{
 "metadata": {
  "name": ""
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# SymPy: Open Source Symbolic Mathematics\n",
      "\n",
      "This notebook uses the [SymPy](http://sympy.org) package to perform symbolic manipulations,\n",
      "and combined with numpy and matplotlib, also displays numerical visualizations of symbolically\n",
      "constructed expressions.\n",
      "\n",
      "We first load sympy printing extensions, as well as all of sympy:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from IPython.display import display\n",
      "\n",
      "from sympy.interactive import printing\n",
      "printing.init_printing()\n",
      "\n",
      "from __future__ import division\n",
      "import sympy as sym\n",
      "from sympy import *\n",
      "x, y, z = symbols(\"x y z\")\n",
      "k, m, n = symbols(\"k m n\", integer=True)\n",
      "f, g, h = map(Function, 'fgh')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<h2>Elementary operations</h2>"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Rational(3,2)*pi + exp(I*x) / (x**2 + y)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{3 \\pi}{2} + \\frac{e^{i x}}{x^{2} + y}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAHAAAAAzBAMAAAC9ApG6AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nQmCC\n",
       "prompt_number": 2,
       "text": [
        "        \u2148\u22c5x \n",
        "3\u22c5\u03c0    \u212f    \n",
        "\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\n",
        " 2     2    \n",
        "      x  + y"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "exp(I*x).subs(x,pi).evalf()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$-1.0$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACkAAAAPBAMAAACLu/vuAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMiK7mat272aJ\nVER1AWP9AAAAiklEQVQYGWNggANFMIuxZroDXIiBIewjmMN0gdEUISpWARHdxcBwCCHKwAkRzWBg\nuC+AEIaK/mBgeH8BXZTxO1B0Aroo6x8GBv8DOESFjEFAhQFqGyNQLaYJDEBz72PYxmDGwLAf02Wz\nGBiKEJaBzZVXYGC+wLgEIcre9bmbgbmBgfHkDAeEKAoLAGTVKXybU7YXAAAAAElFTkSuQmCC\n",
       "prompt_number": 3,
       "text": [
        "-1.00000000000000"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": [
      "e = x + 2*y"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "srepr(e)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 5,
       "text": [
        "\"Add(Symbol('x'), Mul(Integer(2), Symbol('y')))\""
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "exp(pi * sqrt(163)).evalf(50)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$262537412640768743.99999999999925007259719818568888$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAgsAAAAPBAMAAACYf5HCAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\njp/0A2HdO72wUKUV4juHKOv/Qxz4F/nKkQHS8LrfAAAAAElFTkSuQmCC\n",
       "prompt_number": 6,
       "text": [
        "262537412640768743.99999999999925007259719818568888"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<h2>Algebra<h2>"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "eq = ((x+y)**2 * (x+1))\n",
      "eq"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left(x + 1\\right) \\left(x + y\\right)^{2}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAIoAAAAbBAMAAABRkwqxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nFAvU8si/1P588YegT3N5QVRLejL9m/gCd/R8g+k5mUIAAAAASUVORK5CYII=\n",
       "prompt_number": 7,
       "text": [
        "               2\n",
        "(x + 1)\u22c5(x + y) "
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expand(eq)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$x^{3} + 2 x^{2} y + x^{2} + x y^{2} + 2 x y + y^{2}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAASgAAAAYBAMAAAC7JH0zAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nJhQ2HQAAAABJRU5ErkJggg==\n",
       "prompt_number": 8,
       "text": [
        " 3      2      2      2            2\n",
