scientific-python-sympy.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![PHENIX](logoPHENIX.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 4) Scientific python: sympy"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this part we'll focus on the uses of python as a tool for scientists."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**N.B.** The following content borrows heavily from lecture 5 on scientific python. The original source, in its more detailed form can be found [here](https://github.com/jrjohansson/scientific-python-lectures)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Sympy - Symbolic algebra in Python"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib notebook\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Introduction"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There are two notable Computer Algebra Systems (CAS) for Python:\n",
    "\n",
    "* [SymPy](http://sympy.org/en/index.html) - A python module that can be used in any Python program, or in an IPython session, that provides powerful CAS features. \n",
    "* [Sage](http://www.sagemath.org/) - Sage is a full-featured and very powerful CAS enviroment that aims to provide an open source system that competes with Mathematica and Maple. Sage is not a regular Python module, but rather a CAS environment that uses Python as its programming language.\n",
    "\n",
    "Sage is in some aspects more powerful than SymPy, but both offer very comprehensive CAS functionality. The advantage of SymPy is that it is a regular Python module and integrates well with the IPython notebook. \n",
    "\n",
    "In this lecture we will therefore look at how to use SymPy with IPython notebooks. If you are interested in an open source CAS environment I also recommend to read more about Sage.\n",
    "\n",
    "To get started using SymPy in a Python program or notebook, import the module `sympy`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from sympy import *"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To get nice-looking $\\LaTeX$ formatted output run:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "IPython console for SymPy 1.0 (Python 3.5.2-64-bit) (ground types: python)\n",
      "\n",
      "These commands were executed:\n",
      ">>> from __future__ import division\n",
      ">>> from sympy import *\n",
      ">>> x, y, z, t = symbols('x y z t')\n",
      ">>> k, m, n = symbols('k m n', integer=True)\n",
      ">>> f, g, h = symbols('f g h', cls=Function)\n",
      ">>> init_printing()\n",
      "\n",
      "Documentation can be found at http://docs.sympy.org/1.0/\n"
     ]
    }
   ],
   "source": [
    "init_session()\n",
    "\n",
    "# or with older versions of sympy/ipython, load the IPython extension\n",
    "#%load_ext sympy.interactive.ipythonprinting\n",
    "# or\n",
    "#%load_ext sympyprinting"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Symbolic variables"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In SymPy we need to create symbols for the variables we want to work with. We can create a new symbol using the `Symbol` class:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x = Symbol('x')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEgAAAAbBAMAAAAt2dQtAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRK5CYII=\n",
      "text/latex": [
       "$$\\left(x + \\pi\\right)^{2}$$"
      ],
      "text/plain": [
       "       2\n",
       "(x + π) "
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(pi + x)**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# alternative way of defining symbols\n",
    "a, b, c = symbols(\"a, b, c\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sympy.core.symbol.Symbol"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can add assumptions to symbols when we create them:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x = Symbol('x', real=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x.is_imaginary"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x = Symbol('x', positive=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACoAAAAPBAMAAABgjEDtAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMA782r3SJ2ZjIQmUS7\nVIlAnjihAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAv0lEQVQYGWNg/GQs72z0hQEV8Acw5BcwNKIK\nMsxiAIkyo4mGg0XZJqAKR4BFOUCi0Q2c3QFwWaAJ3Iq5j0LXH+A9n8DAuvwxWAooysC4dn4B0wEG\n/gSGLRO4JUEaQKJMDgwMPGBROQYGMaAgRNQAKsrxq7zcHC66ACrK/hckBARgExbA1H4DiyFEmQ8w\nxCcwODEwTIOpZQGqZRdguHiSob+AYSUDA/caeZkV3Of/XGBgeJc2RWQCp1XeBKghaBQAM0c287zN\nvm0AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\mathrm{True}$$"
      ],
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x > 0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Complex numbers"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The imaginary unit is denoted `I` in Sympy. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACoAAAAQBAMAAACSDPCjAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAVO8Qq5l2zWaJ3SK7RPx7\nN2kAAAAJcEhZcwAADsQAAA7EAZUrDhsAAABySURBVBgZY2DABCwHGBgYldHFWRoYTEIeQ0UZE5Ck\n2SgRNQsDmoRhwoRLWES5BcLhopzl5VXq5eUODAzMDM/gokAGzGWMIEdhmMvuYIBF1E5gAhbRe9wN\nDKxOzz2A5gABzFyWDggfQsJEkcWATtoA5wMA/Fcc5MixWvAAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$1 + i$$"
      ],
      "text/plain": [
       "1 + ⅈ"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1+1*I"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABgAAAAPBAMAAAAMihLoAAAAJFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAADHJj5lAAAAC3RSTlMAEM3dMlTvq5l2ZtVdCTcAAAAJcEhZcwAA\nDsQAAA7EAZUrDhsAAAAqSURBVAgdY2DAClgTEcLi7RsRHAZOMjlCxiCgwkC2ATA3cJRtqoKxwTQA\nC0AL2ft3JesAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$-1$$"
      ],
      "text/plain": [
