Lecture 24.ipynb

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    "using Plots, ComplexPhasePortrait, ApproxFun, SingularIntegralEquations,\n",
    "        SpecialFunctions\n",
    "gr();"
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    "# M3M6: Methods of Mathematical Physics\n",
    "\n",
    "$$\n",
    "\\def\\dashint{{\\int\\!\\!\\!\\!\\!\\!-\\,}}\n",
    "\\def\\infdashint{\\dashint_{\\!\\!\\!-\\infty}^{\\,\\infty}}\n",
    "\\def\\D{\\,{\\rm d}}\n",
    "\\def\\E{{\\rm e}}\n",
    "\\def\\dx{\\D x}\n",
    "\\def\\dt{\\D t}\n",
    "\\def\\dz{\\D z}\n",
    "\\def\\C{{\\mathbb C}}\n",
    "\\def\\R{{\\mathbb R}}\n",
    "\\def\\H{{\\mathbb H}}\n",
    "\\def\\CC{{\\cal C}}\n",
    "\\def\\HH{{\\cal H}}\n",
    "\\def\\FF{{\\cal F}}\n",
    "\\def\\I{{\\rm i}}\n",
    "\\def\\qqqquad{\\qquad\\qquad}\n",
    "\\def\\qqand{\\qquad\\hbox{and}\\qquad}\n",
    "\\def\\qqfor{\\qquad\\hbox{for}\\qquad}\n",
    "\\def\\qqwhere{\\qquad\\hbox{where}\\qquad}\n",
    "\\def\\Res_#1{\\underset{#1}{\\rm Res}}\\,\n",
    "\\def\\sech{{\\rm sech}\\,}\n",
    "\\def\\acos{\\,{\\rm acos}\\,}\n",
    "\\def\\erfc{\\,{\\rm erfc}\\,}\n",
    "\\def\\vc#1{{\\mathbf #1}}\n",
    "\\def\\ip<#1,#2>{\\left\\langle#1,#2\\right\\rangle}\n",
    "\\def\\br[#1]{\\left[#1\\right]}\n",
    "\\def\\norm#1{\\left\\|#1\\right\\|}\n",
    "\\def\\half{{1 \\over 2}}\n",
    "\\def\\fL{f_{\\rm L}}\n",
    "\\def\\fR{f_{\\rm R}}\n",
    "\\def\\questionequals{= \\!\\!\\!\\!\\!\\!{\\scriptstyle ? \\atop }\\,\\,\\,}\n",
    "$$\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",
    "\n",
    "\n",
    "# Lecture 24:  Riemann–Hilbert problems on the real line\n",
    "\n",
    "Last lecture we saw that  integral equation\n",
    "$$\n",
    "\\lambda u(x) + \\int_{0}^\\infty K(x-t)u(t) \\dt = f(x)\\qqfor 0 < x < \\infty.\n",
    "$$\n",
    "could be transformmed into a Riemann–Hilbert problem: find $\\Phi(z)$ analytic off $\\R$ such that\n",
    "$$\n",
    "\\Phi_+(s) - g(s)\\Phi_-(s) = h(s) \\qqand \\Phi(\\infty) = C\n",
    "$$\n",
    "where $\\Phi_\\pm(s) = \\lim_{\\epsilon \\rightarrow 0} \\Phi(s \\pm \\I \\epsilon)$ are the limits from above and below.  \n",
    "\n",
    "We now wish to study the solution of this equations, using the Plemelj theorem and Cauchy transform of last lecture.\n",
    "\n",
    "Outline:\n",
    "\n",
    "3. Solving variable coefficient Riemann–Hilbert problems\n",
    "   - Homogeneous problems: $h = 0$\n",
    "   - Inhomogeneous problems: $h \\neq 0$\n",
    "   \n",
    "   \n",
    "  "
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    "### Homogenous Riemann–Hilbert problems on the real line\n",
    "\n",
    "Back to the Riemann–Hilbert problem, but let's assume $h$ is zero. Consider\n",
    "$$\n",
    "\\kappa_+(s) = g(s) \\kappa_-(s) \\qqand \\kappa(\\infty) = I\n",
    "$$\n",
    "\n",
    "Formally, taking logs of both sides reduces this to a subtractive RH problem:\n",
    "$$\n",
    "\\log \\kappa_+(s) - \\log\\kappa_-(s) \\questionequals \\log g(s)\n",
    "$$\n",
    "Assuming that $g(s) \\rightarrow 1$ as $s \\rightarrow \\pm \\infty$ at a sufficient rate, this motivates the guess\n",
    "$$\n",
    "\\kappa(z) = \\E^{\\CC[\\log g](z)}\n",
    "$$\n",
    "\n",
    "Assuming $\\log g(s)$ is \"nice\", we have guaranteed that\n",
    "\n",
    "**Theorem (Homogeneous solution to RH problem)** Suppose $\\log g(s)$ satisfies the conditions of Plemelj on the circle. Then $\\kappa(z) = \\E^{\\CC_\\R[\\log g](z)}$ is the unique solution to the following RH problem:\n",
    "1. $\\kappa(z)$ is analytic off $\\R$\n",
    "2. $\\lim_{z\\rightarrow \\infty}\\kappa(z) = 1$ \n",
    "2. $\\kappa_+(s) = g(s) \\kappa_-(s)$ for $s \\in \\R$\n",
    "\n",
    "\n",
    "**Proof** (1) follows from definition. (2) follows since $\\CC[\\log g](z) \\rightarrow 0$. And (3) follows via:\n",
    "$$\n",
    "\\kappa_+(s) = \\E^{\\CC_+[\\log g](s)} = \\E^{\\CC_-[\\log g](s) + \\log g(s)} = \\kappa_-(s) g(s)\n",
    "$$\n",
    "To see uniqueness, observe that $\\kappa$ must be invertible, as it is an exponential of something finite. Thus $\\kappa(z)^{-1}$ is also analytic off $\\R$. Therefore, if we have another solution $\\tilde \\kappa(z)$ we can consider $r(z) = \\tilde\\kappa(z) \\kappa(z)^{-1}$ which satisfies:\n",
    "$$\n",
    "r_+(s) = {\\tilde\\kappa_+(s) \\over \\kappa_+(s)} = {\\tilde\\kappa_-(s) g(s) \\over \\kappa_-(s) g(s)} = r_-(s)\n",
    "$$\n",
    "Hence $r(z)$ is entire. since both terms tend to $1$, it must be $r(z) = 1$.\n",
    "\n",
    "\n",
    "⬛️\n",
    "\n",
    "When is $\\log g(s)$ nice? It is sufficient that $g(s) = 1 + O(s^{-1})$ at $\\pm \\infty$ but also we need to worry about the image: for example, $g(s) \\neq 0$ is required to avoid a singularity.  We similarly need that the winding number of the image of $g(s)$ to be zero: otherwise, $\\log g(s)$ will extend to another sheet.   For example, if $g(s) = {z-\\I \\over z+\\I}$ it satisfies the right asymptotics, but surrounds the origin:"
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       "  2202.76,510.911 2202.15,509.93 2201.54,508.941 2200.92,507.945 2200.29,506.943 2199.65,505.933 2199.01,504.915 2198.36,503.891 2197.7,502.859 2197.04,501.819 \n",
       "  2196.36,500.772 2195.68,499.717 2194.99,498.655 2194.29,497.584 2193.59,496.506 2192.87,495.42 2192.15,494.326 2191.42,493.224 2190.67,492.113 2189.92,490.994 \n",
       "  2189.16,489.867 2188.39,488.731 2187.62,487.587 2186.83,486.434 2186.03,485.272 2185.22,484.102 2184.4,482.922 2183.57,481.734 2182.73,480.536 2181.88,479.329 \n",
       "  2181.02,478.113 2180.15,476.888 2179.27,475.653 2178.37,474.408 2177.47,473.153 2176.55,471.889 2175.62,470.615 2174.68,469.331 2173.72,468.036 2172.75,466.732 \n",
       "  2171.77,465.417 2170.78,464.091 2169.77,462.755 2168.75,461.408 2167.72,460.05 2166.67,458.682 2165.61,457.302 2164.53,455.911 2163.44,454.509 2162.33,453.095 \n",
       "  2161.21,451.67 2160.07,450.233 2158.91,448.784 2157.74,447.323 2156.56,445.85 2155.35,444.365 2154.13,442.868 2152.89,441.358 2151.64,439.836 2150.36,438.3 \n",
       "  2149.07,436.752 2147.76,435.191 2146.43,433.617 2145.08,432.029 2143.71,430.428 2142.31,428.813 2140.9,427.184 2139.47,425.541 2138.02,423.885 2136.54,422.214 \n",
       "  2135.04,420.528 2133.52,418.828 2131.98,417.114 2130.41,415.384 2128.82,413.64 2127.2,411.88 2125.56,410.105 2123.89,408.314 2122.19,406.507 2120.47,404.685 \n",
       "  2118.72,402.847 2116.95,400.992 2115.14,399.121 2113.31,397.233 2111.45,395.329 2109.55,393.407 2107.63,391.469 2105.67,389.513 2103.68,387.54 2101.66,385.549 \n",
       "  2099.61,383.54 2097.52,381.513 2095.39,379.468 2093.23,377.405 2091.04,375.323 2088.8,373.222 2086.53,371.103 2084.22,368.964 2081.87,366.806 2079.48,364.628 \n",
