Solution Sheet 4.ipynb

{
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  {
   "cell_type": "code",
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   "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:"
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       "-2.220446049250313e-16"
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     "execution_count": 5,
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   "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)"
   ]
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   "source": [
    "Differentiating it satisfies $H u = g'/(-\\pi)$:"
   ]
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     "execution_count": 6,
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   ],
   "source": [
    "hilbert(u,0.1) ≈ g'(0.1)/(-π) ≈ (2*0.1)/(π*(1+0.1^2)^2)"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "The Cauchy transform also works:"
   ]
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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)"
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Therefore the Hilbert transform is given by:"
   ]
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   "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)"
   ]
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   "metadata": {},
   "source": [
    "We thus have an expression for $u$, we are just missing the constant:"
   ]
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   "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)"
   ]
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   "source": [
    "We can find $C$ interms of the relevant integral, which we call $D$:"
   ]
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    {
     "data": {
      "text/plain": [
       "(-0.3247204711377926 + 0.0im, -0.32472047136213555 + 0.0im)"
      ]
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     "execution_count": 92,
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   ],
   "source": [
    "D = quadgk(x -> x == 0 ? 0.0 : ũ(x)*log(abs(x)),-1,1)[1]\n",
    "C, (1-D)/(π*log(1/2))"
   ]
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   "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:"
   ]
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   ],
   "source": [
    "z = 2+im; π*logkernel(x,z) ≈ real((1-z^2)/2 * (log(z-1)-log(z+1)) - z)"
   ]
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   "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:"
   ]
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   "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)"
   ]
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   "cell_type": "code",
   "execution_count": 151,
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       "  1011.27,1428.03 1013.78,1428.21 1014.32,1428.25 1016.29,1428.38 1018.79,1428.55 1021.3,1428.71 1023.81,1428.87 1026.31,1429.02 1028.82,1429.17 1031.33,1429.31 \n",
       "  1033.83,1429.45 1036.34,1429.59 1038.85,1429.72 1041.36,1429.85 1043.86,1429.97 1046.37,1430.09 1047.3,1430.14 1048.88,1430.21 1051.38,1430.32 1053.89,1430.43 \n",
       "  1056.4,1430.53 1058.9,1430.63 1061.41,1430.72 1063.92,1430.81 1066.42,1430.89 1068.93,1430.98 1071.44,1431.05 1073.95,1431.13 1076.45,1431.19 1078.96,1431.26 \n",
       "  1081.47,1431.32 1083.97,1431.37 1086.48,1431.42 1088.99,1431.47 1091.49,1431.51 1094,1431.55 1096.51,1431.59 1099.02,1431.62 1101.52,1431.64 1104.03,1431.67 \n",
       "  1106.54,1431.68 1109.04,1431.7 1111.55,1431.71 1114.06,1431.71 1116.56,1431.71 1119.07,1431.71 1121.58,1431.7 1124.08,1431.69 1126.59,1431.67 1129.1,1431.65 \n",
       "  1131.61,1431.63 1134.11,1431.6 1136.62,1431.56 1139.13,1431.53 1141.63,1431.48 1144.14,1431.44 1146.65,1431.39 1149.15,1431.33 1151.66,1431.27 1154.17,1431.21 \n",
       "  1156.67,1431.14 1159.18,1431.07 1161.69,1431 1164.2,1430.92 1166.7,1430.83 1169.21,1430.74 1171.72,1430.65 1174.22,1430.56 1176.73,1430.45 1179.24,1430.35 \n",
