Legendre_polys.ipynb

{
 "metadata": {
  "name": "Legendre_polys"
 },
 "name": "Legendre_polys",
 "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": "code",
     "collapsed": true,
     "input": "x = Symbol(\"x\")",
     "language": "python",
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "n = Symbol(\"n\", integer=True)\nk = Symbol(\"k\", integer=True)\nl = Symbol(\"l\", integer=True)\nm = Symbol(\"m\", integer=True)",
     "language": "python",
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "kern = (-1)**k*binomial(n, k)**2*((1+x)/2)**(n-k)*((1-x)/2)**k\nkern",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAUsAAAAwBAMAAABqC0UbAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nJd7t17qhiPxNsCtIdd14MEI/0/0Lfbq7J6OVidgAAAAASUVORK5CYII=\n",
       "prompt_number": 5,
       "text": "\n               k        -k + n    2\n    k \u239b  x   1\u239e  \u239bx   1\u239e       \u239bn\u239e \n(-1) \u22c5\u239c- \u2500 + \u2500\u239f \u22c5\u239c\u2500 + \u2500\u239f      \u22c5\u239c \u239f \n      \u239d  2   2\u23a0  \u239d2   2\u23a0       \u239dk\u23a0 "
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "Pn = Sum(kern, (k, 0, n))\nPn",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAWoAAAA1BAMAAACaWbxGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nSaUNIRsxQIEAAAAASUVORK5CYII=\n",
       "prompt_number": 6,
       "text": "\n  n                                      \n ____                                    \n \u2572                                       \n  \u2572                  k        -k + n    2\n   \u2572      k \u239b  x   1\u239e  \u239bx   1\u239e       \u239bn\u239e \n   \u2571  (-1) \u22c5\u239c- \u2500 + \u2500\u239f \u22c5\u239c\u2500 + \u2500\u239f      \u22c5\u239c \u239f \n  \u2571         \u239d  2   2\u23a0  \u239d2   2\u23a0       \u239dk\u23a0 \n \u2571                                       \n \u203e\u203e\u203e\u203e                                    \nk = 0                                    "
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "[ expand(Pn.subs(n,i).doit()) for i in range(5) ]",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 7,
       "text": "\n\u23a1         2         3            4       2    \u23a4\n\u23a2      3\u22c5x    1  5\u22c5x    3\u22c5x  35\u22c5x    15\u22c5x    3\u23a5\n\u23a21, x, \u2500\u2500\u2500\u2500 - \u2500, \u2500\u2500\u2500\u2500 - \u2500\u2500\u2500, \u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500 + \u2500\u23a5\n\u23a3       2     2   2      2     8       4     8\u23a6"
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "legendre(n,x)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAG8AAAASBAMAAACnV51MAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nSUVORK5CYII=\n",
       "prompt_number": 5,
       "text": "legendre(n, x)"
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "expand_func(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAG8AAAASBAMAAACnV51MAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nSUVORK5CYII=\n",
       "prompt_number": 6,
       "text": "legendre(n, x)"
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "legendre(n, -x)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAH0AAAASBAMAAACEDMyZAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIjK7q81UdmaZ7xDd\nRIne2sQcAAAB/klEQVQ4EY2ST0jbUBzHv4ldmprWBAaDyaAP5m6DOnbwKnjRy/QkAy8PbxPUMBkt\nDFyEMVIPq4OhePO+SxjC/lzS2zwGRHDsYMCLl8EslBW6sv3es2leq7D+Dnnf7+/7PrxfXgIMU97N\nm0RbY/S4VoWZQOllHMUo8gvp+w2lkcozFXmT9vuU4ZK9mQ9Vvt5HKWZpKF4eo1CprAzFZ6KU6FcP\nJJ+tvEOufLGJctU5ufT3gO/+OqkXkUgAGzA3P1W5yhprq6+ZSOj99zEezGFfz+1oH6zfKAbmAmqO\n1hrhIgFC4HMx1g9Uftlq2R5GKW3oDeS9b9hGieEILdgs5Cg6VgcyAcaBHyuBUVd5r9C2gLxDvNny\nq3wDT1Dz/V0ax47o4xF/CZkA9wh8j9y05G8/ppoCxhbI5pngf5E6LXPUxIaUb0MmwFdqH2GUi7hX\nIzFJOb/WAZhwxQi65MNInN+GTOT90TAlnYbtVWBzUyYEnMN0KgGNNIktdGj+7CReMUJkAtxiMA6w\nMYaHLOGN9ja7Qz8vMNHk5t5zhH//uKiW2URzef4pnvnri8dNFyIBsi6yMe5uoRSTk6XNrs5ywO9a\naI/wcTExg6tVTzpeIrqr+H9lGdPAbldfX34mLS8RV2sh7vm1ylveM4MiQ5cjylBvkPyh7P7/YblX\ne4yBrS+Bfw0QjzwErWTyAAAAAElFTkSuQmCC\n",
       "prompt_number": 7,
       "text": "legendre(n, -x)"
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "expand_func(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJ8AAAASBAMAAAC3GxHVAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZqsiVN12RBDvzTKJ\nu5m05BpfAAACeUlEQVQ4EY2TP2gTcRTHv9e7JNdemuSsCAVJWhxc/HODDl3MbUVabLt0vixRtMW4\nSUE9HQoimI6OFTp0PAo6FErORadCwFkQLXUrCbUm5AZ9v5f787si1jfc733fe59vfr/fXQCUIAWJ\n9UsXfKnyrzSF0iDrEUtGXgP2MjqA6snlKM9ddaOU1jRKBaKB8+LBoV8GNAd6x1gE9oKonFqfyT+f\noOEM0QDBwzA+/qRkG1o771Pyd8OmbBijQwOmxZbiEIY1FJyx95ScbSijoUeN7qER+0EY7tLN5O/+\nl6GMhia7QDHxY8NYB8jWvqNQ/7CCetlaOzYPgafmAmX3G6LDqLKyXrbZQZufPecJt+Ypw1E3LAQ4\nwox7DUdG4av6Rv+FqqtsomKp3XFbdBh9XvWNDQbm9G6xBKJnQl4s4siZ6NYDI0CmtIxJtDwsoYui\n17RRtfQeuAOBPr7tasNXU8r1daZv0if3RYQTGnrCmiJQumbZXsUVVEzzgN5RsUHfDBkegzsAofiG\nwpSYBvKb9Mh4eClEGGKHyZGVDslHdRsV0U4M++AOGF3CqC3awLhPD6Ljl0BSGMY6UHuAJ6aqDRhs\n2GyIHfbBHR6l7bYMviS3aCtcGouOGBru0coRYB+KVXPpMG1MoEdHzrbx0CMP7gCEahtYzeMdJf1J\n7wb9v4CsM+TpKvdPfgBmKHcGtnL4AM3fJw7KdW9nMHf9Le6YC1tPBg5Eh9Gsj1sTaPlQp2enbab1\nxdAhXLZTUv2MV1upiiQStBRVBf0pErzm/JTUpoCDVEUWMRoZMj3iyjMvZEH5fO2ifaqUyAjVrLDG\ntO4kE8A9WZyVR6gWDRL9B64wrER+FapwAAAAAElFTkSuQmCC\n",
       "prompt_number": 9,
       "text": "\n    n               \n(-1) \u22c5legendre(n, x)"
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "legendre(-n, x)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAH0AAAASBAMAAACEDMyZAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n7yPgH2Snj5ZL7riKAAAAAElFTkSuQmCC\n",
       "prompt_number": 10,
       "text": "legendre(-n, x)"
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "expand_func(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAI4AAAASBAMAAAB/d/sDAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nSUVORK5CYII=\n",
       "prompt_number": 11,
       "text": "legendre(n - 1, x)"
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "legendre(n, 0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAG4AAAASBAMAAABIlfZyAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAElFTkSuQmCC\n",
       "prompt_number": 12,
       "text": "legendre(n, 0)"
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "expand_func(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAALMAAAAuBAMAAACRwjVdAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAABJRU5ErkJggg==\n",
       "prompt_number": 15,
       "text": "\n         ___     \n     2\u22c5\u2572\u2571 \u03c0      \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n   \u239bn\u239e  \u239b  n   1\u239e\nn\u22c5\u0393\u239c\u2500\u239f\u22c5\u0393\u239c- \u2500 + \u2500\u239f\n   \u239d2\u23a0  \u239d  2   2\u23a0"
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "legendre(n, 1)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAG4AAAASBAMAAABIlfZyAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 16,
       "text": "legendre(n, 1)"
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "expand_func(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAcAAAANBAMAAACX52mGAAAAJ1BMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAilU6eAAAADHRSTlMAEKtEMol2VCK73Zk3WuVAAAAAHUlE\nQVQIHWNgEDJgYHDNKWBgYOAhgWidsxqoDwgA7OQHvHltKPsAAAAASUVORK5CYII=\n",
       "prompt_number": 17,
       "text": "1"
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "legendre(n, -1)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAHwAAAASBAMAAABrzqenAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nkTY4RgAAAABJRU5ErkJggg==\n",
       "prompt_number": 18,
       "text": "legendre(n, -1)"
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "expand_func(_, deep=True)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAC4AAAASBAMAAADWL/HSAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZqsiVN12RBDvzTKJ\nu5m05BpfAAAAyUlEQVQYGU2QIQ4CMRBFf3YpFBBAwg0QWBAIDkGyRygoBGKPsMFh2GuQYDAYEhLU\nYuAIWLgAeGY6nU1H9P//Ou00BdBHVBTsarohkkwijD3QMK5HaFhjOybogBJnYpnybvUlewAWeEiD\n7jBfAm8cHZJcKcD8AoxQ5eARWsw1lwpJmbcLAXMRvzI34dkzIH1xOek3J2ncifg1vkfn8EY8txPO\nKb/6k0DTBYP0+fsAgxBtplyU/kHqrsZr66YxKdSxbutgXW3JrEP4A4VcJ15vu9KEAAAAAElFTkSu\nQmCC\n",