       "-1"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEwAAAAbBAMAAAAkMnRXAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left(i x + 1\\right)^{2}$$"
      ],
      "text/plain": [
       "         2\n",
       "(ⅈ⋅x + 1) "
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(x * I + 1)**2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Rational numbers"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There are three different numerical types in SymPy: `Real`, `Rational`, `Integer`: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "r1 = Rational(4,5)\n",
    "r2 = Rational(5,4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAqBAMAAACXcryGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMpndu3bvImbNiRBU\nq0Qb3U6NAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAkklEQVQYGWNgYGAQAmIgMAGTrClgqmIKmFoA\npjgKwNRWBjC1AEzxCICpXQxg6uzdu9+ugnVAtDN8AXOW/L8Bpski/oPAB6K0Tt0gdACoMP//L5Dy\nWVcLQNQFEAGnbrQqgnjeDPUPQDQDiwGYYvrOwMA7gYHrHwMD2wQGpt8MDEwMDMwTGBjYFRjaQMYU\nrdVmYAAALnIpDsFeUO4AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{4}{5}$$"
      ],
      "text/plain": [
       "4/5"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABUAAAAqBAMAAACuFQ3dAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMpndu3bvImbNiRBU\nq0Qb3U6NAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA5UlEQVQYGWNgYGAQAmIYMAEyjm0A81hTGLgv\n3YewK6YAheIh7AUINkcBgr2VAcFegGDzCCDYuxgQ7LN37367CjeTIQVhPsMXBHvJ/xsMuvlzG4Ai\nNAb/4eADZTYVqTgyMGxasRBoCvcFhvUPGOYwbC5gYODdwMCWwJHAwDWBgYHNgIH3I8sFBp7fDAyc\nv4Fs/gsMfH/A1jL/3q/AwPcVzH6/oH4BA+MvMNuWoV4BymZSYICrCWJgAOrlAenlUWAoZ3Fg4ACa\nCYrRcKBdTEC72K1W6Row2DBsEwDaBfSjAcOu0IsMDACGnlMXi4yUmQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{41}{20}$$"
      ],
      "text/plain": [
       "41\n",
       "──\n",
       "20"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r1+r2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABUAAAAqBAMAAACuFQ3dAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVO8Qq5l2zWaJMkS7\nIt2ZnNffAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA/klEQVQYGWNggICZmQtADCEDBgbzDawFDAzM\nrv5A9lEGngMg8XwDBu6/IAYQANk8CmAWmM1fNS0azAOK21cwMAmAOCD2ZwZWZSib/wEDx28om6+A\ngeMHlM2igBDnAqpXgIozeDIYgcwJ0S9pYOAM9QIJ0xj8h4MPlNl0NeYUA0OZwVKg+5kdGOZvYND/\n/xNoIosBA5MCQ6XbBSCb6QEDyy8GB7A97J8RbKAA32cGj7YosAzD/gkMhxnubwBznoBIngcgkjUA\nTIKjIxVocAEDJyhsGQMY7jIVMLB+BrKXMjDksgINK2Bg4H05M+QBbwBDE9BidqAfHzBcnRLMwAAA\nq1hKQiigoh8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{16}{25}$$"
      ],
      "text/plain": [
       "16\n",
       "──\n",
       "25"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r1/r2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Numerical evaluation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "SymPy uses a library for artitrary precision as numerical backend, and has predefined SymPy expressions for a number of mathematical constants, such as: `pi`, `e`, `oo` for infinity.\n",
    "\n",
    "To evaluate an expression numerically we can use the `evalf` function (or `N`). It takes an argument `n` which specifies the number of significant digits."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAgsAAAAPBAMAAACYf5HCAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nuErRtoiwcMr5//oXWhd7due/PDYAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$3.1415926535897932384626433832795028841971693993751$$"
      ],
      "text/plain": [
       "3.1415926535897932384626433832795028841971693993751"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pi.evalf(n=50)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "y = (x + pi)**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHUAAAAbBAMAAACw1N2lAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nZwkTauY/hwp/AUIch6GgpxacAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left(x + 3.1416\\right)^{2}$$"
      ],
      "text/plain": [
       "            2\n",
       "(x + 3.1416) "
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N(y, 5) # same as evalf"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When we numerically evaluate algebraic expressions we often want to substitute a symbol with a numerical value. In SymPy we do that using the `subs` function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFYAAAAbBAMAAAAUvmV2AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIma7zZnddlTvRIkQ\nqzLsm4+cAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABlUlEQVQ4EY2TPUjDUBDH/1ESLekXiq5mEMRN\nC24dKgQUpdLdDzrqINRB1zpZ3Zxd2slFEAU7Kk9wtwriqIvgpoIuKsS7916SJojmoHfv/vfr9V5y\nBRJae0okJGHnM6WkrJXPviVmW8ZnUhbo/07OWrXkbJtQwyH3p/XkqdxTITcYcjPiaTvMwtMNHY9w\nDEzTYU/IQtP7kjHuroDs7G6Re6fvHhQ7f8+/xtZQQXu7hpTnvcLicl1ItaNrcTa9rwrDHP5hMafY\nzS72dmxFicEMdc/zPgB+XmQFdrpvESMtToMZ7IvGwAYNOarUVQ6aBXqrlJHpuz3jtO+R0h36kB2w\nC1iTl+nMdddct8wFTGQE+Rc+RljaUFsvk//Msh2LIc12zZAqwdTL5LOmkxPEHvIXInczAaskRX9e\nXItcixR9N/k4aN70O4wKxvnNkPl9t5A6oXRRirL9crPsYB0YOl9SYsAWkGF2QcnyHWsiDH5fpRhV\nFeVehpA+XUYU2h1lkxH514R2UlnXrvtSPAbtDCdeiufyP0TiD8zVWRHvw7nlAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left(1.5 + \\pi\\right)^{2}$$"
      ],
      "text/plain": [
       "         2\n",
       "(1.5 + π) "
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y.subs(x, 1.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKAAAAAPBAMAAACRq9klAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nQmCC\n",
      "text/latex": [
       "$$21.5443823618587$$"
      ],
      "text/plain": [
       "21.5443823618587"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N(y.subs(x, 1.5))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The `subs` function can of course also be used to substitute Symbols and expressions:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFIAAAAbBAMAAAAdVcUMAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left(a + 2 \\pi\\right)^{2}$$"
      ],