       "  2077.04,362.43 2074.57,360.213 2072.05,357.976 2069.48,355.718 2066.87,353.44 2064.21,351.141 2061.51,348.822 2058.76,346.481 2055.95,344.12 2053.1,341.737 \n",
       "  2050.19,339.332 2047.23,336.906 2044.21,334.458 2041.14,331.988 2038.01,329.496 2034.83,326.982 2031.58,324.445 2028.27,321.886 2024.9,319.304 2021.46,316.699 \n",
       "  2017.96,314.071 2014.39,311.42 2010.75,308.746 2007.03,306.048 2003.25,303.327 1999.39,300.583 1995.45,297.815 1991.44,295.024 1987.35,292.209 1983.17,289.371 \n",
       "  1978.91,286.509 1974.56,283.623 1970.12,280.713 1965.59,277.781 1960.97,274.824 1956.25,271.845 1951.44,268.842 1946.52,265.816 1941.5,262.767 1936.38,259.695 \n",
       "  1931.14,256.6 1925.8,253.484 1920.33,250.345 1914.76,247.185 1909.06,244.004 1903.23,240.801 1897.28,237.578 1891.2,234.336 1884.99,231.074 1878.64,227.793 \n",
       "  1872.14,224.494 1865.51,221.178 1858.72,217.846 1851.78,214.497 1844.69,211.134 1837.43,207.758 1830.01,204.369 1822.42,200.968 1814.65,197.558 1806.71,194.138 \n",
       "  1798.58,190.712 1790.26,187.28 1781.76,183.844 1773.05,180.406 1764.14,176.969 1755.02,173.533 1745.68,170.102 1736.13,166.678 1726.35,163.263 1716.34,159.861 \n",
       "  1706.09,156.474 1695.59,153.106 1684.85,149.76 1673.85,146.439 1662.59,143.148 1651.06,139.891 1639.25,136.671 1627.16,133.494 1614.79,130.365 1602.11,127.289 \n",
       "  1589.14,124.271 1575.85,121.318 1562.25,118.435 1548.33,115.63 1534.07,112.91 1519.48,110.281 1504.55,107.752 1489.26,105.332 1473.62,103.028 1457.62,100.851 \n",
       "  1441.25,98.8107 1424.5,96.9165 1407.37,95.1797 1389.86,93.612 1371.96,92.2252 1353.66,91.0321 1334.96,90.046 1315.87,89.2806 1296.37,88.7507 1276.46,88.4712 \n",
       "  1256.14,88.4582 1235.42,88.728 1214.3,89.2979 1192.77,90.1856 1170.84,91.4097 1148.51,92.9892 1125.79,94.9439 1102.69,97.294 1079.22,100.06 1055.39,103.265 \n",
       "  1031.21,106.928 1006.7,111.073 981.877,115.723 956.757,120.899 931.367,126.625 905.73,132.924 879.876,139.818 853.835,147.331 827.642,155.484 801.335,164.298 \n",
       "  774.953,173.795 748.54,183.995 722.144,194.916 695.815,206.577 669.607,218.991 643.576,232.175 617.783,246.139 592.29,260.892 567.164,276.443 542.472,292.795 \n",
       "  518.285,309.949 494.676,327.902 471.719,346.65 449.489,366.181 428.061,386.483 407.513,407.537 387.918,429.323 369.353,451.813 351.89,474.978 335.599,498.782 \n",
       "  320.549,523.187 306.803,548.149 294.422,573.622 283.46,599.556 273.968,625.896 265.988,652.585 259.558,679.563 254.709,706.77 251.464,734.14 249.837,761.608 \n",
       "  249.837,789.11 251.464,816.579 254.709,843.949 259.558,871.155 265.988,898.134 273.968,924.823 283.46,951.163 294.422,977.096 306.803,1002.57 320.549,1027.53 \n",
       "  335.599,1051.94 351.89,1075.74 369.353,1098.91 387.918,1121.4 407.513,1143.18 428.061,1164.24 449.489,1184.54 471.719,1204.07 494.676,1222.82 518.285,1240.77 \n",
       "  542.472,1257.92 567.164,1274.28 592.29,1289.83 617.783,1304.58 643.576,1318.54 669.607,1331.73 695.815,1344.14 722.144,1355.8 748.54,1366.72 774.953,1376.92 \n",
       "  801.335,1386.42 827.642,1395.24 853.835,1403.39 879.876,1410.9 905.73,1417.79 931.367,1424.09 956.757,1429.82 981.877,1435 1006.7,1439.65 1031.21,1443.79 \n",
       "  1055.39,1447.45 1079.22,1450.66 1102.69,1453.42 1125.79,1455.77 1148.51,1457.73 1170.84,1459.31 1192.77,1460.53 1214.3,1461.42 1235.42,1461.99 1256.14,1462.26 \n",
       "  1276.46,1462.25 1296.37,1461.97 1315.87,1461.44 1334.96,1460.67 1353.66,1459.69 1371.96,1458.49 1389.86,1457.11 1407.37,1455.54 1424.5,1453.8 1441.25,1451.91 \n",
       "  1457.62,1449.87 1473.62,1447.69 1489.26,1445.39 1504.55,1442.97 1519.48,1440.44 1534.07,1437.81 1548.33,1435.09 1562.25,1432.28 1575.85,1429.4 1589.14,1426.45 \n",
       "  1602.11,1423.43 1614.79,1420.35 1627.16,1417.22 1639.25,1414.05 1651.06,1410.83 1662.59,1407.57 1673.85,1404.28 1684.85,1400.96 1695.59,1397.61 1706.09,1394.24 \n",
       "  1716.34,1390.86 1726.35,1387.46 1736.13,1384.04 1745.68,1380.62 1755.02,1377.19 1764.14,1373.75 1773.05,1370.31 1781.76,1366.87 1790.26,1363.44 1798.58,1360.01 \n",
       "  1806.71,1356.58 1814.65,1353.16 1822.42,1349.75 1830.01,1346.35 1837.43,1342.96 1844.69,1339.58 1851.78,1336.22 1858.72,1332.87 1865.51,1329.54 1872.14,1326.22 \n",
       "  1878.64,1322.93 1884.99,1319.65 1891.2,1316.38 1897.28,1313.14 1903.23,1309.92 1909.06,1306.72 1914.76,1303.53 1920.33,1300.37 1925.8,1297.23 1931.14,1294.12 \n",
       "  1936.38,1291.02 1941.5,1287.95 1946.52,1284.9 1951.44,1281.88 1956.25,1278.87 1960.97,1275.89 1965.59,1272.94 1970.12,1270.01 1974.56,1267.1 1978.91,1264.21 \n",
       "  1983.17,1261.35 1987.35,1258.51 1991.44,1255.69 1995.45,1252.9 1999.39,1250.14 2003.25,1247.39 2007.03,1244.67 2010.75,1241.97 2014.39,1239.3 2017.96,1236.65 \n",
       "  2021.46,1234.02 2024.9,1231.42 2028.27,1228.83 2031.58,1226.27 2034.83,1223.74 2038.01,1221.22 2041.14,1218.73 2044.21,1216.26 2047.23,1213.81 2050.19,1211.39 \n",
       "  2053.1,1208.98 2055.95,1206.6 2058.76,1204.24 2061.51,1201.9 2064.21,1199.58 2066.87,1197.28 2069.48,1195 2072.05,1192.74 2074.57,1190.51 2077.04,1188.29 \n",
       "  2079.48,1186.09 2081.87,1183.91 2084.22,1181.76 2086.53,1179.62 2088.8,1177.5 2091.04,1175.4 2093.23,1173.31 2095.39,1171.25 2097.52,1169.21 2099.61,1167.18 \n",
       "  2101.66,1165.17 2103.68,1163.18 2105.67,1161.21 2107.63,1159.25 2109.55,1157.31 2111.45,1155.39 2113.31,1153.49 2115.14,1151.6 2116.95,1149.73 2118.72,1147.87 \n",
       "  2120.47,1146.03 2122.19,1144.21 2123.89,1142.41 2125.56,1140.61 2127.2,1138.84 2128.82,1137.08 2130.41,1135.33 2131.98,1133.61 2133.52,1131.89 2135.04,1130.19 \n",
       "  2136.54,1128.51 2138.02,1126.83 2139.47,1125.18 2140.9,1123.53 2142.31,1121.91 2143.71,1120.29 2145.08,1118.69 2146.43,1117.1 2147.76,1115.53 2149.07,1113.97 \n",
       "  2150.36,1112.42 2151.64,1110.88 2152.89,1109.36 2154.13,1107.85 2155.35,1106.35 2156.56,1104.87 2157.74,1103.4 2158.91,1101.93 2160.07,1100.49 2161.21,1099.05 \n",
       "  2162.33,1097.62 2163.44,1096.21 2164.53,1094.81 2165.61,1093.42 2166.67,1092.04 2167.72,1090.67 2168.75,1089.31 2169.77,1087.96 2170.78,1086.63 2171.77,1085.3 \n",
       "  2172.75,1083.99 2173.72,1082.68 2174.68,1081.39 2175.62,1080.1 2176.55,1078.83 2177.47,1077.57 2178.37,1076.31 2179.27,1075.07 2180.15,1073.83 2181.02,1072.61 \n",
       "  2181.88,1071.39 2182.73,1070.18 2183.57,1068.98 2184.4,1067.8 2185.22,1066.62 2186.03,1065.45 2186.83,1064.28 2187.62,1063.13 2188.39,1061.99 2189.16,1060.85 \n",
       "  2189.92,1059.72 2190.67,1058.61 2191.42,1057.5 2192.15,1056.39 2192.87,1055.3 2193.59,1054.21 2194.29,1053.13 2194.99,1052.06 2195.68,1051 2196.36,1049.95 \n",
       "  2197.04,1048.9 2197.7,1047.86 2198.36,1046.83 2199.01,1045.8 2199.65,1044.79 2200.29,1043.78 2200.92,1042.77 2201.54,1041.78 2202.15,1040.79 2202.76,1039.81 \n",