        "x  + 2\u22c5x \u22c5y + x  + x\u22c5y  + 2\u22c5x\u22c5y + y "
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "a = 1/x + (x*sin(x) - 1)/x\n",
      "a"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{1}{x} \\left(x \\sin{\\left (x \\right )} - 1\\right) + \\frac{1}{x}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAALYAAAAqBAMAAADserwPAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nTkSuQmCC\n",
       "prompt_number": 9,
       "text": [
        "x\u22c5sin(x) - 1   1\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\n",
        "     x         x"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "simplify(a)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\sin{\\left (x \\right )}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADcAAAAVBAMAAAAQkWtIAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMnZUzRC73UTviSKZ\nZqszMyTAAAABLElEQVQoFZWRvUvDUBRHTzRpa2NFxdUSKl0qSBcFdcnUOZOgk4MIIkgGN5fgpqOr\nDiI66yQUEfInqIvgB1TBSYSoIH6g8b0kfYndvEPeeffc/Lgk8K8aTqbPk1PbSl835xPu649B+0yl\n7rTZbkN61hSmpFrbimZjMgdUp+Aq1OtoIxX7+IehncExS/SLPhiVx/W9CGfAZQUalvklZO4M5ri0\nVqFks+GwK+USfAuZr8MBh840dLvk30aRcgFehOzxwKEhCC3A2A99KRdTGZOUJ5TeM1LGoj1jgFh8\nCp4yUi403hVQjRaadFiWmUlssUXhNReYYqbXZ/PiyiuHD+XwbuKjFWUZ99XarQjXPfH4W2vq2lSk\n4EaR+EidlTtKOobdqUD9bDWVnTlNLtfwC1XgR6gJ3g8BAAAAAElFTkSuQmCC\n",
       "prompt_number": 10,
       "text": [
        "sin(x)"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "eq = Eq(x**3 + 2*x**2 + 4*x + 8, 0)\n",
      "eq"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$x^{3} + 2 x^{2} + 4 x + 8 = 0$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAMcAAAAWBAMAAACGSZV/AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nnlCvv+wOXivccChMyHVbD2/5D+LqohDLQZpLAAAAAElFTkSuQmCC\n",
       "prompt_number": 11,
       "text": [
        " 3      2              \n",
        "x  + 2\u22c5x  + 4\u22c5x + 8 = 0"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "solve(eq, x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{bmatrix}-2, & - 2 i, & 2 i\\end{bmatrix}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAIwAAAAZBAMAAAARRdv9AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYlw1phdFB7UYV40JiJ+I/xZj/sZ/avYbjdhWy3er6JMAAAAASUVORK5CYII=\n",
       "prompt_number": 12,
       "text": [
        "[-2, -2\u22c5\u2148, 2\u22c5\u2148]"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "a, b = symbols('a b')\n",
      "Sum(6*n**2 + 2**n, (n, a, b))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\sum_{n=a}^{b} \\left(2^{n} + 6 n^{2}\\right)$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAIMAAAA9BAMAAAB8c8dQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 13,
       "text": [
        "  b              \n",
        " ___             \n",
        " \u2572               \n",
        "  \u2572   \u239b n      2\u239e\n",
        "  \u2571   \u239d2  + 6\u22c5n \u23a0\n",
        " \u2571               \n",
        " \u203e\u203e\u203e             \n",
        "n = a            "
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<h2>Calculus</h2>"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "limit((sin(x)-x)/x**3, x, 0)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- \\frac{1}{6}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAB0AAAAqBAMAAAC9wk0pAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMiK7mat272aJ\nRFQidGHIAAAAiUlEQVQoFWNgQAAmAQQbyGLNR+Ezpvej8BkY5g9ufsV6iw0oHhoQjpAxCKgwMPwH\ngQ+0dMOZOQ7IxgtdYFdA5uswsCUg8bm/InGATLYFqHx+lZO1yCLy2gxcD5AE5D8ysC9E4vMbMHD/\nReJzKTBw/0biA81HkecD6l+AJM+QwyD1AJnPU97CwAAAnlQjKf5L1GYAAAAASUVORK5CYII=\n",
       "prompt_number": 14,
       "text": [