       "prompt_number": 20,
       "text": "\n    n\n(-1) "
      }
     ],
     "prompt_number": 20
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "diff(legendre(n,x), x)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAUAAAAAoBAMAAAB9bj4aAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZnaJEJlUIjLvRN2r\nu80lCBqkAAAEr0lEQVRYCe1XX2hbVRz+kt7k3iaZN6mmdSA2SUV8sC4O8UEYXAf+w0HawirIHi4O\nYavCrorikJHLfBCxW/swfI6ob/7pBqKPHSqCIAsi80laRYr4slbLEjpo/J1z7rnnnDSQvXktnofc\n7/fn+/Lld//kHkCulAS3cyz2NfXHfeVB4TAK1U1L30UqdjhILveDp6ULgRYw+IQZv2mGUfTJT1p6\nj4RW45AkrRktmfKj4EJXyyr4ue5pXOUFylb0jLMW6qHEuY5EdNwjQTnre62BSb6gxfkYDzbY0A2u\nxM0S/C4BP7qhEUaBpRvcKwHn8E2dRpJ3eCqhzA43aM6La1SVEqHhBgdIEM8wSJLpopJV0x1usNBS\nvAhdNDLDDQ6QIAXDIEk663SnTB0t+1TbBLInF+4JgS7S1eeQqT08hVo5+KxTmgM+Lp0h9FCLVWhA\niiZZlNIWGSQq5qbG6sTI3XilXEFudryTu/Hqn7zCJDD2SHVeIxGMDAojrIVcPd6sO8t0fRI8ZW25\nRWbwOM563+K4k5mxL1l/oeml1jEZ2FsHfFYBGjGNNARrlNJquSGjFmYyMyFjNCr2Nu5FqoNm8Tqr\ncAlr4m+8qzgMRQaVJHV+dNrLrgA2mSzmti3q6jpd5ItXcBBLIS5ji66oho9mYO2AV4CzMY3aBSsf\nsC+Qyw0ZNV9JLXMG/eyudRN2B42WEGUSuScv4SXJEMfIoJL8kvInkFklg2QSI+v0gW5qq1T2p/E1\nJkuleRqo26JnDBnsgFeAt6hL0Fg/Z+VDQud+ofUjATdk1HS7UOcMN8QOPWOYwUCIcgn63hM6K56g\nkvyK6pcx6tMppgniQJ0+yOAmfX5Q8zHJw9jgNngFeIbygsYaOGvUY1AuMkgwN3a/YGgGPSHKJfJF\nvC8Z4hhNUEnSKaa5LDkBvxw912d/Ll17BwiZ12YLDp8gnRma4DZ4hd8kkgYIFrui1XJDRh2hBGcw\ng3SBsAl6QpTfJI3A2XxHkQhFBpUkjSq7jGmm9Aey2wfDNwh1cQSpoOrRyWtjAjs0wXQbh0LyxCv0\n/Awj2nsEBOsCEdVyQ0ZN30UZpuW2SGUa6S47xVyUSeA00m02B7XIoCHptOlZWMfrE9RSg3114aoP\nLO76qbkH0OjdqqBcCxd3T137FS+Wzmx8ulsBqxCnEtGW6pJVUt9Cj69rG4xq9XpXGIPCxd1idnZ8\nd2HtN7AKl8AU7NlA49lHbj0PQzJVVOVnFWTI/hlPb5gpFVkrEscC9L+0Z50LrIv+nqxIKIn+Bl1S\n/6sbaRmd2VVg3sjowWMykGq5usxox5eBjK/FBowljCwFuuSHevEbPQBOVu/0zYwWFTwRZIMoeUwr\nxjB7uPRoHPQDKdGf1yWtVb16VA+GYKsiGrKy7z4JbvsoJfoJumTa66/+H+/HCfQSvvbjzP+rv+n8\ng2GirVsVh96NErycTXpjSfTir2Oaw2MtLUgEzBgvDG8f8hPhSjNR1jBB1zfjfz3KVSzDQ5IM8o32\n+afuTqxBttGe6PU2E2sw2vEb/pJ1DY6sS3Px3j1ZBsWOX5oUxyTdJGKjbfpL0gTljt90mKAJRjt+\n099ra1+0zMw+jf4BgXbw8CWykKcAAAAASUVORK5CYII=\n",
       "prompt_number": 21,
       "text": "\nn\u22c5(x\u22c5legendre(n, x) - legendre(n - 1, x))\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\n                   2                     \n                  x  - 1                 "
      }
     ],
     "prompt_number": 21
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "expand_func(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAUAAAAAoBAMAAAB9bj4aAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n099ra1+0zMw+jf4BgXbw8CWykKcAAAAASUVORK5CYII=\n",
       "prompt_number": 23,
       "text": "\nn\u22c5(x\u22c5legendre(n, x) - legendre(n - 1, x))\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\n                   2                     \n                  x  - 1                 "
      }
     ],
     "prompt_number": 23
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "diff(legendre(n,x), n)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAIoAAAAnBAMAAAAiK81nAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nlwAFuyVNAPIX5cuzVjdMhXQAAAAASUVORK5CYII=\n",
       "prompt_number": 32,
       "text": "\nd                 \n\u2500\u2500(legendre(n, x))\ndn                "
      }
     ],
     "prompt_number": 32
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "LP = legendre(n,x).rewrite(\"polynomial\")\nLP",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAWoAAAA1BAMAAACaWbxGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarImaJVN0yu5nN\nRO9S3WZtAAAH+klEQVRoBc1aaYgcRRSunvTM7MzO7C5GMRJ1xqj54Y+4sgoeqANeeGaImiABHRM8\nUNGgeJNk1BhREQcxIirZJSEG1iOLEpQkwqpE8Ye4Kl4gZiIeiKATNfG2rapX79Xrmu6asCJs/ej3\n3ve+r/p1TXV3dfcI8b+1HdPqWavKPuljR7Z86f+YWzotvVb1e6RBtbBbp4Mjz5rv4U0ztXlaOq3y\nVV3oiL91z/m+6uC0duEV7V437M0nJ7VKVp0+kq3gF5BWxJPJfcTRcCoeJ0cBsApvv49V99AdxvoB\nlayaj+SFQEBefwPiu8TVTJnqXjCUmuKJrTrIfEPsHrr+uhYs3b59e12p5o6cOHKKsCMZvg69G57A\nE+Yl8U4VMr5tYNSKE5yayNRwcYXKFT9rGorWzZ4yUdxomJ0BoFLz2o7koXj4wCtXA+jjeLEsudPY\nLkq6Gg0Vlv0aS5kA4GCNCgdFpw6w0j1xrBUbsjIAz6kRBCpVtR3JtzELvMcOvAyB/bHLOSmxaiE0\n/KkinieuaoFA6wYTqxYaLjeBSCpVNY1kbhizmhdGUQeB/bDmMmmYvqqLVd4d6HxViw2cL/0cj9cP\nUeTyKJHuFBs856u6sIczQeetetQWxpXgPyvNoiuPfkUaL69bqZC4xFe1UDuiBjpv1fEBIaV21BAE\nl+wT62Xk48VVFG0kTzneqndxKui8VZc9M7U4LET54N3iDtkn8QaiH1T7MYoic8vhe1R+/6RGgt+1\nwY236oE60uQ4gc5btfja8l2v0pJI/7B4WSWQl43ksai2bsuf4Ljb7IRGMu1Ywlu1OVCu81ftWR8u\nUL301cW3yhJvy14VyhaeDtbdwk1DlMzRmbS36jI7HY3OX/WuIXevFL+pvEqt0HlcWuIdHtUULlum\nDtbZDhxysUIqK2K4t2o+643OX3VKVu4x0DtaJTJjDRkRrxw1ZahbQ22z2jUbFTyzcraKVvPhCK/7\n7TVD4YbgtRYF3dlbHp6yGHkI9zUIcpy8/tnmi3BrTWYs76b4fD6ey4IJIRZcqZEbOd7LP9kS9ks3\na8wK4l6uzWPLK0X8t8/aQK1sbhbiBVjFPs/VvfxjLWG/dJmOFcS94iSPLS+MdrNEH/l6ZVMaEp1B\nPWnuo0S3U7CHCsnVNSIl6lxBPuWqK+fEMPUkHcZb/RdLyMHFpqZ+ph6O5fQa6xuEE2y+7oCjUwQk\n6lxB4W/iO85AgwOM1x+xzKmWpKouTGKMF0iMuXWLEAul1LREnSsIUy5J8qoRGxDOe5Td9jq4N3OZ\noTih3/xtKy9uKb5bhKjYfXHd7dd8dJLu3xUEicemqGrobOO8hf8MYSLooGcEOOf5USLlzmAvDIVb\nhBisIofr8q1KHcrrEvyEfNeyX02lGC8bTSA5bKNnqtbPJRJjM0pGctUSRaJe3hNoOhYBsIQGGxqX\nG9ANqkwnK+9tMD5GALCi7kO+a9kZ4vK2tJAcNuVTyLhscgGuf5wHTQb2HqrUeNVgIjepvG3jzz80\nvlF51NyqMXGDdhIE06g6eAB7FUGbXKjaN0NovYtjTdrkGSJo4d0lYL88daKd9BlSalimO68x5ucB\nsocGV8D9v6sIe+ZzXSDXJ3Ut7hKkno3svFZKxhvBMqR9x/pqhhTGMGZ8A+X3XN46R/tdRbClFtOt\nPui7UPYpmyuANZJOORt7/CrBeJlJxpxHvl7ZZGFsJNj9G4YnrDwhuQixcIq6Ybozlny4BHC36jD1\nLgNzLf/ekBYy3h2AQH9zaHfaKVHu+1hi8bYVLC5jgPCuGmVBl71+MyHSQQHC5dQ7+sCE1q0CteUV\nvsL+atLJTWGk7ccUjZMnnUytwH55yhB8BUECdEeJ0brFyEM42yHIcYpjGlgOsOVdRHVOqNRxkIdt\nMEzRAvKkM1AVp/HY+AS/apOg+1LMaluMPIT7mwQ5TqatgZ0AW979yJtVV94BGCqbGaKIHn4UIlfn\nN9kUcQhmV2/QrRU59lRGAoTVg3hyg4c5fPlLvOKEoZcfTajDdlWZsr7yXoyHGAHMpg/qim2kxCzA\n8SsFJwT6PC238/oGRbwdZ6p27idLf/yZs7t8OY68BezawHCA7UljH5oqVUazLsDs7LUp8NYqkxs+\nQr0PEcjL6LWD3uijUrnElmvGYPqJYqgAmHPRfy5OxAjgazHstjpVugoSyMvOo1bvljCkEL84zWcp\n5gJcaljI6DIMsklh4LcY5LijNQlUNrQ07OE5MgzvRUfZPL765iDBo/wMAd3NgmOkAjj+OpOS2umr\nSvMFvMHy8eIqivTEwugDkUGXWwOzR3ShX9CJclVUOdH4Bp41lpAzUL4pneViZ14aH8/QXSPfWlEr\nbLr0GAqsY2C8TkFC6566dNGY5ZFn4EqNkG5HXZl3ikcOksbL61YqJN+WG9MGougP9Jk1sH27r3L5\nttzIt59j0rjNwObO52YhVjfzNeLuqjReHrC7tpu6kBTgovgU7qkLcRGf2F+RfmPGo9eUZy+uJaoI\nnNMjT8R3ydNOT11pKi6IR/BxSmGMh1/vxOtiMk53o3KPPPJzDfTA9tRdHee70ecIMB6uIAtN4YwR\nksneQp7XudvN9tDlJ1xBPM4PQ8x5+Lk7Oyl6TUB3LR/vHKOwq4geuvNRmWZvhQTjqQ/Xc+W34O3y\niWZDmmzG4fS5W471GzOuurSC6HO3nNePpJFmHK4+XM8dGZF/ErhHNGdcdWkF2c/dTy9upZFmGv4v\nSaUNIRsxQIEAAAAASUVORK5CYII=\n",
       "prompt_number": 31,
       "text": "\n  n                                      \n ____                                    \n \u2572                                       \n  \u2572                  k        -k + n    2\n   \u2572      k \u239b  x   1\u239e  \u239bx   1\u239e       \u239bn\u239e \n   \u2571  (-1) \u22c5\u239c- \u2500 + \u2500\u239f \u22c5\u239c\u2500 + \u2500\u239f      \u22c5\u239c \u239f \n  \u2571         \u239d  2   2\u23a0  \u239d2   2\u23a0       \u239dk\u23a0 \n \u2571                                       \n \u203e\u203e\u203e\u203e                                    \nk = 0                                    "
      }
     ],
     "prompt_number": 31
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "[ legendre(i, x) for i in range(5) ]",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 37,
       "text": "\n[legendre(0, x), legendre(1, x), legendre(2, x), legendre(3, x), legendre(4, x\n)]"
      }
     ],
     "prompt_number": 37
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "[ expand(Pn.subs(n,N).doit()) - legendre(N,x) for N in range(10) ]",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 38,
