{
 "metadata": {
  "name": "sympy"
 }, 
 "nbformat": 2, 
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "markdown", 
     "source": [
      "# SymPy: Open Source Symbolic Mathematics", 
      "", 
      "This notebook uses the [SymPy](http://sympy.org) package to perform symbolic manipulations,", 
      "and combined with numpy and matplotlib, also displays numerical visualizations of symbolically", 
      "constructed expressions.", 
      "", 
      "We first load sympy printing and plotting support, as well as all of sympy:"
     ]
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": [
      "%load_ext sympyprinting", 
      "%pylab inline", 
      "", 
      "from __future__ import division", 
      "import sympy as sym", 
      "from sympy import *", 
      "x, y, z = symbols(\"x y z\")", 
      "k, m, n = symbols(\"k m n\", integer=True)", 
      "f, g, h = map(Function, 'fgh')"
     ], 
     "language": "python", 
     "outputs": [
      {
       "output_type": "stream", 
       "stream": "stdout", 
       "text": [
        "", 
        "Welcome to pylab, a matplotlib-based Python environment [backend: module://IPython.zmq.pylab.backend_inline].", 
        "For more information, type 'help(pylab)'."
       ]
      }
     ], 
     "prompt_number": 1
    }, 
    {
     "cell_type": "markdown", 
     "source": [
      "<h2>Elementary operations</h2>"
     ]
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": [
      "Rational(3,2)*pi + exp(I*x) / (x**2 + y)"
     ], 
     "language": "python", 
     "outputs": [
      {
       "latex": [
        "$$\\frac{3}{2} \\pi + \\frac{e^{\\mathbf{\\imath} x}}{x^{2} + y}$$"
       ], 
       "output_type": "pyout", 
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFAAAAAlCAYAAADV/m7fAAAABHNCSVQICAgIfAhkiAAAA91JREFU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\ndFwsfldTlGRq0aJFi6bjP9GM0XhICUQDAAAAAElFTkSuQmCC\n", 
       "prompt_number": 2, 
       "text": [
        "", 
        "        \u2148\u22c5x ", 
        "3\u22c5\u03c0    \u212f    ", 
        "\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500", 
        " 2     2    ", 
        "      x  + y"
       ]
      }
     ], 
     "prompt_number": 2
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": [
      "exp(I*x).subs(x,pi).evalf()"
     ], 
     "language": "python", 
     "outputs": [
      {
       "latex": [
        "$$-1.0$$"
       ], 
       "output_type": "pyout", 
       "png": "iVBORw0KGgoAAAANSUhEUgAAACsAAAASCAYAAADCKCelAAAABHNCSVQICAgIfAhkiAAAAU1JREFU\nSInt1csrBlEcxvGPWxKFhcsCK5QFJVlYsfJPsLKQjf8CG8qejVKWkhUbsRBZuWzct0okxQK5LGam\npmmU923evMpTp9N5njm/+c7MmXP4QyopQM0WbKM9j3mzuMEjKjCP+yzhIlVjGOf4zHFuGS4xHvOm\nsInSTOhi6sIaZrArd9gRvAgeOFJHWGcsC8DvtCR32ENspfhXWI0Gmb/iPFSOHlykZJcYigbFANsk\n+NGfUrJn1KOS4oBtDvvnlCzy6igO2Jew/0jJKuJZMcCe4fWbrBpvuCNY3JG6seDnB8UhJvIEjOsN\np4K1mVQNboW7Sxz2BAMZ3DwfHaM14ZWhF3uR8RvLoBVVCe8Igwm/H7WYKyTMuuCzNaZkfXjHRsKv\nFxzTkzFvAfuFAGzEjmBj/wzbAw4wGruuDdeYTqnRiWUshm0FDYWA/def1heszTze5axPeQAAAABJ\nRU5ErkJggg==\n", 
       "prompt_number": 4, 
       "text": [
        "-1.00000000000000"
       ]
      }
     ], 
     "prompt_number": 4
    }, 
    {
     "cell_type": "code", 
     "collapsed": true, 
     "input": [
      "e = x + 2*y"
     ], 
     "language": "python", 
     "outputs": [], 
     "prompt_number": 5
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": [
      "srepr(e)"
     ], 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 6, 
       "text": [
        "Add(Symbol('x'), Mul(Integer(2), Symbol('y')))"
       ]
      }
     ], 
     "prompt_number": 6
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": [
      "exp(pi * sqrt(163)).evalf(50)"
     ], 
     "language": "python", 
     "outputs": [
      {
       "latex": [
