ch03-code-listing.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Chapter 3: Symbolic computing"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Robert Johansson\n",
    "\n",
    "Source code listings for [Numerical Python - A Practical Techniques Approach for Industry](http://www.apress.com/9781484205549) (ISBN 978-1-484205-54-9).\n",
    "\n",
    "The source code listings can be downloaded from http://www.apress.com/9781484205549"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import sympy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "sympy.init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from sympy import I, pi, oo"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x = sympy.Symbol(\"x\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "y = sympy.Symbol(\"y\", real=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y.is_real"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x.is_real is None"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.Symbol(\"z\", imaginary=True).is_real"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x = sympy.Symbol(\"x\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "y = sympy.Symbol(\"y\", positive=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACcAAAAVBAMAAAA3P+qgAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAInarRM2ZVBDdiWbv\nuzJCz3LGAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA5UlEQVQYGWNggABG/f9w8AEqxsDsAGMh0aJI\nbDgzHc5CMDgmQNlcuRlwUbYCKHMhgylcsBPGmsjwXgDGXg1jTGGQd4Cy4UYC+fUNUEGmDVAGkJoE\nYroAMSeIAQFcCUCa6SiQUIIIgMhtIEJC34GBAeRKDiVDp0UMLAtYHgA5/RcYuBcA6S6GnQpTGZ6W\n1zYAOWwfGThB9AqG9Q2FDOf//weyGRi/MewC0Q0MsSAKCuwbZkBYn2AiQPr98gQwj+8fkiDr0QYQ\nr4LpA4MGXJj9G4jJ8p33AyOSxwJBghxGGmq7QQwgAAA+Pzg+3uwqMgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\sqrt{x^{2}}$$"
      ],
      "text/plain": [
       "   ____\n",
       "  ╱  2 \n",
       "╲╱  x  "
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.sqrt(x ** 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAANBAMAAACN24kIAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHaZIu+JVM27RDKr\nZt2dj8xZAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAV0lEQVQIHWNgVFZgcBVgYGMPYNBnYHBnbmCY\nycAgwKrA8ImBgWG9APdvIGXBwBQApOYwcB4AUpYM6y8AKZkkewcgBZJlYOBpYPwFpPgucCoAKS4l\nayAJAGRhDUaSCq7HAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$y$$"
      ],
      "text/plain": [
       "y"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.sqrt(y ** 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "n1, n2, n3 = sympy.Symbol(\"n\"), sympy.Symbol(\"n\", integer=True), sympy.Symbol(\"n\", odd=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEYAAAAUBAMAAADCRlZLAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\cos{\\left (\\pi n \\right )}$$"
      ],
      "text/plain": [
       "cos(π⋅n)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.cos(n1 * pi)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADIAAAAXBAMAAAC7cAEHAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIma7zZnddlTvRIkQ\nMqvFy5UvAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA4ElEQVQoFWNgwAKYgq6XYhEGCjGtV+AUwCol\nU8bAf4CBgVGBARkwgRQ/Z1gJJIWRxYHsh0A8lyGUl4HBEUmm8wADwxGgMR8YsnoZmALgMjxP3wNl\nuAsYuB4wVCswsCE7oh4owzMBqlQMrgXIAMkweEFFULwElgmFyhhBaTAFllGBimRgyLRBRWYB7XRL\nA4KUC1B77iNkELrApsFkME1bBlWJ2wVbEWZB7YmGisD0grhx8/0UGBh8oTIooQMSY0yAyiCFKEQE\nGKJQYAhjQGlgLEABeswhVDIqwNSAaXBsg1gAnnUt8qpUaKMAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left(-1\\right)^{n}$$"
      ],
      "text/plain": [
       "    n\n",
       "(-1) "
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.cos(n2 * pi)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABgAAAAPBAMAAAAMihLoAAAAJFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAADHJj5lAAAAC3RSTlMAEM3dMlTvq5l2ZtVdCTcAAAAJcEhZcwAA\nDsQAAA7EAZUrDhsAAAAqSURBVAgdY2DAClgTEcLi7RsRHAZOMjlCxiCgwkC2ATA3cJRtqoKxwTQA\nC0AL2ft3JesAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$-1$$"
      ],
      "text/plain": [
       "-1"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.cos(n3 * pi)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "a, b, c = sympy.symbols(\"a, b, c\", negative=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "d, e, f = sympy.symbols(\"d, e, f\", positive=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Numbers"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "i = sympy.Integer(19)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\"i = 19 [type <class 'sympy.core.numbers.Integer'>]\""
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"i = {} [type {}]\".format(i, type(i))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(True, True, True)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "i.is_Integer, i.is_real, i.is_odd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "f = sympy.Float(2.3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\"f = 2.30000000000000 [type <class 'sympy.core.numbers.Float'>]\""
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"f = {} [type {}]\".format(f, type(f))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(False, True, False)"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.is_Integer, f.is_real, f.is_odd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "i, f = sympy.sympify(19), sympy.sympify(2.3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sympy.core.numbers.Integer"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(i)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sympy.core.numbers.Float"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "n = sympy.Symbol(\"n\", integer=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(True, False, None, True)"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n.is_integer, n.is_Integer, n.is_positive, n.is_Symbol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "i = sympy.Integer(19)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(True, True, True, False)"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "i.is_integer, i.is_Integer, i.is_positive, i.is_Symbol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$8663234049605954426644038200675212212900743262211018069459689001$$"
      ],
      "text/plain": [
       "8663234049605954426644038200675212212900743262211018069459689001"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "i ** 50"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAABmYAAAAPBAMAAAAMv22AAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nIs1wrmlGtoZ0d9/7854D7zlw9g78H92pRuBUjuYeAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000$$"
      ],
      "text/plain": [
       "933262154439441526816992388562667004907159682643816214685929638952175999932299\n",
       "156089414639761565182862536979208272237582511852109168640000000000000000000000\n",
       "00"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.factorial(100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'0.2999999999999999888977698'"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"%.25f\" % 0.3  # create a string represention with 25 decimals"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAARIAAAAOBAMAAAAReo5EAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nspDWMD0hkygG+nK1pNH/ADnZ4dvbyzcpAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$0.2999999999999999888977698$$"
      ],
      "text/plain": [
       "0.2999999999999999888977698"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.Float(0.3, 25)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABoAAAAOBAMAAADDIxFwAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJZjLNVN0i77ur\nRHZ72Yd1AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAApUlEQVQIHWNgEDIxZQCBycbbgWQ6g8QEIMWp\nwFDvwMCygIE5Acjj+87A38DAfYCB/SOQx7uWQT6Bgf8AA+93IA8IgCrlFRh4/4E5jM0MDPMNGFg/\ng3icFopAngKUx8BgdQFJJQPXQpAp7CBTOCcwcH9n4N7AwAKygf8jiAe0nSkByOPawMDxk4GhnUEq\ngPEbA4sDQ/0DBgbhd0cZGHoYGC4p7WUAAA3jJi9CuhBLAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$0.3$$"
      ],
      "text/plain": [
       "0.3000000000000000000000000"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.Float('0.3', 25)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Rationals"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABUAAAArBAMAAABlSd54AAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAVO8Qq5l2zWYiibvdRH6G\nTkUAAAAJcEhZcwAADsQAAA7EAZUrDhsAAACRSURBVCgVY2BAAUIGQC6YYHb1M2AAEyAFeSBxMEFX\ndoheSQMDmAA5gqbgHRw8oII9s1ZtBJsCDErOAAa5BgZIULI9ZOA7AJIABiWvLoNdAZQNpEBqoGHL\nfArEBAc355pgOJuBYdEFmDgDA5MylM05gYH7IZTN9xjBZtrAwPMSKs7RwCCXAGSDg/Jq6F6QMAIA\nAA3gNbSDbGNZAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{11}{13}$$"
      ],
      "text/plain": [
       "11\n",
       "──\n",
       "13"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.Rational(11, 13)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "r1 = sympy.Rational(2, 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "r2 = sympy.Rational(4, 5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABUAAAArBAMAAABlSd54AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHaJmUSrVGYizbvd\n7zLJaKAlAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAzElEQVQoFWNgAAIhZVcQBQYODGwFUCZ7AwPn\nAiibeQMD4wEom/W3AHMDlM2Q/8MCxmRg/r8Qzi6L+RsA5XAkMOhfhLJ5JzDwfYey5YF0BkKcYSeU\nzbqQgUMBymZ45oxwG0yMFvR/OPhABeMXCTwBh2KtAAPD/f+fgSYypuQD2YuTJoBN1weyE6D2oLMz\nmvzgajYyzA8AcUBqGBi4D4BICJvlJ5TNtoCB+SuUzbuAgeUjlM3CwMCzAMrmdGDoBVnsdn9ZA8MU\nM0eQMAIAAJXjNTEg0pUqAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{8}{15}$$"
      ],
      "text/plain": [
       "8/15"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r1 * r2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAqBAMAAACXcryGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAzXYQMplU74mrIma7\nRN0SDTw+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAqklEQVQYGWNgEDJwTWBgYKj//wNIMkhMWgCi\nJoAIODV7UwiId5hh/QUQzcDzAEyx/mJg4BRg4PjLwMAuwMD6mYGBlYGBT4CBgSuAYfsCoJKlncFg\nheQR/0HgA3F6OzQcgAoNL7ACbWc4ysBzgIGBF+g6IOApAFP8Ik2xQIa9NAN7ApD6zMBazMDA/4CB\nG+h4oJO5fwO9UgDmsQDlQHpmMJgDVTJwhE5hYAAAsEYnsoUVooQAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{5}{6}$$"
      ],
      "text/plain": [
       "5/6"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r1 / r2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x, y, z = sympy.symbols(\"x, y, z\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "f = sympy.Function(\"f\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sympy.core.function.UndefinedFunction"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACUAAAAUBAMAAAD4luk4AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJl2IquJVETdZu8y\nu83OyatpAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA/klEQVQYGVWQP04CURCHv4WVP8ouxMJAbLbw\nAIAX2BYbC4kNjcbKCkywhsQLWGgJIWqjrRfgCDT0XMBkiYUUFv7m7YbgJDPze9+bmZc3kNlZlt/S\nfNiO8RcZ24uduBwNCFMJ3sqYt6nDiylnNxaLicKTO1s4tlB4BF+eWTiG/d5zRD7SjNPugxpzklSb\nEMzgiPfmVDUrsfM5lAdwTb821HgbM5IfjKHGlxSVROFWbgy+LTi2lLBeKr/GStb7Kbc37vIJrfQN\nbyNWiCj9BImvq9yVatdiavAuWu0PyXCukoXEzt86cB/OjP3bwaRhiCB2Kd3V64k7bHe6vRTe2f0f\nX/ExQF2OnBgAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$f{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "f(x)"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "g = sympy.Function(\"g\")(x, y, z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAE0AAAAUBAMAAAA6pq28AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$g{\\left (x,y,z \\right )}$$"
      ],
      "text/plain": [