      "text/plain": [
       "         2\n",
       "(a + 2⋅π) "
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y.subs(x, a+pi)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also combine numerical evolution of expressions with NumPy arrays:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x_vec = np.arange(0, 10, 0.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "y_vec = np.array([N(((x + pi)**2).subs(x, xx)) for xx in x_vec])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/javascript": [
       "/* Put everything inside the global mpl namespace */\n",
       "window.mpl = {};\n",
       "\n",
       "\n",
       "mpl.get_websocket_type = function() {\n",
       "    if (typeof(WebSocket) !== 'undefined') {\n",
       "        return WebSocket;\n",
       "    } else if (typeof(MozWebSocket) !== 'undefined') {\n",
       "        return MozWebSocket;\n",
       "    } else {\n",
       "        alert('Your browser does not have WebSocket support.' +\n",
       "              'Please try Chrome, Safari or Firefox ≥ 6. ' +\n",
       "              'Firefox 4 and 5 are also supported but you ' +\n",
       "              'have to enable WebSockets in about:config.');\n",
       "    };\n",
       "}\n",
       "\n",
       "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n",
       "    this.id = figure_id;\n",
       "\n",
       "    this.ws = websocket;\n",
       "\n",
       "    this.supports_binary = (this.ws.binaryType != undefined);\n",
       "\n",
       "    if (!this.supports_binary) {\n",
       "        var warnings = document.getElementById(\"mpl-warnings\");\n",
       "        if (warnings) {\n",
       "            warnings.style.display = 'block';\n",
       "            warnings.textContent = (\n",
       "                \"This browser does not support binary websocket messages. \" +\n",
       "                    \"Performance may be slow.\");\n",
       "        }\n",
       "    }\n",
       "\n",
       "    this.imageObj = new Image();\n",
       "\n",
       "    this.context = undefined;\n",
       "    this.message = undefined;\n",
       "    this.canvas = undefined;\n",
       "    this.rubberband_canvas = undefined;\n",
       "    this.rubberband_context = undefined;\n",
       "    this.format_dropdown = undefined;\n",
       "\n",
       "    this.image_mode = 'full';\n",
       "\n",
       "    this.root = $('<div/>');\n",
       "    this._root_extra_style(this.root)\n",
       "    this.root.attr('style', 'display: inline-block');\n",
       "\n",
       "    $(parent_element).append(this.root);\n",
       "\n",
       "    this._init_header(this);\n",
       "    this._init_canvas(this);\n",
       "    this._init_toolbar(this);\n",
       "\n",
       "    var fig = this;\n",
       "\n",
       "    this.waiting = false;\n",
       "\n",
       "    this.ws.onopen =  function () {\n",
       "            fig.send_message(\"supports_binary\", {value: fig.supports_binary});\n",
       "            fig.send_message(\"send_image_mode\", {});\n",
       "            if (mpl.ratio != 1) {\n",
       "                fig.send_message(\"set_dpi_ratio\", {'dpi_ratio': mpl.ratio});\n",
       "            }\n",
       "            fig.send_message(\"refresh\", {});\n",
       "        }\n",
       "\n",
       "    this.imageObj.onload = function() {\n",
       "            if (fig.image_mode == 'full') {\n",
       "                // Full images could contain transparency (where diff images\n",
       "                // almost always do), so we need to clear the canvas so that\n",
       "                // there is no ghosting.\n",
       "                fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n",
       "            }\n",
       "            fig.context.drawImage(fig.imageObj, 0, 0);\n",
       "        };\n",
       "\n",
       "    this.imageObj.onunload = function() {\n",
       "        this.ws.close();\n",
       "    }\n",
       "\n",
       "    this.ws.onmessage = this._make_on_message_function(this);\n",
       "\n",
       "    this.ondownload = ondownload;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_header = function() {\n",
       "    var titlebar = $(\n",
       "        '<div class=\"ui-dialog-titlebar ui-widget-header ui-corner-all ' +\n",
       "        'ui-helper-clearfix\"/>');\n",
       "    var titletext = $(\n",
       "        '<div class=\"ui-dialog-title\" style=\"width: 100%; ' +\n",
       "        'text-align: center; padding: 3px;\"/>');\n",
       "    titlebar.append(titletext)\n",
       "    this.root.append(titlebar);\n",
       "    this.header = titletext[0];\n",
       "}\n",
       "\n",
       "\n",
       "\n",
       "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n",
       "\n",
       "}\n",
       "\n",
       "\n",
       "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n",
       "\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_canvas = function() {\n",
       "    var fig = this;\n",
       "\n",
       "    var canvas_div = $('<div/>');\n",
       "\n",
       "    canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n",
       "\n",
       "    function canvas_keyboard_event(event) {\n",
       "        return fig.key_event(event, event['data']);\n",
       "    }\n",
       "\n",
       "    canvas_div.keydown('key_press', canvas_keyboard_event);\n",
       "    canvas_div.keyup('key_release', canvas_keyboard_event);\n",
       "    this.canvas_div = canvas_div\n",
       "    this._canvas_extra_style(canvas_div)\n",
       "    this.root.append(canvas_div);\n",
       "\n",
       "    var canvas = $('<canvas/>');\n",
       "    canvas.addClass('mpl-canvas');\n",