        "$$262537412640768743.99999999999925007259719818568888$$"
       ], 
       "output_type": "pyout", 
       "png": "iVBORw0KGgoAAAANSUhEUgAAAfwAAAASCAYAAAC+a2xVAAAABHNCSVQICAgIfAhkiAAACXRJREFU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\nyMjIyMjIyMjIyMjIyMjIyMjI+P/H+4goW6CaW6G1AAAAAElFTkSuQmCC\n", 
       "prompt_number": 7, 
       "text": [
        "262537412640768743.99999999999925007259719818568888"
       ]
      }
     ], 
     "prompt_number": 7
    }, 
    {
     "cell_type": "markdown", 
     "source": [
      "<h2>Algebra<h2>"
     ]
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": [
      "eq = ((x+y)**2 * (x+1))", 
      "eq"
     ], 
     "language": "python", 
     "outputs": [
      {
       "latex": [
        "$$\\left(x + 1\\right) \\left(x + y\\right)^{2}$$"
       ], 
       "output_type": "pyout", 
       "png": "iVBORw0KGgoAAAANSUhEUgAAAHQAAAAbCAYAAACtOKuoAAAABHNCSVQICAgIfAhkiAAAA/tJREFU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\nAElFTkSuQmCC\n", 
       "prompt_number": 8, 
       "text": [
        "", 
        "               2", 
        "(x + 1)\u22c5(x + y) "
       ]
      }
     ], 
     "prompt_number": 8
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": [
      "expand(eq)"
     ], 
     "language": "python", 
     "outputs": [
      {
       "latex": [
        "$$x^{3} + 2 x^{2} y + x^{2} + x y^{2} + 2 x y + y^{2}$$"
       ], 
       "output_type": "pyout", 
       "png": "iVBORw0KGgoAAAANSUhEUgAAAQ0AAAAbCAYAAABm6to6AAAABHNCSVQICAgIfAhkiAAABQhJREFU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\n", 
       "prompt_number": 9, 
       "text": [
        "", 
        " 3      2      2      2            2", 
        "x  + 2\u22c5x \u22c5y + x  + x\u22c5y  + 2\u22c5x\u22c5y + y "
       ]
      }
     ], 
     "prompt_number": 9
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": [
      "a = 1/x + (x*sin(x) - 1)/x", 
      "a"
     ], 
     "language": "python", 
     "outputs": [
      {
       "latex": [
        "$$\\frac{x \\operatorname{sin}\\left(x\\right) -1}{x} + \\frac{1}{x}$$"
       ], 
       "output_type": "pyout", 
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFwAAAAbCAYAAADxsuiMAAAABHNCSVQICAgIfAhkiAAAAolJREFU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\n1YVPJWzHYFHp/atRo0aNquc3S/HNyBXE+1kAAAAASUVORK5CYII=\n", 
       "prompt_number": 10, 
       "text": [
        "", 
        "x\u22c5sin(x) - 1   1", 
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500", 
        "     x         x"
       ]
      }
     ], 
     "prompt_number": 10
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": [
      "simplify(a)"
     ], 
     "language": "python", 
     "outputs": [
      {
       "latex": [
        "$$\\operatorname{sin}\\left(x\\right)$$"
       ], 
       "output_type": "pyout", 
       "png": "iVBORw0KGgoAAAANSUhEUgAAADAAAAASCAYAAAAdZl26AAAABHNCSVQICAgIfAhkiAAAAnNJREFU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\n89kW1gAAAABJRU5ErkJggg==\n", 
       "prompt_number": 11, 
       "text": [
        "sin(x)"
       ]
      }
     ], 
     "prompt_number": 11
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": [
      "eq = Eq(x**3 + 2*x**2 + 4*x + 8, 0)", 
      "eq"
     ], 
     "language": "python", 
     "outputs": [
      {
       "latex": [
        "$$x^{3} + 2 x^{2} + 4 x + 8 = 0$$"
       ], 
       "output_type": "pyout", 
       "png": "iVBORw0KGgoAAAANSUhEUgAAALIAAAAZCAYAAACVUXRFAAAABHNCSVQICAgIfAhkiAAABSRJREFU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\nX/JnRRiZjdxGAAAAAElFTkSuQmCC\n", 
       "prompt_number": 12, 
       "text": [
        "", 
        " 3      2              ", 
        "x  + 2\u22c5x  + 4\u22c5x + 8 = 0"
       ]
      }
     ], 
     "prompt_number": 12
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": [