       "g(x, y, z)"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEcAAAAVBAMAAADm2O7QAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZpkQ3Ynvq81UMrtE\ndiLw+n06AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABS0lEQVQoFbVSMUjDUBB9aQxN2jR2cRTTQShI\nIZtTUEE3hQ7iLLiIVOig4NjFvVNdM7h1sJtTsYN0E0tAnUTdHZQuTuK7/2PI0uDiDfffvff4/+74\nAIwaZkepEYhobs22UHGPRDY+ck2oi2zt55t2/2La+wdT+PT+MFG9hQ14B7rL21apCqjnih2uoWns\nzPFkNHuo6HnNYbxCYknYXsQJA2vqvErldesoDwTBhCfA9oG+LMuErQWiY8xPSEn0A8knLMeRoFRw\np7hSDJfdFsWWLD0hFVDo4FEIxoXKqidrHXCCF1TU3Si3ca1EOD6KROky77s3eIP5Ra4wcD81ulxY\n3M6aVp/DWhXY5GXmafytkW8cjrImYgm9BXaVIEWmz6mK6+J5hrWRRgmpTJn/FJLfwJBZUBLqP+H8\ntwTbwl0cMQvS4bbU6eT+8WUO8wPqJkPkxFO6HAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left\\{x, y, z\\right\\}$$"
      ],
      "text/plain": [
       "set([x, y, z])"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g.free_symbols"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sin"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.sin"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADcAAAAUBAMAAADbzbjtAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMnZUZs0Qu91E7yKJ\nmaurDqYVAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABKUlEQVQoFWNgQAdboALeUJpxEUIFdwGUzbcB\nwmD8g5BkhYoxsAUgBGEsXxiD4RCcBWd0w1lZEBa3IFyEuwHOZJ3AwKjsFLD9H4PoWiE9BaA40wUG\nBjal15OAZvJeYEhnYGhgiGNgaFfg/g2UZHdgYMhh8FBoZWBgCWCYlcCgwFDHwFDFwPAFKMlhwMBw\njOF8gjUDA3MDA8d3bQaQZCkDwyegJNcEBoYEhnAgi4HxAwPb0f8XQJLlCEkICyS5jYHlJ5IkyFgG\nxo8gnZwNDEYMDE+QJEEOMmP6wKAGdpBVAkM1yEyosTwXGDg/s3/gBqrhPcAwx8lngtz/Z3L/H5r/\negA2i+2Rmq4L0FhWASCBChDBtwNVAsTDG/DsG6AasEUZPLLhqpBNR0omACUOR3UP731RAAAAAElF\nTkSuQmCC\n",
      "text/latex": [
       "$$\\sin{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "sin(x)"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.sin(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABgAAAAPBAMAAAAMihLoAAAAJFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAADHJj5lAAAAC3RSTlMAEM3dMlTvq5l2ZtVdCTcAAAAJcEhZcwAA\nDsQAAA7EAZUrDhsAAAAqSURBVAgdY2DAClgTEcLi7RsRHAZOMjlCxiCgwkC2ATA3cJRtqoKxwTQA\nC0AL2ft3JesAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$-1$$"
      ],
      "text/plain": [
       "-1"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.sin(pi * 1.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "n = sympy.Symbol(\"n\", integer=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJZjLNVN0i77ur\nRHZ72Yd1AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAVElEQVQIHWNgEDIxZWBgSGeQmMDAsoCBOYGB\n+wAD+0cG/gMMvN8Z5BUYeP8xzDdgYP3MMF8BREJEgLLs3xm4NzCwfATpYkpgYGhnkApgYBB+d5QB\nAPogE3QldevOAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$0$$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.sin(pi * n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "h = sympy.Lambda(x, x**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAE4AAAAcCAMAAAD4MnnTAAAANlBMVEX///8AAAAAAAAAAAAAAAAA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\nAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left( x \\mapsto x^{2} \\right)$$"
      ],
      "text/plain": [
       "     2\n",
       "x ↦ x "
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAAOBAMAAAA/Njq6AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAjUlEQVQIHWNgVDJ2YOA+yqLLwBDGwP6DgfP/\n/wIGhlQGhpkM3GUHGRgYZjEw5DuwARkMDPsD4Ewgb30A27tDDkAGA8d3BuYAxh8gJlcCiLQCEUog\ngkFfgIGBR4GBIZuBob6BgeESA6PAbKBoAAOLAgOPAFBiNQODVHlRFoMQA99EoBX///9iYFRPCmAA\nAHKeHEmWkOSlAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$25$$"
      ],
      "text/plain": [
       "25"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h(5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEYAAAAbBAMAAAAzEOSeAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIma7zZnddlTvRIkQ\nqzLsm4+cAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABRklEQVQoFWNgIAC2mh4goIKBW4DXgZAaNgG+\njwTVLGD8RUgNAwPHH8Jq2AoIq9kKVMKoACSwAkWQKFMAkBDGKg8S5BUAEqsZ1jEwOGJVwzGHgYHH\ngYGBz6PDBmIWQlk7hHk2FuSb7QwMnP//f2BgAxmIAFA1DOwgNSeh4mIIeRALRQ3LAohkKZA6XitV\nDQ0FFDWsID8BgRHQbRuY0tkcwDxUc5igghkMDNwHuP9wQJ2FYg5HAkTjLKA5DCwGYM4WF5dMFxdf\nEBvsZr4PcDUMDPwFEA6am2FqgHYxMLx/gFUNVwJEGOhmjgP1DOwHIFwU9zA5QASB0bq+wZ5BCMJD\n9RfrBojoMgaGs+XHgxRQ1bDe/HKJgYHlAUQUR1yAJWFxAE4fUEOA1EEEExynEK4hsigKG5Q2IAB3\nGmMWgKlhVICx0GlFqAAAVSNDOSl9DosAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left(x + 1\\right)^{2}$$"
      ],
      "text/plain": [
       "       2\n",
       "(x + 1) "
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "h(1+x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Expressions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x = sympy.Symbol(\"x\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "e = 1 + 2 * x**2 + 3 * x**3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHoAAAAWBAMAAAD9QZX2AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nK020cjRFd2uWKT3f7I3ug+r+A0e+eeqwHljaAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$3 x^{3} + 2 x^{2} + 1$$"
      ],
      "text/plain": [
       "   3      2    \n",
       "3⋅x  + 2⋅x  + 1"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJMAAAAcBAMAAACXLZArAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left ( 1, \\quad 2 x^{2}, \\quad 3 x^{3}\\right )$$"
      ],
      "text/plain": [
       "⎛      2     3⎞\n",
       "⎝1, 2⋅x , 3⋅x ⎠"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e.args"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAB0AAAAUBAMAAACDsiv0AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAuklEQVQYGWNgQAB2ZVUEB8gqZMhG4S9k8BdA\nFljBIN+AzGdguB+Ayl/DwMCoZOwAE2RXYGAIY2D/AeM/AjJSGRhmQvkcBRwODLMYGPIdIALedy8H\nMOwPgPPn//8PllgfwHomsacGqouB4ztDOMOrA6thfK4EhlKG+oCLML4SA0MAw2YYj4FHAcT8A+df\nYmAUYGD5AuOzKDDwCFzl+cBwFCIiVV6UxfGX+QPjAwh/////v1hzjh56B+ECAL4vLTO4aP+SAAAA\nAElFTkSuQmCC\n",
      "text/latex": [
       "$$2 x^{2}$$"
      ],
      "text/plain": [
       "   2\n",
       "2⋅x "
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e.args[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABQAAAAUBAMAAAB/pwA+AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAfElEQVQIHWNggIDcvQJQFpsD8wQok8OB5wuM\nGcDyFcpkYOD6CGdyKMCZuXAW+wU4M4whBsrmmbVqI5TJ9///BwYGIWVXNQWwCGMCexMHxFI2AbaP\nXA4QUQbODWAGiOBXgDPPH4AyuQT0GZgEwJz4BfsZ3CHCwkpC1wvATAAvUhbC3/cGUAAAAABJRU5E\nrkJggg==\n",
      "text/latex": [
       "$$x^{2}$$"
      ],
      "text/plain": [
       " 2\n",
       "x "
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e.args[1].args[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAJBAMAAAAWSsseAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAEHarIkSJZt3NVLsymT3i\nYlMAAAAJcEhZcwAADsQAAA7EAZUrDhsAAABASURBVAgdY2AQUnZVU2BgTGBv4pjAwCbA9pDLgYGR\ngXMDAwjwKYCpcweAFJeAHgOTAEPcgn0M7gwMwkpC1wsYAB0ECeuXDPmiAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$x$$"
      ],
      "text/plain": [
       "x"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e.args[1].args[1].args[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAwAAAAUBAMAAABL3sEiAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIma7zZnddlTvRImr\nEDIioekeAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAYklEQVQIHWNgYFRgEmBgYBBmYDgIpBwZGK4w\nMDAFMDBwFzCwASV4JzCIAcUZvBhKQVQogxGIUmHIAFFtDLNA1HsYBRZcBlOyFSQXzbAMRPmBDWNM\ngBnNYAi2CGQtkMUAdgQAOacPRe5eOPwAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left ( \\right )$$"
      ],
      "text/plain": [
       "()"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e.args[1].args[1].args[0].args"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Simplification"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "expr = 2 * (x**2 - x) - x * (x + 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALUAAAAZBAMAAACFo3IPAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nnTiIJQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$2 x^{2} - x \\left(x + 1\\right) - 2 x$$"
      ],
      "text/plain": [
       "   2                  \n",
       "2⋅x  - x⋅(x + 1) - 2⋅x"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAE0AAAAUBAMAAAA6pq28AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABOUlEQVQoFX2Su07DMBSGv5Q0bVIuVlmYUAUS\na18hA7cFKU8AEQMLSwckJqQIMcCA6MiYR+gbNG9ANwbEyFQJAQswgY0T20GFs/g7//l9bCcHTOwY\n+g0nruCP3Kxi/+g4Zy6uUrmGbmL0PaJ3vKHJ4cxhi/2cDzi0AhcOW9xPvE9YtYKfWa6RPJcwge76\n1kYPGhNZLbnmW0lhYYKXts7b8p6dAYZd3/a1gNaQQASvUQztHMOuj+AJogyPcKT0xUT2K9m7vZRx\nVWj/s2D+ReJST+XKV7GiMjZhnGvfXaE0dS5oVlTGl1C+ZkYk+jSEfkfFlUmuU7iJ1TsO8jFqAoIJ\nhh3fKdGb/C4py2vd+4EsyNaGHV/z8UFePCysNPu/6fqutf0xB9rgzkEndjbVsTZXs+f0Z0O9xb9z\n/w31/EBlQQ+knQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$x \\left(x - 3\\right)$$"
      ],
      "text/plain": [
       "x⋅(x - 3)"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.simplify(expr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAE0AAAAUBAMAAAA6pq28AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NVLsy\nme8Q6PJIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABOUlEQVQoFX2Su07DMBSGv5Q0bVIuVlmYUAUS\na18hA7cFKU8AEQMLSwckJqQIMcCA6MiYR+gbNG9ANwbEyFQJAQswgY0T20GFs/g7//l9bCcHTOwY\n+g0nruCP3Kxi/+g4Zy6uUrmGbmL0PaJ3vKHJ4cxhi/2cDzi0AhcOW9xPvE9YtYKfWa6RPJcwge76\n1kYPGhNZLbnmW0lhYYKXts7b8p6dAYZd3/a1gNaQQASvUQztHMOuj+AJogyPcKT0xUT2K9m7vZRx\nVWj/s2D+ReJST+XKV7GiMjZhnGvfXaE0dS5oVlTGl1C+ZkYk+jSEfkfFlUmuU7iJ1TsO8jFqAoIJ\nhh3fKdGb/C4py2vd+4EsyNaGHV/z8UFePCysNPu/6fqutf0xB9rgzkEndjbVsTZXs+f0Z0O9xb9z\n/w31/EBlQQ+knQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$x \\left(x - 3\\right)$$"
      ],
      "text/plain": [
       "x⋅(x - 3)"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALUAAAAZBAMAAACFo3IPAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nnTiIJQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$2 x^{2} - x \\left(x + 1\\right) - 2 x$$"
      ],
      "text/plain": [
       "   2                  \n",
       "2⋅x  - x⋅(x + 1) - 2⋅x"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "expr = 2 * sympy.cos(x) * sympy.sin(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIMAAAAUBAMAAABc0JOuAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nxWH/z/D+KV5xqNDDsh7KBwP3z7aBH99fTv+a9LX/Ob4AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$2 \\sin{\\left (x \\right )} \\cos{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "2⋅sin(x)⋅cos(x)"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEEAAAAUBAMAAAAgmk0yAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\sin{\\left (2 x \\right )}$$"
      ],
      "text/plain": [
       "sin(2⋅x)"
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.trigsimp(expr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "expr = sympy.exp(x) * sympy.exp(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACcAAAAQBAMAAABn8nsTAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEIl2mSJE3e9UMqtm\nzbsXyEShAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAs0lEQVQYGWNgAANlL+sFYAabNgO7AESMWWAv\newKYycQvsAIixsDFcAfKCkxnsIMyGRi+wlibGDbDmFwfuKDMMoYuKCueqYAJyqxi+wBlSZspK0KZ\n2XYXgCxGZZMAiACIxRjAdwDISw9gVYAIgljcCrlADmsng8gCsCCYxXZIAcjh7jE+CBZDYjHwb4AI\nMSCxGPhBysEAwWLgUGBgMAALIlgMzBcYRCBOQrAYGExsDoAVIrEAvqUgt4PujtMAAAAASUVORK5C\nYII=\n",
      "text/latex": [
       "$$e^{x} e^{y}$$"
      ],
      "text/plain": [
       " x  y\n",
       "ℯ ⋅ℯ "
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACkAAAATBAMAAAD/rzkOAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEIl2mSJE3e9UMqtm\nzbsXyEShAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAv0lEQVQYGWNgIAi4sargQxNV9rJewMAAEmXT\nZmAXgMgyC+xlT4CIMvELrIAIMnAx3AGxQGoD0xnsoKIMDF8ZGPbevXH3rgHDJobNMFGuD1xQtQxl\nDF1Q0XimAiaYaBXbB6iotJmyIkw02+4CkMmobBIAlQTZxhjAdwBIpQewKgApEACJcivkAknWTgaR\nBUAaBFiBmO2QApDk7jE+CKTQAP8GNAEwl18BmygHUNQAQ4L5AoMIzGVIkiY2B5B4ECYAAmIdoxyx\n/gIAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$e^{x + y}$$"
      ],
      "text/plain": [
       " x + y\n",