       "  2203.36,1038.83 2203.95,1037.86 2204.54,1036.9 2205.12,1035.95 2205.69,1035 2206.26,1034.06 2206.82,1033.13 2207.37,1032.2 2207.92,1031.28 2208.46,1030.36 \n",
       "  2209,1029.45 2209.53,1028.55 2210.06,1027.65 2210.57,1026.76 2211.09,1025.87 2211.6,1025 2212.1,1024.12 2212.6,1023.26 2213.09,1022.39 2213.57,1021.54 \n",
       "  2214.06,1020.69 2214.53,1019.84 2215,1019.01 2215.47,1018.17 2215.93,1017.34 2216.39,1016.52 2216.84,1015.7 2217.29,1014.89 2217.73,1014.09 2218.17,1013.29 \n",
       "  2218.6,1012.49 2219.03,1011.7 2219.46,1010.91 2219.88,1010.13 2220.29,1009.36 2220.71,1008.59 2221.11,1007.82 2221.52,1007.06 2221.92,1006.3 2222.31,1005.55 \n",
       "  2222.71,1004.8 2223.09,1004.06 2223.48,1003.32 2223.86,1002.59 2224.24,1001.86 2224.61,1001.14 2224.98,1000.42 2225.34,999.705 2225.71,998.994 2226.07,998.288 \n",
       "  2226.42,997.586 2226.77,996.889 2227.12,996.195 2227.47,995.506 2227.81,994.821 2228.15,994.14 2228.48,993.463 2228.82,992.79 2229.15,992.122 2229.47,991.457 \n",
       "  2229.8,990.796 2230.12,990.139 2230.43,989.486 2230.75,988.837 2231.06,988.192 2231.37,987.55 2231.67,986.912 2231.98,986.278 2232.28,985.648 2232.57,985.021 \n",
       "  2232.87,984.398 2233.16,983.779 2233.45,983.163 2233.74,982.55 2234.02,981.941 2234.3,981.336 2234.58,980.734 2234.86,980.136 2235.13,979.54 2235.41,978.949 \n",
       "  2235.68,978.36 2235.94,977.775 2236.21,977.193 2236.47,976.615 2236.73,976.039 2236.99,975.467 2237.24,974.898 2237.5,974.332 2237.75,973.769 2238,973.21 \n",
       "  2238.25,972.653 2238.49,972.1 2238.73,971.549 2238.97,971.002 2239.21,970.457 2239.45,969.916 2239.68,969.377 2239.92,968.842 2240.15,968.309 2240.38,967.779 \n",
       "  2240.6,967.252 2240.83,966.727 2241.05,966.206 2241.27,965.687 2241.49,965.171 2241.71,964.658 2241.93,964.148 2242.14,963.64 2242.36,963.135 2242.57,962.632 \n",
       "  2242.78,962.132 2242.98,961.635 2243.19,961.14 2243.39,960.648 2243.6,960.159 2243.8,959.672 2244,959.187 2244.19,958.705 2244.39,958.225 2244.58,957.748 \n",
       "  2244.78,957.274 2244.97,956.801 2245.16,956.331 2245.35,955.864 2245.53,955.399 2245.72,954.936 2245.9,954.476 2246.09,954.018 2246.27,953.562 2246.45,953.108 \n",
       "  2246.63,952.657 2246.8,952.208 2246.98,951.761 2247.15,951.316 2247.33,950.874 2247.5,950.434 2247.67,949.996 2247.84,949.56 2248.01,949.126 2248.17,948.694 \n",
       "  2248.34,948.265 2248.5,947.837 2248.66,947.412 2248.83,946.988 2248.99,946.567 2249.15,946.148 2249.3,945.731 2249.46,945.315 2249.62,944.902 2249.77,944.491 \n",
       "  2249.93,944.081 2250.08,943.674 2250.23,943.269 2250.38,942.865 2250.53,942.463 2250.68,942.064 2250.82,941.666 2250.97,941.27 2251.11,940.875 2251.26,940.483 \n",
       "  2251.4,940.093 2251.54,939.704 2251.68,939.317 2251.82,938.932 2251.96,938.549 2252.1,938.167 2252.24,937.787 2252.37,937.409 2252.51,937.033 2252.64,936.658 \n",
       "  2252.77,936.285 2252.91,935.914 2253.04,935.545 2253.17,935.177 2253.3,934.811 2253.43,934.446 2253.55,934.083 2253.68,933.722 2253.81,933.362 2253.93,933.004 \n",
       "  2254.06,932.648 2254.18,932.293 2254.3,931.94 2254.42,931.588 2254.54,931.238 2254.66,930.889 2254.78,930.542 2254.9,930.197 2255.02,929.852 2255.14,929.51 \n",
       "  2255.25,929.169 2255.37,928.829 2255.48,928.491 2255.59,928.155 2255.71,927.819 2255.82,927.486 2255.93,927.153 2256.04,926.823 2256.15,926.493 2256.26,926.165 \n",
       "  2256.37,925.839 2256.48,925.513 2256.58,925.189 2256.69,924.867 2256.8,924.546 2256.9,924.226 2257.01,923.908 2257.11,923.591 2257.21,923.275 2257.32,922.961 \n",
       "  2257.42,922.648 2257.52,922.336 2257.62,922.025 2257.72,921.716 2257.82,921.408 2257.92,921.102 2258.01,920.796 2258.11,920.492 2258.21,920.19 2258.3,919.888 \n",
       "  2258.4,919.588 2258.49,919.289 2258.59,918.991 2258.68,918.694 2258.78,918.399 2258.87,918.104 2258.96,917.811 2259.05,917.519 2259.14,917.229 2259.23,916.939 \n",
       "  2259.32,916.651 2259.41,916.364 2259.5,916.078 2259.59,915.793 2259.68,915.509 2259.76,915.226 2259.85,914.945 2259.94,914.665 2260.02,914.385 2260.11,914.107 \n",
       "  2260.19,913.83 2260.27,913.554 2260.36,913.279 2260.44,913.005 2260.52,912.733 2260.6,912.461 2260.69,912.191 2260.77,911.921 2260.85,911.652 2260.93,911.385 \n",
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      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g = s -> (s-im)/(s+im)\n",
    "\n",
    "ss = linspace(-10.,10.,1000)\n",
    "plot(real.(g.(ss)), imag.(g.(ss)); label=\"image of (s-i)/(s+i)\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Therefore, $\\log g(s)$ has a branch cut if we use the standard branch, which breaks the continuity requirement:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
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       "  1810.62,775.359 1812.67,775.359 1814.72,775.359 1816.77,775.359 1818.82,775.359 1820.87,775.359 1822.93,775.359 1824.98,775.359 1827.03,775.359 1829.08,775.359 \n",
       "  1831.13,775.359 1833.18,775.359 1835.23,775.359 1837.28,775.359 1839.33,775.359 1841.38,775.359 1843.44,775.359 1845.49,775.359 1847.54,775.359 1849.59,775.359 \n",
       "  1851.64,775.359 1853.69,775.359 1855.74,775.359 1857.79,775.359 1859.84,775.359 1861.89,775.359 1863.95,775.359 1866,775.359 1868.05,775.359 1870.1,775.359 \n",
       "  1872.15,775.359 1874.2,775.359 1876.25,775.359 1878.3,775.359 1880.35,775.359 1882.4,775.359 1884.46,775.359 1886.51,775.359 1888.56,775.359 1890.61,775.359 \n",
       "  1892.66,775.359 1894.71,775.359 1896.76,775.359 1898.81,775.359 1900.86,775.359 1902.91,775.359 1904.97,775.359 1907.02,775.359 1909.07,775.359 1911.12,775.359 \n",
       "  1913.17,775.359 1915.22,775.359 1917.27,775.359 1919.32,775.359 1921.37,775.359 1923.42,775.359 1925.48,775.359 1927.53,775.359 1929.58,775.359 1931.63,775.359 \n",
       "  1933.68,775.359 1935.73,775.359 1937.78,775.359 1939.83,775.359 1941.88,775.359 1943.94,775.359 1945.99,775.359 1948.04,775.359 1950.09,775.359 1952.14,775.359 \n",
       "  1954.19,775.359 1956.24,775.359 1958.29,775.359 1960.34,775.359 1962.39,775.359 1964.45,775.359 1966.5,775.359 1968.55,775.359 1970.6,775.359 1972.65,775.359 \n",
       "  1974.7,775.359 1976.75,775.359 1978.8,775.359 1980.85,775.359 1982.9,775.359 1984.96,775.359 1987.01,775.359 1989.06,775.359 1991.11,775.359 1993.16,775.359 \n",
       "  1995.21,775.359 1997.26,775.359 1999.31,775.359 2001.36,775.359 2003.41,775.359 2005.47,775.359 2007.52,775.359 2009.57,775.359 2011.62,775.359 2013.67,775.359 \n",
       "  2015.72,775.359 2017.77,775.359 2019.82,775.359 2021.87,775.359 2023.92,775.359 2025.98,775.359 2028.03,775.359 2030.08,775.359 2032.13,775.359 2034.18,775.359 \n",
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       "  1113.27,468.01 1115.33,464.331 1117.38,460.579 1119.43,456.752 1121.48,452.848 1123.53,448.865 1125.58,444.803 1127.63,440.658 1129.68,436.43 1131.73,432.116 \n",