       "    canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n",
       "\n",
       "    this.canvas = canvas[0];\n",
       "    this.context = canvas[0].getContext(\"2d\");\n",
       "\n",
       "    var backingStore = this.context.backingStorePixelRatio ||\n",
       "\tthis.context.webkitBackingStorePixelRatio ||\n",
       "\tthis.context.mozBackingStorePixelRatio ||\n",
       "\tthis.context.msBackingStorePixelRatio ||\n",
       "\tthis.context.oBackingStorePixelRatio ||\n",
       "\tthis.context.backingStorePixelRatio || 1;\n",
       "\n",
       "    mpl.ratio = (window.devicePixelRatio || 1) / backingStore;\n",
       "\n",
       "    var rubberband = $('<canvas/>');\n",
       "    rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n",
       "\n",
       "    var pass_mouse_events = true;\n",
       "\n",
       "    canvas_div.resizable({\n",
       "        start: function(event, ui) {\n",
       "            pass_mouse_events = false;\n",
       "        },\n",
       "        resize: function(event, ui) {\n",
       "            fig.request_resize(ui.size.width, ui.size.height);\n",
       "        },\n",
       "        stop: function(event, ui) {\n",
       "            pass_mouse_events = true;\n",
       "            fig.request_resize(ui.size.width, ui.size.height);\n",
       "        },\n",
       "    });\n",
       "\n",
       "    function mouse_event_fn(event) {\n",
       "        if (pass_mouse_events)\n",
       "            return fig.mouse_event(event, event['data']);\n",
       "    }\n",
       "\n",
       "    rubberband.mousedown('button_press', mouse_event_fn);\n",
       "    rubberband.mouseup('button_release', mouse_event_fn);\n",
       "    // Throttle sequential mouse events to 1 every 20ms.\n",
       "    rubberband.mousemove('motion_notify', mouse_event_fn);\n",
       "\n",
       "    rubberband.mouseenter('figure_enter', mouse_event_fn);\n",
       "    rubberband.mouseleave('figure_leave', mouse_event_fn);\n",
       "\n",
       "    canvas_div.on(\"wheel\", function (event) {\n",
       "        event = event.originalEvent;\n",
       "        event['data'] = 'scroll'\n",
       "        if (event.deltaY < 0) {\n",
       "            event.step = 1;\n",
       "        } else {\n",
       "            event.step = -1;\n",
       "        }\n",
       "        mouse_event_fn(event);\n",
       "    });\n",
       "\n",
       "    canvas_div.append(canvas);\n",
       "    canvas_div.append(rubberband);\n",
       "\n",
       "    this.rubberband = rubberband;\n",
       "    this.rubberband_canvas = rubberband[0];\n",
       "    this.rubberband_context = rubberband[0].getContext(\"2d\");\n",
       "    this.rubberband_context.strokeStyle = \"#000000\";\n",
       "\n",
       "    this._resize_canvas = function(width, height) {\n",
       "        // Keep the size of the canvas, canvas container, and rubber band\n",
       "        // canvas in synch.\n",
       "        canvas_div.css('width', width)\n",
       "        canvas_div.css('height', height)\n",
       "\n",
       "        canvas.attr('width', width * mpl.ratio);\n",
       "        canvas.attr('height', height * mpl.ratio);\n",
       "        canvas.attr('style', 'width: ' + width + 'px; height: ' + height + 'px;');\n",
       "\n",
       "        rubberband.attr('width', width);\n",
       "        rubberband.attr('height', height);\n",
       "    }\n",
       "\n",
       "    // Set the figure to an initial 600x600px, this will subsequently be updated\n",
       "    // upon first draw.\n",
       "    this._resize_canvas(600, 600);\n",
       "\n",
       "    // Disable right mouse context menu.\n",
       "    $(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n",
       "        return false;\n",
       "    });\n",
       "\n",
       "    function set_focus () {\n",
       "        canvas.focus();\n",
       "        canvas_div.focus();\n",
       "    }\n",
       "\n",
       "    window.setTimeout(set_focus, 100);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_toolbar = function() {\n",
       "    var fig = this;\n",
       "\n",
       "    var nav_element = $('<div/>')\n",
       "    nav_element.attr('style', 'width: 100%');\n",
       "    this.root.append(nav_element);\n",
       "\n",
       "    // Define a callback function for later on.\n",
       "    function toolbar_event(event) {\n",
       "        return fig.toolbar_button_onclick(event['data']);\n",
       "    }\n",
       "    function toolbar_mouse_event(event) {\n",
       "        return fig.toolbar_button_onmouseover(event['data']);\n",
       "    }\n",
       "\n",
       "    for(var toolbar_ind in mpl.toolbar_items) {\n",
       "        var name = mpl.toolbar_items[toolbar_ind][0];\n",
       "        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n",
       "        var image = mpl.toolbar_items[toolbar_ind][2];\n",
       "        var method_name = mpl.toolbar_items[toolbar_ind][3];\n",
       "\n",
       "        if (!name) {\n",
       "            // put a spacer in here.\n",
       "            continue;\n",
       "        }\n",
       "        var button = $('<button/>');\n",
       "        button.addClass('ui-button ui-widget ui-state-default ui-corner-all ' +\n",
       "                        'ui-button-icon-only');\n",
       "        button.attr('role', 'button');\n",
       "        button.attr('aria-disabled', 'false');\n",
       "        button.click(method_name, toolbar_event);\n",
       "        button.mouseover(tooltip, toolbar_mouse_event);\n",
       "\n",
       "        var icon_img = $('<span/>');\n",
       "        icon_img.addClass('ui-button-icon-primary ui-icon');\n",
       "        icon_img.addClass(image);\n",