       "ℯ     "
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.powsimp(expr)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Expand"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "expr = (x + 1) * (x + 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGkAAAAWBAMAAAAx2K8dAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$x^{2} + 3 x + 2$$"
      ],
      "text/plain": [
       " 2          \n",
       "x  + 3⋅x + 2"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.expand(expr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQUAAAAUBAMAAAB2RTUoAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n9J3E/8Nvuf8AjsUp+QcIvw0AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\sin{\\left (x \\right )} \\cos{\\left (y \\right )} + \\sin{\\left (y \\right )} \\cos{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "sin(x)⋅cos(y) + sin(y)⋅cos(x)"
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.sin(x + y).expand(trig=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "a, b = sympy.symbols(\"a, b\", positive=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIgAAAAUBAMAAACkMGhZAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\log{\\left (a \\right )} + \\log{\\left (b \\right )}$$"
      ],
      "text/plain": [
       "log(a) + log(b)"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.log(a * b).expand(log=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALwAAAAZBAMAAAB5tlnFAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nSUVORK5CYII=\n",
      "text/latex": [
       "$$i e^{b} \\sin{\\left (a \\right )} + e^{b} \\cos{\\left (a \\right )}$$"
      ],
      "text/plain": [
       "   b           b       \n",
       "ⅈ⋅ℯ ⋅sin(a) + ℯ ⋅cos(a)"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.exp(I*a + b).expand(complex=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACgAAAAQBAMAAACW+SCeAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpmJdu8QRM1mu90y\nVKvMIHo8AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAy0lEQVQYGV3NOwrCQBAG4D8JhlXJ4wQimMZC\ntI9FUNEbCHZ7gwhinZxAcoQ0gqWdnaSwNzfQIqVFqmDpZMcHOMXyzT/LDKDKmFwe/8Lcjk8c/oRM\nJIC5P4LFczguQdzpUeL0ioz6nBqlOtRKHyMgkh/VoVlsphLwv9J63rIeUO1mAwa2riiZ5lOG9T36\nv4KeKsGq4NBmKr1CO1CCHiPksJEizDgUXUS5Is1unMFJMJQc5jhYvN5Z0wFVImieDaadtLossxh3\nFm96fcYLvlsuMIojjCoAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$a^{x} b^{x}$$"
      ],
      "text/plain": [
       " x  x\n",
       "a ⋅b "
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.expand((a * b)**x, power_exp=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFAAAAAUBAMAAADo9qfkAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEIl2mSJE3e9UMqtm\nzbsXyEShAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABPElEQVQoFWNgAANmBwiNTjI7sN1FEeMKQOHC\nOUBxFTiHAOMmAXm49Fc4C8hgs2ZgELG+xcBksngXmjjbZ2sBhDCTvABjFddHBkPuC2XICoHizIXM\nDgjhwOkMLA5cBQwLeA8gq2MAivMZ8FxAEn7MwDGBeQMDAz/M9/PeAYEAA1A8gWUDQpjx4wL5AL6D\nAgzrGRYgGwkU5xfgmIAQ5rygwLFgfQ7rB3UGBWSFQHF2gXNIwoybAtg2T/Zh3JxkJICsECjOtUkB\nKsyobAJzGFSNkLErsmoYOz2AVQHGBtNspQxBKAIQDmsng8gCFPGICYxeKAIQDneP8UFU4R2HbVAc\nCpXl34CqjIHhN7oAhM+vgC7+CV0AwucAKjRAkWpiYGBGEYBwmC8wiKAGjy0DI9bgMbE5gKqf2fkM\nqgCYBwCM8EXkfZ8t2AAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$e^{i a x} e^{- i b x}$$"
      ],
      "text/plain": [
       " ⅈ⋅a⋅x  -ⅈ⋅b⋅x\n",
       "ℯ     ⋅ℯ      "
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.exp(I*(a-b)*x).expand(power_exp=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Factor"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAH8AAAAUBAMAAABWoP+5AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nSbR/PWk/4gdnfPuAN3gAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left(x - 1\\right) \\left(x + 1\\right)$$"
      ],
      "text/plain": [
       "(x - 1)⋅(x + 1)"
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.factor(x**2 - 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKkAAAAUBAMAAAAErwC3AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n9wUjTv8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$x \\left(\\sin{\\left (z \\right )} + \\cos{\\left (y \\right )}\\right)$$"
      ],
      "text/plain": [
       "x⋅(sin(z) + cos(y))"
      ]
     },
     "execution_count": 86,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.factor(x * sympy.cos(y) + sympy.sin(z) * x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEQAAAAmBAMAAACPASDQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nqCrmLh369TuqRUThDzg7gsBaURuJAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\log{\\left (\\frac{a}{b} \\right )}$$"
      ],
      "text/plain": [
       "   ⎛a⎞\n",
       "log⎜─⎟\n",
       "   ⎝b⎠"
      ]
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.logcombine(sympy.log(a) - sympy.log(b))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "expr = x + y + x * y * z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGsAAAARBAMAAAAoKE+YAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$x y z + x + y$$"
      ],
      "text/plain": [
       "x⋅y⋅z + x + y"
      ]
     },
     "execution_count": 89,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAH0AAAAUBAMAAABSVS+EAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n1xAkyZZUszIC/+v/6xcr8lyJLvu7eAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$x \\left(y z + 1\\right) + y$$"
      ],
      "text/plain": [
       "x⋅(y⋅z + 1) + y"
      ]
     },
     "execution_count": 90,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.collect(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHsAAAAUBAMAAABfS1/DAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ngg==\n",
      "text/latex": [
       "$$x + y \\left(x z + 1\\right)$$"
      ],
      "text/plain": [
       "x + y⋅(x⋅z + 1)"
      ]
     },
     "execution_count": 91,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.collect(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "expr = sympy.cos(x + y) + sympy.sin(x - y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAU0AAAAUBAMAAAD7KHJ8AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nDqMXJkX5zyEu5oUchpgXJoXszIKk/wcJBESbwi7VHgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left(\\sin{\\left (x \\right )} + \\cos{\\left (x \\right )}\\right) \\left(- \\sin{\\left (y \\right )} + \\cos{\\left (y \\right )}\\right)$$"
      ],
      "text/plain": [
       "(sin(x) + cos(x))⋅(-sin(y) + cos(y))"
      ]
     },
     "execution_count": 93,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.expand(trig=True).collect([sympy.cos(x), sympy.sin(x)]).collect(sympy.cos(y) - sympy.sin(y))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Together, apart, cancel"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJAAAAAsBAMAAAB4YCyFAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nBqs5DCjKAE3Pnczn9+KCAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$- \\frac{1}{x + 2} + \\frac{1}{x + 1}$$"
      ],
      "text/plain": [
       "    1       1  \n",
       "- ───── + ─────\n",
       "  x + 2   x + 1"
      ]
     },
     "execution_count": 94,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.apart(1/(x**2 + 3*x + 2), x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAE8AAAAvBAMAAABQ7orvAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nyn+5Cl/yOIwIZ1JIEAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{y + 1}{y \\left(x + 1\\right)}$$"
      ],
      "text/plain": [
       "  y + 1  \n",
       "─────────\n",
       "y⋅(x + 1)"
      ]
     },
     "execution_count": 95,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.together(1 / (y * x + y) + 1 / (1+x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADAAAAAsBAMAAADROCZUAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVO8Qq5l2zWYiRInd\nuzLEnmxuAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAnklEQVQ4EWNgIAIwKmNXZBLyGbsEA9tIlWB1\n+u6BI0yGsvB/HGAI+Qk9ETMmQByPkYhhEhiJGE3CLGVWNsQMVAnmDRzHeArAhqNKcBtwf2acgEWC\nmYH9AkiYu7y8Rr283AHEhuYg/gQQBwhQjWJgeP8ALIwuwWiQz8BmAJZC1bFe4D7DDIgWVAmTNLOI\nBlQJjEQM1wFRhyCZId6BCAAA0AJHIkxuWMIAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{1}{x + 1}$$"
      ],
      "text/plain": [
       "  1  \n",
       "─────\n",
       "x + 1"
      ]
     },
     "execution_count": 96,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.cancel(y / (y * x + y))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Substitutions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABQAAAASBAMAAACp/uMjAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAmklEQVQIHWNgVDJ2YICAMAb2H1BmKgPDTChz\nFgNDvgOEvT8AzgQKrA+I0WFgnApkcXxnKLjJwPMByORKYLygzcBsAGQqMbAyzGHgP8DAwKMAVPOV\nwd+BgeESA6MA0waGkwwMLAoMPALMCQxmDAxS5UVZDEwGHP8YGPb///+LgTX90EcGGACqhoBshvgG\nKHMdgw2UxdBzyIGBAQCs7SBbH+RNaAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$2 y$$"
      ],
      "text/plain": [
       "2⋅y"
      ]
     },
     "execution_count": 97,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(x + y).subs(x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEkAAAAVBAMAAAD4Ed5jAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\sin{\\left (y e^{y} \\right )}$$"
      ],
      "text/plain": [
       "   ⎛   y⎞\n",
       "sin⎝y⋅ℯ ⎠"
      ]
     },
     "execution_count": 98,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.sin(x * sympy.exp(x)).subs(x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEwAAAAVBAMAAAAeOBUnAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n6aFc12b2UFYAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\cos{\\left (y e^{y} \\right )}$$"
      ],
      "text/plain": [
       "   ⎛   y⎞\n",
       "cos⎝y⋅ℯ ⎠"
      ]
     },
     "execution_count": 99,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.sin(x * z).subs({z: sympy.exp(y), x: y, sympy.sin: sympy.cos})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "expr = x * y + z**2 *x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "values = {x: 1.25, y: 0.4, z: 3.2}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACMAAAAPBAMAAACcmWsnAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAVO8Qq5l2zWYiibvdRH6G\nTkUAAAAJcEhZcwAADsQAAA7EAZUrDhsAAACXSURBVBgZY2AAAc5VqwWQaEZlBoapDIwvQEIQ2iTk\nMQNDngDDG5AQlGYDCl26wPwcJASlQUIMUI1QGiJkswAkwcAApsFCs44agEUgNEQVly5YiAFMQ4QY\nlCDKwDRIaCYDwz6QW6E0SOidAUQISoOEtBkYzk0AqoLSIKEOBsZnDHYFEJqB1em5BwNvaPQFBh4H\nCA2xGoUEAL97MkiN6bx0AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$13.3$$"
      ],
      "text/plain": [
       "13.3000000000000"
      ]
     },
     "execution_count": 102,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.subs(values)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Numerical evaluation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKAAAAAPBAMAAACRq9klAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAElFTkSuQmCC\n",
      "text/latex": [
       "$$4.14159265358979$$"
      ],
      "text/plain": [
       "4.14159265358979"
      ]
     },
     "execution_count": 103,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.N(1 + pi)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAgsAAAAPBAMAAACYf5HCAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nuErRtoiwcMr5//oXWhd7due/PDYAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$3.1415926535897932384626433832795028841971693993751$$"
      ],
      "text/plain": [
       "3.1415926535897932384626433832795028841971693993751"
      ]
     },
     "execution_count": 104,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.N(pi, 50)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHwAAAAQBAMAAAAmBgasAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nSUVORK5CYII=\n",