       "  1181.74,1430.24 1184.01,1430.14 1184.25,1430.12 1186.76,1430.01 1189.27,1429.88 1191.77,1429.76 1194.28,1429.63 1196.79,1429.49 1199.29,1429.35 1201.8,1429.21 \n",
       "  1204.31,1429.06 1206.81,1428.91 1209.32,1428.75 1211.83,1428.59 1214.33,1428.43 1216.84,1428.26 1216.99,1428.25 1219.35,1428.08 1221.86,1427.9 1224.36,1427.72 \n",
       "  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",
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       "  1209.32,1223.25 1211.83,1223.04 1214.33,1222.83 1216.84,1222.61 1219.35,1222.38 1220.57,1222.27 1221.86,1222.15 1224.36,1221.92 1226.87,1221.67 1229.38,1221.42 \n",
       "  1231.88,1221.17 1234.39,1220.91 1236.9,1220.65 1239.35,1220.38 1239.4,1220.37 1241.91,1220.1 1244.42,1219.82 1246.92,1219.53 1249.43,1219.23 1251.94,1218.93 \n",
       "  1254.45,1218.63 1255.56,1218.49 1256.95,1218.32 1259.46,1218 1261.97,1217.68 1264.47,1217.35 1266.98,1217.02 1269.49,1216.68 1270.04,1216.6 1271.99,1216.33 \n",
       "  1274.5,1215.98 1277.01,1215.62 1279.51,1215.26 1282.02,1214.89 1283.23,1214.71 1284.53,1214.52 1287.04,1214.14 1289.54,1213.75 1292.05,1213.36 1294.56,1212.96 \n",
       "  1295.42,1212.82 1297.06,1212.56 1299.57,1212.15 1302.08,1211.73 1304.58,1211.31 1306.79,1210.93 1307.09,1210.88 1309.6,1210.45 1312.11,1210.01 1314.61,1209.56 \n",
       "  1317.12,1209.11 1317.49,1209.04 1319.63,1208.65 1322.13,1208.19 1324.64,1207.72 1327.15,1207.24 1327.61,1207.15 1329.65,1206.76 1332.16,1206.27 1334.67,1205.78 \n",
       "  1337.17,1205.28 1337.24,1205.26 1339.68,1204.77 1342.19,1204.26 1344.7,1203.74 1346.44,1203.37 1347.2,1203.21 1349.71,1202.68 1352.22,1202.14 1354.72,1201.6 \n",
       "  1355.25,1201.48 1357.23,1201.05 1359.74,1200.49 1362.24,1199.93 1363.72,1199.59 1364.75,1199.36 1367.26,1198.79 1369.76,1198.2 1371.89,1197.7 1372.27,1197.62 \n",
       "  1374.78,1197.02 1377.29,1196.42 1379.78,1195.81 1379.79,1195.81 1382.3,1195.2 1384.81,1194.58 1387.31,1193.95 1387.41,1193.93 1389.82,1193.32 1392.33,1192.68 \n",
       "  1394.81,1192.04 1394.83,1192.03 1397.34,1191.37 1399.85,1190.71 1401.98,1190.15 1402.36,1190.05 1404.86,1189.37 1407.37,1188.69 1408.97,1188.26 1409.88,1188.01 \n",
       "  1412.38,1187.31 1414.89,1186.61 1415.76,1186.37 1417.4,1185.9 1419.9,1185.19 1422.38,1184.48 1422.41,1184.47 1424.92,1183.74 1427.42,1183.01 1428.84,1182.59 \n",
       "  1429.93,1182.26 1432.44,1181.52 1434.95,1180.76 1435.15,1180.7 1437.45,1180 1439.96,1179.23 1441.31,1178.81 1442.47,1178.45 1444.97,1177.66 1447.34,1176.92 \n",
       "  1447.48,1176.87 1449.99,1176.07 1452.49,1175.27 1453.23,1175.03 1455,1174.45 1457.51,1173.63 1459.01,1173.14 1460.01,1172.81 1462.52,1171.97 1464.67,1171.25 \n",
       "  1465.03,1171.13 1467.54,1170.28 1470.04,1169.42 1470.22,1169.36 1472.55,1168.56 1475.06,1167.68 1475.66,1167.47 1477.56,1166.8 1480.07,1165.91 1481.01,1165.58 \n",
       "  1482.58,1165.02 1485.08,1164.12 1486.26,1163.69 1487.59,1163.21 1490.1,1162.29 1491.41,1161.8 1492.6,1161.36 1495.11,1160.43 1496.48,1159.91 1497.62,1159.48 \n",
       "  1500.13,1158.53 1501.46,1158.02 1502.63,1157.57 1505.14,1156.61 1506.37,1156.13 1507.65,1155.63 1510.15,1154.65 1511.19,1154.24 1512.66,1153.66 1515.17,1152.66 \n",
       "  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",
       "<polyline clip-path=\"url(#clip9303)\" style=\"stroke:#f4de52; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
       "  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",
       "<polyline clip-path=\"url(#clip9303)\" style=\"stroke:#f2ef78; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
       "  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",
       "  \"/>\n",
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       "  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",
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       "  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",
