Lecture 21.ipynb

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   "source": [
    "using Plots, ComplexPhasePortrait, ApproxFun, SingularIntegralEquations, OscillatoryIntegrals, \n",
    "        SpecialFunctions\n",
    "gr();"
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    "# M3M6: Methods of Mathematical Physics\n",
    "\n",
    "$$\n",
    "\\def\\dashint{{\\int\\!\\!\\!\\!\\!\\!-\\,}}\n",
    "\\def\\infdashint{\\dashint_{\\!\\!\\!-\\infty}^{\\,\\infty}}\n",
    "\\def\\D{\\,{\\rm d}}\n",
    "\\def\\E{{\\rm e}}\n",
    "\\def\\dx{\\D x}\n",
    "\\def\\dt{\\D t}\n",
    "\\def\\dz{\\D z}\n",
    "\\def\\C{{\\mathbb C}}\n",
    "\\def\\R{{\\mathbb R}}\n",
    "\\def\\H{{\\mathbb H}}\n",
    "\\def\\CC{{\\cal C}}\n",
    "\\def\\HH{{\\cal H}}\n",
    "\\def\\FF{{\\cal F}}\n",
    "\\def\\I{{\\rm i}}\n",
    "\\def\\qqqquad{\\qquad\\qquad}\n",
    "\\def\\qqand{\\qquad\\hbox{for}\\qquad}\n",
    "\\def\\qqfor{\\qquad\\hbox{for}\\qquad}\n",
    "\\def\\qqwhere{\\qquad\\hbox{where}\\qquad}\n",
    "\\def\\Res_#1{\\underset{#1}{\\rm Res}}\\,\n",
    "\\def\\sech{{\\rm sech}\\,}\n",
    "\\def\\acos{\\,{\\rm acos}\\,}\n",
    "\\def\\vc#1{{\\mathbf #1}}\n",
    "\\def\\ip<#1,#2>{\\left\\langle#1,#2\\right\\rangle}\n",
    "\\def\\norm#1{\\left\\|#1\\right\\|}\n",
    "\\def\\half{{1 \\over 2}}\n",
    "\\def\\fL{f_{\\rm L}}\n",
    "\\def\\fR{f_{\\rm R}}\n",
    "$$\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 21: Laplace transforms\n",
    "\n",
    "Our goal is to solve the integral equation\n",
    "$$\n",
    "\\lambda u(x) + \\int_{0}^\\infty K(x-t)u(t) \\dt = f(x)\\qqfor 0 < x < \\infty.\n",
    "$$\n",
    "A key tool will be the half-Fourier transforms\n",
    "$$\n",
    "\\int_{-\\infty}^0 u(t) \\E^{-\\I s t} \\dt \\qqand \\int_0^\\infty u(t) \\E^{-\\I s t} \\dt\n",
    "$$\n",
    "and the Laplace transform\n",
    "$$\n",
    "\\int_0^\\infty u(t) \\E^{-z t} \\dt\n",
    "$$\n",
    "in particular we are interested in the analyticity properties with respect to $s$/$z$. \n",
    "\n",
    "Outline:\n",
    "\n",
    "2.  Analyticity properties of Fourier transforms\n",
    "   - Inverse Fourier transform on shifted contours\n",
    "1. Half-Fourier transforms\n",
    "   - Inverting the Half-Fourier transform\n",
    "   - Relationship to Laplace transform\n",
    "2. Application: solving differential equations on the half-line\n",
    "\n",
    "## Analyticity properties of Fourier transforms\n",
    "\n",
    "\n",
    "Consider the Fourier transform of \n",
    "$$\n",
    "\\sech x = {2 \\over \\E^x + \\E^{-x}}\n",
    "$$\n",
    "This function has exponential decay in both directions:"
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       "  600.691,1458.25 601.697,1458.21 602.703,1458.17 603.709,1458.13 604.715,1458.09 605.721,1458.04 606.728,1458 607.734,1457.95 608.74,1457.91 609.746,1457.87 \n",
       "  610.752,1457.82 611.759,1457.77 612.765,1457.73 613.771,1457.68 614.777,1457.63 615.783,1457.59 616.789,1457.54 617.796,1457.49 618.802,1457.44 619.808,1457.39 \n",
       "  620.814,1457.34 621.82,1457.29 622.827,1457.24 623.833,1457.19 624.839,1457.13 625.845,1457.08 626.851,1457.03 627.857,1456.97 628.864,1456.92 629.87,1456.86 \n",
       "  630.876,1456.81 631.882,1456.75 632.888,1456.7 633.895,1456.64 634.901,1456.58 635.907,1456.52 636.913,1456.46 637.919,1456.4 638.925,1456.34 639.932,1456.28 \n",
       "  640.938,1456.22 641.944,1456.16 642.95,1456.1 643.956,1456.03 644.962,1455.97 645.969,1455.91 646.975,1455.84 647.981,1455.78 648.987,1455.71 649.993,1455.64 \n",
       "  651,1455.57 652.006,1455.51 653.012,1455.44 654.018,1455.37 655.024,1455.3 656.03,1455.22 657.037,1455.15 658.043,1455.08 659.049,1455.01 660.055,1454.93 \n",
       "  661.061,1454.86 662.068,1454.78 663.074,1454.71 664.08,1454.63 665.086,1454.55 666.092,1454.47 667.098,1454.39 668.105,1454.31 669.111,1454.23 670.117,1454.15 \n",
       "  671.123,1454.07 672.129,1453.98 673.136,1453.9 674.142,1453.81 675.148,1453.73 676.154,1453.64 677.16,1453.55 678.166,1453.46 679.173,1453.37 680.179,1453.28 \n",
       "  681.185,1453.19 682.191,1453.1 683.197,1453.01 684.204,1452.91 685.21,1452.82 686.216,1452.72 687.222,1452.62 688.228,1452.52 689.234,1452.43 690.241,1452.33 \n",
       "  691.247,1452.22 692.253,1452.12 693.259,1452.02 694.265,1451.91 695.272,1451.81 696.278,1451.7 697.284,1451.6 698.29,1451.49 699.296,1451.38 700.302,1451.27 \n",
       "  701.309,1451.16 702.315,1451.04 703.321,1450.93 704.327,1450.81 705.333,1450.7 706.34,1450.58 707.346,1450.46 708.352,1450.34 709.358,1450.22 710.364,1450.1 \n",
       "  711.37,1449.97 712.377,1449.85 713.383,1449.72 714.389,1449.6 715.395,1449.47 716.401,1449.34 717.408,1449.21 718.414,1449.07 719.42,1448.94 720.426,1448.81 \n",
       "  721.432,1448.67 722.438,1448.53 723.445,1448.39 724.451,1448.25 725.457,1448.11 726.463,1447.97 727.469,1447.82 728.476,1447.68 729.482,1447.53 730.488,1447.38 \n",
       "  731.494,1447.23 732.5,1447.07 733.506,1446.92 734.513,1446.77 735.519,1446.61 736.525,1446.45 737.531,1446.29 738.537,1446.13 739.544,1445.96 740.55,1445.8 \n",
       "  741.556,1445.63 742.562,1445.46 743.568,1445.29 744.574,1445.12 745.581,1444.95 746.587,1444.77 747.593,1444.6 748.599,1444.42 749.605,1444.24 750.611,1444.06 \n",
       "  751.618,1443.87 752.624,1443.69 753.63,1443.5 754.636,1443.31 755.642,1443.12 756.649,1442.92 757.655,1442.73 758.661,1442.53 759.667,1442.33 760.673,1442.13 \n",
       "  761.679,1441.92 762.686,1441.72 763.692,1441.51 764.698,1441.3 765.704,1441.09 766.71,1440.88 767.717,1440.66 768.723,1440.44 769.729,1440.22 770.735,1440 \n",
       "  771.741,1439.77 772.747,1439.55 773.754,1439.32 774.76,1439.08 775.766,1438.85 776.772,1438.61 777.778,1438.37 778.785,1438.13 779.791,1437.89 780.797,1437.64 \n",