      "text/latex": [
       "$$x + 0.3183099$$"
      ],
      "text/plain": [
       "x + 0.3183099"
      ]
     },
     "execution_count": 105,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(x + 1/pi).evalf(7)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "expr = sympy.sin(pi * x * sympy.exp(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 107,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAt4AAAAUBAMAAABIVieXAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nFwLfzMzYj8D0AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left [ 0, \\quad 0.774, \\quad 0.642, \\quad 0.722, \\quad 0.944, \\quad 0.205, \\quad 0.974, \\quad 0.977, \\quad -0.87, \\quad -0.695\\right ]$$"
      ],
      "text/plain": [
       "[0, 0.774, 0.642, 0.722, 0.944, 0.205, 0.974, 0.977, -0.87, -0.695]"
      ]
     },
     "execution_count": 107,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[expr.subs(x, xx).evalf(3) for xx in range(0, 10)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "expr_func = sympy.lambdify(x, expr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKoAAAAPBAMAAACGiUnsAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n3XBu51oScu9rkamVy0PSrSgblKf/Pv4BiZcTRKSxARgAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$0.773942685266709$$"
      ],
      "text/plain": [
       "0.773942685266709"
      ]
     },
     "execution_count": 109,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr_func(1.0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 110,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "expr_func = sympy.lambdify(x, expr, 'numpy')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "xvalues = np.arange(0, 10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 113,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.        ,  0.77394269,  0.64198244,  0.72163867,  0.94361635,\n",
       "        0.20523391,  0.97398794,  0.97734066, -0.87034418, -0.69512687])"
      ]
     },
     "execution_count": 113,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr_func(xvalues)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Calculus"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "f = sympy.Function('f')(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEAAAAArBAMAAAA6dJN0AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrkJggg==\n",
      "text/latex": [
       "$$\\frac{d}{d x} f{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "d       \n",
       "──(f(x))\n",
       "dx      "
      ]
     },
     "execution_count": 115,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.diff(f, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEgAAAAvBAMAAACyMpGWAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nayxWv8rUDvcPkHqVnKurwLUAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{d^{2}}{d x^{2}}  f{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "  2      \n",
       " d       \n",
       "───(f(x))\n",
       "  2      \n",
       "dx       "
      ]
     },
     "execution_count": 116,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.diff(f, x, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEgAAAAvBAMAAACyMpGWAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{d^{3}}{d x^{3}}  f{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "  3      \n",
       " d       \n",
       "───(f(x))\n",
       "  3      \n",
       "dx       "
      ]
     },
     "execution_count": 117,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.diff(f, x, 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 118,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "g = sympy.Function('g')(x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGoAAAAzBAMAAACNjoCbAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{\\partial^{2}}{\\partial x\\partial y}  g{\\left (x,y \\right )}$$"
      ],
      "text/plain": [
       "   2          \n",
       "  ∂           \n",
       "─────(g(x, y))\n",
       "∂y ∂x         "
      ]
     },
     "execution_count": 119,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g.diff(x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAH0AAAAzBAMAAABI/BoKAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n3uThA9n6xgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{\\partial^{5}}{\\partial x^{3}\\partial y^{2}}  g{\\left (x,y \\right )}$$"
      ],
      "text/plain": [
       "    5           \n",
       "   ∂            \n",
       "───────(g(x, y))\n",
       "  2   3         \n",
       "∂y  ∂x          "
      ]
     },
     "execution_count": 120,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g.diff(x, 3, y, 2)         # equivalent to s.diff(g, x, x, x, y, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 121,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "expr = x**4 + x**3 + x**2 + x + 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 122,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKoAAAAWBAMAAACiUBq/AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nlaqwoJo/baWpPyMv6VIAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$4 x^{3} + 3 x^{2} + 2 x + 1$$"
      ],
      "text/plain": [
       "   3      2          \n",
       "4⋅x  + 3⋅x  + 2⋅x + 1"
      ]
     },
     "execution_count": 122,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.diff(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 123,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJAAAAAcBAMAAAB8GisoAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nqN8VQoK+KFGk4GAk9fbifzfoqkEUkYf7AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$2 \\left(6 x^{2} + 3 x + 1\\right)$$"
      ],
      "text/plain": [
       "  ⎛   2          ⎞\n",
       "2⋅⎝6⋅x  + 3⋅x + 1⎠"
      ]
     },
     "execution_count": 123,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.diff(x, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 124,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "expr = (x + 1)**3 * y ** 2 * (z - 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 125,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGAAAAAbBAMAAABxU5cJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$6 y \\left(x + 1\\right)^{2}$$"
      ],
      "text/plain": [
       "           2\n",
       "6⋅y⋅(x + 1) "
      ]
     },
     "execution_count": 125,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.diff(x, y, z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 126,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "expr = sympy.sin(x * y) * sympy.cos(x / 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 127,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAVkAAAAqBAMAAADrHTUzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nQmCC\n",
      "text/latex": [
       "$$y \\cos{\\left (\\frac{x}{2} \\right )} \\cos{\\left (x y \\right )} - \\frac{1}{2} \\sin{\\left (\\frac{x}{2} \\right )} \\sin{\\left (x y \\right )}$$"
      ],
      "text/plain": [
       "                       ⎛x⎞         \n",
       "                    sin⎜─⎟⋅sin(x⋅y)\n",
       "     ⎛x⎞               ⎝2⎠         \n",
       "y⋅cos⎜─⎟⋅cos(x⋅y) - ───────────────\n",
       "     ⎝2⎠                   2       "
      ]
     },
     "execution_count": 127,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.diff(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 128,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "expr = sympy.special.polynomials.hermite(x, 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 129,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAZwAAAA6BAMAAAB2XfJNAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nxAplIcIePrDI/wCl06q6zOM6IwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{2^{x} \\sqrt{\\pi} \\operatorname{polygamma}{\\left (0,- \\frac{x}{2} + \\frac{1}{2} \\right )}}{2 \\Gamma{\\left(- \\frac{x}{2} + \\frac{1}{2} \\right)}} + \\frac{2^{x} \\sqrt{\\pi} \\log{\\left (2 \\right )}}{\\Gamma{\\left(- \\frac{x}{2} + \\frac{1}{2} \\right)}}$$"
      ],
      "text/plain": [
       " x   ___          ⎛     x   1⎞                  \n",
       "2 ⋅╲╱ π ⋅polygamma⎜0, - ─ + ─⎟    x   ___       \n",
       "                  ⎝     2   2⎠   2 ⋅╲╱ π ⋅log(2)\n",
       "────────────────────────────── + ───────────────\n",
       "            ⎛  x   1⎞                ⎛  x   1⎞  \n",
       "         2⋅Γ⎜- ─ + ─⎟               Γ⎜- ─ + ─⎟  \n",
       "            ⎝  2   2⎠                ⎝  2   2⎠  "
      ]
     },
     "execution_count": 129,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.diff(x).doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 130,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "d = sympy.Derivative(sympy.exp(sympy.cos(x)), x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 131,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFEAAAArBAMAAADyGHmiAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nuX66mWpSMAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{d}{d x} e^{\\cos{\\left (x \\right )}}$$"
      ],
      "text/plain": [
       "d ⎛ cos(x)⎞\n",
       "──⎝ℯ      ⎠\n",
       "dx         "
      ]
     },
     "execution_count": 131,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 132,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIQAAAAaBAMAAACEBumnAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nXhMMGpZEHr5/9T5/L6HIcLYAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$- e^{\\cos{\\left (x \\right )}} \\sin{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "  cos(x)       \n",
       "-ℯ      ⋅sin(x)"
      ]
     },
     "execution_count": 132,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d.doit()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Integrals"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 133,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "a, b = sympy.symbols(\"a, b\")\n",
    "x, y = sympy.symbols('x, y')\n",
    "f = sympy.Function('f')(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 134,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFkAAAAuBAMAAACxAqjlAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\int f{\\left (x \\right )}\\, dx$$"
      ],
      "text/plain": [
       "⌠        \n",
       "⎮ f(x) dx\n",
       "⌡        "
      ]
     },
     "execution_count": 134,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.integrate(f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 135,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFkAAABKBAMAAAAiZ+WpAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMARM2JmVQidqsQ3btm\n7zKsjR9xAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACmUlEQVRIDeWWPWhTURTH/8kzH+/lmQQndUlc\ndBEJVBHqYNBiHQQzObg0opMKFRppqEOCiCIOfbi0CqUBEXFqJkWoJDiJiy1oBwXNJKJDEkurwc9z\n7819L1WHe0bxwDv3nP/55X68894jgLQtWTWaeStvxikqXODQqSKHzkxeZ+BTXolBP8QTBr2Mowy6\nh5cM+r5bZ9Dvhxjwv4tuGvIYmy/PVc1pu+tcMKdjHXMWcDitxDhjH0AlzdlJw+PQ3zlw8guHjn3m\n0Ikuh071OHS7toGONlXqFtX4m39e1YI9egQ4p7Or/eDjC62IcdpvZXzfGrBT127qgMTAFvxWzm9L\nw83ritWfJTR4Lvun38oxAp2ipu0lFVk1rdAYCVo5TOnZoLRDheF8IMH6qpOnr64A4gM6Uz41QdsY\nEYUz+9sFLVAaXhWiNNHURcDOufecLOS3NLSChqcFKqf8I0d/UHoXsDyrFykCJymda+KQL1A+vk5O\nWqRDwzLNjXBdCG26jnu44wuUjwlGmjw80bQe7VrRr2GLx0gJFBzMkpOWqNJAOyGuJfx2WmYVckUl\nkNbIi4qwzWJGOmXEqyDmyVPaa0hkT2iBym8LghQ2nya3F8gUGjhP4WG6duFZNacFWitoZYbmE8vP\nTs4ca1K4h67ZiXfX0loAQkErD1B1oPPRmsg3Wky/lZHqblH586ka5OPf+pnzqCajB7rqP7FaoNER\nDRTmlopy9N8G/2dSVi7VGUhEGG2SI/vrm9auq6KZr+TMOEVNpzn0AgfGJw6dDN4cg5+5+nYbsEB8\nyQjrQynWDcy0zOcexg1zGI+TKwz69CXGRhjz/keoPXqZcQ9vt9qe+c0ZwQdzmP4cl81pex2Lt8zx\nN8lu05zeerFkTP8CGqydhhTj2cgAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\int_{a}^{b} f{\\left (x \\right )}\\, dx$$"
