16,1294.08 1728.26,1293.53 1730.76,1292.28 1730.95,1292.19 1733.27,1291.02 1734.71,1290.3 1735.78,1289.76 1738.29,1288.49 1738.43,1288.41 1740.79,1287.2 \n",
" 1742.13,1286.52 1743.3,1285.92 1745.79,1284.63 1745.81,1284.62 1748.31,1283.32 1749.41,1282.74 1750.82,1282 1753.01,1280.85 1753.33,1280.68 1755.83,1279.36 \n",
" 1756.58,1278.96 1758.34,1278.02 1760.11,1277.07 1760.85,1276.68 1763.35,1275.33 1763.62,1275.18 1765.86,1273.97 1767.1,1273.29 1768.37,1272.6 1770.55,1271.4 \n",
" 1770.88,1271.22 1773.38,1269.84 1773.97,1269.51 1775.89,1268.45 1777.37,1267.62 1778.4,1267.05 1780.74,1265.73 1780.9,1265.64 1783.41,1264.22 1784.08,1263.84 \n",
" 1785.92,1262.8 1787.39,1261.95 1788.42,1261.36 1790.68,1260.06 1790.93,1259.92 1793.44,1258.47 1793.95,1258.17 1795.94,1257.01 1797.18,1256.28 1798.45,1255.54 \n",
" 1800.4,1254.4 1800.96,1254.06 1803.47,1252.58 1803.59,1252.51 1805.97,1251.08 1806.75,1250.62 1808.48,1249.58 1809.89,1248.73 1810.99,1248.06 1813.01,1246.84 \n",
" 1813.49,1246.54 1816,1245.01 1816.11,1244.95 1818.51,1243.47 1819.18,1243.06 1821.01,1241.92 1822.23,1241.17 1823.52,1240.36 1825.25,1239.28 1826.03,1238.79 \n",
" 1828.26,1237.39 1828.54,1237.21 1831.04,1235.62 1831.24,1235.5 1833.55,1234.03 1834.2,1233.61 1836.06,1232.42 1837.14,1231.72 1838.56,1230.8 1840.06,1229.83 \n",
" 1841.07,1229.18 1842.96,1227.94 1843.58,1227.54 1845.84,1226.05 1846.08,1225.89 1848.59,1224.23 1848.7,1224.16 1851.1,1222.57 1851.54,1222.27 1853.6,1220.89 \n",
" 1854.36,1220.38 1856.11,1219.2 1857.16,1218.49 1858.62,1217.5 1859.94,1216.6 1861.13,1215.79 1862.7,1214.71 1863.63,1214.07 1865.44,1212.82 1866.14,1212.34 \n",
" 1868.17,1210.93 1868.65,1210.6 1870.87,1209.04 1871.15,1208.85 1873.56,1207.15 1873.66,1207.08 1876.17,1205.31 1876.23,1205.26 1878.67,1203.52 1878.88,1203.37 \n",
" 1881.18,1201.72 1881.51,1201.48 1883.69,1199.92 1884.13,1199.59 1886.19,1198.09 1886.73,1197.7 1888.7,1196.26 1889.31,1195.81 1891.21,1194.42 1891.88,1193.93 \n",
" 1893.72,1192.56 1894.43,1192.04 1896.22,1190.7 1896.96,1190.15 1898.73,1188.82 1899.47,1188.26 1901.24,1186.92 1901.97,1186.37 1903.74,1185.02 1904.46,1184.48 \n",
" \n",
" \"/>\n",
"<polyline clip-path=\"url(#clip9303)\" style=\"stroke:#f2ef78; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
" 1904.46,1184.48 1906.25,1183.1 1906.92,1182.59 1908.76,1181.17 1909.37,1180.7 1911.26,1179.23 1911.81,1178.81 1913.77,1177.28 1914.23,1176.92 1916.28,1175.31 \n",
" 1916.64,1175.03 1918.78,1173.33 1919.03,1173.14 1921.29,1171.33 1921.4,1171.25 1923.76,1169.36 1923.8,1169.33 1926.1,1167.47 1926.31,1167.31 1928.43,1165.58 \n",
" 1928.81,1165.27 1930.75,1163.69 1931.32,1163.22 1933.05,1161.8 1933.83,1161.16 1935.34,1159.91 1936.33,1159.08 1937.61,1158.02 1938.84,1156.99 1939.87,1156.13 \n",
" 1941.35,1154.89 1942.11,1154.24 1943.85,1152.77 1944.34,1152.35 1946.36,1150.63 1946.56,1150.46 1948.76,1148.57 1948.87,1148.48 1950.95,1146.68 1951.38,1146.32 \n",
" 1953.13,1144.79 1953.88,1144.14 1955.29,1142.9 1956.39,1141.94 1957.44,1141.01 1958.9,1139.73 1959.58,1139.12 1961.4,1137.5 1961.7,1137.23 1963.81,1135.34 \n",
" 1963.91,1135.26 1965.91,1133.46 1966.42,1133 1968,1131.57 1968.92,1130.72 1970.07,1129.68 1971.43,1128.43 1972.13,1127.79 1973.94,1126.12 1974.18,1125.9 \n",
" 1976.21,1124.01 1976.44,1123.79 1978.23,1122.12 1978.95,1121.44 1980.24,1120.23 1981.46,1119.08 1982.24,1118.34 1983.97,1116.7 1984.23,1116.45 1986.2,1114.56 \n",
" 1986.47,1114.3 1988.17,1112.67 1988.98,1111.88 1990.12,1110.78 1991.49,1109.45 1992.06,1108.89 1993.99,1107 1993.99,1106.99 1995.9,1105.11 1996.5,1104.52 \n",
" 1997.81,1103.22 1999.01,1102.02 1999.7,1101.33 2001.51,1099.51 2001.58,1099.44 2003.45,1097.55 2004.02,1096.98 2005.31,1095.66 2006.53,1094.42 2007.16,1093.77 \n",
" 2009,1091.88 2009.03,1091.85 2010.83,1089.99 2011.54,1089.25 2012.64,1088.1 2014.05,1086.63 2014.45,1086.21 2016.24,1084.32 2016.56,1083.99 2018.03,1082.43 \n",
" 2019.06,1081.33 2019.8,1080.54 2021.56,1078.65 2021.57,1078.65 2023.32,1076.76 2024.08,1075.94 2025.06,1074.87 2026.58,1073.21 \n",
" \"/>\n",
"<polyline clip-path=\"url(#clip9303)\" style=\"stroke:#f5f78c; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
" 2026.58,1269.15 2026.11,1269.51 2024.08,1271.08 2023.66,1271.4 2021.57,1273 2021.18,1273.29 2019.06,1274.9 2018.7,1275.18 2016.56,1276.8 2016.2,1277.07 \n",
" 2014.05,1278.69 2013.68,1278.96 2011.54,1280.56 2011.16,1280.85 2009.03,1282.43 2008.61,1282.74 2006.53,1284.28 2006.06,1284.63 2004.02,1286.13 2003.48,1286.52 \n",
" 2001.51,1287.96 2000.9,1288.41 1999.01,1289.78 1998.3,1290.3 1996.5,1291.6 1995.68,1292.19 1993.99,1293.4 1993.05,1294.08 1991.49,1295.2 1990.4,1295.97 \n",
" 1988.98,1296.98 1987.74,1297.86 1986.47,1298.76 1985.06,1299.75 1983.97,1300.52 1982.37,1301.64 1981.46,1302.28 1979.66,1303.53 1978.95,1304.02 1976.94,1305.42 \n",
" 1976.44,1305.76 1974.2,1307.31 1973.94,1307.48 1971.44,1309.2 1971.43,1309.2 1968.92,1310.91 1968.67,1311.09 1966.42,1312.61 1965.88,1312.98 1963.91,1314.3 \n",
" 1963.07,1314.87 1961.4,1315.98 1960.25,1316.75 1958.9,1317.65 1957.41,1318.64 1956.39,1319.32 1954.55,1320.53 1953.88,1320.97 1951.67,1322.42 1951.38,1322.62 \n",
" 1948.87,1324.26 1948.78,1324.31 1946.36,1325.88 1945.87,1326.2 1943.85,1327.5 1942.94,1328.09 1941.35,1329.11 1939.99,1329.98 1938.84,1330.72 1937.03,1331.87 \n",
" 1936.33,1332.31 1934.04,1333.76 1933.83,1333.9 1931.32,1335.48 1931.04,1335.65 1928.81,1337.04 1928.02,1337.54 1926.31,1338.61 1924.98,1339.43 1923.8,1340.16 \n",
" 1921.92,1341.32 1921.29,1341.7 1918.84,1343.21 1918.78,1343.24 1916.28,1344.77 1915.74,1345.1 1913.77,1346.29 1912.61,1346.99 1911.26,1347.8 1909.47,1348.88 \n",
" 1908.76,1349.31 1906.31,1350.77 1906.25,1350.81 1903.74,1352.3 1903.13,1352.66 1901.24,1353.78 1899.93,1354.55 1898.73,1355.25 1896.7,1356.44 1896.22,1356.72 \n",
" 1893.72,1358.18 1893.46,1358.33 1891.21,1359.63 1890.19,1360.22 1888.7,1361.07 1886.9,1362.11 1886.19,1362.51 1883.69,1363.94 1883.59,1364 1881.18,1365.36 \n",
" 1880.25,1365.89 1878.67,1366.78 1876.89,1367.78 1876.17,1368.18 1873.66,1369.58 1873.51,1369.67 1871.15,1370.98 1870.11,1371.56 1868.65,1372.36 1866.68,1373.45 \n",
" 1866.14,1373.74 1863.63,1375.11 1863.22,1375.34 1861.13,1376.48 1859.75,1377.22 1858.62,1377.83 1856.24,1379.11 1856.11,1379.18 1853.6,1380.53 1852.71,1381 \n",
" 1851.1,1381.87 1849.16,1382.89 1848.59,1383.19 1846.08,1384.52 1845.58,1384.78 1843.58,1385.83 1841.97,1386.67 1841.07,1387.14 1838.56,1388.45 1838.34,1388.56 \n",
" 1836.06,1389.74 1834.68,1390.45 1833.55,1391.03 1831.04,1392.31 1830.99,1392.34 1828.54,1393.59 1827.27,1394.23 1826.03,1394.86 1823.52,1396.12 1823.52,1396.12 \n",
" 1821.01,1397.38 1819.75,1398.01 1818.51,1398.63 1816,1399.87 1815.95,1399.9 1813.49,1401.11 1812.11,1401.79 1810.99,1402.34 1808.48,1403.57 1808.25,1403.68 \n",
" 1805.97,1404.79 1804.35,1405.57 1803.47,1406 1800.96,1407.2 1800.42,1407.46 1798.45,1408.4 1796.46,1409.35 1795.94,1409.6 1793.44,1410.78 1792.47,1411.24 \n",
" 1790.93,1411.96 1788.45,1413.13 1788.42,1413.14 1785.92,1414.31 1784.39,1415.02 1783.41,1415.47 1780.9,1416.63 1780.29,1416.91 1778.4,1417.78 1776.16,1418.8 \n",
" 1775.89,1418.92 1773.38,1420.06 1772,1420.69 1770.88,1421.19 1768.37,1422.32 1767.8,1422.58 1765.86,1423.44 1763.56,1424.47 1763.35,1424.56 1760.85,1425.67 \n",
" 1759.28,1426.36 1758.34,1426.77 1755.83,1427.87 1754.97,1428.25 1753.33,1428.96 1750.82,1430.05 1750.61,1430.14 1748.31,1431.13 1746.22,1432.03 1745.81,1432.2 \n",
" 1743.3,1433.27 1741.78,1433.92 1740.79,1434.33 1738.29,1435.39 1737.3,1435.81 1735.78,1436.45 1733.27,1437.49 1732.78,1437.69 1730.76,1438.53 1728.26,1439.57 \n",
" 1728.22,1439.58 1725.75,1440.6 1723.61,1441.47 1723.24,1441.62 1720.74,1442.64 1718.95,1443.36 1718.23,1443.66 1715.72,1444.66 1714.25,1445.25 1713.22,1445.67 \n",
" 1710.71,1446.66 1709.5,1447.14 1708.2,1447.66 1705.69,1448.64 1704.7,1449.03 1703.19,1449.62 1700.68,1450.6 1699.85,1450.92 1698.17,1451.57 1695.67,1452.54 \n",
" 1694.95,1452.81 1693.16,1453.5 1690.65,1454.45 1689.99,1454.7 1688.15,1455.4 1685.64,1456.35 1684.98,1456.59 1683.13,1457.29 1680.63,1458.22 1679.92,1458.48 \n",
" 1678.12,1459.15 1675.61,1460.07 1674.8,1460.37 1673.1,1460.99 1670.6,1461.9 1669.61,1462.26 \n",
" \"/>\n",
"<polyline clip-path=\"url(#clip9303)\" style=\"stroke:#f2e763; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
" 2026.58,832.603 2026.41,832.995 2025.57,834.885 2024.72,836.774 2024.08,838.192 2023.86,838.664 2023,840.554 2022.12,842.443 2021.57,843.633 2021.24,844.333 \n",
" 2020.36,846.223 2019.46,848.112 2019.06,848.939 2018.55,850.002 2017.64,851.892 2016.72,853.781 2016.56,854.116 2015.79,855.671 2014.86,857.561 2014.05,859.173 \n",
" 2013.91,859.45 2012.96,861.34 2012,863.23 2011.54,864.116 2011.03,865.12 2010.05,867.009 2009.06,868.899 2009.03,868.955 2008.07,870.789 2007.07,872.678 \n",
" 2006.53,873.689 2006.06,874.568 2005.04,876.458 2004.02,878.331 2004.01,878.347 2002.98,880.237 2001.93,882.127 2001.51,882.879 2000.88,884.016 1999.82,885.906 \n",
" 1999.01,887.342 1998.75,887.796 1997.67,889.685 1996.59,891.575 1996.5,891.723 1995.49,893.465 1994.39,895.355 1993.99,896.024 1993.27,897.244 1992.15,899.134 \n",
" 1991.49,900.25 1991.02,901.024 1989.89,902.913 1988.98,904.404 1988.74,904.803 1987.58,906.693 1986.47,908.49 1986.42,908.582 1985.24,910.472 1984.06,912.362 \n",
" 1983.97,912.509 1982.87,914.251 1981.66,916.141 1981.46,916.464 1980.45,918.031 1979.24,919.92 1978.95,920.357 1978.01,921.81 1976.77,923.7 1976.44,924.192 \n",
" 1975.52,925.59 1974.27,927.479 1973.94,927.969 1973,929.369 1971.73,931.259 1971.43,931.692 1970.44,933.148 1969.15,935.038 1968.92,935.362 1967.84,936.928 \n",
" 1966.53,938.817 1966.42,938.98 1965.21,940.707 1963.91,942.548 1963.88,942.597 1962.53,944.486 1961.4,946.068 1961.18,946.376 1959.82,948.266 1958.9,949.542 \n",
" 1958.45,950.155 1957.07,952.045 1956.39,952.97 1955.68,953.935 1954.28,955.825 1953.88,956.354 1952.87,957.714 1951.45,959.604 1951.38,959.697 1950.01,961.494 \n",
" 1948.87,962.996 1948.57,963.383 1947.12,965.273 1946.36,966.256 1945.66,967.163 1944.19,969.052 1943.85,969.477 1942.7,970.942 1941.35,972.66 1941.21,972.832 \n",
" 1939.71,974.721 1938.84,975.805 1938.19,976.611 1936.67,978.501 1936.33,978.914 1935.13,980.39 1933.83,981.987 1933.59,982.28 1932.03,984.17 1931.32,985.026 \n",
" 1930.46,986.06 1928.88,987.949 1928.81,988.032 1927.29,989.839 1926.31,991.004 1925.69,991.729 1924.08,993.618 1923.8,993.945 1922.46,995.508 1921.29,996.854 \n",
" 1920.82,997.398 1919.17,999.287 1918.78,999.732 1917.52,1001.18 1916.28,1002.58 1915.85,1003.07 1914.17,1004.96 1913.77,1005.4 1912.48,1006.85 1911.26,1008.19 \n",
" 1910.77,1008.74 1909.05,1010.63 1908.76,1010.95 1907.33,1012.52 1906.25,1013.69 1905.59,1014.4 1903.84,1016.29 1903.74,1016.39 1902.07,1018.18 1901.24,1019.07 \n",
" 1900.3,1020.07 1898.73,1021.73 1898.51,1021.96 1896.71,1023.85 1896.22,1024.36 1894.89,1025.74 1893.72,1026.96 1893.07,1027.63 1891.23,1029.52 1891.21,1029.54 \n",
" 1889.38,1031.41 1888.7,1032.1 1887.51,1033.3 1886.19,1034.63 1885.64,1035.19 1883.75,1037.08 1883.69,1037.14 1881.84,1038.97 1881.18,1039.62 1879.93,1040.86 \n",
" 1878.67,1042.09 1878,1042.75 1876.17,1044.53 1876.05,1044.64 1874.1,1046.53 1873.66,1046.95 1872.12,1048.42 1871.15,1049.35 1870.14,1050.31 1868.65,1051.72 \n",
" 1868.14,1052.2 1866.14,1054.08 1866.13,1054.09 1864.1,1055.98 1863.63,1056.42 1862.06,1057.87 1861.13,1058.73 1860.01,1059.76 1858.62,1061.03 1857.94,1061.65 \n",
" 1856.11,1063.31 1855.86,1063.54 1853.76,1065.43 1853.6,1065.56 1851.64,1067.32 1851.1,1067.8 1849.52,1069.21 1848.59,1070.02 1847.37,1071.1 1846.08,1072.23 \n",
" 1845.21,1072.99 1843.58,1074.41 1843.04,1074.87 1841.07,1076.58 1840.85,1076.76 1838.65,1078.65 1838.56,1078.72 1836.42,1080.54 1836.06,1080.86 1834.19,1082.43 \n",
" 1833.55,1082.97 1831.93,1084.32 1831.04,1085.07 1829.66,1086.21 1828.54,1087.15 1827.38,1088.1 1826.03,1089.21 1825.08,1089.99 1823.52,1091.26 1822.76,1091.88 \n",
" 1821.01,1093.29 1820.42,1093.77 1818.51,1095.31 1818.07,1095.66 1816,1097.31 1815.7,1097.55 1813.49,1099.3 1813.31,1099.44 1810.99,1101.27 1810.91,1101.33 \n",
" 1808.48,1103.22 1808.48,1103.22 1806.04,1105.11 1805.97,1105.16 1803.58,1107 1803.47,1107.09 1801.1,1108.89 1800.96,1109 1798.61,1110.78 1798.45,1110.9 \n",
" 1796.09,1112.67 1795.94,1112.78 1793.56,1114.56 1793.44,1114.65 1791.01,1116.45 1790.93,1116.5 1788.43,1118.34 1788.42,1118.34 1785.92,1120.17 1785.84,1120.23 \n",
" 1783.41,1121.99 1783.23,1122.12 1780.9,1123.79 1780.6,1124.01 1778.4,1125.57 1777.94,1125.9 1775.89,1127.35 1775.27,1127.79 1773.38,1129.11 1772.58,1129.68 \n",
" 1770.88,1130.86 1769.86,1131.57 1768.37,1132.6 1767.12,1133.46 1765.86,1134.32 1764.36,1135.34 1763.35,1136.03 1761.58,1137.23 1760.85,1137.73 1758.78,1139.12 \n",
" 1758.34,1139.42 1755.96,1141.01 1755.83,1141.1 1753.33,1142.76 1753.11,1142.9 1750.82,1144.41 1750.24,1144.79 1748.31,1146.05 1747.34,1146.68 1745.81,1147.68 \n",
" 1744.42,1148.57 1743.3,1149.3 1741.48,1150.46 1740.79,1150.9 1738.51,1152.35 1738.29,1152.5 1735.78,1154.08 1735.52,1154.24 1733.27,1155.65 1732.5,1156.13 \n",
" 1730.76,1157.21 1729.46,1158.02 1728.26,1158.76 1726.39,1159.91 1725.75,1160.3 1723.3,1161.8 1723.24,1161.83 1720.74,1163.35 1720.17,1163.69 1718.23,1164.86 \n",
" 1717.02,1165.58 1715.72,1166.36 1713.85,1167.47 1713.22,1167.84 1710.71,1169.32 1710.64,1169.36 1708.2,1170.79 1707.41,1171.25 1705.69,1172.24 1704.15,1173.14 \n",
" 1703.19,1173.69 1700.85,1175.03 1700.68,1175.13 1698.17,1176.55 1697.53,1176.92 1695.67,1177.97 1694.18,1178.81 1693.16,1179.38 1690.8,1180.7 1690.65,1180.78 \n",
" 1688.15,1182.16 1687.38,1182.59 1685.64,1183.54 1683.93,1184.48 1683.13,1184.91 1680.63,1186.27 1680.45,1186.37 1678.12,1187.62 1676.94,1188.26 1675.61,1188.96 \n",
" 1673.39,1190.15 1673.1,1190.3 1670.6,1191.62 1669.81,1192.04 1668.09,1192.93 1666.19,1193.93 1665.58,1194.24 1663.08,1195.54 1662.54,1195.81 1660.57,1196.82 \n",
" 1658.84,1197.7 1658.06,1198.1 1655.56,1199.37 1655.12,1199.59 1653.05,1200.63 1651.35,1201.48 1650.54,1201.89 1648.04,1203.13 1647.54,1203.37 1645.53,1204.37 \n",
" 1643.7,1205.26 1643.02,1205.59 1640.51,1206.81 1639.81,1207.15 1638.01,1208.02 1635.88,1209.04 1635.5,1209.22 1632.99,1210.42 1631.91,1210.93 1630.49,1211.6 \n",
" 1627.98,1212.78 1627.89,1212.82 1625.47,1213.95 1623.83,1214.71 1622.97,1215.11 1620.46,1216.27 1619.73,1216.6 1617.95,1217.41 1615.57,1218.49 1615.45,1218.55 \n",
" 1612.94,1219.68 1611.37,1220.38 1610.43,1220.8 1607.92,1221.92 1607.12,1222.27 1605.42,1223.02 1602.91,1224.12 1602.82,1224.16 1600.4,1225.21 1598.46,1226.05 \n",
" 1597.9,1226.29 1595.39,1227.37 1594.06,1227.94 1592.88,1228.44 1590.38,1229.5 1589.59,1229.83 1587.87,1230.55 1585.36,1231.6 1585.07,1231.72 1582.85,1232.64 \n",
" 1580.5,1233.61 1580.35,1233.67 1577.84,1234.69 1575.86,1235.5 1575.33,1235.71 1572.83,1236.72 1571.16,1237.39 1570.32,1237.72 1567.81,1238.72 1566.39,1239.28 \n",
" 1565.31,1239.71 1562.8,1240.69 1561.56,1241.17 1560.29,1241.66 1557.79,1242.63 1556.66,1243.06 1555.28,1243.59 1552.77,1244.54 1551.7,1244.95 1550.26,1245.49 \n",
" 1547.76,1246.43 1546.65,1246.84 1545.25,1247.36 1542.74,1248.28 1541.54,1248.73 1540.24,1249.2 1537.73,1250.11 1536.34,1250.62 1535.22,1251.02 1532.72,1251.92 \n",
" 1531.07,1252.51 1530.21,1252.81 1527.7,1253.7 1525.71,1254.4 1525.2,1254.57 1522.69,1255.45 1520.26,1256.28 1520.18,1256.31 1517.67,1257.17 1515.17,1258.02 \n",
" 1514.72,1258.17 1512.66,1258.87 1510.15,1259.71 1509.08,1260.06 1507.65,1260.54 1505.14,1261.37 1503.35,1261.95 1502.63,1262.19 1500.13,1263 1497.62,1263.81 \n",
" 1497.51,1263.84 1495.11,1264.61 1492.6,1265.41 1491.57,1265.73 1490.1,1266.2 1487.59,1266.98 1485.51,1267.62 1485.08,1267.75 1482.58,1268.52 1480.07,1269.29 \n",
" 1479.33,1269.51 1477.56,1270.05 1475.06,1270.8 1473.02,1271.4 1472.55,1271.54 1470.04,1272.28 1467.54,1273.02 1466.59,1273.29 1465.03,1273.74 1462.52,1274.47 \n",
" 1460.01,1275.18 1460.01,1275.18 1457.51,1275.89 1455,1276.59 1453.28,1277.07 1452.49,1277.29 1449.99,1277.98 1447.48,1278.67 1446.4,1278.96 1444.97,1279.35 \n",
