Fourier_Transforms.ipynb

{
 "metadata": {
  "name": "Fourier_Transforms"
 },
 "name": "Fourier_Transforms",
 "nbformat": 3,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "%load_ext sympyprt",
     "language": "python",
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "from sympy import *",
     "language": "python",
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "source": "Examples of Fourier transformations."
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "t = Symbol(\"t\")\nw = Symbol(\"w\")",
     "language": "python",
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "n = Symbol(\"n\", integer=True)",
     "language": "python",
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "a = Symbol(\"a\")\nb = Symbol(\"b\")",
     "language": "python",
     "outputs": [],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "source": "The following examples are taken from 'Bronstein', 6th edition, page 1114ff"
    },
    {
     "cell_type": "markdown",
     "source": "4"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "1",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 6,
       "text": "1"
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "fourier_transform(_, t, w)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAMBAMAAABCcoqQAAAAJFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAADHJj5lAAAAC3RSTlMAVJlmiUTNuxAydkvU/DYAAABBSURBVAgd\nY2AQMhFgYGBNYChlYGAzYDBjYOgWYIhuYJgdwDB7AkP0BBADQkIYQFnpBgb2BSCVQF2pDAwMIlYC\nDADyLg3mxPBcdQAAAABJRU5ErkJggg==\n",
       "prompt_number": 7,
       "text": "0"
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "source": "5"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "t**n",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAABAAAAANBAMAAABSlfMXAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAIjK7iUSZzd12ZqtUECkr\naBAAAABZSURBVAgdY2BgYOCdLTUFSDEq8HBM4AMymB0Y7jFoABl8AQyHGVyADD0GhkyGogkMYRWZ\nDBsZPAMYGCyBomDwCEozPYAyuAygDO4JPBAW+4G1EAZPpgKYAQCOtg3ZZW3MogAAAABJRU5ErkJg\ngg==\n",
       "prompt_number": 8,
       "text": "\n n\nt "
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "fourier_transform(_, t, w)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAMBAMAAABCcoqQAAAAJFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAADHJj5lAAAAC3RSTlMAVJlmiUTNuxAydkvU/DYAAABBSURBVAgd\nY2AQMhFgYGBNYChlYGAzYDBjYOgWYIhuYJgdwDB7AkP0BBADQkIYQFnpBgb2BSCVQF2pDAwMIlYC\nDADyLg3mxPBcdQAAAABJRU5ErkJggg==\n",
       "prompt_number": 9,
       "text": "0"
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "markdown",
     "source": "9"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "1/sqrt(4*pi*a)*exp(-t**2/(4*a))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAD0AAAA2BAMAAACGi4wZAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ngg==\n",
       "prompt_number": 10,
       "text": "\n        2    \n      -t     \n      \u2500\u2500\u2500    \n      4\u22c5a    \n     \u212f       \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n    ___   ___\n2\u22c5\u2572\u2571 \u03c0 \u22c5\u2572\u2571 a "
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "fourier_transform(_, t, w)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEIAAAAUBAMAAADLrfYxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZnaZIruJ70QQVDLN\n3atI+RUPAAABFklEQVQoFWNgwACLDBFCXMoCCA6MxanA7gBjM0xiKIGz4QzuB6wNcM4+hgg4G8GY\nwD4BzuFkqISzgQy2DRBeIpIgG4o72iEquBYgqViFxGbglIGo6OJugAtzLNgBZzMwMPFsYKuvr2H5\n/38Cg5Kqg7LDPkYD//8/kVQs4dnQzcQEFmBWmM81b0ESywOYNKOxsbEh5wKeDQy72MBi/hP2MLBP\nOMGgAFMBphmV1C0YLCeD2WsZ1BhaGT5wOaCoYGDg3cB5wAcs1sdwQ2AxSwHESgZGRROISk79F1wK\nvGA2s6DyAmZBJRWIRBiDE4SBi2QrYEjBJQcR5w9UdMCvwr8BvzwDw3wHBk78algXMEjgV8FpKOSA\nVwUA/CQyQ1Dp8i8AAAAASUVORK5CYII=\n",
       "prompt_number": 11,
       "text": "\n     2    2\n -4\u22c5\u03c0 \u22c5a\u22c5w \n\u212f          "
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_fourier_transform(_, w, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAD0AAAA2BAMAAACGi4wZAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ngg==\n",
       "prompt_number": 12,
       "text": "\n        2    \n      -t     \n      \u2500\u2500\u2500    \n      4\u22c5a    \n     \u212f       \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n    ___   ___\n2\u22c5\u2572\u2571 \u03c0 \u22c5\u2572\u2571 a "
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "markdown",
     "source": "10"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "1/(2*a)*exp(-a*Abs(t))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACkAAAArBAMAAAAXhrzOAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZnaZIruJ70QQVDLN\n3atI+RUPAAABFElEQVQoFWNgQAacIM4EZBEQm6sBSKwAsZABUHQ2VlFDrKJX0ESZlEK4GtqKGlDN\nNeL8wNXA9QCultHY2NiQ5Sf3Aa4GZgG4KNgZXBeYErgaeDdMQDGB4wFvI1cDP8cGsCijognEyYZC\nVkATpCAmhDE4wT0C9xtbAUMKpih/oKIDQnQDkLkaiP0b4GJIjPkODODwQxICMlkXMEigioB4nIZC\nDpiig1fkPxbwYUCcy5gchcVeFYb9ApjCRQx8BZii1xjYfmKKAmOqgIFJcS26zHwFhiAGW3TRYAbG\nCwzGaKJMCQzsCQy30EQXMUzwb+BEs5FDgUFh/wTGCwtQFG8VFApgd/B5oIAiWv//fwCHoeQjB4go\nAG7aT3Gc/32VAAAAAElFTkSuQmCC\n",
