Lecture-5-Sympy.ipynb

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  "name": "Lecture-5-Sympy"
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   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# Sympy - Symbolic algebra in Python\n",
      "\n",
      "J.R. Johansson (robert@riken.jp) http://dml.riken.jp/~rob/\n",
      "\n",
      "The latest version of this [IPython notebook](http://ipython.org/ipython-doc/dev/interactive/htmlnotebook.html) lecture is available at [http://github.com/jrjohansson/scientific-python-lectures](http://github.com/jrjohansson/scientific-python-lectures).\n",
      "\n",
      "The other notebooks in this lecture series are indexed at [http://jrjohansson.github.com](http://jrjohansson.github.com)."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%pylab inline"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "Welcome to pylab, a matplotlib-based Python environment [backend: module://IPython.zmq.pylab.backend_inline].\n",
        "For more information, type 'help(pylab)'.\n"
       ]
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Introduction\n",
      "\n",
      "There are two notable Computer Algebra Systems (CAS) for Python:\n",
      "\n",
      "* [SymPy](http://sympy.org/en/index.html) - A python module that can be used in any Python program, or in an IPython session, that provides powerful CAS features. \n",
      "* [Sage](http://www.sagemath.org/) - Sage is a full-featured and very power CAS enviroment that aims to provide an open source system that competes with Mathematica and Maple. Sage is not a regular Python module, but rather a CAS environment that use Python as programming language.\n",
      "\n",
      "Sage is in some aspects more powerful than SymPy, but both offer very comprehensive CAS functionality. The advantage of SymPy is that it is a regular Python module and integrates well with the IPython notebook. \n",
      "\n",
      "In this lecture we will therefore look at how to use SymPy with IPython notebooks. If you are interested in open source CAS environment I also recommend to read more about Sage.\n",
      "\n",
      "To get start using SymPy in a Python program or notebook, import the module `sympy`:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sympy import *"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "To get nice-looking $\\LaTeX$ formatted output:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# commands starting with % are IPython commands\n",
      "%load_ext sympyprinting"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Symbolic variables\n",
      "\n",
      "In SymPy we need to create symbols for the variables we want to work with. We can create a new symbol using the `Symbol` class:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = Symbol('x')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(pi + x)**2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left(x + \\pi\\right)^{2}$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAD4AAAAZCAYAAABpaJ3KAAAABHNCSVQICAgIfAhkiAAAAqRJREFU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\nSUVORK5CYII=\n",
       "prompt_number": 5,
       "text": [
        "\n",
        "       2\n",
        "(x + \u03c0) "
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# alternative way of defining symbols\n",
      "a, b, c = symbols(\"a, b, c\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 6
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "type(a)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 7,
       "text": [
        "sympy.core.symbol.Symbol"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can add assumptions to symbols when we create them:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = Symbol('x', real=True)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x.is_imaginary"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADIAAAASCAYAAAAZk42HAAAABHNCSVQICAgIfAhkiAAAArRJREFU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\nx2n3Adczkwoss95+AAAAAElFTkSuQmCC\n",
       "prompt_number": 9,
       "text": [
        "False"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = Symbol('x', positive=True)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x > 0"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACsAAAASCAYAAADCKCelAAAABHNCSVQICAgIfAhkiAAAAjJJREFU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\nSUVORK5CYII=\n",
       "prompt_number": 11,
       "text": [
        "True"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Complex numbers\n",
      "\n",
      "The imaginary unit is denoted `I` in Sympy. "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "1+1*I"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$1 + \\mathbf{\\imath}$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACcAAAAUCAYAAAAOTSQ2AAAABHNCSVQICAgIfAhkiAAAAQxJREFU\nSInt1aFKREEUxvEfchXZB9goskU0aNIHsIhgMSkYNYm4XfANxGb22kwmiyiYRSwq9pUFg6ZNIoqG\ne1eucAVhuDu74FfmzHzMOX+GOTMMoGq4jQ1Rpllc4zM2SFKIJ7GHZ3wE5p3GA94D85QqFXZyKcZD\nIYZCE1SpfoCbwgFOsVo0YsMNo4ltXGCnaMaGW8aJrAGXZE30raRsRw91hTYamMdi0QyFO8JMyfoY\n5vBW4q3jJo9b+biBR5z/pWiqd09Jgifs5vNG14h952ABdRxiBFtd4ze40XysVcsFJmT/eBubMkj8\nhKvjDPdYyddauMRahXDH6GAfL7irsBb6/Pvq4LWi3P8abH0BzAksVA4zGGgAAAAASUVORK5CYII=\n",
       "prompt_number": 12,
       "text": [
        "1 + \u2148"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "I**2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$-1$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAABsAAAASCAYAAACq26WdAAAABHNCSVQICAgIfAhkiAAAAIFJREFU\nOI3t1LEJg2AQhuEnIUVwAHewSJlpXMBhnMDKNawcwEZCtsgMYgotLAwY+BWM+Zorjrv37vg4dq4I\njy1AdzTo55KXQJAEOV7oAvVcpNKHzc5bTvGH7Q82tf4NBU4La1tkwSf6UqVftf51jNEKvUGMCk/D\nCXvD66qRrgU9qN7ZeBCFgVrmLQAAAABJRU5ErkJggg==\n",
       "prompt_number": 13,
       "text": [
        "-1"
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(x * I + 1)**2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left(\\mathbf{\\imath} x + 1\\right)^{2}$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEUAAAAZCAYAAABnweOlAAAABHNCSVQICAgIfAhkiAAAAthJREFU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\nQmCC\n",
       "prompt_number": 14,
       "text": [
        "\n",
        "         2\n",
        "(\u2148\u22c5x + 1) "
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Rational numbers\n",
      "\n",
      "There are three different numerical types in SymPy: `Real`, `Rational`, `Integer`: "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "r1 = Rational(4,5)\n",
      "r2 = Rational(5,4)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 15
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "r1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{4}{5}$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAeCAYAAAD+QjQzAAAABHNCSVQICAgIfAhkiAAAAPNJREFU\nOI3N0rtKQ0EQBuAvUQwWotgExCKNsUxhYyEoxEZBi5DS2tpW8AUsfIA0eQ0b38BWEBVB0sRLJYJE\nFLXIHrKu54CVOrDM7Oz/z3X5gYwV+I/wjF4RcRV3WMsc5QQwjQbOY2cK2kUnDR2DtnGMlyLQHGZw\nlldk1l07AFfC2UQF77jKI8KNqLt0TvM4QBNVPOGyKNJfSinoj99P3cUbBjjFUvYwHoF6hsMsoR+z\nYxDc5qWI17KBBdSxh2s8pIQdTAR7HRe+f8ovqWuGs2uIkMt4xGS4TwX9mtY0wEmwW5jFIaO1wBYW\nA7GOfdynNf0X+QTklCPHiBE8XQAAAABJRU5ErkJggg==\n",
       "prompt_number": 16,
       "text": [
        "4/5"
       ]
      }
     ],
     "prompt_number": 16
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "r1+r2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{41}{20}$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAABAAAAAeCAYAAAAl+Z4RAAAABHNCSVQICAgIfAhkiAAAAXFJREFU\nOI3t1L9LVmEUB/CP9mJDpglhmhVZ0BRNiQSBiUgOEU3vYEND4iDhFPGCDY4tujf5D7hKa2sFNZQU\nCZGLEZQSidjSbXjO671cU1OJIPzC4XCe5znfe+75xV/CFK6Wzo5iFmd2cu7DZ1wrnI1gEhnObufc\ninE8LRHUsYmgsfRgFI93CnErgpt4gh97ITiJY3izG2eohL6ODtTCvoBhHMHcbknhoz9M4qHSg1N4\niAGcwHe8x22M4TI6I9rne4nsf0RD6OyfRrEvFPugF1UMYQKLIaT619COe3iL5SJRMx4V7CrW0IWm\nIOqMux68KEdyCT9xPuwWKbFVDGK+8LYBq+gmn8bXuIIPYZ8OvSD1/tcCQYYVXCwSZHgmL2cN03iF\n4/E7RaxLO3LTRoK7+IT7YX+T90sdzfjyO4IboR/gcIT/Tpq+Oipok1doA33x9Y6QW1JeKliSr/N+\nvCw7n5NmPytJS9wPSMv2DmakjXWAwC84nEJiZ009lwAAAABJRU5ErkJggg==\n",
       "prompt_number": 17,
       "text": [
        "\n",
        "41\n",
        "\u2500\u2500\n",
        "20"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "r1/r2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{16}{25}$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAABAAAAAeCAYAAAAl+Z4RAAAABHNCSVQICAgIfAhkiAAAAYRJREFU\nOI3t1b9L1VEYBvBPPzAELZACI82LgQ5BDQ0aWCoILro0OJREUEvg0hK03bGCZkGqJQj/gSBQWhNs\nMJpEcHLKpUIsKrwN5/1yz/3ilQsFDfUs73nPe857nnPe85zDb+JIye/ES6zgcyl2E9MYwBjelpPd\nQRU1VEqxW3gY7Qq+42QzVuUEbdhGX9bXXzSONsuS4XKs1ocRXMJrbLbK4Hr0XQm/A5/QA4dbYFAc\n5ruwO/iKqVYTrAWDvGI1B2x/vyq8wWS0TwWrM+WJNzAfCRYxl8V68RT3sIDxFpj/MzgUtvZXWfwx\nDEl1rmIJV7PYc/zEN6xKgmpAh7reYQa76retim6cbrb6BezhXPjHpYOdyRLsi0JMHyTdFxrvDbsR\nth13JWk/w/lmCQu8wJPMn5VeJpjAugOUfBuP1e8IjdKtSNu7mG+hwFTY+zgWg4cl+bZHrDPsDxof\niVGcxSupKmPBYksq33KMu4YuPJLR7Mf7mJjjBL5I/8FgLDiAB/joP8AvdSFCFQESh2EAAAAASUVO\nRK5CYII=\n",
       "prompt_number": 18,
       "text": [
        "\n",
        "16\n",
        "\u2500\u2500\n",
        "25"
       ]
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Numerical evaluation\n",
      "\n",
      "SymPy uses a library for artitrary precision as numerical backend, and has predefined SymPy expressions for a number of mathematical constants, such as: `pi`, `e`, `oo` for infinity.\n",
      "\n",
      "To evaluate an expression numerically we can use the `evalf` function (or `N`). It takes an argument `n` which specifies the number of significant digits."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "pi.evalf(n=50)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$3.1415926535897932384626433832795028841971693993751$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAfsAAAASCAYAAABct3csAAAABHNCSVQICAgIfAhkiAAACrFJREFU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\nKKGn/IoAAAAASUVORK5CYII=\n",
       "prompt_number": 19,
       "text": [
        "3.1415926535897932384626433832795028841971693993751"
       ]
      }
     ],
     "prompt_number": 19
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "y = (x + pi)**2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 20
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "N(y, 5) # same as evalf"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$x^{2} + 6.2832 x + 9.8696$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAALsAAAAZCAYAAABpRF+PAAAABHNCSVQICAgIfAhkiAAABfpJREFU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\nRU5ErkJggg==\n",
       "prompt_number": 21,
       "text": [
        "\n",