      "solve(eq, x)"
     ], 
     "language": "python", 
     "outputs": [
      {
       "output_type": "pyout", 
       "prompt_number": 13, 
       "text": [
        "[-2, -2\u22c5\u2148, 2\u22c5\u2148]"
       ]
      }
     ], 
     "prompt_number": 13
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": [
      "a, b = symbols('a b')", 
      "Sum(6*n**2 + 2**n, (n, a, b))"
     ], 
     "language": "python", 
     "outputs": [
      {
       "latex": [
        "$$\\sum_{n=a}^{b} \\left(2^{n} + 6 n^{2}\\right)$$"
       ], 
       "output_type": "pyout", 
       "png": "iVBORw0KGgoAAAANSUhEUgAAAHgAAAA4CAYAAAA2PDy+AAAABHNCSVQICAgIfAhkiAAABt1JREFU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\nTkSuQmCC\n", 
       "prompt_number": 14, 
       "text": [
        "", 
        "  b              ", 
        " ___             ", 
        " \u2572               ",
        "  \u2572   \u239b n      2\u239e",
        "  \u2571   \u239d2  + 6\u22c5n \u23a0",
        " \u2571               ",
        " \u203e\u203e\u203e             ",
        "n = a            "
       ]
      }
     ], 
     "prompt_number": 14
    }, 
    {
     "cell_type": "markdown", 
     "source": [
      "<h2>Calculus</h2>"
     ]
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": [
      "limit((sin(x)-x)/x**3, x, 0)"
     ], 
     "language": "python", 
     "outputs": [
      {
       "latex": [
        "$$- \\frac{1}{6}$$"
       ], 
       "output_type": "pyout", 
       "png": "iVBORw0KGgoAAAANSUhEUgAAABkAAAAeCAYAAADZ7LXbAAAABHNCSVQICAgIfAhkiAAAAPpJREFU\nSInt1aFKBFEUh/Gfq0WDCO6iguhoMwg2k6YFi7DRaBKMvoDZavYhDBaDLyBo2CfQJhgMgm6wjGHu\nLOOAgnIWXHa/cs+ce/n+3GHmXoaMOdxgpT4xFRRwhCbaaAQ5vyVHVm8OPHUc8j9DJoM8hzjBFtaw\niNsg96gxUak3cVHr/UQXx78N+St5gGNIiHhdJR1MYwazOA90o7hHzlKd4SMFhdHAE1YrvfXqgoib\ncQNLih3sYhtXeAhw9zlQfMY76Xker1guF0Scwm9pvE/jC3rYjwzpKnZSPdFzvAe4v3CNvVS38IyF\ncjLqP2niFI9Jfom7IPeYAfAJyood4uaM00cAAAAASUVORK5CYII=\n", 
       "prompt_number": 15, 
       "text": [
        "-1/6"
       ]
      }
     ], 
     "prompt_number": 15
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": [
      "(1/cos(x)).series(x, 0, 6)"
     ], 
     "language": "python", 
     "outputs": [
      {
       "latex": [
        "$$1 + \\frac{1}{2} x^{2} + \\frac{5}{24} x^{4} + \\operatorname{\\mathcal{O}}\\left(x^{6}\\right)$$"
       ], 
       "output_type": "pyout", 
       "png": "iVBORw0KGgoAAAANSUhEUgAAAMEAAAAfCAYAAABedqnDAAAABHNCSVQICAgIfAhkiAAABlFJREFU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\nVzAtLS0tLS0tLS0tLS0t+B+WXJxZxtS3TgAAAABJRU5ErkJggg==\n", 
       "prompt_number": 16, 
       "text": [
        "", 
        "     2      4        ", 
        "    x    5\u22c5x     \u239b 6\u239e", 
        "1 + \u2500\u2500 + \u2500\u2500\u2500\u2500 + O\u239dx \u23a0", 
        "    2     24         "
       ]
      }
     ], 
     "prompt_number": 16
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": [
      "diff(cos(x**2)**2 / (1+x), x)"
     ], 
     "language": "python", 
     "outputs": [
      {
       "latex": [
        "$$- 4 \\frac{x \\operatorname{sin}\\left(x^{2}\\right) \\operatorname{cos}\\left(x^{2}\\right)}{x + 1} - \\frac{\\operatorname{cos}^{2}\\left(x^{2}\\right)}{\\left(x + 1\\right)^{2}}$$"
       ], 
       "output_type": "pyout", 
       "png": "iVBORw0KGgoAAAANSUhEUgAAAMIAAAAoCAYAAACsPiXVAAAABHNCSVQICAgIfAhkiAAABhBJREFU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\nRCKRSCQWBq8BH7XPIH70GuoAAAAASUVORK5CYII=\n", 
       "prompt_number": 17, 
       "text": [
        "", 
        "                           2    ", 
        "         \u239b 2\u239e    \u239b 2\u239e       \u239b 2\u239e", 
        "  4\u22c5x\u22c5sin\u239dx \u23a0\u22c5cos\u239dx \u23a0   cos \u239dx \u23a0", 