       "prompt_number": 13,
       "text": "\n -a\u22c5\u2502t\u2502\n\u212f      \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n  2\u22c5a  "
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "fourier_transform(_, t, w)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFoAAAAmBAMAAAC2ZpGLAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ng/mza3t5k8n2Ot3e/uy+AfKjiHQvE3QKAAAAAElFTkSuQmCC\n",
       "prompt_number": 14,
       "text": "\n     1      \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n 2      2  2\na  + 4\u22c5\u03c0 \u22c5w "
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_fourier_transform(_, w, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAcEAAABHCAMAAABI+wyrAAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAAAElFTkSuQmCC\n",
       "prompt_number": 15,
       "text": "\n        \u239b               \u2502  2  2  -\u2148\u22c5\u03c0\u239e           \u239b               \u2502  2  2  \u2148\u22c5\u03c0\u239e\n\u256d\u2500\u256e3, 1 \u239c    1/2        \u2502 a \u22c5t \u22c5\u212f    \u239f   \u256d\u2500\u256e3, 1 \u239c    1/2        \u2502 a \u22c5t \u22c5\u212f   \u239f\n\u2502\u2576\u2510     \u239c               \u2502 \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f + \u2502\u2576\u2510     \u239c               \u2502 \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\n\u2570\u2500\u256f1, 3 \u239d1/2, 0, 1/2    \u2502      4     \u23a0   \u2570\u2500\u256f1, 3 \u239d1/2, 0, 1/2    \u2502     4     \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                      3/2                                     \n                                   4\u22c5\u03c0   \u22c5a                                   "
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "hyperexpand(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAcEAAABHCAMAAABI+wyrAAAANlBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABHL6OuAAAAEXRSTlMAEImrmSJ2\n3UTvVLvNMmZAoe8m1AUAAAlASURBVHgB7ZyJkoQoDEDR9mpb3fX/f3YDAkm4mvYY3SqcqmmBCLyE\nG1SI065Pmx1VNWeLPljw/wrcDkGljvU734JzHYzjsGcVztuheCO84jbgo5DtVEUU8gALimaJZG63\nd5xX3AV8DLKKGvA2IGacz8ichx0J3vuAD0FOcQ3dVSSZlao+vy1nD0YcCd77LHgEsn5FSMH7ERYU\nQx/P4e8hKd4bgfdDzu9YJ/gYC4pP97uhYk8keW+04H7IT2oM+Yw6KOY1Ucpipor4J3nvtOBeyNRz\nQ7e+sgeCV80mlCHSao/YKuid4hX3Au+E/CR6waAKYp6XWnBYY8n+6n8arzgdOA45jMvLLpg4iw7V\nakN+1YUjfzoQi/+d3RYI8Te851tQRCHfsxjemz68RYdOBzB1aceqr1CY7/fPv+jnFBMM2H3XTPmP\n7uUVdwNHIWE2hRXUGZtMZzWipEh6xUSIueuwGfCcGcYZfhjL/A3vL8AOfow3BYmLNtyC1Rqfzf+o\naNqK8kSEqGA2QEbxjjMGxPyTGWWS0rGP1y1oXrTMIx84mzcKOdcvOxrnyh1TRTs7YQWWAprlQGS1\nS9SOk+kl5uhTkx7noZ28bkFzYnWd+cDZvByyql9NXbetIh/stJ1bsE50gzALg0yj3l0Cx50CEjBc\nqla1NtbKkRM6nUjQWb9X+MPe7/XBsODdcd5LgNO8DiWD7HpV5Ov3OEorrWYsxy34QRUF1JKhaHwq\naUEQa7ZKZCY92onP87thFh384dU584nZaf7P4M0pWZijPOAkr0tJIT96v6FaK5mSnDSAOFzcgv23\ngcwXRSMP6djdRJTQ0myyuk4bpw5r1LWI7bdpoNEHE9GGc9yqsE1wNEVy8zmL1xQ0m0785osFNWGI\nN0pJIBvLCyYau+UFJbaBdshddHhTJQUyyxQdCCdeaaAB6pPsi1XLLPOxOcnz7m1TVdroKmR2mnNu\nwbN4xdnA33g5JUIOK22AjG54qVW+tnE1Mvw3Q9H2AbQgKSZdt4xyY2HuqnaBZqCZ+gaypp1CLN3S\nhLeOqkm0tIlvnXEas+BZvDkl6xdgj9cFdigR8hMcYgas6qjF5m67sYp2/INOtCAGyz2FBizYqrmy\n9Fe7DNYpO4k2kC0QHMGbbipVTmFjFjyJF0sWIsTvsoA5r3CBHUqEXLfS2zWT6UilTuyMwuYKbW69\nyI1VNPGL3waAFmmChraFgi3iqZFwHa6DXkJOs8IseA4vKWhe6gGPLGDGK74CW8jVDFBWpj4vG9ju\nqqCcxbCojF1Vw7Up1cv2dNCouwWdkfUDrWimAcVqu+zmA9fUy/+acyeviMIQTUVlcoA5r/gKbCHf\nmmxOt5IwrrCzbKijGSfTEjJ+kWxl7DAYJupY6KxOhZPA9K2F28RYHdzF+xfAjBfqOFF3kNZCNnqi\nboZothyxZXvZP9EK4sw0gknEZXwLqq04aElJGi/o3axTz/GphcNpKl/bwGwyzII7eeMwJB98+kUC\ncoAZr1nUSAAj5KRaz+GlLEkqDlm2l3nBnlPlLJpZku+oDAOqzdyzhaVzMlSBQ0YLAqjzWcu3cqkT\nT41kxD7eEy0YAea84iswgexeTdeBujZ8q3bodHCbAsLIE+CyYlppoZ+oDLXgACUFrqFZlurVkZ5u\nbDoVssVcQTgNDaVn/Nw1eFYHhdjFez0w5xXfgF1IA89Ng8v2EI61Vgq71hmbpbZtno7NldHeMMez\nIw1RLROxk5U4dOP2IYObMRo7zWSC93HALiQyIRFdtofwNxuroph6dJBzko/TyDkymAaxYCfOt+Bg\nV5kwyegdZjLJ61rwduA4JBJB02a3KaR96NjQBZpkG7zQhRFw06iYBtGCY3uBBRdnZZul7TpoJhO8\nLsztwHFIQ8S3KYCbHrIgi2FKI1ujzPeNXRmiOmvBCtq38+ugXNrJvrJ4vUXi+4HjkIoI9iakltnZ\nJm4frqFtsjjzVQUuQl3WgkvXde/GaXyp5K57M3nPengXr54d3wkcg9QVR+5N6G0Kq4YqMccclO3Y\npN8+F7ixFpRh/ekjGT5qDqSPXjt5YZQuc30n8DfIwN6EwH19XAzTuggAeTKoNmEXmaCaw0nhk+tg\nvI8nWXBu07z+ybTbgb9Ckpm1Za3j3ctWGDcsKx+/YXUwLrYzJJHPaIw/8uradyPwN8jA3oRsM8h8\nm+ti69iTZ6HoA9dacMLZJk00df8rL4wQ5BzzRuAdkJDh3pZU71Cj2tviswlPBlV4qQWTbztgHjLu\nkNc/W3gz8E5ItYUuyfFs4azHIWqCO0GHZjyIjK+rSy3IDnH5af/gY3kJjOG7GXgvZK87fLWVpYam\ns