       "text": "\n\u23a1                                            2                          3     \n\u23a2                                         3\u22c5x                     1  5\u22c5x    3\u22c5\n\u23a2-legendre(0, x) + 1, x - legendre(1, x), \u2500\u2500\u2500\u2500 - legendre(2, x) - \u2500, \u2500\u2500\u2500\u2500 - \u2500\u2500\n\u23a3                                          2                      2   2      2\n\n                        4       2                           5       3         \nx                   35\u22c5x    15\u22c5x                     3  63\u22c5x    35\u22c5x    15\u22c5x  \n\u2500 - legendre(3, x), \u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500 - legendre(4, x) + \u2500, \u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500 -\n                      8       4                      8    8       4      8    \n\n                      6        4        2                             7       \n                 231\u22c5x    315\u22c5x    105\u22c5x                     5   429\u22c5x    693\u22c5\n legendre(5, x), \u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500 - legendre(6, x) - \u2500\u2500, \u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\n                   16       16       16                      16    16       16\n\n 5        3                                8         6         4        2     \nx    315\u22c5x    35\u22c5x                   6435\u22c5x    3003\u22c5x    3465\u22c5x    315\u22c5x      \n\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500 - legendre(7, x), \u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500 - le\n       16      16                      128        32        64       32       \n\n                           9         7         5         3                    \n                35  12155\u22c5x    6435\u22c5x    9009\u22c5x    1155\u22c5x    315\u22c5x            \ngendre(8, x) + \u2500\u2500\u2500, \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500 - legendre(\n               128    128         32        64        32      128             \n\n     \u23a4\n     \u23a5\n9, x)\u23a5\n     \u23a6"
      }
     ],
     "prompt_number": 38
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "legendre(n,x).rewrite(\"polynomial\")",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAWoAAAA1BAMAAACaWbxGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nSaUNIRsxQIEAAAAASUVORK5CYII=\n",
       "prompt_number": 39,
       "text": "\n  n                                      \n ____                                    \n \u2572                                       \n  \u2572                  k        -k + n    2\n   \u2572      k \u239b  x   1\u239e  \u239bx   1\u239e       \u239bn\u239e \n   \u2571  (-1) \u22c5\u239c- \u2500 + \u2500\u239f \u22c5\u239c\u2500 + \u2500\u239f      \u22c5\u239c \u239f \n  \u2571         \u239d  2   2\u23a0  \u239d2   2\u23a0       \u239dk\u23a0 \n \u2571                                       \n \u203e\u203e\u203e\u203e                                    \nk = 0                                    "
      }
     ],
     "prompt_number": 39
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "kern0 = factorial(2*n-2*k)/(2**n*factorial(n-k)*factorial(k)*factorial(n-2*k))*(-1)**k*x**(n-2*k)\nkern0",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAOwAAAAvBAMAAAASgTs9AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nEfnLsGNiVPE3xY/QiJ4kWiwAAAAASUVORK5CYII=\n",
       "prompt_number": 40,
       "text": "\n    k  -n  -2\u22c5k + n              \n(-1) \u22c52  \u22c5x        \u22c5(-2\u22c5k + 2\u22c5n)!\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\n     k!\u22c5(-2\u22c5k + n)!\u22c5(-k + n)!    "
      }
     ],
     "prompt_number": 40
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "Pn0 = Sum(kern0, (k,0,floor(n/2)))\nPn0",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAQ4AAAA+BAMAAAAokuiPAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n6/qhK4B6xBtY2vjpSYUA9pdb/gfZqEo5wxb14AAAAABJRU5ErkJggg==\n",
       "prompt_number": 41,
       "text": "\n \u23a2n\u23a5                                   \n \u23a2\u2500\u23a5                                   \n \u23a32\u23a6                                   \n ____                                  \n \u2572                                     \n  \u2572       k  -n  -2\u22c5k + n              \n   \u2572  (-1) \u22c52  \u22c5x        \u22c5(-2\u22c5k + 2\u22c5n)!\n   \u2571  \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\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  \u2571        k!\u22c5(-2\u22c5k + n)!\u22c5(-k + n)!    \n \u2571                                     \n \u203e\u203e\u203e\u203e                                  \nk = 0                                  "
      }
     ],
     "prompt_number": 41
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "[ expand(Pn0.subs(n,N).doit()) - legendre(N,x) for N in range(10) ]",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 42,
       "text": "\n\u23a1                                            2                          3     \n\u23a2                                         3\u22c5x                     1  5\u22c5x    3\u22c5\n\u23a2-legendre(0, x) + 1, x - legendre(1, x), \u2500\u2500\u2500\u2500 - legendre(2, x) - \u2500, \u2500\u2500\u2500\u2500 - \u2500\u2500\n\u23a3                                          2                      2   2      2\n\n                        4       2                           5       3         \nx                   35\u22c5x    15\u22c5x                     3  63\u22c5x    35\u22c5x    15\u22c5x  \n\u2500 - legendre(3, x), \u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500 - legendre(4, x) + \u2500, \u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500 -\n                      8       4                      8    8       4      8    \n\n                      6        4        2                             7       \n                 231\u22c5x    315\u22c5x    105\u22c5x                     5   429\u22c5x    693\u22c5\n legendre(5, x), \u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500 - legendre(6, x) - \u2500\u2500, \u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\n                   16       16       16                      16    16       16\n\n 5        3                                8         6         4        2     \nx    315\u22c5x    35\u22c5x                   6435\u22c5x    3003\u22c5x    3465\u22c5x    315\u22c5x      \n\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500 - legendre(7, x), \u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500 - le\n       16      16                      128        32        64       32       \n\n                           9         7         5         3                    \n                35  12155\u22c5x    6435\u22c5x    9009\u22c5x    1155\u22c5x    315\u22c5x            \ngendre(8, x) + \u2500\u2500\u2500, \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500 - legendre(\n               128    128         32        64        32      128             \n\n     \u23a4\n     \u23a5\n9, x)\u23a5\n     \u23a6"
      }
     ],
     "prompt_number": 42
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "from numpy import linspace\nfrom matplotlib.pyplot import *",
     "language": "python",
     "outputs": [],
     "prompt_number": 43
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "U = linspace(-1,1,500)",
     "language": "python",
     "outputs": [],
     "prompt_number": 47
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "F = lambda k: lambda u: legendre(k,u).evalf()\npolyvals = [ map(p, U) for p in map(F, xrange(10)) ]",
     "language": "python",
     "outputs": [],
     "prompt_number": 48
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "figure(figsize=(8,8))\nfor V in polyvals:\n    plot(U, V)",
     "language": "python",
     "outputs": [
      {
       "output_type": "display_data",
       "png": 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NiBo7Fnr3bvLSTXXPqw95MXc3o+gVAoES7HY/QSWAqb171haDBWuKtdqz7mnGe9jbvMYo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\nkcX04QypD/sAAAAASUVORK5CYII=\n"
      }
     ],
     "prompt_number": 49
    },
    {
     "cell_type": "markdown",
     "source": "Associated Legendre Polynomials"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "kern2 = factorial(2*l-2*k)/(2**l*factorial(l-k)*factorial(k)*factorial(l-2*k-m))*(-1)**k*x**(l-m-2*k)\nkern2",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAPYAAAAvBAMAAAAiDSocAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 50,