      ],
      "text/plain": [
       "b        \n",
       "⌠        \n",
       "⎮ f(x) dx\n",
       "⌡        \n",
       "a        "
      ]
     },
     "execution_count": 135,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.integrate(f, (x, a, b))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 136,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEwAAAAUBAMAAADVZMaCAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$- \\cos{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "-cos(x)"
      ]
     },
     "execution_count": 136,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.integrate(sympy.sin(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 137,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIoAAAAUBAMAAACgxbhkAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJVCLvMs1Edmar\n3bti/yyrAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACRElEQVQ4EZ2UMWgTYRTH/2dyvVxr4tGCdJD2\nMArioKGlYNXhFh0Uydmhi4sVi4MoXSxYKIk6WEVoQHErPRoV0eXQRQqiCA62S0G6CGIQXDppWwlR\nqP7fJZeel6+CPvjyvfu/33v38u6+A1rsTF3pcFsicaFOKkG90oDfxpPi1yR7LlFUgamwh+54Vvxa\nyDsUVeCzEE54obfFLuQ6lwpcDnO0s6G3xU7S/MaYAtSXmjm3mp7SETIR0AQ7+/dAO1wuYeLkY9Lt\nNn+G+h5ZwO4/k7Vsn4OuJ4MwHswXGRIytTDnCKifx7g76WJFqyDNWMZnn6OYpvuCK2LDjjFqLCPj\n9QDSgpC917TvAqaKyI+9BwpXR9DOWNoBdvo4QPcUV8SmkKymczCq9y141IXMW1gTsNcGDE56R+7T\nistYWwk4aOED3QIXuj+K3eQMV0VygLX0xkOJCLkPGqsUMOsByRqrVLqmhZPYZWhVukEV7nUzf3DP\nM7qhHfpl0xfyOkx2UNjsZa+FV9TZJ8snvxKb4dq0Zi+rp9HB2wb/6Ce28VnPyFzgnANm73nY7gcz\n09aRKR5rme4Um26zYdZOAIusItOtyQw5Xf0iEu74GG4kR5Bx+QbYwG28fJ0DskQjNuxjwLiAVO64\nhS/UhVzEZ3oEd+0fgJ49WjKPPB2kYi5RKl9ZKAEcadS0N2UXQ/3vcHf+OaMB2Tnn04uBktQ8AZG3\nWPRWC0kV+G+nUWqrTqMMJzDVgW+E6ltIqkC90kBlgn+1kFSC//HF/A2d55Ex6VFP2AAAAABJRU5E\nrkJggg==\n",
      "text/latex": [
       "$$\\cos{\\left (a \\right )} - \\cos{\\left (b \\right )}$$"
      ],
      "text/plain": [
       "cos(a) - cos(b)"
      ]
     },
     "execution_count": 137,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.integrate(sympy.sin(x), (x, a, b))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 138,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAB4AAAAtBAMAAABL8MaSAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAInarRM2ZVBDdiWbv\nuzJCz3LGAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA/UlEQVQoFWNggAL5/2AA4zKYwFlgBmMAKp+t\nAJXficplSEflc0xA5XNtYGAC2ZcAFQZqX/VccZMgTJU2A18Bl4AeiMviACRA2nkZLIAkgw7QJUwK\nQMZjsChj0xcGBk6QRDbDDxDF8ZeBYReI/sEAlAGCfAGGGUCKaQLDKTBfPoF7AZDBu4HBCsxn/sHZ\nAGZACb5fZchcBoaYCFR+fQIqn1UAlU8RDxINcJIis3BrFsu9hyzJYcCw3gFJgLmBgecAEp9nAgPz\nPyQ+6w9UPlCKExz4CDX+CxBsEGsyKpctAZWvhsplT2CApwWQzDMGBkUkFYyzViUjpyhWYHhB+ADK\nYEVdhUA7AwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{\\sqrt{\\pi}}{2}$$"
      ],
      "text/plain": [
       "  ___\n",
       "╲╱ π \n",
       "─────\n",
       "  2  "
      ]
     },
     "execution_count": 138,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.integrate(sympy.exp(-x**2), (x, 0, oo))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 139,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "a, b, c = sympy.symbols(\"a, b, c\", positive=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 140,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADEAAAAVBAMAAAAdjxsPAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAInarRM2ZVBDdiWbv\nuzJCz3LGAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABEUlEQVQoFXWQsS4EURSGv52xs7MsoyFEw3ZE\nshMvYAoPsLwAnUoMT3ATiUQiGVEph4KERKZFpdhGIplesyql1SxR4FyrOJOMU9z///5zTm7uhUFV\nvn+r94dK3EhBwU4USMOmBu39I03ae7Em7Q80FPxVgRTYa1ryoD5Mni8aOGt2B20nY/h6d2ZunKF7\n1mE6rrZhRZp1eOHUkbnDjC14wI1xOtJp2tXcNbBtOMb7sDzVisC+ZiyUTR7xv2jk4iDJGUlFvSww\n+J/UeiSWJXmjbkQTE0T474y2d4IMYqj0ubUT8zRSWOMmTL2Q/a5Ey+ZETp5wU9hbeF6KuLybtdHr\nxYaVkqp2TElqo5r8yT+1Wpr/ADT/QXgx8Os4AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\sqrt{\\pi} a c$$"
      ],
      "text/plain": [
       "  ___    \n",
       "╲╱ π ⋅a⋅c"
      ]
     },
     "execution_count": 140,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.integrate(a * sympy.exp(-((x-b)/c)**2), (x, -oo, oo))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 141,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKgAAAAuBAMAAABOjE9CAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n75/Dr/5/ZpHY5TNAArAAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\int \\sin{\\left (x \\cos{\\left (x \\right )} \\right )}\\, dx$$"
      ],
      "text/plain": [
       "⌠                 \n",
       "⎮ sin(x⋅cos(x)) dx\n",
       "⌡                 "
      ]
     },
     "execution_count": 141,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.integrate(sympy.sin(x * sympy.cos(x)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 142,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "expr = sympy.sin(x*sympy.exp(y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 143,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIAAAAAVBAMAAAB8u/sIAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nefgPbBO3AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$- e^{- y} \\cos{\\left (x e^{y} \\right )}$$"
      ],
      "text/plain": [
       "  -y    ⎛   y⎞\n",
       "-ℯ  ⋅cos⎝x⋅ℯ ⎠"
      ]
     },
     "execution_count": 143,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.integrate(expr, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 144,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "expr = (x + y)**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 145,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIkAAAAvBAMAAAAlT/QJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\frac{x^{3}}{3} + x^{2} y + x y^{2}$$"
      ],
      "text/plain": [
       " 3              \n",
       "x     2        2\n",
       "── + x ⋅y + x⋅y \n",
       "3               "
      ]
     },
     "execution_count": 145,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.integrate(expr, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 146,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKcAAAAvBAMAAAB028dqAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n9QB5kZR17CGxjRb/A/oDAJUKyCcFMJkAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{x^{3} y}{3} + \\frac{x^{2} y^{2}}{2} + \\frac{x y^{3}}{3}$$"
      ],
      "text/plain": [
       " 3      2  2      3\n",
       "x ⋅y   x ⋅y    x⋅y \n",
       "──── + ───── + ────\n",
       " 3       2      3  "
      ]
     },
     "execution_count": 146,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.integrate(expr, x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 147,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAqBAMAAACXcryGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMt2rmYlmIkR2uxDN\nVO+L8+I6AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAoElEQVQYGWNgVHYNDWVgYPv///9nBgaWAgYG\nAwYGViASYACBTjDJsABMsSaAqf1gkmEOhNICU9xAzUDA+AVM8UIo9rdgHpkE0EX//38gTnPoVJBz\n9zSwHABSSxi4FjAwcHwHMhkYuB6AKf6DkR5Axv5jDHwXgNRvBpbXDAz8CgxsPxgYmA4wsH1jYGB+\nAOYxAuVAekwZ9gFVMrC6GDIwAADSEChF6yJe4wAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{7}{6}$$"
      ],
      "text/plain": [
       "7/6"
      ]
     },
     "execution_count": 147,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.integrate(expr, (x, 0, 1), (y, 0, 1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Series"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 148,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x = sympy.Symbol(\"x\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 149,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "f = sympy.Function(\"f\")(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 150,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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95okT6iC0zPOdALjE6z1Ydp3jk2D7xbLDld7BDK21vtSXI0LFsgUas9cwgs9J39PiHNaI\ngc8wxqPRicLpCcuS6FPCivUIjKIf7+tLglm09ponjtZD0DLNmQMqy8OM+4NXlunHUe5S/MXpLz7k\naRiPHY2q1XFwRpPNMNVPQ2u4m4Tw4WHz2PEloLU14he95J8bOKDJWRftWN7EVqmSszAePBqxWB6d\n0eQRmernoTXcTUKY5WHKzFpspvWq2pK/8NDEWpmSszAePBqxWB6d0eQQmernoTXcTUKY5UHTdqGF\n6hSF5uGbGP6DkhMxHjkaaejs0IwmW+hUPwmt4W4S0g8vP08JrybTiIXps+gQYhPjmUrOw3joaNTF\n8tiMJkuoVD8LreFuEpLzkrDBn1wi57sOSIeO+LxlE4K0kpMwHj0aBTqzHjuprcxwLloLvFUQxrUX\ncOjBr/LZOiezqZF9+ivhEzCeAFGcgdHkr8/R/h9lXadyED6q1AAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$f{\\left (0 \\right )} + x \\left. \\frac{d}{d x} f{\\left (x \\right )} \\right|_{\\substack{ x=0 }} + \\frac{x^{2}}{2} \\left. \\frac{d^{2}}{d x^{2}}  f{\\left (x \\right )} \\right|_{\\substack{ x=0 }} + \\frac{x^{3}}{6} \\left. \\frac{d^{3}}{d x^{3}}  f{\\left (x \\right )} \\right|_{\\substack{ x=0 }} + \\frac{x^{4}}{24} \\left. \\frac{d^{4}}{d x^{4}}  f{\\left (x \\right )} \\right|_{\\substack{ x=0 }} + \\frac{x^{5}}{120} \\left. \\frac{d^{5}}{d x^{5}}  f{\\left (x \\right )} \\right|_{\\substack{ x=0 }} + \\mathcal{O}\\left(x^{6}\\right)$$"
      ],
      "text/plain": [
       "                             ⎛  2      ⎞│         ⎛  3      ⎞│         ⎛  4   \n",
       "                           2 ⎜ d       ⎟│       3 ⎜ d       ⎟│       4 ⎜ d    \n",
       "                          x ⋅⎜───(f(x))⎟│      x ⋅⎜───(f(x))⎟│      x ⋅⎜───(f(\n",
       "                             ⎜  2      ⎟│         ⎜  3      ⎟│         ⎜  4   \n",
       "         ⎛d       ⎞│         ⎝dx       ⎠│x=0      ⎝dx       ⎠│x=0      ⎝dx    \n",
       "f(0) + x⋅⎜──(f(x))⎟│    + ────────────────── + ────────────────── + ──────────\n",
       "         ⎝dx      ⎠│x=0           2                    6                    24\n",
       "\n",
       "   ⎞│         ⎛  5      ⎞│           \n",
       "   ⎟│       5 ⎜ d       ⎟│           \n",
       "x))⎟│      x ⋅⎜───(f(x))⎟│           \n",
       "   ⎟│         ⎜  5      ⎟│           \n",
       "   ⎠│x=0      ⎝dx       ⎠│x=0    ⎛ 6⎞\n",
       "──────── + ────────────────── + O⎝x ⎠\n",
       "                  120                "
      ]
     },
     "execution_count": 150,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.series(f, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 151,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x0 = sympy.Symbol(\"{x_0}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 152,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAgcAAAA3CAMAAACvmhmNAAAAOVBMVEX///8AAAAAAAAAAAAAAAAA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\n4/ea4a+3bBLnWL+PtcD/GvdCbS9/amAAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$f{\\left ({x_{0}} \\right )} + \\left(x - {x_{0}}\\right) \\left. \\frac{d}{d \\xi_{1}} f{\\left (\\xi_{1} \\right )} \\right|_{\\substack{ \\xi_{1}={x_{0}} }} + \\mathcal{O}\\left(\\left(x - {x_{0}}\\right)^{2}; x\\rightarrow{x_{0}}\\right)$$"
      ],
      "text/plain": [
       "                       ⎛ d        ⎞│            ⎛           2           ⎞\n",
       "f({x_0}) + (x - {x_0})⋅⎜───(f(ξ₁))⎟│         + O⎝(x - {x_0}) ; x → {x_0}⎠\n",
       "                       ⎝dξ₁       ⎠│ξ₁={x_0}                             "
      ]
     },
     "execution_count": 152,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.series(x, x0, n=2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 153,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAR8AAAA3CAMAAAACAb6QAAAAOVBMVEX///8AAAAAAAAAAAAAAAAA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\n7/S8zJC0hzBrx/syC71u3//p63/HryjCti1TEwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left(x - {x_{0}}\\right) \\left. \\frac{d}{d \\xi_{1}} f{\\left (\\xi_{1} \\right )} \\right|_{\\substack{ \\xi_{1}={x_{0}} }} + f{\\left ({x_{0}} \\right )}$$"
      ],
      "text/plain": [