       "  781.803,1437.39 782.809,1437.14 783.815,1436.89 784.822,1436.63 785.828,1436.38 786.834,1436.11 787.84,1435.85 788.846,1435.58 789.853,1435.31 790.859,1435.04 \n",
       "  791.865,1434.77 792.871,1434.49 793.877,1434.21 794.883,1433.93 795.89,1433.64 796.896,1433.35 797.902,1433.06 798.908,1432.77 799.914,1432.47 800.921,1432.17 \n",
       "  801.927,1431.86 802.933,1431.56 803.939,1431.25 804.945,1430.93 805.951,1430.62 806.958,1430.3 807.964,1429.98 808.97,1429.65 809.976,1429.32 810.982,1428.99 \n",
       "  811.989,1428.65 812.995,1428.32 814.001,1427.97 815.007,1427.63 816.013,1427.28 817.019,1426.93 818.026,1426.57 819.032,1426.21 820.038,1425.85 821.044,1425.48 \n",
       "  822.05,1425.11 823.057,1424.73 824.063,1424.36 825.069,1423.97 826.075,1423.59 827.081,1423.2 828.087,1422.8 829.094,1422.41 830.1,1422 831.106,1421.6 \n",
       "  832.112,1421.19 833.118,1420.78 834.125,1420.36 835.131,1419.93 836.137,1419.51 837.143,1419.08 838.149,1418.64 839.155,1418.2 840.162,1417.76 841.168,1417.31 \n",
       "  842.174,1416.86 843.18,1416.4 844.186,1415.94 845.192,1415.47 846.199,1415 847.205,1414.53 848.211,1414.05 849.217,1413.56 850.223,1413.07 851.23,1412.57 \n",
       "  852.236,1412.07 853.242,1411.57 854.248,1411.06 855.254,1410.54 856.26,1410.02 857.267,1409.5 858.273,1408.96 859.279,1408.43 860.285,1407.89 861.291,1407.34 \n",
       "  862.298,1406.79 863.304,1406.23 864.31,1405.66 865.316,1405.09 866.322,1404.52 867.328,1403.94 868.335,1403.35 869.341,1402.76 870.347,1402.16 871.353,1401.56 \n",
       "  872.359,1400.94 873.366,1400.33 874.372,1399.7 875.378,1399.08 876.384,1398.44 877.39,1397.8 878.396,1397.15 879.403,1396.49 880.409,1395.83 881.415,1395.17 \n",
       "  882.421,1394.49 883.427,1393.81 884.434,1393.12 885.44,1392.43 886.446,1391.72 887.452,1391.01 888.458,1390.3 889.464,1389.57 890.471,1388.84 891.477,1388.11 \n",
       "  892.483,1387.36 893.489,1386.61 894.495,1385.85 895.502,1385.08 896.508,1384.3 897.514,1383.52 898.52,1382.73 899.526,1381.93 900.532,1381.12 901.539,1380.31 \n",
       "  902.545,1379.48 903.551,1378.65 904.557,1377.81 905.563,1376.96 906.57,1376.11 907.576,1375.24 908.582,1374.37 909.588,1373.48 910.594,1372.59 911.6,1371.69 \n",
       "  912.607,1370.78 913.613,1369.86 914.619,1368.94 915.625,1368 916.631,1367.05 917.638,1366.1 918.644,1365.13 919.65,1364.16 920.656,1363.17 921.662,1362.18 \n",
       "  922.668,1361.17 923.675,1360.16 924.681,1359.13 925.687,1358.1 926.693,1357.05 927.699,1356 928.706,1354.93 929.712,1353.86 930.718,1352.77 931.724,1351.67 \n",
       "  932.73,1350.56 933.736,1349.44 934.743,1348.31 935.749,1347.17 936.755,1346.02 937.761,1344.85 938.767,1343.67 939.773,1342.48 940.78,1341.28 941.786,1340.07 \n",
       "  942.792,1338.85 943.798,1337.61 944.804,1336.36 945.811,1335.1 946.817,1333.83 947.823,1332.54 948.829,1331.24 949.835,1329.93 950.841,1328.61 951.848,1327.27 \n",
       "  952.854,1325.92 953.86,1324.55 954.866,1323.17 955.872,1321.78 956.879,1320.38 957.885,1318.96 958.891,1317.52 959.897,1316.08 960.903,1314.61 961.909,1313.14 \n",
       "  962.916,1311.65 963.922,1310.14 964.928,1308.62 965.934,1307.08 966.94,1305.53 967.947,1303.97 968.953,1302.39 969.959,1300.79 970.965,1299.18 971.971,1297.55 \n",
       "  972.977,1295.9 973.984,1294.24 974.99,1292.57 975.996,1290.87 977.002,1289.16 978.008,1287.44 979.015,1285.69 980.021,1283.93 981.027,1282.15 982.033,1280.36 \n",
       "  983.039,1278.54 984.045,1276.71 985.052,1274.87 986.058,1273 987.064,1271.11 988.07,1269.21 989.076,1267.29 990.083,1265.35 991.089,1263.39 992.095,1261.41 \n",
       "  993.101,1259.41 994.107,1257.39 995.113,1255.36 996.12,1253.3 997.126,1251.23 998.132,1249.13 999.138,1247.01 1000.14,1244.87 1001.15,1242.72 1002.16,1240.54 \n",
       "  1003.16,1238.34 1004.17,1236.12 1005.18,1233.87 1006.18,1231.61 1007.19,1229.32 1008.19,1227.02 1009.2,1224.69 1010.21,1222.33 1011.21,1219.96 1012.22,1217.56 \n",
       "  1013.22,1215.14 1014.23,1212.7 1015.24,1210.23 1016.24,1207.74 1017.25,1205.23 1018.26,1202.69 1019.26,1200.12 1020.27,1197.54 1021.27,1194.93 1022.28,1192.29 \n",
       "  1023.29,1189.63 1024.29,1186.94 1025.3,1184.23 1026.31,1181.5 1027.31,1178.73 1028.32,1175.95 1029.32,1173.13 1030.33,1170.29 1031.34,1167.42 1032.34,1164.53 \n",
       "  1033.35,1161.61 1034.35,1158.66 1035.36,1155.68 1036.37,1152.68 1037.37,1149.65 1038.38,1146.59 1039.39,1143.5 1040.39,1140.39 1041.4,1137.24 1042.4,1134.07 \n",
       "  1043.41,1130.86 1044.42,1127.63 1045.42,1124.37 1046.43,1121.08 1047.43,1117.76 1048.44,1114.41 1049.45,1111.03 1050.45,1107.61 1051.46,1104.17 1052.47,1100.7 \n",
       "  1053.47,1097.19 1054.48,1093.66 1055.48,1090.09 1056.49,1086.49 1057.5,1082.86 1058.5,1079.19 1059.51,1075.5 1060.52,1071.77 1061.52,1068.01 1062.53,1064.21 \n",
       "  1063.53,1060.39 1064.54,1056.52 1065.55,1052.63 1066.55,1048.7 1067.56,1044.74 1068.56,1040.74 1069.57,1036.71 1070.58,1032.65 1071.58,1028.55 1072.59,1024.42 \n",
       "  1073.6,1020.25 1074.6,1016.05 1075.61,1011.81 1076.61,1007.53 1077.62,1003.22 1078.63,998.877 1079.63,994.495 1080.64,990.078 1081.64,985.625 1082.65,981.135 \n",
       "  1083.66,976.61 1084.66,972.047 1085.67,967.449 1086.68,962.813 1087.68,958.141 1088.69,953.432 1089.69,948.686 1090.7,943.903 1091.71,939.083 1092.71,934.226 \n",
       "  1093.72,929.332 1094.73,924.4 1095.73,919.431 1096.74,914.425 1097.74,909.381 1098.75,904.3 1099.76,899.182 1100.76,894.026 1101.77,888.833 1102.77,883.602 \n",
       "  1103.78,878.334 1104.79,873.029 1105.79,867.687 1106.8,862.307 1107.81,856.89 1108.81,851.437 1109.82,845.946 1110.82,840.419 1111.83,834.854 1112.84,829.254 \n",
       "  1113.84,823.616 1114.85,817.943 1115.86,812.233 1116.86,806.488 1117.87,800.707 1118.87,794.89 1119.88,789.038 1120.89,783.152 1121.89,777.23 1122.9,771.274 \n",