9njGOsF1lZlU6vbMJTxdXalBfUuhp/oDh/DC1Tw9IOAd0OqWb7UBAylTSTeKM54EBlXeVdaMPna\nrZuRL27L+zTg/ZAfYx57ptNsaaAuUCR6HO9CC7Z0IRAztfOO8FoYwrfFSjxiBy5PBj4Aie/gm6N2\nsLDJL+JhZLgAuE4GovHLvvi8C3nt2ULCt6VDPP4K+Aik6sAh5/Zsobdxgx5WxtPodRasf59JeLmj\nHob3UcDHIEd1rCbnbGFC5jILjmwHjJpi7/3Gm3W28K+Aj0LKDe+cs4Upmass2J5uQJi1y24uBWPK\nRkrmTOArIA1D+S0aKBr4XQNmC6H8/k818LvJyxNFA3+sgXHs7Ir8HyddkjtFA7CrmXp//JQ0SiTX\naoC9XHhtUiX2KzSwTO7K3xWplDiv1ECP719cmUyJ+xINyCOphz5rfEmuSqT5Ghhgpe38A4356RfJ\nwxpYlrrMJg5rsURQNJCrgdZ8FsA8oBfLjLP8Pl0DE3vD6um5LfnzNND5x/DGjr3Y6T1SPJ6kgXmQ\n7+ywS55tNW+YsYDieKIGOvVFOFF9dO8HizEzdIzm5WLz4a8nZr3kSWoAqpuqg6+5qqth2FZi4L8+\ngShPysdfnCwqvF8D8pua0oJzJdpF4Jcv6u1EZ+DDX/fnueSAaKCShtL9IHyBzU4rRn0A0v/wF3m4\n3D5AA2MNV9/X8swvfJPBvO0I3yKZpZf6fAf/8NcDMl2y4Gpgm00MUAH1AFQdApQ9YuDDX+7Dxf0A\nDWzfTJHvO9gXj0yuwKTOh79MSPl9jAbmZl0bGM/IDUH5gWB2+R/+YsHFUTRwiwb+A5HTb8wfUqA2\nAAAAAElFTkSuQmCC\n",
       "prompt_number": 16,
       "text": "\n        \u239b               \u2502  2  2  -\u2148\u22c5\u03c0\u239e           \u239b               \u2502  2  2  \u2148\u22c5\u03c0\u239e\n\u256d\u2500\u256e3, 1 \u239c    1/2        \u2502 a \u22c5t \u22c5\u212f    \u239f   \u256d\u2500\u256e3, 1 \u239c    1/2        \u2502 a \u22c5t \u22c5\u212f   \u239f\n\u2502\u2576\u2510     \u239c               \u2502 \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f + \u2502\u2576\u2510     \u239c               \u2502 \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\n\u2570\u2500\u256f1, 3 \u239d1/2, 0, 1/2    \u2502      4     \u23a0   \u2570\u2500\u256f1, 3 \u239d1/2, 0, 1/2    \u2502     4     \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                      3/2                                     \n                                   4\u22c5\u03c0   \u22c5a                                   "
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "markdown",
     "source": "11"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "1/(t**2+a**2)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADcAAAAmBAMAAACSfx5LAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKtEMol2VCK73Zlm\n781XBQ0aAAABCUlEQVQ4EWNgQAZCBsg8VLZrfgGqAAqPf1SSoXX+ajzhhxJcQ47zHzf4QKlnWNJU\n0I1wS3KACokx+AqgyjImsFyEivgz8KOFOMsHhs9QSRYG+QZknRwODA6MP+AiWyEsdwjFGcDAwA3E\nEMCzAULXQCiQJaehUgwMZgwCYDZU0p+BgTmBESjCllPFwJLY44Akaf1qG4ObxBGgyDSGXQz9//+D\n5RigOqcxcPz//4GBgWcBw26IBIiESn6BivAGMCyHMrlnznw1c6YCAwPQkxDQb8D4EcoEUhCdPBOg\nIv4CPAsK4LIQSaYEdogIb0OfQgKaJF/BQYgI856Dag1okuzbHOAiSAxzJDYDAwAVFVLpFBkgTAAA\nAABJRU5ErkJggg==\n",
       "prompt_number": 17,
       "text": "\n   1   \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n 2    2\na  + t "
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "fourier_transform(_, t, w)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAd4AAABGCAMAAABVAgVrAAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\nr17Tysd/1QO6UfSr5h+1+w8R8HPVHBf35gAAAABJRU5ErkJggg==\n",
       "prompt_number": 18,
       "text": "\n\u256d\u2500\u256e3, 1 \u239b    1/2        \u2502  2  2  2  -\u2148\u22c5\u03c0\u239e   \u256d\u2500\u256e3, 1 \u239b    1/2        \u2502  2  2  2\n\u2502\u2576\u2510     \u239c               \u2502 \u03c0 \u22c5a \u22c5w \u22c5\u212f    \u239f + \u2502\u2576\u2510     \u239c               \u2502 \u03c0 \u22c5a \u22c5w \n\u2570\u2500\u256f1, 3 \u239d1/2, 0, 1/2    \u2502               \u23a0   \u2570\u2500\u256f1, 3 \u239d1/2, 0, 1/2    \u2502         \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                         ___                                  \n                                     2\u22c5\u2572\u2571 \u03c0 \u22c5a                                \n\n  \u2148\u22c5\u03c0\u239e\n\u22c5\u212f   \u239f\n     \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\n      \n      "
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "hyperexpand(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAd4AAABGCAMAAABVAgVrAAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\nr17Tysd/1QO6UfSr5h+1+w8R8HPVHBf35gAAAABJRU5ErkJggg==\n",
       "prompt_number": 19,
       "text": "\n\u256d\u2500\u256e3, 1 \u239b    1/2        \u2502  2  2  2  -\u2148\u22c5\u03c0\u239e   \u256d\u2500\u256e3, 1 \u239b    1/2        \u2502  2  2  2\n\u2502\u2576\u2510     \u239c               \u2502 \u03c0 \u22c5a \u22c5w \u22c5\u212f    \u239f + \u2502\u2576\u2510     \u239c               \u2502 \u03c0 \u22c5a \u22c5w \n\u2570\u2500\u256f1, 3 \u239d1/2, 0, 1/2    \u2502               \u23a0   \u2570\u2500\u256f1, 3 \u239d1/2, 0, 1/2    \u2502         \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                         ___                                  \n                                     2\u22c5\u2572\u2571 \u03c0 \u22c5a                                \n\n  \u2148\u22c5\u03c0\u239e\n\u22c5\u212f   \u239f\n     \u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\n      \n      "