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       "  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",
       "  1417.4,1096.92 1415.46,1097.55 1414.89,1097.74 1412.38,1098.55 1409.88,1099.35 1409.61,1099.44 1407.37,1100.15 1404.86,1100.94 1403.6,1101.33 1402.36,1101.72 \n",
       "  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",
       "  1324.64,1122.02 1324.18,1122.12 1322.13,1122.56 1319.63,1123.08 1317.12,1123.61 1315.17,1124.01 1314.61,1124.12 1312.11,1124.63 1309.6,1125.13 1307.09,1125.62 \n",
       "  1305.69,1125.9 1304.58,1126.11 1302.08,1126.59 1299.57,1127.06 1297.06,1127.53 1295.66,1127.79 1294.56,1127.99 1292.05,1128.44 1289.54,1128.89 1287.04,1129.33 \n",
       "  1285,1129.68 1284.53,1129.76 1282.02,1130.18 1279.51,1130.6 1277.01,1131.01 1274.5,1131.42 1273.57,1131.57 1271.99,1131.82 1269.49,1132.21 1266.98,1132.59 \n",
       "  1264.47,1132.97 1261.97,1133.34 1261.19,1133.46 1259.46,1133.71 1256.95,1134.07 1254.45,1134.42 1251.94,1134.76 1249.43,1135.1 1247.6,1135.34 1246.92,1135.43 \n",
       "  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",
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       "  1201.8,1140.31 1199.29,1140.52 1196.79,1140.73 1194.28,1140.92 1193.1,1141.01 1191.77,1141.12 1189.27,1141.3 1186.76,1141.48 1184.25,1141.65 1181.74,1141.82 \n",
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       "  507.785,479.623 507.377,480.74 507.094,481.513 506.412,483.403 505.742,485.292 505.084,487.182 504.87,487.807 504.436,489.072 503.798,490.962 503.172,492.851 \n",
       "  502.558,494.741 502.363,495.349 501.953,496.631 501.359,498.52 500.776,500.41 500.204,502.3 499.856,503.471 499.642,504.189 499.091,506.079 498.55,507.969 \n",
       "  498.021,509.858 497.502,511.748 497.349,512.317 496.993,513.638 496.495,515.527 496.007,517.417 495.53,519.307 495.064,521.197 494.842,522.116 494.608,523.086 \n",
       "  494.161,524.976 493.726,526.866 493.3,528.755 492.886,530.645 492.482,532.535 492.335,533.239 492.088,534.424 491.703,536.314 491.329,538.204 490.966,540.093 \n",
       "  490.613,541.983 490.27,543.873 489.938,545.762 489.828,546.404 489.615,547.652 489.302,549.542 488.999,551.431 488.707,553.321 488.425,555.211 488.153,557.101 \n",
       "  487.891,558.99 487.64,560.88 487.398,562.77 487.321,563.397 487.166,564.659 486.944,566.549 486.732,568.439 486.531,570.328 486.339,572.218 486.157,574.108 \n",
       "  485.986,575.997 485.824,577.887 485.672,579.777 485.531,581.666 485.399,583.556 485.278,585.446 485.166,587.336 485.064,589.225 484.973,591.115 484.891,593.005 \n",
       "  484.819,594.894 484.814,595.03 484.757,596.784 484.705,598.674 484.662,600.563 484.63,602.453 484.608,604.343 484.595,606.232 484.593,608.122 484.6,610.012 \n",
       "  484.618,611.901 484.645,613.791 484.683,615.681 484.73,617.571 484.787,619.46 484.814,620.217 484.855,621.35 484.932,623.24 485.019,625.129 485.116,627.019 \n",
       "  485.222,628.909 485.339,630.798 485.466,632.688 485.603,634.578 485.75,636.467 485.907,638.357 486.074,640.247 486.251,642.136 486.439,644.026 486.636,645.916 \n",
       "  486.843,647.806 487.061,649.695 487.289,651.585 487.321,651.844 487.526,653.475 487.773,655.364 488.031,657.254 488.299,659.144 488.577,661.033 488.865,662.923 \n",
       "  489.163,664.813 489.472,666.702 489.792,668.592 489.828,668.802 490.12,670.482 490.459,672.371 490.809,674.261 491.169,676.151 491.539,678.041 491.92,679.93 \n",
       "  492.312,681.82 492.335,681.931 492.712,683.71 493.124,685.599 493.546,687.489 493.978,689.379 494.422,691.268 494.842,693.017 494.876,693.158 495.34,695.048 \n",
       "  495.814,696.937 496.299,698.827 496.795,700.717 497.303,702.606 497.349,702.776 497.819,704.496 498.347,706.386 498.885,708.276 499.435,710.165 499.856,711.583 \n",
       "  499.996,712.055 500.566,713.945 501.148,715.834 501.741,717.724 502.345,719.614 502.363,719.668 502.959,721.503 503.585,723.393 504.222,725.283 504.87,727.171 \n",