" 1442.47,1280.02 1439.96,1280.69 1439.35,1280.85 1437.45,1281.35 1434.95,1282.01 1432.44,1282.66 1432.12,1282.74 1429.93,1283.3 1427.42,1283.94 1424.92,1284.58 \n",
" 1424.7,1284.63 1422.41,1285.2 1419.9,1285.83 1417.4,1286.44 1417.08,1286.52 1414.89,1287.05 1412.38,1287.66 1409.88,1288.26 1409.23,1288.41 1407.37,1288.85 \n",
" 1404.86,1289.44 1402.36,1290.02 1401.15,1290.3 1399.85,1290.6 1397.34,1291.17 1394.83,1291.73 1392.8,1292.19 1392.33,1292.29 1389.82,1292.85 1387.31,1293.4 \n",
" 1384.81,1293.94 1384.17,1294.08 1382.3,1294.48 1379.79,1295.01 1377.29,1295.54 1375.23,1295.97 1374.78,1296.06 1372.27,1296.58 1369.76,1297.09 1367.26,1297.6 \n",
" 1365.94,1297.86 1364.75,1298.1 1362.24,1298.59 1359.74,1299.08 1357.23,1299.56 1356.26,1299.75 1354.72,1300.04 1352.22,1300.52 1349.71,1300.98 1347.2,1301.45 \n",
" 1346.15,1301.64 1344.7,1301.9 1342.19,1302.36 1339.68,1302.8 1337.17,1303.24 1335.55,1303.53 1334.67,1303.68 1332.16,1304.11 1329.65,1304.54 1327.15,1304.96 \n",
" 1324.64,1305.37 1324.37,1305.42 1322.13,1305.78 1319.63,1306.19 1317.12,1306.59 1314.61,1306.98 1312.53,1307.31 1312.11,1307.37 1309.6,1307.76 1307.09,1308.14 \n",
" 1304.58,1308.51 1302.08,1308.88 1299.89,1309.2 1299.57,1309.24 1297.06,1309.6 1294.56,1309.95 1292.05,1310.3 1289.54,1310.65 1287.04,1310.98 1286.27,1311.09 \n",
" 1284.53,1311.32 1282.02,1311.65 1279.51,1311.97 1277.01,1312.29 1274.5,1312.6 1271.99,1312.91 1271.43,1312.98 1269.49,1313.21 1266.98,1313.51 1264.47,1313.8 \n",
" 1261.97,1314.09 1259.46,1314.37 1256.95,1314.65 1254.96,1314.87 1254.45,1314.92 1251.94,1315.19 1249.43,1315.45 1246.92,1315.71 1244.42,1315.96 1241.91,1316.21 \n",
" 1239.4,1316.45 1236.9,1316.69 1236.21,1316.75 1234.39,1316.92 1231.88,1317.15 1229.38,1317.38 1226.87,1317.59 1224.36,1317.81 1221.86,1318.02 1219.35,1318.22 \n",
" 1216.84,1318.42 1214.33,1318.61 1213.88,1318.64 1211.83,1318.8 1209.32,1318.98 1206.81,1319.16 1204.31,1319.34 1201.8,1319.51 1199.29,1319.67 1196.79,1319.83 \n",
" 1194.28,1319.98 1191.77,1320.13 1189.27,1320.28 1186.76,1320.42 1184.57,1320.53 1184.25,1320.55 1181.74,1320.68 1179.24,1320.81 1176.73,1320.93 1174.22,1321.04 \n",
" 1171.72,1321.15 1169.21,1321.26 1166.7,1321.36 1164.2,1321.46 1161.69,1321.55 1159.18,1321.63 1156.67,1321.72 1154.17,1321.79 1151.66,1321.86 1149.15,1321.93 \n",
" 1146.65,1321.99 1144.14,1322.05 1141.63,1322.1 1139.13,1322.15 1136.62,1322.19 1134.11,1322.23 1131.61,1322.26 1129.1,1322.29 1126.59,1322.32 1124.08,1322.33 \n",
" 1121.58,1322.35 1119.07,1322.36 1116.56,1322.36 1114.06,1322.36 1111.55,1322.35 1109.04,1322.34 1106.54,1322.33 1104.03,1322.31 1101.52,1322.28 1099.02,1322.26 \n",
" 1096.51,1322.22 1094,1322.18 1091.49,1322.14 1088.99,1322.09 1086.48,1322.03 1083.97,1321.98 1081.47,1321.91 1078.96,1321.84 1076.45,1321.77 1073.95,1321.69 \n",
" 1071.44,1321.61 1068.93,1321.52 1066.42,1321.43 1063.92,1321.33 1061.41,1321.23 1058.9,1321.12 1056.4,1321.01 1053.89,1320.89 1051.38,1320.77 1048.88,1320.65 \n",
" 1046.73,1320.53 1046.37,1320.52 1043.86,1320.38 1041.36,1320.24 1038.85,1320.09 1036.34,1319.94 1033.83,1319.79 1031.33,1319.63 1028.82,1319.46 1026.31,1319.29 \n",
" 1023.81,1319.11 1021.3,1318.93 1018.79,1318.75 1017.42,1318.64 1016.29,1318.56 1013.78,1318.36 1011.27,1318.16 1008.77,1317.96 1006.26,1317.75 1003.75,1317.54 \n",
" 1001.24,1317.32 998.737,1317.09 996.231,1316.86 995.092,1316.75 993.724,1316.63 991.217,1316.39 988.71,1316.14 986.203,1315.89 983.696,1315.64 981.189,1315.38 \n",
" 978.682,1315.12 976.339,1314.87 976.175,1314.85 973.668,1314.57 971.161,1314.29 968.654,1314.01 966.147,1313.72 963.64,1313.43 961.133,1313.13 959.878,1312.98 \n",
" 958.626,1312.82 956.119,1312.51 953.613,1312.2 951.106,1311.88 948.599,1311.56 946.092,1311.23 945.034,1311.09 943.585,1310.89 941.078,1310.55 938.571,1310.21 \n",
" 936.064,1309.86 933.557,1309.5 931.417,1309.2 931.05,1309.14 928.543,1308.78 926.036,1308.41 923.529,1308.03 921.022,1307.65 918.775,1307.31 918.515,1307.27 \n",
" 916.008,1306.88 913.501,1306.48 910.994,1306.08 908.488,1305.67 906.932,1305.42 905.981,1305.26 903.474,1304.84 900.967,1304.42 898.46,1303.99 895.953,1303.56 \n",
" 895.754,1303.53 893.446,1303.12 890.939,1302.68 888.432,1302.23 885.925,1301.78 885.15,1301.64 883.418,1301.32 880.911,1300.86 878.404,1300.39 875.897,1299.91 \n",
" 875.039,1299.75 873.39,1299.43 870.883,1298.95 868.376,1298.46 865.87,1297.96 865.363,1297.86 863.363,1297.46 860.856,1296.95 858.349,1296.44 856.074,1295.97 \n",
" 855.842,1295.92 853.335,1295.4 850.828,1294.87 848.321,1294.33 847.133,1294.08 845.814,1293.79 843.307,1293.25 840.8,1292.7 838.502,1292.19 838.293,1292.14 \n",
" 835.786,1291.58 833.279,1291.01 830.772,1290.44 830.158,1290.3 828.265,1289.86 825.758,1289.28 823.252,1288.69 822.073,1288.41 820.745,1288.09 818.238,1287.49 \n",
" 815.731,1286.89 814.228,1286.52 813.224,1286.27 810.717,1285.66 808.21,1285.03 806.603,1284.63 805.703,1284.4 803.196,1283.77 800.689,1283.13 799.184,1282.74 \n",
" 798.182,1282.48 795.675,1281.83 793.168,1281.17 791.955,1280.85 790.661,1280.51 788.154,1279.84 785.647,1279.16 784.905,1278.96 783.14,1278.48 780.634,1277.79 \n",
" 778.127,1277.1 778.021,1277.07 775.62,1276.4 773.113,1275.7 771.296,1275.18 770.606,1274.99 768.099,1274.27 765.592,1273.55 764.719,1273.29 763.085,1272.82 \n",
" 760.578,1272.08 758.28,1271.4 758.071,1271.34 755.564,1270.59 753.057,1269.84 751.976,1269.51 750.55,1269.08 748.043,1268.32 745.796,1267.62 745.536,1267.54 \n",
" 743.029,1266.77 740.522,1265.98 739.737,1265.73 738.016,1265.19 735.509,1264.39 733.791,1263.84 733.002,1263.59 730.495,1262.78 727.988,1261.97 727.952,1261.95 \n",
" 725.481,1261.14 722.974,1260.32 722.219,1260.06 720.467,1259.48 717.96,1258.64 716.585,1258.17 715.453,1257.79 712.946,1256.94 711.046,1256.28 710.439,1256.08 \n",
" 707.932,1255.21 705.597,1254.4 705.425,1254.34 702.918,1253.46 700.411,1252.57 700.236,1252.51 697.904,1251.67 695.398,1250.77 694.961,1250.62 692.891,1249.87 \n",
" 690.384,1248.95 689.766,1248.73 687.877,1248.03 685.37,1247.1 684.649,1246.84 682.863,1246.17 680.356,1245.23 679.608,1244.95 677.849,1244.28 675.342,1243.33 \n",
" 674.639,1243.06 672.835,1242.36 670.328,1241.4 669.741,1241.17 667.821,1240.42 665.314,1239.44 664.911,1239.28 662.807,1238.45 660.3,1237.45 660.147,1237.39 \n",
" 657.793,1236.45 655.447,1235.5 655.286,1235.43 652.78,1234.42 650.809,1233.61 650.273,1233.39 647.766,1232.36 646.231,1231.72 645.259,1231.31 642.752,1230.27 \n",
" 641.711,1229.83 640.245,1229.21 637.738,1228.15 637.248,1227.94 635.231,1227.08 632.84,1226.05 632.724,1226 630.217,1224.91 628.486,1224.16 627.71,1223.82 \n",
" 625.203,1222.72 624.184,1222.27 622.696,1221.61 620.189,1220.5 619.932,1220.38 617.682,1219.37 615.731,1218.49 615.175,1218.24 612.668,1217.1 611.578,1216.6 \n",
" 610.162,1215.95 607.655,1214.8 607.471,1214.71 605.148,1213.63 603.411,1212.82 602.641,1212.46 600.134,1211.28 599.395,1210.93 597.627,1210.09 595.423,1209.04 \n",
" 595.12,1208.9 592.613,1207.69 591.495,1207.15 590.106,1206.48 587.607,1205.26 587.599,1205.26 585.092,1204.03 583.762,1203.37 582.585,1202.79 580.078,1201.55 \n",
" 579.955,1201.48 577.571,1200.29 576.189,1199.59 575.064,1199.03 572.557,1197.75 572.46,1197.7 570.05,1196.47 568.769,1195.81 567.544,1195.18 565.115,1193.93 \n",
" 565.037,1193.88 562.53,1192.58 561.497,1192.04 560.023,1191.26 557.914,1190.15 557.516,1189.93 555.009,1188.6 554.367,1188.26 552.502,1187.26 550.853,1186.37 \n",
" 549.995,1185.9 547.488,1184.54 547.372,1184.48 544.981,1183.17 543.925,1182.59 542.474,1181.79 540.509,1180.7 539.967,1180.4 537.46,1179 537.125,1178.81 \n",
" 534.953,1177.59 533.773,1176.92 532.446,1176.17 530.45,1175.03 529.939,1174.74 527.432,1173.3 527.158,1173.14 524.926,1171.85 523.896,1171.25 522.419,1170.39 \n",
" 520.662,1169.36 519.912,1168.92 517.457,1167.47 517.405,1167.44 514.898,1165.95 514.28,1165.58 512.391,1164.45 511.131,1163.69 509.884,1162.94 508.008,1161.8 \n",
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" 1515.94,1152.35 1517.67,1151.65 1520.18,1150.64 1520.61,1150.46 1522.69,1149.61 1525.2,1148.58 1525.22,1148.57 1527.7,1147.54 1529.75,1146.68 1530.21,1146.49 \n",
" 1532.72,1145.43 1534.22,1144.79 1535.22,1144.37 1537.73,1143.29 1538.63,1142.9 1540.24,1142.21 1542.74,1141.12 1542.98,1141.01 1545.25,1140.02 1547.26,1139.12 \n",
" 1547.76,1138.9 1550.26,1137.79 1551.49,1137.23 1552.77,1136.66 1555.28,1135.52 1555.66,1135.34 1557.79,1134.37 1559.78,1133.46 1560.29,1133.22 1562.8,1132.05 \n",
" 1563.84,1131.57 1565.31,1130.88 1567.81,1129.69 1567.85,1129.68 1570.32,1128.5 1571.81,1127.79 1572.83,1127.3 1575.33,1126.09 1575.72,1125.9 1577.84,1124.86 \n",
" 1579.59,1124.01 1580.35,1123.63 1582.85,1122.39 1583.4,1122.12 1585.36,1121.14 1587.17,1120.23 1587.87,1119.88 1590.38,1118.61 1590.9,1118.34 1592.88,1117.32 \n",
" 1594.58,1116.45 1595.39,1116.03 1597.9,1114.73 1598.23,1114.56 1600.4,1113.42 1601.82,1112.67 1602.91,1112.09 1605.38,1110.78 1605.42,1110.76 1607.92,1109.42 \n",
" 1608.9,1108.89 1610.43,1108.06 1612.38,1107 1612.94,1106.7 1615.45,1105.32 1615.83,1105.11 1617.95,1103.93 1619.23,1103.22 1620.46,1102.54 1622.6,1101.33 \n",
" 1622.97,1101.13 1625.47,1099.7 1625.94,1099.44 1627.98,1098.27 1629.23,1097.55 1630.49,1096.83 1632.5,1095.66 1632.99,1095.37 1635.5,1093.91 1635.73,1093.77 \n",
" 1638.01,1092.43 1638.93,1091.88 1640.51,1090.94 1642.09,1089.99 1643.02,1089.43 1645.23,1088.1 1645.53,1087.92 1648.04,1086.39 1648.33,1086.21 1650.54,1084.85 \n",
" 1651.4,1084.32 1653.05,1083.3 1654.44,1082.43 1655.56,1081.74 1657.45,1080.54 1658.06,1080.16 1660.44,1078.65 1660.57,1078.57 1663.08,1076.97 1663.39,1076.76 \n",
" 1665.58,1075.35 1666.32,1074.87 1668.09,1073.72 1669.21,1072.99 1670.6,1072.08 1672.09,1071.1 1673.1,1070.42 1674.93,1069.21 1675.61,1068.75 1677.75,1067.32 \n",
" 1678.12,1067.07 1680.54,1065.43 1680.63,1065.37 1683.13,1063.65 1683.3,1063.54 1685.64,1061.93 1686.05,1061.65 1688.15,1060.19 1688.76,1059.76 1690.65,1058.43 \n",
" 1691.45,1057.87 1693.16,1056.66 1694.12,1055.98 1695.67,1054.87 1696.76,1054.09 1698.17,1053.07 1699.38,1052.2 1700.68,1051.26 1701.98,1050.31 1703.19,1049.42 \n",
" 1704.55,1048.42 1705.69,1047.58 1707.1,1046.53 1708.2,1045.71 1709.63,1044.64 1710.71,1043.83 1712.14,1042.75 1713.22,1041.93 1714.63,1040.86 1715.72,1040.02 \n",
" 1717.09,1038.97 1718.23,1038.09 1719.53,1037.08 1720.74,1036.14 1721.95,1035.19 1723.24,1034.18 1724.35,1033.3 1725.75,1032.2 1726.74,1031.41 1728.26,1030.2 \n",
" 1729.1,1029.52 1730.76,1028.18 1731.44,1027.63 1733.27,1026.14 1733.76,1025.74 1735.78,1024.09 1736.06,1023.85 1738.29,1022.01 1738.34,1021.96 1740.61,1020.07 \n",
" 1740.79,1019.92 1742.85,1018.18 1743.3,1017.8 1745.08,1016.29 1745.81,1015.67 1747.28,1014.4 1748.31,1013.52 1749.47,1012.52 1750.82,1011.34 1751.64,1010.63 \n",
" 1753.33,1009.15 1753.8,1008.74 1755.83,1006.94 1755.93,1006.85 1758.05,1004.96 1758.34,1004.7 1760.15,1003.07 1760.85,1002.44 1762.24,1001.18 1763.35,1000.16 \n",
" 1764.31,999.287 1765.86,997.854 1766.36,997.398 1768.37,995.528 1768.39,995.508 1770.41,993.618 1770.88,993.178 1772.41,991.729 1773.38,990.803 1774.39,989.839 \n",
" 1775.89,988.405 1776.36,987.949 1778.32,986.06 1778.4,985.982 1780.25,984.17 1780.9,983.533 1782.18,982.28 1783.41,981.059 1784.08,980.39 1785.92,978.558 \n",
" 1785.97,978.501 1787.85,976.611 1788.42,976.03 1789.71,974.721 1790.93,973.475 1791.56,972.832 1793.39,970.942 1793.44,970.893 1795.21,969.052 1795.94,968.281 \n",
" 1797.01,967.163 1798.45,965.64 1798.8,965.273 1800.57,963.383 1800.96,962.97 1802.33,961.494 1803.47,960.268 1804.08,959.604 1805.81,957.714 1805.97,957.536 \n",
" 1807.53,955.825 1808.48,954.771 1809.23,953.935 1810.92,952.045 1810.99,951.974 1812.6,950.155 1813.49,949.143 1814.26,948.266 1815.92,946.376 1816,946.278 \n",
" 1817.55,944.486 1818.51,943.377 1819.18,942.597 1820.79,940.707 1821.01,940.44 1822.39,938.817 1823.52,937.466 1823.97,936.928 1825.54,935.038 1826.03,934.453 \n",
" 1827.11,933.148 1828.54,931.402 1828.65,931.259 1830.19,929.369 1831.04,928.309 1831.71,927.479 1833.22,925.59 1833.55,925.176 1834.72,923.7 1836.06,921.999 \n",
" 1836.2,921.81 1837.68,919.92 1838.56,918.778 1839.14,918.031 1840.59,916.141 1841.07,915.512 1842.03,914.251 1843.45,912.362 1843.58,912.199 1844.87,910.472 \n",
" 1846.08,908.837 1846.27,908.582 1847.66,906.693 1848.59,905.425 1849.04,904.803 1850.41,902.913 1851.1,901.961 1851.77,901.024 1853.12,899.134 1853.6,898.443 \n",
" 1854.45,897.244 1855.77,895.355 1856.11,894.87 1857.09,893.465 1858.39,891.575 1858.62,891.238 1859.68,889.685 1860.96,887.796 1861.13,887.547 1862.23,885.906 \n",
" 1863.48,884.016 1863.63,883.793 1864.73,882.127 1865.97,880.237 1866.14,879.973 1867.19,878.347 1868.41,876.458 1868.65,876.086 1869.61,874.568 1870.81,872.678 \n",
" 1871.15,872.128 1871.99,870.789 1873.17,868.899 1873.66,868.096 1874.33,867.009 1875.48,865.12 1876.17,863.986 1876.62,863.23 1877.76,861.34 1878.67,859.795 \n",
" 1878.88,859.45 1879.99,857.561 1881.09,855.671 1881.18,855.519 1882.19,853.781 1883.27,851.892 1883.69,851.152 1884.34,850.002 1885.4,848.112 1886.19,846.691 \n",
" 1886.46,846.223 1887.5,844.333 1888.53,842.443 1888.7,842.129 1889.56,840.554 1890.57,838.664 1891.21,837.461 1891.57,836.774 1892.57,834.885 1893.55,832.995 \n",
" 1893.72,832.681 1894.53,831.105 1895.5,829.215 1896.22,827.78 1896.45,827.326 1897.4,825.436 1898.34,823.546 1898.73,822.751 1899.27,821.657 1900.19,819.767 \n",
" 1901.1,817.877 1901.24,817.585 1902,815.988 1902.89,814.098 1903.74,812.272 1903.77,812.208 1904.65,810.319 1905.51,808.429 1906.25,806.798 1906.37,806.539 \n",
" 1907.21,804.65 1908.05,802.76 1908.76,801.151 1908.88,800.87 1909.7,798.98 1910.51,797.091 1911.26,795.316 1911.31,795.201 1912.11,793.311 1912.89,791.422 \n",
" 1913.67,789.532 1913.77,789.274 1914.43,787.642 1915.19,785.753 1915.94,783.863 1916.28,783.004 1916.68,781.973 1917.41,780.084 1918.14,778.194 1918.78,776.483 \n",
" 1918.85,776.304 1919.56,774.415 1920.26,772.525 1920.95,770.635 1921.29,769.676 1921.63,768.746 1922.3,766.856 1922.96,764.966 1923.62,763.076 1923.8,762.551 \n",
" 1924.27,761.187 1924.91,759.297 1925.54,757.407 1926.16,755.518 1926.31,755.059 1926.77,753.628 1927.38,751.738 1927.97,749.849 1928.56,747.959 1928.81,747.142 \n",
" 1929.14,746.069 1929.72,744.18 1930.28,742.29 1930.84,740.4 1931.32,738.727 1931.38,738.511 1931.92,736.621 1932.45,734.731 1932.98,732.841 1933.49,730.952 \n",
" 1933.83,729.705 1934,729.062 1934.5,727.172 1934.99,725.283 1935.47,723.393 1935.95,721.503 1936.33,719.939 1936.41,719.614 1936.87,717.724 1937.32,715.834 \n",
" 1937.77,713.945 1938.2,712.055 1938.63,710.165 1938.84,709.212 1939.05,708.276 1939.46,706.386 1939.86,704.496 1940.26,702.606 1940.65,700.717 1941.03,698.827 \n",
" 1941.35,697.192 1941.4,696.937 1941.76,695.048 1942.12,693.158 1942.47,691.268 1942.81,689.379 1943.14,687.489 1943.47,685.599 1943.78,683.71 1943.85,683.266 \n",
" 1944.09,681.82 1944.39,679.93 1944.69,678.041 1944.97,676.151 1945.25,674.261 1945.52,672.371 1945.79,670.482 1946.04,668.592 1946.29,666.702 1946.36,666.129 \n",
" 1946.53,664.813 1946.76,662.923 1946.99,661.033 1947.2,659.144 1947.41,657.254 1947.61,655.364 1947.81,653.475 1947.99,651.585 1948.17,649.695 1948.34,647.806 \n",
" 1948.51,645.916 1948.66,644.026 1948.81,642.136 1948.87,641.365 1948.95,640.247 1949.08,638.357 1949.21,636.467 1949.33,634.578 1949.44,632.688 1949.54,630.798 \n",
" 1949.64,628.909 1949.72,627.019 1949.8,625.129 1949.88,623.24 1949.94,621.35 1950,619.46 1950.05,617.571 1950.09,615.681 1950.12,613.791 1950.15,611.901 \n",
" 1950.17,610.012 1950.18,608.122 1950.19,606.232 1950.18,604.343 1950.17,602.453 1950.15,600.563 1950.13,598.674 1950.09,596.784 1950.05,594.894 1950,593.005 \n",
" 1949.95,591.115 1949.88,589.225 1949.81,587.336 1949.73,585.446 1949.65,583.556 1949.55,581.666 1949.45,579.777 1949.34,577.887 1949.22,575.997 1949.1,574.108 \n",