       "text": "\n    k  -l  -2\u22c5k + l - m              \n(-1) \u22c52  \u22c5x            \u22c5(-2\u22c5k + 2\u22c5l)!\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     k!\u22c5(-k + l)!\u22c5(-2\u22c5k + l - m)!    "
      }
     ],
     "prompt_number": 50
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "pref = (1-x**2)**(m/2)\npref",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAGUAAAAeBAMAAADHt83+AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIjKruxB23VSJRO9m\nmc3kD4mRAAABwklEQVQ4EdWTvUvDQBjGn35Ea5vaopMgtNBBq0PVxUnoEAddLAgdRGgR7CZOioNC\ncOog2NFJAoI4KEbBxSmTdCyd9B/QVQehgxDfS3rJ3dmhqy8keX7PPW/uLh/AsFUcNhjm9JVQD63u\nhk6Gwf/ZE20shVsQVXLZQXXxpftSZ+6EOFTFfEVkpre4YR6ip9lEkQVusWsJGUtkpkdMOrmu+6nZ\nETtZJohL942iliVTrlXCxJ6DRFnvpB2ChjyON4UJC3RU8YB0PWamqgQHckbvyLxpAaM0dQHriGXH\nK+kdmrMsZ67hr3XWt4v7Jm2oBWh4CoIxYqGiu9N5D8/6ZsYEkjbBmNN3aF4rkEzkXNdnsUf7Jq/p\n++ycyQa6eXEbvlCxB880lzkXBHMVLhP5mvPDAVLPMVsAm8yvEhfQcV8JbiD3vAchT1BPxGBVBy75\nUNowPgxjgSF7BlB6wrUBayzUL2ltR9z1r+GnoqEHJxiUeugZiJWyOJ1vP0ZMDvJ+vgLbE/E658n2\nVZtr8J6pjRMLUTv0mdK9ncoeoxvBUjPePySMD5LjecV9VXgAzqheXDX+cle1ki3VUXlA4lTNqMz+\nTLF+AXGuV4WFo/2oAAAAAElFTkSuQmCC\n",
       "prompt_number": 51,
       "text": "\n          m\n          \u2500\n          2\n\u239b   2    \u239e \n\u239d- x  + 1\u23a0 "
      }
     ],
     "prompt_number": 51
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "lim = floor((l-m)/2)\nlim",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFEAAAAlBAMAAADIEhjSAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIruJVBCrRDJ23Zlm\n782xNQueAAABXklEQVQ4Ee2Sv0oDQRCHfzF3MWrwAjZ2LsHOwmBhY3PYWQRSCYKQF7CwsRSCPkBO\nbAUF7UyRShAs0kosUqQJFr7CpUh9zu7GZf9MkQdwit2Zb7+dO+YOkHF6qTZ34eDtY+pKsmIhEsb8\nh/70lppSXd5awqxDkLi1/0qfs/ItL5nQ0JRCmYvyxWAmsc14zggG2Wb0ZDCT2ObGgBEMss31ocEm\nie7PRunDB9W22VEDM5JKznea1Sl6ntlyJVVdtdrRHPJE9UwKihxvVJcaMjLKDCwLHP6ZtMvo681d\n+9jMMCFmvWc823YlWcUzJN1qTieWWcrlQ72IxrhGTdCJZeI99TQq15oYIb7oumbo2UT3rB3t2lDn\nHtTmHXrDQPWgNqeojAPTgwIHjRTPKId/nAOPG4NFq5WwJ8DCDk0sCBbuBRoBDtKAw2DhCdqhycHV\nDOF7svDm61MEPVn4UxSh6cNfaGNXBkP9T7kAAAAASUVORK5CYII=\n",
       "prompt_number": 52,
       "text": "\n\u23a2l   m\u23a5\n\u23a2\u2500 - \u2500\u23a5\n\u23a32   2\u23a6"
      }
     ],
     "prompt_number": 52
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "Plm = pref * Sum(kern2, (k, 0, lim))\nPlm",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAaAAAAA+BAMAAACfA9LtAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nOsKZif+WwAS8pe9cImn8D3Xf98Cseps9AAAAAElFTkSuQmCC\n",
       "prompt_number": 53,
       "text": "\n            \u23a2l   m\u23a5                                      \n            \u23a2\u2500 - \u2500\u23a5                                      \n            \u23a32   2\u23a6                                      \n          m   ____                                       \n          \u2500   \u2572                                          \n          2    \u2572        k  -l  -2\u22c5k + l - m              \n\u239b   2    \u239e      \u2572   (-1) \u22c52  \u22c5x            \u22c5(-2\u22c5k + 2\u22c5l)!\n\u239d- x  + 1\u23a0 \u22c5    \u2571   \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\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               \u2571         k!\u22c5(-k + l)!\u22c5(-2\u22c5k + l - m)!    \n              \u2571                                          \n              \u203e\u203e\u203e\u203e                                       \n             k = 0                                       "
      }
     ],
     "prompt_number": 53
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "L = 4\nM = 4",
     "language": "python",
     "outputs": [],
     "prompt_number": 54
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "expand(Plm.subs(l,L).subs(m,M).doit())",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAKIAAAARBAMAAACsgmrzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKtEMol2VCK73Zlm\nze8l1wreAAACBklEQVQ4EZXSMUgbURgH8H+8s5foSYJDdBAy1MExVOhkyC0dujQdxFBUuA61DkJv\nqQ4dFMGCIFVaGglF7KCW4lIQB8HhHBqXCoE6OmQIXRzUVpCKcH7vvbvzvUsg+Ia797773++9u/eA\nJm24SU0u6RP98rB1X6u3yKTxLNUioj7uO1bHDaMCktmGYpNCbzCv3UrUkZlrAlCJG9003bufDpa9\n/4I05lTRHF0FYg8BY78YOLtBR9wV48kbB1oVn/B+0F9jD1RxEgVXH7kCckjQ7KyZP8Rdd/g9YiQd\ndLk4wpIIAZWJU/9FUfmDthpA4gnMqig9Rop3DFeMVYNGGQf5VCgCGyLnXzfRfsHE2BU06lDTXy5Z\nvCOJkkFiIYuCtTBFf2mmONSP9F9ljUCixkT9H/RLnlj2PA5CEu8MkJi3CJ1Hh2NYGfdGhOVrxmai\ndkmompDE0EAoAp1fTfp68YdkEF9o5ItqIioyg4u0YpoC7efAeEgtlqmV2PDBc7rwr6ZlSom98tr3\n8keWEH8uMMTOzKa2uPiZBSJtmu0s3xmDdkZOSGuknQkMEhM2nZ5vaKvG6EU34kGzYTMRJTo9akIS\nQ4OvmE74bxzggzXbV487UfHpr8oKF3PoyqoJSQwNLKxvZ/G24sB8NYD5w0eHURBnnrcSf3G9A2Ps\ndSThi4rR8P69ClrDBwG3Wy+srhFHom0AAAAASUVORK5CYII=\n",
       "prompt_number": 55,
       "text": "\n     4        2      \n105\u22c5x  - 210\u22c5x  + 105"
      }
     ],
     "prompt_number": 55
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "Pml2 = (-1)**M*(1-x**2)**(Rational(M,2))*diff(Pn.subs(n,L).doit(), x, M)\nexpand(Pml2)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAKIAAAARBAMAAACsgmrzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ndSThi4rR8P69ClrDBwG3Wy+srhFHom0AAAAASUVORK5CYII=\n",
       "prompt_number": 56,
       "text": "\n     4        2      \n105\u22c5x  - 210\u22c5x  + 105"
      }
     ],
     "prompt_number": 56
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "expand(assoc_legendre(L, M, x))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAKIAAAARBAMAAACsgmrzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ndSThi4rR8P69ClrDBwG3Wy+srhFHom0AAAAASUVORK5CYII=\n",
       "prompt_number": 57,
       "text": "\n     4        2      \n105\u22c5x  - 210\u22c5x  + 105"
      }
     ],
     "prompt_number": 57
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "assoc_legendre(m,n,x)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAALYAAAASBAMAAAAEUznPAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAElFTkSuQmCC\n",
       "prompt_number": 24,
       "text": "assoc_legendre(m, n, x)"
      }
     ],
     "prompt_number": 24
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "expand_func(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAALYAAAASBAMAAAAEUznPAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEGZ2mbvNRCLdMlSr\nie9ysnNlAAAC0ElEQVQ4EaWTP2gTYRjGn7sL3vUu8Y7qIgo5k1hKIzQkav1XCKmDUIe4KtJCB8EO\nPdyEFkPFRaUVHZRO3yQKYkM7qLTYlA6KSyK1LnVIVl3aSpsjpcT3S3rJ0TRB7Dfk+73v877P9+e+\nAAcYYoveVtqjfftGr7vTX9zBHhbiexL1UB1g9aBOsl1niGlX0ICzDZlaQmc1dIHg9m5zCY14yGjM\n7Wb+wbvFzshEMg/i/blpMxfUQQiBMwbCM4PiXAe0aBTQ+ugHoH3HQin09QZzUvSqPDIeikBOhG15\n5MHtigKsAbPfov1SyOQNCBJXAHOrd79y2ZMUB+UcFsfgNb7Dl8M5TDAq0ZknrhS8cU+cXcGE4Y8o\n6+iAaGPSvMkVQFiD0LmDdxGVFqGQMwdozG9uAAXIprIuv4B1z5IVG4otbOMo/wo6m2Z41hYR82oJ\nbaZuokQSlfgtcAVQ8pCPF7BgaHnuyPkjB4i4b5DFS6A9ZiO7ybzlGyKtvyGXKjp5dwcC/VLSmxM3\nAqE0XVGRHiD3ToEr5P0U8GSwAl+m0kJ8rdqLH3yegjgl2IIybxuPs+Pk/cflTbocvAS+JL9+x9tA\nN2+FkAcdCEM4nK7ExL8rANzicwF6WrFnGJa74DtfgsIPjhQpOpu0oPqIlCLAuLdQrOzbAFdIoDXp\nECVMK4wiYnXtNQcB2zC57Lc0O5pBz2lI1k944rgAMUUFOvMl0SmdJFyhjG6hiLOQStyOK5S/A3yC\nnMcpCW8YZymZU6eAhfZfShpqEtpS1+WZWNAM0vsT+xL0lZd6UC6Xs8MIxZhQLj+n9EU1O/x2y9QS\n4a33A0PgCoAY0AsxjhNdmM5xVhIpdQd4Eh4N0xc1qaTVeJgSXlVvc5+qY66c6bDoQIv/fLVknp5C\nU2+f5fiA32911LzHnEyzWVsOrDbTgMWapKUc/LALQsbJ/N98pNam1cg5pWTgL//oylHf2kniAAAA\nAElFTkSuQmCC\n",
       "prompt_number": 25,
       "text": "assoc_legendre(m, n, x)"
      }
     ],
     "prompt_number": 25
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "assoc_legendre(m,n,-x)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAMUAAAASBAMAAAAZLQdUAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nx8UrKZWNSJVQERwBfgPXpNHwLpDbgAAAAABJRU5ErkJggg==\n",
       "prompt_number": 26,
       "text": "assoc_legendre(m, n, -x)"
      }
     ],
     "prompt_number": 26