       "  504.87,727.172 505.529,729.062 506.198,730.952 506.88,732.841 507.377,734.195 507.573,734.731 508.276,736.621 508.992,738.511 509.719,740.4 509.884,740.822 \n",
       "  510.456,742.29 511.206,744.18 511.967,746.069 512.391,747.105 512.74,747.959 513.524,749.849 514.32,751.738 514.898,753.089 515.128,753.628 515.947,755.518 \n",
       "  516.778,757.407 517.405,758.81 517.622,759.297 518.476,761.187 519.344,763.076 519.912,764.296 520.223,764.966 521.114,766.856 522.018,768.746 522.419,769.571 \n",
       "  522.934,770.635 523.862,772.525 524.803,774.415 524.926,774.657 525.755,776.304 526.721,778.194 527.432,779.568 527.699,780.084 528.69,781.973 529.694,783.863 \n",
       "  529.939,784.32 530.71,785.753 531.739,787.642 532.446,788.925 532.781,789.532 533.836,791.422 534.905,793.311 534.953,793.395 535.986,795.201 537.081,797.091 \n",
       "  537.46,797.737 538.189,798.98 539.311,800.87 539.967,801.962 540.446,802.76 541.595,804.65 542.474,806.078 542.758,806.539 543.934,808.429 544.981,810.09 \n",
       "  545.125,810.319 546.329,812.208 547.488,814.005 547.548,814.098 548.78,815.988 549.995,817.827 550.028,817.877 551.289,819.767 552.502,821.562 552.566,821.657 \n",
       "  553.856,823.546 555.009,825.214 555.162,825.436 556.482,827.326 557.516,828.788 557.818,829.215 559.169,831.105 560.023,832.287 560.535,832.995 561.916,834.885 \n",
       "  562.53,835.715 563.313,836.774 564.726,838.664 565.037,839.075 566.154,840.554 567.544,842.371 567.599,842.443 569.059,844.333 570.05,845.602 570.536,846.223 \n",
       "  572.029,848.112 572.557,848.775 573.538,850.002 575.064,851.89 575.066,851.892 576.608,853.781 577.571,854.949 578.168,855.671 579.746,857.561 580.078,857.955 \n",
       "  581.34,859.45 582.585,860.91 582.953,861.34 584.583,863.23 585.092,863.815 586.231,865.12 587.599,866.672 587.897,867.009 589.581,868.899 590.106,869.482 \n",
       "  591.284,870.789 592.613,872.248 593.006,872.678 594.746,874.568 595.12,874.97 596.505,876.458 597.627,877.65 598.284,878.347 600.083,880.237 600.134,880.29 \n",
       "  601.901,882.127 602.641,882.889 603.739,884.016 605.148,885.45 605.598,885.906 607.477,887.796 607.655,887.973 609.377,889.685 610.162,890.459 611.297,891.575 \n",
       "  612.668,892.911 613.24,893.465 615.175,895.328 615.204,895.355 617.189,897.244 617.682,897.71 619.197,899.134 620.189,900.06 621.227,901.024 622.696,902.378 \n",
       "  623.28,902.913 625.203,904.664 625.356,904.803 627.455,906.693 627.71,906.92 629.578,908.582 630.217,909.146 631.726,910.472 632.724,911.343 633.898,912.362 \n",
       "  635.231,913.511 636.094,914.251 637.738,915.651 638.316,916.141 640.245,917.764 640.563,918.031 642.752,919.85 642.837,919.92 645.136,921.81 645.259,921.91 \n",
       "  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",
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       "  705.715,657.254 706.091,659.144 706.483,661.033 706.89,662.923 707.312,664.813 707.75,666.702 707.932,667.463 708.202,668.592 708.668,670.482 709.15,672.371 \n",
       "  709.647,674.261 710.16,676.151 710.439,677.147 710.688,678.041 711.23,679.93 711.788,681.82 712.363,683.71 712.946,685.577 712.953,685.599 713.557,687.489 \n",
       "  714.176,689.379 714.813,691.268 715.453,693.122 715.466,693.158 716.132,695.048 716.815,696.937 717.515,698.827 717.96,700.002 718.23,700.717 718.961,702.606 \n",
       "  719.708,704.496 720.467,706.37 720.473,706.386 721.252,708.276 722.049,710.165 722.863,712.055 722.974,712.307 723.692,713.945 724.538,715.834 725.403,717.724 \n",
       "  725.481,717.891 726.282,719.614 727.18,721.503 727.988,723.169 728.096,723.393 729.027,725.283 729.978,727.172 730.495,728.181 730.946,729.062 731.931,730.952 \n",
       "  732.936,732.841 733.002,732.964 733.956,734.731 734.997,736.621 735.509,737.534 736.055,738.511 737.133,740.4 738.016,741.92 738.23,742.29 739.345,744.18 \n",