        " 2                    \n",
        "x  + 6.2832\u22c5x + 9.8696"
       ]
      }
     ],
     "prompt_number": 21
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "When we numerically evaluate algebraic expressions we often want to substitute a symbol with a numerical value. In SymPy we do that using the `subs` function:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "y.subs(x, 1.5)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left(1.5 + \\pi\\right)^{2}$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAE4AAAAZCAYAAACfIRhSAAAABHNCSVQICAgIfAhkiAAAAw1JREFU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\n",
       "prompt_number": 22,
       "text": [
        "\n",
        "         2\n",
        "(1.5 + \u03c0) "
       ]
      }
     ],
     "prompt_number": 22
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "N(y.subs(x, 1.5))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$21.5443823618587$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJ4AAAASCAYAAACq92fYAAAABHNCSVQICAgIfAhkiAAABaNJREFU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\n7D7BBBNMMMEEE3zweA99PyzFs2zkKAAAAABJRU5ErkJggg==\n",
       "prompt_number": 23,
       "text": [
        "21.5443823618587"
       ]
      }
     ],
     "prompt_number": 23
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The `subs` function can of course also be used to substitute Symbols and expressions:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "y.subs(x, a+pi)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left(a + 2 \\pi\\right)^{2}$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEkAAAAZCAYAAAB9/QMrAAAABHNCSVQICAgIfAhkiAAAA1FJREFU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\nAABJRU5ErkJggg==\n",
       "prompt_number": 24,
       "text": [
        "\n",
        "         2\n",
        "(a + 2\u22c5\u03c0) "
       ]
      }
     ],
     "prompt_number": 24
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can also combine numerical evolution of expressions with NumPy arrays:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 25
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x_vec = numpy.arange(0, 10, 0.1)\n",
      "y_vec = numpy.array([N(((x + pi)**2).subs(x, xx)) for xx in x_vec])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 26
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "fig, ax = subplots()\n",
      "ax.plot(x_vec, y_vec);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "display_data",
       "png": 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LyZMnU19f33KVi4hH7d0Lt9xipjrm5EBiotUVSXM0Gu7Dhw8nKCjomGOZmZkkHv5bT0xM\nZPny5QBkZGSQkJBAYGAgYWFhhIeHk5eX10Jli4gnffqpacOccQbk5YHdbnVF0lxNvqFaWVmJzWYD\nwGazUVlZCcD27dsZOnRow+tCQkIoLy8/7v3Tp09v+N7hcOBwOJpagoh4iMsFzz0Hjz5qRuy33GJ1\nRQLgdDpxOp3NOkezZssEBAQQEBDQ6M9/7OhwFxHrVFebtdeLiswSvRddZHVF8oMfD3xnzJjR5HM0\nebaMzWajoqICgB07dtC9e3cAgoODKS0tbXhdWVkZwcHBTS5IRFrehg0wYAB0726+V7D7nyaHe1xc\nHGlpaQCkpaUxZsyYhuPp6enU1tZSXFxMUVERQ7T+p4hXqa8368Jcfz389a/wt7+Z6Y7ifxptyyQk\nJJCTk8POnTsJDQ3l8ccf56GHHiI+Pp6FCxcSFhbG0qVLAbDb7cTHx2O322nfvj3z589vtGUjIq2r\nogImToR9++DDD+H8862uSFpSgMvlcrXaxQICaMXLichhK1fCpElw++3m5ml7PZvuU9zJTv0Vi/ix\nmhrzMNKyZWa3pCuusLoiaS0KdxE/tWWL2SnpggvMqo5du1pdkbQmrS0j4md+WMlx+HCYPBlef13B\n3hZp5C7iR3buNHPXi4vNSo69elldkVhFI3cRP/HOO9C/P1x4oVnJUcHetmnkLuLjDhyA3/8eliwx\n+5qOHGl1ReINFO4iPmzzZvjVryAiwiz+pd66/EBtGREfVF9vFvq66iq45x4z1VHBLkfTyF3Ex5SW\nwm9+A/v3m976z39udUXijTRyF/Eh//oXDBwII0bAe+8p2OWnaeQu4gN27YI77zSbVWdlwcUXW12R\neDuN3EW83FtvQb9+0LMnfPSRgl1OjUbuIl5q7164/34T7i+9BFdeaXVF4ks0chfxQmvXmtH6gQNm\niqOCXZpKI3cRL7J/v1mS91//MnubxsVZXZH4Ko3cRbzEBx+Yfvq2bebGqYJdmsPtcE9JSaF3795E\nR0czYcIEampqqKqqIiYmhsjISGJjY6murvZkrSJ+qaYGHnnEbH03YwYsXQrdulldlfg6t8K9pKSE\n559/nvz8fDZv3kxdXR3p6emkpqYSExNDYWEhI0aMIDU11dP1iviVjRvNvPUtW0xvPT7e6orEX7gV\n7meffTaBgYHs27ePQ4cOsW/fPs477zwyMzNJTEwEIDExkeXLl3u0WBF/UVNjFvu69lozan/9dbDZ\nrK5K/IlbN1S7dOnCvffey/nnn0/Hjh25+uqriYmJobKyEtvhf0NtNhuVlZXHvXf69OkN3zscDhwO\nh1uFi/iqvDxITobwcDNa79HD6orE2zidTpxOZ7PO4dYG2V999RWjR49m7dq1dO7cmZtuuokbb7yR\nqVOnsnv37obXdenShaqqqiMX0wbZ0obt3w+PPQYvvmgW/Ro/HgICrK5KfIE72elWW2bjxo0MGzaM\nrl270r59e8aOHcv69evp0aMHFRUVAOzYsYPu3bu7c3oRv7NundlI45tvzEyYhAQFu7Qst8I9KiqK\nDRs2sH//flwuF6tWrcJutzN69GjS0tIASEtLY8yYMR4tVsTX7N0LU6fCzTdDSorZUENjHmkNbvXc\n+/Xrx8SJExk0aBDt2rXj4osv5vbbb+d///sf8fHxLFy4kLCwMJYuXerpekV8RnY23HEHOBzw2WcQ\nFGR1RdKWuNVzd/ti6rlLG7Brl9lAIycH/vEPuPpqqysSX9dqPXcROZ7LBenp0KcPnHOOGa0r2MUq\nWltGxAO2bYPJk6GkBN54A4YOtboiaes0chdphro6eOYZsybM0KGQn69gF++gkbuImz79FG6/HTp0\ngPffh4susroikSM0chdpou+/hwcegJgYE+5r1ijYxfso3EWaYOVKc8O0vNzcMJ00CdrpvyLxQmrL\niJyC7dvhrrvg44/N9MbYWKsrEmmcxhwijTh0CObNg759ISoKNm9WsItv0Mhd5Cfk5cH/+3/QubPZ\n07RXL6srEjl1GrmL/EhVFfz2t2ZnpHvugdWrFeziexTuIofV18PixWC3w2mnmd2Rfv1rrd4ovklt\nGRHMnPU774TaWvjPf8zWdyK+TCN3adOqq+F3vzM3SSdOhPXrFeziHxTu0ib90ILp1cvsZ1pQYB5I\nOu00qysT8Qy1ZaTN2bgRpkwx37/5JgwaZG09Ii3B7ZF7dXU148aNo1evXtjtdj744AOqqqqIiYkh\nMjKS2NhYqqurPVmrSLP8979w220werTZRCM3V8Eu/svtcL/rrru45ppr2LJlC5s2bSIqKorU1FRi\nYmIoLCxkxIgRpKamerJWEbccPAhz50Lv3nDWWWYWTFKSlg0Q/+bWTkx79uxhwIABfP3118ccj4qK\nIicnB5vNRkVFBQ6Hgy+++OLIxbQTk7Sy7GyYNg1CQmDOHDPNUcTXuJOdbvXci4uLOffcc0lKSuLT\nTz9l4MCBzJkzh8rKSmw2GwA2m43Kysrj3jt9+vSG7x0OBw6Hw50SRBpVVAT33mtulP71r6YVo/nq\n4iucTidOp7NZ53Br5L5x40YuvfRScnNzGTx4MNOmTeOss87i2WefZffu3Q2v69KlC1VVVUcuppG7\ntLDqavjzn+GFF+DBB800xzPOsLoqkeZptT1UQ0JCCAkJYfDgwQCMGzeO/Px8evToQUVFBQA7duyg\ne/fu7pxepMkOHYLnnjOLe333HXz+Odx/v4Jd2i63wr1Hjx6EhoZSWFgIwKpVq+jduzejR48mLS0N\ngLS0NMaMGeO5SkV+QlYW9OsHS5aY7xcsgMPdQZE2y622DMCnn37KrbfeSm1tLRdeeCGLFy+mrq6O\n+Ph4tm3bRlhYGEuXLuWcc845cjG1ZcSDPvsM7rsPvv4annpKfXXxX+5kp9vh7g6Fu3hCRQX88Y+w\nfDn8/vdmBcfTT7e6KpGW02o9dxErfP89/OlPZpu7zp1h61azO5KCXeR4CnfxeocOwT//CZGR5kbp\nhx/C7NkQFGR1ZSLeS2vLiNdyuczyuw8+COeea9owhydoichJKNzFK+XmmlCvqoLUVLjuOt0sFWkK\ntWXEq2zZAjfcAOPHQ3IybNqkWTAi7lC4i1fYts2E+eWXw7Bh5mZpUpLWVxdxl8JdLPXtt2YT6v79\noUcPsybM/fdDx45WVybi2xTuYok9e8xc9agos2/p55/DzJlw1DNvItIMCndpVd9/D7NmQUQEfPON\n2RXp2WehZ0+rKxPxL5otI63iwAF4/nlISYHLLoOcHLN/qYi0DIW7tKjaWli0CJ54wizu9Z//wIAB\nVlcl4v8U7tIiDh6EF180a6tfdBH8+98wZIjVVYm0HQp38aiDB+Gll0yoX3ghvPwy/OIXVlcl0vYo\n3MUjfhipP/EEhIVBWhoMH251VSJtl8JdmqW21gT5zJlmpK5QF/EOCndxy4EDsHChmdbYq5faLyLe\nxu157nV1dQwYMIDRo0cDUFVVRUxMDJGRkcTGxlJdXe2xIsV7fP89/PWvZpSelQXLlsHbbyvYRbyN\n2+E+d+5c7HY7AYdXdEpNTSUmJobCwkJGjBhBamqqx4oU61VXm5ukF1wA69fDihXw5puaASPirdwK\n97KyMlauXMmtt97asPVTZmYmiYmJACQmJrJ8+XLPVSmWqayEhx4yI/WiInjvPXjtNc1VF/F2bvXc\n7777bmbPns13333XcKyyshLb4S3nbTYblZWVJ3zv9OnTG753OBw4HA53SpAW9vXX8Je/wKuvwoQJ\n8NFHZhaMiLQ8p9OJ0+ls1jmaHO4rVqyge/fuDBgw4CcvHhAQ0NCu+bGjw128zyefmJuk77wDt99u\n1lc//P9sEWklPx74zpgxo8nnaHK45+bmkpmZycqVKzlw4ADfffcdt9xyCzabjYqKCnr06MGOHTvo\n3r17k4sRa7hcsHo1PPkkfPYZ3H03/OMfcPbZVlcmIu4KcP3QNHdDTk4OTz31FG+++SYPPPAAXbt2\n5cEHHyQ1NZXq6urjbqoGBATQjMuJhx08aGa7zJ5tpjbef79pwZxxhtWVicjR3MnOZs9z/6H98tBD\nDxEfH8/ChQsJCwtj6dKlzT21tJDvvoN//hPmzDGzX2bMgGuvhXZaAFrEbzRr5N7ki2nkbqlvvoF5\n8+CFFyAmBu69FwYPtroqETkZd7JTYzU/53KZeek33wwXX2w2ms7Ph/R0BbuIP9PyA37qh376nDlm\nn9Lf/c5slqGbpCJtg9oyfua//4UFC+C558xWdtOmwXXXwWmnWV2ZiLhLbZk2bONGSEw0G2N88w2s\nXAlr1sD11yvYRdoijdx9WE2NWQrg2WehogJ++1u49Vbo2tXqykTEk9zJToW7DyopMQ8ZLVoEffvC\nlClqvYj4M7Vl/FhdndlcevRoGDjQPHT03ntmmQC1XkTkxzRbxsvt2GFG6AsWmDVe7rgDliyBn/3M\n6spExJsp3L1QfT1kZ5tAX7MGbroJXn/djNhFRE6Fwt2LlJWZUfqiReam6B13mD1JzzrL6spExNco\n3C1WW2t2NFq0CDZsgPHj4Y03tBmGiDSPwt0imzfD4sVmY2m7HZKTzbRG9dJFxBMU7q1o1y6zs9Hi\nxWb7uokTITcXwsOtrkxE/I3mubewgwfN06JpafDuu3DNNZCUBCNGaPqiiJwaPcTkJVwuyMszLZcl\nS8ySAImJZtZL585WVycivsaSzTrkiC+/hFdeMV8At9xibpL+/OfW1iUibY9bT6iWlpZy5ZVX0rt3\nb/r06cO8efMAqKqqIiYmhsjISGJjY6murvZosd5oxw6YOxcuuQR+8QvTV3/pJdi6FR59VMEuItZw\nqy1TUVFBRUUF/fv3Z+/evQwcOJDly5ezePFiunXrxgMPPMCsWbPYvXv3Mfuo+ktbpqrKTFd89VX4\n6COzJMCECTByJLTX70Ii4mGW9dzHjBnDlClTmDJlCjk5OdhsNioqKnA4HHzxxRfNKtBbVFdDZqbp\noa9bZ7apS0gwN0g7drS6OhHxZ5b03EtKSvj444+55JJLqKysxGazAWCz2aisrDzu9dOnT2/43uFw\n4HA4mltCi/kh0F97DXJy4Mor4Ve/MlvU6alREWkpTqcTp9PZrHM0a+S+d+9errjiCh599FHGjBlD\nUFAQu3fvbvh5ly5dqKqqOnIxHxi579wJGRlmi7r334errjKzXEaP1hZ1ImKNVh25Hzx4kBtvvJFb\nbrmFMWPGADS0Y3r06MGOHTvo3r27u6dvVaWlsHy56aN/9BHExsJvfgNLl2qELiK+ya3ZMi6Xi0mT\nJmG325k2bVrD8bi4ONLS0gBIS0trCH1v43KZx////GcYPBj69zehftddZkej116Dm29WsIuI73Kr\nLbNu3Touv/xy+vbtS0BAAAApKSkMGTKE+Ph4tm3bRlhYGEuXLuWcc845cjEL2zIHD8LataaHnplp\nltW9/nrzNXw4BAZaUpaIyEnpCdUf2bkT3noLVqwwOxaFh0NcnOmf9+0Lh/+/JCLi1dp8uNfXw8cf\nm7VcVq6EggKzhsu115opiz17ttilRURaTJsM92+/NaPyrCx4+20ICjJBfs01pt1yxhkevZyISKtr\nE+FeUwPr15tt6LKzoajIzD//5S/h6qvhggs8VKyIiJfwy3Cvr4dNm8xyuatWmbnnF11kgjw2Fi69\nVDdDRcS/+UW4u1xm0S2n0wT6mjXQpYtZt2XkSHA4zJ9FRNoKnwx3lwu++MI83p+TY0L99NNNiI8Y\nYVouoaGtVaGIiPfxiXA/eNDFJ5+YxbfWrjVfnTrB5ZfDFVeYMA8La62KRES8n0+E+9lnuwgNNTNZ\nLrvMhLpG5iIiP80nwn33bhdHPbQqIiIn4RPh7u2rQoqIeBt3stOthcNERMS7KdxFRPyQwl1ExA8p\n3EVE/JDCXUTEDyncLdLczW/9iT6LI/RZHKHPonk8Hu5ZWVlERUURERHBrFmzPH16v6F/cY/QZ3GE\nPosj9Fk0j0fDva6ujilTppCVlUVBQQGvvvoqW7Zs8eQlRETkFHg03PPy8ggPDycsLIzAwEDGjx9P\nRkaGJy8hIiKnwKNPqC5btoy3336b559/HoCXX36ZDz74gGeeecZcTJuWioi4palR3d6TFz9ZeGvp\nARGR1uEOKhbYAAAEPUlEQVTRtkxwcDClpaUNfy4tLSUkJMSTlxARkVPg0XAfNGgQRUVFlJSUUFtb\ny5IlS4iLi/PkJURE5BR4tC3Tvn17nn32Wa6++mrq6uqYNGkSvXr18uQlRETkFHh8nvuoUaPYunUr\nX375JQ8//HDDcc1/P6K0tJQrr7yS3r1706dPH+bNm2d1SZaqq6tjwIABjB492upSLFVdXc24cePo\n1asXdrudDRs2WF2SZVJSUujduzfR0dFMmDCBmpoaq0tqNcnJydhsNqKjoxuOVVVVERMTQ2RkJLGx\nsVRXV5/0PK3yhKrmvx8rMDCQp59+ms8//5wNGzbwt7/9rU1/HnPnzsVut7f52VR33XUX11xzDVu2\nbGHTpk1t9rfekpISnn/+efLz89m8eTN1dXWkp6dbXVarSUpKIisr65hjqampxMTEUFhYyIgRI0hN\nTT3peVol3DX//Vg9evSgf//+AHTq1IlevXqxfft2i6uyRllZGStXruTWW29t07Op9uzZw9q1a0lO\nTgZMi7Nz584WV2WNs88+m8DAQPbt28ehQ4fYt28fwcHBVpfVaoYPH05QUNAxxzIzM0lMTAQgMTGR\n5cuXn/Q8rRLu5eXlhB61UWpISAjl5eWtcWmvV1JSwscff8wll1xidSmWuPvuu5k9ezbt2rXtZY6K\ni4s599xzSUpK4uKLL+a2225j3759VpdliS5dunDvvfdy/vnnc95553HOOecwcuRIq8uyVGVlJTab\nDQCbzUZlZeVJ39Mq/0W19V+3f8revXsZN24cc+fOpVOnTlaX0+pWrFhB9+7dGTBgQJsetQMcOnSI\n/Px8Jk+eTH5+PmeeeeYp/ertj7766ivmzJlDSUkJ27dvZ+/evbzyyitWl+U1AgICTilTWyXcNf/9\neAcPHuTGG2/k17/+NWPGjLG6HEvk5uaSmZnJBRdcQEJCAqtXr2bixIlWl2WJkJAQQkJCGDx4MADj\nxo0jPz/f4qqssXHjRoYNG0bXrl1p3749Y8eOJTc31+qyLGWz2aioqABgx44ddO/e/aTvaZVw1/z3\nY7lcLiZNmoTdbmfatGlWl2OZmTNnUlpaSnFxMenp6Vx11VW8+OKLVpdliR49ehAaGkphYSEAq1at\nonfv3hZXZY2oqCg2bNjA/v37cblcrFq1CrvdbnVZloqLiyMtLQ2AtLS0UxsQulrJypUrXZGRka4L\nL7zQNXPmzNa6rFdau3atKyAgwNWvXz9X//79Xf3793e99dZbVpdlKafT6Ro9erTVZVjqk08+cQ0a\nNMjVt29f1w033OCqrq62uiTLzJo1y2W32119+vRxTZw40VVbW2t1Sa1m/Pjxrp49e7oCAwNdISEh\nrkWLFrl27drlGjFihCsiIsIVExPj2r1790nP49GFw0RExDu07SkKIiJ+SuEuIuKHFO4iIn5I4S4i\n4ocU7iIifkjhLiLih/4/gplMippgtscAAAAASUVORK5CYII=\n"
      }
     ],
     "prompt_number": 27
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Algebraic manipulations\n",
      "\n",