       "        icon_img.addClass('ui-corner-all');\n",
       "\n",
       "        var tooltip_span = $('<span/>');\n",
       "        tooltip_span.addClass('ui-button-text');\n",
       "        tooltip_span.html(tooltip);\n",
       "\n",
       "        button.append(icon_img);\n",
       "        button.append(tooltip_span);\n",
       "\n",
       "        nav_element.append(button);\n",
       "    }\n",
       "\n",
       "    var fmt_picker_span = $('<span/>');\n",
       "\n",
       "    var fmt_picker = $('<select/>');\n",
       "    fmt_picker.addClass('mpl-toolbar-option ui-widget ui-widget-content');\n",
       "    fmt_picker_span.append(fmt_picker);\n",
       "    nav_element.append(fmt_picker_span);\n",
       "    this.format_dropdown = fmt_picker[0];\n",
       "\n",
       "    for (var ind in mpl.extensions) {\n",
       "        var fmt = mpl.extensions[ind];\n",
       "        var option = $(\n",
       "            '<option/>', {selected: fmt === mpl.default_extension}).html(fmt);\n",
       "        fmt_picker.append(option)\n",
       "    }\n",
       "\n",
       "    // Add hover states to the ui-buttons\n",
       "    $( \".ui-button\" ).hover(\n",
       "        function() { $(this).addClass(\"ui-state-hover\");},\n",
       "        function() { $(this).removeClass(\"ui-state-hover\");}\n",
       "    );\n",
       "\n",
       "    var status_bar = $('<span class=\"mpl-message\"/>');\n",
       "    nav_element.append(status_bar);\n",
       "    this.message = status_bar[0];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.request_resize = function(x_pixels, y_pixels) {\n",
       "    // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,\n",
       "    // which will in turn request a refresh of the image.\n",
       "    this.send_message('resize', {'width': x_pixels, 'height': y_pixels});\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.send_message = function(type, properties) {\n",
       "    properties['type'] = type;\n",
       "    properties['figure_id'] = this.id;\n",
       "    this.ws.send(JSON.stringify(properties));\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.send_draw_message = function() {\n",
       "    if (!this.waiting) {\n",
       "        this.waiting = true;\n",
       "        this.ws.send(JSON.stringify({type: \"draw\", figure_id: this.id}));\n",
       "    }\n",
       "}\n",
       "\n",
       "\n",
       "mpl.figure.prototype.handle_save = function(fig, msg) {\n",
       "    var format_dropdown = fig.format_dropdown;\n",
       "    var format = format_dropdown.options[format_dropdown.selectedIndex].value;\n",
       "    fig.ondownload(fig, format);\n",
       "}\n",
       "\n",
       "\n",
       "mpl.figure.prototype.handle_resize = function(fig, msg) {\n",
       "    var size = msg['size'];\n",
       "    if (size[0] != fig.canvas.width || size[1] != fig.canvas.height) {\n",
       "        fig._resize_canvas(size[0], size[1]);\n",
       "        fig.send_message(\"refresh\", {});\n",
       "    };\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_rubberband = function(fig, msg) {\n",
       "    var x0 = msg['x0'] / mpl.ratio;\n",
       "    var y0 = (fig.canvas.height - msg['y0']) / mpl.ratio;\n",
       "    var x1 = msg['x1'] / mpl.ratio;\n",
       "    var y1 = (fig.canvas.height - msg['y1']) / mpl.ratio;\n",
       "    x0 = Math.floor(x0) + 0.5;\n",
       "    y0 = Math.floor(y0) + 0.5;\n",
       "    x1 = Math.floor(x1) + 0.5;\n",
       "    y1 = Math.floor(y1) + 0.5;\n",
       "    var min_x = Math.min(x0, x1);\n",
       "    var min_y = Math.min(y0, y1);\n",
       "    var width = Math.abs(x1 - x0);\n",
       "    var height = Math.abs(y1 - y0);\n",
       "\n",
       "    fig.rubberband_context.clearRect(\n",
       "        0, 0, fig.canvas.width, fig.canvas.height);\n",
       "\n",
       "    fig.rubberband_context.strokeRect(min_x, min_y, width, height);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_figure_label = function(fig, msg) {\n",
       "    // Updates the figure title.\n",
       "    fig.header.textContent = msg['label'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_cursor = function(fig, msg) {\n",
       "    var cursor = msg['cursor'];\n",
       "    switch(cursor)\n",
       "    {\n",
       "    case 0:\n",
       "        cursor = 'pointer';\n",
       "        break;\n",
       "    case 1:\n",
       "        cursor = 'default';\n",
       "        break;\n",
       "    case 2:\n",
       "        cursor = 'crosshair';\n",
       "        break;\n",
       "    case 3:\n",
       "        cursor = 'move';\n",
       "        break;\n",
       "    }\n",
       "    fig.rubberband_canvas.style.cursor = cursor;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_message = function(fig, msg) {\n",
       "    fig.message.textContent = msg['message'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_draw = function(fig, msg) {\n",
       "    // Request the server to send over a new figure.\n",
       "    fig.send_draw_message();\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_image_mode = function(fig, msg) {\n",
       "    fig.image_mode = msg['mode'];\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function() {\n",
       "    // Called whenever the canvas gets updated.\n",
       "    this.send_message(\"ack\", {});\n",
       "}\n",
       "\n",
       "// A function to construct a web socket function for onmessage handling.\n",
       "// Called in the figure constructor.\n",
       "mpl.figure.prototype._make_on_message_function = function(fig) {\n",
       "    return function socket_on_message(evt) {\n",
       "        if (evt.data instanceof Blob) {\n",