       "  740.482,746.069 740.522,746.136 741.635,747.959 742.811,749.849 743.029,750.194 744.006,751.738 745.222,753.628 745.536,754.108 746.458,755.518 747.716,757.407 \n",
       "  748.043,757.891 748.995,759.297 750.297,761.187 750.55,761.549 751.619,763.076 752.966,764.966 753.057,765.092 754.334,766.856 755.564,768.524 755.727,768.746 \n",
       "  757.143,770.635 758.071,771.853 758.584,772.525 760.05,774.415 760.578,775.085 761.541,776.304 763.06,778.194 763.085,778.225 764.602,780.084 765.592,781.274 \n",
       "  766.174,781.973 767.775,783.863 768.099,784.24 769.403,785.753 770.606,787.124 771.062,787.642 772.751,789.532 773.113,789.931 774.47,791.422 775.62,792.662 \n",
       "  776.223,793.311 778.01,795.201 778.127,795.322 779.83,797.091 780.634,797.911 781.686,798.98 783.14,800.434 783.58,800.87 785.511,802.76 785.647,802.891 \n",
       "  787.481,804.65 788.154,805.285 789.492,806.539 790.661,807.618 791.546,808.429 793.168,809.892 793.645,810.319 795.675,812.109 795.789,812.208 797.98,814.098 \n",
       "  798.182,814.269 800.221,815.988 800.689,816.376 802.514,817.877 803.196,818.43 804.861,819.767 805.703,820.434 807.264,821.657 808.21,822.388 809.726,823.546 \n",
       "  810.717,824.294 812.25,825.436 813.224,826.153 814.837,827.326 815.731,827.967 817.492,829.215 818.238,829.737 820.218,831.105 820.745,831.465 823.017,832.995 \n",
       "  823.252,833.151 825.758,834.797 825.894,834.885 828.265,836.404 828.853,836.774 830.772,837.973 831.898,838.664 833.279,839.505 835.032,840.554 835.786,841.001 \n",
       "  838.26,842.443 838.293,842.462 840.8,843.891 841.591,844.333 843.307,845.287 845.026,846.223 845.814,846.65 848.321,847.982 848.571,848.112 850.828,849.285 \n",
       "  852.237,850.002 853.335,850.558 855.842,851.802 856.025,851.892 858.349,853.021 859.949,853.781 860.856,854.211 863.363,855.376 864.012,855.671 865.87,856.516 \n",
       "  868.223,857.561 868.376,857.629 870.883,858.721 872.598,859.45 873.39,859.788 875.897,860.832 877.143,861.34 878.404,861.855 880.911,862.855 881.87,863.23 \n",
       "  883.418,863.836 885.925,864.794 886.793,865.12 888.432,865.734 890.939,866.654 891.93,867.009 893.446,867.555 895.953,868.437 897.296,868.899 898.46,869.301 \n",
       "  900.967,870.147 902.91,870.789 903.474,870.975 905.981,871.788 908.488,872.583 908.796,872.678 910.994,873.363 913.501,874.126 914.985,874.568 916.008,874.874 \n",
       "  918.515,875.607 921.022,876.324 921.499,876.458 923.529,877.028 926.036,877.717 928.38,878.347 928.543,878.391 931.05,879.054 933.557,879.702 935.677,880.237 \n",
       "  936.064,880.335 938.571,880.958 941.078,881.567 943.437,882.127 943.585,882.162 946.092,882.747 948.599,883.319 951.106,883.878 951.738,884.016 953.613,884.427 \n",
       "  956.119,884.965 958.626,885.49 960.661,885.906 961.133,886.003 963.64,886.508 966.147,887 968.654,887.481 970.33,887.796 971.161,887.952 973.668,888.414 \n",
       "  976.175,888.865 978.682,889.305 980.9,889.685 981.189,889.735 983.696,890.157 986.203,890.568 988.71,890.97 991.217,891.361 992.623,891.575 993.724,891.744 \n",
       "  996.231,892.118 998.737,892.482 1001.24,892.837 1003.75,893.183 1005.85,893.465 1006.26,893.52 1008.77,893.849 1011.27,894.169 1013.78,894.481 1016.29,894.783 \n",
       "  1018.79,895.077 1021.23,895.355 1021.3,895.363 1023.81,895.641 1026.31,895.91 1028.82,896.172 1031.33,896.425 1033.83,896.67 1036.34,896.907 1038.85,897.135 \n",
       "  1040.08,897.244 1041.36,897.357 1043.86,897.571 1046.37,897.777 1048.88,897.975 1051.38,898.165 1053.89,898.348 1056.4,898.523 1058.9,898.69 1061.41,898.85 \n",
       "  1063.92,899.002 1066.19,899.134 1066.42,899.147 1068.93,899.285 1071.44,899.416 1073.95,899.539 1076.45,899.655 1078.96,899.763 1081.47,899.864 1083.97,899.958 \n",
       "  1086.48,900.045 1088.99,900.124 1091.49,900.197 1094,900.262 1096.51,900.32 1099.02,900.37 1101.52,900.414 1104.03,900.451 1106.54,900.48 1109.04,900.502 \n",