      }
     ],
     "prompt_number": 19
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_fourier_transform(_, w, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAckAAABCCAMAAAAL4d3sAAAAOVBMVEX///8AAAAAAAAAAAAAAAAA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\nTkSuQmCC\n",
       "prompt_number": 20,
       "text": "\n  \u239b          \u239b               \u2502  2  -2\u22c5\u2148\u22c5\u03c0\u239e           \u239b               \u2502  2  2\u22c5\u2148\n  \u239c  \u256d\u2500\u256e3, 3 \u239c1/2, 0, 1/2    \u2502 a \u22c5\u212f      \u239f   \u256d\u2500\u256e3, 3 \u239c1/2, 0, 1/2    \u2502 a \u22c5\u212f   \n\u2148\u22c5\u239c- \u2502\u2576\u2510     \u239c               \u2502 \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f + \u2502\u2576\u2510     \u239c               \u2502 \u2500\u2500\u2500\u2500\u2500\u2500\u2500\n  \u239c  \u2570\u2500\u256f3, 3 \u239c1/2, 0, 1/2    \u2502      2    \u239f   \u2570\u2500\u256f3, 3 \u239c1/2, 0, 1/2    \u2502      2 \n  \u239d          \u239d               \u2502     t     \u23a0           \u239d               \u2502     t  \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                        2                                     \n                                     4\u22c5\u03c0 \u22c5a\u22c5t                                 \n\n\u22c5\u03c0\u239e\u239e\n  \u239f\u239f\n\u2500\u2500\u239f\u239f\n  \u239f\u239f\n  \u23a0\u23a0\n\u2500\u2500\u2500\u2500\n    \n    "
      }
     ],
     "prompt_number": 20
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "hyperexpand(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAckAAABCCAMAAAAL4d3sAAAAOVBMVEX///8AAAAAAAAAAAAAAAAA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\nTkSuQmCC\n",
       "prompt_number": 21,
       "text": "\n  \u239b          \u239b               \u2502  2  -2\u22c5\u2148\u22c5\u03c0\u239e           \u239b               \u2502  2  2\u22c5\u2148\n  \u239c  \u256d\u2500\u256e3, 3 \u239c1/2, 0, 1/2    \u2502 a \u22c5\u212f      \u239f   \u256d\u2500\u256e3, 3 \u239c1/2, 0, 1/2    \u2502 a \u22c5\u212f   \n\u2148\u22c5\u239c- \u2502\u2576\u2510     \u239c               \u2502 \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f + \u2502\u2576\u2510     \u239c               \u2502 \u2500\u2500\u2500\u2500\u2500\u2500\u2500\n  \u239c  \u2570\u2500\u256f3, 3 \u239c1/2, 0, 1/2    \u2502      2    \u239f   \u2570\u2500\u256f3, 3 \u239c1/2, 0, 1/2    \u2502      2 \n  \u239d          \u239d               \u2502     t     \u23a0           \u239d               \u2502     t  \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                        2                                     \n                                     4\u22c5\u03c0 \u22c5a\u22c5t                                 \n\n\u22c5\u03c0\u239e\u239e\n  \u239f\u239f\n\u2500\u2500\u239f\u239f\n  \u239f\u239f\n  \u23a0\u23a0\n\u2500\u2500\u2500\u2500\n    \n    "
      }
     ],
     "prompt_number": 21
    },
    {
     "cell_type": "markdown",
     "source": "12"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "t/(t**2+a**2)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADcAAAAmBAMAAACSfx5LAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIjK7iUSZzd12ZqtU\nEO//KStoAAABJUlEQVQ4EWNgQAaMCsg8NDazA5oAMpc/AJmHxtZH4yNzwyoykblobEs0Pgr3EwoP\nlcP0AZWPwuMyQOGicrgn8KAKIPPYD6xF5qKyeTIVUAVGCu8/boAvGogKHqaZm9DVac5WgArdZNC7\ngCrLO4HpLVREnwE1UTIqABPFV6gkE8N9ATBTByIATNwKvB+hkgwMKRDWKQgFMofNASbJlQBhQSVB\niXsZSIRz1gkGhkgGiIMgkqDEzTqBFyhpyZDBwDRXSgGkkAGqE5i4NS8uYmDgamBIY5D//x8sB5P8\nxMD4//8HBgY+B4YeiAQDm7FxhbHxBgYGeOKWD+D9C5WEGwtP3PoXuBoOwGUhdsITN5+A+IYJaJLw\nxM2atXanAJokjsQdA1cGYgAAMNlT4la2fVAAAAAASUVORK5CYII=\n",
       "prompt_number": 22,
       "text": "\n   t   \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n 2    2\na  + t "
      }
     ],
     "prompt_number": 22
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "fourier_transform(_, t, w)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAegAAABGCAMAAAAw7/cUAAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\n0g0Ss4B+TT925pb9LQuYodXfgvpM8x8vQnTiurRSDgAAAABJRU5ErkJggg==\n",
       "prompt_number": 23,