       "            ⎛ d        ⎞│                   \n",
       "(x - {x_0})⋅⎜───(f(ξ₁))⎟│         + f({x_0})\n",
       "            ⎝dξ₁       ⎠│ξ₁={x_0}           "
      ]
     },
     "execution_count": 153,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f.series(x, x0, n=2).removeO()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 154,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMQAAAAwBAMAAAC8i8hXAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYII=\n",
      "text/latex": [
       "$$1 - \\frac{x^{2}}{2} + \\frac{x^{4}}{24} + \\mathcal{O}\\left(x^{6}\\right)$$"
      ],
      "text/plain": [
       "     2    4        \n",
       "    x    x     ⎛ 6⎞\n",
       "1 - ── + ── + O⎝x ⎠\n",
       "    2    24        "
      ]
     },
     "execution_count": 154,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.cos(x).series()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 155,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAANEAAAAwBAMAAAB9DIL7AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nMep64CsAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$x - \\frac{x^{3}}{6} + \\frac{x^{5}}{120} + \\mathcal{O}\\left(x^{6}\\right)$$"
      ],
      "text/plain": [
       "     3     5        \n",
       "    x     x     ⎛ 6⎞\n",
       "x - ── + ─── + O⎝x ⎠\n",
       "    6    120        "
      ]
     },
     "execution_count": 155,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.sin(x).series()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 156,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAVoAAAAwBAMAAACiZ6/NAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$1 + x + \\frac{x^{2}}{2} + \\frac{x^{3}}{6} + \\frac{x^{4}}{24} + \\frac{x^{5}}{120} + \\mathcal{O}\\left(x^{6}\\right)$$"
      ],
      "text/plain": [
       "         2    3    4     5        \n",
       "        x    x    x     x     ⎛ 6⎞\n",
       "1 + x + ── + ── + ── + ─── + O⎝x ⎠\n",
       "        2    6    24   120        "
      ]
     },
     "execution_count": 156,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.exp(x).series()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 157,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAATwAAAAcBAMAAAAO8B63AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVO8Qq5l2zWbdMiJE\nibtHdKBDAAAACXBIWXMAAA7EAAAOxAGVKw4bAAADx0lEQVRYCc1WTWgTQRR+cbv52WTbSLGXXioi\nHvSw1FIPUhoUiuKlYAUR1IXa3sSKYqgU3YOCIlRRtNRTUW+CzdGDaATrQdQWtBVPBsWzEav1D+LM\nzszmzeyzxot2IDPvffPNN9++2UwC0Hibf+9R5IUBEgaoUOxbt89RsIm5C4EJ/SF3S00jBMUO7AoB\nAyQ+UfBsrUDBJjaW7TChP+SZkvODoNhz6SoBA/SQ9h7eJMkSdMoiSBYsH3wRN9hnytmfJPU31Rsk\n7T0lJRR4Rwar8ix4pNAGR/q4oHOKWu+0kvZmhndTbIkdk2NL25AHbrAMk5jK+AQIyRMUCq5F2rsH\n4zH6zObXS2WO2j7vWXvQn7sEcEUkjfbzNJE+3Au0PYDpkiGT7H/+5GD43jzw5FRvHhgwxLLEOoP9\n2zTdT085Hwnc8ml726G3bND7cusgucTB6PlbyvCV1RCga4A8A6HQpgk9gydaLpLOqSxT4i3hh4Po\nEnv2fA5QrsKNoEqkxO1CZxmgxglrFSsV8OplAoBko/aco6vfqeVozJTSsv6aPYAsKfwiKpGyN+sd\nYHKn2bk60YpcJTHCXsXKX9hbVatVkS0VOgsvSyI27I0v5hUHje78nMyUvavAeR/YJ90hpwDOznvM\nGstj1bOGdz1eEDyl0H3o3mE/WhoFBqzsIYGIygITluK57yHpLevtEdYh0RyrZsxeD5z3J8IlIBWs\nufS1DF+qNxNW9pAAXmDCUtwWJ7qeUZsr7CHQXg5zHrP3At547UJXKrie+ylRwluFsQkre0gArzFh\nKd5c5SSHXyypAgAW5b9Swp41OsnaxBQjeXCK9QCvisXLxeJxFlmQqrBBtDpTg91icXRDsVhgJCUQ\n8rdw3cmTJlwXT4XVa+aHk+nX98qyGytWPYAvoTDr5AMCtPgK0kYdVtVDAhq7rhvCUrwpfPfCu5pX\nD+9Vr164QHb8yEWL7E1PKUgbdTiyVxfQ2AYsxd1FRrLGOFPYq4vyf0ix6rUmqzAodKVCwjsCSU9A\nqDdhZQ8JIDaYsBS3agHA/TJnNlXY3Y72or65uY/N1WzA2dHhzuY/wF2B4N6EpT0sgOgxWB3NxcX2\nmV0hkd9zWJTdg/a2pR1IhNW5b3BI/d+WCl3D3XsDjRQmJiztYQG0KAYre87Ob/sEj1cLi9oVtJ4I\nlQIxRUDqcIkpCiLEN+m8lK/nZrbVBJbNrdKy0+YkIb5f5/TG33ed8I+zG/p+Z/T0v2dNec3CdS37\n/4k1gj2k53C2EuI12MR9nKyIOFFGNnajeGWGvwADMwMucnfhMgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$1 - x + x^{2} - x^{3} + x^{4} - x^{5} + \\mathcal{O}\\left(x^{6}\\right)$$"
      ],
      "text/plain": [
       "         2    3    4    5    ⎛ 6⎞\n",
       "1 - x + x  - x  + x  - x  + O⎝x ⎠"
      ]
     },
     "execution_count": 157,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(1/(1+x)).series()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 158,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "expr = sympy.cos(x) / (1 + sympy.sin(x * y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 159,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAcgAAAA1BAMAAAA39GaqAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nQmCC\n",
      "text/latex": [
       "$$1 - x y + x^{2} \\left(y^{2} - \\frac{1}{2}\\right) + x^{3} \\left(- \\frac{5 y^{3}}{6} + \\frac{y}{2}\\right) + \\mathcal{O}\\left(x^{4}\\right)$$"
      ],
      "text/plain": [
       "                           ⎛     3    ⎞        \n",
       "           2 ⎛ 2   1⎞    3 ⎜  5⋅y    y⎟    ⎛ 4⎞\n",
       "1 - x⋅y + x ⋅⎜y  - ─⎟ + x ⋅⎜- ──── + ─⎟ + O⎝x ⎠\n",
       "             ⎝     2⎠      ⎝   6     2⎠        "
      ]
     },
     "execution_count": 159,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.series(x, n=4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 160,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAg8AAAAvBAMAAACWODSOAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n8csBMlRz/w8SvECCUAANXgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\cos{\\left (x \\right )} - x y \\cos{\\left (x \\right )} + x^{2} y^{2} \\cos{\\left (x \\right )} - \\frac{5 x^{3}}{6} y^{3} \\cos{\\left (x \\right )} + \\mathcal{O}\\left(y^{4}\\right)$$"
      ],
      "text/plain": [
       "                                        3  3               \n",
       "                       2  2          5⋅x ⋅y ⋅cos(x)    ⎛ 4⎞\n",
       "cos(x) - x⋅y⋅cos(x) + x ⋅y ⋅cos(x) - ────────────── + O⎝y ⎠\n",
       "                                           6               "
      ]
     },
     "execution_count": 160,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.series(y, n=4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 161,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAskAAAAwBAMAAAARGfomAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nzC37t5eC1QAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$- \\frac{61 x^{5}}{120} y^{5} + \\frac{5 x^{5}}{12} y^{3} - \\frac{x^{5} y}{24} + \\frac{2 x^{4}}{3} y^{4} - \\frac{x^{4} y^{2}}{2} + \\frac{x^{4}}{24} - \\frac{5 x^{3}}{6} y^{3} + \\frac{x^{3} y}{2} + x^{2} y^{2} - \\frac{x^{2}}{2} - x y + 1$$"
      ],
      "text/plain": [
       "      5  5      5  3    5        4  4    4  2    4      3  3    3             \n",
       "  61⋅x ⋅y    5⋅x ⋅y    x ⋅y   2⋅x ⋅y    x ⋅y    x    5⋅x ⋅y    x ⋅y    2  2   \n",
       "- ──────── + ─────── - ──── + ─────── - ───── + ── - ─────── + ──── + x ⋅y  - \n",
       "    120         12      24       3        2     24      6       2             \n",
       "\n",
       " 2          \n",
       "x           \n",
       "── - x⋅y + 1\n",
       "2           "
      ]
     },
     "execution_count": 161,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.series(y).removeO().series(x).removeO().expand()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Limits"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 162,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAPBAMAAAArJJMAAAAAHlBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAACGjDitAAAACXRSTlMAVO8Qq5l2zWYZcMvdAAAACXBIWXMAAA7EAAAOxAGV\nKw4bAAAAHUlEQVQIHWNgAANGZQYGk5DJQDYbqQSr03QPsBkAJYgIYEZbtZEAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$1$$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 162,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.limit(sympy.sin(x) / x, x, 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 163,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "f = sympy.Function('f')\n",
    "x, h = sympy.symbols(\"x, h\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 164,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "diff_limit = (f(x + h) - f(x))/h"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 165,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEoAAAAUBAMAAADYerbFAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMnZUZrtE7yKJ\nmatywa77AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABQklEQVQoFWNgwAm4CyBS7BvQlXgFIER2wJiv\nYQwY7X8AxmJg0IYxhWAMLDT7BZggD5IFMDEYzQx3DuMCmBiU3uIAF9gMZzFoIpgMrulB3P0PWLSu\nhD4FicYCsWjczhigdXcRqhgbGJgZmB4wMLcy7AdZ1s7AwHiA3ZRZgYGhCKGKcyEDKwPHAwaOBQz8\nBUBhMwYGHgGej2xAHckMQsYgoAIUtm/dAFalwMAPtIVhJdAsBqYGIIvBH0RAgYz+X7CqBrgqBggL\nWRW3AEP9BJCNUFVAG4GmOIDIySACAtgCGPgKkFQBXc8mEM/AKoDieraFDBwbgAbBzDrLwDA/oZ9h\nO9CMOzCTGBg4j4efZHn/T+T9P7f+dUBhoDVyoaLXQd4F+Q0HQMQQtwEOJUBh4mKboRtmAr6UwwBP\nhXDlMG3INFKKBgDEpkPH0/6h/gAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$- \\sin{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "-sin(x)"
      ]
     },
     "execution_count": 165,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.limit(diff_limit.subs(f, sympy.cos), h, 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 166,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADkAAAAUBAMAAADFBIheAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJVCLvMs1Edmar\n3bti/yyrAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABOUlEQVQoFWNgwADRUJGtGDJAAdYHUFGeACzS\nHDBBxgYsspvgYlZwFoJxFc4Ug7PgDNYLcCbHBAYGIWMVBkarxRMYqrxXACW4FBgYGFcdnQU0lkmB\ngTWFoSygNoDhFeMDBl6gLN8GBgZhhr0LbjAwcDcwcDQw+BdcZmCor0xk4ALK8jowMBgy2Au4MjCw\nXGCQB5rE/oWBgd/g/iuQX9iAlgkw9AFZDOwfGOYvACr6CZR9INz/CSrLwPALKgvXqyrAsB+oD2Qy\nA/t3kCznBZC9DA5JDAzzpyxg4AG6COQqR64PDAvBrmLNZmAOKCtgaGZJZOALYGBgVmDg/ML3gRWo\niMmAgUFGy5SBVdlmAqf1akuIcYynFi7aDWRyFAAJNIAIyVg0GRAXfyyALAcDrDEIj324MqhqCIWU\ncgBCYEPSNlhnwgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\cos{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "cos(x)"
      ]
     },
     "execution_count": 166,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.limit(diff_limit.subs(f, sympy.sin), h, 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 167,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "expr = (x**2 - 3*x) / (2*x - 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 168,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "p = sympy.limit(expr/x, x, oo)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 169,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "q = sympy.limit(expr - p*x, x, oo)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 170,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAF4AAAAyBAMAAAAnynF8AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nR70wlmbhz4jYHaMMhFjVCuNQAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left ( \\frac{1}{2}, \\quad -1\\right )$$"
      ],
      "text/plain": [
       "(1/2, -1)"
      ]
     },
     "execution_count": 170,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p, q"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Sums and products"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 171,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "n = sympy.symbols(\"n\", integer=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 172,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x = sympy.Sum(1/(n**2), (n, 1, oo))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 173,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADsAAAA5BAMAAAB6w06LAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nSUVORK5CYII=\n",
      "text/latex": [
       "$$\\sum_{n=1}^{\\infty} \\frac{1}{n^{2}}$$"
      ],
      "text/plain": [
       "  ∞     \n",
       " ____   \n",
       " ╲      \n",
       "  ╲   1 \n",
       "   ╲  ──\n",
       "   ╱   2\n",
       "  ╱   n \n",
       " ╱      \n",
       " ‾‾‾‾   \n",
       "n = 1   "