        "- \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500", 
        "         x + 1                 2", 
        "                        (x + 1) "
       ]
      }
     ], 
     "prompt_number": 17
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": [
      "integrate(x**2 * cos(x), (x, 0, pi/2))"
     ], 
     "language": "python", 
     "outputs": [
      {
       "latex": [
        "$$-2 + \\frac{1}{4} \\pi^{2}$$"
       ], 
       "output_type": "pyout", 
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEwAAAAfCAYAAABNjStyAAAABHNCSVQICAgIfAhkiAAAArVJREFU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\nTp1YfAdxIoFGVphKVAAAAABJRU5ErkJggg==\n", 
       "prompt_number": 18, 
       "text": [
        "", 
        "      2", 
        "     \u03c0 ", 
        "-2 + \u2500\u2500", 
        "     4 "
       ]
      }
     ], 
     "prompt_number": 18
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": [
      "eqn = Eq(Derivative(f(x),x,x) + 9*f(x), 1)", 
      "display(eqn)", 
      "dsolve(eqn, f(x))"
     ], 
     "language": "python", 
     "outputs": [
      {
       "latex": [
        "$$9 \\operatorname{f}\\left(x\\right) + \\frac{\\partial^{2}}{\\partial^{2} x}  \\operatorname{f}\\left(x\\right) = 1$$"
       ], 
       "output_type": "display_data", 
       "png": "iVBORw0KGgoAAAANSUhEUgAAAI4AAAAnCAYAAADZ7nAuAAAABHNCSVQICAgIfAhkiAAABX5JREFU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\n", 
       "text": [
        "", 
        "           2          ", 
        "          d           ", 
        "9\u22c5f(x) + \u2500\u2500\u2500(f(x)) = 1", 
        "           2          ", 
        "         dx           "
       ]
      }, 
      {
       "latex": [
        "$$\\operatorname{f}\\left(x\\right) = C_{1} \\operatorname{sin}\\left(3 x\\right) + C_{2} \\operatorname{cos}\\left(3 x\\right) + \\frac{1}{9}$$"
       ], 
       "output_type": "pyout", 
       "png": "iVBORw0KGgoAAAANSUhEUgAAAQUAAAAeCAYAAAAl8At9AAAABHNCSVQICAgIfAhkiAAACFtJREFU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\n", 
       "prompt_number": 19, 
       "text": [
        "f(x) = C\u2081\u22c5sin(3\u22c5x) + C\u2082\u22c5cos(3\u22c5x) + 1/9"
       ]
      }
     ], 
     "prompt_number": 19
    }, 
    {
     "cell_type": "markdown", 
     "source": [
      "# Illustrating Taylor series", 
      "", 
      "We will define a function to compute the Taylor series expansions of a symbolically defined expression at", 
      "various orders and visualize all the approximations together with the original function"
     ]
    }, 
    {
     "cell_type": "code", 
     "collapsed": true, 
     "input": [
      "# You can change the default figure size to be a bit larger if you want,", 
      "# uncomment the next line for that:", 
      "#plt.rc('figure', figsize=(10, 6))"
     ], 
     "language": "python", 
     "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):", 
      "    \"\"\"Plot the Taylor series approximations to a function at various orders.", 
      "", 
      "    Parameters", 
      "    ----------", 
      "    func : a sympy function", 
      "    x0 : float", 
      "      Origin of the Taylor series expansion.  If not given, x0=xrange[0].", 
      "    orders : list", 
      "      List of integers with the orders of Taylor series to show.  Default is (2, 4).", 
      "    xrange : 2-tuple or array.", 
      "      Either an (xmin, xmax) tuple indicating the x range for the plot (default is (0, 1)),", 
      "      or the actual array of values to use.", 
      "    yrange : 2-tuple", 
      "      (ymin, ymax) tuple indicating the y range for the plot.  If not given,", 
      "      the full range of values will be automatically used. ", 
      "    npts : int", 
      "      Number of points to sample the x range with.  Default is 200.", 