        "-1/6"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(1/cos(x)).series(x, 0, 6)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$1 + \\frac{x^{2}}{2} + \\frac{5 x^{4}}{24} + \\mathcal{O}\\left(x^{6}\\right)$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAM4AAAAwBAMAAACrqVieAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nL/nNVv6/x/8HW/gXypJ/ihwAAAAASUVORK5CYII=\n",
       "prompt_number": 15,
       "text": [
        "     2      4        \n",
        "    x    5\u22c5x     \u239b 6\u239e\n",
        "1 + \u2500\u2500 + \u2500\u2500\u2500\u2500 + O\u239dx \u23a0\n",
        "    2     24         "
       ]
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "diff(cos(x**2)**2 / (1+x), x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- \\frac{4 x \\cos{\\left (x^{2} \\right )}}{x + 1} \\sin{\\left (x^{2} \\right )} - \\frac{\\cos^{2}{\\left (x^{2} \\right )}}{\\left(x + 1\\right)^{2}}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAASoAAAA4BAMAAAC4fag4AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMpm7du8iZolU\nq0RaI+fpAAAF+klEQVRoBcVYXYgbVRQ+k0w2v5sNfWiVIqap1heFWpV1fVqKqAh1tz7UhyJEQdEH\nafogW1mLRRQpaM2Ly+qDG3+K1iLEQiltQSNorbiwq/hbrOZBUCzSFLf1pz/xnPszc+9k5s5MNq0H\ndu75zs93T++9Mzm9AHHl4COluCkB8fktmz2e2Nx2mzPk68NlqHnY+oOH4Sue2D/3XWLmZN3+B/7o\nrwxP1hhM1Jmpf+41gjLTzl6CfMMzQV9wD8y1WaKR254MJk+5m1Y4A3BzcGQcz0KJos3cySrF+Msc\ny2e+JBb4pH9UXOsoSzBz76gGsx50XaTOuXAZWn6SJRu5s29Wg2d40XHliCrZcPAylL0818idT1QD\nZ7DxLAk5Bb/gUahKuIyx2CyOY7qZex9WZU398OshjLS2Hm3B6qkPwX56qow4twsfq3a+Pl2zb5x/\nCCBNOL7otBvnj9FpNXJbNaxqNbxV24CRsy17tz0GQ7WVAOsRp8pYaTM3mixnut0OQNFdO/RGFp32\n5W6XMo3cecCqPoCPSp9jZAUKS8km2Eu/laCJeKgKkC/lzxTqCFDsJT7GfHppWbqR+xOqqgTrKNI6\ni4+JFsD55LmnyJBYRCMFCMEvaR/SQ8s4TNxWk0/6F0UW6bnQBjhnHehOop6kB4zU6EmSvcDHeM8e\nWpZu4i6cPPnzSzXaGktZq7OvQo6Whf49uHrjuJhMwtaqtx2gNGetJC3jCuHOVOHjdAe2Y2wFoJCZ\nhOKlLwBOIKbMQukZSIuq8BfaKE47oEd5aZk3hHukWrww1Mk2MHa2AYft5yDRPF6CxxEPVwFOt7bB\na6iTBL2D93E3UDuQFroyeGmZy8yd33bxje+2T79HodY303V45ehPcOqd/W3E9H26ZmrVkQbqJOwb\nw1X1aVcFonagyHZddffQMmdEbo2IA+/aBHzbE2KDMQnbgSM+RD6miNw+mbBHNyZqOhboAdc6ii+u\nW6Nr99GicfskwlbdeK0zof2C68m6X3xqB9L4F0UCuUOT79Ej3negfd5RQXQmZKB2INtxXSYtkNuU\nxHzDLS3kVg1JkHGCeDtwh3SYx0jcvhRWWTXnmipy9ImSVHk7sEVC82iVVX8Atxri6p+6KsCMC7J3\nuvq7jsrbgR0ONitB3OYs8hZaSsxmrlPzNXMRNj5/984aWe7HP9mhET7trB2hYPHjDo4O87Dm6wmA\ndbUs+wV6FBNkh0a5+Lv5PwhrvrCqxwD+pelvwj/ZoREW/90jNaasuIXkOhjpxpQOfiap+cKqHgZg\nDdBtOLfo0FgVcy0crLU0wYZxVCPPgbH9C2u+sCrcOVYVrZVQmdb/WrH0Ph+s+fJWxTs0xiiuEeKw\n+zdmcRiANV9KVd9ituzQiCjqO6hMGtCYKRGhKmu+cPvkDuL3yenQKNn9foVSyQDnnkYa4o/UfG3q\nfrmp+/WxvxcxHdfGkh0asUX8tqsTO/c0qtGkrzE5mS/T1kNu0GE0tBDx08vZZtw2JYg+1dQ8VniG\nFs8BNmYR5Goes/LB8DnsXRpfjna1R+7tsUjD26Qo3ZC0+42fCWM6vCr4USNIjmuQg2zVx4gmPB/D\n4zhSYxZB4lS1QuPbryEBlNZedbPzUcZ3mN/TqC5XZ9cvHMapqjDpUoD/qkwrEfhzVOeQnw/0iXsa\nLUgCfv3CUZyqYF4y4JgqKcBRdzsaKYWagOx8XAUg7mm0IAnU65dYVUmCwLG4XnPpVSVamtML1OuX\nwVZFr5jSGOpVhb5/4vrlQKVye6VyPVUd5R30/ut68VBDawz1qnLl3gTNMuFcvwx2rZItrTHUqyro\n26sVhEC9fhlsVfSjpDSGelV2x1uIhtXrl8FXJRvDfKWy9tlKBU+aOB8hVanXL6Kq1Pd/ntAq7w/Q\nDoqrOyLQ16po3kFKkCLXSuLljXTalcZQryr0tLtz/+6qA9Dyi1pjqFeVYps5gFniUuAmqY2hrIqf\nj0Q7Lt2g4sc0IlkVN85qvisJ9mqTFRsqPKSCK6oP1QOns8qBrsvt8O9v2KyGgi93Vc7leO9E+xzT\nfxAXF9fGMlsTAAAAAElFTkSuQmCC\n",
       "prompt_number": 16,
       "text": [
        "         \u239b 2\u239e    \u239b 2\u239e      2\u239b 2\u239e\n",
        "  4\u22c5x\u22c5sin\u239dx \u23a0\u22c5cos\u239dx \u23a0   cos \u239dx \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\n",
        "         x + 1                 2\n",
        "                        (x + 1) "
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "integrate(x**2 * cos(x), (x, 0, pi/2))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$-2 + \\frac{\\pi^{2}}{4}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEwAAAAwBAMAAABJWYGiAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMiKZu6uJRO92\nVGZ6zyUAAAABVUlEQVQ4EY3SMUvDQBgG4C8mkVyTmODkpkRX6eAPsH9ACLhLprrWRTetuHRrBw24\ndXBx1UVwCYiTg4s4KQShuyKKupwXrq1Rr5f3W3LcPffe8eWIKivdDisNEYudBYCZsfUGMLdvfwGM\nyH+BmJlBLIXUTAKxUzoBnLX8uKVnt5zzd5fzZy1jhw8b17GWFItrdOQ1KpUALQf561bbRMKm8wBJ\nmw+DPhB3RW4PYLvk/GV2c6cB7Jwjhry+faJLmWbEmtRzoq481c80rB5CTCSsyl5q04hqn/KwCma2\nNGx2paglITalonGaeJSjGi6Jj9cuxiyKFs+iKCnGyronWzZsnKZiVkIewNaP0wO5XZtW5/wVYD8X\n+Z0mLqOuWl6eH3WzPKcY3wybrlgqTdlPEGNTEBtAzMggxghidxAzehDzO53uRVbqz8ShCzWEAoix\nvY984kn/Fr4Bq5lS2/Bs4I0AAAAASUVORK5CYII=\n",
