diff --git a/docs/examples/notebooks/sympy.ipynb b/docs/examples/notebooks/sympy.ipynb index fc5e434..35cc91d 100644 --- a/docs/examples/notebooks/sympy.ipynb +++ b/docs/examples/notebooks/sympy.ipynb @@ -1,401 +1,625 @@ { - "metadata": { - "name": "sympy" + "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:" + ] }, - "nbformat": 2, - "worksheets": [ - { - "cells": [ - { - "cell_type": "markdown", - "source": [ - "

SymPy: Open Source Symbolic Mathematics

" - ] - }, - { - "cell_type": "code", - "collapsed": true, - "input": [ - "%load_ext sympyprinting" - ], - "language": "python", - "outputs": [], - "prompt_number": 1 - }, - { - "cell_type": "code", - "collapsed": true, - "input": [ - "from __future__ import division", - "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": [], - "prompt_number": 2 - }, - { - "cell_type": "markdown", - "source": [ - "

Elementary operations

" - ] - }, - { - "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": "iVBORw0KGgoAAAANSUhEUgAAAGkAAAApCAYAAAA22gdWAAAABHNCSVQICAgIfAhkiAAABGRJREFU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\n", - 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Algebra

" - ] - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "((x+y)**2 * (x+1)).expand()" - ], - "language": "python", - "outputs": [ - { - "latex": [ - "$$x^{3} + 2 x^{2} y + x^{2} + x y^{2} + 2 x y + y^{2}$$" - ], - "output_type": "pyout", - "png": "iVBORw0KGgoAAAANSUhEUgAAAQ0AAAAbCAYAAABm6to6AAAABHNCSVQICAgIfAhkiAAABXNJREFU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\nOFM1l1Xqnqo5TyQSiUQikUgkEv8v/gNI9gkr+UdTSgAAAABJRU5ErkJggg==\n", - "prompt_number": 8, - "text": [ - "3 2 2 2 2", - "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", - "display(a)", - "simplify(a)" - ], - "language": "python", - "outputs": [ - { - "latex": [ - "$$\\frac{x \\operatorname{sin}\\left(x\\right) -1}{x} + \\frac{1}{x}$$" - ], - "output_type": "display_data", - "png": "iVBORw0KGgoAAAANSUhEUgAAAHUAAAAeCAYAAAASN1ElAAAABHNCSVQICAgIfAhkiAAAAyFJREFU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\noZik/6S2SgAAAABJRU5ErkJggg==\n", - "text": [ - "x\u22c5sin(x) - 