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "expand_func(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAMUAAAASBAMAAAAZLQdUAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nx8UrKZWNSJVQERwBfgPXpNHwLpDbgAAAAABJRU5ErkJggg==\n",
       "prompt_number": 28,
       "text": "assoc_legendre(m, n, -x)"
      }
     ],
     "prompt_number": 28
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "assoc_legendre(m,-n,x)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAMUAAAASBAMAAAAZLQdUAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\np1crti5EqihHgd+I79GlTiwrcQAAAABJRU5ErkJggg==\n",
       "prompt_number": 34,
       "text": "assoc_legendre(m, -n, x)"
      }
     ],
     "prompt_number": 34
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "expand_func(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAATUAAAArBAMAAADrmgJoAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nB4wAAAAASUVORK5CYII=\n",
       "prompt_number": 35,
       "text": "\n    n                                 \n(-1) \u22c5assoc_legendre(m, n, x)\u22c5(m - n)!\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\n               (m + n)!               "
      }
     ],
     "prompt_number": 35
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "assoc_legendre(-m,n,x)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAMUAAAASBAMAAAAZLQdUAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nVY4BvwEJnNGl1+5YWQAAAABJRU5ErkJggg==\n",
       "prompt_number": 38,
       "text": "assoc_legendre(-m, n, x)"
      }
     ],
     "prompt_number": 38
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "expand_func(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAMUAAAASBAMAAAAZLQdUAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nVY4BvwEJnNGl1+5YWQAAAABJRU5ErkJggg==\n",
       "prompt_number": 39,
       "text": "assoc_legendre(-m, n, x)"
      }
     ],
     "prompt_number": 39
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "assoc_legendre(n,m,0)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAALYAAAASBAMAAAAEUznPAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nSwsk+yzp+AOoK9cwYYhJLAAAAABJRU5ErkJggg==\n",
       "prompt_number": 40,
       "text": "assoc_legendre(n, m, 0)"
      }
     ],
     "prompt_number": 40
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "expand_func(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAXcAAAAuBAMAAAAhGERGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZomZu1R2IjLvq90Q\nRM0XT9VLAAAFeElEQVRoBe1ZTWhcVRT+kvl7ybwx0YgVV4O40FJoI22gXU1lqChSRyliceFrrBWK\n0im60G4yEqwECgmIuok0LYgVNxEENxWfoJW4SRANVBR3IugiSjVSKvGce++5P2/eiAsDM9C7ePec\n73zfOd+83HmZTIDtX9GWWts/6P+ZEJ88/cyrJ6RXMZWo//fb9rZK1WSstSJWZyXo/73cjn7DCl7B\nN+L1QQn6fx9J8D6OYAKTxmu01P+mxWG1jt3pFN7FYqKh0baUBmJ/CO/hV0x1tNkziB+lx82AvIT4\nz+AWX8LouV0/pAHWv0lxj++tsoRzlfHnfKif42lrjoyj0AGG2xQMxColsfhc/wP4nJJbsCxQn+/r\nKFuHh4F7KfkOb1uor4Po4K3HrcEfO3GDkoP40EJ9HYxsbV2zBquN4TVKLmPKQoMTlDb5yA/qeuOF\nQXVOvk81Btj8amuAzUcD7P2m9d53QP9xPJDXjd6v6mZl++/Ay3ZEsW3DMJC/oWP+Ze+tSscmQrGA\nBFLIat1YCEUksuf7IdT24g/depXu62H+cfuh8Fnh6v2MTR3FQjpwhVDrxsJRQm2+H4VKr2JqJSM9\nzLtPtB6ZVY9YqaM4SEf2hYdaP7OUjDbfD6Oitt9sAflk6mi7h99jlBM3zVIcpCNbCLXeWNc+o833\nw6j0uuwU+WSqWwP4mLJo//W7P7hBQTWli1mOIkhXgbWz5y/u/DSlwBvrtTcas+X7USj3AkoNtalL\nlhy1TM05m2ZkmJxfov2IKfPmKBp8TWquoLS7U4x2wrFdWpGGfqSjQlUvjNaF2n1sSuOm5gxcZYTN\nc6uLpsybo2jwMam5gtKSeb4n/tgurUhD89JRoaoXqmtC/U/mV1Ois3n+Q/UTJ+0yIKO8gtKSeSR0\n4ryxHsVrSOG/mFe9MNa2gtqpC8ATu47tK0+MK1Du/Pr1cQxNX5loUT8eyuZ5beqNr0wpMeXYpAbF\nfFbL5ml5Y7u1igEoP3eQH52bjgrVPjCTGqre4h2/Yy6JNlQm5jn5bKYeLdAdG6fYmK+Y16C4hvIW\n5nUq5jnztcZ8ZqyiWK3uoK7Kj879jtqHzBJ+7YtFPJ2WyCYt3/z3R9PSMn0FkxBuzJuXyEy1mHID\nR3Xij/K1xrx5iUYJBFqL0t0nP091d9Q+2HylySsBf7ZEYQn3YHiJ+Hubhz9qPkzBGOEbOMClLvMv\nsfSQpQw3sJMUhWbzl2aT6LZgtZ75fK24gfVzIOioXBrz2Z8fnYufMNIifnjn6asXRqtc6XFsiEI/\nmDeVFP6d97U9jk2g1R30lfzkdVQ+sNq21OgEPbwp/wvz+hOXHJuzJzuINzEftYM3LEF2GcpYJ9rQ\nn7jEfFZrzHtj0aU1XY2faON1BZiOCjVvWPX40OwrmEvpyNYWcDs/CO2dj5NoEfQ2OD5snhLytHlH\n6+gqlKsYaiQKNaOkYLXGvDdWKE5r2ho/5Ubd76hQ87QqJoYKOv5ja5jG0B58dacCzZ2nN+bfKNfx\n4g5gjgrR/mv8DSRwSF35IpQJxBc6CpX7lNHOnn9A1b2x3VrVwPqpTOrDYToql8oHMLRkqOTAP0OM\n1viA0+q4A3KXRvTVSzyKrn1riF7Bo/tj/fZ+96wf01G71L3iBU9wvxf7YaEu2c8S8L7iJY7igRy6\ngq8NxjpKRpvvh1HTa58T0IHJXxMCR8sS8V4wPxmOLYUTf9lCqPXG9tTm+2FUehXdpHWkLvGimv1/\nQaHjwUDDZo5iIR24Qqj1xjpKRpvvh1HpVauLJHr+dEfiYD/7pX4DA18HuJc6SsiAK4RaN9ajhNp8\nPwq1vZ4UyQz9gs1bFf4Fq1achPVSx+SOEhLgClmtHesoGW2+H0Z1r38AVzxjk+vrBsUAAAAASUVO\nRK5CYII=\n",
       "prompt_number": 44,
       "text": "\n               m   ___             \n              2 \u22c5\u2572\u2571 \u03c0              \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\n\u239b  m   n\u239e  \u239b  m   n\u239e  \u239b  m   n   1\u239e\n\u239c- \u2500 + \u2500\u239f\u22c5\u0393\u239c- \u2500 + \u2500\u239f\u22c5\u0393\u239c- \u2500 - \u2500 + \u2500\u239f\n\u239d  2   2\u23a0  \u239d  2   2\u23a0  \u239d  2   2   2\u23a0"
      }
     ],
     "prompt_number": 44
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "for i in xrange(5):\n    for j in xrange(-i, i+1):\n        Poly = expand(assoc_legendre(i,j,x))\n        print((i, j, Poly))",
     "language": "python",
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "(0, 0, legendre(0, x))\n(1, -1, sqrt(-x**2 + 1)/2)\n(1, 0, legendre(1, x))\n(1, 1, -sqrt(-x**2 + 1))\n(2, -2, -x**2/8 + 1/8)\n(2, -1, x*sqrt(-x**2 + 1)/2)\n(2, 0, legendre(2, x))\n(2, 1, -3*x*sqrt(-x**2 + 1))\n(2, 2, -3*x**2 + 3)\n(3, -3, -x**2*sqrt(-x**2 + 1)/48 + sqrt(-x**2 + 1)/48)\n(3, -2, -x**3/8 + x/8)\n(3, -1, 5*x**2*sqrt(-x**2 + 1)/8 - sqrt(-x**2 + 1)/8)\n(3, 0, legendre(3, x))\n(3, 1, -15*x**2*sqrt(-x**2 + 1)/2 + 3*sqrt(-x**2 + 1)/2)\n(3, 2, -15*x**3 + 15*x)\n(3, 3, 15*x**2*sqrt(-x**2 + 1) - 15*sqrt(-x**2 + 1))\n(4, -4, x**4/384 - x**2/192 + 1/384)\n(4, -3, -x**3*sqrt(-x**2 + 1)/48 + x*sqrt(-x**2 + 1)/48)\n(4, -2, -7*x**4/48 + x**2/6 - 1/48)\n(4, -1, 7*x**3*sqrt(-x**2 + 1)/8 - 3*x*sqrt(-x**2 + 1)/8)\n(4, 0, legendre(4, x))\n(4, 1, -35*x**3*sqrt(-x**2 + 1)/2 + 15*x*sqrt(-x**2 + 1)/2)\n(4, 2, -105*x**4/2 + 60*x**2 - 15/2)"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(4, 3, 105*x**3*sqrt(-x**2 + 1) - 105*x*sqrt(-x**2 + 1))\n(4, 4, 105*x**4 - 210*x**2 + 105)"
      }
     ],
     "prompt_number": 63
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "diff(assoc_legendre(n,m,x),x)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAgEAAAAoBAMAAACV9DSFAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nLwccB+N5HiuZAAAAAElFTkSuQmCC\n",
       "prompt_number": 45,
       "text": "\nn\u22c5x\u22c5assoc_legendre(n, m, x) - (m + n)\u22c5assoc_legendre(n - 1, m, x)\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\n                               2                                 \n                              x  - 1                             "
      }
     ],
     "prompt_number": 45
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "expand_func(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAgEAAAAoBAMAAACV9DSFAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nLwccB+N5HiuZAAAAAElFTkSuQmCC\n",
       "prompt_number": 47,
       "text": "\nn\u22c5x\u22c5assoc_legendre(n, m, x) - (m + n)\u22c5assoc_legendre(n - 1, m, x)\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\n                               2                                 \n                              x  - 1                             "
      }
     ],
     "prompt_number": 47
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "diff(assoc_legendre(n,m,x),m)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAANYAAAAnBAMAAACBA6uhAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nu31ouVJ6rIX/96Gr9QR/AYKb/zjb3LxOAAAAAElFTkSuQmCC\n",
       "prompt_number": 48,
       "text": "\nd                          \n\u2500\u2500(assoc_legendre(n, m, x))\ndm                         "
      }
     ],
     "prompt_number": 48
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "diff(assoc_legendre(n,m,x),n)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAANEAAAAnBAMAAABj37DYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nTkSuQmCC\n",