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       "  1154.3,378.524 1156.35,373.05 1158.4,367.469 1160.45,361.78 1162.5,355.981 1164.55,350.071 1166.6,344.05 1168.65,337.917 1170.7,331.67 1172.75,325.309 \n",
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       "  1215.83,166.768 1217.88,158.234 1219.93,149.644 1221.98,141.003 1224.03,132.318 1226.08,123.595 1228.13,114.84 1230.18,106.062 1232.23,97.2651 1234.28,88.4582 \n",
       "  1236.34,1462.26 1238.39,1453.45 1240.44,1444.66 1242.49,1435.88 1244.54,1427.12 1246.59,1418.4 1248.64,1409.72 1250.69,1401.07 1252.74,1392.48 1254.79,1383.95 \n",
       "  1256.85,1375.48 1258.9,1367.08 1260.95,1358.75 1263,1350.5 1265.05,1342.33 1267.1,1334.24 1269.15,1326.25 1271.2,1318.36 1273.25,1310.56 1275.3,1302.86 \n",
       "  1277.36,1295.27 1279.41,1287.78 1281.46,1280.4 1283.51,1273.13 1285.56,1265.98 1287.61,1258.93 1289.66,1252 1291.71,1245.18 1293.76,1238.47 1295.82,1231.89 \n",
       "  1297.87,1225.41 1299.92,1219.05 1301.97,1212.8 1304.02,1206.67 1306.07,1200.65 1308.12,1194.74 1310.17,1188.94 1312.22,1183.25 1314.27,1177.67 1316.33,1172.2 \n",
       "  1318.38,1166.83 1320.43,1161.56 1322.48,1156.4 1324.53,1151.34 1326.58,1146.38 1328.63,1141.52 1330.68,1136.75 1332.73,1132.07 1334.78,1127.49 1336.84,1123 \n",
       "  1338.89,1118.6 1340.94,1114.29 1342.99,1110.06 1345.04,1105.92 1347.09,1101.85 1349.14,1097.87 1351.19,1093.97 1353.24,1090.14 1355.29,1086.39 1357.35,1082.71 \n",
       "  1359.4,1079.1 1361.45,1075.57 1363.5,1072.1 1365.55,1068.7 1367.6,1065.36 1369.65,1062.09 1371.7,1058.88 1373.75,1055.73 1375.8,1052.65 1377.86,1049.62 \n",
       "  1379.91,1046.64 1381.96,1043.73 1384.01,1040.86 1386.06,1038.05 1388.11,1035.3 1390.16,1032.59 1392.21,1029.93 1394.26,1027.32 1396.31,1024.76 1398.37,1022.24 \n",
       "  1400.42,1019.77 1402.47,1017.34 1404.52,1014.96 1406.57,1012.62 1408.62,1010.31 1410.67,1008.05 1412.72,1005.83 1414.77,1003.65 1416.82,1001.5 1418.88,999.388 \n",
       "  1420.93,997.313 1422.98,995.273 1425.03,993.267 1427.08,991.295 1429.13,989.354 1431.18,987.446 1433.23,985.569 1435.28,983.722 1437.33,981.905 1439.39,980.118 \n",
       "  1441.44,978.358 1443.49,976.627 1445.54,974.922 1447.59,973.245 1449.64,971.593 1451.69,969.967 1453.74,968.366 1455.79,966.79 1457.85,965.237 1459.9,963.708 \n",
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       "  1502.97,936.24 1505.02,935.122 1507.07,934.018 1509.12,932.929 1511.17,931.854 1513.22,930.793 1515.27,929.746 1517.32,928.712 1519.38,927.691 1521.43,926.684 \n",
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       "  1564.5,908.143 1566.55,907.37 1568.6,906.605 1570.65,905.849 1572.7,905.101 1574.75,904.361 1576.8,903.629 1578.85,902.906 1580.91,902.19 1582.96,901.482 \n",
       "  1585.01,900.782 1587.06,900.089 1589.11,899.404 1591.16,898.726 1593.21,898.055 1595.26,897.392 1597.31,896.735 1599.36,896.085 1601.42,895.442 1603.47,894.805 \n",
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       "  1626.03,888.215 1628.08,887.651 1630.13,887.092 1632.18,886.539 1634.23,885.992 1636.28,885.449 1638.33,884.912 1640.39,884.379 1642.44,883.852 1644.49,883.33 \n",
       "  1646.54,882.813 1648.59,882.3 1650.64,881.793 1652.69,881.29 1654.74,880.791 1656.79,880.297 1658.84,879.808 1660.9,879.323 1662.95,878.843 1665,878.367 \n",
       "  1667.05,877.895 1669.1,877.428 1671.15,876.964 1673.2,876.505 1675.25,876.05 1677.3,875.598 1679.35,875.151 1681.41,874.708 1683.46,874.268 1685.51,873.833 \n",
       "  1687.56,873.401 1689.61,872.972 1691.66,872.548 1693.71,872.127 1695.76,871.71 1697.81,871.296 1699.86,870.885 1701.92,870.479 1703.97,870.075 1706.02,869.675 \n",
       "  1708.07,869.278 1710.12,868.884 1712.17,868.494 1714.22,868.107 1716.27,867.723 1718.32,867.342 1720.37,866.964 1722.43,866.59 1724.48,866.218 1726.53,865.849 \n",
       "  1728.58,865.483 1730.63,865.12 1732.68,864.76 1734.73,864.403 1736.78,864.049 1738.83,863.697 1740.88,863.348 1742.94,863.002 1744.99,862.659 1747.04,862.318 \n",
       "  1749.09,861.98 1751.14,861.644 1753.19,861.311 1755.24,860.98 1757.29,860.652 1759.34,860.327 1761.39,860.004 1763.45,859.683 1765.5,859.365 1767.55,859.049 \n",
       "  1769.6,858.735 1771.65,858.424 1773.7,858.115 1775.75,857.808 1777.8,857.503 1779.85,857.201 1781.91,856.901 1783.96,856.603 1786.01,856.307 1788.06,856.014 \n",
       "  1790.11,855.722 1792.16,855.433 1794.21,855.145 1796.26,854.86 1798.31,854.576 1800.36,854.295 1802.42,854.015 1804.47,853.738 1806.52,853.462 1808.57,853.189 \n",
       "  1810.62,852.917 1812.67,852.647 1814.72,852.379 1816.77,852.113 1818.82,851.849 1820.87,851.586 1822.93,851.325 1824.98,851.066 1827.03,850.809 1829.08,850.553 \n",
       "  1831.13,850.299 1833.18,850.047 1835.23,849.797 1837.28,849.548 1839.33,849.301 1841.38,849.055 1843.44,848.811 1845.49,848.569 1847.54,848.328 1849.59,848.089 \n",
       "  1851.64,847.851 1853.69,847.615 1855.74,847.38 1857.79,847.147 1859.84,846.916 1861.89,846.686 1863.95,846.457 1866,846.23 1868.05,846.004 1870.1,845.78 \n",
       "  1872.15,845.557 1874.2,845.335 1876.25,845.115 1878.3,844.896 1880.35,844.679 1882.4,844.462 1884.46,844.248 1886.51,844.034 1888.56,843.822 1890.61,843.611 \n",
       "  1892.66,843.402 1894.71,843.193 1896.76,842.986 1898.81,842.781 1900.86,842.576 1902.91,842.373 1904.97,842.171 1907.02,841.97 1909.07,841.77 1911.12,841.571 \n",
       "  1913.17,841.374 1915.22,841.178 1917.27,840.983 1919.32,840.789 1921.37,840.596 1923.42,840.405 1925.48,840.214 1927.53,840.025 1929.58,839.836 1931.63,839.649 \n",
       "  1933.68,839.463 1935.73,839.278 1937.78,839.094 1939.83,838.911 1941.88,838.729 1943.94,838.548 1945.99,838.368 1948.04,838.189 1950.09,838.012 1952.14,837.835 \n",
       "  1954.19,837.659 1956.24,837.484 1958.29,837.31 1960.34,837.137 1962.39,836.965 1964.45,836.794 1966.5,836.624 1968.55,836.455 1970.6,836.286 1972.65,836.119 \n",
       "  1974.7,835.953 1976.75,835.787 1978.8,835.623 1980.85,835.459 1982.9,835.296 1984.96,835.134 1987.01,834.973 1989.06,834.813 1991.11,834.653 1993.16,834.495 \n",
       "  1995.21,834.337 1997.26,834.18 1999.31,834.024 2001.36,833.869 2003.41,833.715 2005.47,833.561 2007.52,833.408 2009.57,833.256 2011.62,833.105 2013.67,832.954 \n",
       "  2015.72,832.805 2017.77,832.656 2019.82,832.508 2021.87,832.36 2023.92,832.214 2025.98,832.068 2028.03,831.923 2030.08,831.779 2032.13,831.635 2034.18,831.492 \n",
       "  2036.23,831.35 2038.28,831.208 2040.33,831.068 2042.38,830.927 2044.43,830.788 2046.49,830.649 2048.54,830.511 2050.59,830.374 2052.64,830.237 2054.69,830.102 \n",
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       "  2077.25,828.649 2079.3,828.521 2081.35,828.393 2083.4,828.266 2085.45,828.14 2087.51,828.014 2089.56,827.889 2091.61,827.764 2093.66,827.64 2095.71,827.517 \n",