       "text": "\n  \u256d\u2500\u256e3, 1 \u239b    0        \u2502  2  2  2  -\u2148\u22c5\u03c0\u239e   \u256d\u2500\u256e3, 1 \u239b    0        \u2502  2  2  2  \n- \u2502\u2576\u2510     \u239c             \u2502 \u03c0 \u22c5a \u22c5w \u22c5\u212f    \u239f + \u2502\u2576\u2510     \u239c             \u2502 \u03c0 \u22c5a \u22c5w \u22c5\u212f\n  \u2570\u2500\u256f1, 3 \u239d0, 0, 1/2    \u2502               \u23a0   \u2570\u2500\u256f1, 3 \u239d0, 0, 1/2    \u2502           \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                         ___                                  \n                                     2\u22c5\u2572\u2571 \u03c0                                   \n\n\u2148\u22c5\u03c0\u239e\n   \u239f\n   \u23a0\n\u2500\u2500\u2500\u2500\n    \n    "
      }
     ],
     "prompt_number": 23
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "hyperexpand(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAegAAABGCAMAAAAw7/cUAAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\n0g0Ss4B+TT925pb9LQuYodXfgvpM8x8vQnTiurRSDgAAAABJRU5ErkJggg==\n",
       "prompt_number": 24,
       "text": "\n  \u256d\u2500\u256e3, 1 \u239b    0        \u2502  2  2  2  -\u2148\u22c5\u03c0\u239e   \u256d\u2500\u256e3, 1 \u239b    0        \u2502  2  2  2  \n- \u2502\u2576\u2510     \u239c             \u2502 \u03c0 \u22c5a \u22c5w \u22c5\u212f    \u239f + \u2502\u2576\u2510     \u239c             \u2502 \u03c0 \u22c5a \u22c5w \u22c5\u212f\n  \u2570\u2500\u256f1, 3 \u239d0, 0, 1/2    \u2502               \u23a0   \u2570\u2500\u256f1, 3 \u239d0, 0, 1/2    \u2502           \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                         ___                                  \n                                     2\u22c5\u2572\u2571 \u03c0                                   \n\n\u2148\u22c5\u03c0\u239e\n   \u239f\n   \u23a0\n\u2500\u2500\u2500\u2500\n    \n    "
      }
     ],
     "prompt_number": 24
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_fourier_transform(_, w, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAn8AAABCCAMAAAAR60CTAAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\nSUVORK5CYII=\n",
       "prompt_number": 25,
       "text": "\n  \u239b            \u239b             \u2502  2\u239e           \u239b             \u2502  2  -2\u22c5\u2148\u22c5\u03c0\u239e      \n  \u239c    \u256d\u2500\u256e3, 3 \u239c1/2, 0, 0    \u2502 a \u239f   \u256d\u2500\u256e3, 3 \u239c1/2, 0, 0    \u2502 a \u22c5\u212f      \u239f   \u256d\u2500\u256e\n\u2148\u22c5\u239c- 2\u22c5\u2502\u2576\u2510     \u239c             \u2502 \u2500\u2500\u239f + \u2502\u2576\u2510     \u239c             \u2502 \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f + \u2502\u2576\u2510\n  \u239c    \u2570\u2500\u256f3, 3 \u239c0, 0, 1/2    \u2502  2\u239f   \u2570\u2500\u256f3, 3 \u239c0, 0, 1/2    \u2502      2    \u239f   \u2570\u2500\u256f\n  \u239d            \u239d             \u2502 t \u23a0           \u239d             \u2502     t     \u23a0      \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                       2                      \n                                                    4\u22c5\u03c0 \u22c5t                    \n\n     \u239b             \u2502  2  2\u22c5\u2148\u22c5\u03c0\u239e\u239e\n3, 3 \u239c1/2, 0, 0    \u2502 a \u22c5\u212f     \u239f\u239f\n     \u239c             \u2502 \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u239f\n3, 3 \u239c0, 0, 1/2    \u2502      2   \u239f\u239f\n     \u239d             \u2502     t    \u23a0\u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                \n                                "
      }
     ],
     "prompt_number": 25
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "hyperexpand(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAn8AAABCCAMAAAAR60CTAAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\nSUVORK5CYII=\n",
       "prompt_number": 26,
       "text": "\n  \u239b            \u239b             \u2502  2\u239e           \u239b             \u2502  2  -2\u22c5\u2148\u22c5\u03c0\u239e      \n  \u239c    \u256d\u2500\u256e3, 3 \u239c1/2, 0, 0    \u2502 a \u239f   \u256d\u2500\u256e3, 3 \u239c1/2, 0, 0    \u2502 a \u22c5\u212f      \u239f   \u256d\u2500\u256e\n\u2148\u22c5\u239c- 2\u22c5\u2502\u2576\u2510     \u239c             \u2502 \u2500\u2500\u239f + \u2502\u2576\u2510     \u239c             \u2502 \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f + \u2502\u2576\u2510\n  \u239c    \u2570\u2500\u256f3, 3 \u239c0, 0, 1/2    \u2502  2\u239f   \u2570\u2500\u256f3, 3 \u239c0, 0, 1/2    \u2502      2    \u239f   \u2570\u2500\u256f\n  \u239d            \u239d             \u2502 t \u23a0           \u239d             \u2502     t     \u23a0      \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                                       2                      \n                                                    4\u22c5\u03c0 \u22c5t                    \n\n     \u239b             \u2502  2  2\u22c5\u2148\u22c5\u03c0\u239e\u239e\n3, 3 \u239c1/2, 0, 0    \u2502 a \u22c5\u212f     \u239f\u239f\n     \u239c             \u2502 \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u239f\n3, 3 \u239c0, 0, 1/2    \u2502      2   \u239f\u239f\n     \u239d             \u2502     t    \u23a0\u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                                \n                                "