       "prompt_number": 49,
       "text": "\nd                          \n\u2500\u2500(assoc_legendre(n, m, x))\ndn                         "
      }
     ],
     "prompt_number": 49
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "expand_func(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAANEAAAAnBAMAAABj37DYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nTkSuQmCC\n",
       "prompt_number": 51,
       "text": "\nd                          \n\u2500\u2500(assoc_legendre(n, m, x))\ndn                         "
      }
     ],
     "prompt_number": 51
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "assoc_legendre(n,m,x).rewrite(\"polynomial\")",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAb4AAAA+BAMAAACmZGO2AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 52,
       "text": "\n            \u23a2  m   n\u23a5                                      \n            \u23a2- \u2500 + \u2500\u23a5                                      \n            \u23a3  2   2\u23a6                                      \n          m    ____                                        \n          \u2500    \u2572                                           \n          2     \u2572         k  -n  -2\u22c5k - m + n              \n\u239b   2    \u239e       \u2572    (-1) \u22c52  \u22c5x            \u22c5(-2\u22c5k + 2\u22c5n)!\n\u239d- x  + 1\u23a0 \u22c5     \u2571    \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\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                \u2571          k!\u22c5(-k + n)!\u22c5(-2\u22c5k - m + n)!    \n               \u2571                                           \n               \u203e\u203e\u203e\u203e                                        \n              k = 0                                        "
      }
     ],
     "prompt_number": 52
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "expand_func(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAb4AAAA+BAMAAACmZGO2AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 54,
       "text": "\n            \u23a2  m   n\u23a5                                      \n            \u23a2- \u2500 + \u2500\u23a5                                      \n            \u23a3  2   2\u23a6                                      \n          m    ____                                        \n          \u2500    \u2572                                           \n          2     \u2572         k  -n  -2\u22c5k - m + n              \n\u239b   2    \u239e       \u2572    (-1) \u22c52  \u22c5x            \u22c5(-2\u22c5k + 2\u22c5n)!\n\u239d- x  + 1\u23a0 \u22c5     \u2571    \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\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                \u2571          k!\u22c5(-k + n)!\u22c5(-2\u22c5k - m + n)!    \n               \u2571                                           \n               \u203e\u203e\u203e\u203e                                        \n              k = 0                                        "
      }
     ],
     "prompt_number": 54
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "for i in range(5):\n    for j in range(-i, i+1):\n        Pml2 = (-S.One)**(Abs(j))*(S.One-x**2)**(Rational(Abs(j),2))*diff(legendre(i,x), x, Abs(j))\n        if j < 0:\n            Pml2 = (-S.One)**Abs(j)*factorial(i-Abs(j))/factorial(Abs(j)+i) * Pml2\n            \n        A = expand(Pml2.doit())\n        B = assoc_legendre(i, j, x)\n        print((i, j, simplify(A - B)))",
     "language": "python",
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "(0, 0, 0)\n(1, -1, sqrt(-x**2 + 1)*(-x**2 + x*legendre(1, x) - legendre(0, x) + 1)/(2*(x**2 - 1)))\n(1, 0, 0)\n(1, 1, sqrt(-x**2 + 1)*(x**2 - x*legendre(1, x) + legendre(0, x) - 1)/(x**2 - 1))"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(2, -2, (3*x**4 - 2*x**2*legendre(2, x) - 6*x**2 + 2*x*legendre(1, x) - 2*legendre(0, x) + 2*legendre(2, x) + 3)/(24*(x**2 - 1)))"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(2, -1, sqrt(-x**2 + 1)*(-3*x**3 + 2*x*legendre(2, x) + 3*x - 2*legendre(1, x))/(6*(x**2 - 1)))\n(2, 0, 0)\n(2, 1, sqrt(-x**2 + 1)*(3*x**3 - 2*x*legendre(2, x) - 3*x + 2*legendre(1, x))/(x**2 - 1))"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(2, 2, (3*x**4 - 2*x**2*legendre(2, x) - 6*x**2 + 2*x*legendre(1, x) - 2*legendre(0, x) + 2*legendre(2, x) + 3)/(x**2 - 1))\n(3, -3, sqrt(-x**2 + 1)*(5*x**6 - 15*x**4 - 2*x**3*legendre(3, x) + 3*x**2*legendre(2, x) + 15*x**2 + 3*x*legendre(3, x) + 2*legendre(0, x) - 6*legendre(2, x) - 5)/(240*(x**4 - 2*x**2 + 1)))"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(3, -2, (5*x**5 - 10*x**3 - 2*x**2*legendre(3, x) + 3*x*legendre(2, x) + 5*x - 2*legendre(1, x) + legendre(3, x))/(40*(x**2 - 1)))"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(3, -1, sqrt(-x**2 + 1)*(-5*x**4 + 6*x**2 + 2*x*legendre(3, x) - 2*legendre(2, x) - 1)/(8*(x**2 - 1)))\n(3, 0, 0)\n(3, 1, 3*sqrt(-x**2 + 1)*(5*x**4 - 6*x**2 - 2*x*legendre(3, x) + 2*legendre(2, x) + 1)/(2*(x**2 - 1)))"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(3, 2, 3*(5*x**5 - 10*x**3 - 2*x**2*legendre(3, x) + 3*x*legendre(2, x) + 5*x - 2*legendre(1, x) + legendre(3, x))/(x**2 - 1))"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(3, 3, 3*sqrt(-x**2 + 1)*(-5*x**6 + 15*x**4 + 2*x**3*legendre(3, x) - 3*x**2*legendre(2, x) - 15*x**2 - 3*x*legendre(3, x) - 2*legendre(0, x) + 6*legendre(2, x) + 5)/(x**4 - 2*x**2 + 1))"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(4, -4, (-35*x**8 + 140*x**6 + 8*x**4*legendre(4, x) - 210*x**4 - 16*x**3*legendre(3, x) + 12*x**2*legendre(2, x) - 16*x**2*legendre(4, x) + 140*x**2 + 16*x*legendre(1, x) + 28*x*legendre(3, x) + 8*legendre(0, x) - 48*legendre(2, x) + 8*legendre(4, x) - 35)/(13440*(x**4 - 2*x**2 + 1)))"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(4, -3, sqrt(-x**2 + 1)*(35*x**7 - 105*x**5 - 8*x**3*legendre(4, x) + 105*x**3 + 16*x**2*legendre(3, x) - 12*x*legendre(2, x) + 8*x*legendre(4, x) - 35*x + 8*legendre(1, x) - 12*legendre(3, x))/(1680*(x**4 - 2*x**2 + 1)))"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(4, -2, (105*x**6 - 225*x**4 - 24*x**2*legendre(4, x) + 135*x**2 + 40*x*legendre(3, x) - 24*legendre(2, x) + 8*legendre(4, x) - 15)/(720*(x**2 - 1)))"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(4, -1, sqrt(-x**2 + 1)*(-35*x**5 + 50*x**3 + 8*x*legendre(4, x) - 15*x - 8*legendre(3, x))/(40*(x**2 - 1)))\n(4, 0, 0)\n(4, 1, sqrt(-x**2 + 1)*(35*x**5 - 50*x**3 - 8*x*legendre(4, x) + 15*x + 8*legendre(3, x))/(2*(x**2 - 1)))"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(4, 2, (105*x**6 - 225*x**4 - 24*x**2*legendre(4, x) + 135*x**2 + 40*x*legendre(3, x) - 24*legendre(2, x) + 8*legendre(4, x) - 15)/(2*(x**2 - 1)))"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(4, 3, 3*sqrt(-x**2 + 1)*(-35*x**7 + 105*x**5 + 8*x**3*legendre(4, x) - 105*x**3 - 16*x**2*legendre(3, x) + 12*x*legendre(2, x) - 8*x*legendre(4, x) + 35*x - 8*legendre(1, x) + 12*legendre(3, x))/(x**4 - 2*x**2 + 1))"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": "\n(4, 4, 3*(-35*x**8 + 140*x**6 + 8*x**4*legendre(4, x) - 210*x**4 - 16*x**3*legendre(3, x) + 12*x**2*legendre(2, x) - 16*x**2*legendre(4, x) + 140*x**2 + 16*x*legendre(1, x) + 28*x*legendre(3, x) + 8*legendre(0, x) - 48*legendre(2, x) + 8*legendre(4, x) - 35)/(x**4 - 2*x**2 + 1))"
      },
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": ""
      }
     ],
     "prompt_number": 68
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "P11 = assoc_legendre(1,1,x)\nP11",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAF8AAAAWBAMAAABUNV1iAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIrt2MkQQmauJzVTv\nZt272hcsAAABLklEQVQ4EWNgQAX+//GDD6jKGRjU0AUI8FkKCChAl+YQQBchwJcgII8h3Ygi4mzh\ngMLH5DA2IIuxGbAcQ+ZjYbM7IAuyfGD4isxnyAhA4QI5SagCDmwfwQIsG8CUpz2ERlI0FYkNYrJP\nAAtwX4CI86NrYFOASMDJ6xAWVg2SQDnOALhS7p1bbziwGrCBBbBpYO1PYGDggqtn2ML2l/+Cc+IV\nnBoY+YHOtUZouMD6jY3x//8PmBqElIBAGeicAwwMC4DScD6IAwZFSlqPlDRATBRPM/5l4L0AUQEm\nmYA2wgA2PzAwaCF7gSGBfwMvTD0Ddg3xDnfhKhi4v91xkIZzsWvga1gAV8HA2Lu1dwOcC9Ugqf88\nAC7GwMD72QCJh8JkRehFFj+DzCGGPZkYRQwAhABd9ZaSMiYAAAAASUVORK5CYII=\n",