" 1948.97,572.218 1948.87,570.875 1948.83,570.328 1948.68,568.439 1948.53,566.549 1948.36,564.659 1948.19,562.77 1948.02,560.88 1947.83,558.99 1947.64,557.101 \n",
" 1947.44,555.211 1947.23,553.321 1947.01,551.431 1946.79,549.542 1946.56,547.652 1946.36,546.081 1946.32,545.762 1946.08,543.873 1945.82,541.983 1945.56,540.093 \n",
" 1945.29,538.204 1945.01,536.314 1944.73,534.424 1944.44,532.535 1944.14,530.645 1943.85,528.913 1943.83,528.755 1943.51,526.866 1943.19,524.976 1942.86,523.086 \n",
" 1942.52,521.197 1942.17,519.307 1941.82,517.417 1941.46,515.527 1941.35,514.962 1941.09,513.638 1940.71,511.748 1940.33,509.858 1939.93,507.969 1939.53,506.079 \n",
" 1939.12,504.189 1938.84,502.909 1938.71,502.3 1938.28,500.41 1937.85,498.52 1937.41,496.631 1936.96,494.741 1936.51,492.851 1936.33,492.153 1936.04,490.962 \n",
" 1935.57,489.072 1935.09,487.182 1934.6,485.292 1934.11,483.403 1933.83,482.356 1933.6,481.513 1933.09,479.623 1932.57,477.734 1932.04,475.844 1931.51,473.954 \n",
" 1931.32,473.304 1930.96,472.065 1930.41,470.175 1929.85,468.285 1929.28,466.396 1928.81,464.855 1928.71,464.506 1928.12,462.616 1927.53,460.727 1926.93,458.837 \n",
" 1926.32,456.947 1926.31,456.906 1925.7,455.057 1925.08,453.168 1924.44,451.278 1923.8,449.388 1923.8,449.383 1923.15,447.499 1922.49,445.609 1921.83,443.719 \n",
" 1921.29,442.228 1921.15,441.83 1920.47,439.94 1919.77,438.05 1919.07,436.161 1918.78,435.392 1918.36,434.271 1917.65,432.381 1916.92,430.492 1916.28,428.837 \n",
" 1916.19,428.602 1915.44,426.712 1914.69,424.822 1913.93,422.933 1913.77,422.537 1913.16,421.043 1912.39,419.153 1911.6,417.264 1911.26,416.464 1910.81,415.374 \n",
" 1910,413.484 1909.19,411.595 1908.76,410.598 1908.37,409.705 1907.54,407.815 1906.7,405.926 1906.25,404.919 1905.85,404.036 1905,402.146 1904.13,400.257 \n",
" 1903.74,399.414 1903.26,398.367 1902.38,396.477 1901.48,394.587 1901.24,394.067 1900.58,392.698 1899.67,390.808 1898.75,388.918 1898.73,388.868 1897.83,387.029 \n",
" 1896.89,385.139 1896.22,383.808 1895.94,383.249 1894.99,381.36 1894.02,379.47 1893.72,378.875 1893.05,377.58 1892.06,375.691 1891.21,374.062 1891.07,373.801 \n",
" 1890.07,371.911 1889.06,370.022 1888.7,369.362 1888.04,368.132 1887.01,366.242 1886.19,364.768 1885.97,364.352 1884.92,362.463 1883.86,360.573 1883.69,360.274 \n",
" 1882.79,358.683 1881.71,356.794 1881.18,355.875 1880.62,354.904 1879.52,353.014 1878.67,351.566 1878.42,351.125 1877.3,349.235 1876.17,347.345 1876.17,347.341 \n",
" 1875.03,345.456 1873.88,343.566 1873.66,343.199 1872.73,341.676 1871.56,339.787 1871.15,339.135 1870.38,337.897 1869.19,336.007 1868.65,335.144 1868,334.117 \n",
" 1866.79,332.228 1866.14,331.224 1865.57,330.338 1864.34,328.448 1863.63,327.371 1863.1,326.559 1861.85,324.669 1861.13,323.584 1860.59,322.779 1859.32,320.89 \n",
" 1858.62,319.859 1858.04,319 1856.74,317.11 1856.11,316.194 1855.44,315.221 1854.13,313.331 1853.6,312.587 1852.8,311.441 1851.46,309.552 1851.1,309.036 \n",
" 1850.12,307.662 1848.76,305.772 1848.59,305.538 1847.39,303.882 1846.08,302.093 1846.01,301.993 1844.62,300.103 1843.58,298.698 1843.22,298.213 1841.8,296.324 \n",
" 1841.07,295.351 1840.38,294.434 1838.94,292.544 1838.56,292.051 1837.49,290.655 1836.06,288.796 1836.03,288.765 1834.56,286.875 1833.55,285.586 1833.08,284.986 \n",
" 1831.58,283.096 1831.04,282.418 1830.07,281.206 1828.56,279.317 1828.54,279.292 1827.02,277.427 1826.03,276.207 1825.48,275.537 1823.92,273.647 1823.52,273.16 \n",
" 1822.36,271.758 1821.01,270.152 1820.78,269.868 1819.18,267.978 1818.51,267.181 1817.58,266.089 1816,264.246 1815.96,264.199 1814.33,262.309 1813.49,261.347 \n",
" 1812.69,260.42 1811.03,258.53 1810.99,258.481 1809.36,256.64 1808.48,255.65 1807.68,254.751 1805.98,252.861 1805.97,252.851 1804.27,250.971 1803.47,250.085 \n",
" 1802.55,249.082 1800.96,247.349 1800.81,247.192 1799.06,245.302 1798.45,244.644 1797.3,243.413 1795.94,241.969 1795.52,241.523 1793.73,239.633 1793.44,239.323 \n",
" \n",
" \"/>\n",
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" 1793.44,239.323 1791.93,237.743 1790.93,236.706 1790.11,235.854 1788.42,234.116 1788.28,233.964 1786.43,232.074 1785.92,231.555 1784.57,230.185 1783.41,229.02 \n",
" 1782.69,228.295 1780.9,226.511 1780.8,226.405 1778.89,224.516 1778.4,224.028 1776.97,222.626 1775.89,221.57 1775.03,220.736 1773.38,219.137 1773.08,218.847 \n",
" 1771.11,216.957 1770.88,216.728 1769.13,215.067 1768.37,214.344 1767.13,213.178 1765.86,211.983 1765.12,211.288 1763.35,209.644 1763.09,209.398 1761.04,207.508 \n",
" 1760.85,207.329 1758.98,205.619 1758.34,205.035 1756.9,203.729 1755.83,202.764 1754.81,201.839 1753.33,200.514 1752.69,199.95 1750.82,198.285 1750.57,198.06 \n",
" 1748.42,196.17 1748.31,196.077 1746.26,194.281 1745.81,193.89 1744.08,192.391 1743.3,191.722 1741.88,190.501 1740.79,189.575 1739.66,188.612 1738.29,187.447 \n",
" 1737.43,186.722 1735.78,185.338 1735.17,184.832 1733.27,183.248 1732.9,182.943 1730.76,181.177 1730.61,181.053 1728.31,179.163 1728.26,179.124 1725.98,177.273 \n",
" 1725.75,177.09 1723.63,175.384 1723.24,175.073 1721.27,173.494 1720.74,173.074 1718.88,171.604 1718.23,171.092 1716.47,169.715 1715.72,169.127 1714.05,167.825 \n",
" 1713.22,167.18 1711.6,165.935 1710.71,165.248 1709.14,164.046 1708.2,163.334 1706.65,162.156 1705.69,161.436 1704.14,160.266 1703.19,159.553 1701.61,158.377 \n",
" 1700.68,157.687 1699.06,156.487 1698.17,155.836 1696.48,154.597 1695.67,154.001 1693.89,152.708 1693.16,152.181 1691.27,150.818 1690.65,150.376 1688.63,148.928 \n",
" 1688.15,148.586 1685.96,147.038 1685.64,146.811 1683.27,145.149 1683.13,145.05 1680.63,143.304 1680.56,143.259 1678.12,141.572 1677.82,141.369 1675.61,139.854 \n",
" 1675.06,139.48 1673.1,138.15 1672.28,137.59 1670.6,136.46 1669.46,135.7 1668.09,134.783 1666.63,133.811 1665.58,133.12 1663.76,131.921 1663.08,131.47 \n",
" 1660.87,130.031 1660.57,129.833 1658.06,128.21 1657.96,128.142 1655.56,126.599 1655.01,126.252 1653.05,125.001 1652.04,124.362 1650.54,123.416 1649.04,122.473 \n",
" 1648.04,121.844 1646.01,120.583 1645.53,120.284 1643.02,118.736 1642.95,118.693 1640.51,117.2 1639.86,116.803 1638.01,115.677 1636.74,114.914 1635.5,114.165 \n",
" 1633.59,113.024 1632.99,112.666 1630.49,111.178 1630.41,111.134 1627.98,109.701 1627.2,109.245 1625.47,108.237 1623.95,107.355 1622.97,106.783 1620.67,105.465 \n",
" 1620.46,105.342 1617.95,103.911 1617.36,103.576 1615.45,102.491 1614.01,101.686 1612.94,101.083 1610.63,99.7963 1610.43,99.6854 1607.92,98.2986 1607.21,97.9066 \n",
" 1605.42,96.9226 1603.76,96.0169 1602.91,95.5575 1600.4,94.2031 1600.26,94.1272 1597.9,92.8589 1596.73,92.2375 1595.39,91.5253 1593.16,90.3479 1592.88,90.2023 \n",
" 1590.38,88.8892 1589.55,88.4582 \n",
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" 1786.34,88.4582 1788.42,89.9971 1788.9,90.3479 1790.93,91.8611 1791.43,92.2375 1793.44,93.7389 1793.95,94.1272 1795.94,95.6306 1796.45,96.0169 1798.45,97.5364 \n",
" 1798.94,97.9066 1800.96,99.4564 1801.4,99.7963 1803.47,101.391 1803.85,101.686 1805.97,103.34 1806.28,103.576 1808.48,105.303 1808.69,105.465 1810.99,107.282 \n",
" 1811.08,107.355 1813.46,109.245 1813.49,109.275 1815.81,111.134 1816,111.284 1818.16,113.024 1818.51,113.308 1820.48,114.914 1821.01,115.348 1822.79,116.803 \n",
" 1823.52,117.404 1825.08,118.693 1826.03,119.475 1827.36,120.583 1828.54,121.563 1829.62,122.473 1831.04,123.668 1831.87,124.362 1833.55,125.789 1834.09,126.252 \n",
" 1836.06,127.927 1836.31,128.142 1838.5,130.031 1838.56,130.081 1840.69,131.921 1841.07,132.254 1842.85,133.811 1843.58,134.444 1845.01,135.7 1846.08,136.651 \n",
" 1847.14,137.59 1848.59,138.877 1849.27,139.48 1851.1,141.121 1851.37,141.369 1853.47,143.259 1853.6,143.384 1855.55,145.149 1856.11,145.665 1857.61,147.038 \n",
" 1858.62,147.966 1859.66,148.928 1861.13,150.286 1861.7,150.818 1863.63,152.626 1863.72,152.708 1865.73,154.597 1866.14,154.986 1867.72,156.487 1868.65,157.366 \n",
" 1869.7,158.377 1871.15,159.767 1871.67,160.266 1873.63,162.156 1873.66,162.189 1875.57,164.046 1876.17,164.633 1877.5,165.935 1878.67,167.098 1879.41,167.825 \n",
" 1881.18,169.585 1881.31,169.715 1883.2,171.604 1883.69,172.095 1885.08,173.494 1886.19,174.627 1886.94,175.384 1888.7,177.183 1888.79,177.273 1890.63,179.163 \n",
" 1891.21,179.762 1892.45,181.053 1893.72,182.366 1894.27,182.943 1896.07,184.832 1896.22,184.994 1897.86,186.722 1898.73,187.648 1899.64,188.612 1901.24,190.326 \n",
" 1901.4,190.501 1903.15,192.391 1903.74,193.031 1904.89,194.281 1906.25,195.762 1906.62,196.17 1908.34,198.06 1908.76,198.52 1910.05,199.95 1911.26,201.306 \n",
" 1911.74,201.839 1913.42,203.729 1913.77,204.121 1915.1,205.619 1916.28,206.963 1916.76,207.508 1918.41,209.398 1918.78,209.836 1920.04,211.288 1921.29,212.738 \n",
" 1921.67,213.178 1923.29,215.067 1923.8,215.671 1924.89,216.957 1926.31,218.635 1926.48,218.847 1928.07,220.736 1928.81,221.632 1929.64,222.626 1931.2,224.516 \n",
" 1931.32,224.661 1932.75,226.405 1933.83,227.725 1934.29,228.295 1935.82,230.185 1936.33,230.823 1937.34,232.074 1938.84,233.955 1938.85,233.964 1940.35,235.854 \n",
" 1941.35,237.125 1941.83,237.743 1943.31,239.633 1943.85,240.332 1944.78,241.523 1946.24,243.413 1946.36,243.576 1947.68,245.302 1948.87,246.86 1949.12,247.192 \n",
" 1950.55,249.082 1951.38,250.185 1951.96,250.971 1953.37,252.861 1953.88,253.55 1954.77,254.751 1956.16,256.64 1956.39,256.959 1957.53,258.53 1958.9,260.411 \n",
" 1958.9,260.42 1960.26,262.309 1961.4,263.909 1961.61,264.199 1962.95,266.089 1963.91,267.453 1964.28,267.978 1965.6,269.868 1966.42,271.046 1966.91,271.758 \n",
" 1968.21,273.647 1968.92,274.688 1969.5,275.537 1970.79,277.427 1971.43,278.381 1972.06,279.317 1973.33,281.206 1973.94,282.128 1974.58,283.096 1975.83,284.986 \n",
" 1976.44,285.93 1977.06,286.875 1978.29,288.765 1978.95,289.788 1979.51,290.655 1980.72,292.544 1981.46,293.706 1981.92,294.434 1983.11,296.324 1983.97,297.685 \n",
" 1984.3,298.213 1985.47,300.103 1986.47,301.727 1986.64,301.993 1987.79,303.882 1988.94,305.772 1988.98,305.836 1990.08,307.662 1991.21,309.552 1991.49,310.015 \n",
" 1992.33,311.441 1993.45,313.331 1993.99,314.266 1994.55,315.221 1995.65,317.11 1996.5,318.591 1996.74,319 1997.82,320.89 1998.89,322.779 1999.01,322.995 \n",
" 1999.95,324.669 2001,326.559 2001.51,327.482 2002.05,328.448 2003.09,330.338 2004.02,332.055 2004.12,332.228 2005.14,334.117 2006.15,336.007 2006.53,336.719 \n",
" 2007.15,337.897 2008.15,339.787 2009.03,341.478 2009.14,341.676 2010.12,343.566 2011.09,345.456 2011.54,346.34 2012.05,347.345 2013.01,349.235 2013.96,351.125 \n",
" 2014.05,351.307 2014.9,353.014 2015.83,354.904 2016.56,356.388 2016.75,356.794 2017.67,358.683 2018.58,360.573 2019.06,361.589 2019.48,362.463 2020.37,364.352 \n",
" 2021.26,366.242 2021.57,366.918 2022.13,368.132 2023,370.022 2023.86,371.911 2024.08,372.384 2024.72,373.801 2025.56,375.691 2026.4,377.58 2026.58,377.996 \n",
" \n",
" \"/>\n",
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" 2026.58,135.564 2024.99,133.811 2024.08,132.817 2023.25,131.921 2021.57,130.093 2021.51,130.031 2019.76,128.142 2019.06,127.393 2018,126.252 2016.56,124.714 \n",
" 2016.22,124.362 2014.44,122.473 2014.05,122.058 2012.65,120.583 2011.54,119.424 2010.84,118.693 2009.03,116.811 2009.03,116.803 2007.2,114.914 2006.53,114.22 \n",
" 2005.36,113.024 2004.02,111.649 2003.52,111.134 2001.66,109.245 2001.51,109.099 1999.79,107.355 1999.01,106.569 1997.91,105.465 1996.5,104.059 1996.02,103.576 \n",
" 1994.11,101.686 1993.99,101.568 1992.2,99.7963 1991.49,99.097 1990.27,97.9066 1988.98,96.6447 1988.33,96.0169 1986.47,94.211 1986.39,94.1272 1984.43,92.2375 \n",
" 1983.97,91.796 1982.45,90.3479 1981.46,89.3991 1980.47,88.4582 \n",
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" 1348.95,88.4582 1349.71,88.6523 1352.22,89.3005 1354.72,89.9571 1356.2,90.3479 1357.23,90.6213 1359.74,91.2929 1362.24,91.973 1363.21,92.2375 1364.75,92.6605 \n",
" 1367.26,93.3558 1369.76,94.0598 1370,94.1272 1372.27,94.7706 1374.78,95.4901 1376.59,96.0169 1377.29,96.2177 1379.79,96.9528 1382.3,97.6967 1383,97.9066 \n",
" 1384.81,98.4481 1387.31,99.2079 1389.23,99.7963 1389.82,99.9762 1392.33,100.752 1394.83,101.537 1395.31,101.686 1397.34,102.329 1399.85,103.131 1401.23,103.576 \n",
" 1402.36,103.94 1404.86,104.758 1407.01,105.465 1407.37,105.585 1409.88,106.42 1412.38,107.264 1412.65,107.355 1414.89,108.115 1417.4,108.977 1418.17,109.245 \n",
" 1419.9,109.846 1422.41,110.724 1423.57,111.134 1424.92,111.611 1427.42,112.507 1428.86,113.024 1429.93,113.412 1432.44,114.325 1434.04,114.914 1434.95,115.248 \n",
" 1437.45,116.18 1439.11,116.803 1439.96,117.12 1442.47,118.07 1444.09,118.693 1444.97,119.029 1447.48,119.997 1448.98,120.583 1449.99,120.975 1452.49,121.962 \n",
" 1453.78,122.473 1455,122.958 1457.51,123.964 1458.49,124.362 1460.01,124.979 1462.52,126.005 1463.12,126.252 1465.03,127.039 1467.54,128.084 1467.67,128.142 \n",
" 1470.04,129.137 1472.15,130.031 1472.55,130.201 1475.06,131.275 1476.55,131.921 1477.56,132.359 1480.07,133.453 1480.88,133.811 1482.58,134.556 1485.08,135.671 \n",
" 1485.15,135.7 1487.59,136.795 1489.35,137.59 1490.1,137.929 1492.6,139.074 1493.49,139.48 1495.11,140.23 1497.56,141.369 1497.62,141.396 1500.13,142.573 \n",
" 1501.58,143.259 1502.63,143.76 1505.14,144.959 1505.53,145.149 1507.65,146.167 1509.44,147.038 1510.15,147.388 1512.66,148.619 1513.29,148.928 1515.17,149.861 \n",
" 1517.08,150.818 1517.67,151.115 1520.18,152.379 1520.83,152.708 1522.69,153.656 1524.52,154.597 1525.2,154.943 1527.7,156.243 1528.17,156.487 1530.21,157.554 \n",
" 1531.77,158.377 1532.72,158.877 1535.22,160.212 1535.32,160.266 1537.73,161.559 1538.83,162.156 1540.24,162.918 1542.3,164.046 1542.74,164.289 1545.25,165.673 \n",
" 1545.72,165.935 1547.76,167.069 1549.11,167.825 1550.26,168.478 1552.45,169.715 1552.77,169.899 1555.28,171.334 1555.75,171.604 1557.79,172.781 1559.01,173.494 \n",
" 1560.29,174.241 1562.24,175.384 1562.8,175.715 1565.31,177.202 1565.43,177.273 1567.81,178.703 1568.58,179.163 1570.32,180.217 1571.69,181.053 1572.83,181.745 \n",
" 1574.78,182.943 1575.33,183.287 1577.82,184.832 1577.84,184.843 1580.35,186.413 1580.84,186.722 1582.85,187.998 1583.82,188.612 1585.36,189.597 1586.77,190.501 \n",
" 1587.87,191.212 1589.69,192.391 1590.38,192.841 1592.57,194.281 1592.88,194.485 1595.39,196.144 1595.43,196.17 1597.9,197.819 1598.25,198.06 1600.4,199.51 \n",
" 1601.05,199.95 1602.91,201.216 1603.82,201.839 1605.42,202.939 1606.56,203.729 1607.92,204.678 1609.27,205.619 1610.43,206.433 1611.96,207.508 1612.94,208.205 \n",
" 1614.61,209.398 1615.45,209.994 1617.24,211.288 1617.95,211.8 1619.85,213.178 1620.46,213.623 1622.43,215.067 1622.97,215.464 1624.98,216.957 1625.47,217.323 \n",
" 1627.51,218.847 1627.98,219.2 1630.01,220.736 1630.49,221.096 1632.49,222.626 1632.99,223.01 1634.95,224.516 1635.5,224.943 1637.38,226.405 1638.01,226.895 \n",
" 1639.79,228.295 1640.51,228.867 1642.18,230.185 1643.02,230.859 1644.54,232.074 1645.53,232.871 1646.88,233.964 1648.04,234.903 1649.2,235.854 1650.54,236.956 \n",
" 1651.5,237.743 1653.05,239.031 1653.77,239.633 1655.56,241.126 1656.03,241.523 1658.06,243.244 1658.26,243.413 1660.47,245.302 1660.57,245.384 1662.67,247.192 \n",
" 1663.08,247.547 1664.84,249.082 1665.58,249.733 1666.99,250.971 1668.09,251.943 1669.13,252.861 1670.6,254.176 1671.24,254.751 1673.1,256.434 1673.33,256.64 \n",
" 1675.41,258.53 1675.61,258.716 1677.46,260.42 1678.12,261.024 1679.5,262.309 1680.63,263.359 1681.52,264.199 1683.13,265.719 1683.52,266.089 1685.51,267.978 \n",
" 1685.64,268.106 1687.47,269.868 1688.15,270.521 1689.42,271.758 1690.65,272.965 1691.35,273.647 1693.16,275.436 1693.26,275.537 1695.16,277.427 1695.67,277.938 \n",
" 1697.04,279.317 1698.17,280.469 1698.9,281.206 1700.68,283.031 1700.74,283.096 1702.57,284.986 1703.19,285.625 1704.39,286.875 1705.69,288.251 1706.18,288.765 \n",
" 1707.96,290.655 1708.2,290.911 1709.73,292.544 1710.71,293.604 1711.48,294.434 1713.21,296.324 1713.22,296.332 1714.93,298.213 1715.72,299.096 1716.63,300.103 \n",