       "  1111.55,900.517 1114.06,900.525 1116.56,900.526 1119.07,900.52 1121.58,900.507 1124.08,900.487 1126.59,900.459 1129.1,900.425 1131.61,900.383 1134.11,900.334 \n",
       "  1136.62,900.278 1139.13,900.215 1141.63,900.145 1144.14,900.067 1146.65,899.983 1149.15,899.891 1151.66,899.792 1154.17,899.685 1156.67,899.571 1159.18,899.45 \n",
       "  1161.69,899.322 1164.2,899.186 1165.1,899.134 1166.7,899.043 1169.21,898.892 1171.72,898.734 1174.22,898.569 1176.73,898.396 1179.24,898.216 1181.74,898.027 \n",
       "  1184.25,897.831 1186.76,897.628 1189.27,897.416 1191.22,897.244 1191.77,897.196 1194.28,896.97 1196.79,896.735 1199.29,896.492 1201.8,896.242 1204.31,895.982 \n",
       "  1206.81,895.715 1209.32,895.439 1210.07,895.355 1211.83,895.156 1214.33,894.864 1216.84,894.564 1219.35,894.255 1221.86,893.937 1224.36,893.61 1225.45,893.465 \n",
       "  1226.87,893.276 1229.38,892.932 1231.88,892.58 1234.39,892.218 1236.9,891.847 1238.68,891.575 1239.4,891.466 1241.91,891.077 1244.42,890.679 1246.92,890.27 \n",
       "  1249.43,889.851 1250.4,889.685 1251.94,889.423 1254.45,888.986 1256.95,888.538 1259.46,888.079 1260.97,887.796 1261.97,887.611 1264.47,887.133 1266.98,886.643 \n",
       "  1269.49,886.142 1270.64,885.906 1271.99,885.631 1274.5,885.109 1277.01,884.575 1279.51,884.029 1279.57,884.016 1282.02,883.473 1284.53,882.905 1287.04,882.323 \n",
       "  1287.86,882.127 1289.54,881.73 1292.05,881.125 1294.56,880.507 1295.62,880.237 1297.06,879.876 1299.57,879.232 1302.08,878.574 1302.92,878.347 1304.58,877.903 \n",
       "  1307.09,877.218 1309.6,876.517 1309.81,876.458 1312.11,875.804 1314.61,875.076 1316.32,874.568 1317.12,874.331 1319.63,873.573 1322.13,872.797 1322.51,872.678 \n",
       "  1324.64,872.007 1327.15,871.199 1328.39,870.789 1329.65,870.375 1332.16,869.533 1334.01,868.899 1334.67,868.674 1337.17,867.797 1339.38,867.009 1339.68,866.9 \n",
       "  1342.19,865.987 1344.51,865.12 1344.7,865.052 1347.2,864.099 1349.44,863.23 1349.71,863.124 1352.22,862.13 1354.16,861.34 1354.72,861.113 1357.23,860.075 \n",
       "  1358.71,859.45 1359.74,859.014 1362.24,857.929 1363.08,857.561 1364.75,856.822 1367.26,855.688 1367.29,855.671 1369.76,854.532 1371.35,853.781 1372.27,853.348 \n",
       "  1374.78,852.137 1375.28,851.892 1377.29,850.901 1379.07,850.002 1379.79,849.635 1382.3,848.34 1382.73,848.112 1384.81,847.016 1386.28,846.223 1387.31,845.661 \n",
       "  1389.71,844.333 1389.82,844.274 1392.33,842.856 1393.04,842.443 1394.83,841.404 1396.27,840.554 1397.34,839.917 1399.41,838.664 1399.85,838.394 1402.36,836.835 \n",
       "  1402.45,836.774 1404.86,835.239 1405.41,834.885 1407.37,833.604 1408.29,832.995 1409.88,831.929 1411.09,831.105 1412.38,830.213 1413.81,829.215 1414.89,828.454 \n",
       "  1416.47,827.326 1417.4,826.652 1419.05,825.436 1419.9,824.805 1421.58,823.546 1422.41,822.912 1424.04,821.657 1424.92,820.971 1426.44,819.767 1427.42,818.981 \n",
       "  1428.79,817.877 1429.93,816.941 1431.08,815.988 1432.44,814.849 1433.32,814.098 1434.95,812.703 1435.52,812.208 1437.45,810.502 1437.66,810.319 1439.76,808.429 \n",
       "  1439.96,808.244 1441.81,806.539 1442.47,805.927 1443.82,804.65 1444.97,803.55 1445.79,802.76 1447.48,801.11 1447.72,800.87 1449.62,798.98 1449.99,798.606 \n",
       "  1451.47,797.091 1452.49,796.034 1453.29,795.201 1455,793.395 1455.08,793.311 1456.83,791.422 1457.51,790.683 1458.55,789.532 1460.01,787.898 1460.24,787.642 \n",
       "  1461.9,785.753 1462.52,785.034 1463.53,783.863 1465.03,782.092 1465.13,781.973 1466.7,780.084 1467.54,779.064 1468.25,778.194 1469.76,776.304 1470.04,775.95 \n",
       "  1471.25,774.415 1472.55,772.745 1472.72,772.525 1474.16,770.635 1475.06,769.441 1475.58,768.746 1476.97,766.856 1477.56,766.037 1478.34,764.966 1479.68,763.076 \n",
       "  1480.07,762.526 1481.01,761.187 1482.31,759.297 1482.58,758.9 1483.59,757.407 1484.85,755.518 1485.08,755.153 1486.08,753.628 1487.3,751.738 1487.59,751.275 \n",