       "prompt_number": 69,
       "text": "\n    __________\n   \u2571    2     \n-\u2572\u2571  - x  + 1 "
      }
     ],
     "prompt_number": 69
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "assoc_legendre(1,-1,x)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAF0AAAAlBAMAAADSLvhcAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nwS/NX3ueiDnqjAAAAABJRU5ErkJggg==\n",
       "prompt_number": 70,
       "text": "\n   __________\n  \u2571    2     \n\u2572\u2571  - x  + 1 \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n      2      "
      }
     ],
     "prompt_number": 70
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "-P11/2",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAF0AAAAlBAMAAADSLvhcAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nwS/NX3ueiDnqjAAAAABJRU5ErkJggg==\n",
       "prompt_number": 71,
       "text": "\n   __________\n  \u2571    2     \n\u2572\u2571  - x  + 1 \n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n      2      "
      }
     ],
     "prompt_number": 71
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "simplify(assoc_legendre(n, l, x) + assoc_legendre(n, -l, x))",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAZQAAAA5BAMAAADjHsAXAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRK5CYII=\n",
       "prompt_number": 72,
       "text": "\n\u239b    l                     \u239e                        \n\u239d(-1) \u22c5(-l + n)! + (l + n)!\u23a0\u22c5assoc_legendre(n, l, x)\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\n                      (l + n)!                      "
      }
     ],
     "prompt_number": 72
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "dl = diff(legendre(n,x),x)\ndl",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAUAAAAAoBAMAAAB9bj4aAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n099ra1+0zMw+jf4BgXbw8CWykKcAAAAASUVORK5CYII=\n",
       "prompt_number": 73,
       "text": "\nn\u22c5(x\u22c5legendre(n, x) - legendre(n - 1, x))\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\n                   2                     \n                  x  - 1                 "
      }
     ],
     "prompt_number": 73
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "# Takes long time but works\n#integrate(dl, x)",
     "language": "python",
     "outputs": [],
     "prompt_number": 74
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "dlm = diff(legendre(m,x),x)\ndln = diff(legendre(n,x),x)",
     "language": "python",
     "outputs": [],
     "prompt_number": 75
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "# Takes an even longer time\n#integrate(dlm*dln, (x, -1,1))",
     "language": "python",
     "outputs": [],
     "prompt_number": 76
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "assoc_legendre(S(3)/4, S(1)/4, x)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAMAAAAAsBAMAAADBdKrNAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEGZ2mbvNRCLdMlSr\nie9ysnNlAAAEJklEQVRYCaVXXYgbVRT+ZpLdmZ1JzFAFEYSMyVKkKTTsqvXfsKsi+hIr9cEquyLS\nhyIdfFMqG+rPg9a2WEEsPsyD6Iuyy76otNAtiy/6sIHaltKubMUnQehmsTtkW+I5c2cmM3EmGyfn\n4d7vnu+c8839mTsJMJypVnL+3cnU4MxHRnJs1kzmBmWk1wG5vByE3xuZ0IHAnxqMLgAXUax6Bb5/\nisZdu6fRxSnRNcq7gjvqfnohIqCc9P1pe+26mxnMAFEBvJS2sJ830nTRs/64V2DeCJh0QBTYTTvt\nWc8MvAfw2f/fvypS5i0/tUdAueET6XqpLfJG1/z8HgH84RPp+oxb+FfkPCH07gG6u5NKYXSJ0tRb\nGFn303tnMDfcLhcXuPB5FJtJAr2CftyA/Tn3+e5c3efHfzrzueVjtx8LpCNuHnz8Hw87jrwYdv8e\nHsTiXD3WTU5txo6jFCfsfTk8iMWZG7FudhbsOEqKCHwVFxLx6ZH4CDWIwN+RjLiBthnndX2DCLQS\ns31C3YJUesBAZXFWPrMT+sQEoE9T4y7R5HgD00+Um5mJ55TDR8erUKYqjnL4/YNghmyLm3g7c/Xt\nX4iRWsjW5VmlifPvIWf8hnwTD+G4TVTBztbU67latmY/jeNGsaquYydkByfMA8xQCD1ekul20Wwx\nuQHFVNeVL2C9YymqA9WRtnCXQUzBnrfx2VhVXtPaGDMLJtpEUUjRAjMUIha40CETh6ULZcwZXAW3\ngR2TDlb+sXOdV2QKaynevVKw95RKL2TquabcKo0v0JZs0hllgQaYCQS4TIxdEr7bkE9LjqSedYxP\nVo6SwEZIgEKU8mNgXT61voCBPSK5zxIBr4mYDbr/VGfRxoVdyD/chsrrgIZb8YQFLU9IpaNms4C0\n6c7AADNE0A4mmUQHwGSyRUuqOxNL2LsbGesasjU8ArlBTMHO13F/5j6CF8lTsLCJB5Fp8xIxQ37K\nDkw7HYHndvylLpCHHldf3vX44mTZLNMRlaen6Jgu7wVt28ohjE/aUqdzityPaiuHvrtp6lOVmz/M\nvAFmuN4GN8I02s0QPFY5UuEhz76vfdiQvuEnibc/Pfcx6mWBQ5Ad0btLhETas0A2WWC/F/sm9Z5A\nCDLpHhAvKrbTL5SuxhKu80lBZX6m/kcXh6A7zp4UISlb74P47TOUL+YZgm7NkaWUpUUavdRsVRYQ\nFoKugy6AYUx8EDMIBEJQ1J1rDFMfeXeF3+0KhKAofGWo+tD4eyWZgUAIEsH2luhSt6coU19d/fKy\nqBCCwqEFP5hSSvwk8viYehaC5En+UeHHb9OPmW6A/8LRIARpRPfWcKavcf7Xnef9MiHILv4DNJzt\n65vuflj7RmxL9v+bNypexG2r9AlQZvuQWO1HDsjRdZtoei2RGpzQzeTYD5j6F1iNEDN0uErHAAAA\nAElFTkSuQmCC\n",
       "prompt_number": 77,
       "text": "assoc_legendre(3/4, 1/4, x)"
      }
     ],
     "prompt_number": 77
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "expand_func(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAMAAAAAsBAMAAADBdKrNAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAElFTkSuQmCC\n",
       "prompt_number": 78,
       "text": "assoc_legendre(3/4, 1/4, x)"
      }
     ],
     "prompt_number": 78
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "legendre(10, x)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAHYAAAASBAMAAAB87DduAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\noe4lbi7kH6zLqBcupqlYAAAAAElFTkSuQmCC\n",
       "prompt_number": 79,
       "text": "legendre(10, x)"
      }
     ],
     "prompt_number": 79
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "expand_func(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAeQAAAAnBAMAAAAsiyZkAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 80,
       "text": "\n       10           8          6          4         2      \n46189\u22c5x     109395\u22c5x    45045\u22c5x    15015\u22c5x    3465\u22c5x     63\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\n   256         256        128        128        256     256"
      }
     ],
     "prompt_number": 80
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "factor(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAfUAAAAlBAMAAACpL225AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAAAElFTkSuQmCC\n",
       "prompt_number": 81,
       "text": "\n       10           8          6          4         2     \n46189\u22c5x   - 109395\u22c5x  + 90090\u22c5x  - 30030\u22c5x  + 3465\u22c5x  - 63\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\n                           256                            "
      }
     ],
     "prompt_number": 81
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "assoc_legendre(10, 2, x)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAh4AAAAsBAMAAADYwKz2AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nypP5L5Mjh1qks53SAAAAAElFTkSuQmCC\n",
       "prompt_number": 82,
       "text": "\n           \u239b         8           6           4          2       \u239e\n\u239b   2    \u239e \u239c2078505\u22c5x    765765\u22c5x    675675\u22c5x    45045\u22c5x    3465\u239f\n\u239d- x  + 1\u23a0\u22c5\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u239f\n           \u239d   128           32          64         32      128 \u23a0"
      }
     ],
     "prompt_number": 82
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "factor(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAecAAAAnBAMAAADHvJ1nAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 83,
       "text": "\n                     \u239b      8         6         4        2    \u239e\n-495\u22c5(x - 1)\u22c5(x + 1)\u22c5\u239d4199\u22c5x  - 6188\u22c5x  + 2730\u22c5x  - 364\u22c5x  + 7\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\n                              128                              "
      }
     ],
     "prompt_number": 83
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "assoc_legendre(Rational(3,2), Rational(1,2), x)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAMAAAAAsBAMAAADBdKrNAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nonYAAAAASUVORK5CYII=\n",
       "prompt_number": 84,
       "text": "assoc_legendre(3/2, 1/2, x)"
      }