1 1", - "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500", - " x x" - ] - }, - { - "latex": [ - "$$\\operatorname{sin}\\left(x\\right)$$" - ], - "output_type": "pyout", - "png": "iVBORw0KGgoAAAANSUhEUgAAADAAAAASCAYAAAAdZl26AAAABHNCSVQICAgIfAhkiAAAAnVJREFU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\nfj0LzBdrAAAAAElFTkSuQmCC\n", - "prompt_number": 9, - "text": [ - "sin(x)" - ] - } - ], - "prompt_number": 9 - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "solve(Eq(x**3 + 2*x**2 + 4*x + 8, 0), x)" - ], - "language": "python", - "outputs": [ - { - "output_type": "pyout", - "prompt_number": 10, - "text": [ - "[-2\u22c5\u2148, 2\u22c5\u2148, -2]" - ] - } - ], - "prompt_number": 10 - }, - { - "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": "iVBORw0KGgoAAAANSUhEUgAAAHgAAAA5CAYAAAD9YO8bAAAABHNCSVQICAgIfAhkiAAAB5NJREFU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\nCnsQh3MAAAAASUVORK5CYII=\n", - "prompt_number": 11, - "text": [ - "b ", - " ___ ", - " \\ ` ", - " \\ \u239b n 2\u239e", - " / \u239d2 + 6\u22c5n \u23a0", - " /__, ", - "n = a" - ] - } - ], - "prompt_number": 11 - }, - { - "cell_type": "markdown", - "source": [ - "

Calculus

" - ] - }, - { - "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": "iVBORw0KGgoAAAANSUhEUgAAACYAAAAiCAYAAAAzrKu4AAAABHNCSVQICAgIfAhkiAAAASpJREFU\nWIXt178rRlEYwPEPXikZLH4NShTKoAyymw3KPyJlsBssVqvBYDIwCcl/gHpXr8liEYWN4bx0k3Ku\nzsmr937rDGd47vPtuc+5zz20OYN/LfCVUWxjp2xgLb3LJzOYxxCeygbnFKs31xT6ywZ3JtdJRCVW\nlrYW62iuUnRlEPlgHOtYwLBwOu9wnzFnRetRuim/4S3BM/4PxYpNYlV8FRvYSm7UpCjRg2nxYg+4\nTS1U8VtSnMqfmBP6tw97eI0Jyj0rNzGCfQxgLXO+KBZxUNgvC3Mzipyvchc3uMYYrnCeMV80dZyi\nVyjAMVZig3P2WDcu8SyMrRNsxAbnFGvgsbB/Ef7LosgpdoiJwn4YFxnzRVPDEZaEi++ZcPltGWaF\nz0TLXnwq2pN3yugjnSzSytcAAAAASUVORK5CYII=\n", - "prompt_number": 12, - "text": [ - "-1/6" - ] - } - ], - "prompt_number": 12 - }, - { - "cell_type": "code", - "collapsed": false, - "input": [ - "(1/cos(x)).series(x, 0, 6)" - ], - "language": "python", - "outputs": [ - { - "latex": [ - "$$1 + \\frac{1}{2} x^{2} + \\frac{5}{24} x^{4} + \\operatorname{\\mathcal{O}}\\left(x^{6}\\right)$$" - 