       "  1123.9,765.284 1124.91,759.26 1125.92,753.203 1126.92,747.112 1127.93,740.989 1128.94,734.833 1129.94,728.645 1130.95,722.426 1131.95,716.176 1132.96,709.895 \n",
       "  1133.97,703.584 1134.97,697.244 1135.98,690.874 1136.98,684.476 1137.99,678.05 1139,671.597 1140,665.117 1141.01,658.611 1142.02,652.08 1143.02,645.524 \n",
       "  1144.03,638.944 1145.03,632.341 1146.04,625.715 1147.05,619.068 1148.05,612.4 1149.06,605.712 1150.07,599.006 1151.07,592.281 1152.08,585.538 1153.08,578.78 \n",
       "  1154.09,572.006 1155.1,565.218 1156.1,558.417 1157.11,551.603 1158.11,544.779 1159.12,537.945 1160.13,531.102 1161.13,524.251 1162.14,517.394 1163.15,510.532 \n",
       "  1164.15,503.666 1165.16,496.798 1166.16,489.929 1167.17,483.059 1168.18,476.192 1169.18,469.327 1170.19,462.467 1171.2,455.612 1172.2,448.765 1173.21,441.927 \n",
       "  1174.21,435.1 1175.22,428.285 1176.23,421.483 1177.23,414.697 1178.24,407.928 1179.24,401.178 1180.25,394.448 1181.26,387.741 1182.26,381.058 1183.27,374.4 \n",
       "  1184.28,367.77 1185.28,361.17 1186.29,354.601 1187.29,348.065 1188.3,341.564 1189.31,335.101 1190.31,328.676 1191.32,322.293 1192.32,315.952 1193.33,309.657 \n",
       "  1194.34,303.408 1195.34,297.209 1196.35,291.06 1197.36,284.965 1198.36,278.925 1199.37,272.942 1200.37,267.019 1201.38,261.157 1202.39,255.359 1203.39,249.626 \n",
       "  1204.4,243.961 1205.41,238.365 1206.41,232.842 1207.42,227.393 1208.42,222.019 1209.43,216.724 1210.44,211.509 1211.44,206.376 1212.45,201.328 1213.45,196.366 \n",
       "  1214.46,191.492 1215.47,186.709 1216.47,182.018 1217.48,177.422 1218.49,172.922 1219.49,168.52 1220.5,164.218 1221.5,160.018 1222.51,155.922 1223.52,151.931 \n",
       "  1224.52,148.048 1225.53,144.274 1226.54,140.611 1227.54,137.06 1228.55,133.624 1229.55,130.303 1230.56,127.099 1231.57,124.014 1232.57,121.05 1233.58,118.206 \n",
       "  1234.58,115.486 1235.59,112.89 1236.6,110.419 1237.6,108.075 1238.61,105.859 1239.62,103.771 1240.62,101.814 1241.63,99.9867 1242.63,98.2914 1243.64,96.7287 \n",
       "  1244.65,95.2993 1245.65,94.0039 1246.66,92.843 1247.66,91.8174 1248.67,90.9275 1249.68,90.1738 1250.68,89.5566 1251.69,89.0762 1252.7,88.7329 1253.7,88.5269 \n",
       "  1254.71,88.4582 1255.71,88.5269 1256.72,88.7329 1257.73,89.0762 1258.73,89.5566 1259.74,90.1738 1260.75,90.9275 1261.75,91.8174 1262.76,92.843 1263.76,94.0039 \n",
       "  1264.77,95.2993 1265.78,96.7287 1266.78,98.2914 1267.79,99.9867 1268.79,101.814 1269.8,103.771 1270.81,105.859 1271.81,108.075 1272.82,110.419 1273.83,112.89 \n",
       "  1274.83,115.486 1275.84,118.206 1276.84,121.05 1277.85,124.014 1278.86,127.099 1279.86,130.303 1280.87,133.624 1281.87,137.06 1282.88,140.611 1283.89,144.274 \n",
       "  1284.89,148.048 1285.9,151.931 1286.91,155.922 1287.91,160.018 1288.92,164.218 1289.92,168.52 1290.93,172.922 1291.94,177.422 1292.94,182.018 1293.95,186.709 \n",
       "  1294.96,191.492 1295.96,196.366 1296.97,201.328 1297.97,206.376 1298.98,211.509 1299.99,216.724 1300.99,222.019 1302,227.393 1303,232.842 1304.01,238.365 \n",
       "  1305.02,243.961 1306.02,249.626 1307.03,255.359 1308.04,261.157 1309.04,267.019 1310.05,272.942 1311.05,278.925 1312.06,284.965 1313.07,291.06 1314.07,297.209 \n",
       "  1315.08,303.408 1316.09,309.657 1317.09,315.952 1318.1,322.293 1319.1,328.676 1320.11,335.101 1321.12,341.564 1322.12,348.065 1323.13,354.601 1324.13,361.17 \n",
       "  1325.14,367.77 1326.15,374.4 1327.15,381.058 1328.16,387.741 1329.17,394.448 1330.17,401.178 1331.18,407.928 1332.18,414.697 1333.19,421.483 1334.2,428.285 \n",
       "  1335.2,435.1 1336.21,441.927 1337.21,448.765 1338.22,455.612 1339.23,462.467 1340.23,469.327 1341.24,476.192 1342.25,483.059 1343.25,489.929 1344.26,496.798 \n",
       "  1345.26,503.666 1346.27,510.532 1347.28,517.394 1348.28,524.251 1349.29,531.102 1350.3,537.945 1351.3,544.779 1352.31,551.603 1353.31,558.417 1354.32,565.218 \n",
       "  1355.33,572.006 1356.33,578.78 1357.34,585.538 1358.34,592.281 1359.35,599.006 1360.36,605.712 1361.36,612.4 1362.37,619.068 1363.38,625.715 1364.38,632.341 \n",
       "  1365.39,638.944 1366.39,645.524 1367.4,652.08 1368.41,658.611 1369.41,665.117 1370.42,671.597 1371.43,678.05 1372.43,684.476 1373.44,690.874 1374.44,697.244 \n",
       "  1375.45,703.584 1376.46,709.895 1377.46,716.176 1378.47,722.426 1379.47,728.645 1380.48,734.833 1381.49,740.989 1382.49,747.112 1383.5,753.203 1384.51,759.26 \n",
       "  1385.51,765.284 1386.52,771.274 1387.52,777.23 1388.53,783.152 1389.54,789.038 1390.54,794.89 1391.55,800.707 1392.55,806.488 1393.56,812.233 1394.57,817.943 \n",
       "  1395.57,823.616 1396.58,829.254 1397.59,834.854 1398.59,840.419 1399.6,845.946 1400.6,851.437 1401.61,856.89 1402.62,862.307 1403.62,867.687 1404.63,873.029 \n",
       "  1405.64,878.334 1406.64,883.602 1407.65,888.833 1408.65,894.026 1409.66,899.182 1410.67,904.3 1411.67,909.381 1412.68,914.425 1413.68,919.431 1414.69,924.4 \n",
       "  1415.7,929.332 1416.7,934.226 1417.71,939.083 1418.72,943.903 1419.72,948.686 1420.73,953.432 1421.73,958.141 1422.74,962.813 1423.75,967.449 1424.75,972.047 \n",
       "  1425.76,976.61 1426.77,981.135 1427.77,985.625 1428.78,990.078 1429.78,994.495 1430.79,998.877 1431.8,1003.22 1432.8,1007.53 1433.81,1011.81 1434.81,1016.05 \n",
       "  1435.82,1020.25 1436.83,1024.42 1437.83,1028.55 1438.84,1032.65 1439.85,1036.71 1440.85,1040.74 1441.86,1044.74 1442.86,1048.7 1443.87,1052.63 1444.88,1056.52 \n",
       "  1445.88,1060.39 1446.89,1064.21 1447.89,1068.01 1448.9,1071.77 1449.91,1075.5 1450.91,1079.19 1451.92,1082.86 1452.93,1086.49 1453.93,1090.09 1454.94,1093.66 \n",