      }
     ],
     "prompt_number": 26
    },
    {
     "cell_type": "markdown",
     "source": "14"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "exp(I*a*t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAB0AAAAQBAMAAAAYI2niAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZnaZIruJ70QQVDLN\n3atI+RUPAAAAeElEQVQIHWNgQAazkTlAtiEa/woSn0kppK2oASFgxPmB6wGEawwEU35yH2AWQEhz\nXWBK4N0wAS7A8YC3kZ9jAwMDo6IJWNBQyIpZCsgKY3CCKwIy2AoYUpD5/IGKDsh8/wZkHgPDfAcG\nTmQR1gUMEsh8TkMhByQ+AMPIE8ZavNhRAAAAAElFTkSuQmCC\n",
       "prompt_number": 27,
       "text": "\n \u2148\u22c5a\u22c5t\n\u212f     "
      }
     ],
     "prompt_number": 27
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "fourier_transform(_, t, w)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAMBAMAAABCcoqQAAAAJFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAADHJj5lAAAAC3RSTlMAVJlmiUTNuxAydkvU/DYAAABBSURBVAgd\nY2AQMhFgYGBNYChlYGAzYDBjYOgWYIhuYJgdwDB7AkP0BBADQkIYQFnpBgb2BSCVQF2pDAwMIlYC\nDADyLg3mxPBcdQAAAABJRU5ErkJggg==\n",
       "prompt_number": 28,
       "text": "0"
      }
     ],
     "prompt_number": 28
    },
    {
     "cell_type": "markdown",
     "source": "15"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "cos(a*t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADkAAAASBAMAAAATXWtDAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVJl2Zs0i7zIQRN2r\nibuk4LwrAAABKklEQVQoFWWQMUvDQBTHf6Fpampqi7rZQVoXcQm2S51SFaGTXcUliwg6ZHW8QVcb\nXFyEBr+AcXUKODkZcBMEwS/gJLqIr7G5Yn1wd7/7/9893j2YCk/fJ6Qlx9d4oUlDd0yWwnC1mkM8\nhrkU9nIxP3V+NYJ2ruanI2IWA9nPZM23PBa2PLqLfaiKUGgvcb3fzC6FvnVSWealnPIAQ3HfWBtt\nUAoZ1uzVwCc4uiOCQzBjGvAp7qzPjYJdqeI9fgsdy4uUW2beR67Sbseqf8AlBJH9hfkqblYZK1AE\np4qnrJFBaMY1xzV+u4rZqSQ0zIQNKCpK/kHiFms9OJf39VbE5rpntuRXFFzKK71732gq2Bb3T9j5\nJEfqv0nyPEmWCUyHE2rlSpMG29XYgR8bajv4246+AgAAAABJRU5ErkJggg==\n",
       "prompt_number": 29,
       "text": "cos(a\u22c5t)"
      }
     ],
     "prompt_number": 29
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "fourier_transform(_, t, w)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAMBAMAAABCcoqQAAAAJFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAADHJj5lAAAAC3RSTlMAVJlmiUTNuxAydkvU/DYAAABBSURBVAgd\nY2AQMhFgYGBNYChlYGAzYDBjYOgWYIhuYJgdwDB7AkP0BBADQkIYQFnpBgb2BSCVQF2pDAwMIlYC\nDADyLg3mxPBcdQAAAABJRU5ErkJggg==\n",
       "prompt_number": 30,
       "text": "0"
      }
     ],
     "prompt_number": 30
    },
    {
     "cell_type": "markdown",
     "source": "16"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "sin(a*t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADcAAAASBAMAAAANlFvwAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZlTNRImrEN27MiJ2\n75lZwTvPAAABJElEQVQoFV2QvUrDUBTHf0ljorElVsGCKBRdxE0QdAyI4CQdfAAfQEgmN23wCRxF\nHC4IglODLi6lyeRzuDkUURSEKuhJmpu2Hrjn/D8u5557YDLmSjpCWrJjjTgr0EqopfkCzCrMZIjX\nSjMqzJkWbBdYF30bz4dUq0W1RcujI/kGjNUF91Ltftf7wj051XSdgx1hQvaww7ZyfujJw20xr3nJ\nEtRC9n0LT/GeJbrSKeIVBmJOx9i/V5k+yGd4lPstznHfMlMR3gdxaZ5Cz3e+MC6GbZeYSkpTZuiE\nRtS0EzMfaJGqL+Zn3raiqMUPR0ml2YA7WE5TN9i6/dgINuUbCdZz4zg2+wrq0no8nGiM/V8fhyPT\nkvVOhi2bKOJEg7I6SQmf4A9IJ0QQle9nVwAAAABJRU5ErkJggg==\n",
       "prompt_number": 31,
       "text": "sin(a\u22c5t)"
      }
     ],
     "prompt_number": 31
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "fourier_transform(_, t, w)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAMBAMAAABCcoqQAAAAJFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAADHJj5lAAAAC3RSTlMAVJlmiUTNuxAydkvU/DYAAABBSURBVAgd\nY2AQMhFgYGBNYChlYGAzYDBjYOgWYIhuYJgdwDB7AkP0BBADQkIYQFnpBgb2BSCVQF2pDAwMIlYC\nDADyLg3mxPBcdQAAAABJRU5ErkJggg==\n",
       "prompt_number": 32,
       "text": "0"
      }
     ],
     "prompt_number": 32
    },
    {
     "cell_type": "markdown",
     "source": "17"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "1/cosh(t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADsAAAApBAMAAAB5FUwQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKtEMol2VCK73Zlm\nze8l1wreAAABXklEQVQ4Ee2SvUsDQRDFf0c+OOUkh4WFjQmC2OUaaw/BQhC00VpRrrEwBEmjhaDW\nZ2NtQAshTWxsRLgqKgqmEGwsIvgHBEyhqOjsZYP5Iq0ITrFv3nu7O7NzBy0x6LTQNjK9nmlTWmni\n324dSMj2Dgs9p9rlyF+VvnpF9RdeZama4dJc3HQ1G1doFDVrwEytnlnyO+9CrqE3UNv9wg9gwG7o\nGrWtjr1APBBsDm2PQlQmEi2Lt+kF5EoBl1tlardnaneVyEVhSSHxsvkcK7IQmSfFO74tLYvcJy5P\nMJIxTv1t/J0jHD5IuGBWCJETmBR+JzRIv7nUSDhi52EOieMfe9nMvmrbqMC1ssPLMX0Xf9ZlUduq\npTEy9dbyXMWSnFtJVuSx6nLu4cGyie5LnvUcbtYCyysFU58b6UeRPHnuqnyTQPIuMVzXOoaqt8ZU\nBZjQtANSSjGSsnSNIaXGbfgG9VFwkmFyk04AAAAASUVORK5CYII=\n",