       "            /* FIXME: We get \"Resource interpreted as Image but\n",
       "             * transferred with MIME type text/plain:\" errors on\n",
       "             * Chrome.  But how to set the MIME type?  It doesn't seem\n",
       "             * to be part of the websocket stream */\n",
       "            evt.data.type = \"image/png\";\n",
       "\n",
       "            /* Free the memory for the previous frames */\n",
       "            if (fig.imageObj.src) {\n",
       "                (window.URL || window.webkitURL).revokeObjectURL(\n",
       "                    fig.imageObj.src);\n",
       "            }\n",
       "\n",
       "            fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n",
       "                evt.data);\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        }\n",
       "        else if (typeof evt.data === 'string' && evt.data.slice(0, 21) == \"data:image/png;base64\") {\n",
       "            fig.imageObj.src = evt.data;\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        var msg = JSON.parse(evt.data);\n",
       "        var msg_type = msg['type'];\n",
       "\n",
       "        // Call the  \"handle_{type}\" callback, which takes\n",
       "        // the figure and JSON message as its only arguments.\n",
       "        try {\n",
       "            var callback = fig[\"handle_\" + msg_type];\n",
       "        } catch (e) {\n",
       "            console.log(\"No handler for the '\" + msg_type + \"' message type: \", msg);\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        if (callback) {\n",
       "            try {\n",
       "                // console.log(\"Handling '\" + msg_type + \"' message: \", msg);\n",
       "                callback(fig, msg);\n",
       "            } catch (e) {\n",
       "                console.log(\"Exception inside the 'handler_\" + msg_type + \"' callback:\", e, e.stack, msg);\n",
       "            }\n",
       "        }\n",
       "    };\n",
       "}\n",
       "\n",
       "// from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas\n",
       "mpl.findpos = function(e) {\n",
       "    //this section is from http://www.quirksmode.org/js/events_properties.html\n",
       "    var targ;\n",
       "    if (!e)\n",
       "        e = window.event;\n",
       "    if (e.target)\n",
       "        targ = e.target;\n",
       "    else if (e.srcElement)\n",
       "        targ = e.srcElement;\n",
       "    if (targ.nodeType == 3) // defeat Safari bug\n",
       "        targ = targ.parentNode;\n",
       "\n",
       "    // jQuery normalizes the pageX and pageY\n",
       "    // pageX,Y are the mouse positions relative to the document\n",
       "    // offset() returns the position of the element relative to the document\n",
       "    var x = e.pageX - $(targ).offset().left;\n",
       "    var y = e.pageY - $(targ).offset().top;\n",
       "\n",
       "    return {\"x\": x, \"y\": y};\n",
       "};\n",
       "\n",
       "/*\n",
       " * return a copy of an object with only non-object keys\n",
       " * we need this to avoid circular references\n",
       " * http://stackoverflow.com/a/24161582/3208463\n",
       " */\n",
       "function simpleKeys (original) {\n",
       "  return Object.keys(original).reduce(function (obj, key) {\n",
       "    if (typeof original[key] !== 'object')\n",
       "        obj[key] = original[key]\n",
       "    return obj;\n",
       "  }, {});\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.mouse_event = function(event, name) {\n",
       "    var canvas_pos = mpl.findpos(event)\n",
       "\n",
       "    if (name === 'button_press')\n",
       "    {\n",
       "        this.canvas.focus();\n",
       "        this.canvas_div.focus();\n",
       "    }\n",
       "\n",
       "    var x = canvas_pos.x * mpl.ratio;\n",
       "    var y = canvas_pos.y * mpl.ratio;\n",
       "\n",
       "    this.send_message(name, {x: x, y: y, button: event.button,\n",
       "                             step: event.step,\n",
       "                             guiEvent: simpleKeys(event)});\n",
       "\n",
       "    /* This prevents the web browser from automatically changing to\n",
       "     * the text insertion cursor when the button is pressed.  We want\n",
       "     * to control all of the cursor setting manually through the\n",
       "     * 'cursor' event from matplotlib */\n",
       "    event.preventDefault();\n",
       "    return false;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._key_event_extra = function(event, name) {\n",
       "    // Handle any extra behaviour associated with a key event\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.key_event = function(event, name) {\n",
       "\n",
       "    // Prevent repeat events\n",
       "    if (name == 'key_press')\n",
       "    {\n",
       "        if (event.which === this._key)\n",
       "            return;\n",
       "        else\n",
       "            this._key = event.which;\n",
       "    }\n",
       "    if (name == 'key_release')\n",
       "        this._key = null;\n",
       "\n",
       "    var value = '';\n",
       "    if (event.ctrlKey && event.which != 17)\n",
       "        value += \"ctrl+\";\n",
       "    if (event.altKey && event.which != 18)\n",
       "        value += \"alt+\";\n",
       "    if (event.shiftKey && event.which != 16)\n",
       "        value += \"shift+\";\n",
       "\n",
       "    value += 'k';\n",
       "    value += event.which.toString();\n",
       "\n",
       "    this._key_event_extra(event, name);\n",
       "\n",
       "    this.send_message(name, {key: value,\n",
       "                             guiEvent: simpleKeys(event)});\n",