" 1718.23,301.896 1718.32,301.993 1719.99,303.882 1720.74,304.735 1721.65,305.772 1723.24,307.612 1723.29,307.662 1724.92,309.552 1725.75,310.529 1726.53,311.441 \n",
" 1728.13,313.331 1728.26,313.487 1729.71,315.221 1730.76,316.488 1731.28,317.11 1732.84,319 1733.27,319.533 1734.38,320.89 1735.78,322.622 1735.91,322.779 \n",
" 1737.42,324.669 1738.29,325.759 1738.92,326.559 1740.41,328.448 1740.79,328.943 1741.88,330.338 1743.3,332.177 1743.34,332.228 1744.79,334.117 1745.81,335.463 \n",
" 1746.22,336.007 1747.64,337.897 1748.31,338.803 1749.04,339.787 1750.44,341.676 1750.82,342.198 1751.82,343.566 1753.19,345.456 1753.33,345.65 1754.54,347.345 \n",
" 1755.83,349.163 1755.88,349.235 1757.22,351.125 1758.34,352.739 1758.53,353.014 1759.84,354.904 1760.85,356.38 1761.13,356.794 1762.41,358.683 1763.35,360.089 \n",
" 1763.68,360.573 1764.94,362.463 1765.86,363.869 1766.18,364.352 1767.41,366.242 1768.37,367.723 1768.63,368.132 1769.84,370.022 1770.88,371.656 1771.04,371.911 \n",
" 1772.22,373.801 1773.38,375.671 1773.39,375.691 1774.56,377.58 1775.71,379.47 1775.89,379.773 1776.85,381.36 1777.97,383.249 1778.4,383.967 1779.09,385.139 \n",
" 1780.19,387.029 1780.9,388.257 1781.29,388.918 1782.37,390.808 1783.41,392.648 1783.44,392.698 1784.5,394.587 1785.55,396.477 1785.92,397.15 1786.59,398.367 \n",
" 1787.61,400.257 1788.42,401.767 1788.63,402.146 1789.63,404.036 1790.63,405.926 1790.93,406.509 1791.61,407.815 1792.58,409.705 1793.44,411.382 1793.55,411.595 \n",
" 1794.5,413.484 1795.44,415.374 1795.94,416.401 1796.37,417.264 1797.29,419.153 1798.2,421.043 1798.45,421.573 1799.1,422.933 1799.99,424.822 1800.87,426.712 \n",
" 1800.96,426.913 1801.74,428.602 1802.59,430.492 1803.44,432.381 1803.47,432.436 1804.28,434.271 1805.11,436.161 1805.92,438.05 1805.97,438.162 1806.73,439.94 \n",
" 1807.53,441.83 1808.32,443.719 1808.48,444.112 1809.1,445.609 1809.86,447.499 1810.62,449.388 1810.99,450.31 1811.37,451.278 1812.11,453.168 1812.84,455.057 \n",
" 1813.49,456.785 1813.56,456.947 1814.26,458.837 1814.96,460.727 1815.65,462.616 1816,463.58 1816.33,464.506 1817,466.396 1817.67,468.285 1818.32,470.175 \n",
" 1818.51,470.737 1818.96,472.065 1819.59,473.954 1820.21,475.844 1820.83,477.734 1821.01,478.319 1821.43,479.623 1822.03,481.513 1822.61,483.403 1823.19,485.292 \n",
" 1823.52,486.407 1823.75,487.182 1824.31,489.072 1824.86,490.962 1825.4,492.851 1825.93,494.741 1826.03,495.108 1826.45,496.631 1826.96,498.52 1827.46,500.41 \n",
" 1827.95,502.3 1828.44,504.189 1828.54,504.582 1828.91,506.079 1829.38,507.969 1829.83,509.858 1830.28,511.748 1830.72,513.638 1831.04,515.066 1831.15,515.527 \n",
" 1831.57,517.417 1831.98,519.307 1832.38,521.197 1832.77,523.086 1833.16,524.976 1833.53,526.866 1833.55,526.952 1833.9,528.755 1834.26,530.645 1834.61,532.535 \n",
" 1834.94,534.424 1835.28,536.314 1835.6,538.204 1835.91,540.093 1836.06,541.008 1836.21,541.983 1836.51,543.873 1836.8,545.762 1837.07,547.652 1837.34,549.542 \n",
" 1837.6,551.431 1837.85,553.321 1838.09,555.211 1838.33,557.101 1838.55,558.99 1838.56,559.087 1838.77,560.88 1838.98,562.77 1839.17,564.659 1839.36,566.549 \n",
" 1839.55,568.439 1839.72,570.328 1839.88,572.218 1840.04,574.108 1840.18,575.997 1840.32,577.887 1840.45,579.777 1840.57,581.666 1840.68,583.556 1840.78,585.446 \n",
" 1840.88,587.336 1840.96,589.225 1841.04,591.115 1841.07,592.031 1841.1,593.005 1841.16,594.894 1841.21,596.784 1841.26,598.674 1841.29,600.563 1841.31,602.453 \n",
" 1841.33,604.343 1841.34,606.232 1841.34,608.122 1841.33,610.012 1841.31,611.901 1841.28,613.791 1841.24,615.681 1841.2,617.571 1841.14,619.46 1841.08,621.35 \n",
" 1841.07,621.658 1841.01,623.24 1840.93,625.129 1840.84,627.019 1840.75,628.909 1840.64,630.798 1840.53,632.688 1840.4,634.578 1840.27,636.467 1840.13,638.357 \n",
" 1839.98,640.247 1839.82,642.136 1839.65,644.026 1839.48,645.916 1839.29,647.806 1839.1,649.695 1838.9,651.585 1838.69,653.475 1838.56,654.545 1838.47,655.364 \n",
" 1838.24,657.254 1838,659.144 1837.76,661.033 1837.5,662.923 1837.24,664.813 1836.97,666.702 1836.68,668.592 1836.39,670.482 1836.09,672.371 1836.06,672.599 \n",
" 1835.79,674.261 1835.47,676.151 1835.14,678.041 1834.81,679.93 1834.46,681.82 1834.11,683.71 1833.75,685.599 1833.55,686.611 1833.38,687.489 1833,689.379 \n",
" 1832.61,691.268 1832.21,693.158 1831.8,695.048 1831.39,696.937 1831.04,698.466 1830.96,698.827 1830.53,700.717 1830.08,702.606 1829.63,704.496 1829.17,706.386 \n",
" 1828.7,708.276 1828.54,708.912 1828.22,710.165 1827.73,712.055 1827.23,713.945 1826.72,715.834 1826.2,717.724 1826.03,718.348 1825.68,719.614 1825.14,721.503 \n",
" 1824.59,723.393 1824.04,725.283 1823.52,727.015 1823.47,727.172 1822.9,729.062 1822.32,730.952 1821.73,732.841 1821.12,734.731 1821.01,735.064 1820.51,736.621 \n",
" 1819.89,738.511 1819.26,740.4 1818.62,742.29 1818.51,742.609 1817.97,744.18 1817.31,746.069 1816.64,747.959 1816,749.729 1815.96,749.849 1815.27,751.738 \n",
" 1814.57,753.628 1813.86,755.518 1813.49,756.481 1813.14,757.407 1812.41,759.297 1811.67,761.187 1810.99,762.921 1810.92,763.076 1810.17,764.966 1809.4,766.856 \n",
" 1808.62,768.746 1808.48,769.078 1807.83,770.635 1807.03,772.525 1806.22,774.415 1805.97,774.987 1805.4,776.304 1804.57,778.194 1803.73,780.084 1803.47,780.673 \n",
" 1802.88,781.973 1802.02,783.863 1801.15,785.753 1800.96,786.158 1800.27,787.642 1799.37,789.532 1798.47,791.422 1798.45,791.46 1797.56,793.311 1796.63,795.201 \n",
" 1795.94,796.59 1795.7,797.091 1794.75,798.98 1793.79,800.87 1793.44,801.567 1792.83,802.76 1791.85,804.65 1790.93,806.401 1790.86,806.539 1789.86,808.429 \n",
" 1788.85,810.319 1788.42,811.1 1787.82,812.208 1786.79,814.098 1785.92,815.677 1785.74,815.988 1784.69,817.877 1783.62,819.767 1783.41,820.136 1782.54,821.657 \n",
" 1781.45,823.546 1780.9,824.486 1780.35,825.436 1779.23,827.326 1778.4,828.733 1778.11,829.215 1776.97,831.105 1775.89,832.885 1775.82,832.995 1774.66,834.885 \n",
" 1773.49,836.774 1773.38,836.944 1772.3,838.664 1771.11,840.554 1770.88,840.916 1769.9,842.443 1768.68,844.333 1768.37,844.805 1767.44,846.223 1766.2,848.112 \n",
" 1765.86,848.616 1764.94,850.002 1763.67,851.892 1763.35,852.353 1762.38,853.781 1761.09,855.671 1760.85,856.018 1759.78,857.561 1758.46,859.45 1758.34,859.615 \n",
" 1757.12,861.34 1755.83,863.146 1755.77,863.23 1754.41,865.12 1753.33,866.614 1753.04,867.009 1751.65,868.899 1750.82,870.022 1750.25,870.789 1748.84,872.678 \n",
" 1748.31,873.372 1747.41,874.568 1745.97,876.458 1745.81,876.667 1744.51,878.347 1743.3,879.907 1743.04,880.237 1741.56,882.127 1740.79,883.095 1740.06,884.016 \n",
" 1738.55,885.906 1738.29,886.234 1737.02,887.796 1735.78,889.324 1735.48,889.685 1733.93,891.575 1733.27,892.368 1732.36,893.465 1730.77,895.355 1730.76,895.366 \n",
" 1729.18,897.244 1728.26,898.32 1727.56,899.134 1725.93,901.024 1725.75,901.232 1724.29,902.913 1723.24,904.102 1722.63,904.803 1720.95,906.693 1720.74,906.932 \n",
" 1719.26,908.582 1718.23,909.723 1717.55,910.472 1715.83,912.362 1715.72,912.476 1714.09,914.251 1713.22,915.192 1712.33,916.141 1710.71,917.873 1710.56,918.031 \n",
" 1708.77,919.92 1708.2,920.518 1706.97,921.81 1705.69,923.129 1705.14,923.7 1703.3,925.59 1703.19,925.708 1701.45,927.479 1700.68,928.253 1699.57,929.369 \n",
" 1698.17,930.767 1697.68,931.259 1695.77,933.148 1695.67,933.25 1693.84,935.038 1693.16,935.703 1691.9,936.928 1690.65,938.126 1689.93,938.817 1688.15,940.52 \n",
" 1687.95,940.707 1685.95,942.597 1685.64,942.886 1683.93,944.486 1683.13,945.224 1681.89,946.376 1680.63,947.535 1679.83,948.266 1678.12,949.82 1677.75,950.155 \n",
" 1675.65,952.045 1675.61,952.079 1673.53,953.935 1673.1,954.312 1671.39,955.825 1670.6,956.52 1669.23,957.714 1668.09,958.704 1667.05,959.604 1665.58,960.864 \n",
" 1664.85,961.494 1663.08,963 1662.62,963.383 1660.57,965.113 1660.38,965.273 1658.11,967.163 1658.06,967.203 1655.82,969.052 1655.56,969.27 1653.51,970.942 \n",
" 1653.05,971.316 1651.17,972.832 1650.54,973.34 1648.82,974.721 1648.04,975.343 1646.44,976.611 1645.53,977.326 1644.03,978.501 1643.02,979.287 1641.6,980.39 \n",
" 1640.51,981.228 1639.15,982.28 1638.01,983.15 1636.67,984.17 1635.5,985.052 1634.16,986.06 1632.99,986.934 1631.63,987.949 1630.49,988.798 1629.08,989.839 \n",
" 1627.98,990.643 1626.49,991.729 1625.47,992.47 1623.88,993.618 1622.97,994.278 1621.25,995.508 1620.46,996.069 1618.58,997.398 1617.95,997.841 1615.89,999.287 \n",
" 1615.45,999.597 1613.17,1001.18 1612.94,1001.34 1610.43,1003.06 1610.42,1003.07 1607.92,1004.76 1607.64,1004.96 1605.42,1006.45 1604.82,1006.85 1602.91,1008.12 \n",
" 1601.98,1008.74 1600.4,1009.78 1599.11,1010.63 1597.9,1011.42 1596.21,1012.52 1595.39,1013.04 1593.27,1014.4 1592.88,1014.65 1590.38,1016.25 1590.3,1016.29 \n",
" 1587.87,1017.82 1587.29,1018.18 1585.36,1019.39 1584.25,1020.07 1582.85,1020.94 1581.18,1021.96 1580.35,1022.47 1578.07,1023.85 1577.84,1023.99 1575.33,1025.5 \n",
" 1574.92,1025.74 1572.83,1026.99 1571.74,1027.63 1570.32,1028.47 1568.52,1029.52 1567.81,1029.93 1565.31,1031.38 1565.26,1031.41 1562.8,1032.82 1561.95,1033.3 \n",
" 1560.29,1034.25 1558.61,1035.19 1557.79,1035.66 1555.28,1037.05 1555.23,1037.08 1552.77,1038.44 1551.8,1038.97 1550.26,1039.81 1548.33,1040.86 1547.76,1041.17 \n",
" 1545.25,1042.52 1544.81,1042.75 1542.74,1043.85 1541.25,1044.64 1540.24,1045.17 1537.73,1046.48 1537.64,1046.53 1535.22,1047.78 1533.99,1048.42 1532.72,1049.07 \n",
" 1530.28,1050.31 1530.21,1050.34 1527.7,1051.61 1526.52,1052.2 1525.2,1052.86 1522.71,1054.09 1522.69,1054.1 1520.18,1055.33 1518.84,1055.98 1517.67,1056.54 \n",
" 1515.17,1057.75 1514.92,1057.87 1512.66,1058.94 1510.94,1059.76 1510.15,1060.13 1507.65,1061.3 1506.9,1061.65 1505.14,1062.46 1502.8,1063.54 1502.63,1063.61 \n",
" 1500.13,1064.76 1498.64,1065.43 1497.62,1065.89 1495.11,1067 1494.41,1067.32 1492.6,1068.11 1490.11,1069.21 1490.1,1069.21 1487.59,1070.3 1485.75,1071.1 \n",
" 1485.08,1071.38 1482.58,1072.45 1481.31,1072.99 1480.07,1073.51 1477.56,1074.56 1476.8,1074.87 1475.06,1075.6 1472.55,1076.62 1472.2,1076.76 1470.04,1077.64 \n",
" 1467.54,1078.65 1467.53,1078.65 1465.03,1079.65 1462.77,1080.54 1462.52,1080.64 1460.01,1081.63 1457.93,1082.43 1457.51,1082.6 1455,1083.56 1452.99,1084.32 \n",
" 1452.49,1084.51 1449.99,1085.46 1447.96,1086.21 1447.48,1086.39 1444.97,1087.32 1442.82,1088.1 1442.47,1088.23 1439.96,1089.14 1437.58,1089.99 1437.45,1090.04 \n",
" 1434.95,1090.93 1432.44,1091.81 1432.23,1091.88 1429.93,1092.68 1427.42,1093.55 1426.77,1093.77 1424.92,1094.4 1422.41,1095.25 1421.18,1095.66 1419.9,1096.09 \n",
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" 1399.85,1102.49 1397.45,1103.22 1397.34,1103.25 1394.83,1104.01 1392.33,1104.76 1391.12,1105.11 1389.82,1105.49 1387.31,1106.23 1384.81,1106.95 1384.63,1107 \n",
" 1382.3,1107.66 1379.79,1108.37 1377.94,1108.89 1377.29,1109.07 1374.78,1109.76 1372.27,1110.45 1371.04,1110.78 1369.76,1111.12 1367.26,1111.79 1364.75,1112.45 \n",
" 1363.93,1112.67 1362.24,1113.11 1359.74,1113.75 1357.23,1114.39 1356.57,1114.56 1354.72,1115.02 1352.22,1115.65 1349.71,1116.26 1348.94,1116.45 1347.2,1116.87 \n",
" 1344.7,1117.47 1342.19,1118.07 1341.03,1118.34 1339.68,1118.65 1337.17,1119.23 1334.67,1119.8 1332.79,1120.23 1332.16,1120.37 1329.65,1120.93 1327.15,1121.48 \n",
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" 1244.42,1135.76 1241.91,1136.08 1239.4,1136.39 1236.9,1136.7 1234.39,1137 1232.37,1137.23 1231.88,1137.29 1229.38,1137.58 1226.87,1137.86 1224.36,1138.13 \n",
" \n",
" \"/>\n",
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" 672.835,1035.27 672.691,1035.19 670.328,1033.86 669.35,1033.3 667.821,1032.43 666.048,1031.41 665.314,1030.99 662.807,1029.53 662.786,1029.52 660.3,1028.07 \n",
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" 618.137,1001.18 617.682,1000.86 615.415,999.287 615.175,999.12 612.722,997.398 612.668,997.36 610.162,995.582 610.058,995.508 607.655,993.787 607.421,993.618 \n",
" 605.148,991.973 604.811,991.729 602.641,990.142 602.229,989.839 600.134,988.292 599.672,987.949 597.627,986.423 597.142,986.06 595.12,984.535 594.638,984.17 \n",
" 592.613,982.628 592.159,982.28 590.106,980.701 589.704,980.39 587.599,978.754 587.275,978.501 585.092,976.787 584.869,976.611 582.585,974.799 582.487,974.721 \n",
" 580.129,972.832 580.078,972.791 577.794,970.942 577.571,970.76 575.482,969.052 575.064,968.709 573.193,967.163 572.557,966.635 570.925,965.273 570.05,964.539 \n",
" 568.68,963.383 567.544,962.419 566.456,961.494 565.037,960.277 564.254,959.604 562.53,958.111 562.073,957.714 560.023,955.921 559.913,955.825 557.774,953.935 \n",
" 557.516,953.706 555.654,952.045 555.009,951.465 553.555,950.155 552.502,949.2 551.476,948.266 549.995,946.908 549.417,946.376 547.488,944.589 547.377,944.486 \n",
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" 534.953,932.576 533.623,931.259 532.446,930.084 531.731,929.369 529.939,927.562 529.857,927.479 528,925.59 527.432,925.008 526.16,923.7 524.926,922.421 \n",
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" 513.752,910.472 512.391,908.966 512.045,908.582 510.354,906.693 509.884,906.164 508.678,904.803 507.377,903.323 507.018,902.913 505.373,901.024 504.87,900.442 \n",
" 503.743,899.134 502.363,897.519 502.129,897.244 500.529,895.355 499.856,894.552 498.945,893.465 497.376,891.575 497.349,891.543 495.82,889.685 494.842,888.486 \n",
" 494.28,887.796 492.754,885.906 492.335,885.383 491.242,884.016 489.828,882.231 489.746,882.127 488.262,880.237 487.321,879.028 486.793,878.347 485.337,876.458 \n",
" 484.814,875.773 483.896,874.568 482.468,872.678 482.308,872.464 481.053,870.789 479.801,869.098 479.653,868.899 478.265,867.009 477.294,865.674 476.891,865.12 \n",
" 475.53,863.23 474.787,862.188 474.182,861.34 472.847,859.45 472.28,858.64 471.525,857.561 470.216,855.671 469.773,855.025 468.92,853.781 467.637,851.892 \n",
" 467.266,851.341 466.365,850.002 465.107,848.112 464.759,847.584 463.861,846.223 462.628,844.333 462.252,843.752 461.406,842.443 460.197,840.554 459.745,839.84 \n",
" 459,838.664 457.815,836.774 457.238,835.845 456.642,834.885 455.481,832.995 454.731,831.761 454.332,831.105 453.195,829.215 452.224,827.585 452.07,827.326 \n",
" 450.955,825.436 449.854,823.546 449.717,823.31 448.762,821.657 447.683,819.767 447.21,818.93 446.615,817.877 445.559,815.988 444.703,814.44 444.514,814.098 \n",
" 443.48,812.208 442.457,810.319 442.196,809.831 441.445,808.429 440.445,806.539 439.69,805.096 439.456,804.65 438.477,802.76 437.51,800.87 437.183,800.224 \n",
" 436.553,798.98 435.606,797.091 434.676,795.208 434.672,795.201 433.747,793.311 432.833,791.422 432.169,790.03 431.931,789.532 431.037,787.642 430.156,785.753 \n",
" 429.662,784.681 429.284,783.863 428.423,781.973 427.573,780.084 427.155,779.143 426.733,778.194 425.902,776.304 425.083,774.415 424.648,773.398 424.274,772.525 \n",
" 423.474,770.635 422.685,768.746 422.141,767.424 421.907,766.856 421.137,764.966 420.379,763.076 419.634,761.195 419.631,761.187 418.891,759.297 418.162,757.407 \n",
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" 647.463,923.7 647.766,923.944 649.817,925.59 650.273,925.953 652.198,927.479 652.78,927.937 654.608,929.369 655.286,929.897 657.047,931.259 657.793,931.833 \n",
" 659.514,933.148 660.3,933.746 662.012,935.038 662.807,935.635 664.539,936.928 665.314,937.503 667.098,938.817 667.821,939.348 669.687,940.707 670.328,941.171 \n",
" 672.309,942.597 672.835,942.973 674.963,944.486 675.342,944.754 677.651,946.376 677.849,946.515 680.356,948.255 680.372,948.266 682.863,949.975 683.128,950.155 \n",
" 685.37,951.675 685.919,952.045 687.877,953.356 688.747,953.935 690.384,955.018 691.611,955.825 692.891,956.661 694.513,957.714 695.398,958.286 697.453,959.604 \n",
" 697.904,959.892 700.411,961.481 700.432,961.494 702.918,963.052 703.452,963.383 705.425,964.605 706.513,965.273 707.932,966.141 709.616,967.163 710.439,967.66 \n",
" 712.762,969.052 712.946,969.162 715.453,970.648 715.952,970.942 717.96,972.118 719.189,972.832 720.467,973.572 722.471,974.721 722.974,975.009 725.481,976.431 \n",