       "  2097.76,827.394 2099.81,827.271 2101.86,827.15 2103.91,827.029 2105.97,826.908 2108.02,826.788 2110.07,826.668 2112.12,826.549 2114.17,826.431 2116.22,826.313 \n",
       "  2118.27,826.196 2120.32,826.079 2122.37,825.963 2124.42,825.847 2126.48,825.732 2128.53,825.617 2130.58,825.503 2132.63,825.39 2134.68,825.276 2136.73,825.164 \n",
       "  2138.78,825.052 2140.83,824.94 2142.88,824.829 2144.93,824.718 2146.99,824.608 2149.04,824.499 2151.09,824.39 2153.14,824.281 2155.19,824.173 2157.24,824.065 \n",
       "  2159.29,823.958 2161.34,823.851 2163.39,823.745 2165.44,823.639 2167.5,823.533 2169.55,823.429 2171.6,823.324 2173.65,823.22 2175.7,823.116 2177.75,823.013 \n",
       "  2179.8,822.911 2181.85,822.808 2183.9,822.707 2185.95,822.605 2188.01,822.504 2190.06,822.404 2192.11,822.304 2194.16,822.204 2196.21,822.105 2198.26,822.006 \n",
       "  2200.31,821.908 2202.36,821.81 2204.41,821.712 2206.46,821.615 2208.52,821.518 2210.57,821.422 2212.62,821.326 2214.67,821.23 2216.72,821.135 2218.77,821.04 \n",
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      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(ss, real.(log.(g.(ss))))\n",
    "plot!(ss, imag.(log.(g.(ss))))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We could have analytically continued $\\log g(z)$ using\n",
    "$$\n",
    "\\log_1 z = \\begin{cases} \\log z & \\Im z > 0 \\\\\n",
    "                            \\log_+ z & z < 0 \\\\\n",
    "                            \\log z + 2\\pi \\I & \\Im z < 0 \n",
    "                            \\end{cases}\n",
    "                            $$\n",
    "But then $\\lim_{s\\rightarrow + \\infty} \\log_1 g(s) = 2 \\pi \\I$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
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       "  1492.38,1462.26 1494.45,1462.26 1496.53,1462.26 1498.61,1462.26 1500.69,1462.26 1502.76,1462.26 1504.84,1462.26 1506.92,1462.26 1509,1462.26 1511.07,1462.26 \n",
       "  1513.15,1462.26 1515.23,1462.26 1517.31,1462.26 1519.38,1462.26 1521.46,1462.26 1523.54,1462.26 1525.62,1462.26 1527.69,1462.26 1529.77,1462.26 1531.85,1462.26 \n",
       "  1533.93,1462.26 1536.01,1462.26 1538.08,1462.26 1540.16,1462.26 1542.24,1462.26 1544.32,1462.26 1546.39,1462.26 1548.47,1462.26 1550.55,1462.26 1552.63,1462.26 \n",
       "  1554.7,1462.26 1556.78,1462.26 1558.86,1462.26 1560.94,1462.26 1563.01,1462.26 1565.09,1462.26 1567.17,1462.26 1569.25,1462.26 1571.32,1462.26 1573.4,1462.26 \n",
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       "  1596.25,1462.26 1598.33,1462.26 1600.41,1462.26 1602.49,1462.26 1604.56,1462.26 1606.64,1462.26 1608.72,1462.26 1610.8,1462.26 1612.87,1462.26 1614.95,1462.26 \n",
       "  1617.03,1462.26 1619.11,1462.26 1621.19,1462.26 1623.26,1462.26 1625.34,1462.26 1627.42,1462.26 1629.5,1462.26 1631.57,1462.26 1633.65,1462.26 1635.73,1462.26 \n",
       "  1637.81,1462.26 1639.88,1462.26 1641.96,1462.26 1644.04,1462.26 1646.12,1462.26 1648.19,1462.26 1650.27,1462.26 1652.35,1462.26 1654.43,1462.26 1656.5,1462.26 \n",
       "  1658.58,1462.26 1660.66,1462.26 1662.74,1462.26 1664.81,1462.26 1666.89,1462.26 1668.97,1462.26 1671.05,1462.26 1673.12,1462.26 1675.2,1462.26 1677.28,1462.26 \n",
       "  1679.36,1462.26 1681.43,1462.26 1683.51,1462.26 1685.59,1462.26 1687.67,1462.26 1689.74,1462.26 1691.82,1462.26 1693.9,1462.26 1695.98,1462.26 1698.05,1462.26 \n",
       "  1700.13,1462.26 1702.21,1462.26 1704.29,1462.26 1706.36,1462.26 1708.44,1462.26 1710.52,1462.26 1712.6,1462.26 1714.68,1462.26 1716.75,1462.26 1718.83,1462.26 \n",
       "  1720.91,1462.26 1722.99,1462.26 1725.06,1462.26 1727.14,1462.26 1729.22,1462.26 1731.3,1462.26 1733.37,1462.26 1735.45,1462.26 1737.53,1462.26 1739.61,1462.26 \n",
       "  1741.68,1462.26 1743.76,1462.26 1745.84,1462.26 1747.92,1462.26 1749.99,1462.26 1752.07,1462.26 1754.15,1462.26 1756.23,1462.26 1758.3,1462.26 1760.38,1462.26 \n",
       "  1762.46,1462.26 1764.54,1462.26 1766.61,1462.26 1768.69,1462.26 1770.77,1462.26 1772.85,1462.26 1774.92,1462.26 1777,1462.26 1779.08,1462.26 1781.16,1462.26 \n",
       "  1783.23,1462.26 1785.31,1462.26 1787.39,1462.26 1789.47,1462.26 1791.54,1462.26 1793.62,1462.26 1795.7,1462.26 1797.78,1462.26 1799.86,1462.26 1801.93,1462.26 \n",
       "  1804.01,1462.26 1806.09,1462.26 1808.17,1462.26 1810.24,1462.26 1812.32,1462.26 1814.4,1462.26 1816.48,1462.26 1818.55,1462.26 1820.63,1462.26 1822.71,1462.26 \n",
       "  1824.79,1462.26 1826.86,1462.26 1828.94,1462.26 1831.02,1462.26 1833.1,1462.26 1835.17,1462.26 1837.25,1462.26 1839.33,1462.26 1841.41,1462.26 1843.48,1462.26 \n",
       "  1845.56,1462.26 1847.64,1462.26 1849.72,1462.26 1851.79,1462.26 1853.87,1462.26 1855.95,1462.26 1858.03,1462.26 1860.1,1462.26 1862.18,1462.26 1864.26,1462.26 \n",
       "  1866.34,1462.26 1868.41,1462.26 1870.49,1462.26 1872.57,1462.26 1874.65,1462.26 1876.72,1462.26 1878.8,1462.26 1880.88,1462.26 1882.96,1462.26 1885.03,1462.26 \n",
       "  1887.11,1462.26 1889.19,1462.26 1891.27,1462.26 1893.35,1462.26 1895.42,1462.26 1897.5,1462.26 1899.58,1462.26 1901.66,1462.26 1903.73,1462.26 1905.81,1462.26 \n",
       "  1907.89,1462.26 1909.97,1462.26 1912.04,1462.26 1914.12,1462.26 1916.2,1462.26 1918.28,1462.26 1920.35,1462.26 1922.43,1462.26 1924.51,1462.26 1926.59,1462.26 \n",
       "  1928.66,1462.26 1930.74,1462.26 1932.82,1462.26 1934.9,1462.26 1936.97,1462.26 1939.05,1462.26 1941.13,1462.26 1943.21,1462.26 1945.28,1462.26 1947.36,1462.26 \n",
       "  1949.44,1462.26 1951.52,1462.26 1953.59,1462.26 1955.67,1462.26 1957.75,1462.26 1959.83,1462.26 1961.9,1462.26 1963.98,1462.26 1966.06,1462.26 1968.14,1462.26 \n",
       "  1970.21,1462.26 1972.29,1462.26 1974.37,1462.26 1976.45,1462.26 1978.53,1462.26 1980.6,1462.26 1982.68,1462.26 1984.76,1462.26 1986.84,1462.26 1988.91,1462.26 \n",
       "  1990.99,1462.26 1993.07,1462.26 1995.15,1462.26 1997.22,1462.26 1999.3,1462.26 2001.38,1462.26 2003.46,1462.26 2005.53,1462.26 2007.61,1462.26 2009.69,1462.26 \n",
       "  2011.77,1462.26 2013.84,1462.26 2015.92,1462.26 2018,1462.26 2020.08,1462.26 2022.15,1462.26 2024.23,1462.26 2026.31,1462.26 2028.39,1462.26 2030.46,1462.26 \n",
       "  2032.54,1462.26 2034.62,1462.26 2036.7,1462.26 2038.77,1462.26 2040.85,1462.26 2042.93,1462.26 2045.01,1462.26 2047.08,1462.26 2049.16,1462.26 2051.24,1462.26 \n",
       "  2053.32,1462.26 2055.39,1462.26 2057.47,1462.26 2059.55,1462.26 2061.63,1462.26 2063.7,1462.26 2065.78,1462.26 2067.86,1462.26 2069.94,1462.26 2072.02,1462.26 \n",
       "  2074.09,1462.26 2076.17,1462.26 2078.25,1462.26 2080.33,1462.26 2082.4,1462.26 2084.48,1462.26 2086.56,1462.26 2088.64,1462.26 2090.71,1462.26 2092.79,1462.26 \n",
       "  2094.87,1462.26 2096.95,1462.26 2099.02,1462.26 2101.1,1462.26 2103.18,1462.26 2105.26,1462.26 2107.33,1462.26 2109.41,1462.26 2111.49,1462.26 2113.57,1462.26 \n",