      ]
     },
     "execution_count": 173,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 174,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABYAAAAvBAMAAAAV7ydtAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAdpmJMlQiZrurEN1E\n782PMUhmAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA1klEQVQoFWNggACjpgVQFgNzAucDGJs7geUb\nnH2A7Q+MzcDA+gvB5hZAsI0QTHYFBPswwxEYh+VpaBeQLf////9PvP//f2BgYDYJTBdLgMi7Mpgx\nOkCYQLKBcwGMzTKBG8ZkYCrgh4uvX8B/ACYhzcBrAGO3M3DC2TAxGtFAn8AA5TYYCzvADFl3gOkB\njN3EwNEAZfPAo4SBgeMCTAEDA/8TQx0Yb/1zBt4AKGf9LwamW1A2/wYGdlgscj9gYP8OFee8gBBn\nAaqHGzuZYS3MHAZmpSkg5QADkTn1RYfm4gAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{\\pi^{2}}{6}$$"
      ],
      "text/plain": [
       " 2\n",
       "π \n",
       "──\n",
       "6 "
      ]
     },
     "execution_count": 174,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x.doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 175,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x = sympy.Product(n, (n, 1, 7))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 176,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAC8AAAA9BAMAAADPFy0PAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAiVQQq7tEMpnN72bd\nInbXH9raAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABQ0lEQVQ4EWNgAANGZWMDCAuNZPn//yuaEITL\nkpYWjFWCiYEpAasEA4MhDnEGZxwS7AE4JBgbcEhwCuCQ4NqAQ4InAYfEYBPm+o8A35Edx5vy/+td\nENj/f1IBsgQDQ/wXMF//vwGqOIM8VOIPmvgQl0j0vffkAdhLaB5sCH3A/BeLBLvCSQZeSGii6mBm\n/sbA/QGLDgamDwwsE7BJcExg4N8ADmhUoxi4BBj0DQRAWtAk7icwSHIXYJHIYWC4sgmbHWCxgZeI\nh6QbebR0xVvx/3/YBQbmwv//J11AOJWBAZx2PzCwglIwStpFVkQPdiUOSyp8cEgw6BAjwd1d9wKu\nDlkHd/4GrgssM4FgNgOKUZffMPCD0wpIG7IOBj+GLKxGMSxn6GAHGzUPVQfvB4Zj6jAtWjAGkGZz\nYHi4Acq/cd4PLgMAq3SO5ejkdgIAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\prod_{n=1}^{7} n$$"
      ],
      "text/plain": [
       "  7    \n",
       "┬───┬  \n",
       "│   │ n\n",
       "│   │  \n",
       "n = 1  "
      ]
     },
     "execution_count": 176,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 177,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACcAAAAPBAMAAACVcstdAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAzXYQMplU74mrIma7\nRN0SDTw+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA3klEQVQYGWWOsU4CQRRFz2RIWAYDv7AJMUSr\nqaz3D8DG0LGJ9hsSGqhWOypCYUFlYey1kkDhtCQE/ANia28ihYG3s9M51ZmT+967qFblBt1bJECl\nG6h6POaoVH+LbLhAanIJ73Avsu0C1eTDGjKLeXOBvDzAPiVSjpJqq/NE/4iMWSoX6MyaP/MLnVzH\nypVUbNwaSXbyCJGeCtl/9uN3IsP4DKafsj67lbr+UJYykKTdwMfVfP40iguydOGrqHwte6q+vNAL\n9TGNVA9FNl0g03u06IeLBKLd4bUkCfx7JyKQSaHRsUD4AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$5040$$"
      ],
      "text/plain": [
       "5040"
      ]
     },
     "execution_count": 177,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x.doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 178,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x = sympy.Symbol(\"x\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 179,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADYAAAARBAMAAABkwkJgAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAEIl2mSJE3e9UMqtmzXCQ\nkgMAAAAJcEhZcwAADsQAAA7EAZUrDhsAAACYSURBVBgZY2AAA2Uv6wUQFgbJLLCXPQFDFCLAxXAH\nhwxI+Cm6HGcjTITrAReMCaGlvR9CBeKYCphQ5RiYYXLSZsqK2OQYlU0C0MRBXLC+9ABWBRxyrJ0M\nIgtwyHH3GB+ES8mWg8AtEB9kJt8GuAwKAyyngCIE54DkOIByBnARBAMkx3yBQQSnH0xsDiBUw1ks\nO97ugXOwMADcXB8K98p9oQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$e^{x} - 1$$"
      ],
      "text/plain": [
       " x    \n",
       "ℯ  - 1"
      ]
     },
     "execution_count": 179,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.Sum((x)**n/(sympy.factorial(n)), (n, 1, oo)).doit().simplify()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Equations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 180,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x = sympy.symbols(\"x\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 181,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEoAAAAUBAMAAADYerbFAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAdt3NMolEECK7VJlm\n76txUg1gAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAvElEQVQoFWMQMmEgBA6rMYTB1XCERi6Ac1AY\nYUiquhm4/qBIgjlrCxiAahBmyS9g+ISuiv31ezRVuQnsn9FVMTDIo6liwGojpqqiACLMatUGmo4O\nMM1ivYauBuGuYiUQAAfuXUzD0M3qYGCYvwDDMHRV/wuIUXWTgUG/gaBZZxi4PjLUG6CpQ7eRbffO\nBAYmdVRV++7bHUCJR4j0ClRVYB5ybEOksQQ/plnsAsSYxYNFEdAsIRVs4ihiQmoAsgIxph5f0UEA\nAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left [ -3, \\quad 1\\right ]$$"
      ],
      "text/plain": [
       "[-3, 1]"
      ]
     },
     "execution_count": 181,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.solve(x**2 + 2*x - 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 182,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "a, b, c = sympy.symbols(\"a, b, c\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 183,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAd0AAAAyBAMAAADrdhy+AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n2d97/wsKDVkvv1rNcAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left [ \\frac{1}{2 a} \\left(- b + \\sqrt{- 4 a c + b^{2}}\\right), \\quad - \\frac{1}{2 a} \\left(b + \\sqrt{- 4 a c + b^{2}}\\right)\\right ]$$"
      ],
      "text/plain": [
       "⎡        _____________   ⎛       _____________⎞ ⎤\n",
       "⎢       ╱           2    ⎜      ╱           2 ⎟ ⎥\n",
       "⎢-b + ╲╱  -4⋅a⋅c + b    -⎝b + ╲╱  -4⋅a⋅c + b  ⎠ ⎥\n",
       "⎢─────────────────────, ────────────────────────⎥\n",
       "⎣         2⋅a                     2⋅a           ⎦"
      ]
     },
     "execution_count": 183,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.solve(a * x**2 + b * x + c, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 184,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGgAAAAyBAMAAABCJ4MDAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrkJggg==\n",
      "text/latex": [
       "$$\\left [ - \\frac{3 \\pi}{4}, \\quad \\frac{\\pi}{4}\\right ]$$"
      ],
      "text/plain": [
       "⎡-3⋅π   π⎤\n",
       "⎢─────, ─⎥\n",
       "⎣  4    4⎦"
      ]
     },
     "execution_count": 184,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.solve(sympy.sin(x) - sympy.cos(x), x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 185,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALQAAAAyBAMAAAAHCuzEAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n/gX5FSSLrXWHZgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left [ - \\operatorname{LambertW}{\\left (\\frac{1}{2} \\right )}\\right ]$$"
      ],
      "text/plain": [
       "[-LambertW(1/2)]"
      ]
     },
     "execution_count": 185,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.solve(sympy.exp(x) + 2 * x, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 186,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAABLEAAAAcBAMAAABsXDBEAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nqyA/Pl4K9qxb1legFyvwPyFZCd5pqmHzAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left [ \\operatorname{RootOf} {\\left(x^{5} - x^{2} + 1, 0\\right)}, \\quad \\operatorname{RootOf} {\\left(x^{5} - x^{2} + 1, 1\\right)}, \\quad \\operatorname{RootOf} {\\left(x^{5} - x^{2} + 1, 2\\right)}, \\quad \\operatorname{RootOf} {\\left(x^{5} - x^{2} + 1, 3\\right)}, \\quad \\operatorname{RootOf} {\\left(x^{5} - x^{2} + 1, 4\\right)}\\right ]$$"
      ],
      "text/plain": [
       "⎡      ⎛ 5    2       ⎞        ⎛ 5    2       ⎞        ⎛ 5    2       ⎞       \n",
       "⎣RootOf⎝x  - x  + 1, 0⎠, RootOf⎝x  - x  + 1, 1⎠, RootOf⎝x  - x  + 1, 2⎠, RootO\n",
       "\n",
       " ⎛ 5    2       ⎞        ⎛ 5    2       ⎞⎤\n",
       "f⎝x  - x  + 1, 3⎠, RootOf⎝x  - x  + 1, 4⎠⎦"
      ]
     },
     "execution_count": 186,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.solve(x**5 - x**2 + 1, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 187,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAPBAMAAAArJJMAAAAAHlBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAACGjDitAAAACXRSTlMAVO8Qq5l2zWYZcMvdAAAACXBIWXMAAA7EAAAOxAGV\nKw4bAAAAHUlEQVQIHWNgAANGZQYGk5DJQDYbqQSr03QPsBkAJYgIYEZbtZEAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$1$$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 187,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1 #s.solve(s.tan(x) - x, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 188,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "eq1 = x + 2 * y - 1\n",
    "eq2 = x - y + 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 189,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALEAAAAyBAMAAADhIyeAAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left [ \\left \\{ x : - \\frac{1}{3}, \\quad y : \\frac{2}{3}\\right \\}\\right ]$$"
      ],
      "text/plain": [
       "[{x: -1/3, y: 2/3}]"
      ]
     },
     "execution_count": 189,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.solve([eq1, eq2], [x, y], dict=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 190,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "eq1 = x**2 - y\n",
    "eq2 = y**2 - x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 191,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "sols = sympy.solve([eq1, eq2], [x, y], dict=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 192,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAABAYAAAA/BAMAAACBTtecAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nToIUefw/r6/8Pw2QhWZp8jauAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left [ \\left \\{ x : 0, \\quad y : 0\\right \\}, \\quad \\left \\{ x : 1, \\quad y : 1\\right \\}, \\quad \\left \\{ x : - \\frac{1}{2} + \\frac{\\sqrt{3} i}{2}, \\quad y : - \\frac{1}{2} - \\frac{\\sqrt{3} i}{2}\\right \\}, \\quad \\left \\{ x : \\frac{1}{4} \\left(1 - \\sqrt{3} i\\right)^{2}, \\quad y : - \\frac{1}{2} + \\frac{\\sqrt{3} i}{2}\\right \\}\\right ]$$"
      ],
      "text/plain": [
       "⎡                                                                  ⎧          \n",
       "⎢                            ⎧           ___               ___  ⎫  ⎪   ⎛      \n",
       "⎢                            ⎪     1   ╲╱ 3 ⋅ⅈ       1   ╲╱ 3 ⋅ⅈ⎪  ⎨   ⎝1 - ╲╱\n",
       "⎢{x: 0, y: 0}, {x: 1, y: 1}, ⎨x: - ─ + ───────, y: - ─ - ───────⎬, ⎪x: ───────\n",
       "⎢                            ⎪     2      2          2      2   ⎪  ⎩         4\n",
       "⎣                            ⎩                                  ⎭             \n",
       "\n",
       "      2                  ⎫⎤\n",
       "___  ⎞              ___  ⎪⎥\n",
       " 3 ⋅ⅈ⎠        1   ╲╱ 3 ⋅ⅈ⎬⎥\n",
       "───────, y: - ─ + ───────⎪⎥\n",
       "              2      2   ⎭⎥\n",
       "                          ⎦"
      ]
     },
     "execution_count": 192,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sols"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 193,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[True, True, True, True]"
      ]
     },
     "execution_count": 193,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[eq1.subs(sol).simplify() == 0 and eq2.subs(sol).simplify() == 0 for sol in sols]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Linear algebra"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 194,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABYAAAAyBAMAAACqpzYoAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMA74lUMhCrmXbNZiK7\nRN38cqFhAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAnElEQVQoFWOQ//+JAQT0/39kEHZxBbNDXAwZ\nRMAsEOGIyTYMgImzpufD2QwM/QPDLpOftgCrO0HOx+Z+GotvrzkCs5c1gUH/AtQNbAEMjAJQNuMD\nBrYvUDbnVwQbqI/vK8LN9xUQ7GcwMxkY2AsQ7HYgE+pH5gKG3TC2OwNDN5TN/Uip7AGUzfn//38Y\nG6QRCLDELwniyGkGKS0BAEoGOmJJTwriAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1\\\\2\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1⎤\n",