" 725.801,976.611 727.988,977.838 729.181,978.501 730.495,979.229 732.61,980.39 733.002,980.605 735.509,981.966 736.092,982.28 738.016,983.312 739.628,984.17 \n",
" 740.522,984.644 743.029,985.961 743.218,986.06 745.536,987.265 746.865,987.949 748.043,988.554 750.55,989.829 750.57,989.839 753.057,991.091 754.337,991.729 \n",
" 755.564,992.339 758.071,993.573 758.164,993.618 760.578,994.794 762.058,995.508 763.085,996.002 765.592,997.197 766.017,997.398 768.099,998.379 770.045,999.287 \n",
" 770.606,999.548 773.113,1000.71 774.146,1001.18 775.62,1001.85 778.127,1002.98 778.319,1003.07 780.634,1004.1 782.571,1004.96 783.14,1005.21 785.647,1006.3 \n",
" 786.903,1006.85 788.154,1007.39 790.661,1008.46 791.318,1008.74 793.168,1009.52 795.675,1010.57 795.819,1010.63 798.182,1011.6 800.412,1012.52 800.689,1012.63 \n",
" 803.196,1013.64 805.101,1014.4 805.703,1014.65 808.21,1015.64 809.888,1016.29 810.717,1016.62 813.224,1017.59 814.78,1018.18 815.731,1018.55 818.238,1019.5 \n",
" 819.782,1020.07 820.745,1020.43 823.252,1021.36 824.899,1021.96 825.758,1022.28 828.265,1023.18 830.137,1023.85 830.772,1024.08 833.279,1024.97 835.503,1025.74 \n",
" 835.786,1025.84 838.293,1026.71 840.8,1027.56 841.006,1027.63 843.307,1028.41 845.814,1029.25 846.655,1029.52 848.321,1030.07 850.828,1030.89 852.457,1031.41 \n",
" 853.335,1031.69 855.842,1032.49 858.349,1033.28 858.422,1033.3 860.856,1034.06 863.363,1034.83 864.567,1035.19 865.87,1035.59 868.376,1036.34 870.883,1037.08 \n",
" 870.898,1037.08 873.39,1037.81 875.897,1038.53 877.439,1038.97 878.404,1039.25 880.911,1039.95 883.418,1040.65 884.2,1040.86 885.925,1041.33 888.432,1042.01 \n",
" 890.939,1042.68 891.203,1042.75 893.446,1043.34 895.953,1043.99 898.46,1044.64 898.473,1044.64 900.967,1045.27 903.474,1045.9 905.981,1046.52 906.039,1046.53 \n",
" 908.488,1047.13 910.994,1047.73 913.501,1048.32 913.934,1048.42 916.008,1048.9 918.515,1049.48 921.022,1050.05 922.197,1050.31 923.529,1050.61 926.036,1051.16 \n",
" 928.543,1051.7 930.874,1052.2 931.05,1052.24 933.557,1052.76 936.064,1053.28 938.571,1053.79 940.036,1054.09 941.078,1054.3 943.585,1054.79 946.092,1055.28 \n",
" 948.599,1055.76 949.755,1055.98 951.106,1056.23 953.613,1056.7 956.119,1057.15 958.626,1057.6 960.138,1057.87 961.133,1058.04 963.64,1058.48 966.147,1058.9 \n",
" 968.654,1059.32 971.161,1059.73 971.326,1059.76 973.668,1060.13 976.175,1060.53 978.682,1060.92 981.189,1061.3 983.531,1061.65 983.696,1061.67 986.203,1062.04 \n",
" 988.71,1062.4 991.217,1062.75 993.724,1063.09 996.231,1063.43 997.067,1063.54 998.737,1063.76 1001.24,1064.08 1003.75,1064.39 1006.26,1064.7 1008.77,1065 \n",
" 1011.27,1065.29 1012.45,1065.43 1013.78,1065.58 1016.29,1065.86 1018.79,1066.13 1021.3,1066.39 1023.81,1066.65 1026.31,1066.9 1028.82,1067.14 1030.67,1067.32 \n",
" 1031.33,1067.38 1033.83,1067.61 1036.34,1067.83 1038.85,1068.04 1041.36,1068.25 1043.86,1068.45 1046.37,1068.65 1048.88,1068.83 1051.38,1069.01 1053.89,1069.18 \n",
" 1054.22,1069.21 1056.4,1069.35 1058.9,1069.51 1061.41,1069.66 1063.92,1069.81 1066.42,1069.94 1068.93,1070.08 1071.44,1070.2 1073.95,1070.32 1076.45,1070.43 \n",
" 1078.96,1070.53 1081.47,1070.63 1083.97,1070.72 1086.48,1070.8 1088.99,1070.88 1091.49,1070.95 1094,1071.01 1096.51,1071.06 1098.14,1071.1 1099.02,1071.11 \n",
" 1101.52,1071.15 1104.03,1071.19 1106.54,1071.22 1109.04,1071.24 1111.55,1071.25 1114.06,1071.26 1116.56,1071.26 1119.07,1071.26 1121.58,1071.24 1124.08,1071.22 \n",
" 1126.59,1071.2 1129.1,1071.16 1131.61,1071.12 1133.17,1071.1 1134.11,1071.08 1136.62,1071.02 1139.13,1070.96 1141.63,1070.9 1144.14,1070.82 1146.65,1070.74 \n",
" 1149.15,1070.65 1151.66,1070.56 1154.17,1070.46 1156.67,1070.35 1159.18,1070.23 1161.69,1070.11 1164.2,1069.98 1166.7,1069.84 1169.21,1069.7 1171.72,1069.55 \n",
" 1174.22,1069.39 1176.73,1069.23 1177.09,1069.21 1179.24,1069.06 1181.74,1068.88 1184.25,1068.7 1186.76,1068.51 1189.27,1068.31 1191.77,1068.1 1194.28,1067.89 \n",
" 1196.79,1067.67 1199.29,1067.44 1200.63,1067.32 1201.8,1067.21 1204.31,1066.97 1206.81,1066.72 1209.32,1066.46 1211.83,1066.2 1214.33,1065.93 1216.84,1065.66 \n",
" 1218.86,1065.43 1219.35,1065.37 1221.86,1065.08 1224.36,1064.78 1226.87,1064.48 1229.38,1064.16 1231.88,1063.84 1234.24,1063.54 1234.39,1063.52 1236.9,1063.18 \n",
" 1239.4,1062.84 1241.91,1062.49 1244.42,1062.14 1246.92,1061.77 1247.77,1061.65 1249.43,1061.4 1251.94,1061.02 1254.45,1060.64 1256.95,1060.24 1259.46,1059.84 \n",
" 1259.98,1059.76 1261.97,1059.43 1264.47,1059.02 1266.98,1058.59 1269.49,1058.16 1271.17,1057.87 1271.99,1057.72 1274.5,1057.28 1277.01,1056.82 1279.51,1056.36 \n",
" 1281.55,1055.98 1282.02,1055.89 1284.53,1055.41 1287.04,1054.93 1289.54,1054.43 1291.27,1054.09 1292.05,1053.93 1294.56,1053.42 1297.06,1052.91 1299.57,1052.38 \n",
" 1300.43,1052.2 1302.08,1051.85 1304.58,1051.31 1307.09,1050.76 1309.11,1050.31 1309.6,1050.2 1312.11,1049.64 1314.61,1049.06 1317.12,1048.48 1317.37,1048.42 \n",
" 1319.63,1047.89 1322.13,1047.29 1324.64,1046.68 1325.26,1046.53 1327.15,1046.07 1329.65,1045.44 1332.16,1044.81 1332.83,1044.64 1334.67,1044.17 1337.17,1043.52 \n",
" 1339.68,1042.86 1340.1,1042.75 1342.19,1042.2 1344.7,1041.52 1347.11,1040.86 1347.2,1040.83 1349.71,1040.14 1352.22,1039.44 1353.87,1038.97 1354.72,1038.73 \n",
" 1357.23,1038.01 1359.74,1037.28 1360.4,1037.08 1362.24,1036.54 1364.75,1035.79 1366.74,1035.19 1367.26,1035.03 1369.76,1034.27 1372.27,1033.49 1372.88,1033.3 \n",
" 1374.78,1032.71 1377.29,1031.91 1378.85,1031.41 1379.79,1031.11 1382.3,1030.3 1384.65,1029.52 1384.81,1029.47 1387.31,1028.64 1389.82,1027.79 1390.3,1027.63 \n",
" 1392.33,1026.94 1394.83,1026.08 1395.8,1025.74 1397.34,1025.21 1399.85,1024.32 1401.17,1023.85 1402.36,1023.43 1404.86,1022.53 1406.41,1021.96 1407.37,1021.61 \n",
" 1409.88,1020.69 1411.52,1020.07 1412.38,1019.75 1414.89,1018.81 1416.52,1018.18 1417.4,1017.85 1419.9,1016.88 1421.42,1016.29 1422.41,1015.91 1424.92,1014.92 \n",
" 1426.2,1014.4 1427.42,1013.92 1429.93,1012.91 1430.89,1012.52 1432.44,1011.88 1434.95,1010.85 1435.48,1010.63 1437.45,1009.81 1439.96,1008.75 1439.99,1008.74 \n",
" 1442.47,1007.68 1444.4,1006.85 1444.97,1006.6 1447.48,1005.51 1448.73,1004.96 1449.99,1004.4 1452.49,1003.29 1452.98,1003.07 1455,1002.16 1457.16,1001.18 \n",
" 1457.51,1001.02 1460.01,999.865 1461.26,999.287 1462.52,998.699 1465.03,997.52 1465.29,997.398 1467.54,996.329 1469.25,995.508 1470.04,995.125 1472.55,993.907 \n",
" 1473.14,993.618 1475.06,992.677 1476.97,991.729 1477.56,991.432 1480.07,990.175 1480.73,989.839 1482.58,988.903 1484.44,987.949 1485.08,987.618 1487.59,986.318 \n",
" 1488.09,986.06 1490.1,985.005 1491.68,984.17 1492.6,983.677 1495.11,982.334 1495.21,982.28 1497.62,980.978 1498.69,980.39 1500.13,979.606 1502.12,978.501 \n",
" 1502.63,978.218 1505.14,976.816 1505.5,976.611 1507.65,975.399 1508.83,974.721 1510.15,973.965 1512.12,972.832 1512.66,972.516 1515.17,971.051 1515.35,970.942 \n",
" 1517.67,969.57 1518.54,969.052 1520.18,968.072 1521.69,967.163 1522.69,966.557 1524.79,965.273 1525.2,965.026 1527.7,963.477 1527.85,963.383 1530.21,961.911 \n",
" 1530.87,961.494 1532.72,960.328 1533.85,959.604 1535.22,958.726 1536.79,957.714 1537.73,957.106 1539.69,955.825 1540.24,955.468 1542.56,953.935 1542.74,953.811 \n",
" 1545.25,952.135 1545.38,952.045 1547.76,950.44 1548.18,950.155 1550.26,948.726 1550.93,948.266 1552.77,946.991 1553.65,946.376 1555.28,945.236 1556.34,944.486 \n",
" 1557.79,943.461 1559,942.597 1560.29,941.665 1561.62,940.707 1562.8,939.847 1564.21,938.817 1565.31,938.008 1566.76,936.928 1567.81,936.147 1569.29,935.038 \n",
" 1570.32,934.263 1571.79,933.148 1572.83,932.357 1574.26,931.259 1575.33,930.427 1576.7,929.369 1577.84,928.474 1579.11,927.479 1580.35,926.496 1581.49,925.59 \n",
" 1582.85,924.494 1583.84,923.7 1585.36,922.467 1586.17,921.81 1587.87,920.414 1588.47,919.92 1590.38,918.336 1590.74,918.031 1592.88,916.23 1592.99,916.141 \n",
" 1595.21,914.251 1595.39,914.098 1597.41,912.362 1597.9,911.937 1599.58,910.472 1600.4,909.748 1601.73,908.582 1602.91,907.53 1603.85,906.693 1605.42,905.283 \n",
" 1605.95,904.803 1607.92,903.005 1608.02,902.913 1610.08,901.024 1610.43,900.695 1612.11,899.134 1612.94,898.354 1614.12,897.244 1615.45,895.98 1616.1,895.355 \n",
" 1617.95,893.573 1618.06,893.465 1620.01,891.575 1620.46,891.132 1621.93,889.685 1622.97,888.654 1623.83,887.796 1625.47,886.142 1625.71,885.906 1627.56,884.016 \n",
" 1627.98,883.591 1629.4,882.127 1630.49,881.002 1631.22,880.237 1632.99,878.374 1633.02,878.347 1634.8,876.458 1635.5,875.705 1636.56,874.568 1638.01,872.995 \n",
" 1638.3,872.678 1640.02,870.789 1640.51,870.241 1641.72,868.899 1643.02,867.443 1643.41,867.009 1645.07,865.12 1645.53,864.599 1646.72,863.23 1648.04,861.707 \n",
" 1648.35,861.34 1649.96,859.45 1650.54,858.766 1651.56,857.561 1653.05,855.775 1653.14,855.671 1654.7,853.781 1655.56,852.729 1656.24,851.892 1657.77,850.002 \n",
" 1658.06,849.63 1659.28,848.112 1660.57,846.474 1660.77,846.223 1662.25,844.333 1663.08,843.257 1663.71,842.443 1665.15,840.554 1665.58,839.98 1666.58,838.664 \n",
" 1667.99,836.774 1668.09,836.639 1669.39,834.885 1670.6,833.229 1670.77,832.995 1672.14,831.105 1673.1,829.749 1673.49,829.215 1674.82,827.326 1675.61,826.196 \n",
" 1676.14,825.436 1677.45,823.546 1678.12,822.565 1678.74,821.657 1680.01,819.767 1680.63,818.853 1681.28,817.877 1682.52,815.988 1683.13,815.055 1683.76,814.098 \n",
" \n",
" \"/>\n",
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" 1683.76,814.098 1684.98,812.208 1685.64,811.166 1686.18,810.319 1687.37,808.429 1688.15,807.182 1688.55,806.539 1689.71,804.65 1690.65,803.096 1690.86,802.76 \n",
" 1691.99,800.87 1693.11,798.98 1693.16,798.902 1694.22,797.091 1695.32,795.201 1695.67,794.591 1696.4,793.311 1697.47,791.422 1698.17,790.157 1698.52,789.532 \n",
" 1699.57,787.642 1700.59,785.753 1700.68,785.591 1701.61,783.863 1702.61,781.973 1703.19,780.879 1703.6,780.084 1704.58,778.194 1705.55,776.304 1705.69,776.013 \n",
" 1706.5,774.415 1707.44,772.525 1708.2,770.977 1708.37,770.635 1709.29,768.746 1710.19,766.856 1710.71,765.755 1711.08,764.966 1711.96,763.076 1712.83,761.187 \n",
" 1713.22,760.328 1713.68,759.297 1714.53,757.407 1715.36,755.518 1715.72,754.674 1716.18,753.628 1716.98,751.738 1717.78,749.849 1718.23,748.765 1718.56,747.959 \n",
" 1719.34,746.069 1720.1,744.18 1720.74,742.568 1720.85,742.29 1721.59,740.4 1722.31,738.511 1723.03,736.621 1723.24,736.04 1723.73,734.731 1724.42,732.841 \n",
" 1725.11,730.952 1725.75,729.131 1725.77,729.062 1726.43,727.172 1727.08,725.283 1727.72,723.393 1728.26,721.765 1728.34,721.503 1728.96,719.614 1729.56,717.724 \n",
" 1730.16,715.834 1730.74,713.945 1730.76,713.855 1731.31,712.055 1731.87,710.165 1732.42,708.276 1732.96,706.386 1733.27,705.259 1733.48,704.496 1734,702.606 \n",
" 1734.51,700.717 1735,698.827 1735.49,696.937 1735.78,695.787 1735.96,695.048 1736.43,693.158 1736.88,691.268 1737.33,689.379 1737.76,687.489 1738.18,685.599 \n",
" 1738.29,685.114 1738.59,683.71 1738.99,681.82 1739.38,679.93 1739.77,678.041 1740.14,676.151 1740.49,674.261 1740.79,672.651 1740.84,672.371 1741.18,670.482 \n",
" 1741.51,668.592 1741.83,666.702 1742.14,664.813 1742.44,662.923 1742.73,661.033 1743,659.144 1743.27,657.254 1743.3,657.058 1743.53,655.364 1743.78,653.475 \n",
" 1744.02,651.585 1744.24,649.695 1744.46,647.806 1744.67,645.916 1744.87,644.026 1745.05,642.136 1745.23,640.247 1745.4,638.357 1745.55,636.467 1745.7,634.578 \n",
" 1745.81,633.114 1745.84,632.688 1745.96,630.798 1746.08,628.909 1746.19,627.019 1746.29,625.129 1746.37,623.24 1746.45,621.35 1746.52,619.46 1746.57,617.571 \n",
" 1746.62,615.681 1746.66,613.791 1746.69,611.901 1746.7,610.012 1746.71,608.122 1746.71,606.232 1746.7,604.343 1746.67,602.453 1746.64,600.563 1746.6,598.674 \n",
" 1746.55,596.784 1746.49,594.894 1746.41,593.005 1746.33,591.115 1746.24,589.225 1746.14,587.336 1746.03,585.446 1745.9,583.556 1745.81,582.149 1745.77,581.666 \n",
" 1745.63,579.777 1745.48,577.887 1745.32,575.997 1745.15,574.108 1744.97,572.218 1744.77,570.328 1744.57,568.439 1744.36,566.549 1744.14,564.659 1743.91,562.77 \n",
" 1743.66,560.88 1743.41,558.99 1743.3,558.174 1743.15,557.101 1742.88,555.211 1742.6,553.321 1742.3,551.431 1742,549.542 1741.69,547.652 1741.37,545.762 \n",
" 1741.03,543.873 1740.79,542.543 1740.69,541.983 1740.34,540.093 1739.97,538.204 1739.6,536.314 1739.22,534.424 1738.82,532.535 1738.42,530.645 1738.29,530.042 \n",
" 1738,528.755 1737.58,526.866 1737.14,524.976 1736.7,523.086 1736.24,521.197 1735.78,519.328 1735.77,519.307 1735.3,517.417 1734.81,515.527 1734.31,513.638 \n",
" 1733.8,511.748 1733.28,509.858 1733.27,509.819 1732.75,507.969 1732.21,506.079 1731.66,504.189 1731.1,502.3 1730.76,501.191 1730.53,500.41 1729.95,498.52 \n",
" 1729.35,496.631 1728.75,494.741 1728.26,493.239 1728.13,492.851 1727.51,490.962 1726.87,489.072 1726.22,487.182 1725.75,485.836 1725.56,485.292 1724.89,483.403 \n",
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" 572.682,668.592 573.053,670.482 573.436,672.371 573.83,674.261 574.237,676.151 574.656,678.041 575.064,679.829 575.087,679.93 575.529,681.82 575.983,683.71 \n",
" 576.449,685.599 576.928,687.489 577.419,689.379 577.571,689.95 577.922,691.268 578.436,693.158 578.963,695.048 579.502,696.937 580.055,698.827 580.078,698.906 \n",
" 580.618,700.717 581.193,702.606 581.782,704.496 582.384,706.386 582.585,707.005 582.997,708.276 583.623,710.165 584.262,712.055 584.914,713.945 585.092,714.451 \n",
" 585.578,715.834 586.254,717.724 586.944,719.614 587.599,721.372 587.648,721.503 588.362,723.393 589.091,725.283 589.833,727.172 590.106,727.855 590.588,729.062 \n",
" 591.355,730.952 592.137,732.841 592.613,733.972 592.932,734.731 593.739,736.621 594.561,738.511 595.12,739.774 595.397,740.4 596.245,742.29 597.107,744.18 \n",
" 597.627,745.299 597.984,746.069 598.873,747.959 599.778,749.849 600.134,750.58 600.696,751.738 601.628,753.628 602.576,755.518 602.641,755.645 603.536,757.407 \n",
" 604.511,759.297 605.148,760.511 605.502,761.187 606.505,763.076 607.526,764.966 607.655,765.202 608.559,766.856 609.608,768.746 610.162,769.728 610.672,770.635 \n",
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" 619.742,785.753 620.189,786.454 620.947,787.642 622.169,789.532 622.696,790.337 623.407,791.422 624.662,793.311 625.203,794.116 625.934,795.201 627.223,797.091 \n",
" 627.71,797.797 628.528,798.98 629.852,800.87 630.217,801.386 631.192,802.76 632.551,804.65 632.724,804.888 633.926,806.539 635.231,808.307 635.321,808.429 \n",
" 636.733,810.319 637.738,811.646 638.164,812.208 639.614,814.098 640.245,814.911 641.082,815.988 642.57,817.877 642.752,818.106 644.076,819.767 645.259,821.231 \n",
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" 687.877,866.151 688.827,867.009 690.384,868.403 690.94,868.899 692.891,870.621 693.082,870.789 695.253,872.678 695.398,872.803 697.454,874.568 697.904,874.951 \n",
" 699.685,876.458 700.411,877.067 701.948,878.347 702.918,879.15 704.241,880.237 705.425,881.202 706.567,882.127 707.932,883.223 708.926,884.016 710.439,885.215 \n",
" 711.318,885.906 712.946,887.177 713.745,887.796 715.453,889.11 716.206,889.685 717.96,891.015 718.704,891.575 720.467,892.893 721.237,893.465 722.974,894.744 \n",
" 723.809,895.355 725.481,896.569 726.418,897.244 727.988,898.368 729.066,899.134 730.495,900.142 731.755,901.024 733.002,901.891 734.484,902.913 735.509,903.615 \n",
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" 778.127,929.613 780.634,930.966 781.18,931.259 783.14,932.303 784.746,933.148 785.647,933.621 788.154,934.922 788.379,935.038 790.661,936.208 792.083,936.928 \n",
" 793.168,937.476 795.675,938.727 795.858,938.817 798.182,939.963 799.711,940.707 800.689,941.182 803.196,942.386 803.64,942.597 805.703,943.574 807.652,944.486 \n",
" 808.21,944.747 810.717,945.905 811.749,946.376 813.224,947.048 815.731,948.175 815.934,948.266 818.238,949.289 820.213,950.155 820.745,950.388 823.252,951.473 \n",