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"prompt_number": 13, - "text": [ - "2 4 ", - " x 5\u22c5x ", - "1 + \u2500\u2500 + \u2500\u2500\u2500\u2500 + O(x**6)", - " 2 24" - ] - } - ], - "prompt_number": 13 - }, - { - "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": "iVBORw0KGgoAAAANSUhEUgAAANsAAAArCAYAAADxEf1ZAAAABHNCSVQICAgIfAhkiAAABtFJREFU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\n", - "prompt_number": 14, - "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": 14 - }, - { - "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": "iVBORw0KGgoAAAANSUhEUgAAAFgAAAAiCAYAAADbLB6TAAAABHNCSVQICAgIfAhkiAAAA1BJREFU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\nAElFTkSuQmCC\n", - "prompt_number": 15, - "text": [ - "2", - " \u03c0 ", - "-2 + \u2500\u2500", - " 4" - ] - } - ], - "prompt_number": 15 - }, - { - "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": "iVBORw0KGgoAAAANSUhEUgAAAJoAAAAqCAYAAABLNUJiAAAABHNCSVQICAgIfAhkiAAABadJREFU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\nlH4rFJOrgoKCgoKCgoKCluFfJCE9lRbmMyMAAAAASUVORK5CYII=\n", - "text": [ - "2 ", - " d ", - "9\u22c5f(x) + \u2500\u2500\u2500(f(x)) = 1", - " 2 ", - " dx" - ] - }, - { - "latex": [ - "$$\\operatorname{f}\\left(x\\right) = C_{1} \\operatorname{cos}\\left(3 x\\right) + C_{2} \\operatorname{sin}\\left(3 x\\right) + \\frac{1}{9}$$" - ], - "output_type": "pyout", - "png": "iVBORw0KGgoAAAANSUhEUgAAAREAAAAiCAYAAAB4De05AAAABHNCSVQICAgIfAhkiAAACTtJREFU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\nHF+toMfj8Xg8Ho/H4/F4PB7Pns7/AW58iu5gsm0fAAAAAElFTkSuQmCC\n", - "prompt_number": 16, - "text": [ - "f(x) = C\u2081\u22c5cos(3\u22c5x) + C\u2082\u22c5sin(3\u22c5x) + 1/9" - ] - } - ], - "prompt_number": 16 - } - ] - } - ] + { + "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": [ + "

Elementary operations

" + ] + }, + { + "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": 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"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": 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Calculus