       "prompt_number": 17,
       "text": [
        "      2\n",
        "     \u03c0 \n",
        "-2 + \u2500\u2500\n",
        "     4 "
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "eqn = Eq(Derivative(f(x),x,x) + 9*f(x), 1)\n",
      "display(eqn)\n",
      "dsolve(eqn, f(x))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$9 f{\\left (x \\right )} + \\frac{d^{2}}{d x^{2}}  f{\\left (x \\right )} = 1$$"
       ],
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAL4AAAAvBAMAAACvYG1IAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n70az7X2tO1K2hH8DDpT5LbY9x+oAAAAASUVORK5CYII=\n",
       "text": [
        "           2          \n",
        "          d           \n",
        "9\u22c5f(x) + \u2500\u2500\u2500(f(x)) = 1\n",
        "           2          \n",
        "         dx           "
       ]
      },
      {
       "latex": [
        "$$f{\\left (x \\right )} = C_{1} \\sin{\\left (3 x \\right )} + C_{2} \\cos{\\left (3 x \\right )} + \\frac{1}{9}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAUMAAAAqBAMAAADbkSQSAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nVkj/JFxZAv4ByMpfji3f9pUAAAAASUVORK5CYII=\n",
       "prompt_number": 18,
       "text": [
        "f(x) = C\u2081\u22c5sin(3\u22c5x) + C\u2082\u22c5cos(3\u22c5x) + 1/9"
       ]
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Illustrating Taylor series\n",
      "\n",
      "We will define a function to compute the Taylor series expansions of a symbolically defined expression at\n",
      "various orders and visualize all the approximations together with the original function"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%matplotlib inline\n",
      "import numpy as np\n",
      "import matplotlib.pyplot as plt"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 19
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": [
      "# You can change the default figure size to be a bit larger if you want,\n",
      "# uncomment the next line for that:\n",
      "#plt.rc('figure', figsize=(10, 6))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 20
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": [
      "def plot_taylor_approximations(func, x0=None, orders=(2, 4), xrange=(0,1), yrange=None, npts=200):\n",
      "    \"\"\"Plot the Taylor series approximations to a function at various orders.\n",
      "\n",
      "    Parameters\n",
      "    ----------\n",
      "    func : a sympy function\n",
      "    x0 : float\n",
      "      Origin of the Taylor series expansion.  If not given, x0=xrange[0].\n",
      "    orders : list\n",
      "      List of integers with the orders of Taylor series to show.  Default is (2, 4).\n",
      "    xrange : 2-tuple or array.\n",
      "      Either an (xmin, xmax) tuple indicating the x range for the plot (default is (0, 1)),\n",
      "      or the actual array of values to use.\n",
      "    yrange : 2-tuple\n",
      "      (ymin, ymax) tuple indicating the y range for the plot.  If not given,\n",
      "      the full range of values will be automatically used. \n",
      "    npts : int\n",
      "      Number of points to sample the x range with.  Default is 200.\n",
      "    \"\"\"\n",
      "    if not callable(func):\n",
      "        raise ValueError('func must be callable')\n",
      "    if isinstance(xrange, (list, tuple)):\n",
      "        x = np.linspace(float(xrange[0]), float(xrange[1]), npts)\n",
      "    else:\n",
      "        x = xrange\n",