     ],
     "prompt_number": 84
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "legendre(S(3)/2, 1.5).evalf()",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAIwAAAAOBAMAAAAPluneAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nzf8YfwG/MMaMkD0SvgAAAABJRU5ErkJggg==\n",
       "prompt_number": 85,
       "text": "2.03434274858116"
      }
     ],
     "prompt_number": 85
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "legendre(S(3)/2+I, 1.5-I).evalf()",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAUQAAAAOBAMAAAClcHhLAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYII=\n",
       "prompt_number": 86,
       "text": "5.20465949508608 + 0.299479454976305\u22c5\u2148"
      }
     ],
     "prompt_number": 86
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "legendre(S(3)/2, 2).evalf(50)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAcoAAAAOBAMAAABdvEH5AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n/v/Xn890sVBtKBTQAAAAAElFTkSuQmCC\n",
       "prompt_number": 87,
       "text": "3.2439396660408049154502287929704557672075154110177"
      }
     ],
     "prompt_number": 87
    },
    {
     "cell_type": "markdown",
     "source": "Different branch cuts!"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "assoc_legendre(2, 1, x) #type=1",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAHMAAAAWBAMAAAABVL48AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nGxztIGHy1/+sUOpPwV/ukXc6LYLrOgAAAABJRU5ErkJggg==\n",
       "prompt_number": 88,
       "text": "\n        __________\n       \u2571    2     \n-3\u22c5x\u22c5\u2572\u2571  - x  + 1 "
      }
     ],
     "prompt_number": 88
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "assoc_legendre(2, 1, x) #type=2\n# -3*sqrt(1-z)*z*sqrt(1+z)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAHMAAAAWBAMAAAABVL48AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nGxztIGHy1/+sUOpPwV/ukXc6LYLrOgAAAABJRU5ErkJggg==\n",
       "prompt_number": 89,
       "text": "\n        __________\n       \u2571    2     \n-3\u22c5x\u22c5\u2572\u2571  - x  + 1 "
      }
     ],
     "prompt_number": 89
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "assoc_legendre(2, 1, x) #type=3\n# -3(1-z)*z*sqrt(1+z) / sqrt(z-1)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAHMAAAAWBAMAAAABVL48AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nGxztIGHy1/+sUOpPwV/ukXc6LYLrOgAAAABJRU5ErkJggg==\n",
       "prompt_number": 90,
       "text": "\n        __________\n       \u2571    2     \n-3\u22c5x\u22c5\u2572\u2571  - x  + 1 "
      }
     ],
     "prompt_number": 90
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "s = series(sin(x), x, n=5)",
     "language": "python",
     "outputs": [],
     "prompt_number": 91
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "s = sin(x) + O(x**5)",
     "language": "python",
     "outputs": [],
     "prompt_number": 92
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "s",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAHsAAAAWBAMAAAASg/7IAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n/B8sNPq+f52w7/gbTWYSd/oHPR5+80J0fJ0AAAAASUVORK5CYII=\n",
       "prompt_number": 93,
       "text": "\n          \u239b 5\u239e\nsin(x) + O\u239dx \u23a0"
      }
     ],
     "prompt_number": 93
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "legendre(2, s)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAOEAAAAWBAMAAADECuboAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nKikVsN3JBfsAAAAASUVORK5CYII=\n",
       "prompt_number": 94,
       "text": "\n        \u239b             \u239b 5\u239e\u239e\nlegendre\u239d2, sin(x) + O\u239dx \u23a0\u23a0"
      }
     ],
     "prompt_number": 94
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "expand_func(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAMYAAAAlBAMAAADrZYtCAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n3Tvq9X+i81Dd3wCDfO2rUN1qlQAAAABJRU5ErkJggg==\n",
       "prompt_number": 95,
       "text": "\n                  2    \n  \u239b          \u239b 5\u239e\u239e     \n3\u22c5\u239dsin(x) + O\u239dx \u23a0\u23a0    1\n\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\n         2            2"
      }
     ],
     "prompt_number": 95
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "expand(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAMMAAAAlBAMAAAANTEAGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n9Sf8BeJw41gBcPd+AAAAAElFTkSuQmCC\n",
       "prompt_number": 96,
       "text": "\n           2           \n  1   3\u22c5sin (x)    \u239b 6\u239e\n- \u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + O\u239dx \u23a0\n  2       2            "
      }
     ],
     "prompt_number": 96
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "assoc_legendre(S(3)/2, S(1)/2, s)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAATMAAAAsBAMAAAD7gUKXAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAABJRU5ErkJggg==\n",
       "prompt_number": 97,
       "text": "\n              \u239b                    \u239b 5\u239e\u239e\nassoc_legendre\u239d3/2, 1/2, sin(x) + O\u239dx \u23a0\u23a0"
      }
     ],
     "prompt_number": 97
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "expand_func(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAATMAAAAsBAMAAAD7gUKXAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAABJRU5ErkJggg==\n",
       "prompt_number": 98,
       "text": "\n              \u239b                    \u239b 5\u239e\u239e\nassoc_legendre\u239d3/2, 1/2, sin(x) + O\u239dx \u23a0\u23a0"
      }
     ],
     "prompt_number": 98
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "1+x**2+O(x**3)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAHoAAAAWBAMAAAD9QZX2AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nVstShspbZJR/wf2pwQ5SPCRyVd9YB/4CczpzdTII4vYAAAAASUVORK5CYII=\n",
       "prompt_number": 99,
       "text": "\n     2    \u239b 3\u239e\n1 + x  + O\u239dx \u23a0"
      }
     ],
     "prompt_number": 99
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "_**2",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJIAAAAaBAMAAACuthgIAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIjKruxB23VSJRO9m\nmc3kD4mRAAACZklEQVQ4Ea1UO2gUURS9M+ZlJ/vJBm2FTBUMAbMsCGLjitPYSGy0EXZAEgwoRNBS\nslgJCbiFiJUuKKJESBBEsBqMRRohiIW/yhRpY2NUhPXce2dn32SnmGIPnPube07mzcyGaCiYuuIP\nxYdM6L4ejpO7R3+H40S++Qmnw5lu7kI9c54Iiqt1Mh0SElXaRE6NsnCRZuay5omg5K8RXcUKkz6A\nI9mCWap2eGUQscChdRVzXwwN0cLgLk9cak6krrjtp29CniSC79iqKacuvCO6haOeSmni5kt6uDg2\nf/wXjyAQTJ9AuiF0ut098hrkPvit19yWZonlbW2OaaqEm2T2UUMQYw3HWkTNBA61EWInL+JJjGeE\nRWBV+/f87Z1GLQLkFapGRHjqQqRCByHDyZ0/6uNKz8m8LaN9jF4EyNPUxKSJP8cEqhMIB5zuPnxZ\nn+x25Xp8T6NL3H0DWcAL3rWT6EodJcpJNkw7eX4z+oepQk833uBuB4TAWihEuEsQmOWQdirT+hz7\nK9SptM0d/7ggsBbGQyIm0Hf6FJy/GZyV4SOJCJUg2A2CGk7QQDfGQQTJguWUcTqic1D0EJ9uA/0l\nvlMRJAuFFk4HAvIlp09ncNpILnJQp9E/+IDPcA+BtWA9cS4PPKd7l185LUwV6kTLH811viV+W9YC\nv0l5/fgNhuTc3/8qKi+SdGTryZYUEmKnyvJtX3oIrIUZzJhAuaaZY7HVr5PqeVJpYQuINjFkAmZD\nc+6YFvyAjsn4rCl/tAXuEv6rgIKR/B6DgoqPbw4UFNtxkTfZghcQMRV3ekXe3BeYEA8aZPwHMsuO\ndCFbTY8AAAAASUVORK5CYII=\n",
       "prompt_number": 100,
       "text": "\n                2\n\u239b     2    \u239b 3\u239e\u239e \n\u239d1 + x  + O\u239dx \u23a0\u23a0 "
      }
     ],
     "prompt_number": 100
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "expand(_)",
     "language": "python",
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAIIAAAAWBAMAAAD+2lmbAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nNa/tTcA21wuYbJt2Af8AKKt6dt2LzkoAAAAASUVORK5CYII=\n",
       "prompt_number": 101,
       "text": "\n       2    \u239b 3\u239e\n1 + 2\u22c5x  + O\u239dx \u23a0"
      }
     ],
     "prompt_number": 101
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "# Takes long time\n#integrate(legendre(n, x)*sin(pi*x), (x, -1, 1))",
     "language": "python",
     "outputs": [],
     "prompt_number": 102
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": "",
     "language": "python",
     "outputs": []
    }
   ]
  }
 ]
}