" 824.589,952.045 825.758,952.544 828.265,953.601 829.067,953.935 830.772,954.645 833.279,955.674 833.65,955.825 835.786,956.691 838.293,957.693 838.347,957.714 \n",
" 840.8,958.684 843.162,959.604 843.307,959.66 845.814,960.625 848.103,961.494 848.321,961.576 850.828,962.516 853.177,963.383 853.335,963.442 855.842,964.357 \n",
" 858.349,965.258 858.39,965.273 860.856,966.149 863.363,967.027 863.755,967.163 865.87,967.894 868.376,968.748 869.28,969.052 870.883,969.592 873.39,970.424 \n",
" 874.975,970.942 875.897,971.244 878.404,972.054 880.851,972.832 880.911,972.851 883.418,973.639 885.925,974.414 886.931,974.721 888.432,975.18 890.939,975.935 \n",
" 893.221,976.611 893.446,976.678 895.953,977.412 898.46,978.135 899.749,978.501 900.967,978.847 903.474,979.55 905.981,980.241 906.53,980.39 908.488,980.923 \n",
" 910.994,981.595 913.501,982.256 913.593,982.28 916.008,982.909 918.515,983.551 920.971,984.17 921.022,984.183 923.529,984.806 926.036,985.419 928.543,986.022 \n",
" 928.701,986.06 931.05,986.617 933.557,987.202 936.064,987.776 936.83,987.949 938.571,988.343 941.078,988.9 943.585,989.447 945.411,989.839 946.092,989.985 \n",
" 948.599,990.515 951.106,991.036 953.613,991.547 954.519,991.729 956.119,992.05 958.626,992.544 961.133,993.029 963.64,993.505 964.249,993.618 966.147,993.973 \n",
" 968.654,994.432 971.161,994.883 973.668,995.324 974.734,995.508 976.175,995.757 978.682,996.183 981.189,996.599 983.696,997.007 986.148,997.398 986.203,997.406 \n",
" 988.71,997.799 991.217,998.182 993.724,998.557 996.231,998.924 998.737,999.282 998.776,999.287 1001.24,999.633 1003.75,999.976 1006.26,1000.31 1008.77,1000.64 \n",
" 1011.27,1000.96 1013.07,1001.18 1013.78,1001.27 1016.29,1001.57 1018.79,1001.86 1021.3,1002.15 1023.81,1002.43 1026.31,1002.7 1028.82,1002.96 1029.84,1003.07 \n",
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" 1078.96,1006.63 1081.47,1006.73 1083.97,1006.83 1084.42,1006.85 1086.48,1006.92 1088.99,1007 1091.49,1007.08 1094,1007.14 1096.51,1007.2 1099.02,1007.26 \n",
" 1101.52,1007.3 1104.03,1007.34 1106.54,1007.37 1109.04,1007.39 1111.55,1007.41 1114.06,1007.42 1116.56,1007.42 1119.07,1007.41 1121.58,1007.4 1124.08,1007.38 \n",
" 1126.59,1007.35 1129.1,1007.31 1131.61,1007.27 1134.11,1007.22 1136.62,1007.16 1139.13,1007.1 1141.63,1007.02 1144.14,1006.94 1146.65,1006.86 1146.89,1006.85 \n",
" 1149.15,1006.76 1151.66,1006.66 1154.17,1006.55 1156.67,1006.43 1159.18,1006.31 1161.69,1006.17 1164.2,1006.04 1166.7,1005.89 1169.21,1005.73 1171.72,1005.57 \n",
" 1174.22,1005.4 1176.73,1005.22 1179.24,1005.04 1180.32,1004.96 1181.74,1004.85 1184.25,1004.65 1186.76,1004.44 1189.27,1004.23 1191.77,1004 1194.28,1003.77 \n",
" 1196.79,1003.53 1199.29,1003.29 1201.47,1003.07 1201.8,1003.03 1204.31,1002.77 1206.81,1002.5 1209.32,1002.23 1211.83,1001.94 1214.33,1001.65 1216.84,1001.35 \n",
" 1218.24,1001.18 1219.35,1001.04 1221.86,1000.72 1224.36,1000.4 1226.87,1000.07 1229.38,999.727 1231.88,999.379 1232.52,999.287 1234.39,999.022 1236.9,998.658 \n",
" 1239.4,998.285 1241.91,997.904 1244.42,997.514 1245.15,997.398 1246.92,997.117 1249.43,996.711 1251.94,996.297 1254.45,995.874 1256.57,995.508 1256.95,995.443 \n",
" 1259.46,995.004 1261.97,994.556 1264.47,994.099 1266.98,993.633 1267.06,993.618 1269.49,993.16 1271.99,992.677 1274.5,992.186 1276.79,991.729 1277.01,991.685 \n",
" 1279.51,991.176 1282.02,990.658 1284.53,990.131 1285.89,989.839 1287.04,989.595 1289.54,989.05 1292.05,988.496 1294.48,987.949 1294.56,987.931 1297.06,987.359 \n",
" 1299.57,986.777 1302.08,986.185 1302.6,986.06 1304.58,985.585 1307.09,984.974 1309.6,984.354 1310.33,984.17 1312.11,983.724 1314.61,983.085 1317.12,982.435 \n",
" 1317.71,982.28 1319.63,981.777 1322.13,981.108 1324.64,980.428 1324.78,980.39 1327.15,979.739 1329.65,979.04 1331.56,978.501 1332.16,978.33 1334.67,977.61 \n",
" 1337.17,976.879 1338.08,976.611 1339.68,976.138 1342.19,975.387 1344.37,974.721 1344.7,974.624 1347.2,973.851 1349.71,973.067 1350.45,972.832 1352.22,972.272 \n",
" 1354.72,971.466 1356.33,970.942 1357.23,970.648 1359.74,969.82 1362.03,969.052 1362.24,968.979 1364.75,968.128 1367.26,967.264 1367.55,967.163 1369.76,966.39 \n",
" 1372.27,965.502 1372.91,965.273 1374.78,964.604 1377.29,963.693 1378.13,963.383 1379.79,962.77 1382.3,961.834 1383.2,961.494 1384.81,960.886 1387.31,959.925 \n",
" 1388.14,959.604 1389.82,958.952 1392.33,957.965 1392.96,957.714 1394.83,956.966 1397.34,955.952 1397.65,955.825 1399.85,954.927 1402.24,953.935 1402.36,953.887 \n",
" 1404.86,952.834 1406.72,952.045 1407.37,951.767 1409.88,950.686 1411.09,950.155 1412.38,949.591 1414.89,948.481 1415.37,948.266 1417.4,947.357 1419.56,946.376 \n",
" 1419.9,946.218 1422.41,945.064 1423.65,944.486 1424.92,943.896 1427.42,942.711 1427.66,942.597 1429.93,941.512 1431.59,940.707 1432.44,940.297 1434.95,939.065 \n",
" 1435.45,938.817 1437.45,937.819 1439.22,936.928 1439.96,936.555 1442.47,935.275 1442.92,935.038 1444.97,933.978 1446.56,933.148 1447.48,932.664 1449.99,931.332 \n",
" 1450.12,931.259 1452.49,929.984 1453.62,929.369 1455,928.617 1457.06,927.479 1457.51,927.232 1460.01,925.829 1460.44,925.59 1462.52,924.408 1463.76,923.7 \n",
" 1465.03,922.967 1467.02,921.81 1467.54,921.507 1470.04,920.027 1470.22,919.92 1472.55,918.528 1473.37,918.031 1475.06,917.009 1476.47,916.141 1477.56,915.469 \n",
" 1479.52,914.251 1480.07,913.908 1482.52,912.362 1482.58,912.325 1485.08,910.721 1485.47,910.472 1487.59,909.096 1488.37,908.582 1490.1,907.447 1491.23,906.693 \n",
" 1492.6,905.776 1494.05,904.803 1495.11,904.082 1496.82,902.913 1497.62,902.363 1499.55,901.024 1500.13,900.621 1502.24,899.134 1502.63,898.854 1504.89,897.244 \n",
" 1505.14,897.062 1507.5,895.355 1507.65,895.244 1510.07,893.465 1510.15,893.401 1512.6,891.575 1512.66,891.53 1515.1,889.685 1515.17,889.632 1517.56,887.796 \n",
" 1517.67,887.707 1519.99,885.906 1520.18,885.753 1522.38,884.016 1522.69,883.769 1524.74,882.127 1525.2,881.756 1527.06,880.237 1527.7,879.713 1529.36,878.347 \n",
" 1530.21,877.638 1531.62,876.458 1532.72,875.531 1533.85,874.568 1535.22,873.392 1536.05,872.678 1537.73,871.219 1538.22,870.789 1540.24,869.012 1540.36,868.899 \n",
" 1542.48,867.009 1542.74,866.769 1544.56,865.12 1545.25,864.49 1546.62,863.23 1547.76,862.173 1548.65,861.34 1550.26,859.819 1550.65,859.45 1552.63,857.561 \n",
" 1552.77,857.425 1554.58,855.671 1555.28,854.99 1556.51,853.781 1557.79,852.513 1558.41,851.892 1560.28,850.002 1560.29,849.994 1562.14,848.112 1562.8,847.429 \n",
" 1563.96,846.223 1565.31,844.819 1565.77,844.333 1567.55,842.443 1567.81,842.161 1569.31,840.554 1570.32,839.454 1571.04,838.664 1572.76,836.774 1572.83,836.697 \n",
" 1574.45,834.885 1575.33,833.885 1576.12,832.995 1577.77,831.105 1577.84,831.021 1579.4,829.215 1580.35,828.097 1581,827.326 1582.59,825.436 1582.85,825.116 \n",
" 1584.16,823.546 1585.36,822.072 1585.7,821.657 1587.23,819.767 1587.87,818.964 1588.73,817.877 1590.22,815.988 1590.38,815.79 1591.69,814.098 1592.88,812.544 \n",
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" 704.707,793.311 705.425,794.215 706.211,795.201 707.737,797.091 707.932,797.329 709.286,798.98 710.439,800.366 710.86,800.87 712.457,802.76 712.946,803.332 \n",
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" 1048.88,948.273 1051.38,948.474 1053.89,948.667 1056.4,948.852 1058.9,949.03 1061.41,949.199 1063.92,949.36 1066.42,949.514 1068.93,949.659 1071.44,949.797 \n",
" 1073.95,949.928 1076.45,950.05 1078.75,950.155 1078.96,950.165 1081.47,950.272 1083.97,950.372 1086.48,950.465 1088.99,950.549 1091.49,950.626 1094,950.695 \n",
" 1096.51,950.757 1099.02,950.811 1101.52,950.857 1104.03,950.896 1106.54,950.927 1109.04,950.951 1111.55,950.967 1114.06,950.976 1116.56,950.977 1119.07,950.97 \n",
" 1121.58,950.956 1124.08,950.935 1126.59,950.905 1129.1,950.869 1131.61,950.824 1134.11,950.772 1136.62,950.713 1139.13,950.646 1141.63,950.571 1144.14,950.488 \n",
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" 765.592,674.987 765.946,676.151 766.539,678.041 767.149,679.93 767.777,681.82 768.099,682.764 768.421,683.71 769.08,685.599 769.758,687.489 770.454,689.379 \n",
" 770.606,689.782 771.164,691.268 771.892,693.158 772.639,695.048 773.113,696.221 773.402,696.937 774.181,698.827 774.978,700.717 775.62,702.204 775.793,702.606 \n",
" 776.623,704.496 777.472,706.386 778.127,707.812 778.339,708.276 779.221,710.165 780.122,712.055 780.634,713.106 781.041,713.945 781.976,715.834 782.93,717.724 \n",
" 783.14,718.133 783.901,719.614 784.889,721.503 785.647,722.925 785.897,723.393 786.92,725.283 787.964,727.172 788.154,727.512 789.024,729.062 790.103,730.952 \n",
" 790.661,731.913 791.2,732.841 792.315,734.731 793.168,736.151 793.45,736.621 794.602,738.511 795.675,740.238 795.776,740.4 796.965,742.29 798.178,744.18 \n",
" 798.182,744.186 799.406,746.069 800.658,747.959 800.689,748.004 801.928,749.849 803.196,751.702 803.221,751.738 804.532,753.628 805.703,755.283 805.869,755.518 \n",
" 807.224,757.407 808.21,758.755 808.606,759.297 810.009,761.187 810.717,762.121 811.438,763.076 812.894,764.966 813.224,765.386 814.374,766.856 815.731,768.551 \n",
" 815.886,768.746 817.424,770.635 818.238,771.613 818.995,772.525 820.6,774.415 820.745,774.581 822.237,776.304 823.252,777.446 823.914,778.194 825.633,780.084 \n",
" 825.758,780.218 827.391,781.973 828.265,782.888 829.197,783.863 830.772,785.465 831.055,785.753 832.966,787.642 833.279,787.944 834.934,789.532 835.786,790.327 \n",
" 836.967,791.422 838.293,792.617 839.07,793.311 840.8,794.815 841.249,795.201 843.307,796.923 843.51,797.091 845.814,798.944 845.86,798.98 848.308,800.87 \n",
" 848.321,800.88 850.828,802.735 850.863,802.76 853.335,804.512 853.534,804.65 855.842,806.215 856.332,806.539 858.349,807.848 859.27,808.429 860.856,809.413 \n",
" 862.359,810.319 863.363,810.915 865.611,812.208 865.87,812.355 868.376,813.74 869.046,814.098 870.883,815.072 872.677,815.988 873.39,816.35 875.897,817.581 \n",
" 876.522,817.877 878.404,818.768 880.599,819.767 880.911,819.909 883.418,821.013 884.938,821.657 885.925,822.076 888.432,823.103 889.556,823.546 890.939,824.095 \n",
" 893.446,825.054 894.481,825.436 895.953,825.983 898.46,826.881 899.746,827.326 900.967,827.751 903.474,828.595 905.384,829.215 905.981,829.411 908.488,830.206 \n",
" 910.994,830.973 911.439,831.105 913.501,831.722 916.008,832.448 917.962,832.995 918.515,833.151 921.022,833.838 923.529,834.503 925.01,834.885 926.036,835.151 \n",
" 928.543,835.782 931.05,836.395 932.65,836.774 933.557,836.991 936.064,837.574 938.571,838.139 940.966,838.664 941.078,838.689 943.585,839.227 946.092,839.75 \n",
" 948.599,840.259 950.093,840.554 951.106,840.755 953.613,841.24 956.119,841.711 958.626,842.17 960.161,842.443 961.133,842.618 963.64,843.055 966.147,843.481 \n",
" 968.654,843.895 971.161,844.299 971.38,844.333 973.668,844.694 976.175,845.078 978.682,845.452 981.189,845.816 983.696,846.171 984.073,846.223 986.203,846.517 \n",
" 988.71,846.855 991.217,847.183 993.724,847.503 996.231,847.813 998.714,848.112 998.737,848.115 1001.24,848.41 1003.75,848.697 1006.26,848.976 1008.77,849.246 \n",
" 1011.27,849.509 1013.78,849.764 1016.19,850.002 1016.29,850.012 1018.79,850.253 1021.3,850.486 1023.81,850.713 1026.31,850.932 1028.82,851.144 1031.33,851.35 \n",
" 1033.83,851.548 1036.34,851.739 1038.41,851.892 1038.85,851.924 1041.36,852.103 1043.86,852.276 1046.37,852.441 1048.88,852.601 1051.38,852.754 1053.89,852.9 \n",
" 1056.4,853.041 1058.9,853.175 1061.41,853.303 1063.92,853.425 1066.42,853.541 1068.93,853.65 1071.44,853.754 1072.13,853.781 1073.95,853.853 1076.45,853.945 \n",
" 1078.96,854.031 1081.47,854.112 1083.97,854.187 1086.48,854.256 1088.99,854.319 1091.49,854.376 1094,854.428 1096.51,854.474 1099.02,854.515 1101.52,854.549 \n",
" 1104.03,854.578 1106.54,854.602 1109.04,854.619 1111.55,854.631 1114.06,854.638 1116.56,854.638 1119.07,854.634 1121.58,854.623 1124.08,854.607 1126.59,854.585 \n",
" 1129.1,854.558 1131.61,854.525 1134.11,854.486 1136.62,854.441 1139.13,854.391 1141.63,854.335 1144.14,854.274 1146.65,854.206 1149.15,854.133 1151.66,854.054 \n",
" 1154.17,853.969 1156.67,853.878 1159.18,853.782 1159.19,853.781 1161.69,853.679 1164.2,853.571 1166.7,853.457 1169.21,853.337 1171.72,853.21 1174.22,853.078 \n",
" 1176.73,852.939 1179.24,852.794 1181.74,852.643 1184.25,852.485 1186.76,852.321 1189.27,852.151 1191.77,851.974 1192.89,851.892 1194.28,851.79 1196.79,851.601 \n",
" 1199.29,851.404 1201.8,851.201 1204.31,850.991 1206.81,850.773 1209.32,850.549 1211.83,850.317 1214.33,850.078 1215.11,850.002 1216.84,849.832 1219.35,849.579 \n",
" 1221.86,849.319 1224.36,849.05 1226.87,848.774 1229.38,848.489 1231.88,848.197 1232.58,848.112 1234.39,847.896 1236.9,847.588 1239.4,847.271 1241.91,846.945 \n",
" 1244.42,846.61 1246.92,846.266 1247.23,846.223 1249.43,845.914 1251.94,845.552 1254.45,845.181 1256.95,844.799 1259.46,844.407 1259.92,844.333 1261.97,844.006 \n",
" 1264.47,843.595 1266.98,843.173 1269.49,842.738 1271.14,842.443 1271.99,842.293 1274.5,841.838 1277.01,841.37 1279.51,840.889 1281.21,840.554 1282.02,840.395 \n",
" 1284.53,839.89 1287.04,839.371 1289.54,838.837 1290.33,838.664 1292.05,838.291 1294.56,837.73 1297.06,837.152 1298.65,836.774 1299.57,836.559 1302.08,835.951 \n",
" 1304.58,835.325 1306.29,834.885 1307.09,834.681 1309.6,834.022 1312.11,833.341 1313.34,832.995 1314.61,832.641 1317.12,831.922 1319.63,831.179 1319.87,831.105 \n",
" 1322.13,830.418 1324.64,829.63 1325.92,829.215 1327.15,828.82 1329.65,827.984 1331.56,827.326 1332.16,827.12 1334.67,826.231 1336.83,825.436 1337.17,825.309 \n",
" 1339.68,824.36 1341.75,823.546 1342.19,823.375 1344.7,822.36 1346.37,821.657 1347.2,821.306 1349.71,820.215 1350.7,819.767 1352.22,819.084 1354.72,817.908 \n",
" 1354.79,817.877 1357.23,816.691 1358.63,815.988 1359.74,815.425 1362.24,814.107 1362.26,814.098 1364.75,812.739 1365.69,812.208 1367.26,811.314 1368.95,810.319 \n",
" 1369.76,809.829 1372.04,808.429 1372.27,808.281 1374.78,806.667 1374.97,806.539 1377.29,804.984 1377.77,804.65 1379.79,803.227 1380.44,802.76 1382.3,801.393 \n",
" 1383,800.87 1384.81,799.479 1385.45,798.98 1387.31,797.482 1387.79,797.091 1389.82,795.399 1390.05,795.201 1392.23,793.311 1392.33,793.227 1394.34,791.422 \n",
" 1394.83,790.961 1396.37,789.532 1397.34,788.602 1398.34,787.642 1399.85,786.151 1400.25,785.753 1402.11,783.863 1402.36,783.602 1403.91,781.973 1404.86,780.954 \n",
" 1405.67,780.084 1407.37,778.214 1407.39,778.194 1409.07,776.304 1409.88,775.37 1410.71,774.415 1412.31,772.525 1412.38,772.434 1413.88,770.635 1414.89,769.395 \n",
" 1415.42,768.746 1416.93,766.856 1417.4,766.258 1418.41,764.966 1419.86,763.076 1419.9,763.024 1421.3,761.187 1422.41,759.684 1422.7,759.297 1424.08,757.407 \n",
" 1424.92,756.239 1425.44,755.518 1426.77,753.628 1427.42,752.687 1428.08,751.738 1429.38,749.849 1429.93,749.022 1430.65,747.959 1431.9,746.069 1432.44,745.238 \n",
" 1433.13,744.18 1434.34,742.29 1434.95,741.326 1435.53,740.4 1436.7,738.511 1437.45,737.279 1437.85,736.621 1438.99,734.731 1439.96,733.086 1440.1,732.841 \n",
" 1441.2,730.952 1442.28,729.062 1442.47,728.73 1443.34,727.172 1444.38,725.283 1444.97,724.194 1445.41,723.393 1446.41,721.503 1447.4,719.614 1447.48,719.463 \n",
" 1448.37,717.724 1449.33,715.834 1449.99,714.502 1450.26,713.945 1451.18,712.055 1452.08,710.165 1452.49,709.284 1452.97,708.276 1453.83,706.386 1454.68,704.496 \n",
" 1455,703.767 1455.51,702.606 1456.33,700.717 1457.12,698.827 1457.51,697.893 1457.9,696.937 1458.67,695.048 1459.41,693.158 1460.01,691.59 1460.14,691.268 \n",
" 1460.85,689.379 1461.55,687.489 1462.22,685.599 1462.52,684.744 1462.88,683.71 1463.53,681.82 1464.16,679.93 1464.76,678.041 1465.03,677.198 1465.36,676.151 \n",
" 1465.94,674.261 1466.49,672.371 1467.04,670.482 1467.54,668.689 1467.56,668.592 1468.07,666.702 1468.57,664.813 1469.04,662.923 1469.5,661.033 1469.94,659.144 \n",
" 1470.04,658.711 1470.37,657.254 1470.78,655.364 1471.18,653.475 1471.55,651.585 1471.91,649.695 1472.26,647.806 1472.55,646.112 1472.58,645.916 1472.9,644.026 \n",