       "  1488.49,749.849 1489.67,747.959 1490.1,747.256 1490.82,746.069 1491.96,744.18 1492.6,743.085 1493.07,742.29 1494.17,740.4 1495.11,738.749 1495.25,738.511 \n",
       "  1496.31,736.621 1497.35,734.731 1497.62,734.229 1498.37,732.841 1499.37,730.952 1500.13,729.508 1500.36,729.062 1501.33,727.172 1502.28,725.283 1502.63,724.559 \n",
       "  1503.21,723.393 1504.12,721.503 1505.02,719.614 1505.14,719.358 1505.9,717.724 1506.77,715.834 1507.61,713.945 1507.65,713.864 1508.44,712.055 1509.26,710.165 \n",
       "  1510.05,708.276 1510.15,708.027 1510.83,706.386 1511.6,704.496 1512.34,702.606 1512.66,701.785 1513.07,700.717 1513.79,698.827 1514.49,696.937 1515.17,695.058 \n",
       "  1515.17,695.048 1515.84,693.158 1516.49,691.268 1517.13,689.379 1517.67,687.709 1517.75,687.489 1518.35,685.599 1518.94,683.71 1519.52,681.82 1520.07,679.93 \n",
       "  1520.18,679.552 1520.62,678.041 1521.14,676.151 1521.66,674.261 1522.15,672.371 1522.63,670.482 1522.69,670.266 1523.1,668.592 1523.56,666.702 1523.99,664.813 \n",
       "  1524.41,662.923 1524.82,661.033 1525.2,659.223 1525.21,659.144 1525.59,657.254 1525.95,655.364 1526.3,653.475 1526.63,651.585 1526.95,649.695 1527.25,647.806 \n",
       "  1527.54,645.916 1527.7,644.802 1527.81,644.026 1528.07,642.136 1528.32,640.247 1528.55,638.357 1528.76,636.467 1528.96,634.578 1529.15,632.688 1529.32,630.798 \n",
       "  1529.48,628.909 1529.62,627.019 1529.75,625.129 1529.86,623.24 1529.96,621.35 1530.04,619.46 1530.11,617.571 1530.17,615.681 1530.21,613.791 1530.21,613.5 \n",
       "  1530.23,611.901 1530.24,610.012 1530.24,608.122 1530.22,606.232 1530.21,605.645 1530.19,604.343 1530.14,602.453 1530.08,600.563 1530,598.674 1529.91,596.784 \n",
       "  1529.8,594.894 1529.68,593.005 1529.55,591.115 1529.4,589.225 1529.23,587.336 1529.05,585.446 1528.86,583.556 1528.65,581.666 1528.42,579.777 1528.18,577.887 \n",
       "  1527.93,575.997 1527.7,574.42 1527.66,574.108 1527.37,572.218 1527.08,570.328 1526.76,568.439 1526.43,566.549 1526.09,564.659 1525.73,562.77 1525.35,560.88 \n",
       "  1525.2,560.132 1524.96,558.99 1524.55,557.101 1524.13,555.211 1523.7,553.321 1523.24,551.431 1522.78,549.542 1522.69,549.202 1522.29,547.652 1521.79,545.762 \n",
       "  1521.28,543.873 1520.75,541.983 1520.2,540.093 1520.18,540.031 1519.64,538.204 1519.06,536.314 1518.47,534.424 1517.85,532.535 1517.67,531.995 1517.23,530.645 \n",
       "  1516.58,528.755 1515.92,526.866 1515.25,524.976 1515.17,524.761 1514.55,523.086 1513.85,521.197 1513.12,519.307 1512.66,518.145 1512.37,517.417 1511.61,515.527 \n",
       "  1510.84,513.638 1510.15,512.014 1510.04,511.748 1509.23,509.858 1508.4,507.969 1507.65,506.286 1507.55,506.079 1506.69,504.189 1505.81,502.3 1505.14,500.897 \n",
       "  1504.91,500.41 1503.99,498.52 1503.05,496.631 1502.63,495.797 1502.1,494.741 1501.13,492.851 1500.13,490.962 1500.13,490.945 1499.13,489.072 1498.1,487.182 \n",
       "  1497.62,486.321 1497.05,485.292 1495.98,483.403 1495.11,481.891 1494.89,481.513 1493.79,479.623 1492.66,477.734 1492.6,477.638 1491.52,475.844 1490.35,473.954 \n",
       "  1490.1,473.548 1489.17,472.065 1487.96,470.175 1487.59,469.605 1486.73,468.285 1485.49,466.396 1485.08,465.795 1484.22,464.506 1482.93,462.616 1482.58,462.11 \n",
       "  1481.62,460.727 1480.28,458.837 1480.07,458.539 1478.93,456.947 1477.56,455.075 1477.55,455.057 1476.15,453.168 1475.06,451.712 1474.73,451.278 1473.28,449.388 \n",
       "  1472.55,448.443 1471.81,447.499 1470.32,445.609 1470.04,445.26 1468.81,443.719 1467.54,442.161 1467.26,441.83 1465.7,439.94 1465.03,439.14 1464.11,438.05 \n",
       "  1462.52,436.192 1462.49,436.161 1460.85,434.271 1460.01,433.317 1459.19,432.381 1457.51,430.507 1457.49,430.492 1455.78,428.602 1455,427.762 1454.03,426.712 \n",
       "  1452.49,425.077 1452.25,424.822 1450.45,422.933 1449.99,422.451 1448.62,421.043 1447.48,419.881 1446.76,419.153 1444.97,417.364 1444.87,417.264 1442.95,415.374 \n",