       "  1455.94,1097.19 1456.95,1100.7 1457.96,1104.17 1458.96,1107.61 1459.97,1111.03 1460.98,1114.41 1461.98,1117.76 1462.99,1121.08 1463.99,1124.37 1465,1127.63 \n",
       "  1466.01,1130.86 1467.01,1134.07 1468.02,1137.24 1469.02,1140.39 1470.03,1143.5 1471.04,1146.59 1472.04,1149.65 1473.05,1152.68 1474.06,1155.68 1475.06,1158.66 \n",
       "  1476.07,1161.61 1477.07,1164.53 1478.08,1167.42 1479.09,1170.29 1480.09,1173.13 1481.1,1175.95 1482.1,1178.73 1483.11,1181.5 1484.12,1184.23 1485.12,1186.94 \n",
       "  1486.13,1189.63 1487.14,1192.29 1488.14,1194.93 1489.15,1197.54 1490.15,1200.12 1491.16,1202.69 1492.17,1205.23 1493.17,1207.74 1494.18,1210.23 1495.19,1212.7 \n",
       "  1496.19,1215.14 1497.2,1217.56 1498.2,1219.96 1499.21,1222.33 1500.22,1224.69 1501.22,1227.02 1502.23,1229.32 1503.23,1231.61 1504.24,1233.87 1505.25,1236.12 \n",
       "  1506.25,1238.34 1507.26,1240.54 1508.27,1242.72 1509.27,1244.87 1510.28,1247.01 1511.28,1249.13 1512.29,1251.23 1513.3,1253.3 1514.3,1255.36 1515.31,1257.39 \n",
       "  1516.32,1259.41 1517.32,1261.41 1518.33,1263.39 1519.33,1265.35 1520.34,1267.29 1521.35,1269.21 1522.35,1271.11 1523.36,1273 1524.36,1274.87 1525.37,1276.71 \n",
       "  1526.38,1278.54 1527.38,1280.36 1528.39,1282.15 1529.4,1283.93 1530.4,1285.69 1531.41,1287.44 1532.41,1289.16 1533.42,1290.87 1534.43,1292.57 1535.43,1294.24 \n",
       "  1536.44,1295.9 1537.44,1297.55 1538.45,1299.18 1539.46,1300.79 1540.46,1302.39 1541.47,1303.97 1542.48,1305.53 1543.48,1307.08 1544.49,1308.62 1545.49,1310.14 \n",
       "  1546.5,1311.65 1547.51,1313.14 1548.51,1314.61 1549.52,1316.08 1550.53,1317.52 1551.53,1318.96 1552.54,1320.38 1553.54,1321.78 1554.55,1323.17 1555.56,1324.55 \n",
       "  1556.56,1325.92 1557.57,1327.27 1558.57,1328.61 1559.58,1329.93 1560.59,1331.24 1561.59,1332.54 1562.6,1333.83 1563.61,1335.1 1564.61,1336.36 1565.62,1337.61 \n",
       "  1566.62,1338.85 1567.63,1340.07 1568.64,1341.28 1569.64,1342.48 1570.65,1343.67 1571.66,1344.85 1572.66,1346.02 1573.67,1347.17 1574.67,1348.31 1575.68,1349.44 \n",
       "  1576.69,1350.56 1577.69,1351.67 1578.7,1352.77 1579.7,1353.86 1580.71,1354.93 1581.72,1356 1582.72,1357.05 1583.73,1358.1 1584.74,1359.13 1585.74,1360.16 \n",
       "  1586.75,1361.17 1587.75,1362.18 1588.76,1363.17 1589.77,1364.16 1590.77,1365.13 1591.78,1366.1 1592.78,1367.05 1593.79,1368 1594.8,1368.94 1595.8,1369.86 \n",
       "  1596.81,1370.78 1597.82,1371.69 1598.82,1372.59 1599.83,1373.48 1600.83,1374.37 1601.84,1375.24 1602.85,1376.11 1603.85,1376.96 1604.86,1377.81 1605.87,1378.65 \n",
       "  1606.87,1379.48 1607.88,1380.31 1608.88,1381.12 1609.89,1381.93 1610.9,1382.73 1611.9,1383.52 1612.91,1384.3 1613.91,1385.08 1614.92,1385.85 1615.93,1386.61 \n",
       "  1616.93,1387.36 1617.94,1388.11 1618.95,1388.84 1619.95,1389.57 1620.96,1390.3 1621.96,1391.01 1622.97,1391.72 1623.98,1392.43 1624.98,1393.12 1625.99,1393.81 \n",
       "  1627,1394.49 1628,1395.17 1629.01,1395.83 1630.01,1396.49 1631.02,1397.15 1632.03,1397.8 1633.03,1398.44 1634.04,1399.08 1635.04,1399.7 1636.05,1400.33 \n",
       "  1637.06,1400.94 1638.06,1401.56 1639.07,1402.16 1640.08,1402.76 1641.08,1403.35 1642.09,1403.94 1643.09,1404.52 1644.1,1405.09 1645.11,1405.66 1646.11,1406.23 \n",
       "  1647.12,1406.79 1648.12,1407.34 1649.13,1407.89 1650.14,1408.43 1651.14,1408.96 1652.15,1409.5 1653.16,1410.02 1654.16,1410.54 1655.17,1411.06 1656.17,1411.57 \n",
       "  1657.18,1412.07 1658.19,1412.57 1659.19,1413.07 1660.2,1413.56 1661.21,1414.05 1662.21,1414.53 1663.22,1415 1664.22,1415.47 1665.23,1415.94 1666.24,1416.4 \n",
       "  1667.24,1416.86 1668.25,1417.31 1669.25,1417.76 1670.26,1418.2 1671.27,1418.64 1672.27,1419.08 1673.28,1419.51 1674.29,1419.93 1675.29,1420.36 1676.3,1420.78 \n",
       "  1677.3,1421.19 1678.31,1421.6 1679.32,1422 1680.32,1422.41 1681.33,1422.8 1682.33,1423.2 1683.34,1423.59 1684.35,1423.97 1685.35,1424.36 1686.36,1424.73 \n",
       "  1687.37,1425.11 1688.37,1425.48 1689.38,1425.85 1690.38,1426.21 1691.39,1426.57 1692.4,1426.93 1693.4,1427.28 1694.41,1427.63 1695.42,1427.97 1696.42,1428.32 \n",
       "  1697.43,1428.65 1698.43,1428.99 1699.44,1429.32 1700.45,1429.65 1701.45,1429.98 1702.46,1430.3 1703.46,1430.62 1704.47,1430.93 1705.48,1431.25 1706.48,1431.56 \n",
       "  1707.49,1431.86 1708.5,1432.17 1709.5,1432.47 1710.51,1432.77 1711.51,1433.06 1712.52,1433.35 1713.53,1433.64 1714.53,1433.93 1715.54,1434.21 1716.55,1434.49 \n",
       "  1717.55,1434.77 1718.56,1435.04 1719.56,1435.31 1720.57,1435.58 1721.58,1435.85 1722.58,1436.11 1723.59,1436.38 1724.59,1436.63 1725.6,1436.89 1726.61,1437.14 \n",
       "  1727.61,1437.39 1728.62,1437.64 1729.63,1437.89 1730.63,1438.13 1731.64,1438.37 1732.64,1438.61 1733.65,1438.85 1734.66,1439.08 1735.66,1439.32 1736.67,1439.55 \n",
       "  1737.67,1439.77 1738.68,1440 1739.69,1440.22 1740.69,1440.44 1741.7,1440.66 1742.71,1440.88 1743.71,1441.09 1744.72,1441.3 1745.72,1441.51 1746.73,1441.72 \n",
       "  1747.74,1441.92 1748.74,1442.13 1749.75,1442.33 1750.76,1442.53 1751.76,1442.73 1752.77,1442.92 1753.77,1443.12 1754.78,1443.31 1755.79,1443.5 1756.79,1443.69 \n",
       "  1757.8,1443.87 1758.8,1444.06 1759.81,1444.24 1760.82,1444.42 1761.82,1444.6 1762.83,1444.77 1763.84,1444.95 1764.84,1445.12 1765.85,1445.29 1766.85,1445.46 \n",
       "  1767.86,1445.63 1768.87,1445.8 1769.87,1445.96 1770.88,1446.13 1771.89,1446.29 1772.89,1446.45 1773.9,1446.61 1774.9,1446.77 1775.91,1446.92 1776.92,1447.07 \n",
       "  1777.92,1447.23 1778.93,1447.38 1779.93,1447.53 1780.94,1447.68 1781.95,1447.82 1782.95,1447.97 1783.96,1448.11 1784.97,1448.25 1785.97,1448.39 1786.98,1448.53 \n",
       "  1787.98,1448.67 1788.99,1448.81 1790,1448.94 1791,1449.07 1792.01,1449.21 1793.01,1449.34 1794.02,1449.47 1795.03,1449.6 1796.03,1449.72 1797.04,1449.85 \n",