" + ] + }, + { + "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": 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+ "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": 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+ "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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\nnD9fO428zKjZfSQzPJKiEEL28/FC+9a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4q3ZtmpQtq7YUq+fyZejXD5KTlTN6c+eqrUhSGMhcN9bOqlXKqamjR3Ot2SaE\nUqR59WpB2THR1B0ay0EPN2qWKh5VIY48fMgrZ89y3sMDe5nuIF/i4qBLF7h0CXr1UrJRFpOPSZFG\n5ropCgwbpuTD+fnnXN+2sVFS5Tz3nA0pixsRt7EG3Y+d5FYxWNnrhGBcZCQLGjeWRr4AkpOV066X\nLikVojZulEa+OCFX9GYiMDAwO+bV7Jw8qZxNv3ABKlXKtUtiIjz7LBw7BrU/iKb8gNv85d6GBmXK\nFHh7i2q3ID/dvMmKmzf59OFDnn32WbXlZOPl5UV8fLze/VNTUymjx/8nU4iNVYx9iRJK4RBzfi8W\nhn5LohX9FStWZN++fU+9ro/ttPiBKYkZcHNTHKvz5sGCBbl2sbdXqgB16QJRCx1pVsKO7oSxu00b\nmhfBY463Hj1i5pUr/OnqSvzx42rLyUF8fDzHjh3Tu39CQgIV/pXywpwIAVevKobezk4pbmPurQxL\n6i8MtKK/ffv2Rl8rV/Ra4cYNcHVVluyNGuXZLSpKMfZ37kCnWTeJ7nUFf1dX2mrgg6wvQghePHMG\nN3t75uYzF2rRvn17gwy9Jbl5E65fV1x8zZrlSKEk0Rh5fa6kj74oUacOTJwIH32Ub7cmTZQEmOXL\nw9FZtel4pCl9Tp9m/4MHhSTU8iy/dYvraWn8X8OGakuxamJjFSMPytpAGvniizT0ZuLf+TIsxpQp\ncPiw8pMP7drBpk2KP3bL1Oq8fK4Fr0RE4Bcbm2v/QtFuJq6kpPDR5cusdnGh1OMoJC3pz42EhASz\n3/P+ffj7ccnhBg3AkmVzLaG/MNG6fn2Qhl5LlCun+OknTVLK/+RD795KUWeAH0dVZuLN1oyLjGTh\n1auadZGl63S8ef48H9SvT6vy5dWWY7XExyvx8qA8CNaoYbmxfvnll+wU5PmRkpJCkyZNDNqklpgP\naejNRKFFrbz2GmRmwvr1BXYdNuyfvdtZ/63AVyltWXvnDqMuXuTRv74otBJxMyUqCns7OybXr5/j\nda3ozwtzbgQmJUFkpLIJW6MG1K5ttls/hRCChQsXMmXKlAL7li1blkGDBvHjjz9aTpCRaGEj1lSk\nodcatrawaJHiq09JKbD7++/DhAlKzvFRA0qz2M6d248e0ef0ae6npxeCYPPw882b7L5/n7UtWmAn\nyx7lSmqqEiev0ymumvr1LVshatu2bVStWjVHvYn8GDNmDN9++y0ZGRmWEyXJFWnozUSh+omfeQba\nt4fvvy/ZBdqRAAAgAElEQVSwq42N8r3w6qtKFaFBfe34xr4Vbezt6XjiBGeSkqzex3344UOmXb7M\nllatcCjxdESwtesvCHP4iB89gosXISMDKlYER8fcjXxcXByLFi3C1dWVatWqMW7cOAC2bt1Kr169\ncHV1ZcmSJSQnJ2dfM3nyZJydnalcuTIeHh7EPt7r+euvv+jcuXO2/t9++43GjRtnt3fs2EHt2rW5\ne/cuoORO1+l0REREmPz3mhPpo5dYL599Bl98AXpE09jawurVyoGqW7eg3/M2zHBwYmbDhjx78iR7\n7ltvHdqLycm8HBHBSmfnInMewMYm50/FihWeek3fH1CM+6VLirEvX16JvMolWwYAI0eO5OTJk/j5\n+XHjxg2GDBlCQEAA48aN48MPP2TTpk1s3