      "One of the main uses of an CAS is to perform algebraic manipulations of expressions. For example, we might want to expand a product, factor an expression, or simply an expression. The functions for doing these basic operations in SymPy are demonstrated in this section."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Expand and factor\n",
      "\n",
      "The first steps in an algebraic manipulation "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(x+1)*(x+2)*(x+3)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left(x + 1\\right) \\left(x + 2\\right) \\left(x + 3\\right)$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAKIAAAAUCAYAAAAOYeZzAAAABHNCSVQICAgIfAhkiAAAA/tJREFU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\nAElFTkSuQmCC\n",
       "prompt_number": 28,
       "text": [
        "(x + 1)\u22c5(x + 2)\u22c5(x + 3)"
       ]
      }
     ],
     "prompt_number": 28
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expand((x+1)*(x+2)*(x+3))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$x^{3} + 6 x^{2} + 11 x + 6$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJ4AAAAZCAYAAADAMJcbAAAABHNCSVQICAgIfAhkiAAAA6ZJREFU\naIHt2l2oZXMYx/HPnmYwx8toiKkTzhjMcLzEeBlGTIo0XLgwyg1l5CXKhcwNZSQliStKSUve0lyo\niTuRueB4SyalRIOUvJSJwngZLp6191m2dY6118vZezvrW6e91jrP+u/n1/Os5//8/2vT0jIEOiXv\nuwjH4UBsQoJX63FpKJyHC7ACG3E/dg3Vo5Zcvsf16fEW/IxDh+dOJQ7Bg5nza4SeyeG40zIfp+Lg\n9Phq/G58E+907Mea9Pww/CUSsGWEeR53D9uJCnTEVNttO6ZF4p05NI8WAWV7PDgLl2At7hDT00Jy\nErbjF+xLP7fjp4rjPoNvcWfFccowgRlRheuwGyZNxafHTXhf9EoLxSn4UiwIYBX2CGFV2IqHVHsg\ny3IO3hXVtg67YfKf8VlSYtAN+Aar0/NdovpdXtZL8eQuLWi7FC/hEbyZXlsmEv+HCj5cmX5uE6v1\nqQpjDaLnZLyC2/BnDXZNMArxcTZeE8GBzfgNJ1QYM1E80Dek33d4he/r52JR7Valf1fh/ArjJcol\nbqJYJStqVxeJmuNTNIuzvIencLtYDV4oqsWnJcYqw7Xpd+2tabzj8bJ/tworahp/sVEoPv2Jtx7X\niVI+hRtxs8jeSdyLz/Bs5p5Ha3G3GB2R6O+ITezLRMJM4T580GdfVM+4bgXNRVHddTNofBBP/mNm\n+74En4gGcaOobk2t9BLFSvmRYor5GLdkrm8Sq6XpzLVx0JN3X9Wptgndifrj0+Nx/3zyd+Dt9PgY\nPIwjBnB2EBLFhB0thP2K5X3/+0pMmV3GQU/efVUTrwndifrj02N1juEDA7lXnkQxYcuEsI9y/jcj\n9ou6i55x0JN3X9XEa0J3oub4ZHu8PZnjtaIXeL2Ek/PxNM7IuX4szhWroX62in1C4tXcd/KX5ftw\nAFbia+Ohpwmq6F7I+ORya2o4kbm2Zg7bOkgUrxA7RQ/Rz4wo8Xkr9VHW039fndspdelO1ByfbgO6\nXOzYn5aeX4rdZl+DLcFdg3rbEC+In2Rle4gO1gnRfxgvPXUyCrqLxKeXeJtTh6ZTgxPFk9LlHlGG\nR4EX8aHYwe+yRZT5ben5OOnJclD6OTGv1dx2o6C7SHx609IbopyuF6+/NojV0RNiXt+Jtxp2uCj7\ncYV4JbNDCOoInz9PbcZJz1HihwmTZrcbvhAN+pN4bgC7UdBdJD4jQ6Lau9FRI9HqmZcyPxJogh9F\n4/l/odXT0tLS0tLS0tKy6Pgb7xIXAWRkl90AAAAASUVORK5CYII=\n",
       "prompt_number": 29,
       "text": [
        "\n",
        " 3      2           \n",
        "x  + 6\u22c5x  + 11\u22c5x + 6"
       ]
      }
     ],
     "prompt_number": 29
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The `expand` function takes a number of keywords arguments which we can tell the functions what kind of expansions we want to have performed. For example, to expand trigonometric expressions, use the `trig=True` keyword argument:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "sin(a+b)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\operatorname{sin}\\left(a + b\\right)$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFEAAAAUCAYAAAD1GtHpAAAABHNCSVQICAgIfAhkiAAAA79JREFU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\nTkSuQmCC\n",
       "prompt_number": 30,
       "text": [
        "sin(a + b)"
       ]
      }
     ],
     "prompt_number": 30
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expand(sin(a+b), trig=True)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\operatorname{sin}\\left(a\\right) \\operatorname{cos}\\left(b\\right) + \\operatorname{sin}\\left(b\\right) \\operatorname{cos}\\left(a\\right)$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAM4AAAAUCAYAAADFugKNAAAABHNCSVQICAgIfAhkiAAABZNJREFU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\nHFrGBPWAAAAAAElFTkSuQmCC\n",
       "prompt_number": 31,
       "text": [
        "sin(a)\u22c5cos(b) + sin(b)\u22c5cos(a)"
       ]
      }
     ],
     "prompt_number": 31
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "See `help(expand)` for a detailed explanation of the various types of expansions the `expand` functions can perform."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The opposite a product expansion is of course factoring. The factor an expression in SymPy use the `factor` function: "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "factor(x**3 + 6 * x**2 + 11*x + 6)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left(x + 1\\right) \\left(x + 2\\right) \\left(x + 3\\right)$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAKIAAAAUCAYAAAAOYeZzAAAABHNCSVQICAgIfAhkiAAAA/tJREFU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\nAElFTkSuQmCC\n",
       "prompt_number": 32,
       "text": [
        "(x + 1)\u22c5(x + 2)\u22c5(x + 3)"
       ]
      }
     ],
     "prompt_number": 32
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Simplify\n",
      "\n",
      "The `simplify` tries to simplify an expression into a nice looking expression, using various techniques. More specific alternatives to the `simplify` functions also exists: `trigsimp`, `powsimp`, `logcombine`, etc. \n",
      "\n",
      "The basic usages of these functions are as follows:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# simplify expands a product\n",
      "simplify((x+1)*(x+2)*(x+3))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$x^{3} + 6 x^{2} + 11 x + 6$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJ4AAAAZCAYAAADAMJcbAAAABHNCSVQICAgIfAhkiAAAA6ZJREFU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\n4/l/odXT0tLS0tLS0tKy6Pgb7xIXAWRkl90AAAAASUVORK5CYII=\n",
       "prompt_number": 33,
       "text": [
        "\n",
        " 3      2           \n",
        "x  + 6\u22c5x  + 11\u22c5x + 6"
       ]
      }
     ],
     "prompt_number": 33
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# simplify uses trigonometric identities\n",
      "simplify(sin(a)**2 + cos(a)**2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$1$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAASCAYAAACNdSR1AAAABHNCSVQICAgIfAhkiAAAAF5JREFU\nKJFjYKAC4GJgYLhEjEJTBgaG0wwMDP/RJViQ2JoMDAw9DAwMrxkYGP6S4owF2ExmIsWEUcVUV8wB\npblwaRRjYGDYycDAcIUBEnv/GSBRv5+BgSGaFBfQGAAA/84M5lOscPUAAAAASUVORK5CYII=\n",
       "prompt_number": 34,
       "text": [
        "1"
       ]
      }
     ],
     "prompt_number": 34
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "simplify(cos(x)/sin(x))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\operatorname{tan}^{-1}\\left(x\\right)$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEcAAAAXCAYAAABZPlLoAAAABHNCSVQICAgIfAhkiAAAAvhJREFU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\nnYHv8VODtbVo0aJFVf4FZIyQ7JRZKA8AAAAASUVORK5CYII=\n",
       "prompt_number": 35,
       "text": [
        "\n",
        "  1   \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "tan(x)"
       ]
      }
     ],
     "prompt_number": 35
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## apart and together\n",
      "\n",
      "To manipulate symbolic expressions of fractions, we can the `apart` and `together` functions:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f1 = 1/((a+1)*(a+2))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 36
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{1}{\\left(a + 1\\right) \\left(a + 2\\right)}$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEwAAAAgCAYAAAC4oZ4KAAAABHNCSVQICAgIfAhkiAAAAjdJREFU\naIHt2LuPDWEYx/GPxbLBigjWbohLK1sIEZdQkLhEopBssBEFlSA0LlHYRiIKlUL8ASiUVBKFrIJI\nFErXRIFC1iVWI7uKmbUzszN79uw5x+zh/TYzz8yceb7nNzPvTF4CTc883MXyskWagWPowzBWlGrS\nZEzZwFrKFmg2QmBVEgKrkhBYlYTAqmR62QIZenEc67AUHXhaqlEgEAj8Q0yLl8OlWgT+L9qmWM8y\nfAr7Zj9cd6KzwSJ5811tOJJz7N/w2YAzommlB9hawekPi0XzUbVwosL+8ea79mFtnX0qOc3FlUTd\ng0F0FTiluIj2GuX6JnhcXmAtuFpnH8Z36sYQVsd1e+zWU+BkRmK9E98y+y7hPWZiD07h9eS8KzIk\nPW5kfRrh9AIb8Saul8XLlwVOqTFsVuZkN/AZN3EHm/B2kmIT5RdaC3wa4TSMJ0Y/q87jGp4XOKXu\nsGR43TiARYn6kSjxEZbgtNFvOdiC2Yn6Oy5X8Qd+YCE+GPtCarTT0bjvuXGcUoElG+9AP37G9XY8\nxAIMxNs+4ULm5H0mPo7l0Y4vOT6NdtobL8+Kwu3Auxyn1FUcTKwPiBMVvUn24zEOVWhcD0YCGcxs\nb5TTNtGdeV8U1C7R1FKeU+oOexX/4CNuYzMOiga9W9iNZzXK9YoeEaLXeT+ux3Urvhb4aJDTKtwT\nXYAk8wucUszByRqaM/b5r4bDWJmo6+FDfZ3GsB5ramgwWbpEj0KWsnwodgoEAoGm5Te5DnYrH1eP\n7QAAAABJRU5ErkJggg==\n",
       "prompt_number": 37,
       "text": [
        "\n",
        "       1       \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "(a + 1)\u22c5(a + 2)"
       ]
      }
     ],
     "prompt_number": 37
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "apart(f1)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- \\frac{1}{a + 2} + \\frac{1}{a + 1}$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAGIAAAAfCAYAAAAcGRgRAAAABHNCSVQICAgIfAhkiAAAAhlJREFU\naIHt2U2oDWEcx/EPXUV5SRLC7cbasZSILbJTN9zltbDQxd1YmgVlYyOlZHGk2FhSyk7sKGUpNhbs\n7kJZWLgWz1GTzsu8PHNmzj3zrenMM/PM//n9zn/meZmhZarZgueYrVtIJCbSzyUkWMVcrUriMPF+\nJlb4AEr5WR9PR0sZ2kQ0hGlKRAczdYsYxDQlYhn76hYxiGlKRKOpIxELeNDbv4MrNWiIyVrzUzld\nDZ4ut11TQ2gT0RDWpfYP4eF/x4bxEZejKyrPYxzuc3wWP/C7z7lFfKhS1DhYrXHLQ1e2MaIWLzEW\nOFmfoEmhFj/tGNEQmp6II7guvGZ+jRO1qilPY79bDFv8bBYWSP+Yxy/sLdhWV7XriFELuUZ/t0iG\nnOvgDw72ylsFE/MF27qH3QWvzUKSsV7fRMR6GzmDm/iGDTiDJXwpEfMTjuJrr7y/9/u5YLylHHWr\n8DMWHuFab38HVmQbf5IcbTzB3XyyClPET5IxdmVPRAfnsTNVfiN0K2l2CebS08Pj2Jgq/8TtPm0s\n4jtuRNA7iix+ynipjGW8SpVv4Sq2Z7g2yVDnrJAIgtG5HNqKUNRPkjF+3ycixvR1RbhbCTOdc3iH\nixFinxTuvpfCQHsKeyLEHUaVfgYSo2t6hmO4gE14itN4XzLuAbwQ/ow020rGHUVVfhaE7oswLX+L\n+yVjRmMcff64WEteWlpaWloaxF9ZTJTg2sWVZgAAAABJRU5ErkJggg==\n",
       "prompt_number": 38,
       "text": [
        "\n",
        "    1       1  \n",
        "- \u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\n",
        "  a + 2   a + 1"
       ]
      }
     ],
     "prompt_number": 38
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f2 = 1/(a+2) + 1/(a+3)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 39
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{1}{a + 3} + \\frac{1}{a + 2}$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFMAAAAfCAYAAACbKPEXAAAABHNCSVQICAgIfAhkiAAAAl5JREFU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\n",
       "prompt_number": 40,
       "text": [
        "\n",
        "  1       1  \n",
        "\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\n",
        "a + 3   a + 2"
       ]
      }
     ],
     "prompt_number": 40