       "  1798.05,1449.97 1799.05,1450.1 1800.06,1450.22 1801.06,1450.34 1802.07,1450.46 1803.08,1450.58 1804.08,1450.7 1805.09,1450.81 1806.1,1450.93 1807.1,1451.04 \n",
       "  1808.11,1451.16 1809.11,1451.27 1810.12,1451.38 1811.13,1451.49 1812.13,1451.6 1813.14,1451.7 1814.14,1451.81 1815.15,1451.91 1816.16,1452.02 1817.16,1452.12 \n",
       "  1818.17,1452.22 1819.18,1452.33 1820.18,1452.43 1821.19,1452.52 1822.19,1452.62 1823.2,1452.72 1824.21,1452.82 1825.21,1452.91 1826.22,1453.01 1827.23,1453.1 \n",
       "  1828.23,1453.19 1829.24,1453.28 1830.24,1453.37 1831.25,1453.46 1832.26,1453.55 1833.26,1453.64 1834.27,1453.73 1835.27,1453.81 1836.28,1453.9 1837.29,1453.98 \n",
       "  1838.29,1454.07 1839.3,1454.15 1840.31,1454.23 1841.31,1454.31 1842.32,1454.39 1843.32,1454.47 1844.33,1454.55 1845.34,1454.63 1846.34,1454.71 1847.35,1454.78 \n",
       "  1848.35,1454.86 1849.36,1454.93 1850.37,1455.01 1851.37,1455.08 1852.38,1455.15 1853.39,1455.22 1854.39,1455.3 1855.4,1455.37 1856.4,1455.44 1857.41,1455.51 \n",
       "  1858.42,1455.57 1859.42,1455.64 1860.43,1455.71 1861.44,1455.78 1862.44,1455.84 1863.45,1455.91 1864.45,1455.97 1865.46,1456.03 1866.47,1456.1 1867.47,1456.16 \n",
       "  1868.48,1456.22 1869.48,1456.28 1870.49,1456.34 1871.5,1456.4 1872.5,1456.46 1873.51,1456.52 1874.52,1456.58 1875.52,1456.64 1876.53,1456.7 1877.53,1456.75 \n",
       "  1878.54,1456.81 1879.55,1456.86 1880.55,1456.92 1881.56,1456.97 1882.56,1457.03 1883.57,1457.08 1884.58,1457.13 1885.58,1457.19 1886.59,1457.24 1887.6,1457.29 \n",
       "  1888.6,1457.34 1889.61,1457.39 1890.61,1457.44 1891.62,1457.49 1892.63,1457.54 1893.63,1457.59 1894.64,1457.63 1895.65,1457.68 1896.65,1457.73 1897.66,1457.77 \n",
       "  1898.66,1457.82 1899.67,1457.87 1900.68,1457.91 1901.68,1457.95 1902.69,1458 1903.69,1458.04 1904.7,1458.09 1905.71,1458.13 1906.71,1458.17 1907.72,1458.21 \n",
       "  1908.73,1458.25 1909.73,1458.3 1910.74,1458.34 1911.74,1458.38 1912.75,1458.42 1913.76,1458.46 1914.76,1458.49 1915.77,1458.53 1916.78,1458.57 1917.78,1458.61 \n",
       "  1918.79,1458.65 1919.79,1458.68 1920.8,1458.72 1921.81,1458.76 1922.81,1458.79 1923.82,1458.83 1924.82,1458.87 1925.83,1458.9 1926.84,1458.93 1927.84,1458.97 \n",
       "  1928.85,1459 1929.86,1459.04 1930.86,1459.07 1931.87,1459.1 1932.87,1459.14 1933.88,1459.17 1934.89,1459.2 1935.89,1459.23 1936.9,1459.26 1937.9,1459.29 \n",
       "  1938.91,1459.32 1939.92,1459.36 1940.92,1459.39 1941.93,1459.42 1942.94,1459.44 1943.94,1459.47 1944.95,1459.5 1945.95,1459.53 1946.96,1459.56 1947.97,1459.59 \n",
       "  1948.97,1459.62 1949.98,1459.64 1950.99,1459.67 1951.99,1459.7 1953,1459.72 1954,1459.75 1955.01,1459.78 1956.02,1459.8 1957.02,1459.83 1958.03,1459.85 \n",
       "  1959.03,1459.88 1960.04,1459.9 1961.05,1459.93 1962.05,1459.95 1963.06,1459.98 1964.07,1460 1965.07,1460.03 1966.08,1460.05 1967.08,1460.07 1968.09,1460.1 \n",
       "  1969.1,1460.12 1970.1,1460.14 1971.11,1460.16 1972.12,1460.19 1973.12,1460.21 1974.13,1460.23 1975.13,1460.25 1976.14,1460.27 1977.15,1460.29 1978.15,1460.31 \n",
       "  1979.16,1460.33 1980.16,1460.35 1981.17,1460.37 1982.18,1460.39 1983.18,1460.41 1984.19,1460.43 1985.2,1460.45 1986.2,1460.47 1987.21,1460.49 1988.21,1460.51 \n",
       "  1989.22,1460.53 1990.23,1460.55 1991.23,1460.57 1992.24,1460.58 1993.24,1460.6 1994.25,1460.62 1995.26,1460.64 1996.26,1460.65 1997.27,1460.67 1998.28,1460.69 \n",
       "  1999.28,1460.71 2000.29,1460.72 2001.29,1460.74 2002.3,1460.76 2003.31,1460.77 2004.31,1460.79 2005.32,1460.8 2006.33,1460.82 2007.33,1460.83 2008.34,1460.85 \n",
       "  2009.34,1460.87 2010.35,1460.88 2011.36,1460.9 2012.36,1460.91 2013.37,1460.93 2014.37,1460.94 2015.38,1460.95 2016.39,1460.97 2017.39,1460.98 2018.4,1461 \n",
       "  2019.41,1461.01 2020.41,1461.02 2021.42,1461.04 2022.42,1461.05 2023.43,1461.06 2024.44,1461.08 2025.44,1461.09 2026.45,1461.1 2027.46,1461.12 2028.46,1461.13 \n",
       "  2029.47,1461.14 2030.47,1461.15 2031.48,1461.17 2032.49,1461.18 2033.49,1461.19 2034.5,1461.2 2035.5,1461.21 2036.51,1461.23 2037.52,1461.24 2038.52,1461.25 \n",
       "  2039.53,1461.26 2040.54,1461.27 2041.54,1461.28 2042.55,1461.29 2043.55,1461.3 2044.56,1461.31 2045.57,1461.33 2046.57,1461.34 2047.58,1461.35 2048.58,1461.36 \n",
       "  2049.59,1461.37 2050.6,1461.38 2051.6,1461.39 2052.61,1461.4 2053.62,1461.41 2054.62,1461.42 2055.63,1461.43 2056.63,1461.44 2057.64,1461.44 2058.65,1461.45 \n",
       "  2059.65,1461.46 2060.66,1461.47 2061.67,1461.48 2062.67,1461.49 2063.68,1461.5 2064.68,1461.51 2065.69,1461.52 2066.7,1461.53 2067.7,1461.53 2068.71,1461.54 \n",
       "  2069.71,1461.55 2070.72,1461.56 2071.73,1461.57 2072.73,1461.58 2073.74,1461.58 2074.75,1461.59 2075.75,1461.6 2076.76,1461.61 2077.76,1461.62 2078.77,1461.62 \n",
       "  2079.78,1461.63 2080.78,1461.64 2081.79,1461.65 2082.79,1461.65 2083.8,1461.66 2084.81,1461.67 2085.81,1461.67 2086.82,1461.68 2087.83,1461.69 2088.83,1461.7 \n",
       "  2089.84,1461.7 2090.84,1461.71 2091.85,1461.72 2092.86,1461.72 2093.86,1461.73 2094.87,1461.74 2095.88,1461.74 2096.88,1461.75 2097.89,1461.76 2098.89,1461.76 \n",
       "  2099.9,1461.77 2100.91,1461.77 2101.91,1461.78 2102.92,1461.79 2103.92,1461.79 2104.93,1461.8 2105.94,1461.8 2106.94,1461.81 2107.95,1461.82 2108.96,1461.82 \n",
       "  2109.96,1461.83 2110.97,1461.83 2111.97,1461.84 2112.98,1461.84 2113.99,1461.85 2114.99,1461.85 2116,1461.86 2117.01,1461.86 2118.01,1461.87 2119.02,1461.87 \n",
       "  2120.02,1461.88 2121.03,1461.88 2122.04,1461.89 2123.04,1461.89 2124.05,1461.9 2125.05,1461.9 2126.06,1461.91 2127.07,1461.91 2128.07,1461.92 2129.08,1461.92 \n",
       "  2130.09,1461.93 2131.09,1461.93 2132.1,1461.94 2133.1,1461.94 2134.11,1461.95 2135.12,1461.95 2136.12,1461.95 2137.13,1461.96 2138.13,1461.96 2139.14,1461.97 \n",