LiRhQsXAuDv709YWBiHDh3i3r17LF26NPtw0YULF2jS\npEn2vQcPHkyXLl0YP348d+/e5a233uLnn3+matWq2X2cnJw4e/asZSZWkjeWyKZmDFYkRTuMGCHE\n9Ol6d3/wQIjWrZWshZ06CZGUJMSphATR9OhRMfrCBZGUS6ETNfk7JUU0PHJE/HTjhtpSDKJdu3b5\nvq940M3zk5EhxNmzSkbK8HAh8stW/eDBA1GuXDkR90Tq6/Hjx4tp06Zlt/fs2SNat24thBBiy5Yt\nom3btiI0NPSp+7Vo0UL4+/s/NUaDBg2Eq6ureOedd566ZvDgweLzzz/Pd34kuZPX50of2ylX9Frm\nk09gyRLlVIweVKoEO3ZAw4ZKjrTBg6FFGXtC27UjISMD92PHCLWSqIiolBSeOXmSyfXqMdKSO4oq\nYC4zn5mpbLwmJSnpkJo2Vf6bF4cOHaJhw4Y5VtgAhw8fpl27dtntdu3aER4eTkJCAv369WPEiBEM\nHz6cxo0b88UXX6B7vJHfsGFDrmcF6j+mUqVKvPzyy5w5cybXTdpr167RUJ5/KHSkoTcTqviJGzSA\nN9+ETz/V+5I6dWDnTmWzbvt2JVXt8aCDrGnRgjmNGtEvPJxZ0dGkFRC+aUmOJyTQ4+RJpjdowPh6\n9QrsXxx99DqdYuQTEhTj3rw5lC6d/zVdunTh77//zvaZZ9G1a9ccJy6PHTuGq6srFSpUwM7OjrFj\nxxIeHo6/vz8//vgjO3fuBMDFxYWoqKgc+k+ePMmKFSv473//m+3//zeRkZG4uLgY/PdaEumjl1g/\n06YpoZZRUXpf4uysnJ61t4d165QQzMxMGFyjBmHt23MiIYE2x47xlwq++w137tDn9Gm+b9qUt+vU\nKfTxtYBOp8TJx8crh+KaNQN9cnI5ODjQq1cvJk+eTGRkJKmpqRw+fJj+/fuzbt069u3bR2RkJF98\n8QUvvfQSoHyJhoeHk5mZib29Pba2ttjb2wNK8rbg4ODs+6empvL6668zf/58li9fzvXr13OEU0ZH\nR2NjY6N3lI7EjJjbj2QsViRFe8yeLcRrrxl82YEDQpQvrzgCRo4UIjPzn/e2xMaKhkeOiCEREYVS\nuSkpI0OMvXhRND5yRByPj7f4eJakIB+9Keh0QkRGKj75EyeUfRZDiIuLEwsXLhTNmzcX1apVExMm\nTBA6nU5s2rRJ/Oc//xEtW7YU33//vUh6fON169aJ5s2bC3t7e+Hu7i7mzp2b436urq4i4nFB9okT\nJ4q+fftmv3fq1ClRpUoVERkZKYQQ4v333xcLFiww4a8v3pjio5dJzYoCCQmKg3bXLmjTxqBLAwOh\nb18lJP+dd2Dx4n+iOZIyM/kqJobvrl9nYLVqzGzYUK+0x4ay59493r10iY4VKvBDs2a5hlBqCUsl\nNRMCoqPh7l0lqqZ5cyXKRk3WrFnDgQMHWLJkSb79UlNTadmyJSdPniwWB5QsgSpJzRISEujfvz8N\nGjRgwIABJCYm5trP0dGR1q1b4+7ujoeHh7HDWT2q+okrVIDp02HGDIMv9fSEuXMDKV1a2dedMEEx\nKADl7ez42NGRCx4eVC1ZkjbHjjHk7FmCHjwwy5dycHw8/U6f5t1Ll/jGyYk1LVoYZeSLg48+K91w\nlpFv2lR9Iw/w2muv8cUXXxTYr0yZMkRFRVmlkZc++nz48ccfadCgAZcuXaJevXp5fqPb2NgQGBhI\nWFgYISEhRguVFMDo0XDqFBgxx+3awebNSsUhX1+YOvUfYw9QtWRJ5jduzJVOnehasSKjL16kZWgo\nMy5f5mh8PDoDjH7so0f8fPMmXU+c4NWICJ6vWpWIDh144YlIEMk/CKEkKIuNVZ62nJxkJkqJYRjt\nunn55ZeZOXMmbm5unDhxgvnz57Nhw4an+jVq1Ihjx449FdL1lBDpujGdxYuV6iP+/kZdvn07DByo\npEuYPBm+/DL305VCCI7Ex7P17l22373LrUePcLO3x7V8eVqUK0elEiUob2dHaVtb4jMyuJ6Wxvnk\nZI7GxxOZkkLPypV5o1Yt+lapQsm8TvZoGHO6bv7trsky8nkUGZMUcUxx3RjtDA0NDcXZ2RkAZ2fn\nPFfrNjY2eHl50ahRI0aMGMGLL76Y5z19fHxwdHQElAgBNze37IRVWY/nsp1Pu2lTPMPDITiYwMd5\ncAy53t4efv/dk1dfhUWLAomMBD8/T2xtc/a3sbHhUVgYfYDPPT25npbGr7t2cTklhSNt2pCQmcnV\no0dJF4KGnTpRu1QpbE+dYni5crz1/POUsbUlMDCQQ2rPlwXbWe6ALFeFMW0hIC6uAvfugY1NAvXq\nQaVKxt9PtrXdTk1NBZTP2sqVKwGy7WWB5LdT27NnT9GqVaunfrZs2SLq168vUlJShBBCJCUliQYN\nGuR6jxuPTzWePXtWNGnSRNy8edPonWNrJiAgQG0JCosXC/H88wZd8qR2f38hSpdWonF8fJTTl9aM\n1cz9YwyNuonPJcpIpxMiKkqJrjl+XAhrDkTKTb+W0Ip+i52M3bNnD+Hh4U/9vPjii3To0IFz584B\ncO7cOTp06JDrPWo/PtXo4uLCiy++yLZt2/T7BpIYx4gREBGhHH01kr59Fe9PuXKwcqWSGVlDiS41\nj06nHIu4d0/ZeJUlACWmYrSDtGPHjixfvpyUlBSWL19Op06dnuqTnJyc/QgSGxvLrl276NOnj/Fq\nrRiryYleurQSgTNrlt6X5Kb9P/9RojUrVIDffoOXX9YrK7IqWM3cG8m/I1GyEpQ9eKAU827WTDnY\nZs1YYySNIWhdvz4Ybejfffddrl69SvPmzbl+/TrvvPMOADdu3KBfv34A3Lp1i+7du+Pm5saQIUOY\nMmUK9Z8oGiGxAMOHw7lzcOSISbfp1g3++gsqV4atW5WqVVac6FLzpKcrqYaz0ho4O1u/kZdoA3lg\nykwEBgZa18ryf/+DP/5QluUFUJD2iAjo0weuXYOWLZVcOXqkoCk0rG3uDY26SUhIoFSpCly8CGlp\nykNZs2YF566xFhISEjS9KtaKflUOTEmsHB8fuHDB5FU9KMb98GFo0UIx+l26gEwpbj5SU+H8ecXI\nly2rrOS1YuQl2kAaejNhTStKQDn99NFHSjHxAtBHe/36cOAAdO0KMTGKWycoyAw6zYDVzb0B3L8P\nMTEVSE9X9kOaN88/1bC18euvv/Lll18ybNgwduzYYdQ9Dh48qFeBcUuhhdW8qUhDX5Tx8YHjx5UT\ns2agShXYswf691cMVM+e8PPPZrl1sUMIuH1bia7R6aBqVSWtgZbS/ERGRnL//n1mz57N119/zeuv\nv86dO3cMuseiRYvw9fXl4cOHFlIpAWnozYZV5lspU0Y54vr55/l2M0R72bKK63/yZGXz8K23lN/V\nrPdslXOfD1l5a2JilHa1agk4OuZd/s9aiYiIYOHChSQkJFCtWjUaN26cI22xPkyePJm+fftaSKF+\nFIdcNxpaP0iMYvRoaNxYidlr2tQst7Szg6++Unz2774LX3+tBPmsXy+P5xdEerqSSz4hQUlp4Oio\neNlySzWhFpcvX2bZsmV5vt+pUyf69+9P3759s901Qghu3rz5VFSdq6srq1atom3btnneT8tBGFpB\nRt0UB2bNguvXIZ9/vMYSFASDBkFcnOJf3rABXF3NPoymyCs6IjFRcdWkp0MHf/NYdvGJYf9mjh8/\nTkBAABkZGbRq1QqdTsfmzZtZvny5STq2b9/OTz/9xObNm3O8vnnzZnr27JldrCQ3Vq1aRWBgICtW\nrDBJQ1FHlVw3Eg0xbpyymv/kE7PHRT7zjJIws39/CA+Hjh3hhx+UUH6JghBK5smYGOX38uUhbZqg\nVKnC1xIbG0vbtm3x9fXlo48+QgjBpEmTTLrngwcPWLFiBb/++utT7w0YMKDA6+UCz/JIQ28mrC2W\nOwdVqyqW96uvFD/LE5iqvVEjJePCe+/BihVKFob9+xWDXxg506157jMzlRTD9+4p7Ro1lO/af/vj\nCzOOu0+fPkybNo1hw4YBcOTIkafSl+jrugHFSM+ZM4effvoJe3t7/v77b4OLf9uo7LfSShy9KUhD\nX1yYPFnxqUyfDtWrm/325crB8uXKCn/MGFi1CkJDFb99cXXlJCTAlSvw6JFi2Bs2VL5z1SYgIICP\nPvoIgNWrVzNq1Ch27tyZnZ6kcePGzJ8/X697+fr6MmDAANLS0ggKCkIIkcPQ+/n50bt3b8rn840v\nV/SWR2P7/NaLta4os6lbF155Bb777qm3zKndx0dx5Tg7K4eq2rWDzz6zbFSOtc29EMop4gsXFCNf\nrhy4uORt5AtzNZmcnIyDgwOVHu+a16pVi9u3b1OzZk2D73Xw4EEmTZqEp6cnderU4dlnn8XJySlH\nnzlz5hCVT+H6b775hiVLlrBnzx5mzJhBfHy8wTpMpaiv5kFuxhYvoqIUJ3p0tMWTqCQmwvvvK+UJ\nAdq3VzJhtmxp0WFV58wZeO659mzZomya1aoFdepoL3RSYn3IFAhWgCZiuZs0gWefVXws/8IS2u3t\n4ccflQNWDRrAsWPQti3Mn6+scs2JNcx9SoriFXN3V/6+UqWUKKQn/fG5ofU4bqnf+pGGvrgxdaqy\nIVtIJ5x69lSicUaNUgzg9OnQurXyBVAUEAL+/FPZh5g/X9l8rVBBeXIpBh4BiUaQht5MWJufOE86\ndlSWmZs2Zb9kae0VKyrJNHfvVrIyXrigpDweOFD53VTUmvvwcHjuOejXT/GKtWoFhw4pqSLs7PS/\nj9Z9xFK/9SMNfXFk6lSl8nch74n06qUYx88/V8Iu/fyUle/bbyvnubRCVJQSrermpjyZVKqkRK6e\nOAGdO6utTiJ5GmnozYQ1+In1xttbKWF04ABQuNpLlYIPP1QKbLz9tvLasmXK9sE770BkpOH3LCz9\nFy8qZwSaN1c2lm1slLMDkZFK9KqxWSe17iOW+q0faeiLI7a2MGWKsqpXiTp1YOlSJb/9K68o/vul\nSxUj+uqrEBBQ6A8cuSIE7N0LL7ygaMs6pT98uOJ28vWFatXU1SiRFIQMryyupKQoGbX271eC3lXm\n/HlYuBB++eWffeLmzWHkSBg8WIncKUyuXIHVqxU9WWHgZcrA668rTyRPhIvnwNAKUxKJPqgSXrlh\nwwZatmyJnZ0dJ06cyLNfUFAQLi4uNG3aFF9fX2OHk5ibsmWVI6yLFqmtBFC+a5YvVwzsxx8rK/4L\nF+CDD5QTpZ07Kw8gp09bZqUvhJK2f+5c8PBQEn7OmqUY+bp14dNPlVw1y5blb+QlEqtEGMm5c+fE\nhQsXhKenpzh+/Hie/dzc3MT+/ftFdHS0aN68uYiNjc21nwlSrIKAgAC1JRhObKwQlSuLgE2b1Fby\nFOnpQmzeLMSrrwpRrpwQiilWfmrUUF5fuFCIvXuF2Lw5QOh0+t9bpxPi2jUhdu8WYsECIV58UYhq\n1XKOUbasEK+9JsSuXUJkZBimvV27dgb1j4+PN2wAK0PqLxzy+lzpYzuNznXjrMfjflbVmGeeeQaA\n3r17ExwcTL9+/YwdVmJOqlVT/CKbN8NLL6mtJgclSigZMfv3h6Qk8PdX4tX37IEbN+D335WfLMqX\nVzxRdesqUTAVKyoPLenpyk9yslLR6c4dZWWe20n72rWVUElvbyX+v1y5QvtzJXnw4MEDNmzYwJ07\nd5gxY4Ze11y6dIkzZ85w+vRpvL29882FX1ywaFKz0NDQHF8ILVq04OjRo3kaeh8fHxwdHQFwcHDA\nzc0tO0Y6K7LCWttZr1mLHr3b48bh6eVF4J49ULKk+npyaZcvDzVqBOLjAytWeHL+PKxYEcilS3Dr\nlidnz3oSHx9IRARERCjXQ+Dj/+berlgxkIYNoUsXT7p0gRIlAqldG5591jz6syI5smK082tXqFDB\noP7W1rakfgcHB3r37s3SpUtzZJnM7/rt27fj5ubGqFGjmDp1KmvXri0S85+amgoon7WVK1cCZNvL\ngsh3M7ZXr17cunXrqdfnzZuHt7c3AM8++yxfffVVrt+ae/fu5eeff2bdunUALFmyhOvXrzN37tyn\nhcjNWPXo3RuGDVN+NMqDB0oKnxs3lKyR8fHKfnPJkspPmTJQs6byU6eOZSNliuNmbEhICH/99RfT\npk0z+73//vtvVq5cySeffGLQdWfPnmXNmjV89tlnJo1/8OBBNm7cyDfffGPSfUzFYoVH9ph4Tr1D\nhw68//772e2IiIjsVKhFDWvOiV4QgV5