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "together(f2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{2 a + 5}{\\left(a + 2\\right) \\left(a + 3\\right)}$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEwAAAAiCAYAAAD1aT8BAAAABHNCSVQICAgIfAhkiAAAAylJREFU\naIHt2V+IFWUYx/FPWtqSKWF//ENQK0hEdVGUlqJBQRbd9MeyCIKCIJD+SKTgRXtREEUGehNS3QTV\nRZcFQXRTFESWiX8g2opIKqEws00pcbt45rDvOc7snJzz7mbNFw77Pmdmn+c3L8/7zszv0PKvYBke\nxwjew6rM9V7FMRzFp7gqc72BMgfPJvFd+AOLG+RcX3N8BAuwsEGNvpiRIecwnsSSIn4XQ1jRIOe5\nfZzzE35sUKMvTs+QczeuxTdFfGHx96uk5lP4HmfgFjyCrxvUHMLDOIQbsAV7G+SbVl7DC0n8Mh4r\nxvNxUH2nj9Qcvw+zivGN+LKPnCdFjg5LeVAsk41FfAXW4bwk/gDHk/+5QEzoacl3K3FmEh/GM0n8\nptj0YRRLcTl2Nb6CKeRWMWHExV6EDWJP6/A0HsU5NblGJjm2HGNiWRITNY5L/5HaPsnStlgtOuUd\ncfdaI+5gB01szHNwBz7CvQ1q7cfzOFLEK/Ax9jXIWUmOJTmMt8WEpMzDTnFB94iOeB03Y0eDevvx\nGZ7ATLEcb2uQ7z/BxvpTWlpaWlpaBkznaXp8WlWc4gzVnzKlNadDT2Xd3if9m7Aov5YTGML9Jd9P\nlx4qNM1MxufjGryfUcQyYSiuwWZ8V3yOiHfNs0y8Ok2FHsINXlXU2oDfhDVVpqmLzZjbsPhkzmid\nEzsDzw1YT50m+NlEJ60tNJ1doalrSS4Ss9uEyZzROif2uO59YxB66jTB9XirGI8LU7NDr6aul+/Z\nPYkG7YzWObGEpzULf5boyaEJ9iTj24WVdLhCU1eH9d4AXsIv2C4MuuvwbQNh4/jExCPMJmEl70zO\nGRMubJmeHJo6XCncjjG82HMs1dTVYanrmcsZ7dDrxHaYi19L9OTW9HnxeQgfCj/v9xJNXWxNxjmc\n0Q5lTmyZhnScS9NyHMDFRXyJWAF3VulI235UuKPkcUapdmKJfeJQhZ5cmo6JX5d+KOJh/IUvKjR1\nLclX8AC24Q2Dd0Ync2LhbvELdpkemTTtKGquF0t7pVgBoxWaTuBqXNZAwMk6o4tFt/XSVA+D19TS\n0tLS8n/hb9AQu6EDGI8CAAAAAElFTkSuQmCC\n",
       "prompt_number": 41,
       "text": [
        "\n",
        "    2\u22c5a + 5    \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "(a + 2)\u22c5(a + 3)"
       ]
      }
     ],
     "prompt_number": 41
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Simplify usually combines fractions but does not factor: "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "simplify(f2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{2 a + 5}{a^{2} + 5 a + 6}$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEMAAAAjCAYAAADPPrcpAAAABHNCSVQICAgIfAhkiAAAAuhJREFU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\nTJ/agREE2gAAAABJRU5ErkJggg==\n",
       "prompt_number": 42,
       "text": [
        "\n",
        "  2\u22c5a + 5   \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        " 2          \n",
        "a  + 5\u22c5a + 6"
       ]
      }
     ],
     "prompt_number": 42
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Calculus\n",
      "\n",
      "In addition to algebraic manipulations, the other main use of CAS is to do calculus, like derivatives and integrals of algebraic expressions."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Differentiation\n",
      "\n",
      "Differentiation is usually simple. Use the `diff` function. The first argument is the expression to take the derivative of, and the second argument is the symbol by which to take the deriative:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "y"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left(x + \\pi\\right)^{2}$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAD4AAAAZCAYAAABpaJ3KAAAABHNCSVQICAgIfAhkiAAAAqRJREFU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\nSUVORK5CYII=\n",
       "prompt_number": 43,
       "text": [
        "\n",
        "       2\n",
        "(x + \u03c0) "
       ]
      }
     ],
     "prompt_number": 43
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "diff(y**2, x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$4 \\left(x + \\pi\\right)^{3}$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEgAAAAZCAYAAACSP2gVAAAABHNCSVQICAgIfAhkiAAAAx1JREFU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\n9V9pAAAAAElFTkSuQmCC\n",
       "prompt_number": 44,
       "text": [
        "\n",
        "         3\n",
        "4\u22c5(x + \u03c0) "
       ]
      }
     ],
     "prompt_number": 44
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "For higher order derivatives we can do:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "diff(y**2, x, x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$12 \\left(x + \\pi\\right)^{2}$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFIAAAAZCAYAAACis3k0AAAABHNCSVQICAgIfAhkiAAAA2tJREFU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\nktpJbU2P9x0wYMCAA5J/AQgsssKpt7epAAAAAElFTkSuQmCC\n",
       "prompt_number": 45,
       "text": [
        "\n",
        "          2\n",
        "12\u22c5(x + \u03c0) "
       ]
      }
     ],
     "prompt_number": 45
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "diff(y**2, x, 2) # same as above"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$12 \\left(x + \\pi\\right)^{2}$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFIAAAAZCAYAAACis3k0AAAABHNCSVQICAgIfAhkiAAAA2tJREFU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\nktpJbU2P9x0wYMCAA5J/AQgsssKpt7epAAAAAElFTkSuQmCC\n",
       "prompt_number": 46,
       "text": [
        "\n",
        "          2\n",
        "12\u22c5(x + \u03c0) "
       ]
      }
     ],
     "prompt_number": 46
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "To calculate the derivative of a multivariate expression, we can do:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x, y, z = symbols(\"x,y,z\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 47
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f = sin(x*y) + cos(y*z)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 48
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$\\frac{d^3f}{dxdy^2}$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "diff(f, x, 1, y, 2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- x^{2} y \\operatorname{cos}\\left(x y\\right) - 2 x \\operatorname{sin}\\left(x y\\right)$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAMsAAAAbCAYAAADSxXscAAAABHNCSVQICAgIfAhkiAAABi9JREFU\neJzt23/slVUdwPEXyI8Ao0JAAgv6ogkybKBpKGIWxNYKJIoaJY1oOaczmP3QTecSS1amNRXtF1Za\nuGr92kqWtqj1wx9lDaebLbKUlVPJygTKqf3xOdf73Ifvvd/7fO/3ex++X573dnfv+ZznOefz+Zzn\nnPM55zyXioqKthhRQp2n4jS8DKdjM35Rgh4VFYc0R2JLJr0G+zCjHHUqKg5dTsTzmJ3SE/GC6DQV\nFRUZRogwrBb+zROdZUFpGlVUDBFuwWfLVuIw4xR8F7fjfnxF98Lgo/AnfKRL9Q0bNuDTytlkOFxZ\niJ/g5Sl9pNhceRyzulD/DPwey7pQ17DhbaKzwEt0p6Eq+BGOzckWiFD4tu6rU9EXZ4qOMi19zsai\nUjU6fPgPHsHUnPwpPNl9dSpa0YOnxUiW/UwsU6l+Ml3MioPBZLxiEMq9H//Fa3Lyx/DMINQ3mHTd\n/6MGsIKTsA7PibDqgzhXxMczcDl246UFypyFi0Qn+wa+mck7H6uwNCObn67/N/aLB+NTOJDy56T8\nZzBaOGUj/p6r99W4TJwB7U/3f0b9gZqI1bguc0+79rdj1zvwK3Fg+2xvjuknbxD+fywjm46jsTN3\nbRF7aO2zE4RdM3GrxpBvpWjD1+H9mIR3pbzFov1+nNNtqPqfVOkNGJnSX8UfxUn96eJs5aJ+lLsV\nY/Bh7Mrl3a3R6WvwN+F0WCJmsZUpvVJ0ipMz92zCbzN6w7ik+5yUni1ClLMz11wuFsc1itrfjl09\n+JjBZ4t4wE7LyIra08pno/EFHCEGpqy9Y/C59Pte/DKVW9v0uViEjXmGtP+3apwxvp0qhVfharFt\nWITFWJt+347vZPImiB5/XkovELPIusw1K/A7HCNmnP34QK6Ok0QYuCh33z8xPqWPwqWpTpiCq3Ll\nFLG/iF3bRKMOFseKdcyVOXnR9mzlszVYnuR3ahzgluGdonPs1egLuMTB4eGQ938+Bt6DT3ZY5jSM\nFVPocyLkqrFMPOTzUvoOMQId0aSsHXjUwWHnWamcd2dkS5Jst5jmz8jd8z68PScrYn8RuzbhzU3K\n6ZSxuAfX9JJXtD1b+WymaJceMcIvz+S9UsxKJ6b7876+Db/OyQ4J/8/Hb3BXm5+bmihzfKpwaZP8\nomzCv4SBNa7AE2JEmiIa4YYm909ukf/xpOspvcj3qG9AnJ/J2yoegGa0a39fdhGd+eJM/kC10QgR\np1/Rh45F7GnlM2Lt8ReNIW+NjWKtkx3FR4vZ6pLctWX6f8A5T4RE4zOy2U2ubYcd+H5OtlOcQhOv\nz7yA9U3ub5X/gDhRbnYwOk9M5w9mZN9TD8l6o137+7KLGHGvb1FXf7lSLMazrOvtQsXbszefjRLr\nxVqdPbl7foCf5mQrxCA3S3SiY5K8NP/31suLMk6cxM9P6WVi0bQvU8dHOyh/Jh7KpMeKDlB7rf/x\n9P10k3sXNsl/C+biQtGZCEdlF3wPiFdy/pyRTRCNUaO/9vdlF7HOGuiYeb14CDfn5IvTd1F72vHZ\ncnG2c7Ow54JM3kgRxu3M6bMOPxOz0Socl+Sl+X8gto7fmpS5TyyOjhPTW41L8bUOyv+r2E6ssUXs\nr/88pR8Wr3As0bhYO0M8GBeKXZlsfg++lPKy25In58qYKhaDGzOyveK/OHtTur/292WXVM9AHha+\nSWzn7hBbuDVGqQ8YRe1px2fHiwd4T5LfnMlbILZ3d+Z0fa14b22MCKk+lOSl+X8g3suaLBrgSeHw\nT4i48gD+hx+KBXh/mYMv4w9itFgkRpFJYoQknH1tSv9DOHiXaJTnc/n7xM7I59V3TGqcmT7jxO7K\nBNwoFsI1NuNb4oCvE/vbsWtFKn9bSw+1z1Pq74XlqYVmRe1px2fTsV3sTt6nsaOuEOuZhan8Gmtx\njhgMr1afqYay/7vKSHGg9vUSdThLfZQbKJrZdZWD4/vDncr/TdiuMR5eJUaLueWogwhZOl10t2PX\nCHyxw3qGI5X/m/CE+isN08VU/N7y1HmR96hvHPSHduxaiTd2UMdwphT/H+r/JVmN14vRZJowML/O\nKIsN4r2iA31d2At92TVFLGq3d6jjcKbyf0VFRUVFRUVFRUVFRYb/A5O294K0O2/cAAAAAElFTkSu\nQmCC\n",
       "prompt_number": 49,
       "text": [
        "\n",
        "   2                          \n",
        "- x \u22c5y\u22c5cos(x\u22c5y) - 2\u22c5x\u22c5sin(x\u22c5y)"
       ]
      }
     ],
     "prompt_number": 49
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Integration\n",
      "\n",
      "Integration is done in a similar fashion:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\operatorname{sin}\\left(x y\\right) + \\operatorname{cos}\\left(y z\\right)$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAIkAAAAWCAYAAAD0FL9fAAAABHNCSVQICAgIfAhkiAAABWFJREFU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\nh+7E/wGPQ05qov6qWAAAAABJRU5ErkJggg==\n",
       "prompt_number": 50,
       "text": [
        "sin(x\u22c5y) + cos(y\u22c5z)"
       ]
      }
     ],
     "prompt_number": 50
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "integrate(f, x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$x \\operatorname{cos}\\left(y z\\right) - \\frac{\\operatorname{cos}\\left(x y\\right)}{y}$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAH8AAAAeCAYAAAAFFcHsAAAABHNCSVQICAgIfAhkiAAABatJREFU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\nYuGHi4DTCSLyWaNGjRo1atSoscb4FMb3Y970yHt8AAAAAElFTkSuQmCC\n",
       "prompt_number": 51,
       "text": [
        "\n",
        "             cos(x\u22c5y)\n",
        "x\u22c5cos(y\u22c5z) - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "                y    "
       ]
      }
     ],
     "prompt_number": 51
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "By providing limits for the integration variable we can evaluate definite integrals:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "integrate(f, (x, -1, 1))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$2 \\operatorname{cos}\\left(y z\\right)$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEQAAAAWCAYAAAB5VTpOAAAABHNCSVQICAgIfAhkiAAAA4xJREFU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\nAAAAAElFTkSuQmCC\n",
       "prompt_number": 52,
       "text": [
        "2\u22c5cos(y\u22c5z)"
       ]
      }
     ],
     "prompt_number": 52
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "and also improper integrals"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "integrate(exp(-x**2), (x, -oo, oo))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\sqrt{\\pi}$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACAAAAAVCAYAAAAnzezqAAAABHNCSVQICAgIfAhkiAAAAUdJREFU\nSInt088rZ1EYBvDP13fkR5JIKRYWmliws8SKmkYWdlJKTU3NajbKYhY2En+AslIWiIUk/8KYsmCL\nFFmxMbsZxMziXnW7+fI9t1sWPHXqfd9znuc8977v4a2jkJH3L1cXgejG5GsaWEBlXmIVgee7cIq7\nvAyEYh5VeQqG/IEOXOAmTwMhmENNib16rOOP6IU8tR7QlyZ+SOWNaMJJqt6Oy/iCNApYwQGWMYp9\nnOE7FnGLv/hZ6uugJxZYe2JvFnUleBP4lMg3UYzjnecuTKOIBlyjNVFvw1SZGg3YiuNa/HqJkBzC\ne/zGKr4l6l+xVKaBcezFcWesGYyPuEI1WjAdwD1Ebxx/xlEWA7CLL5gRDWY56Be177H/Y6Khrc5i\nYBDH+BHA2cZGIh8QPb/hLAaIetkccP4MQ4m8HucYyWrgHW8D/wHjuDhKPb2iEAAAAABJRU5ErkJg\ngg==\n",