       "  2140.15,1461.97 2141.15,1461.98 2142.16,1461.98 2143.17,1461.98 2144.17,1461.99 2145.18,1461.99 2146.18,1462 2147.19,1462 2148.2,1462 2149.2,1462.01 \n",
       "  2150.21,1462.01 2151.22,1462.01 2152.22,1462.02 2153.23,1462.02 2154.23,1462.03 2155.24,1462.03 2156.25,1462.03 2157.25,1462.04 2158.26,1462.04 2159.26,1462.04 \n",
       "  2160.27,1462.05 2161.28,1462.05 2162.28,1462.05 2163.29,1462.06 2164.3,1462.06 2165.3,1462.06 2166.31,1462.07 2167.31,1462.07 2168.32,1462.07 2169.33,1462.08 \n",
       "  2170.33,1462.08 2171.34,1462.08 2172.35,1462.08 2173.35,1462.09 2174.36,1462.09 2175.36,1462.09 2176.37,1462.1 2177.38,1462.1 2178.38,1462.1 2179.39,1462.1 \n",
       "  2180.39,1462.11 2181.4,1462.11 2182.41,1462.11 2183.41,1462.12 2184.42,1462.12 2185.43,1462.12 2186.43,1462.12 2187.44,1462.13 2188.44,1462.13 2189.45,1462.13 \n",
       "  2190.46,1462.13 2191.46,1462.14 2192.47,1462.14 2193.47,1462.14 2194.48,1462.14 2195.49,1462.15 2196.49,1462.15 2197.5,1462.15 2198.51,1462.15 2199.51,1462.16 \n",
       "  2200.52,1462.16 2201.52,1462.16 2202.53,1462.16 2203.54,1462.16 2204.54,1462.17 2205.55,1462.17 2206.56,1462.17 2207.56,1462.17 2208.57,1462.18 2209.57,1462.18 \n",
       "  2210.58,1462.18 2211.59,1462.18 2212.59,1462.18 2213.6,1462.19 2214.6,1462.19 2215.61,1462.19 2216.62,1462.19 2217.62,1462.19 2218.63,1462.2 2219.64,1462.2 \n",
       "  2220.64,1462.2 2221.65,1462.2 2222.65,1462.2 2223.66,1462.2 2224.67,1462.21 2225.67,1462.21 2226.68,1462.21 2227.69,1462.21 2228.69,1462.21 2229.7,1462.22 \n",
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     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xx = -10:0.01:10\n",
    "plot(xx,sech.(xx))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now the Fourier transform of $\\sech x$ is\n",
    "$$\n",
    "\\FF\\sech(s) = \\int_{-\\infty}^\\infty \\sech t \\, \\E^{-\\I s t} \\dt = \\pi \\sech{\\pi s \\over 2}\n",
    "$$\n",
    "This is calculated via Residue theorem with a bit of work."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.2710149513994194 + 0.0im"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = Fun(sech, Line())\n",
    "fourier(f, 2.0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.2710149513994184"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "π*sech(π*2.0/2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that\n",
    "$$\n",
    "\\pi \\sech {\\pi z \\over 2} = {2 \\pi \\over \\E^{\\pi z \\over 2} + \\E^{-{\\pi z \\over 2}}}\n",
    "$$\n",
    "is analytic for $-1 < \\Im z < 1$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
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       "</g>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "phaseplot(-3..3, -3..3, z -> π*sech(π*z/2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is because of the expontial decay. \n",
    "\n",
    "**Theorem (Analyticity of Fourier transforms)** Suppose $|f(x) \\E^{\\gamma x}| < |M(x)|$ where $M$ is absolutely integrable for all $a < \\gamma < b$. Then \n",
    "$$\n",
    "\\widehat f(z) = \\int_{-\\infty}^\\infty f(t) \\E^{-\\I z t} \\dt\n",
    "$$\n",
    "is analytic for $a < \\Im z < b$.\n",
    "\n",
    "**Proof** Let $z = s + \\I \\gamma$ and note that $|f(t) \\E^{\\I z t}| = |f(t) \\E^{\\gamma t}|$. Thus for $a < \\gamma < b$, we can exchange differentiation and integration to get\n",
    "$$\n",
    "{\\D \\widehat f \\over \\dz} = -\\I z \\hat f(z)\n",
    "$$\n",
    "⬛️\n",
    "\n",
    "**Remark** We don't need $f$ to be analytic at all!  Decay in $f$ gives analyticity.\n",
    "\n",
    "In the case of $\\sech x$, we get exponential decay in both directions: that is $\\sech x \\E^{|\\gamma| x}$ is absolutely integrable for $\\gamma < 1$.\n",
    "\n",
    "Another example is $\\E^{-x^2/2}$, which is absolutely integrable for any $\\gamma$. Therefore, it's Fourier transform is in fact entire:\n",
    "$$\n",
    "\\FF[\\E^{-\\diamond^2/2}](z) = \\sqrt{2\\pi} \\E^{-{z^2 \\over 2}}\n",
    "$$\n",
    "\n",
    "\n",
    "### Inverse Fourier transform on shifted contours\n",
    "\n",
    "A neglected fact of the Fourier transform is that we can think of $\\hat f(z)$ living on any line $(-\\infty + \\I \\gamma, \\infty+\\I \\gamma)$, and in fact we can recover $f$ from the Fourier transform only on this line. This works even if $\\hat f(s)$ is not defined on the real-axis, the real-axis is NOT special!\n",
    "\n",
    "**Theorem** Suppose $f(x) \\E^{\\gamma x}$ is square integrable. Then \n",
    "$$\n",
    "f(x) = {1 \\over 2 \\pi} \\int_{-\\infty+\\I \\gamma}^{\\infty + \\I \\gamma} \\widehat f(\\zeta) \\E^{\\I x \\zeta} \\D \\zeta\n",
    "$$\n",
    "\n",
    "**Proof**  Note for $g(x) = f(x) \\E^{\\gamma x}$ \n",
    "$$\n",
    "\\widehat g(s) = \\int_{-\\infty}^\\infty f(t) \\E^{\\gamma t - \\I s t} \\dt = \\widehat f(s + \\I \\gamma).\n",
    "$$\n",
    "Therefore we have\n",
    "\\begin{align*}\n",
    "\\E^{\\gamma x} f(x) &= g(x) = \\FF^{-1} \\widehat g(x) = {1 \\over 2 \\pi} \\int_{-\\infty}^{\\infty} \\widehat g(s) \\E^{\\I x s} \\D s  ={1 \\over 2 \\pi} \\int_{-\\infty}^{\\infty} \\widehat f(s + \\I \\gamma) \\E^{\\I x s} \\D s \\\\\n",
    "   & = {1 \\over 2 \\pi} \\int_{-\\infty+ \\I \\gamma}^{\\infty+ \\I \\gamma} \\widehat f(\\zeta) \\E^{\\I x (\\zeta - \\I \\gamma)} \\D \\zeta \\\\\n",
    "    &= {1 \\over 2 \\pi} \n",
    "   \\E^{\\gamma x}  \\int_{-\\infty+ \\I \\gamma}^{\\infty+ \\I \\gamma} \\widehat f(\\zeta) \\E^{\\I x \\zeta} \\D \\zeta\n",
    "\\end{align*}\n",
    "Which shows the result by cancelling out $\\E^{\\gamma x}$.\n",
    "\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Half-Fourier transforms\n",
    "\n",
    "Consider now \n",
    "$$\n",
    "\\int_0^\\infty f(t) \\E^{-\\I s t} \\dt\n",