      "    \"\"\"", 
      "    if not callable(func):", 
      "        raise ValueError('func must be callable')", 
      "    if isinstance(xrange, (list, tuple)):", 
      "        x = np.linspace(float(xrange[0]), float(xrange[1]), npts)", 
      "    else:", 
      "        x = xrange", 
      "    if x0 is None: x0 = x[0]", 
      "    xs = sym.Symbol('x')", 
      "    # Make a numpy-callable form of the original function for plotting", 
      "    fx = func(xs)", 
      "    f = sym.lambdify(xs, fx, modules=['numpy'])", 
      "    # We could use latex(fx) instead of str(), but matploblib gets confused", 
      "    # with some of the (valid) latex constructs sympy emits.  So we play it safe.", 
      "    plot(x, f(x), label=str(fx), lw=2)", 
      "    # Build the Taylor approximations, plotting as we go", 
      "    apps = {}", 
      "    for order in orders:", 
      "        app = fx.series(xs, x0, n=order).removeO()", 
      "        apps[order] = app", 
      "        # Must be careful here: if the approximation is a constant, we can't", 
      "        # blindly use lambdify as it won't do the right thing.  In that case, ", 
      "        # evaluate the number as a float and fill the y array with that value.", 
      "        if isinstance(app, sym.numbers.Number):", 
      "            y = np.zeros_like(x)", 
      "            y.fill(app.evalf())", 
      "        else:", 
      "            fa = sym.lambdify(xs, app, modules=['numpy'])", 
      "            y = fa(x)", 
      "        tex = sym.latex(app).replace('$', '')", 
      "        plot(x, y, label=r'$n=%s:\\, %s$' % (order, tex) )", 
      "        ", 
      "    # Plot refinements", 
      "    if yrange is not None:", 
      "        plt.ylim(*yrange)", 
      "    grid()", 
      "    legend(loc='best').get_frame().set_alpha(0.8)"
     ], 
     "language": "python", 
     "outputs": [], 
     "prompt_number": 21
    }, 
    {
     "cell_type": "markdown", 
     "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", 
     "outputs": [
      {
       "output_type": "display_data", 
       "png": 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MwoMQYO5cIDSUbroyMlKom7t3aZROTAzQqBGwaxcwaBB1dx1NTcUPT5/Crn59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\nfgk2a5wJG34AAAAASUVORK5CYII=\n"
      }
     ], 
     "prompt_number": 22
    }, 
    {
     "cell_type": "code", 
     "collapsed": false, 
     "input": [
      "plot_taylor_approximations(cos, 0, [2, 4, 6], (0, 2*pi), (-2,2))"
     ], 
     "language": "python", 
     "outputs": [
      {
       "output_type": "display_data", 
       "png": 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4q3ZtmpQtq7YUq+fyZejXD5KTlTN6c+eqrUhSGMhcN9bOqlXKqamjR3Ot2SaE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\ndow29M7OzjRr1kyvvsUhK6WW/cR6ax8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      }
     ], 
     "prompt_number": 23
    }, 
    {
     "cell_type": "markdown", 
     "source": [
      "This shows easily how a Taylor series is useless beyond its convergence radius, illustrated by ", 
      "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", 
      "# a callable function, the simplest way is to use the Python lambda construct.", 
      "plot_taylor_approximations(lambda x: 1/cos(x), 0, [2,4,6], (0, 2*pi), (-5,5))"
     ], 
     "language": "python", 
     "outputs": [
      {
       "output_type": "display_data", 
       "png": 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GloMfwPXLDcs9QJa1A+zrFwIP4BwOh8MoPICbQaPRyC3BKrh+eWF5LjLL2gH2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\nIQwG86sqpdRfVsvmzZslK7cy5H7+1mK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      }
     ], 
     "prompt_number": 24
    }
   ]
  }
 ]
}