       "prompt_number": 53,
       "text": [
        "\n",
        "  \u23bd\u23bd\u23bd\n",
        "\u2572\u2571 \u03c0 "
       ]
      }
     ],
     "prompt_number": 53
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Remember, `oo` is the SymPy notation for inifinity."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Sums and products\n",
      "\n",
      "We can evaluate sums and products using the functions: 'Sum'"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "n = Symbol(\"n\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 54
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Sum(1/n**2, (n, 1, 10))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\sum_{n=1}^{10} n^{-2}$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAD4AAAA8CAYAAAA+CQlPAAAABHNCSVQICAgIfAhkiAAAA/tJREFU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\nAElFTkSuQmCC\n",
       "prompt_number": 55,
       "text": [
        "\n",
        "  10    \n",
        "____    \n",
        "\\   `   \n",
        " \\    1 \n",
        "  \\   \u2500\u2500\n",
        "  /    2\n",
        " /    n \n",
        "/___,   \n",
        "n = 1   "
       ]
      }
     ],
     "prompt_number": 55
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Sum(1/n**2, (n,1, 10)).evalf()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$1.54976773116654$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJ4AAAASCAYAAACq92fYAAAABHNCSVQICAgIfAhkiAAABPBJREFU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\no4/Ni38AjkHuiDMOCCIAAAAASUVORK5CYII=\n",
       "prompt_number": 56,
       "text": [
        "1.54976773116654"
       ]
      }
     ],
     "prompt_number": 56
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Sum(1/n**2, (n, 1, oo)).evalf()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$1.64493406684823$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJ0AAAASCAYAAABBwNzbAAAABHNCSVQICAgIfAhkiAAABdRJREFU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\nJZTw3+BvyrwyawW4KQsAAAAASUVORK5CYII=\n",
       "prompt_number": 57,
       "text": [
        "1.64493406684823"
       ]
      }
     ],
     "prompt_number": 57
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Products work much the same way:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Product(n, (n, 1, 10)) # 10!"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$3628800$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEgAAAASCAYAAAD4+JjWAAAABHNCSVQICAgIfAhkiAAAA5BJREFU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\n8iFqDAAAAABJRU5ErkJggg==\n",
       "prompt_number": 58,
       "text": [
        "3628800"
       ]
      }
     ],
     "prompt_number": 58
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Limits\n",
      "\n",
      "Limits can be evaluated using the `limit` function. For example, "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "limit(sin(x)/x, x, 0)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$1$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAASCAYAAACNdSR1AAAABHNCSVQICAgIfAhkiAAAAF5JREFU\nKJFjYKAC4GJgYLhEjEJTBgaG0wwMDP/RJViQ2JoMDAw9DAwMrxkYGP6S4owF2ExmIsWEUcVUV8wB\npblwaRRjYGDYycDAcIUBEnv/GSBRv5+BgSGaFBfQGAAA/84M5lOscPUAAAAASUVORK5CYII=\n",
       "prompt_number": 59,
       "text": [
        "1"
       ]
      }
     ],
     "prompt_number": 59
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can use 'limit' to check the result of derivation using the `diff` function:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\operatorname{sin}\\left(x y\\right) + \\operatorname{cos}\\left(y z\\right)$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAIkAAAAWCAYAAAD0FL9fAAAABHNCSVQICAgIfAhkiAAABWFJREFU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\nh+7E/wGPQ05qov6qWAAAAABJRU5ErkJggg==\n",
       "prompt_number": 60,
       "text": [
        "sin(x\u22c5y) + cos(y\u22c5z)"
       ]
      }
     ],
     "prompt_number": 60
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "diff(f, x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$y \\operatorname{cos}\\left(x y\\right)$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEYAAAASCAYAAADmMahlAAAABHNCSVQICAgIfAhkiAAAA1VJREFU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\nzR0AAAAASUVORK5CYII=\n",
       "prompt_number": 61,
       "text": [
        "y\u22c5cos(x\u22c5y)"
       ]
      }
     ],
     "prompt_number": 61
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$\\displaystyle \\frac{\\mathrm{d}f(x,y)}{\\mathrm{d}x} = \\frac{f(x+h,y)-f(x,y)}{h}$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "h = Symbol(\"h\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 62
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "limit((f.subs(x, x+h) - f)/h, h, 0)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$y \\operatorname{cos}\\left(x y\\right)$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEYAAAASCAYAAADmMahlAAAABHNCSVQICAgIfAhkiAAAA1VJREFU\nWIXt11uMnVMUB/DfVKkxaFNTqRbDNKJNNUKLFB2GlpDomDRBkHlAiEjQiMtDpWkb6paIYCRI3UIb\nxIMHBInQROKWyIgHkhI0iLaCRjtuHQ9rf2afb845c46Zpi/zT07yrcte3/qvvdfa32ECVbHfvk6g\nQczCX/h7L8RuxxQM5srJ2fMxuAWdeB4vZLYb0IulSV6QfH/DbvyBu0vB5yaf37F/SuBm/FBK7Gjc\niV0p1iDuT+vgUKzAw9mahejDPynva3AdpmE2VmNLg5yWYQ3WieKPQD8OwE0YKNk+wKb0fAm+x4lJ\n7sJO9GT+PakAizLdSnyMSZmuFV+KIsIcbMfFmc9qHJzJnXg0i/N0inE6zsAeUYxmOHXiNlVwJi5P\nz6/j5czWJip5PU4Sp6Mvsy/HJzgyyQvEzl9VesdCDGFxae0vOCjJh2FVeifMwPpSnH4ckskvJZJw\nFB5IcRrlVGCDKGIFZoo+my2OZ29mW5YIzcdb+Fb92fQGvlPZptCd4lya6bqSbotolSWlNVfiopLu\n2JK8FXdVyaNRTgVW4twqcf4z/poCFliLbWL39ohjXAvtdXxuT8mcWkW/NdmGRO8X6EdHnfcdn9Ys\nreNTj1NLpuvGHYWQ9zucj3dEuxTowmbRhy1iTtTCnDo+feJkfFTS3yva8AR8qLIwR4iZUwvn4E+8\nX8ohRz1OQ5luh+FxMKIwHfgik6fgNLyHn5JuZ5UEO3BBHZ/zMA83Zsm8onIgfo7n8FWmaysRasV9\nYo4RLTEgbrSCz61NcMqxWzZjyoX5BtMz+R4ciHfxNd4U1c6xRNwcm2v4dOIJUZTXMv0ivJ3Jh4th\nuTbT7cDUTL5QEJ8vbrLjVBZuFZ5pglOOqbLT2VIyzsWT+FTswmKxO9PF7JiGB9Pzz6LCA3gq6ZR8\ndokb4iHDN0eBs9KvVdwybXhMtFOBdXgRnyW5XXzjbBcnb42YQ4OipV4VF0QznAosT/E3GAWT8COe\nHc1xL6Ib145jvHqc1ovTPQIbVfZ8r9iFeeOYWLOYjEfGsL5RTi14vFaQbYY/u2eJIXjFGJIaL1yG\nk//n2kY59eDsXJHPmBU4RezSzBSwPBf2Fa4W/3UGR3MsoRFOM8R30MYx5jiBCUxgJP4FsUPSXqCQ\nzR0AAAAASUVORK5CYII=\n",
       "prompt_number": 63,
       "text": [
        "y\u22c5cos(x\u22c5y)"
       ]
      }
     ],
     "prompt_number": 63
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "OK!"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can change the direction from which we approach the limiting point using the `dir` keywork argument:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "limit(1/x, x, 0, dir=\"+\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\infty$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAABUAAAANCAYAAABGkiVgAAAABHNCSVQICAgIfAhkiAAAAPdJREFU\nOI3l0q8vhVEcx/HXzK97Z9zZmCmMCcZmV7Ypd1TlBlF3i/+B8ugUBQVFJDCVIGhXkMRrCGY2v8Lz\nhGdn57lR8W3ns/fe53N+8AfTj200cY89TEa4cezgAidIUIkJu3GONUyghht8oJHjlnGHpVw2hn30\nhNIGFoOshEv8YAMzuMVopNQ01sMwidWXHquJLzxgvoCDLejIBe8F4AtW8IYBtNpIP0NpVxv4G68Y\nxCnKBVw5lLYwFQErOMQqNlHN1p0Bt4DH2C4HGM5lVVyjnssS6cOdYQ590uu5Qm+s/gh2cST9XseY\njXC1TPqMp6zMUEz4D+cX89gs5uxY/fAAAAAASUVORK5CYII=\n",
       "prompt_number": 64,
       "text": [
        "\u221e"
       ]
      }
     ],
     "prompt_number": 64
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "limit(1/x, x, 0, dir=\"-\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$-\\infty$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACUAAAANCAYAAAAuYadYAAAABHNCSVQICAgIfAhkiAAAASBJREFU\nOI3t1L8rxHEcx/EHnV+XuIUkRWQQ0tnUleVitRiMNoNb/A8sZydlwYLFyEBWBomJwWQ8YSHl1/C9\n07fre9elk5LX9P68Pq/Pu+en96cPf1wtWMYVrrGOvohcD1ZwiF1kkfgJoHocYBa9SOMUL8iEcpO4\nwETI68YGGqoNlcF4kdeEI3xgAYM4Q2fE+QHMVxsqW8JPCMb5hhuMlumxVChiIXMYa6ipEOQcc/n6\nuUTmAVM4QStyZfq9RkFdYqxCoGLVldl7xyO6sIcUniJy8UJR+02IYuXQH+EnsIUZLCKZX8eKcinc\nVonlS3Fsoj3kJQVjmw55WcHD38cImgXjPUZjtaGgA6vYFnwPOxiKyKXzUPe4E1ym7SeA/vUr+gTW\nkDD8mDETQgAAAABJRU5ErkJggg==\n",
       "prompt_number": 65,
       "text": [
        "-\u221e"
       ]
      }
     ],
     "prompt_number": 65
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Series\n",
      "\n",
      "Series expansion is also one of the most useful features of a CAS. In SymPy we can perform a series expansion of an expression using the `series` function:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "series(exp(x), x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$1 + x + \\frac{1}{2} x^{2} + \\frac{1}{6} x^{3} + \\frac{1}{24} x^{4} + \\frac{1}{120} x^{5} + \\operatorname{\\mathcal{O}}\\left(x^{6}\\right)$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAVYAAAAfCAYAAAC8j+WTAAAABHNCSVQICAgIfAhkiAAACHhJREFU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\nqodNpyQpAAAAAElFTkSuQmCC\n",
       "prompt_number": 66,
       "text": [
        "\n",
        "         2    3    4     5          \n",
        "        x    x    x     x           \n",
        "1 + x + \u2500\u2500 + \u2500\u2500 + \u2500\u2500 + \u2500\u2500\u2500 + O(x**6)\n",
        "        2    6    24   120          "
       ]
      }
     ],
     "prompt_number": 66
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "By default it expands the expression around $x=0$, but we can expand around any value of $x$ by explicitly include a value in the function call:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "series(exp(x), x, 1)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$e + e x + \\frac{1}{2} e x^{2} + \\frac{1}{6} e x^{3} + \\frac{1}{24} e x^{4} + \\frac{1}{120} e x^{5} + \\operatorname{\\mathcal{O}}\\left(x^{6}\\right)$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAYQAAAAfCAYAAADna7rnAAAABHNCSVQICAgIfAhkiAAACNNJREFU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\n",
       "prompt_number": 67,
       "text": [
        "\n",
        "             2      3      4      5          \n",
        "          \u212f\u22c5x    \u212f\u22c5x    \u212f\u22c5x    \u212f\u22c5x           \n",
        "\u212f + \u212f\u22c5x + \u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500 + O(x**6)\n",
        "           2      6      24    120           "
       ]
      }
     ],
     "prompt_number": 67