       "  2115.64,1462.26 2117.72,1462.26 2119.8,1462.26 2121.88,1462.26 2123.95,1462.26 2126.03,1462.26 2128.11,1462.26 2130.19,1462.26 2132.26,1462.26 2134.34,1462.26 \n",
       "  2136.42,1462.26 2138.5,1462.26 2140.57,1462.26 2142.65,1462.26 2144.73,1462.26 2146.81,1462.26 2148.88,1462.26 2150.96,1462.26 2153.04,1462.26 2155.12,1462.26 \n",
       "  2157.2,1462.26 2159.27,1462.26 2161.35,1462.26 2163.43,1462.26 2165.51,1462.26 2167.58,1462.26 2169.66,1462.26 2171.74,1462.26 2173.82,1462.26 2175.89,1462.26 \n",
       "  2177.97,1462.26 2180.05,1462.26 2182.13,1462.26 2184.2,1462.26 2186.28,1462.26 2188.36,1462.26 2190.44,1462.26 2192.51,1462.26 2194.59,1462.26 2196.67,1462.26 \n",
       "  2198.75,1462.26 2200.82,1462.26 2202.9,1462.26 2204.98,1462.26 2207.06,1462.26 2209.13,1462.26 2211.21,1462.26 2213.29,1462.26 2215.37,1462.26 2217.44,1462.26 \n",
       "  2219.52,1462.26 2221.6,1462.26 2223.68,1462.26 2225.75,1462.26 2227.83,1462.26 2229.91,1462.26 2231.99,1462.26 2234.06,1462.26 2236.14,1462.26 2238.22,1462.26 \n",
       "  2240.3,1462.26 2242.37,1462.26 2244.45,1462.26 2246.53,1462.26 2248.61,1462.26 2250.69,1462.26 2252.76,1462.26 2254.84,1462.26 2256.92,1462.26 2259,1462.26 \n",
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       "  204.291,1416.33 206.368,1416.24 208.446,1416.15 210.523,1416.05 212.601,1415.96 214.678,1415.86 216.756,1415.77 218.833,1415.67 220.911,1415.58 222.989,1415.48 \n",
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       "  266.617,1413.36 268.695,1413.25 270.772,1413.15 272.85,1413.04 274.927,1412.93 277.005,1412.83 279.083,1412.72 281.16,1412.61 283.238,1412.5 285.315,1412.39 \n",
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       "  1076.86,1180.82 1078.94,1177.71 1081.02,1174.55 1083.1,1171.32 1085.18,1168.02 1087.25,1164.67 1089.33,1161.24 1091.41,1157.75 1093.49,1154.19 1095.56,1150.56 \n",
       "  1097.64,1146.86 1099.72,1143.09 1101.8,1139.24 1103.87,1135.31 1105.95,1131.3 1108.03,1127.22 1110.11,1123.05 1112.18,1118.8 1114.26,1114.46 1116.34,1110.03 \n",
       "  1118.42,1105.51 1120.49,1100.91 1122.57,1096.2 1124.65,1091.41 1126.73,1086.52 1128.8,1081.53 1130.88,1076.43 1132.96,1071.24 1135.04,1065.94 1137.11,1060.54 \n",
       "  1139.19,1055.03 1141.27,1049.42 1143.35,1043.69 1145.42,1037.85 1147.5,1031.9 1149.58,1025.84 1151.66,1019.66 1153.73,1013.36 1155.81,1006.95 1157.89,1000.43 \n",
       "  1159.97,993.781 1162.04,987.019 1164.12,980.139 1166.2,973.142 1168.28,966.028 1170.36,958.798 1172.43,951.453 1174.51,943.993 1176.59,936.42 1178.67,928.737 \n",
       "  1180.74,920.945 1182.82,913.046 1184.9,905.044 1186.98,896.941 1189.05,888.741 1191.13,880.448 1193.21,872.065 1195.29,863.597 1197.36,855.049 1199.44,846.426 \n",
       "  1201.52,837.734 1203.6,828.977 1205.67,820.162 1207.75,811.294 1209.83,802.382 1211.91,793.43 1213.98,784.447 1216.06,775.438 1218.14,766.411 1220.22,757.374 \n",
       "  1222.29,748.332 1224.37,739.295 1226.45,730.268 1228.53,721.259 1230.6,712.276 1232.68,703.324 1234.76,694.412 1236.84,685.545 1238.91,676.73 1240.99,667.973 \n",
       "  1243.07,659.28 1245.15,650.657 1247.22,642.109 1249.3,633.641 1251.38,625.258 1253.46,616.965 1255.53,608.765 1257.61,600.662 1259.69,592.66 1261.77,584.761 \n",
       "  1263.85,576.969 1265.92,569.286 1268,561.713 1270.08,554.254 1272.16,546.908 1274.23,539.678 1276.31,532.564 1278.39,525.567 1280.47,518.687 1282.54,511.925 \n",
       "  1284.62,505.28 1286.7,498.753 1288.78,492.342 1290.85,486.048 1292.93,479.869 1295.01,473.805 1297.09,467.855 1299.16,462.016 1301.24,456.289 1303.32,450.672 \n",
       "  1305.4,445.163 1307.47,439.761 1309.55,434.464 1311.63,429.271 1313.71,424.18 1315.78,419.189 1317.86,414.297 1319.94,409.501 1322.02,404.8 1324.09,400.193 \n",
       "  1326.17,395.677 1328.25,391.25 1330.33,386.911 1332.4,382.657 1334.48,378.488 1336.56,374.402 1338.64,370.395 1340.71,366.468 1342.79,362.618 1344.87,358.843 \n",
       "  1346.95,355.142 1349.03,351.512 1351.1,347.954 1353.18,344.464 1355.26,341.041 1357.34,337.684 1359.41,334.391 1361.49,331.161 1363.57,327.992 1365.65,324.883 \n",
       "  1367.72,321.833 1369.8,318.839 1371.88,315.901 1373.96,313.018 1376.03,310.188 1378.11,307.41 1380.19,304.682 1382.27,302.004 1384.34,299.375 1386.42,296.792 \n",
       "  1388.5,294.256 1390.58,291.765 1392.65,289.318 1394.73,286.914 1396.81,284.552 1398.89,282.231 1400.96,279.951 1403.04,277.709 1405.12,275.506 1407.2,273.34 \n",
       "  1409.27,271.211 1411.35,269.118 1413.43,267.059 1415.51,265.035 1417.58,263.044 1419.66,261.086 1421.74,259.159 1423.82,257.264 1425.89,255.4 1427.97,253.565 \n",
       "  1430.05,251.76 1432.13,249.983 1434.2,248.234 1436.28,246.512 1438.36,244.818 1440.44,243.149 1442.52,241.506 1444.59,239.888 1446.67,238.295 1448.75,236.726 \n",
       "  1450.83,235.18 1452.9,233.658 1454.98,232.158 1457.06,230.68 1459.14,229.223 1461.21,227.788 1463.29,226.374 1465.37,224.98 1467.45,223.606 1469.52,222.252 \n",
       "  1471.6,220.917 1473.68,219.6 1475.76,218.302 1477.83,217.022 1479.91,215.76 1481.99,214.515 1484.07,213.287 1486.14,212.076 1488.22,210.881 1490.3,209.702 \n",
       "  1492.38,208.539 1494.45,207.391 1496.53,206.258 1498.61,205.141 1500.69,204.038 1502.76,202.949 1504.84,201.874 1506.92,200.813 1509,199.766 1511.07,198.732 \n",
       "  1513.15,197.711 1515.23,196.703 1517.31,195.707 1519.38,194.724 1521.46,193.753 1523.54,192.794 1525.62,191.846 1527.69,190.911 1529.77,189.986 1531.85,189.073 \n",
       "  1533.93,188.17 1536.01,187.278 1538.08,186.397 1540.16,185.526 1542.24,184.665 1544.32,183.815 1546.39,182.974 1548.47,182.143 1550.55,181.321 1552.63,180.509 \n",
       "  1554.7,179.706 1556.78,178.912 1558.86,178.128 1560.94,177.351 1563.01,176.584 1565.09,175.825 1567.17,175.074 1569.25,174.332 1571.32,173.597 1573.4,172.871 \n",
       "  1575.48,172.152 1577.56,171.442 1579.63,170.738 1581.71,170.043 1583.79,169.354 1585.87,168.673 1587.94,167.999 1590.02,167.332 1592.1,166.672 1594.18,166.019 \n",
       "  1596.25,165.372 1598.33,164.733 1600.41,164.099 1602.49,163.472 1604.56,162.852 1606.64,162.237 1608.72,161.629 1610.8,161.027 1612.87,160.431 1614.95,159.841 \n",
       "  1617.03,159.256 1619.11,158.677 1621.19,158.104 1623.26,157.537 1625.34,156.974 1627.42,156.418 1629.5,155.866 1631.57,155.32 1633.65,154.779 1635.73,154.243 \n",
       "  1637.81,153.712 1639.88,153.187 1641.96,152.666 1644.04,152.149 1646.12,151.638 1648.19,151.131 1650.27,150.629 1652.35,150.132 1654.43,149.639 1656.5,149.15 \n",
       "  1658.58,148.666 1660.66,148.186 1662.74,147.711 1664.81,147.239 1666.89,146.772 1668.97,146.309 1671.05,145.85 1673.12,145.395 1675.2,144.944 1677.28,144.497 \n",
       "  1679.36,144.054 1681.43,143.614 1683.51,143.179 1685.59,142.747 1687.67,142.319 1689.74,141.894 1691.82,141.473 1693.9,141.055 1695.98,140.641 1698.05,140.231 \n",