       "    return false;\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onclick = function(name) {\n",
       "    if (name == 'download') {\n",
       "        this.handle_save(this, null);\n",
       "    } else {\n",
       "        this.send_message(\"toolbar_button\", {name: name});\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.toolbar_button_onmouseover = function(tooltip) {\n",
       "    this.message.textContent = tooltip;\n",
       "};\n",
       "mpl.toolbar_items = [[\"Home\", \"Reset original view\", \"fa fa-home icon-home\", \"home\"], [\"Back\", \"Back to  previous view\", \"fa fa-arrow-left icon-arrow-left\", \"back\"], [\"Forward\", \"Forward to next view\", \"fa fa-arrow-right icon-arrow-right\", \"forward\"], [\"\", \"\", \"\", \"\"], [\"Pan\", \"Pan axes with left mouse, zoom with right\", \"fa fa-arrows icon-move\", \"pan\"], [\"Zoom\", \"Zoom to rectangle\", \"fa fa-square-o icon-check-empty\", \"zoom\"], [\"\", \"\", \"\", \"\"], [\"Download\", \"Download plot\", \"fa fa-floppy-o icon-save\", \"download\"]];\n",
       "\n",
       "mpl.extensions = [\"eps\", \"jpeg\", \"pdf\", \"png\", \"ps\", \"raw\", \"svg\", \"tif\"];\n",
       "\n",
       "mpl.default_extension = \"png\";var comm_websocket_adapter = function(comm) {\n",
       "    // Create a \"websocket\"-like object which calls the given IPython comm\n",
       "    // object with the appropriate methods. Currently this is a non binary\n",
       "    // socket, so there is still some room for performance tuning.\n",
       "    var ws = {};\n",
       "\n",
       "    ws.close = function() {\n",
       "        comm.close()\n",
       "    };\n",
       "    ws.send = function(m) {\n",
       "        //console.log('sending', m);\n",
       "        comm.send(m);\n",
       "    };\n",
       "    // Register the callback with on_msg.\n",
       "    comm.on_msg(function(msg) {\n",
       "        //console.log('receiving', msg['content']['data'], msg);\n",
       "        // Pass the mpl event to the overriden (by mpl) onmessage function.\n",
       "        ws.onmessage(msg['content']['data'])\n",
       "    });\n",
       "    return ws;\n",
       "}\n",
       "\n",
       "mpl.mpl_figure_comm = function(comm, msg) {\n",
       "    // This is the function which gets called when the mpl process\n",
       "    // starts-up an IPython Comm through the \"matplotlib\" channel.\n",
       "\n",
       "    var id = msg.content.data.id;\n",
       "    // Get hold of the div created by the display call when the Comm\n",
       "    // socket was opened in Python.\n",
       "    var element = $(\"#\" + id);\n",
       "    var ws_proxy = comm_websocket_adapter(comm)\n",
       "\n",
       "    function ondownload(figure, format) {\n",
       "        window.open(figure.imageObj.src);\n",
       "    }\n",
       "\n",
       "    var fig = new mpl.figure(id, ws_proxy,\n",
       "                           ondownload,\n",
       "                           element.get(0));\n",
       "\n",
       "    // Call onopen now - mpl needs it, as it is assuming we've passed it a real\n",
       "    // web socket which is closed, not our websocket->open comm proxy.\n",
       "    ws_proxy.onopen();\n",
       "\n",
       "    fig.parent_element = element.get(0);\n",
       "    fig.cell_info = mpl.find_output_cell(\"<div id='\" + id + \"'></div>\");\n",
       "    if (!fig.cell_info) {\n",
       "        console.error(\"Failed to find cell for figure\", id, fig);\n",
       "        return;\n",
       "    }\n",
       "\n",
       "    var output_index = fig.cell_info[2]\n",
       "    var cell = fig.cell_info[0];\n",
       "\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_close = function(fig, msg) {\n",
       "    var width = fig.canvas.width/mpl.ratio\n",
       "    fig.root.unbind('remove')\n",
       "\n",
       "    // Update the output cell to use the data from the current canvas.\n",
       "    fig.push_to_output();\n",
       "    var dataURL = fig.canvas.toDataURL();\n",
       "    // Re-enable the keyboard manager in IPython - without this line, in FF,\n",
       "    // the notebook keyboard shortcuts fail.\n",
       "    IPython.keyboard_manager.enable()\n",
       "    $(fig.parent_element).html('<img src=\"' + dataURL + '\" width=\"' + width + '\">');\n",
       "    fig.close_ws(fig, msg);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.close_ws = function(fig, msg){\n",
       "    fig.send_message('closing', msg);\n",
       "    // fig.ws.close()\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.push_to_output = function(remove_interactive) {\n",
       "    // Turn the data on the canvas into data in the output cell.\n",
       "    var width = this.canvas.width/mpl.ratio\n",
       "    var dataURL = this.canvas.toDataURL();\n",
       "    this.cell_info[1]['text/html'] = '<img src=\"' + dataURL + '\" width=\"' + width + '\">';\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function() {\n",
       "    // Tell IPython that the notebook contents must change.\n",
       "    IPython.notebook.set_dirty(true);\n",
       "    this.send_message(\"ack\", {});\n",
       "    var fig = this;\n",
       "    // Wait a second, then push the new image to the DOM so\n",
       "    // that it is saved nicely (might be nice to debounce this).\n",
       "    setTimeout(function () { fig.push_to_output() }, 1000);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._init_toolbar = function() {\n",
       "    var fig = this;\n",
       "\n",
       "    var nav_element = $('<div/>')\n",
       "    nav_element.attr('style', 'width: 100%');\n",
       "    this.root.append(nav_element);\n",
       "\n",
       "    // Define a callback function for later on.\n",