       "prompt_number": 33,
       "text": "\n   1   \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\ncosh(t)"
      }
     ],
     "prompt_number": 33
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "fourier_transform(_, t, w)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAIUAAAAsBAMAAAC552YpAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 34,
       "text": "\n                \u239b   1         \u239e\nFourierTransform\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500, t, w\u239f\n                \u239dcosh(t)      \u23a0"
      }
     ],
     "prompt_number": 34
    },
    {
     "cell_type": "markdown",
     "source": "18"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "1/sinh(t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADkAAAApBAMAAAB94JwtAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKtEMol2VCK73Zlm\nze8l1wreAAABRUlEQVQ4EWNgQAaMysg8NDZL2Hc0EVTuqCwiPDgif21G8IYn6z8e8IHOXuYB2Qcm\nkCzmcIByNEA04wYo75wAmOH5BcLnKWBgaGFgKIPK1kJkGaCyXEDhqQwMvFBRqCKYLEjTZwYGtgNA\nGglA9QIzE8sHIL4AdFr5WZZZDrY/zmxjYPgCIhgYPjAw712dAKIZTBmYBOQdGL8x9Asw/AYRDIxA\nUU6gJMNDBgYrA2YGfgeGP3CCgeMBA4jDwLCCgYHp/xQQ+wsDvwGEYOBYwMDgD5RkWMLAINCu34Aq\ny/iAgeEkSBZo8hEG1gRUWZBr1BgKwK46xMBmAJT9DDIZTDAwXGFguM4jwMAygYHhWE4Oi/5lp78l\n+nfABNDENAaG6hSgXw+AjMcAUhAR9JCEqmM1ADNMMbRBBBRBFKMCkMAGxECCbAIMDACfrHBd89SJ\nFwAAAABJRU5ErkJggg==\n",
       "prompt_number": 35,
       "text": "\n   1   \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\nsinh(t)"
      }
     ],
     "prompt_number": 35
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "fourier_transform(_, t, w)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAIMAAAAsBAMAAAC0+RZuAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nWXzMweVQrSX1vfID/YH+BTriwPYc6orRAAAAAElFTkSuQmCC\n",
       "prompt_number": 36,
       "text": "\n                \u239b   1         \u239e\nFourierTransform\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500, t, w\u239f\n                \u239dsinh(t)      \u23a0"
      }
     ],
     "prompt_number": 36
    },
    {
     "cell_type": "markdown",
     "source": "19"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "sin(a*t**2)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAD8AAAAWBAMAAACF0lkSAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 37,
       "text": "\n   \u239b   2\u239e\nsin\u239da\u22c5t \u23a0"
      }
     ],
     "prompt_number": 37
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "fourier_transform(_, t, w)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAR0AAAA6BAMAAAB/mpByAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\no8iwrjnUY6OuDRs5kBbp+Y+OfwEnSRl4uFEAQAAAAABJRU5ErkJggg==\n",
       "prompt_number": 38,
       "text": "\n            \u239b     \u239b 2  2\u239e      \u239b 2  2\u239e\u239e\n  ___   ___ \u239c     \u239c\u03c0 \u22c5w \u239f      \u239c\u03c0 \u22c5w \u239f\u239f\n\u2572\u2571 2 \u22c5\u2572\u2571 \u03c0 \u22c5\u239c- sin\u239c\u2500\u2500\u2500\u2500\u2500\u239f + cos\u239c\u2500\u2500\u2500\u2500\u2500\u239f\u239f\n            \u239d     \u239d  a  \u23a0      \u239d  a  \u23a0\u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                    ___                \n                2\u22c5\u2572\u2571 a                 "
      }
     ],
     "prompt_number": 38
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "simplify(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAR0AAAA6BAMAAAB/mpByAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMrsQmSKrRHZUzYnd\n72ZiHOLWAAAGDUlEQVRoBe1YXYgbVRQ+k9lJJpudbLCy0hc3xr8qWoNI6YvsaqusKEtQqViQTOsP\nVtCNPmhbix10q0FxG7GWrUjNi/ZFZKUFC4oGfagiyJYqVh+6UWtZf7pmW5aqDxvP/Z258xd8SQr2\nQu79znfOd++ZO3fuTC5AfMlEu2eCLt0Jcj4mUrXWFxhuXhNOI5uoMdf0OGvTk02AKsPRdbTKdKJV\n0tMXcjnceSdrdccqUbQWvnWTlB34QYwq+tLdTra70IdSdUZodqJC0RBcDaAf84X5zDjVzUVfcIh5\nJIRj1ADLAqBosX50ILmcihRQR5wqa3u0w20sLQ9BoVn3M9J+WaJPOMo0EfCbJ50+EKvC+y3L5RJ5\ngSUmwUsy/J2g0jWO9pE2M8qt8CZWdcjVaN5uNDJX7TZ6d7oRPpSSc7lNY0mnapdgjCF5n4Ca8aqy\nK0mUXAyJojA2CxBok3mkMnNzPxjtdnGyNJQdLbfPkqgTpIoqisq8TbNPelVWU+qGIHtIPm8XS/oL\niRg4LTMdcJB50jQJn15TO2zYBGG5nTXhtVcFr2/JOBt4HFX12VL1MOyC4SY3rxS0viQQb9+U+YzQ\nCV3Jtm+reAAcHvIHb1fyVmkUFRzrr9DrwRCqyjZEMF7dcUjOMlOzBZ1uCRRo36bMupto8xS00jQ/\ntMSS2x+QIKGoUvlBeINHUZW7upI5+Aoy9P4DJEXXYPIMQ7p+lXB6fSN17TFmxYXCSI5Fh+ajqNKj\nD8hZpSp9kUkB6G1M8NE3gH4DPl6YVaYhArDN7vtNuy63fWlhjJAfkyrtDJAG+haquyjAarDGEM9n\nT7UJ0/NN+HV/HXlFBePzVRyEFqb6m2CSyy0EjDikBtgEyRUHnykiStaJzcuzkCyO5PS/oExcjwna\n3w42GcPyMevGUdySPkrNwIPId1AtY0hqdw30POnjClLhDmLDCqNJrzeRpwyrdlRSMJiDZVIBFDwe\nBVoOM1k+IzV9XbkE5fuOQAX5DqrjRJqoQ4aMYM7QXKAPjUzJJq5+Ut9dwLIKJ6t9GUllEQZJz+/i\nD+i26alayA1gN9BXKLxXKNi4urGzP/EmNifOIOqgwjWMJbUIN5J2GoqwFe8gMfqhQaiETWpeivdP\nlNx8rhK0v1XnR+TzgrGX7B0dVHR+4DP4EENTDlns1wJ8gMbT5FsGpyRPal5+hIzj5rNZ0P5WXT8j\nNTDKOShfmoPDGNlBRdYPbgr35LG+aGEem505nRir4Wus1XxOgVnBfM6x+xXZM2ZAC1/PDfglY8P6\nrA0vIR2v0unzBdZYBUPn2u083v18hhifwxjW6vM+PzWlTRwY+ue1iffRRZ9cEuMvZaLHwvKBvdUK\nnJxsZqv4zHdSGeeIEMwztKFVeomuJU503J9ZnFmVbxiA4SIjt/E+lIbtz1EqjW3IfJpY1DdkWkUx\n/O8v4QDAh1iWMb0lMZx2YRDFq7KzTJHzCHfnPQZ87zUUbNWkaZzV6tKAR1wYRPGqpB1UiOXIPA8F\nAziTHJWudINsXaKcECCsjVd5vn+kWJOIgB2K5TVSDWml7IHnpGHwl7IkFBCvoi8iJd5vWN5bqTqP\nuub4/B3SMPMShoFY1fNhCoUzbcX0Gi96DRfH/AMgQbEq+bHvdudHdJv2k9S2SqF0h2uMU3WYWjrc\nT6GDEjKbD3VFXwANj1NF5KoMk3Qfa4VHY4ufIHayGcZ6uBjVE56wSPhOpKc/7IZFrA+3k2hVzFp1\n5bDGg1Wo26pNLGM0yKmMbqs2sZjq3qDjAhOcAc/n5nkAW8EELzD/kxlgx0bnz8W6x0a9zOn3W8Ur\nxT026mE+Zklb5sN7Pup7l5DlwONsdM3uXRbuyPh3/dMiNd1jI9fbE3Q9GxXPbcypt3qSgTIongjR\nsglgFdyluHpiJEbpsIYN2Qas70kKyqCvyGMjawa+VFy9MNKOLo6NyhU99t9XV9LbCqY4NhouZhti\nc+zK2CGDaKsX8C8OOzayShttJySmm5TVbuNxDTs2So3//Gipm4NHjqUeG0WGdc+hHBt1b9jIkdRj\no8iwrjnUY6OuDRs5kBbp+Y+OfwEnSRl4uFEAQAAAAABJRU5ErkJggg==\n",
       "prompt_number": 39,
       "text": "\n            \u239b     \u239b 2  2\u239e      \u239b 2  2\u239e\u239e\n  ___   ___ \u239c     \u239c\u03c0 \u22c5w \u239f      \u239c\u03c0 \u22c5w \u239f\u239f\n\u2572\u2571 2 \u22c5\u2572\u2571 \u03c0 \u22c5\u239c- sin\u239c\u2500\u2500\u2500\u2500\u2500\u239f + cos\u239c\u2500\u2500\u2500\u2500\u2500\u239f\u239f\n            \u239d     \u239d  a  \u23a0      \u239d  a  \u23a0\u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                    ___                \n                2\u22c5\u2572\u2571 a                 "
      }
     ],
     "prompt_number": 39
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_fourier_transform(_, w, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAD8AAAAWBAMAAACF0lkSAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 40,
       "text": "\n   \u239b   2\u239e\nsin\u239da\u22c5t \u23a0"
      }
     ],
     "prompt_number": 40
    },
    {
     "cell_type": "markdown",
     "source": "20"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "cos(a*t**2)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEEAAAAWBAMAAABtUuw5AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nQmCC\n",
       "prompt_number": 41,
       "text": "\n   \u239b   2\u239e\ncos\u239da\u22c5t \u23a0"
      }
     ],
     "prompt_number": 41
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "fourier_transform(_, t, w)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAQwAAAA6BAMAAAC39nqkAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nJ5zCAAAAAElFTkSuQmCC\n",
       "prompt_number": 42,
       "text": "\n            \u239b   \u239b 2  2\u239e      \u239b 2  2\u239e\u239e\n  ___   ___ \u239c   \u239c\u03c0 \u22c5w \u239f      \u239c\u03c0 \u22c5w \u239f\u239f\n\u2572\u2571 2 \u22c5\u2572\u2571 \u03c0 \u22c5\u239csin\u239c\u2500\u2500\u2500\u2500\u2500\u239f + cos\u239c\u2500\u2500\u2500\u2500\u2500\u239f\u239f\n            \u239d   \u239d  a  \u23a0      \u239d  a  \u23a0\u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                   ___               \n               2\u22c5\u2572\u2571 a                "
      }
     ],
     "prompt_number": 42
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "simplify(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAQwAAAA6BAMAAAC39nqkAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nJ5zCAAAAAElFTkSuQmCC\n",
       "prompt_number": 43,
       "text": "\n            \u239b   \u239b 2  2\u239e      \u239b 2  2\u239e\u239e\n  ___   ___ \u239c   \u239c\u03c0 \u22c5w \u239f      \u239c\u03c0 \u22c5w \u239f\u239f\n\u2572\u2571 2 \u22c5\u2572\u2571 \u03c0 \u22c5\u239csin\u239c\u2500\u2500\u2500\u2500\u2500\u239f + cos\u239c\u2500\u2500\u2500\u2500\u2500\u239f\u239f\n            \u239d   \u239d  a  \u23a0      \u239d  a  \u23a0\u23a0\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n                   ___               \n               2\u22c5\u2572\u2571 a                "
      }
     ],
     "prompt_number": 43
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inverse_fourier_transform(_, w, t)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEEAAAAWBAMAAABtUuw5AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nQmCC\n",
       "prompt_number": 44,
       "text": "\n   \u239b   2\u239e\ncos\u239da\u22c5t \u23a0"
      }
     ],
     "prompt_number": 44
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "",
     "language": "python",
     "outputs": [],
     "prompt_number": "&nbsp;"
    }
   ]
  }
 ]
}