" 1473.19,642.136 1473.47,640.247 1473.73,638.357 1473.98,636.467 1474.2,634.578 1474.42,632.688 1474.61,630.798 1474.79,628.909 1474.95,627.019 1475.06,625.663 \n",
" 1475.1,625.129 1475.23,623.24 1475.34,621.35 1475.44,619.46 1475.52,617.571 1475.58,615.681 1475.63,613.791 1475.66,611.901 1475.68,610.012 1475.67,608.122 \n",
" 1475.65,606.232 1475.62,604.343 1475.57,602.453 1475.5,600.563 1475.41,598.674 1475.31,596.784 1475.19,594.894 1475.06,593.05 1475.05,593.005 1474.9,591.115 \n",
" 1474.73,589.225 1474.55,587.336 1474.34,585.446 1474.12,583.556 1473.89,581.666 1473.63,579.777 1473.36,577.887 1473.07,575.997 1472.76,574.108 1472.55,572.859 \n",
" 1472.44,572.218 1472.1,570.328 1471.74,568.439 1471.37,566.549 1470.97,564.659 1470.56,562.77 1470.13,560.88 1470.04,560.492 1469.69,558.99 1469.23,557.101 \n",
" 1468.74,555.211 1468.24,553.321 1467.73,551.431 1467.54,550.763 1467.19,549.542 1466.64,547.652 1466.06,545.762 1465.47,543.873 1465.03,542.501 1464.86,541.983 \n",
" 1464.23,540.093 1463.59,538.204 1462.92,536.314 1462.52,535.213 1462.24,534.424 1461.53,532.535 1460.81,530.645 1460.07,528.755 1460.01,528.629 1459.3,526.866 \n",
" 1458.52,524.976 1457.72,523.086 1457.51,522.596 1456.9,521.197 1456.06,519.307 1455.2,517.417 1455,516.999 1454.31,515.527 1453.41,513.638 1452.49,511.762 \n",
" 1452.49,511.748 1451.54,509.858 1450.58,507.969 1449.99,506.836 1449.59,506.079 1448.58,504.189 1447.55,502.3 1447.48,502.169 1446.5,500.41 1445.43,498.52 \n",
" 1444.97,497.736 1444.33,496.631 1443.21,494.741 1442.47,493.504 1442.07,492.851 1440.91,490.962 1439.96,489.453 1439.72,489.072 1438.51,487.182 1437.45,485.567 \n",
" 1437.27,485.292 1436.01,483.403 1434.95,481.83 1434.73,481.513 1433.42,479.623 1432.44,478.229 1432.09,477.734 1430.73,475.844 1429.93,474.753 1429.35,473.954 \n",
" 1427.94,472.065 1427.42,471.391 1426.5,470.175 1425.03,468.285 1424.92,468.136 1423.55,466.396 1422.41,464.982 1422.03,464.506 1420.48,462.616 1419.9,461.921 \n",
" 1418.91,460.727 1417.4,458.947 1417.3,458.837 1415.67,456.947 1414.89,456.056 1414.01,455.057 1412.38,453.241 1412.32,453.168 1410.59,451.278 1409.88,450.502 \n",
" 1408.84,449.388 1407.37,447.831 1407.05,447.499 1405.23,445.609 1404.86,445.227 1403.38,443.719 1402.36,442.687 1401.5,441.83 1399.85,440.206 1399.58,439.94 \n",
" 1397.62,438.05 1397.34,437.783 1395.63,436.161 1394.83,435.415 1393.6,434.271 1392.33,433.1 1391.54,432.381 1389.82,430.836 1389.43,430.492 1387.31,428.621 \n",
" 1387.29,428.602 1385.11,426.712 1384.81,426.454 1382.88,424.822 1382.3,424.331 1380.62,422.933 1379.79,422.253 1378.31,421.043 1377.29,420.217 1375.96,419.153 \n",
" 1374.78,418.222 1373.56,417.264 1372.27,416.267 1371.11,415.374 1369.76,414.351 1368.61,413.484 1367.26,412.472 1366.07,411.595 1364.75,410.629 1363.47,409.705 \n",
" 1362.24,408.822 1360.83,407.815 1359.74,407.049 1358.12,405.926 1357.23,405.309 1355.36,404.036 1354.72,403.603 1352.55,402.146 1352.22,401.928 1349.71,400.284 \n",
" 1349.67,400.257 1347.2,398.67 1346.73,398.367 1344.7,397.086 1343.72,396.477 1342.19,395.531 1340.64,394.587 1339.68,394.004 1337.5,392.698 1337.17,392.505 \n",
" 1334.67,391.033 1334.28,390.808 1332.16,389.588 1330.98,388.918 1329.65,388.168 1327.61,387.029 1327.15,386.775 1324.64,385.406 1324.14,385.139 1322.13,384.062 \n",
" 1320.59,383.249 1319.63,382.742 1317.12,381.446 1316.95,381.36 1314.61,380.173 1313.21,379.47 1312.11,378.923 1309.6,377.696 1309.36,377.58 1307.09,376.49 \n",
" 1305.4,375.691 1304.58,375.307 1302.08,374.145 1301.32,373.801 1299.57,373.004 1297.12,371.911 1297.06,371.885 1294.56,370.785 1292.78,370.022 1292.05,369.706 \n",
" 1289.54,368.647 1288.3,368.132 1287.04,367.608 1284.53,366.588 1283.66,366.242 1282.02,365.588 1279.51,364.607 1278.85,364.352 1277.01,363.644 1274.5,362.7 \n",
" 1273.86,362.463 1271.99,361.774 1269.49,360.867 1268.66,360.573 1266.98,359.977 1264.47,359.105 1263.24,358.683 1261.97,358.25 1259.46,357.413 1257.56,356.794 \n",
" 1256.95,356.594 1254.45,355.791 1251.94,355.006 1251.61,354.904 1249.43,354.236 1246.92,353.483 1245.33,353.014 1244.42,352.747 1241.91,352.027 1239.4,351.324 \n",
" 1238.68,351.125 1236.9,350.635 1234.39,349.963 1231.88,349.308 1231.6,349.235 1229.38,348.667 1226.87,348.041 1224.36,347.433 1223.99,347.345 1221.86,346.838 \n",
" 1219.35,346.258 1216.84,345.695 1215.75,345.456 1214.33,345.145 1211.83,344.61 1209.32,344.091 1206.81,343.588 1206.7,343.566 1204.31,343.096 1201.8,342.62 \n",
" 1199.29,342.16 1196.79,341.714 1196.57,341.676 1194.28,341.281 1191.77,340.862 1189.27,340.458 1186.76,340.069 1184.87,339.787 1184.25,339.693 1181.74,339.331 \n",
" 1179.24,338.982 1176.73,338.648 1174.22,338.328 1171.72,338.023 1170.64,337.897 1169.21,337.73 1166.7,337.45 1164.2,337.184 1161.69,336.933 1159.18,336.694 \n",
" 1156.67,336.47 1154.17,336.259 1151.66,336.062 1150.91,336.007 1149.15,335.878 1146.65,335.706 1144.14,335.548 1141.63,335.404 1139.13,335.273 1136.62,335.156 \n",
" 1134.11,335.051 1131.61,334.96 1129.1,334.883 1126.59,334.818 1124.08,334.767 1121.58,334.73 1119.07,334.705 1116.56,334.693 1114.06,334.695 1111.55,334.71 \n",
" 1109.04,334.739 1106.54,334.78 1104.03,334.835 1101.52,334.903 1099.02,334.984 1096.51,335.078 1094,335.186 1091.49,335.307 1088.99,335.442 1086.48,335.59 \n",
" 1083.97,335.752 1081.47,335.927 1080.4,336.007 1078.96,336.115 1076.45,336.315 1073.95,336.53 1071.44,336.758 1068.93,337 1066.42,337.256 1063.92,337.525 \n",
" 1061.41,337.809 1060.67,337.897 1058.9,338.105 1056.4,338.414 1053.89,338.738 1051.38,339.076 1048.88,339.428 1046.43,339.787 1046.37,339.795 1043.86,340.173 \n",
" 1041.36,340.567 1038.85,340.975 1036.34,341.397 1034.74,341.676 1033.83,341.834 1031.33,342.284 1028.82,342.749 1026.31,343.229 1024.61,343.566 1023.81,343.723 \n",
" 1021.3,344.231 1018.79,344.755 1016.29,345.294 1015.55,345.456 1013.78,345.847 1011.27,346.414 1008.77,346.998 1007.31,347.345 1006.26,347.597 1003.75,348.21 \n",
" 1001.24,348.84 999.71,349.235 998.737,349.485 996.231,350.145 993.724,350.822 992.627,351.125 991.217,351.514 988.71,352.222 986.203,352.947 985.975,353.014 \n",
" 983.696,353.687 981.189,354.444 979.699,354.904 978.682,355.218 976.175,356.008 973.738,356.794 973.668,356.816 971.161,357.64 968.654,358.482 968.066,358.683 \n",
" 966.147,359.341 963.64,360.218 962.643,360.573 961.133,361.112 958.626,362.025 957.445,362.463 956.119,362.955 953.613,363.904 952.451,364.352 951.106,364.872 \n",
" 948.599,365.859 947.641,366.242 946.092,366.864 943.585,367.89 943.002,368.132 941.078,368.934 938.571,369.999 938.518,370.022 936.064,371.082 934.183,371.911 \n",
" 933.557,372.188 931.05,373.313 929.981,373.801 928.543,374.459 926.036,375.628 925.903,375.691 923.529,376.816 921.946,377.58 921.022,378.028 918.515,379.262 \n",
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"source": [
"xx = yy = -4:0.011:4\n",
"V = v.(xx' .+ im*yy)\n",
"\n",
"contour(xx, yy, V; nlevels=50)\n",
"plot!(domain(x); color=:black, legend=false)"
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"\n",
"#### Problem 2.3\n",
"\n",
"As usual for logarithmic singular integral equations we want to solve\n",
"$$\n",
"\\dashint_{-1}^1 {u(t) \\over t-x} \\dt = f'(x) \n",
"$$\n",
"we need to be a bit careful differentiating $f$:\n",
"$$\n",
"{\\D \\over \\dx} \\log|x-\\I| = {\\D \\over \\dx} \\log\\sqrt{x^2 + 1} = { x \\over x^2 + 1}\n",
"$$\n",
"Now we do our usual game and solve for $u$ using the inverse Hilbert transform formula. That is, first calculate\n",
"$$\n",
"C[\\sqrt{1-x^2} {x \\over x^2+1}](z) = {z \\sqrt{z-1} \\sqrt{z+1} \\over 2\\I (z^2+1)} - {1 \\over 2 \\I} + {\\I \\sqrt{\\I - 1} \\sqrt{\\I+1} \\over 4 (z-\\I)} + {\\I \\sqrt{-\\I - 1} \\sqrt{-\\I+1} \\over 4 (z+\\I)}\n",
"$$\n",
"Therefore, we know that\n",
"$$\n",
"u(x) = {\\sqrt{\\I - 1} \\sqrt{\\I+1} \\over 2 \\pi (x-\\I)\\sqrt{1-x^2}} + {\\sqrt{-\\I - 1} \\sqrt{-\\I+1} \\over 2 \\pi (x+\\I) \\sqrt{1-x^2}} + {C \\over \\sqrt{1-x^2}}\n",
"$$\n",
"We can show (e.g. by finding its Cauchy transform and lookin at the asymptotic behaviour) that\n",
"$$\n",
"\\int_{-1}^1\\left[ {\\sqrt{\\I - 1} \\sqrt{\\I+1} \\over 2 \\pi (x-\\I)\\sqrt{1-x^2}} + {\\sqrt{-\\I - 1} \\sqrt{-\\I+1} \\over 2 \\pi (x+\\I) \\sqrt{1-x^2}} \\right] = -1\n",
"$$\n",
"Combined with the fact that\n",
"$$\n",
"\\int_{-1}^1 {1 \\over \\sqrt{1-x^2}} = \\pi\n",
"$$\n",
"we find that $C = 1/\\pi$.\n",
"\n",
"_Verification_ Let's check our work. First we see that the Hilbert transform of $u$ does indeed satisfy the specified equation (using the numerical `u` as calculated above):"
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"source": [
"fp = x -> x/(x^2+1)\n",
"\n",
"π*hilbert(u, 0.1) ≈ fp(0.1)"
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"source": [
"And the Cauchy transform of $\\sqrt{1-x^2} f'(x)$ satisfies the derived formula:"
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"source": [
"C = z -> z * sqrt(z-1)sqrt(z+1)/(2im*(z^2+1)) - 1/(2im) + im*sqrt(im-1)sqrt(im+1)/(4(z-im)) +\n",
" im*sqrt(-im-1)sqrt(-im+1)/(4(z+im))\n",
"cauchy(sqrt(1-x^2)fp(x), z) ≈ C(z)"
]
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"source": [
"Therefore we can invert to the Hilbert transform for $f'$:"
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"source": [
"w = -hilbert(sqrt(1-x^2)fp(x))/sqrt(1-x^2)\n",
"w̃ = sqrt(im-1)sqrt(im+1)/(2(x-im) * sqrt(1-x^2)) +\n",
" sqrt(-im-1)sqrt(-im+1)/(2(x+im) * sqrt(1-x^2))\n",
"hilbert(w,0.1) ≈ hilbert(w̃,0.1) ≈ fp(0.1)"
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"source": [
"And we have recovered $u$ up to $C/\\sqrt{1-x^2}$:"
]
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"source": [
"C = 1/π\n",
"\n",
"u(0.1) ≈ w̃(0.1)/π + C/sqrt(1-0.1^2)"
]
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"## Problem 3\n",
"\n",
"### Problem 3.1 \n",
"\n",
"To be analytic at all we need decay at either $\\pm \\infty$, this has neither so is not defined."
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"### Problem 3.2\n",
"\n",
"It has exponential decay in the right-half plane, therefore\n",
"$$\n",
"\\E^{\\gamma x} f(x) = {\\E^{\\gamma x } \\over 1 + \\E^x}\n",
"$$\n",
"has exponential decay at both $\\pm \\infty$, provided $0 < \\gamma < 1$. Therefore, we can take the strip $0 < \\Im s < 1$. (Note in each case the contour for the inverse Fourier transform can be any contour in the domain of analyticity.)\n",
"\n",
"We can verify this by exact computation using Residue calculus: for $0 < \\Im s < 1$, we can integrate over a rectangle to get:\n",
"$$\n",
"\\left(\\int_{-R}^R + \\int_R^{2\\I \\pi + R} + \\int_{2 \\I \\pi + R}^{2\\I \\pi - R} + \\int_{2 \\I \\pi - R}^{-R} \\right) {\\E^{-\\I s x} \\over 1 + \\E^x} \\dx = 2 \\pi \\I \\Res_{z = \\I \\pi } {\\E^{-\\I s z} \\over 1 + \\E^z} = \n",
"- 2 \\pi \\I \\E^{\\pi s}\n",
"$$\n",
"Note that \n",
"$$\n",
"{\\E^{-\\I s (R + \\I t)} \\over 1 + \\E^{R + \\I t}} = \n",
"{\\E^{-\\I R \\Re s + R \\Im s + t} \\over 1 + \\E^{R + \\I t}} \\rightarrow 0\n",
"$$\n",
"and\n",
"$$\n",
"{\\E^{-\\I s (-R + \\I t)} \\over 1 + \\E^{R + \\I t}} = \n",
"{\\E^{\\I R \\Re s - R \\Im s + t} \\over 1 + \\E^{R + \\I t}} \\rightarrow 0\n",
"$$\n",
"uniformly in $t$ as $R \\rightarrow \\infty$, hence we deduce that\n",
"$$\n",
"\\left(\\int_{-\\infty}^\\infty + \\int_{2 \\I \\pi + \\infty}^{2\\I \\pi - \\infty}\\right) {\\E^{-\\I s x} \\over 1 + \\E^x} \\dx = \n",
"- 2 \\pi \\I \\E^{\\pi s}\n",
"$$\n",
"Now note that\n",
"$$\n",
"\\int_{2 \\I \\pi + \\infty}^{2\\I \\pi - \\infty} {\\E^{-\\I s t} \\over 1 + \\E^t} \\dt = \\int_{\\infty}^{-\\infty} {\\E^{-\\I s (x+2 \\I \\pi)} \\over 1 + \\E^x} \\dx = -\\E^{2 \\pi s} \\int_{-\\infty}^\\infty {\\E^{-\\I s x} \\over 1 + \\E^x} \\dx \n",
"$$\n",
"Therefore, we have\n",
"$$\n",
"\\int_{-\\infty}^\\infty {\\E^{-\\I s x} \\over 1 + \\E^x} \\dx = - 2 \\I \\pi {\\E^{\\pi s} \\over 1 -\\E^{2 \\pi s}} = \\I \\pi {\\rm csch}\\, \\pi x\n",
"$$\n",
"which has poles at $0$ and $\\I$:"
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},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"phaseplot(-3..3, -10..10, z -> 1/(1+exp(z))) #integrand"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [
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"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
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],
"source": [
"phaseplot(-3..3, -3..3, z -> im*π*csch(π*z)) # transform"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Problem 3.3 \n",
"\n",
"Here $\\E^{\\gamma x } f(x) = \\E^{(\\gamma+2) x}$ has decay at $+\\infty$ proved $\\gamma < -2$, hence we have the strip $\\Im s < -2$.\n",
"\n",
"Indeed, its Fourier transform is \n",
"$$\n",
"-{\\I \\over 2 \\I +s}\n",
"$$\n",
"by integration by parts.\n",
"\n",
"### Problem 3.4 \n",
"\n",
"Here it's $\\Im s > 0$: unlike 1.1, we now have decay at $x \\rightarrow \\infty$ since $f_{\\rm L}(x)$ is identically zero.\n",
"\n",
"It's Fourier transform is determinable by integration-by-parts:\n",
"$$\n",
"\\hat f(s) = \\int_{-\\infty}^0 x \\E^{-\\I s x} \\dx = {1 \\over \\I s} \\int_{-\\infty}^0\\E^{-\\I s x} \\dx = {1 \\over s^2}\n",
"$$\n",
"\n",
"### Problem 3.5\n",
"\n",
"The Fourier transforms are given above.\n",
"\n",
"### Problem 3.6\n",
"\n",
"$$\\int_{-\\infty}^\\infty \\delta(x) \\E^{\\I s x} \\dx = 1$$\n",
"\n",
"It's actually an entire function, but non-decaying. This is hinting at the relationship between smoothness of a function and decay of its Fourier transform, and vice-versa: since $\\delta(x)$ \"decays\" to all orders, we expect its Fourier transform to be entire, but since its n ot smooth at all, we expect no decay, so on a formal level we can predict the analyticity properties."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Problem 4\n",
"\n",
"### Problem 4.1\n",
"\n",
"Note that\n",
"$$\n",
"K(z) = {3\\over 2} \\E^{-|x|} \\Rightarrow \\hat K(s) = {3 \\over 1+s^2}\n",
"$$\n",
"Provided $-1 < \\Im s < 1$, and \n",
"$$\n",
"\\widehat f_{\\rm R}(s) = -{\\I \\over s} - {\\alpha \\over s^2}\n",
"$$\n",
"for $\\Im s < 0$. Define\n",
"$$\n",
"h(s) = - \\widehat f_{\\rm R}(s) = {\\I \\over s} + {\\alpha \\over s^2}\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"5.773159728050814e-15 + 7.549516567451064e-15im"
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"source": [
"α = 0.3\n",
"x = Fun(0..100)\n",
"f = 1 + α*x\n",
"h = s -> (im/s + α/s^2)\n",
"\n",
"γ = -0.5 # we take the Fourier transform on R + im*γ\n",
"s = -0.5 + im*γ\n",
"\n",
"-sum(f*exp(-im*s*x)) - h(s)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"Transforming the equation, we have\n",
"$$\n",
"\\Phi_+(s) - (1 + \\hat K(s)) \\Phi_-(s) = {\\I \\over s} + {\\alpha \\over s^2}\n",
"$$\n",
"where\n",
"$$\n",
"1 + \\hat K(s) = {4 + s^2 \\over 1 + s^2} = {(s-2 \\I) (s+2 \\I) \\over (s+\\I)(s-\\I)}\n",
"$$\n",
"This is very close to the the example we did in lectures, so we already know the homogenous solution:\n",
"$$\n",
"\\kappa(z) = \\begin{cases} {z + 2\\I \\over z + \\I} & \\Im z > \\gamma \\\\\n",
" {z - \\I \\over z - 2\\I} & \\Im z < \\gamma\n",
" \\end{cases}\n",
" $$\n",
"which is valid for $-1 < \\Im s < 0$.\n",
"\n"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
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"\" transform=\"translate(29, 12)\"/>\n",
"</g>\n",
"</svg>\n"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"\n",
"g = s -> (4+s^2)/(1+s^2)\n",
"\n",
"κ = z -> imag(z) > γ ? (z+im*2)/(z+im) :\n",
" (z-im)/(z-im*2)\n",
"\n",
"\n",
"phaseplot(-3..3, -3..3, κ)"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"-1.3322676295501878e-15 - 2.7755575615628914e-16im"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"s = 0.1 + γ*im \n",
"κ₊ = κ(s + eps()*im)\n",
"κ₋ = κ(s - eps()*im)\n",
"\n",
"κ₊ - κ₋*g(s)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\n",
"We thus get the RH problem\n",
"$$\n",
"Y_+(s) - Y_-(s) = h(s)/\\kappa_+(s) = ({\\I \\over s} + {\\alpha \\over s^2}) {s + \\I \\over s + 2 \\I}\n",
"$$\n",