    "$$\n",
    "This is in fact the Fourier transform of $f$ extended to the negative real axis by zero:\n",
    "$$\n",
    "\\int_{-\\infty}^\\infty \\begin{cases}f(t) & t \\geq 0 \\\\ 0 & \\hbox{otherwise} \\end{cases} \\E^{-\\I s t} \\dt\n",
    "$$\n",
    "To make sure we remember the domain of definition, we introduce the notation:\n",
    "$$\n",
    "f_{\\rm R}(x) = \\begin{cases}f(t) & t \\geq 0 \\\\ 0 & \\hbox{otherwise} \\end{cases}\n",
    "$$\n",
    "and\n",
    "$$\n",
    "f_{\\rm L}(x) = \\begin{cases}f(t) & t < 0 \\\\ 0 & \\hbox{otherwise} \\end{cases}\n",
    "$$\n",
    "Therefore \n",
    "$$\n",
    "\\widehat\\fR(s) = \\int_0^\\infty f(t) \\E^{-\\I s t} \\dt \\qqand \\widehat\\fL(s) = \\int_{-\\infty}^0 f(t) \\E^{-\\I s t} \\dt\n",
    "$$\n",
    "\n",
    "Because it is identically zero on the negative real axis, we immediately get the following:\n",
    "\n",
    "**Corollary (analyticity of Half-Fourier transform)**  Suppose $f(x)$ is bounded for $x \\geq 0$. Then $\\widehat{\\fR}(z)$ is analytic in the lower half plane \n",
    "$$\\H_+ = \\{ z : \\Im z < 0 \\}.$$\n",
    "\n",
    "More generally, $f$ can even have exponential decay: if $f(x) \\E^{\\gamma x}$ is bounded then $\\widehat\\fR(z)$ is analytic in $\\{z : \\Im z < \\gamma \\}$.  As before, the same inversion formula follows:\n",
    "\n",
    "**Corollary  (inverting Half-Fourier transform)**  Suppose  $f(x) \\E^{\\gamma x}$ is square integrable for $x \\geq 0$. Then \n",
    "$$\n",
    "f(x) = {1 \\over 2 \\pi} \\int_{-\\infty + \\I M}^{\\infty + \\I M} \\widehat{\\fR}(\\zeta) \\E^{\\I x \\zeta} \\D \\zeta\n",
    "$$\n",
    "for any choice of $-\\infty < M \\leq \\gamma$.\n",
    "\n",
    "\n",
    "**Example** Consider $f(x)  = x \\E^{-x}$ for $0 \\leq x < \\infty$.  Note that $f(x) \\E^{\\gamma x}$ is square integrable for any $\\gamma < 1$, and we have\n",
    "$$\n",
    "\\widehat\\fR(z) = \\int_0^\\infty t \\E^{-t -\\I s t} \\dt = {1 \\over (1+\\I s)^2}\n",
    "$$\n",
    "is analytic for $\\Im z < 1$. Thus for any $M < 1$ we have\n",
    "\\begin{align*}\n",
    "f(x) &= {1 \\over 2 \\pi} \\int_{-\\infty+\\I M}^{\\infty +\\I M}\\widehat\\fR(\\zeta) \\E^{\\I x \\zeta}\\D \\zeta \\\\\n",
    "     &={1 \\over 2 \\pi} \\int_{-\\infty+\\I M}^{\\infty +\\I M}{1 \\over (1+\\I \\zeta)^2} \\E^{\\I x \\zeta}\\D \\zeta\n",
    "\\end{align*}\n",
    "\n",
    "Since $x > 0$, we can use Residue calculus in the upper-half plane, which confirms the result:\n",
    "$$\n",
    "\\Res_{z = -\\I} {\\E^{\\I x z} \\over  (1+\\I z)^2}  = \\Res_{z = -\\I} {\\E^{- x} + \\I x \\E^{-x} (z +\\I) + O(z+\\I)^2 \\over  -(z-\\I)^2}  = -\\I x \\E^{-x}.\n",
    "$$\n",
    "**Example** Consider $f(x) = x$.  This function is not square-integrable, but we have $f(x) \\E^{\\gamma x}$ is square integrable for any $\\gamma < 0$, and we find for $\\Im z < 0$\n",
    "$$\n",
    "\\widehat{\\fR}(z) = -{1 \\over z^2}\n",
    "$$\n",
    "Thus we can still use the result to say, for any $M < 0$, \n",
    "$$\n",
    "f(x) =-{1 \\over 2 \\pi} \\int_{-\\infty+\\I M}^{\\infty +\\I M} {1 \\over \\zeta^2} \\E^{\\I x \\zeta}\\D \\zeta\n",
    "$$\n",
    "\n",
    "\n",
    "Note that the results have corresponding analogues for $\\fL$:\n",
    "\n",
    "**Corollary (analyticity of left Half-Fourier transform)**  Suppose  $f(x) \\E^{\\gamma x}$  is bounded for $x \\leq 0$. Then $\\widehat{\\fL}(z)$ is analytic for $\\{z : \\Im z > \\gamma \\}$.\n",
    "\n",
    "\n",
    "**Corollary  (inverting left Half-Fourier transform)**  Suppose  $f(x) \\E^{\\gamma x}$ is square integrable for $x < 0$. Then \n",
    "$$\n",
    "f(x) = {1 \\over 2 \\pi} \\int_{-\\infty + \\I M}^{\\infty + \\I M} \\widehat{\\fL}(\\zeta) \\E^{\\I x \\zeta} \\D \\zeta\n",
    "$$\n",
    "for any choice of $\\gamma \\leq M < \\infty $.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Laplace transforms\n",
    "\n",
    "Now consider the Laplace transform\n",
    "$$\n",
    "\\check f(z) = \\int_0^\\infty f(t) \\E^{-z t} \\dt\n",
    "$$\n",
    "but this is just the half Fourier transform evaluated on the negative imaginary axis!\n",
    "$$\n",
    "\\check f(z) = \\widehat\\fR (-\\I z)\n",
    "$$\n",
    "Thus if $f(x) \\E^{\\gamma x}$ is square integrable, then $\\check f(z)$ is well-defined for $\\Re z \\geq \\gamma$.    \n",
    "\n",
    "<font color=red>NEVER</font> think of the Laplace transform as a real-valued object: it only makes sense as a complex object. This is seen from the inverse Laplace transform\n",
    "$$\n",
    "f(x) = {1 \\over 2 \\pi \\I} \\int_{-\\I \\infty - M}^{\\I \\infty - M}  \\check f(\\zeta) \\E^{\\zeta x} \\D \\zeta\n",
    "$$\n",
    "which is of course just the inverse Fourier transform in disguise.\n",
    "\n",
    "\n",
    "## Application: solving differential equations on the half-line\n",
    "\n",
    "Consider the following ODE for $x \\geq 0$:\n",
    "\\begin{align*}\n",
    "u''(x) + 2u'(x) + u(x) = f(x)\n",
    "\\end{align*}\n",
    "with initial conditions $u(0) = u'(0) = 0$. Note that we have by integration-by-parts\n",
    "\\begin{align*}\n",
    "\\check{u'}(z) = \\int_0^\\infty u'(t) \\E^{-z t} \\dt = u(0) + z \\int_0^\\infty u(t) \\E^{-z t} \\dt = u(0) + z \\check u(z) \\\\\n",
    "\\check{u''}(z) = u'(0) + z \\check{u'}(z) = u'(0) + z u(0) + z^2 \\check u(0)\n",
    "\\end{align*}\n",
    "Thus taking into account the initial conditions, are equation in Laplace space becomes\n",
    "$$\n",
    "(z^2 + 2z + 1) \\check u(z) = \\check f(z)\n",
    "$$\n",
    "Hence we have\n",
    "$$\n",
    "\\check u(z) = {1 \\over 2 \\pi \\I}\\int_{-\\I\\infty-M}^{\\I \\infty-M} {\\check f(\\zeta) \\over \\zeta^2 + 2\\zeta+1} \\E^{x \\zeta} \\D \\zeta \n",
    "$$\n",
    "Consider the case $f(x) = x$, so that\n",
    "$$\n",