       "  1700.13,139.823 1702.21,139.419 1704.29,139.019 1706.36,138.622 1708.44,138.228 1710.52,137.837 1712.6,137.449 1714.68,137.065 1716.75,136.683 1718.83,136.305 \n",
       "  1720.91,135.929 1722.99,135.557 1725.06,135.187 1727.14,134.821 1729.22,134.457 1731.3,134.096 1733.37,133.738 1735.45,133.383 1737.53,133.031 1739.61,132.681 \n",
       "  1741.68,132.334 1743.76,131.989 1745.84,131.648 1747.92,131.308 1749.99,130.972 1752.07,130.638 1754.15,130.306 1756.23,129.977 1758.3,129.65 1760.38,129.326 \n",
       "  1762.46,129.004 1764.54,128.685 1766.61,128.368 1768.69,128.053 1770.77,127.74 1772.85,127.43 1774.92,127.122 1777,126.816 1779.08,126.513 1781.16,126.212 \n",
       "  1783.23,125.912 1785.31,125.615 1787.39,125.32 1789.47,125.027 1791.54,124.737 1793.62,124.448 1795.7,124.161 1797.78,123.876 1799.86,123.594 1801.93,123.313 \n",
       "  1804.01,123.034 1806.09,122.757 1808.17,122.482 1810.24,122.209 1812.32,121.937 1814.4,121.668 1816.48,121.4 1818.55,121.135 1820.63,120.871 1822.71,120.608 \n",
       "  1824.79,120.348 1826.86,120.089 1828.94,119.832 1831.02,119.577 1833.1,119.323 1835.17,119.071 1837.25,118.821 1839.33,118.572 1841.41,118.325 1843.48,118.079 \n",
       "  1845.56,117.835 1847.64,117.593 1849.72,117.352 1851.79,117.113 1853.87,116.875 1855.95,116.639 1858.03,116.405 1860.1,116.171 1862.18,115.94 1864.26,115.71 \n",
       "  1866.34,115.481 1868.41,115.253 1870.49,115.027 1872.57,114.803 1874.65,114.58 1876.72,114.358 1878.8,114.138 1880.88,113.918 1882.96,113.701 1885.03,113.484 \n",
       "  1887.11,113.269 1889.19,113.056 1891.27,112.843 1893.35,112.632 1895.42,112.422 1897.5,112.213 1899.58,112.006 1901.66,111.8 1903.73,111.595 1905.81,111.391 \n",
       "  1907.89,111.189 1909.97,110.987 1912.04,110.787 1914.12,110.588 1916.2,110.391 1918.28,110.194 1920.35,109.998 1922.43,109.804 1924.51,109.611 1926.59,109.419 \n",
       "  1928.66,109.228 1930.74,109.038 1932.82,108.849 1934.9,108.661 1936.97,108.474 1939.05,108.289 1941.13,108.104 1943.21,107.921 1945.28,107.738 1947.36,107.557 \n",
       "  1949.44,107.376 1951.52,107.197 1953.59,107.018 1955.67,106.841 1957.75,106.664 1959.83,106.489 1961.9,106.314 1963.98,106.141 1966.06,105.968 1968.14,105.796 \n",
       "  1970.21,105.625 1972.29,105.456 1974.37,105.287 1976.45,105.119 1978.53,104.951 1980.6,104.785 1982.68,104.62 1984.76,104.456 1986.84,104.292 1988.91,104.129 \n",
       "  1990.99,103.967 1993.07,103.806 1995.15,103.646 1997.22,103.487 1999.3,103.329 2001.38,103.171 2003.46,103.014 2005.53,102.858 2007.61,102.703 2009.69,102.549 \n",
       "  2011.77,102.395 2013.84,102.242 2015.92,102.09 2018,101.939 2020.08,101.789 2022.15,101.639 2024.23,101.49 2026.31,101.342 2028.39,101.195 2030.46,101.048 \n",
       "  2032.54,100.902 2034.62,100.757 2036.7,100.612 2038.77,100.469 2040.85,100.326 2042.93,100.183 2045.01,100.042 2047.08,99.9007 2049.16,99.7605 2051.24,99.621 \n",
       "  2053.32,99.4822 2055.39,99.344 2057.47,99.2066 2059.55,99.0698 2061.63,98.9336 2063.7,98.7982 2065.78,98.6633 2067.86,98.5292 2069.94,98.3957 2072.02,98.2628 \n",
       "  2074.09,98.1306 2076.17,97.999 2078.25,97.868 2080.33,97.7376 2082.4,97.6079 2084.48,97.4788 2086.56,97.3503 2088.64,97.2224 2090.71,97.0951 2092.79,96.9684 \n",
       "  2094.87,96.8423 2096.95,96.7168 2099.02,96.5919 2101.1,96.4675 2103.18,96.3438 2105.26,96.2206 2107.33,96.098 2109.41,95.9759 2111.49,95.8544 2113.57,95.7335 \n",
       "  2115.64,95.6131 2117.72,95.4933 2119.8,95.374 2121.88,95.2552 2123.95,95.137 2126.03,95.0194 2128.11,94.9023 2130.19,94.7856 2132.26,94.6696 2134.34,94.554 \n",
       "  2136.42,94.439 2138.5,94.3245 2140.57,94.2104 2142.65,94.0969 2144.73,93.9839 2146.81,93.8714 2148.88,93.7594 2150.96,93.6479 2153.04,93.5369 2155.12,93.4264 \n",
       "  2157.2,93.3163 2159.27,93.2068 2161.35,93.0977 2163.43,92.9891 2165.51,92.8809 2167.58,92.7733 2169.66,92.6661 2171.74,92.5593 2173.82,92.453 2175.89,92.3472 \n",
       "  2177.97,92.2419 2180.05,92.1369 2182.13,92.0325 2184.2,91.9285 2186.28,91.8249 2188.36,91.7217 2190.44,91.619 2192.51,91.5168 2194.59,91.4149 2196.67,91.3135 \n",
       "  2198.75,91.2125 2200.82,91.112 2202.9,91.0118 2204.98,90.9121 2207.06,90.8128 2209.13,90.7139 2211.21,90.6155 2213.29,90.5174 2215.37,90.4197 2217.44,90.3225 \n",
       "  2219.52,90.2256 2221.6,90.1291 2223.68,90.0331 2225.75,89.9374 2227.83,89.8421 2229.91,89.7472 2231.99,89.6527 2234.06,89.5586 2236.14,89.4648 2238.22,89.3715 \n",
       "  2240.3,89.2785 2242.37,89.1859 2244.45,89.0936 2246.53,89.0018 2248.61,88.9102 2250.69,88.8191 2252.76,88.7283 2254.84,88.6379 2256.92,88.5479 2259,88.4582 \n",
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      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "log₁ = z -> imag(z) > 0 ? log(z) : log(z)+2π*im\n",
    "plot(ss, real.(log₁.(g.(ss))))\n",
    "plot!(ss, imag.(log₁.(g.(ss))))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "\n",
    "\n",
    "**Example** Consider \n",
    "$$\n",
    "g(s) = {s^2+3 \\over s^2+1} = 1 + O(s^{-1})\n",
    "$$\n",
    "Before we do anything **Verify that the winding number is zero**.\n",
    "\n",
    "\n",
    "We provide two methods for calculating $\\kappa$: one guesses the solution, the other uses the solution formula.\n",
    "\n",
    "\n",
    "*Method 1 (Guess and check / kernel factorization)* If we can guess the solution, we can check it satisfies the right criteria.   Factoring $g$ we see immediately that\n",
    "$$\n",
    "g(s) = {s-\\I \\over s-\\sqrt{3} \\I} {s+\\I \\over s+\\sqrt{3} \\I}\n",
    "$$\n",
    "Note that the first term is analytic in  the upper half plane. The second term is analytic in the lower half plane _and_ invertible. Therefore we can guess the solution is \n",
    "$$\n",
    "\\kappa(z)  = \\begin{cases}\n",
    "        {z+\\sqrt 3\\I \\over z+\\I} & \\Im z > 0 \\\\\n",
    "            {z-\\I \\over z-\\sqrt 3\\I} & \\Im z < 0\n",
    "           \\end{cases}\n",
    "           $$\n",
    "This satisfies the three conditions:\n",
    "1. $\\kappa$ is analytic off $\\R$\n",
    "2. $\\lim_{z\\rightarrow \\infty}\\kappa(z) = 1$ \n",
    "3. It has the right jump\n",
    "$$\n",
    "g(s) \\kappa_-(s) =  {s^2+3 \\over s^2+1} {s-\\I \\over s-\\sqrt 3\\I}  = {s+\\I \\over s+\\sqrt 3 \\I} = \\kappa_+(s)\n",
    "$$\n",
    "This function is indeed analytic off the real line."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's check that it has the right jump:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "ename": "UndefVarError",
     "evalue": "UndefVarError: κ not defined",
     "output_type": "error",
     "traceback": [
      "UndefVarError: κ not defined",
      "",
      "Stacktrace:",
      " [1] top-level scope at In[6]:2"
     ]
    }
   ],
   "source": [
    "g = s -> (s^2+3)/(s^2+1)\n",