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "And we can explicitly define to which order the series expansion should be carried out:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "series(exp(x), x, 1, 10)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$e + e x + \\frac{1}{2} e x^{2} + \\frac{1}{6} e x^{3} + \\frac{1}{24} e x^{4} + \\frac{1}{120} e x^{5} + \\frac{1}{720} e x^{6} + \\frac{1}{5040} e x^{7} + \\frac{1}{40320} e x^{8} + \\frac{1}{362880} e x^{9} + \\operatorname{\\mathcal{O}}\\left(x^{10}\\right)$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAuEAAAAfCAYAAABJRxPSAAAABHNCSVQICAgIfAhkiAAADpVJREFU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\nhmEYhmEYhmEYhmEYhmEYhmEYhmGUmv8D+XCbaXJVq+AAAAAASUVORK5CYII=\n",
       "prompt_number": 68,
       "text": [
        "\n",
        "             2      3      4      5      6      7       8       9            \n",
        "          \u212f\u22c5x    \u212f\u22c5x    \u212f\u22c5x    \u212f\u22c5x    \u212f\u22c5x    \u212f\u22c5x     \u212f\u22c5x     \u212f\u22c5x             \n",
        "\u212f + \u212f\u22c5x + \u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500 + O(x**10)\n",
        "           2      6      24    120    720    5040   40320   362880           "
       ]
      }
     ],
     "prompt_number": 68
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The series expansion includes the order of the approximation, which is very useful for keeping track of the order of validity when we do calculations with series expansions of different order:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "s1 = cos(x).series(x, 0, 5)\n",
      "s1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$1 - \\frac{1}{2} x^{2} + \\frac{1}{24} x^{4} + \\operatorname{\\mathcal{O}}\\left(x^{5}\\right)$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAALwAAAAfCAYAAABdwafrAAAABHNCSVQICAgIfAhkiAAABexJREFU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\n",
       "prompt_number": 69,
       "text": [
        "\n",
        "     2    4          \n",
        "    x    x           \n",
        "1 - \u2500\u2500 + \u2500\u2500 + O(x**5)\n",
        "    2    24          "
       ]
      }
     ],
     "prompt_number": 69
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "s2 = sin(x).series(x, 0, 2)\n",
      "s2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$x + \\operatorname{\\mathcal{O}}\\left(x^{2}\\right)$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFQAAAAZCAYAAACvrQlzAAAABHNCSVQICAgIfAhkiAAAA2RJREFU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\njx49evTo0cB/Msqs+GlH4wsAAAAASUVORK5CYII=\n",
       "prompt_number": 70,
       "text": [
        "x + O(x**2)"
       ]
      }
     ],
     "prompt_number": 70
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expand(s1 * s2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$x + \\operatorname{\\mathcal{O}}\\left(x^{2}\\right)$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFQAAAAZCAYAAACvrQlzAAAABHNCSVQICAgIfAhkiAAAA2RJREFU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\njx49evTo0cB/Msqs+GlH4wsAAAAASUVORK5CYII=\n",
       "prompt_number": 71,
       "text": [
        "x + O(x**2)"
       ]
      }
     ],
     "prompt_number": 71
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "If we want to get rid of the order information we can use the `removeO` method:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expand(s1.removeO() * s2.removeO())"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{1}{24} x^{5} - \\frac{1}{2} x^{3} + x$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAHcAAAAfCAYAAADdnlK9AAAABHNCSVQICAgIfAhkiAAAA3xJREFU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\n",
       "prompt_number": 72,
       "text": [
        "\n",
        " 5    3    \n",
        "x    x     \n",
        "\u2500\u2500 - \u2500\u2500 + x\n",
        "24   2     "
       ]
      }
     ],
     "prompt_number": 72
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "But note that this is not the correct expansion of $\\cos(x)\\sin(x)$ to $5$th order:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(cos(x)*sin(x)).series(x, 0, 6)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$x - \\frac{2}{3} x^{3} + \\frac{2}{15} x^{5} + \\operatorname{\\mathcal{O}}\\left(x^{6}\\right)$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAL4AAAAfCAYAAABZNHfWAAAABHNCSVQICAgIfAhkiAAABixJREFU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\n",
       "prompt_number": 73,
       "text": [
        "\n",
        "       3      5          \n",
        "    2\u22c5x    2\u22c5x           \n",
        "x - \u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500 + O(x**6)\n",
        "     3      15           "
       ]
      }
     ],
     "prompt_number": 73
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Linear algebra"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Matrices\n",
      "\n",
      "Matrices are defined using the `Matrix` class:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "m11, m12, m21, m22 = symbols(\"m11, m12, m21, m22\")\n",
      "b1, b2 = symbols(\"b1, b2\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 74
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A = Matrix([[m11, m12],[m21, m22]])\n",
      "A"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left(\\begin{smallmatrix}m_{11} & m_{12}\\\\m_{21} & m_{22}\\end{smallmatrix}\\right)$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFUAAAAXBAMAAACISx4OAAAAMFBMVEX///9UVFSqqqpmZmYiIiK6\nuroQEBB2dnaIiIju7u4yMjJERETc3NyYmJjMzMwAAADbJP84AAAAAXRSTlMAQObYZgAAAV5JREFU\nKM+Vkj1Lw1AUht/apE3SNnY0WwZ/QHBWCJ0FOwiOTm4iFcEPHBUcOqgICi5mFTr4AyrERSgOdlfE\nSTsWdHGq5+Qm5nBd0gNJLs95uTk3eQCuxtLoyOq+I6//5K+zPqx4VhPwgNKZRty9fpZzh8DgGLPt\ncgdrmzB3Y6aC1LBK4I3pE9/m8AHDB5YBI8kK4uCGXlXieW65s4Uunl2ZFQTzDCLA6tCzPqZ9Xqoi\nK4ndY3ICVLhlNtHDw4HISnKenPYSydx55TPkxF684oXjY0c2Rl++e7qgEWPyzatqgHsUrEaEVtHs\nTIiNoll3PFW2VdQdM1Rny0z5PIw1d4ik7tDZtqUpd2aouUNEuYNykP6L1JTI/tHcIaLcgdNOW5kp\n9GE0d4godx5pjECach3r7hBR7uzT1Rem2J7uDpPEHTvKXE/rYtDR3CGi3Kkxrg/zzsok1Nwhotx5\nxVT1CzRChhzD5LsSAAAAAElFTkSuQmCC\n",
       "prompt_number": 75,
       "text": [
        "\n",
        "\u23a1m\u2081\u2081  m\u2081\u2082\u23a4\n",
        "\u23a2        \u23a5\n",
        "\u23a3m\u2082\u2081  m\u2082\u2082\u23a6"
       ]
      }
     ],
     "prompt_number": 75
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "b = Matrix([[b1], [b2]])\n",
      "b"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left(\\begin{smallmatrix}b_{1}\\\\b_{2}\\end{smallmatrix}\\right)$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACQAAAAcBAMAAAD7BwBrAAAAMFBMVEX///8iIiJUVFSYmJiqqqoQ\nEBB2dnbc3NyIiIhmZmbu7u4yMjJERES6urrMzMwAAADhNkqMAAAAAXRSTlMAQObYZgAAARNJREFU\nKM9jYAAB3t0MGCATlfsWiLfAOA0ggu8AkPgOE7oEIrgXAM360bQAIsRmACRmADGHFMcEBi2wmBQD\nA/svIM3TwLzhWjhYqBOocgKQZr3AqcAgDhaqB/IcgDT/AiAFEWI0YHgPMpdxgQ0DVIjzAIMp2CHT\nEmBCHAEMs+FOhgjx/mI4AxN58dMBrOEfwyY037L/ZZBEF/oCUYUUOEBVm9ACh+87gw6YAQ8ckI2t\nYAY8cBiYAxjug1yPCByQ61kvQAPnWU4BOPgaGDgToIFzgFcAJLR+AQNfADRwNnCBQo7BCIhVYIHD\nAlYlB8SrYYGzGmQWM8gclgmQwOECh30N2FoXiO0rq0AKZiH7V/6/AIQBADONQXXouV7VAAAAAElF\nTkSuQmCC\n",
       "prompt_number": 76,
       "text": [
        "\n",
        "\u23a1b\u2081\u23a4\n",
        "\u23a2  \u23a5\n",
        "\u23a3b\u2082\u23a6"
       ]
      }
     ],
     "prompt_number": 76
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "With `Matrix` class instances we can do the usual matrix algebra operations:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A**2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left(\\begin{smallmatrix}m_{11}^{2} + m_{12} m_{21} & m_{11} m_{12} + m_{12} m_{22}\\\\m_{11} m_{21} + m_{21} m_{22} & m_{12} m_{21} + m_{22}^{2}\\end{smallmatrix}\\right)$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAScAAAAnBAMAAAC/A5PGAAAAMFBMVEX///92dnaqqqpEREQiIiLc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\nAAAASUVORK5CYII=\n",
       "prompt_number": 77,
       "text": [
        "\n",
        "\u23a1    2                               \u23a4\n",
        "\u23a2 m\u2081\u2081  + m\u2081\u2082\u22c5m\u2082\u2081    m\u2081\u2081\u22c5m\u2081\u2082 + m\u2081\u2082\u22c5m\u2082\u2082\u23a5\n",
        "\u23a2                                    \u23a5\n",
        "\u23a2                                 2  \u23a5\n",
        "\u23a3m\u2081\u2081\u22c5m\u2082\u2081 + m\u2082\u2081\u22c5m\u2082\u2082   m\u2081\u2082\u22c5m\u2082\u2081 + m\u2082\u2082   \u23a6"
       ]
      }
     ],
     "prompt_number": 77
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A * b"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left(\\begin{smallmatrix}b_{1} m_{11} + b_{2} m_{12}\\\\b_{1} m_{21} + b_{2} m_{22}\\end{smallmatrix}\\right)$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAHwAAAAdBAMAAACamBVyAAAAMFBMVEX///8iIiJUVFSYmJiqqqoQ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\nQmCC\n",
       "prompt_number": 78,
       "text": [
        "\n",
        "\u23a1b\u2081\u22c5m\u2081\u2081 + b\u2082\u22c5m\u2081\u2082\u23a4\n",
        "\u23a2               \u23a5\n",
        "\u23a3b\u2081\u22c5m\u2082\u2081 + b\u2082\u22c5m\u2082\u2082\u23a6"
       ]
      }
     ],
     "prompt_number": 78
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "And calculate determinants and inverses, and the like:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A.det()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$m_{11} m_{22} - m_{12} m_{21}$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAI8AAAATCAYAAACpx16rAAAABHNCSVQICAgIfAhkiAAAAm5JREFU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\nAAAASUVORK5CYII=\n",
       "prompt_number": 79,
       "text": [
        "m\u2081\u2081\u22c5m\u2082\u2082 - m\u2081\u2082\u22c5m\u2082\u2081"
       ]
      }
     ],
     "prompt_number": 79