    "\\check f(z) = {1 \\over z^2}\n",
    "$$\n",
    "Here we need $M < 0$ hence we are integrating on a contour in the right-half plane.  Using Residue calculus, we have\n",
    "$$\n",
    "u(z) = \\left(\\Res_{z = -1} + \\Res_{z=0}\\right) {\\E^{z x} \\over z^2 (z +1)^2} = \\Res_{z = -1}  {\\E^{-x} +\\E^{-x}(x+2) (z+1) +O(z+1)^2 \\over  (z +1)^2} + \\Res_{z=0} {1 + (x-2) z +O(z)^2 \\over  z^2}  \\\\\n",
    " = (x+2)\\E^{-x} + x-2\n",
    "$$\n",
    "\n",
    "\n",
    "## Laplace transform of rational functions\n",
    "\n",
    "\n",
    "We now consider the question of calculating Laplace transforms (or equivalently, half-Fourier transforms)\n",
    "$$\n",
    "\\check f(s) = \\int_0^\\infty f(t) \\E^{-s t} \\D t\n",
    "$$\n",
    "where $f$ is rational.\n",
    "We're going to do something seemingly crazy: we'll first calculate the Cauchy transform\n",
    "$$\n",
    "\\CC[f \\E^{-s \\diamond}](z) = {1 \\over 2 \\pi \\I} \\int_0^\\infty {f(t) \\E^{-s t} \\over t- z} \\D t\n",
    "$$\n",
    "so that\n",
    "$$\n",
    "\\check f(s) = -2 \\pi \\I\\lim_{z \\rightarrow \\infty}  z \\CC[f \\E^{-s \\diamond}](z) \n",
    "$$\n",
    "Note that the exponential decay in the integrand allows us to use Plemelj's lemma: if we find a function $\\phi(z)$ such that\n",
    "1. $\\phi(z)$ is analytic off $[0,\\infty)$\n",
    "2. $\\lim_{z \\rightarrow \\infty} \\phi(z) = 0$ for any angle of approach\n",
    "3. $\\phi$ has weaker than pole singularities at 0\n",
    "4. $\\phi_+(x) - \\phi_-(x) = f(x) \\E^{-s x}$\n",
    "\n",
    "\n",
    "Then we have calculated the Cauchy transform:\n",
    "$$\\phi(z) = \\CC[f \\E^{-s \\diamond}](z).$$\n",
    "\n",
    "\n",
    "Let's start with $f(x) = 1$ and $s = 1$, that is, what is the Cauchy transform of $\\E^{-x}$? Consider the exponential integral:\n",
    "$$\n",
    "{\\rm Ei}(z) = \\int_{-\\infty}^z {\\E^\\zeta \\over \\zeta} \\D \\zeta\n",
    "$$\n",
    "Without loss of generality, the contour of integration is\n",
    "$$\n",
    "(\\infty,-1) \\cup [-1, z)\n",
    "$$\n",
    "that is, a straight line to $-1$ and a straightline from $-1$ to $z$.  Thus we have a branch cut on $[0,\\infty)$ which has the jump\n",
    "$$\n",
    "{\\rm Ei}^+(x) - {\\rm Ei}^-(x) = -\\oint {\\E^\\zeta \\over \\zeta} \\D \\zeta = -2 \\pi \\I\n",
    "$$\n",
    "where \n",
    "$$\n",
    "{\\rm Ei}^\\pm(x) = \\lim_{\\epsilon \\rightarrow 0} {\\rm Ei}(x\\pm\\I \\epsilon)\n",
    "$$\n",
    "Consider\n",
    "$$\n",
    "\\phi(z) = -{\\E^{-z} {\\rm Ei}(z) \\over 2 \\pi \\I }\n",
    "$$\n",
    "1. This is analytic off $[0,\\infty)$\n",
    "2. Integrating by parts we have decay at $\\infty$ in all directions:\n",
    "$$\n",
    "\\E^{-z} {\\rm Ei}(z) = {1  \\over z}  - \\int_{-\\infty}^z {\\E^{\\zeta-z} \\over \\zeta^2} \\D \\zeta = O(z^{-1})\n",
    "$$\n",
    "3. $\\phi$ has a logarithmic singularity at 0\n",
    "4. \n",
    "$$\\phi_+(x) - \\phi_-(x) = \\E^{-x}.$$\n",
    "\n",
    "Thus \n",
    "$$\n",
    "\\CC[ \\E^{-s \\diamond}](z) = \\phi(s z) = - {\\E^{-s z} {\\rm Ei}(s z)  \\over 2 \\pi \\I}\n",
    "$$\n",
    "\n",
    "Let's make sure I didn't make a mistake. Here we first define Ei:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "ei (generic function with 1 method)"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "const ei₋₁ = let ζ = Fun(-50 .. -1)\n",
    "    sum(exp(ζ)/ζ)\n",
    "end\n",
    "function ei(z) \n",
    "    ζ = Fun(Segment(-1 , z))\n",
    "    ei₋₁ + sum(exp(ζ)/ζ)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "#9 (generic function with 1 method)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "φ = (z) -> -exp(-z)*ei(z)/(2π*im)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The expression matches the Cauchy transform:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.02534853710699088 + 0.017828329563678025im"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t = Fun(0 .. 50)\n",
    "s = 2.0\n",
    "z = 2.0+2.0im\n",
    "sum(exp(-s*t)/(t-z))/(2π*im)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.02534853710699058 + 0.017828329563677824im"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "φ(s*z)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We then recover the Laplace transform by taking the limit:\n",
    "$$\n",
    "\\check 1(s) = -2 \\pi \\I  \\lim_{z\\rightarrow \\infty} z \\phi(s z) = \\lim_{z\\rightarrow \\infty} {z \\over s z} = {1 \\over s}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What about rational $f$? Do the same trick of subtracting off the singularities. For example, consider $f(z) = 1/(z+1)$. Then\n",
    "$$\n",
    "{\\phi(s z) - \\phi(-s) \\over z+1}\n",
    "$$\n",
    "satisfies all the necessary properties."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0174397493357667 + 0.013485355450154557im"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t = Fun(0 .. 50)\n",
    "s = 2.0\n",
    "z = 2.0+2.0im\n",
    "f = 1/(t+1)\n",
    "sum(exp(-s*t)*f/(t-z))/(2π*im)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.017439749335766603 + 0.01348535545015462im"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(φ(s*z)-φ(-s))/(z+1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " Therefore, \n",
    "$$\n",
    "\\check f(s) = -2 \\pi \\I  \\lim_{z\\rightarrow \\infty} z {\\phi(s z) - \\phi(-s) \\over z+1}  = \n",
    "2 \\pi \\I \\phi(-s) = -\\E^{s} {\\rm Ei}(-s)\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "sum(exp(-s * t) * f) = 0.3613286168882254\n",
      "(2π) * im * φ(-s) = 0.3613286168882283 + 0.0im\n",
      "-(exp(s)) * ei(-s) = 0.3613286168882283\n"
     ]
    }
   ],
   "source": [
    "@show sum(exp(-s*t)*f)\n",
    "@show 2π*im*φ(-s)\n",
    "@show -exp(s)*ei(-s);"
   ]
  }
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