       "⎢ ⎥\n",
       "⎣2⎦"
      ]
     },
     "execution_count": 194,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.Matrix([1,2])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 195,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADMAAAAZBAMAAABuuAtJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAu90iVO8Qq5l2zWYy\niUQbaetyAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAmUlEQVQoFWMQUjJgwAIeKV1gUMAiDhJiLiBR\nyqJRBq6LJQzZVLYChvsPoAamt39DlmJPYGANgNnFiSLFqsDA/ge7FNc3nFJAw5m+YdcFlHp/AaeU\nOtzxqM5gYOBowCk1A2gmNKDQdLE0MBjjkDrBwDAZuxSz7t12BagUR+mvaqDpMMD1//9/mBRMDIUm\nOb7AuumsSwlXOiwAABrCMVhd5YweAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 2\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "[1  2]"
      ]
     },
     "execution_count": 195,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.Matrix([[1,2]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 196,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADUAAAAyBAMAAAAOzY77AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMA74lUMhCrmXbNZiK7\nRN38cqFhAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABPElEQVQ4Ee3Uv0vEMBQH8O/FK1TjtSf+2PT0\nFjdBEBzvwHPv5KqL52a7aNf6F+jkel3FRfwD5MBV0c1V/AcED11EY4QmfQ+amx3M0uZ9eHlpkgYt\n9YaKJpRqYra3XUHweltNzFXJb6xeaTK965a2HtHsHYgvY95BzGwfmDEGnDJbAuKurcdtELlN136N\nHHlAQ6+H+QY+JhDsuS3Vwzry/MRtm5BXjrx6At9lN+3VQ1vvuHWe6wqmDZT6tmaC7GnmyYJF599w\n3b7lK6N3otgjmaCTMwyGxvwRwmdmJ9Yai9g9oyYvrOkwH1NMEvNeaBouicm1I2peRgxYeSQowKw2\nT2yDmD6M06PSvGVi4TuziX4/XsiKdakNEXyWefpNRwqbytHJmIXW8JTeMxIPH7n9H5gUnT92Psfd\nkWPu1h85h1EamwQVaAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 2\\\\3 & 4\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1  2⎤\n",
       "⎢    ⎥\n",
       "⎣3  4⎦"
      ]
     },
     "execution_count": 196,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.Matrix([[1, 2], [3, 4]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 197,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKEAAABLCAMAAAAvUBdgAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nXwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & 1 & 2 & 3\\\\10 & 11 & 12 & 13\\\\20 & 21 & 22 & 23\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡0   1   2   3 ⎤\n",
       "⎢              ⎥\n",
       "⎢10  11  12  13⎥\n",
       "⎢              ⎥\n",
       "⎣20  21  22  23⎦"
      ]
     },
     "execution_count": 197,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.Matrix(3, 4, lambda m,n: 10 * m + n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 198,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "a, b, c, d = sympy.symbols(\"a, b, c, d\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 199,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "M = sympy.Matrix([[a, b], [c, d]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 200,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADcAAAAyBAMAAAAKOF7GAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMA74lUMhAimXZEzWa7\n3asFX9bSAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABSklEQVQ4Ee3UPUsDMRwG8Kdpc6WnxYJ2suIN\nbiLcN6hg3c/JRbj7BFKKgzhoh0J1s59AFyk4iKNjnQQnwd3FpeAiSgvS4UzIi3/hLrODWS7J75K7\nJA/BavqBrMLStIbF1naWgbe2aljKJNlZysa9FQfixoWfDqxMHMieCHqdw7psmuKP+6H920HivxuQ\nz/iKTw3yZZRHFJsRxLapdZYnqHYpHoFbLIwQb1IcoiIWo0bGAZrUMIOYzGCIx2JC+As7twb97twL\nI4Y3PIimmpaP1/bPKO72xUCNtJ/U1UjSQav/KHYjdxNK9+18xELowIvIgRuOb2L4g+y0Rw/L6/Rs\nwOA945riIJm3AUMhwCtBGfJLO+1JRAgyl9XAYuOXQYb83KDMPi065Hr7ZkCRaKxCrvEY/ICgDrlG\ntn5HDDrkuaci3/2L6LxSXZfxN0SVSvCGgXwsAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}a & b\\\\c & d\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡a  b⎤\n",
       "⎢    ⎥\n",
       "⎣c  d⎦"
      ]
     },
     "execution_count": 200,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 201,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKMAAAAzBAMAAAAJJKXwAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}a^{2} + b c & a b + b d\\\\a c + c d & b c + d^{2}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡ 2                  ⎤\n",
       "⎢a  + b⋅c   a⋅b + b⋅d⎥\n",
       "⎢                    ⎥\n",
       "⎢                  2 ⎥\n",
       "⎣a⋅c + c⋅d  b⋅c + d  ⎦"
      ]
     },
     "execution_count": 201,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M * M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 202,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x = sympy.Matrix(sympy.symbols(\"x_1, x_2\"))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 203,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGQAAAAyBAMAAABYG2ONAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nxc9V9F+4TRojw2xE0gpyAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}a x_{1} + b x_{2}\\\\c x_{1} + d x_{2}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡a⋅x₁ + b⋅x₂⎤\n",
       "⎢           ⎥\n",
       "⎣c⋅x₁ + d⋅x₂⎦"
      ]
     },
     "execution_count": 203,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M * x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 204,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "p, q = sympy.symbols(\"p, q\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 205,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "M = sympy.Matrix([[1, p], [q, 1]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 206,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADUAAAAyBAMAAAAOzY77AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMA74lUMhCrmXbNZkS7\nIt3RmMWfAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABLUlEQVQ4EWOQ//+JAQtg+v9fgEHYxRWLFAOr\ni7MAgwg2GZAYCxFyhgGYuiH6WNPzccoxMPRTRW7TWY+2ALg7Uc1UyDNgOoBdjtdgGgPXB+xyrAyf\nGZi+Ypdj4P7KwCOAQ47pAUN8AQ45zgKGhbjcyX+ABRgD0LAuk5+2AClA9+9ovwCXQxIHMfXAfOxx\nNB23HOtn3HJ2PxtAktjNBGsbznLMojj9Hlb2FaccAweRcrxard2QQISQyPo2XmADmQIHSHKsUgxs\nE+ASQAaSHPsHBkYFoBAcIMkxHmC4vwEuAWQgycUrMKxHlkKRM2A4iUuOs4ELtYxBMpM1o/k3sj72\nxJ9ZiLQEzG4YAJbOUL0HUQaTY1TA0AZLg0zzn+OUw5QAisDMxCYJlMNXRuIpWwGwrk/cug0U6gAA\nAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & p\\\\q & 1\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1  p⎤\n",
       "⎢    ⎥\n",
       "⎣q  1⎦"
      ]
     },
     "execution_count": 206,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 207,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "b = sympy.Matrix(sympy.symbols(\"b_1, b_2\"))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 208,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAB4AAAAyBAMAAAC5cHbcAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMA74lUMhC73c2rRHaZ\nImaqCQggAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA9klEQVQoFWOQ//+JAQb0/39kEHZxhXEZQlwM\nGUTgPBDDEcRnTSyACYL5DBwTUPn8Bqj89QGofHUYF2IeQ+fuw1ARiPk/AuIvMHiAhMB85q8M/Ao+\n0+F8ngaGeAUGczif4wDDegMkPv8DhvMByPwFrHIMSHzOB2wHkPmsp88BzUKYB+Rg4RuCBCH+BbHu\n1M9G4YPEkOXpyKdLfDAdOwrzHzg+dBlmwPjg+Khl8L8A9T84PqoY7i+A8iHxwWAPTAXg8OMHxwdD\nNUw/JD6YDsD4kPh4A+QihTeXApcDMt/D2AxmHkghg/z//0ASET9gQSL46OkdLT8AAPfcTiuXVltr\nAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}b_{1}\\\\b_{2}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡b₁⎤\n",
       "⎢  ⎥\n",
       "⎣b₂⎦"
      ]
     },
     "execution_count": 208,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 209,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAOkAAABDBAMAAACLnj9YAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}b_{1} \\left(\\frac{p q}{- p q + 1} + 1\\right) - \\frac{b_{2} p}{- p q + 1}\\\\- \\frac{b_{1} q}{- p q + 1} + \\frac{b_{2}}{- p q + 1}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡   ⎛  p⋅q       ⎞     b₂⋅p  ⎤\n",
       "⎢b₁⋅⎜──────── + 1⎟ - ────────⎥\n",
       "⎢   ⎝-p⋅q + 1    ⎠   -p⋅q + 1⎥\n",
       "⎢                            ⎥\n",
       "⎢       b₁⋅q        b₂       ⎥\n",
       "⎢   - ──────── + ────────    ⎥\n",
       "⎣     -p⋅q + 1   -p⋅q + 1    ⎦"
      ]
     },
     "execution_count": 209,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = M.solve(b)\n",
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 210,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x = M.LUsolve(b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 211,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJUAAAA/BAMAAAAbC5f0AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nqPptiOfK52H6beiY6rchOYPv9ufz1j+iBFtT32lTOAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}b_{1} - \\frac{p \\left(- b_{1} q + b_{2}\\right)}{- p q + 1}\\\\\\frac{- b_{1} q + b_{2}}{- p q + 1}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡     p⋅(-b₁⋅q + b₂)⎤\n",
       "⎢b₁ - ──────────────⎥\n",
       "⎢        -p⋅q + 1   ⎥\n",
       "⎢                   ⎥\n",
       "⎢    -b₁⋅q + b₂     ⎥\n",
       "⎢    ──────────     ⎥\n",
       "⎣     -p⋅q + 1      ⎦"
      ]
     },
     "execution_count": 211,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 212,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x = M.inv() * b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 213,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$\\left[\\begin{matrix}b_{1} \\left(\\frac{p q}{- p q + 1} + 1\\right) - \\frac{b_{2} p}{- p q + 1}\\\\- \\frac{b_{1} q}{- p q + 1} + \\frac{b_{2}}{- p q + 1}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡   ⎛  p⋅q       ⎞     b₂⋅p  ⎤\n",
       "⎢b₁⋅⎜──────── + 1⎟ - ────────⎥\n",
       "⎢   ⎝-p⋅q + 1    ⎠   -p⋅q + 1⎥\n",
       "⎢                            ⎥\n",
       "⎢       b₁⋅q        b₂       ⎥\n",
       "⎢   - ──────── + ────────    ⎥\n",
       "⎣     -p⋅q + 1   -p⋅q + 1    ⎦"
      ]
     },
     "execution_count": 213,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Versions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "application/json": {
       "Software versions": [
        {
         "module": "Python",
         "version": "3.4.3 64bit [GCC 4.2.1 (Apple Inc. build 5577)]"
        },
        {
         "module": "IPython",
         "version": "3.2.0"
        },
        {
         "module": "OS",
         "version": "Darwin 14.5.0 x86_64 i386 64bit"
        },
        {
         "module": "sympy",
         "version": "0.7.6"
        },
        {
         "module": "numpy",
         "version": "1.9.2"
        }
       ]
      },
      "text/html": [
       "<table><tr><th>Software</th><th>Version</th></tr><tr><td>Python</td><td>3.4.3 64bit [GCC 4.2.1 (Apple Inc. build 5577)]</td></tr><tr><td>IPython</td><td>3.2.0</td></tr><tr><td>OS</td><td>Darwin 14.5.0 x86_64 i386 64bit</td></tr><tr><td>sympy</td><td>0.7.6</td></tr><tr><td>numpy</td><td>1.9.2</td></tr></table>"
      ],
      "text/latex": [
       "\\begin{tabular}{|l|l|}\\hline\n",
       "{\\bf Software} & {\\bf Version} \\\\ \\hline\\hline\n",
       "Python & 3.4.3 64bit [GCC 4.2.1 (Apple Inc. build 5577)] \\\\ \\hline\n",
       "IPython & 3.2.0 \\\\ \\hline\n",
       "OS & Darwin 14.5.0 x86\\_64 i386 64bit \\\\ \\hline\n",
       "sympy & 0.7.6 \\\\ \\hline\n",
       "numpy & 1.9.2 \\\\ \\hline\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "Software versions\n",
       "Python 3.4.3 64bit [GCC 4.2.1 (Apple Inc. build 5577)]\n",
       "IPython 3.2.0\n",
       "OS Darwin 14.5.0 x86_64 i386 64bit\n",
       "sympy 0.7.6\n",
       "numpy 1.9.2\n"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%reload_ext version_information\n",
    "%version_information sympy, numpy"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.4.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}