SymPy Examples.ipynb

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   "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": "iVBORw0KGgoAAAANSUhEUgAAAGUAAAAoCAYAAADnujR9AAAABHNCSVQICAgIfAhkiAAABN9JREFU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\nAElFTkSuQmCC\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": "iVBORw0KGgoAAAANSUhEUgAAACsAAAASCAYAAADCKCelAAAABHNCSVQICAgIfAhkiAAAAVxJREFU\nSInt1b9LHEEYxvGP0UT00EQRIUiwUJEgsTCEkEZS+keYXv8AG/tgay0pDiSQIl2wsLEOpFBQm4CF\nIRhOm6Bco/ijmBH3zolkZY+cxKd5medhdr47M7svd0gtDXhmJ75iPOe8USxgF+fowxwqhdJl9Arf\n4mJ59Bg/MZ3x5rGFR8WgXek5VlAWdjUv7Hvsoy3j9eIEswXw/VFl+WG/40vC38Ta5eDB7ZkKUxdG\n8COR7eHl5aAZYAdjPUxkVXSjneaA7Y71OJFVY31Cc8Cexpq65w9jbaU5YA9uyEqxHlH7q3iBJX/f\nKDYwkxvtuirCrvYkshJ+S8Bu4k0Bi+dVFet4lsiGhU3Bv7kGI+io81bwWu2pDgkv8LmRMJ+EY+1M\nZJM4w2qd/1Q47ncZbxHbMu22TTHqxzIGMBa9XaG3f8DH6FWED2qnbv4vvBXa7oTQKHoxJf1Lu9f/\nqwtgsDsUgp/FvwAAAABJRU5ErkJggg==\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": "iVBORw0KGgoAAAANSUhEUgAAAfwAAAASCAYAAAC+a2xVAAAABHNCSVQICAgIfAhkiAAACXpJREFU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\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": "iVBORw0KGgoAAAANSUhEUgAAAHQAAAAbCAYAAACtOKuoAAAABHNCSVQICAgIfAhkiAAAA+lJREFU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\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": "iVBORw0KGgoAAAANSUhEUgAAAQ0AAAAbCAYAAABm6to6AAAABHNCSVQICAgIfAhkiAAABP5JREFU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\n7l6NeSaTyWQymUwmk8lkMpke518QvOZokU+z1gAAAABJRU5ErkJggg==\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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\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": 10
    },
    {
     "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": 16
    },
    {
     "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": 12
    },
    {
     "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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374dgawslSrTl/n2IiQkhNRUyM9sSG6ugFNlDumMq6WVe4J5rOhF2ezhUJp2l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\n",
       "text": [
        "<matplotlib.figure.Figure at 0xb0a1f76c>"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "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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F1axZ4OEB8+ZB375GRyUclSwQ7SIujBebTDBzJvj4wIABhNUrYuN8X2p8GMnR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\nYJFFREREJIH/B046RrQ37WNaAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0xb0d0edcc>"
       ]
      }
     ],
     "prompt_number": 18
    },
    {
     "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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weDCwdavLiZNVnfvNgtzD2GnD+GmjR/wqVZJrXMqVAz7/XLZ9Ac8942lOnv7+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\n",
       "text": [
        "<matplotlib.figure.Figure at 0xb082f12c>"
       ]
      }
     ],
     "prompt_number": 19
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}