    "\n",
    "κ(0.1+eps()im) - g(0.1)*κ(0.1-eps()im)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "*Method 2 (evaluate explicit form)* This is real valued and positive, hence the winding number of its image is zero.  We have \n",
    "$$\n",
    "\\log g(s) = \\log({s+\\sqrt 3\\I \\over s+\\I}{s-\\sqrt 3\\I \\over s-\\I}) \n",
    "$$\n",
    "Because they are complex conjugates, we know $\\log a \\bar a = \\log a + \\log \\bar a$ as $[1, \\bar a, a]$ lies in the same half plane for $a = {s+\\sqrt 3 \\I \\over s+ \\I}$, therefore we can expand:\n",
    "$$\n",
    "\\log g(s) = \\log{s+\\sqrt 3\\I \\over s+\\I} + \\log {s-\\sqrt 3\\I \\over s-\\I}\n",
    "$$\n",
    "Now we note that $\\log{s+3\\I \\over s+\\I}$ is analytic in the upper-half plane, therefore it's Cauchy transform, by Plemelj, is\n",
    "$$\n",
    "\\CC\\br[\\log{s+\\sqrt 3\\I \\over s+\\I}](z) = \\begin{cases}\n",
    "        \\log{z+\\sqrt 3\\I \\over z+\\I} & \\Im z > 0 \\\\\n",
    "           0 & \\Im z < 0\n",
    "           \\end{cases}\n",
    "$$\n",
    "Similarly,\n",
    "$$\n",
    "\\CC\\br[\\log{s-\\sqrt 3\\I \\over s-\\I}](z) = \\begin{cases}\n",
    "        -\\log{z-\\sqrt 3\\I \\over z-\\I} & \\Im z > 0 \\\\\n",
    "           0 & \\Im z < 0\n",
    "           \\end{cases}\n",
    "$$\n",
    "We thus get:\n",
    "$$\n",
    "\\kappa(z) = \\E^{\\CC\\log g(z)} = \\E^{\\begin{cases}\n",
    "        \\log{z+\\sqrt 3 \\I \\over z+\\I} & \\Im z > 0 \\\\\n",
    "            -\\log{z- \\sqrt 3\\I \\over z-\\I} & \\Im z < 0\n",
    "           \\end{cases}} = \\begin{cases}\n",
    "        {z+ \\sqrt 3 \\I \\over z+\\I} & \\Im z > 0 \\\\\n",
    "            {z-\\I \\over z-\\sqrt 3\\I} & \\Im z < 0\n",
    "           \\end{cases}\n",
    "$$\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "### Inhomogenous Riemann–Hilbert problem\n",
    "\n",
    "Consider now the  Riemann–Hilbert problem with zero at infinity:\n",
    "$$\n",
    "\\Phi_+(s) - g(s)\\Phi_-(s) = h(s) \\qqand \\Phi(\\infty) =0\n",
    "$$\n",
    "Consider writing $\\Phi(z) =  \\kappa(z) \\Psi(z)$. Then we can reduce the Riemann–Hilbert problem to a subtractive problem: \n",
    "\n",
    "$$\n",
    "\\Phi_+(s) - g(s)\\Phi_-(s) = \\kappa_+(s)(\\Psi_+(s) - \\Psi_-(s))\\qqand \\Psi(\\infty) = 0\n",
    "$$\n",
    "Thus once we have $\\kappa$ we can construct $\\Phi$.\n",
    "\n",
    "What if we don't have decay? Just add in a constant times $\\kappa$:\n",
    "\n",
    "**Corollary** Suppose $\\log g$ satisfies the conditions of Plemelj's theorem. Then\n",
    "$$\n",
    "\\Phi(z) = \\kappa(z) \\CC_\\R\\br[{h \\over \\kappa_+}](z) + D \\kappa(z)\n",
    "$$\n",
    "is the unique solution to\n",
    "$$\n",
    "\\Phi_+(s) - g(s)\\Phi_-(s) = h(s) \\qqand \\Phi(\\infty) =D\n",
    "$$\n",
    "\n",
    "\n",
    "*Example* Suppose $h(s) = {\\I \\over \\I-s}$. \n",
    "\n",
    "To decompose this as a sum of things analytic in half planes, we just use partial fraction expansion!\n",
    "\\begin{align*}\n",
    "{\\I \\over \\I-s} {s + \\I \\over s+ \\I \\sqrt{3}} &= {\\I \\over \\I-s}{2\\I \\over \\I(1+ \\sqrt{3})}  +{\\I \\over \\I(1+\\sqrt 3)} {\\I(1-\\sqrt 3) \\over s+ \\I \\sqrt{3}}\\\\\n",
    "&= \\underbrace{{-2 \\I \\over s-\\I}{1 \\over 1+ \\sqrt{3}}}_{-Y_-(s)}  + \\underbrace{{\\I \\over 1+\\sqrt 3} {1-\\sqrt 3 \\over s+ \\I \\sqrt{3}}}_{Y_+(s)}\n",
    "\\end{align*}\n",
    "Thus we get\n",
    "$$\n",
    "Y(z) = \\begin{cases} \n",
    "{\\I \\over 1+\\sqrt 3} {1-\\sqrt 3 \\over z+ \\I \\sqrt{3}} & \\Im z > 0 \\\\\n",
    "{2 \\I \\over z-\\I}{1 \\over 1+ \\sqrt{3}} & \\Im z < 0\n",
    "\\end{cases}\n",
    "$$\n",
    "Let's double check:\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5706161966689216 - 0.08138224450306746im"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h = s -> im/(im-s)\n",
    "\n",
    "Y = z -> imag(z) > 0 ? im*(1-sqrt(3))/(1+sqrt(3))/(z+im*sqrt(3)) :\n",
    "                       2im/((z-im)*(1+sqrt(3)))\n",
    "\n",
    "Y(0.1+eps()im) - Y(0.1-eps()im)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "ename": "UndefVarError",
     "evalue": "UndefVarError: κ not defined",
     "output_type": "error",
     "traceback": [
      "UndefVarError: κ not defined",
      "",
      "Stacktrace:",
      " [1] top-level scope at In[8]:1"
     ]
    }
   ],
   "source": [
    "h(0.1)/κ(0.1+eps()im)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We thus have the solution:\n",
    "\\begin{align*}\n",
    "\\Phi(z)  &= \\kappa(z) \\CC_\\R\\br[{h \\over \\kappa_+}](z) = \\begin{cases} \n",
    "{\\I \\over 1+\\sqrt 3} {1-\\sqrt 3 \\over z+ \\I \\sqrt{3}} {z+\\sqrt{3}\\I \\over z+\\I} & \\Im z > 0 \\\\\n",
    "{2 \\I \\over z-\\I}{1 \\over 1+ \\sqrt{3}} {z-\\I \\over z-\\sqrt{3}\\I} & \\Im z < 0\n",
    "\\end{cases} \\\\\n",
    "    &= \\begin{cases} \n",
    "{\\I \\over 1+\\sqrt 3} {1-\\sqrt 3 \\over z+\\I} & \\Im z > 0 \\\\\n",
    "{2 \\I  \\over 1+ \\sqrt{3}} {1 \\over z-\\sqrt{3}\\I} & \\Im z < 0\n",
    "\\end{cases}\n",
    "\\end{align*}\n",
    "\n",
    "Let's verify it's the right thing:\n",
    "\n",
    "1. It's analytic off $\\R$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
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       "\" transform=\"translate(149, 47)\"/>\n",
       "</g>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "φ =  z -> imag(z) > 0 ? im*(1-sqrt(3))/(1+sqrt(3))/(z+im) :\n",
    "                       2im/((z-sqrt(3)*im)*(1+sqrt(3)))\n",
    "\n",
    "phaseplot(-3..3, -3..3, φ)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2\\. It goes to zero at infinity"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.0004465795146304169 - 0.0004450958617578906im"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "φ(300.0+300.0im)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "\n",
    "3\\. It satisfies the right jump:\n",
    "\\begin{align*}\n",
    " \\Phi_+(s) - g(s) \\Phi_-(s) = {\\I \\over 1+\\sqrt 3} {1-\\sqrt 3 \\over s+\\I}  - {s^2 + 3 \\over s^2 + 1} \n",
    "{2 \\I  \\over 1+ \\sqrt{3}} {1 \\over s-\\sqrt{3}\\I}  \\\\\n",
    " = {\\I (s-\\I) \\over 1+\\sqrt 3} {1-\\sqrt 3 \\over s^2+1}  - {s + \\sqrt{3} \\I \\over s^2 + 1} \n",
    "{2 \\I  \\over 1+ \\sqrt{3}}  \\\\\n",
    "= {1\\over s^2 + 1} \n",
    "{1 \\over 1+ \\sqrt{3}}  \\left(\\I(1-\\sqrt 3) s+1-\\sqrt{3}  - 2\\I s + 2 \\sqrt{3}    \\right) \\\\\n",
    "= {1\\over s^2 + 1} \n",
    " \\left(-\\I s+1    \\right) = {\\I \\over \\I - s}\n",
    "\\end{align*}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-1.1102230246251565e-16 + 2.7755575615628914e-17im"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "φ(0.1+eps()im) - φ(0.1-eps()im)*g(0.1) - h(0.1)"
   ]
  }
 ],
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   "mimetype": "application/julia",
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