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A.inv()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left(\\begin{smallmatrix}\\frac{1}{m_{11}} + \\frac{m_{12} m_{21}}{m_{11}^{2} \\left(m_{22} - \\frac{m_{12} m_{21}}{m_{11}}\\right)} & - \\frac{m_{12}}{m_{11} \\left(m_{22} - \\frac{m_{12} m_{21}}{m_{11}}\\right)}\\\\- \\frac{m_{21}}{m_{11} \\left(m_{22} - \\frac{m_{12} m_{21}}{m_{11}}\\right)} & \\frac{1}{m_{22} - \\frac{m_{12} m_{21}}{m_{11}}}\\end{smallmatrix}\\right)$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAYkAAABEBAMAAACG8tRQAAAAMFBMVEX///9mZmbu7u5UVFRERES6\nuroyMjJ2dnbMzMwQEBDc3NwiIiKqqqqIiIiYmJgAAAAmjM/GAAAAAXRSTlMAQObYZgAABmdJREFU\naN7lWj1sHEUUft4737/v7AIKFImLLBkiCluu0iAslMoS4ToXRuKEhCUgwodAAiSiXEdF7CZBshA5\noTRuyNWnIB9VCgROA5FoYjcRAkEc58chf2ZmZ2d39mZn5s3cmYvhFfaOvfNmv9mdb7733gDELVUF\njJWP/PXMlSbALOQ+nuetzz9qYftrbPun7Tc+8R+E+nOy0U9xHUu/wFelBiysQRoWeWtutI3tr7Ef\n6/nuFf9BiD+rnn+GV0XkUxz32tkuQAcyMF0NWnVvF91fbanmS/CZ/yDUn2nWx4WZbSeiqGm6t0vj\nxRJFAfBD2CJ++kdRhO+h3fUdlczv4rXocr2ZiGJO3dmbGaulfvVReO+GranWAFBMws/wMnM0ZXZ2\nLJrq78AWBTeCYjJcz94sDABF9CDUn/GTWuNX2XofKLznXuCNo980BoqC+jPas/ziUvRWcteP2KD4\n4la3uP+Qtzb227H+/Rh1RPyZb5zuBhcnFDfMwSGw7GrAUPf9X/sxg9TS0vml93v/7G7+SCeIvQqV\nAbnc8RfjY4YiUz/M7wLOMHZZ2bJBkduyH6jc1P//rMPD/xauavZArwACRahpLmLUD2tRiQU+Wx7X\nPlBhzsFnis8M+5S8270Cj0s60Xmoad7CqB/WohJrwefzjBZFpurik6+DsQf+09/tFXhc0gkaJdQ0\nYw2M+mEtX2J1dL2YXQMXn7yXt0d/pnd6BV7i1hVomnwLo36CFpVY/ojeqg7FGrj4PMZve4f+qNQl\ngZeEItA0iyj1E7SIxGIjwimdiGg7+czw7e4MXSFvNiSBl4CCa5oPbNQPlVgdY7fsuJPPsNvmVQql\nJgm8BBRc05y0UD++xGIobmj6pLecfJb5NzRNWWi5qkURUBfXNJdBiE992lOrHyqxiNICFoMkUihl\nxGIXBBqVwlSVzzD4yNNF91ASeDFJF1AXt3MgxKfpcJlUDEJhRUGhlDTzNRBoNApTDT5znJVShGoL\nd3CxabCNnAv/QtiP0p43MTFh1rErCgqlb56iiGiUkupR4rNrVL0cRfYJmdlbhrsZdfFt5DKI8WnP\nl6K2RRWFdtgXJdBoCZkyCN9ZmWwYY4/1N/dQ10kQ41NKeyi7oaLQDlvdAo0iwlSIre5RIj5GnuA6\nBSg+jHHvZFUpfSAWbk4oyIe6zc6INCqHqQqfI5xpC/fITNy1QrEocq8QrSau28je1qCIdj3qTw5T\nFT4z4Q7xN2k8sEIRkTBhPyFaTVy3kWpdVQSlPmmGd1J/cpiq8LkeXpEILL9rhaI8jk0miSNmr2pz\nmuDi83R4daEKlR0UiHAbmccmk8QRp7Sd0i0Hn14UoW60sChCu4hMw8RQzDtFSVqfAnQHFIUt+xHL\nBvL81gHF79HlchOmLVFgk0mBgPo3fC7XYHoVDru9/p9BsfP0PdasNYqKC4reTM1gbWHN7v5lRxS9\nWZUBW8cSRdMNRW9WZbgoNppoBZIkCsKsypBRkI3dBYWUqRkqigtkJ7/vPlqUqRkqCqJpRx45DyZk\nagZqltu+99AcsepWR5SpGabl7pBgfS/xXwOt9yisMhhvJRJ35273/I1VrYZnQZgdy+lpjX5N3r2n\nTH6wHTWovQQZPO05mRHKTxs452cRM9fTiqpW9jsqr72wDJ72nEyGfmfPN1HB0Rxi5oKWXLVy2FGD\n2gvL4GnPyVRoumETE7wpq1bxmWMtuWrlsKOy2gvP4OnOyazQhMalGYzva6iZYy25auXySlpimlB3\nTuY65fo8Ki+6hpk53pKrVk4oojSh9pyMXyPOYtJqplNj8ZmTq1amMEU+XBervWjPyZz31+0eAkW/\nVStTmCIdrovXXnTnZEaZ+HhPR6FBjpfmtdEzl1C1MoUpUdWiYrt1p5gov9lUzxPP8dIaA3rmEqpW\nxjCl1HZdQBX2visN9TyZ6j3YqpUxTMFWLWQLXkJ2V0+hHIXDzOVQ4bCuamG2peD3lwYKDapWDjOH\nOhCrq1rI9RxpCB7l3azqKbQTq0nbzFwZs3nrqhZyPUcSFZwGiw0jiqhqZTVzI+Puu15y1UKyTf4K\ncup9j+d4o0/DauYyNfeYPrkSItnX4dUffQf0iplb7zvaMKFIRSowZX7x204zd/rAUYgF91NGb+mW\nw5he/aBRFMTv90XnKEk7Jg66dsMx1HMKliP0WbU6RHYQlbD/kf0DZMqWDlnTcFQAAAAASUVORK5C\nYII=\n",
       "prompt_number": 80,
       "text": [
        "\n",
        "\u23a1 1          m\u2081\u2082\u22c5m\u2082\u2081                 -m\u2081\u2082       \u23a4\n",
        "\u23a2\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500  \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5\n",
        "\u23a2m\u2081\u2081      2 \u239b      m\u2081\u2082\u22c5m\u2082\u2081\u239e      \u239b      m\u2081\u2082\u22c5m\u2082\u2081\u239e\u23a5\n",
        "\u23a2      m\u2081\u2081 \u22c5\u239cm\u2082\u2082 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f  m\u2081\u2081\u22c5\u239cm\u2082\u2082 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u23a5\n",
        "\u23a2           \u239d        m\u2081\u2081  \u23a0      \u239d        m\u2081\u2081  \u23a0\u23a5\n",
        "\u23a2                                               \u23a5\n",
        "\u23a2           -m\u2082\u2081                      1         \u23a5\n",
        "\u23a2   \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500         \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500   \u23a5\n",
        "\u23a2       \u239b      m\u2081\u2082\u22c5m\u2082\u2081\u239e               m\u2081\u2082\u22c5m\u2082\u2081   \u23a5\n",
        "\u23a2   m\u2081\u2081\u22c5\u239cm\u2082\u2082 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f         m\u2082\u2082 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500   \u23a5\n",
        "\u23a3       \u239d        m\u2081\u2081  \u23a0                 m\u2081\u2081     \u23a6"
       ]
      }
     ],
     "prompt_number": 80
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Solving equations\n",
      "\n",
      "For solving equations and systems of equations we can use the `solve` function:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "solve(x**2 - 1, x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{bmatrix}1, & -1\\end{bmatrix}$$"
       ],
       "output_type": "pyout",
       "prompt_number": 81,
       "text": [
        "[1, -1]"
       ]
      }
     ],
     "prompt_number": 81
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "solve(x**4 - x**2 - 1, x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{bmatrix}\\mathbf{\\imath} \\sqrt{- \\frac{1}{2} + \\frac{1}{2} \\sqrt{5}}, & - \\sqrt{\\frac{1}{2} + \\frac{1}{2} \\sqrt{5}}, & - \\mathbf{\\imath} \\sqrt{- \\frac{1}{2} + \\frac{1}{2} \\sqrt{5}}, & \\sqrt{\\frac{1}{2} + \\frac{1}{2} \\sqrt{5}}\\end{bmatrix}$$"
       ],
       "output_type": "pyout",
       "prompt_number": 82,
       "text": [
        "[\n",
        "       \u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\n",
        "      \u2571         \u23bd\u23bd\u23bd \n",
        "     \u2571    1   \u2572\u2571 5  \n",
        "\u2148\u22c5  \u2571   - \u2500 + \u2500\u2500\u2500\u2500\u2500 \n",
        "  \u2572\u2571      2     2   ,\n",
        " \n",
        "      \u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\n",
        "     \u2571       \u23bd\u23bd\u23bd \n",
        "    \u2571  1   \u2572\u2571 5  \n",
        "-  \u2571   \u2500 + \u2500\u2500\u2500\u2500\u2500 \n",
        " \u2572\u2571    2     2   ,\n",
        " \n",
        "        \u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\n",
        "       \u2571         \u23bd\u23bd\u23bd \n",
        "      \u2571    1   \u2572\u2571 5  \n",
        "-\u2148\u22c5  \u2571   - \u2500 + \u2500\u2500\u2500\u2500\u2500 \n",
        "   \u2572\u2571      2     2   ,\n",
        " \n",
        "     \u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\n",
        "    \u2571       \u23bd\u23bd\u23bd \n",
        "   \u2571  1   \u2572\u2571 5  \n",
        "  \u2571   \u2500 + \u2500\u2500\u2500\u2500\u2500 \n",
        "\u2572\u2571    2     2   ]"
       ]
      }
     ],
     "prompt_number": 82
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "System of equations:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "solve([x + y - 1, x - y - 1], [x,y])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 83,
       "text": [
        "{x: 1, y: 0}"
       ]
      }
     ],
     "prompt_number": 83
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In terms of other symbolic expressions:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "solve([x + y - a, x - y - c], [x,y])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 84,
       "text": [
        "{x: \n",
        "a   c\n",
        "\u2500 + \u2500\n",
        "2   2, y: \n",
        "a   c\n",
        "\u2500 - \u2500\n",
        "2   2}"
       ]
      }
     ],
     "prompt_number": 84
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Quantum mechanics: noncommuting variables\n",
      "\n",
      "How about non-commuting symbols? In quantum mechanics we need to work with noncommuting operators, and SymPy has a nice support for noncommuting symbols and even a subpackage for quantum mechanics related calculations!"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sympy.physics.quantum import *"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 85
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### States\n",
      "\n",
      "We can define symbol states, kets and bras:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Ket('psi')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$${\\left|\\psi\\right\\rangle }$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAABoAAAAXCAYAAAAV1F8QAAAABHNCSVQICAgIfAhkiAAAAZtJREFU\nSInt1T1oVEEUBeDPRWPwr4gGJYIIabcSSSwELWysFFJYBYtUCirW2ghWIgQSQhptNCCCIkpIvZ2J\npgqxCkFEQZNGUPA/xGJms5PHe/t2ZcXGAw9mzp17z507d+bxj/D4bwWuZOZ72/C9ncONYWsrQq2i\nB59z+BqGOil0IgbN4inOdFJoEM9z+DW8xECnhLbhR4HtLkbaETqMcUzjXMLvw2ocb8cirif2T8L5\nHWyWaS0ZT6ALl7GQ8ENC6epCS5jJxOnHzZQo2tExzArlOYXXie0I5uP4O65gJeO/jD7sKBN6g0c4\ngNN4kEluLZm/Exogi3sYLhN6j69CmX7hWeR78SGzdgBPCuJsKROq4zhe4Eucn7T5HLtxKCaWxXlh\nVy0J7bK5/lWNxqjgGkZz/PqjeD3BUqGHQoftTLh17Mct3MfHHL8LmEyJ3AcwwVRM5o7QaYPCnfmG\nG/Lfuz3xe9sscK2J7ZJQujJc1bhnG2jnCariVcmaCo5irkzoZ0GAepuulwid1bgKTbG7gK/iYgv+\nhT++//hj/AZMWUeGKO+s+AAAAABJRU5ErkJggg==\n",
       "prompt_number": 86,
       "text": [
        "\u2758\u03c8\u27e9"
       ]
      }
     ],
     "prompt_number": 86
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Bra('psi')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$${\\left\\langle \\psi\\right|}$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAABkAAAAXCAYAAAD+4+QTAAAABHNCSVQICAgIfAhkiAAAAZ1JREFU\nSInt1T9IVlEYx/GPb/YHLIIyKoIIanyXStJByKHFLXFoya2loBoca6kxggiRhtxyaVAjovndkgrC\n0KkggshqkRCsLLHh3PC813vee981fJZ7zvP8zvme85znnMt/YFNQayG438ZkdxP+/a0gA5ivCNiH\n5VaCFOQiJitCzqLRLuQ4PuNHRUgvXrQLuYwHFQGwHavtQPaiC4s5/zGM4RkuRP5ufMvaO4VzvFm2\nqlH0FPjHsQPX8DbyDwvp+gd5h+dRvEHzTrbhJF7nAH2YFVJyDh+i2KlI/wvX8TW/ws6ofR5PCnbx\nEXM4hEGMRLEa1qL+J7zKT1BLtGNbFCptGH/wNPMfwJec9gxmWkGmMZQAQT9eYiXrD2i+H7tw1Oai\naYKs4Q1OJyC7Nee7bqMIariBe0UD8yl6iEsJyGOhkroi3zoO4g4eYaloYGeu/11Ix2Gbtz2ZLWpC\nqKhe4U78xC0l71feTuB2ieaqkK4ya1BcUe9xRDjIlNWxUAEiBSHkdyQR68i+6xXm/10mSP206rhS\nAQB7Kuq2bMP+AqRpSD7IVgl6AAAAAElFTkSuQmCC\n",
       "prompt_number": 87,
       "text": [
        "\u27e8\u03c8\u2758"
       ]
      }
     ],
     "prompt_number": 87
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "u = Ket('0')\n",
      "d = Ket('1')\n",
      "\n",
      "a, b = symbols('alpha beta', complex=True)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 88
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "phi = a * u + sqrt(1-abs(a)**2) * d; phi"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\alpha {\\left|0\\right\\rangle } + \\sqrt{- \\lvert{\\alpha}\\rvert^{2} + 1} {\\left|1\\right\\rangle }$$"
       ],
       "output_type": "pyout",
       "prompt_number": 89,
       "text": [
        "\n",
        "           \u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd    \n",
        "          \u2571      2         \n",
        "\u03b1\u22c5\u27580\u27e9 + \u2572\u2571  - \u2502\u03b1\u2502  + 1 \u22c5\u27581\u27e9"
       ]
      }
     ],
     "prompt_number": 89
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Dagger(phi)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\sqrt{- \\lvert{\\alpha}\\rvert^{2} + 1} {\\left\\langle 1\\right|} + \\overline{\\alpha} {\\left\\langle 0\\right|}$$"
       ],
       "output_type": "pyout",
       "prompt_number": 90,
       "text": [
        "\n",
        "   \u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd            \n",
        "  \u2571      2            \u23bd    \n",
        "\u2572\u2571  - \u2502\u03b1\u2502  + 1 \u22c5\u27e81\u2758 + \u03b1\u22c5\u27e80\u2758"
       ]
      }
     ],
     "prompt_number": 90