      "    if x0 is None: x0 = x[0]\n",
      "    xs = sym.Symbol('x')\n",
      "    # Make a numpy-callable form of the original function for plotting\n",
      "    fx = func(xs)\n",
      "    f = sym.lambdify(xs, fx, modules=['numpy'])\n",
      "    # We could use latex(fx) instead of str(), but matploblib gets confused\n",
      "    # with some of the (valid) latex constructs sympy emits.  So we play it safe.\n",
      "    plt.plot(x, f(x), label=str(fx), lw=2)\n",
      "    # Build the Taylor approximations, plotting as we go\n",
      "    apps = {}\n",
      "    for order in orders:\n",
      "        app = fx.series(xs, x0, n=order).removeO()\n",
      "        apps[order] = app\n",
      "        # Must be careful here: if the approximation is a constant, we can't\n",
      "        # blindly use lambdify as it won't do the right thing.  In that case, \n",
      "        # evaluate the number as a float and fill the y array with that value.\n",
      "        if isinstance(app, sym.numbers.Number):\n",
      "            y = np.zeros_like(x)\n",
      "            y.fill(app.evalf())\n",
      "        else:\n",
      "            fa = sym.lambdify(xs, app, modules=['numpy'])\n",
      "            y = fa(x)\n",
      "        tex = sym.latex(app).replace('$', '')\n",
      "        plt.plot(x, y, label=r'$n=%s:\\, %s$' % (order, tex) )\n",
      "        \n",
      "    # Plot refinements\n",
      "    if yrange is not None:\n",
      "        plt.ylim(*yrange)\n",
      "    plt.grid()\n",
      "    plt.legend(loc='best').get_frame().set_alpha(0.8)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 21
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "With this function defined, we can now use it for any sympy function or expression"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot_taylor_approximations(sin, 0, [2, 4, 6], (0, 2*pi), (-2,2))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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aFwc3U4ci9DTj7V6Ndq3vnzlS53Ojo6O5//77CQoK4vHHH2fEiBFUVVURFRV1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\nRERkJWy0iIiIiKzk/wMTwbog9cTfIgAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x108046710>"
       ]
      }
     ],
     "prompt_number": 22
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot_taylor_approximations(cos, 0, [2, 4, 6], (0, 2*pi), (-2,2))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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7785ot3I6qt77EqpjMZgx6nU02o/G6bOzMGHdz9eaLVOp11S3z507Z1L1mNu2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\noGFhYcjPz8epU8Kbbd3YUgyXkAAUHjiJ3m8v4DscQgghOkKFlpbl5OT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       "text": [
        "<matplotlib.figure.Figure at 0x107e1a050>"
       ]
      }
     ],
     "prompt_number": 23
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This shows easily how a Taylor series is useless beyond its convergence radius, illustrated by \n",
      "a simple function that has singularities on the real axis:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# For an expression made from elementary functions, we must first make it into\n",
      "# a callable function, the simplest way is to use the Python lambda construct.\n",
      "plot_taylor_approximations(lambda x: 1/cos(x), 0, [2,4,6], (0, 2*pi), (-5,5))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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plCAI+L9NxQCAywbFIiH0xLuIl9QU4O2fn8T9F/0HcRHdZWmLq9mGnLlvYvvt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\nDBs2bEBNTc0j3z906BC++eYbm85VVVWFkydP4uOPP+Z9E83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       "text": [
        "<matplotlib.figure.Figure at 0x107e4a810>"
       ]
      }
     ],
     "prompt_number": 24
    }
   ],
   "metadata": {}
  }
 ]
}