"We see this has poles at $0$ and $-2 \\I$, so using partial fraction expansion we get\n",
"$$\n",
"({\\I \\over s} + {\\alpha \\over s^2}) {s + \\I \\over s + \\sqrt{3} \\I} = \n",
"{\\alpha \\over 2 s^2}- {\\I (\\alpha-2) \\over 4 s} +{\\I (2+\\alpha) \\over 4 (s+2 \\I)}\n",
"$$\n",
"Therefore, splitting the poles between those above and below $\\gamma$, we have\n",
"$$\n",
"Y(z) = \\begin{cases} \n",
" {\\I (2+\\alpha) \\over 4 (z+2 \\I)} & \\Im z > \\gamma \\\\\n",
"-{\\alpha \\over 2 z^2}+{\\I (\\alpha-2) \\over 4 z} & \\Im z < \\gamma\n",
"\\end{cases}\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 91,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"5.551115123125783e-16 - 4.440892098500626e-16im"
]
},
"execution_count": 91,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"\n",
"s = 0.1 + γ*im \n",
"Y = z -> imag(z) > γ ? im*(2+α)/(4*(z+2im)) :\n",
" - α/(2z^2) + im*(α-2)/(4z)\n",
"\n",
"\n",
"Y₊ = Y(s + eps()*im)\n",
"Y₋ = Y(s - eps()*im)\n",
"\n",
"Y₊ - Y₋ - h(s)/κ₊"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We therefore have \n",
"\n",
"$$\n",
"\\Phi(z) = \\kappa(z) Y(z) = \\begin{cases} \n",
" {\\I (2+\\alpha) \\over 4 (z+ \\I)} & \\Im z > \\gamma \\\\\n",
"(-{\\alpha \\over 2 z^2}+{\\I (\\alpha-2) \\over 4 z} ) {z - \\I \\over z - 2\\I} & \\Im z < \\gamma\n",
"\\end{cases}\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 92,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"8.881784197001252e-16 - 1.1102230246251565e-15im"
]
},
"execution_count": 92,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Φ = z -> imag(z) > γ ? im*(2+α)/(4*(z+im)) :\n",
" (-α/(2z^2) + im*(α-2)/(4z))*(z-im)/(z-2im)\n",
"Φ₊ = Φ(s+eps()im) \n",
"Φ₋ = Φ(s-eps()im) \n",
"\n",
"Φ₊ - Φ₋*g(s) - h(s)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Finally, we recover the solution by inverting \n",
"$\\Phi_-$, using Residue calculus in the upper half plane: for $x > 0$ we have\n",
"\\begin{align*}\n",
"u(x) &= {1 \\over 2 \\pi} \\int_{-\\infty+ \\I \\gamma}^{\\infty + \\I \\gamma} (-{\\alpha \\over 2 z^2}+{\\I (\\alpha-2) \\over 4 z} ) {z - \\I \\over z - 2\\I} \\E^{\\I z x} \\dz \\\\\n",
"&= \\I (\\Res_{z = 0} + \\Res_{z = 2\\I}) (-{\\alpha \\over 2 z^2}+{\\I (\\alpha-2) \\over 4 z} ) {z - \\I \\over z - 2\\I} \\E^{\\I z x} = {1+x \\alpha \\over 4} - {\\alpha+1 \\over 4} \\E^{-2 x}\n",
"\\end{align*}\n",
"\n",
"Did it work? yes:"
]
},
{
"cell_type": "code",
"execution_count": 93,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(1.0300000000000011, 1.03)"
]
},
"execution_count": 93,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"t = Fun(0 .. 50)\n",
"\n",
"u = (1+t*α)/4 - (α-1)/4*exp(-2t)\n",
"\n",
"x = 0.1\n",
"\n",
"u(x) + 3/2*sum(exp(-abs(t-x))*u) , f(x)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Problem 4.2\n",
"\n",
"Setting up the problem as above, we arrive at a degenerate RH problem:\n",
"$$\n",
"\\Phi_+(s) - g(s) \\Phi_-(s) = h(s)\n",
"$$\n",
"where \n",
"$$g(s) = \\widehat K(s) = {2\\alpha \\over \\alpha^2 +s^2}= {2 \\alpha \\over (s-\\I \\alpha)(s+\\I \\alpha)} $$\n",
"and\n",
"$$ \n",
"h(s) = {\\I \\over s} + {\\alpha \\over s^2} = \\I {s -\\I \\alpha \\over s^2}\n",
"$$\n",
"\n",
"Suppose we allow $\\kappa_-(s) \\sim s$ to have growth, then we can write\n",
"$$\n",
"\\kappa(z) = \\begin{cases} {1 \\over z+\\I \\alpha }& \\Im z > \\gamma \\\\\n",
" {z-\\I \\alpha \\over 2 \\alpha} & \\Im z < \\gamma\n",
"\\end{cases}\n",
"$$\n",
"so that \n",
"$$\n",
"\\kappa_+(s) = \\kappa_-(s) g(s)\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
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"SUVORK5CYII=\n",
"\" transform=\"translate(29, 12)\"/>\n",
"</g>\n",
"</svg>\n"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"\n",
"α = 0.3\n",
"\n",
"\n",
"g = s -> (2α)/(α^2+s^2)\n",
"h = s -> (im/s + α/s^2)\n",
"\n",
"κ = z -> imag(z) > γ ? 1/(z + im*α) :\n",
" (z-im*α)/(2α)\n",
"\n",
"\n",
"phaseplot(-3..3, -3..3, κ)"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2.6645352591003757e-15 + 1.7763568394002505e-15im"
]
},
"execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"s = 0.1 + γ*im \n",
"κ₊ = κ(s + eps()*im)\n",
"κ₋ = κ(s - eps()*im)\n",
"\n",
"κ₊ - κ₋*g(s)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Then we have \n",
"$$h(s)/\\kappa_+(s) = \\I {s^2 + \\alpha^2 \\over s^2} = \\I + \\I {\\alpha^2 \\over s^2}\n",
"$$\n",
"and then we can write\n",
"$$\n",
"Y(z) = \\begin{cases}\n",
"\\I & \\Im z > \\gamma \\\\\n",
"-{\\I \\alpha^2 \\over z^2} & \\Im z < \\gamma\n",
"\\end{cases}\n",
"$$\n"
]
},
{
"cell_type": "code",
"execution_count": 25,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1.3877787807814457e-16 + 7.771561172376096e-16im"
]
},
"execution_count": 25,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"\n",
"s = 0.1 + γ*im \n",
"Y = z -> imag(z) > γ ? im :\n",
" -im*α^2/s^2\n",
"\n",
"\n",
"Y₊ = Y(s + eps()*im)\n",
"Y₋ = Y(s - eps()*im)\n",
"\n",
"Y₊ - Y₋ - h(s)/κ₊"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Putting things together, we get\n",
"$$\n",
"\\Phi(z) = \\kappa(z) Y(z) = \\begin{cases} {\\I \\over z+\\I \\alpha }& \\Im z > \\gamma \\\\\n",
" -\\I {\\alpha^2 \\over z^2} {z-\\I \\alpha \\over 2 \\alpha} & \\Im z < \\gamma\n",
"\\end{cases}\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"-3.9968028886505635e-15 + 4.218847493575595e-15im"
]
},
"execution_count": 26,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Φ = z -> imag(z) > γ ? im/(z + im*α) :\n",
" -im*α^2/z^2* (z-im*α)/(2α)\n",
"Φ₊ = Φ(s+eps()im) \n",
"Φ₋ = Φ(s-eps()im) \n",
"\n",
"Φ₊ - Φ₋*g(s) - h(s)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We now invert the Fourier transform of $\\Phi_-(s)$ using Jordan's lemma:\n",
"$$\n",
"u(x) = {1 \\over 2 \\pi} \\int_{-\\infty + \\I \\gamma}^{\\infty + \\I \\gamma} \\Phi_-(s) \\E^{\\I s x} \\D s = {\\alpha \\over 2}\\Res_{z = 0} {z- \\I \\alpha \\over z^2} \\E^{\\I z x} = {\\alpha \\over 2} (1+x \\alpha)\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 122,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1.199040866595169e-14"
]
},
"execution_count": 122,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"t = Fun(0 .. 200)\n",
"\n",
"u = α*(1+t*α)/2\n",
"\n",
"x = 0.1\n",
"\n",
"sum(exp(-α*abs(t-x))*u) - (1 + α*x)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 4.3\n",
"\n",
"1. From the same logic as 2.2, we know we need to solve\n",
"$$\n",
"\\Phi_+(s) - g(s) \\Phi_-(s) = h(s)\n",
"$$\n",
"where\n",
"$$\n",
"g(s) = 1 - {2 \\lambda \\over s^2 + 1} = {s^2 +1-2 \\lambda \\over s^2 + 1} = {(s- \\I\\gamma)(s+\\I \\gamma) \\over (s+\\I)(s-\\I)}\n",
"$$\n",
"and \n",
"$$\n",
"h(s) = {1 \\over s^2}\n",
"$$\n",
"where $-1 < \\Im s < 0$, let's say $\\Im s = \\delta$ because I annoyingly used $\\gamma$ in the statement of the problem. \n",
"Writing $s = t + \\I \\delta$, we see that\n",
"$$\n",
"g(s) = {t^2 +2 \\I \\delta t -\\delta^2 +\\gamma^2 \\over s^2 + 1}\n",
"$$\n",
"By ensuring its real part is positive, this has trivial winding number provided $\\gamma^2 = 1 - 2\\lambda > 0$, which is true for $0 < \\lambda < {1 \\over 2}$, and restricting the contour $s$ lives on to be $- {\\gamma} < \\delta < 0$. Factorizing the kernel we get\n",
"$$\n",
"\\kappa(z) = \\begin{cases}\n",
" {z+\\I \\gamma \\over z + \\I} & \\Im z > \\delta\\\\\n",
" {z-\\I \\over z-\\I \\gamma} & \\Im z < \\delta \n",
" \\end{cases}\n",
"$$\n",
"Thus we want to solve\n",
"$$\n",
"Y_+(s) - Y_-(s) = h(s) \\kappa_+(s)^{-1} = {s + \\I \\over s + \\I \\gamma} { 1 \\over s^2} = {1 \\over \\gamma s^2} - {\\I ( \\gamma - 1) \\over \\gamma^2 s} + {\\I \\over \\gamma^2 }{ \\gamma - 1 \\over s+ \\I \\gamma}\n",
"$$\n",
"Which has solution, (since $\\delta > - \\gamma$), \n",
"$$\n",
"Y(z) = \\begin{cases}\n",
" {\\I \\over \\gamma^2 }{ \\gamma - 1 \\over s+ \\I \\gamma} & \\Im z > \\delta\\\\\n",
" {\\I ( \\gamma - 1) \\over \\gamma^2 z} - {1 \\over \\gamma z^2} & \\Im z < \\delta \n",
" \\end{cases}\n",
"$$\n",
"We thus get\n",
"$$\n",
"\\Phi_-(z) = ({\\I ( \\gamma - 1) \\over \\gamma^2 z} - {1 \\over \\gamma z^2}) {z-\\I \\over z-\\I \\gamma} \n",
"$$\n",
"and Jordan's lemma gives us\n",
"$$\n",
"u(x) = {x \\over \\gamma^2} - \\E^{-x \\gamma}(\\gamma-1)/\\gamma^2\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 55,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"-3.239075674343894e-14"
]
},
"execution_count": 55,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"t = Fun(0 .. 200)\n",
"λ = 0.1\n",
"γ = sqrt(1-2λ)\n",
"\n",
"u = t/γ^2 - exp(-t*γ)*(γ-1)/γ^2\n",
"\n",
"x = 0.1\n",
"\n",
"u(x) - λ*sum(exp(-abs(t-x))*u) - x"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Oddly, this is definitely a solution, but not in the form the question asked for. To get the other solution, consider now the bad winding number case of $-1 < \\delta < - \\gamma$. Motivated by 2.2, what if we allow $\\kappa$ to have different behaviour? Consider\n",
"$$\n",
"\\kappa(z) = \\begin{cases}\n",
" {1 \\over z + \\I} & \\Im z > \\delta\\\\\n",
" {(z-\\I) \\over (z-\\I \\gamma) (z+\\I \\gamma)} & \\Im z < \\delta \n",
" \\end{cases}\n",
"$$\n",
"Chosen so that both $\\kappa_+$ and $\\kappa_+^{-1}$ are analytic. \n",
"\n",
"Thus we want to solve\n",
"$$\n",
"Y_+(s) - Y_-(s) = h(s) \\kappa_+(s)^{-1} = { s + \\I \\over s^2} = {1 \\over s} + {\\I \\over s^2}\n",
"$$\n",
"but now we only need $Y_+(s) = O(1)$ and $Y_-(s) = O(1)$. Here is where the non-uniqueness comes in, as we can add an arbitrary constant:\n",
"$$\n",
"Y(z) = \\begin{cases}\n",
" A & \\Im z > 0 \\\\\n",
" A - {1 \\over z} - {\\I \\over z^2} & \\Im z < 0\n",
"\\end{cases}\n",
"$$\n",
"Thus we have\n",
"$$\n",
"\\Phi_-(z) = Y_-(z) \\kappa_-(z) = -( A + {1 \\over z} + {\\I \\over z^2} ){(z-\\I) \\over (z-\\I \\gamma) (z+\\I \\gamma)} \n",
"$$\n",
"Using Jordan's lemma, and now since $\\delta < - \\gamma$, we get\n",
"\\begin{align*}\n",
"u(x) &= \\I (\\Res_{z = 0} + \\Res_{z = \\I \\gamma} + \\Res_{z = - \\I \\gamma}) \\Phi_-(z) \\E^{\\I x z} \\\\\n",
"&= {x \\over \\gamma^2} - \\E^{-x \\gamma}( {\\gamma^2-1 \\over 2 \\gamma^3} + {\\gamma-1 \\over 2\\gamma^3} A) - \\E^{x \\gamma} ({1-\\gamma^2 \\over 2 \\gamma^3} + {\\gamma+1 \\over 2\\gamma^3} A)\\\\\n",
"&= {x \\over \\gamma^2} + {\\E^{x \\gamma} - \\E^{-x \\gamma} \\over 2} {\\gamma-\\gamma^{-1} \\over 2 \\gamma^2} - {A \\over \\gamma^3} ( {\\E^{x \\gamma} - \\E^{-x \\gamma} \\over 2} + \\gamma {\\E^{x \\gamma} + \\E^{-x \\gamma} \\over 2} )\n",
"\\end{align*}\n",
"Redefining $A$ and using the definition of $\\sinh$ and $\\cosh$ gives the form in the assignment.\n",
"\n",
"What's the moral of the story? \n",
"\n",
"1. Different choices of contours can give different solutions\n",
"2. When the winding number is non-trivial, the solution may not be unique"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": true
},
"source": [
"### 4.4\n",
"\n",
"\n",
"1. Integrating by parts we have\n",
"\\begin{align*}\n",
"\\widehat{u_\\rR'}(s) &= \\I s \\widehat{u_\\rR}(s) - u(0) = \\I s \\widehat{u_\\rR}(s) \\\\\n",
"\\widehat{u_\\rR''}(s) &= \\I s \\widehat{u_\\rR'}(s) - u'(0) = -s^2 \\widehat{u_\\rR}(s) - u'(0)\n",
"\\end{align*}\n",
"2. Our integral equation when cast on the whole real line is:\n",
"$$\n",
"u_\\rR''(x) - {72 \\over 5} \\int_{-\\infty}^\\infty \\E^{-5 |x-t|} u_\\rR(t)\\dt = 1_\\rR(x) + p_\\rL(x)\n",
"$$\n",
"where \n",
"$$\n",
"p(x) = {72 \\over 5} \\int_{-\\infty}^\\infty \\E^{-5 |x-t|} u_\\rR(t)\\dt = {72 \\over 5} \\int_0^\\infty \\E^{-5 |x-t|} u_\\rR(t)\\dt.\n",
"$$\n",
"Note that, for $-5 <\\Im s < 5$,\n",
"$$\n",
"\\hat K(s) = {10 \\over s^2 + 25}\n",
"$$\n",
"provided $s$ is in the lower half plane,\n",
"$$\n",
"\\widehat{1_\\rR}(s) = \\int_0^\\infty \\E^{-\\I s x} \\dx = {1 \\over \\I s}\n",
"$$\n",
"Thus our integral equation in frequency space is \n",
"\\begin{align*}\n",
"-\\alpha - s^2 \\widehat{u_\\rR}(s) - {72 \\over 5} \\widehat K(s) \\widehat{u_\\rR}(s) &=\\widehat{p_\\rL}(x) + \\widehat{1_\\rR}(s)\\\\\n",
"\\Phi_+(s) - (s^2 + {144 \\over s^2+25}) \\Phi_-(s) &= \\alpha + {1 \\over \\I s}\\\\\n",
"\\Phi_+(s) - {(s^2 + 9) (s^2+16) \\over s^2+25} \\Phi_-(s) &= \\alpha + {1 \\over \\I s}\n",
"\\end{align*}\n",
"where $s \\in \\R +\\I \\gamma$ for any $-5 < \\gamma < 0$.\n",
"3. We can factorize this to construct $g(s)$ as \n",
"$$\n",
"g(s) = \\kappa_+(s) \\kappa_-(s)^{-1} = {(s +3 \\I) (s+4 \\I) \\over s+5 \\I} {(s -3 \\I) (s-4 \\I) \\over s-5 \\I}\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 33,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"-5.551115123125783e-17 - 2.220446049250313e-16im"
]
},
"execution_count": 33,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"κ = z -> imag(z) > γ ? \n",
" (z+3im)*(z+4im)/(z+5im) :\n",
" (z-5im)/((z-3im)*(z-4im)) \n",
"\n",
"γ = -1.0\n",
"s = 0.1+γ*im\n",
"g = s -> (s^2+9)*(s^2+16)/(s^2+25)\n",
"\n",
"κ(s+eps()im) - g(s)κ(s-eps()im)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Writing $\\Phi(z) = \\kappa(z) Y(z)$ we get the subtractive RH problem\n",
"$$\n",
"Y_+(s) - Y_-(s) = {h(s) \\over \\kappa_+(s)} = (\\alpha + {1 \\over \\I s}) {s + 5 \\I \\over (s + 3 \\I)(s + 4 \\I)}\n",
"$$\n",
"We use partial fraction expansion to write\n",
"$$\n",
"{h(s) \\over \\kappa_+(s)} = -{\\alpha + 1/4 \\over s+4 \\I} + {2/3 + 2 \\alpha \\over s+3 \\I} - {5 \\over 12 s}\n",
"$$\n",
"Therefore we have\n",
"$$\n",
"Y(z) = \\begin{cases} \n",
"-{\\alpha + 1/4 \\over s+4 \\I} + {2/3 + 2 \\alpha \\over s+3 \\I} & 2 \\\\\n",
" {5 \\over 12 s} &1\n",
"\\end{cases}\n",
"$$\n",
"and hence\n",
"$$\n",
"\\Phi(z) = \\begin{cases}\n",
"{(z+3\\I)(z+4\\I) \\over z+5\\I} (-{α+1/4 \\over z+4\\I} + (2/3 + 2α)/(z+3\\I)) & \\Im z > \\gamma \\\\\n",
" {z-5\\I \\over (z-3\\I)(z-4\\I)} {5 \\over 12z}& \\Im z < \\gamma\n",
" \\end{cases}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We can now invert the Fourier transform of \n",
"$$\n",
"\\Phi_-(s) = {s-5\\I \\over (s-3\\I)(s-4\\I)} {5 \\over 12s}\n",
"$$\n",
"This actually decays so fast that we don't need Jordan's lemma to justify here. This has three poles above our contour, so we sum over each residue to get\n",
"$$\n",
"u(x) = \\I (\\Res_{z = 0} +\\Res_{z = 3 \\I } +\\Res_{z = 4\\I} ) \\E^{\\I z x } {z-5\\I \\over (z-3\\I)(z-4\\I)} {5 \\over 12z} = -{25 \\over 144} - {5 \\E^{-4 x} \\over 48} + {5 \\E^{-3 x} \\over 18}\n",
"$$\n",
"\n",
"Here's we check the solution:"
]
},
{
"cell_type": "code",
"execution_count": 42,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1.0000000000000018"
]
},
"execution_count": 42,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"t = Fun(0 .. 200)\n",
"\n",
"u = -25/144 - 5exp(-4t)/48 + 5exp(-3t)/18\n",
"\n",
"x = 1.1\n",
"u''(x) - 72/5*sum(exp(-5abs(x-t))*u)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Here we check the jump of $Y$:"
]
},
{
"cell_type": "code",
"execution_count": 45,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2.7755575615628914e-17 + 5.551115123125783e-17im"
]
},
"execution_count": 45,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"α = u'(0)\n",
"\n",
"h = s -> α + 1/(im*s)\n",
"\n",
"Y = z -> imag(z) > γ ? \n",
" (-(α+1/4)/(z+4im) + (2/3 + 2α)/(z+3im)) :\n",
" 5/(12z)\n",
"\n",
"Y(s+eps()im) - Y(s-eps()im) - h(s)/κ(s + eps()im)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Here we check the jump of $\\Phi$:"
]
},
{
"cell_type": "code",
"execution_count": 46,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"-1.1102230246251565e-16 + 1.3877787807814457e-17im"
]
},
"execution_count": 46,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"γ = -1.0\n",
"\n",
"\n",
"Φ = z -> imag(z) > γ ? \n",
" (z+3im)*(z+4im)/(z+5im) * (-(α+1/4)/(z+4im) + (2/3 + 2α)/(z+3im)) :\n",
" (z-5im)/((z-3im)*(z-4im)) * 5/(12z)\n",
"\n",
"\n",
"Φ(s + eps()*im) - g(s)*Φ(s - eps()*im) - h(s)"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Julia 1.0.0",
"language": "julia",
"name": "julia-1.0"
},
"language_info": {
"file_extension": ".jl",
"mimetype": "application/julia",
"name": "julia",
"version": "1.0.0"
}
},
"nbformat": 4,
"nbformat_minor": 2
}