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Dagger(phi) * d"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left(\\sqrt{- \\lvert{\\alpha}\\rvert^{2} + 1} {\\left\\langle 1\\right|} + \\overline{\\alpha} {\\left\\langle 0\\right|}\\right) {\\left|1\\right\\rangle }$$"
       ],
       "output_type": "pyout",
       "prompt_number": 91,
       "text": [
        "\n",
        "\u239b   \u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd            \u239e    \n",
        "\u239c  \u2571      2            \u23bd    \u239f    \n",
        "\u239d\u2572\u2571  - \u2502\u03b1\u2502  + 1 \u22c5\u27e81\u2758 + \u03b1\u22c5\u27e80\u2758\u23a0\u22c5\u27581\u27e9"
       ]
      }
     ],
     "prompt_number": 91
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Use `qapply` to distribute a mutiplication:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "qapply(Dagger(phi) * d)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\sqrt{- \\lvert{\\alpha}\\rvert^{2} + 1} \\left\\langle 1 \\right. {\\left|1\\right\\rangle } + \\overline{\\alpha} \\left\\langle 0 \\right. {\\left|1\\right\\rangle }$$"
       ],
       "output_type": "pyout",
       "prompt_number": 92,
       "text": [
        "\n",
        "   \u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd                \n",
        "  \u2571      2              \u23bd      \n",
        "\u2572\u2571  - \u2502\u03b1\u2502  + 1 \u22c5\u27e81\u27581\u27e9 + \u03b1\u22c5\u27e80\u27581\u27e9"
       ]
      }
     ],
     "prompt_number": 92
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "qapply(Dagger(phi) * u)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\sqrt{- \\lvert{\\alpha}\\rvert^{2} + 1} \\left\\langle 1 \\right. {\\left|0\\right\\rangle } + \\overline{\\alpha} \\left\\langle 0 \\right. {\\left|0\\right\\rangle }$$"
       ],
       "output_type": "pyout",
       "prompt_number": 93,
       "text": [
        "\n",
        "   \u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd\u23bd                \n",
        "  \u2571      2              \u23bd      \n",
        "\u2572\u2571  - \u2502\u03b1\u2502  + 1 \u22c5\u27e81\u27580\u27e9 + \u03b1\u22c5\u27e80\u27580\u27e9"
       ]
      }
     ],
     "prompt_number": 93
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Operators"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A = Operator('A')\n",
      "B = Operator('B')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 94
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Check if they are commuting!"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A * B == B * A"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADIAAAASCAYAAAAZk42HAAAABHNCSVQICAgIfAhkiAAAArRJREFU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\nx2n3Adczkwoss95+AAAAAElFTkSuQmCC\n",
       "prompt_number": 95,
       "text": [
        "False"
       ]
      }
     ],
     "prompt_number": 95
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expand((A+B)**3)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$A B A + A \\left(B\\right)^{2} + \\left(A\\right)^{2} B + \\left(A\\right)^{3} + B A B + B \\left(A\\right)^{2} + \\left(B\\right)^{2} A + \\left(B\\right)^{3}$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAfsAAAAZCAYAAAA2cIfvAAAABHNCSVQICAgIfAhkiAAABzdJREFU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\noVAoFAqFQqFQKBQKhUKhUGjkfz6hW3tDMIC5AAAAAElFTkSuQmCC\n",
       "prompt_number": 96,
       "text": [
        "\n",
        "           2    2      3              2    2      3\n",
        "A\u22c5B\u22c5A + A\u22c5B  + A \u22c5B + A  + B\u22c5A\u22c5B + B\u22c5A  + B \u22c5A + B "
       ]
      }
     ],
     "prompt_number": 96
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "c = Commutator(A,B)\n",
      "c"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[A,B\\right]$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAC4AAAAWCAYAAAC/kK73AAAABHNCSVQICAgIfAhkiAAAAj5JREFU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\nMid3rp42fI8AAAAASUVORK5CYII=\n",
       "prompt_number": 97,
       "text": [
        "[A,B]"
       ]
      }
     ],
     "prompt_number": 97
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can use the `doit` method to evaluate the commutator:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "c.doit()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$A B - B A$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAE4AAAASCAYAAAD15uiRAAAABHNCSVQICAgIfAhkiAAAAitJREFU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\ngg==\n",
       "prompt_number": 98,
       "text": [
        "A\u22c5B - B\u22c5A"
       ]
      }
     ],
     "prompt_number": 98
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can mix quantum operators with C-numbers:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "c = Commutator(a * A, b * B)\n",
      "c"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\alpha \\beta \\left[A,B\\right]$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEYAAAAWCAYAAAB9oOpzAAAABHNCSVQICAgIfAhkiAAAA61JREFU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\n",
       "prompt_number": 99,
       "text": [
        "\u03b1\u22c5\u03b2\u22c5[A,B]"
       ]
      }
     ],
     "prompt_number": 99
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "To expand the commutator, use the `expand` method with the `commutator=True` keyword argument:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "c = Commutator(A+B, A*B)\n",
      "c.expand(commutator=True)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- \\left[A,B\\right] B + A \\left[A,B\\right]$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJ4AAAAWCAYAAAAxZiXOAAAABHNCSVQICAgIfAhkiAAAA4NJREFU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\nQmCC\n",
       "prompt_number": 100,
       "text": [
        "-[A,B]\u22c5B + A\u22c5[A,B]"
       ]
      }
     ],
     "prompt_number": 100
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Dagger(Commutator(A, B))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- \\left[A^{\\dagger},B^{\\dagger}\\right]$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFUAAAAcCAYAAAAQovP+AAAABHNCSVQICAgIfAhkiAAAAupJREFU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\nB42um8D6ha2RAAAAAElFTkSuQmCC\n",
       "prompt_number": 101,
       "text": [
        "\n",
        " \u23a1 \u2020  \u2020\u23a4\n",
        "-\u23a3A ,B \u23a6"
       ]
      }
     ],
     "prompt_number": 101
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "ac = AntiCommutator(A,B)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 102
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "ac.doit()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$A B + B A$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFMAAAAUCAYAAADx7wHUAAAABHNCSVQICAgIfAhkiAAAAmxJREFU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\nAElFTkSuQmCC\n",
       "prompt_number": 103,
       "text": [
        "A\u22c5B + B\u22c5A"
       ]
      }
     ],
     "prompt_number": 103
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "#### Example: Quadrature commutator"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's look at the commutator of the electromagnetic field quadatures $x$ and $p$. We can write the quadrature operators in terms of the creation and annihilation operators as:\n",
      "\n",
      "$\\displaystyle x = (a + a^\\dagger)/\\sqrt{2}$\n",
      "\n",
      "$\\displaystyle p = -i(a - a^\\dagger)/\\sqrt{2}$\n"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "X = (A + Dagger(A))/sqrt(2)\n",
      "X"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{1}{2} \\sqrt{2} \\left(A^{\\dagger} + A\\right)$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAHUAAAAjCAYAAACq00VWAAAABHNCSVQICAgIfAhkiAAABKlJREFU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\nAAAAAElFTkSuQmCC\n",
       "prompt_number": 104,
       "text": [
        "\n",
        "  \u23bd\u23bd\u23bd \u239b \u2020    \u239e\n",
        "\u2572\u2571 2 \u22c5\u239dA  + A\u23a0\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "      2       "
       ]
      }
     ],
     "prompt_number": 104
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "P = -I * (A - Dagger(A))/sqrt(2)\n",
      "P"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- \\frac{1}{2} \\sqrt{2} \\mathbf{\\imath} \\left(- A^{\\dagger} + A\\right)$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJsAAAAjCAYAAACD+HiUAAAABHNCSVQICAgIfAhkiAAABUJJREFU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\n",
       "prompt_number": 105,
       "text": [
        "\n",
        "   \u23bd\u23bd\u23bd   \u239b   \u2020    \u239e\n",
        "-\u2572\u2571 2 \u22c5\u2148\u22c5\u239d- A  + A\u23a0\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "         2         "
       ]
      }
     ],
     "prompt_number": 105
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's expand the commutator $[x,p]$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Commutator(X, P).expand(commutator=True)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- \\mathbf{\\imath} \\left[A^{\\dagger},A\\right]$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFIAAAAcCAYAAADyfuiHAAAABHNCSVQICAgIfAhkiAAAAmdJREFU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\nrkJggg==\n",
       "prompt_number": 106,
       "text": [
        "\n",
        "   \u23a1 \u2020  \u23a4\n",
        "-\u2148\u22c5\u23a3A ,A\u23a6"
       ]
      }
     ],
     "prompt_number": 106
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Here we see directly that the well known commutation relation for the quadratures\n",
      "\n",
      "$[x,p]=i$\n",
      "\n",
      "is a directly related to\n",
      "\n",
      "$[A, A^\\dagger]=1$ \n",
      "\n",
      "(which SymPy does not know about, and does not simplify)."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "For more details on the quantum module in SymPy, see:\n",
      "\n",
      "* http://docs.sympy.org/0.7.2/modules/physics/quantum/index.html\n",
      "* http://nbviewer.ipython.org/urls/raw.github.com/ipython/ipython/master/docs/examples/notebooks/sympy_quantum_computing.ipynb"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Further reading\n",
      "\n",
      "* http://sympy.org/en/index.html - The SymPy projects web page.\n",
      "* https://github.com/sympy/sympy - The source code of SymPy.\n",
      "* http://live.sympy.org - Online version of SymPy for testing and demonstrations.\n"
     ]
    }
   ],
   "metadata": {}
  }
 ]
}