       "  1442.47,414.9 1441,413.484 1439.96,412.485 1439.02,411.595 1437.45,410.117 1437.01,409.705 1434.97,407.815 1434.95,407.796 1432.89,405.926 1432.44,405.519 \n",
       "  1430.78,404.036 1429.93,403.285 1428.64,402.146 1427.42,401.093 1426.46,400.257 1424.92,398.941 1424.24,398.367 1422.41,396.829 1421.99,396.477 1419.9,394.754 \n",
       "  1419.7,394.587 1417.4,392.716 1417.37,392.698 1415.01,390.808 1414.89,390.714 1412.6,388.918 1412.38,388.746 1410.16,387.029 1409.88,386.813 1407.67,385.139 \n",
       "  1407.37,384.912 1405.14,383.249 1404.86,383.044 1402.56,381.36 1402.36,381.207 1399.94,379.47 1399.85,379.401 1397.34,377.624 1397.28,377.58 1394.83,375.877 \n",
       "  1394.56,375.691 1392.33,374.159 1391.8,373.801 1389.82,372.468 1388.99,371.911 1387.31,370.804 1386.12,370.022 1384.81,369.168 1383.2,368.132 1382.3,367.557 \n",
       "  1380.22,366.242 1379.79,365.972 1377.29,364.413 1377.19,364.352 1374.78,362.878 1374.09,362.463 1372.27,361.367 1370.94,360.573 1369.76,359.88 1367.72,358.683 \n",
       "  1367.26,358.417 1364.75,356.976 1364.43,356.794 1362.24,355.558 1361.07,354.904 1359.74,354.162 1357.64,353.014 1357.23,352.788 1354.72,351.435 1354.14,351.125 \n",
       "  1352.22,350.104 1350.56,349.235 1349.71,348.793 1347.2,347.503 1346.89,347.345 1344.7,346.233 1343.14,345.456 1342.19,344.983 1339.68,343.752 1339.3,343.566 \n",
       "  1337.17,342.541 1335.36,341.676 1334.67,341.349 1332.16,340.176 1331.32,339.787 1329.65,339.02 1327.17,337.897 1327.15,337.885 1324.64,336.766 1322.91,336.007 \n",
       "  1322.13,335.665 1319.63,334.582 1318.53,334.117 1317.12,333.517 1314.61,332.469 1314.03,332.228 1312.11,331.438 1309.6,330.424 1309.38,330.338 1307.09,329.426 \n",
       "  1304.59,328.448 1304.58,328.445 1302.08,327.48 1299.64,326.559 1299.57,326.532 1297.06,325.598 1294.56,324.682 1294.52,324.669 1292.05,323.78 1289.54,322.895 \n",
       "  1289.21,322.779 1287.04,322.024 1284.53,321.169 1283.69,320.89 1282.02,320.329 1279.51,319.504 1277.95,319 1277.01,318.694 1274.5,317.898 1271.99,317.118 \n",
       "  1271.97,317.11 1269.49,316.351 1266.98,315.599 1265.69,315.221 1264.47,314.862 1261.97,314.137 1259.46,313.429 1259.11,313.331 1256.95,312.732 1254.45,312.05 \n",
       "  1252.16,311.441 1251.94,311.383 1249.43,310.728 1246.92,310.087 1244.78,309.552 1244.42,309.46 1241.91,308.845 1239.4,308.245 1236.91,307.662 1236.9,307.658 \n",
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     },
     "execution_count": 151,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "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)"
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "\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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   "cell_type": "markdown",
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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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   "metadata": {},
   "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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   "cell_type": "markdown",
   "metadata": {},
   "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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   "source": [
    "## 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"
    }
   ],
   "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"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
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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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     "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"
      ]
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     "execution_count": 91,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "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)"
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     },
     "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)"
   ]
  }
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