eeH77rRJSYk017fQka+7d3JRDTFpD6/nQHz58yMcff0yX\nLl3UlpIDPz8/vdw9X331FVOmTMn1vUWLFhEcHEw5jfvxzBJHn9e3SVYq1KCgIKKjo9mzZw8dO3Y0\nx5ASc+LhAQ8fwpEjaiuRaBA/Pz969uxpsSdyY+67detWxo0bx9WrVwvse/fu3Tzfs4bi5ebAaB+9\nn58f48ePJy4ujn79+uHu7s6OHTu4ceMGo0aNwt/fH1DyTY8ePZr09HTGjx9PtSJ6ukSrq3kATy8v\npdzgt9+Cla3K9EHLcw/Wl2vFkApTsbGx2NvbY2NjQ1JS0lN99SkOnh8JCQmsX7+ekJAQTp8+TevW\nrQu8xs/Pj3nz5uHr60uPHj2YOXNmvv1Lly6d7/tFwaUsD0xJFOLjlbCV06fNXle2uGHNPnpzFwdf\nunQpb7/9NqtXryY6OvopP7o+xcGNZevWrdjZ2REUFESzZs0ICAhg5syZekUE/pvZs2fn6/+3luLl\nsji4FaBpH32W9mHDYPFimDdPbUkGoeW5h8L10ZuzOPjRo0fp2LEjiYmJeRqagoqDL1y4kJSUlFzf\ne/PNN/OMKrl69SotWrTAycmJmTNnMm3aNGrWrEkDPY5Qnzt3jtWrV2e39+/fnx3RAtC9e/cc7pqi\nsACVhl7yD++9B127wv/9nxKELrEc/9r0NsnEG2iEzFkcPDQ0lOTkZNLS0jh27BgpKSls3bqVF198\nUW89H3zwQb7v2+ZStcXGxobMzEwAbt++TaVKlXBwcOCFF17Qa0wXF5ccNXGnT5/OvHwWN2oXLzcH\n0tCbCS2vKLO1N22qbMyuXaskmdEImpx7FVeJ5ioOPm7cuOzfZ82ahY2NzVNGXp/i4Pmh0+lyff38\n+fOkpqYSFhaWfSDzzz//NGrjtDj46GX2SklOxo9XNmWLwIdb8jTmLA6exe+//86GDRvYuHEjGzZs\nyPFeQcXBC+LSpUts2rSJ2bNn58iptXv3bvz8/NDpdKSmprJt2zbq1q1r9Dh5YQ3Fy82CqfkXzIUV\nSTEKTea6eUwO7ZmZQjRrJsSBA6rpMRRrm3uZ68Z8LFq0SAQHB4v4+HgxdOhQi4wxZ84ci9zX3KiS\n60ZSRLG1hXffVSp7d+umthpJMSdro/js2bM0atTIImNMnDjRIve1JmR4peRp7t+HRo2Ukko1aqit\nRnNYc3ilVvnss8+YNGmS5k+omoIq+eglRZjKlWHQIPj5Z7WVSCQGnXKV5I409GZCUzVjnyBX7WPG\nwJIl8DiMzZrR8tyD9muWWlK/n58fc+fOZdCgQWzcuNEiY2h9/vVB+ugludOunZLqcccOpWCqRKIC\nL730Ei9ZWa0ELSJX9GZCk7Hcj8lT+5gxyklZK0fLcw/Wl+vGUKR+60caekneDB4MISFw+bLaSiQS\niQlIQ28mtOwnzlN72bLg4wNLlxamHIPR8tyD9n3EUr/1Iw29JH9Gj4YVK+BfSZ8kEom2kIbeTGjZ\nT5yv9qZNwd0dnjjabk1oee5B+z5iqd/6kYZeUjCjR8P//qe2ColEYiTS0JsJLfuJC9Tu7Q2XLsG5\nc4Wix1C0PPegfR+x1G/9SEMvKZiSJZVN2Z9+UluJRCIxAqMN/YYNG2jZsiV2dnY50oc+iaOjI61b\nt8bd3R0PDw9jh7N6tOwn1kv7W2/BL79AWprF9RiKlucetO8jlvqtH6MNvaurK35+ftlJ//PCxsaG\nwMBAwsLCCAkJMXY4ido4OYGrK2zerLYSSRHjwYMHLFu2jM8++8zoe0yZMqVQxr106RJ+fn5P5ce3\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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Dd+wA7tyRrlyOZfAAbgZTPlpMDD2OGUODeFm2ZGbiUG4ulolIS//E1HCDAZg/n5rrP/wA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\nYII=\n" 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