       "    function toolbar_event(event) {\n",
       "        return fig.toolbar_button_onclick(event['data']);\n",
       "    }\n",
       "    function toolbar_mouse_event(event) {\n",
       "        return fig.toolbar_button_onmouseover(event['data']);\n",
       "    }\n",
       "\n",
       "    for(var toolbar_ind in mpl.toolbar_items){\n",
       "        var name = mpl.toolbar_items[toolbar_ind][0];\n",
       "        var tooltip = mpl.toolbar_items[toolbar_ind][1];\n",
       "        var image = mpl.toolbar_items[toolbar_ind][2];\n",
       "        var method_name = mpl.toolbar_items[toolbar_ind][3];\n",
       "\n",
       "        if (!name) { continue; };\n",
       "\n",
       "        var button = $('<button class=\"btn btn-default\" href=\"#\" title=\"' + name + '\"><i class=\"fa ' + image + ' fa-lg\"></i></button>');\n",
       "        button.click(method_name, toolbar_event);\n",
       "        button.mouseover(tooltip, toolbar_mouse_event);\n",
       "        nav_element.append(button);\n",
       "    }\n",
       "\n",
       "    // Add the status bar.\n",
       "    var status_bar = $('<span class=\"mpl-message\" style=\"text-align:right; float: right;\"/>');\n",
       "    nav_element.append(status_bar);\n",
       "    this.message = status_bar[0];\n",
       "\n",
       "    // Add the close button to the window.\n",
       "    var buttongrp = $('<div class=\"btn-group inline pull-right\"></div>');\n",
       "    var button = $('<button class=\"btn btn-mini btn-primary\" href=\"#\" title=\"Stop Interaction\"><i class=\"fa fa-power-off icon-remove icon-large\"></i></button>');\n",
       "    button.click(function (evt) { fig.handle_close(fig, {}); } );\n",
       "    button.mouseover('Stop Interaction', toolbar_mouse_event);\n",
       "    buttongrp.append(button);\n",
       "    var titlebar = this.root.find($('.ui-dialog-titlebar'));\n",
       "    titlebar.prepend(buttongrp);\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._root_extra_style = function(el){\n",
       "    var fig = this\n",
       "    el.on(\"remove\", function(){\n",
       "\tfig.close_ws(fig, {});\n",
       "    });\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._canvas_extra_style = function(el){\n",
       "    // this is important to make the div 'focusable\n",
       "    el.attr('tabindex', 0)\n",
       "    // reach out to IPython and tell the keyboard manager to turn it's self\n",
       "    // off when our div gets focus\n",
       "\n",
       "    // location in version 3\n",
       "    if (IPython.notebook.keyboard_manager) {\n",
       "        IPython.notebook.keyboard_manager.register_events(el);\n",
       "    }\n",
       "    else {\n",
       "        // location in version 2\n",
       "        IPython.keyboard_manager.register_events(el);\n",
       "    }\n",
       "\n",
       "}\n",
       "\n",
       "mpl.figure.prototype._key_event_extra = function(event, name) {\n",
       "    var manager = IPython.notebook.keyboard_manager;\n",
       "    if (!manager)\n",
       "        manager = IPython.keyboard_manager;\n",
       "\n",
       "    // Check for shift+enter\n",
       "    if (event.shiftKey && event.which == 13) {\n",
       "        this.canvas_div.blur();\n",
       "        // select the cell after this one\n",
       "        var index = IPython.notebook.find_cell_index(this.cell_info[0]);\n",
       "        IPython.notebook.select(index + 1);\n",
       "    }\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.handle_save = function(fig, msg) {\n",
       "    fig.ondownload(fig, null);\n",
       "}\n",
       "\n",
       "\n",
       "mpl.find_output_cell = function(html_output) {\n",
       "    // Return the cell and output element which can be found *uniquely* in the notebook.\n",
       "    // Note - this is a bit hacky, but it is done because the \"notebook_saving.Notebook\"\n",
       "    // IPython event is triggered only after the cells have been serialised, which for\n",
       "    // our purposes (turning an active figure into a static one), is too late.\n",
       "    var cells = IPython.notebook.get_cells();\n",
       "    var ncells = cells.length;\n",
       "    for (var i=0; i<ncells; i++) {\n",
       "        var cell = cells[i];\n",
       "        if (cell.cell_type === 'code'){\n",
       "            for (var j=0; j<cell.output_area.outputs.length; j++) {\n",
       "                var data = cell.output_area.outputs[j];\n",
       "                if (data.data) {\n",
       "                    // IPython >= 3 moved mimebundle to data attribute of output\n",
       "                    data = data.data;\n",
       "                }\n",
       "                if (data['text/html'] == html_output) {\n",
       "                    return [cell, data, j];\n",
       "                }\n",
       "            }\n",
       "        }\n",
       "    }\n",
       "}\n",
       "\n",
       "// Register the function which deals with the matplotlib target/channel.\n",
       "// The kernel may be null if the page has been refreshed.\n",
       "if (IPython.notebook.kernel != null) {\n",
       "    IPython.notebook.kernel.comm_manager.register_target('matplotlib', mpl.mpl_figure_comm);\n",
       "}\n"
      ],
      "text/plain": [
       "<IPython.core.display.Javascript object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
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\" width=\"639.9999861283738\">"
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "ax.plot(x_vec, y_vec);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "However, this kind of numerical evolution can be very slow, and there is a much more efficient way to do it: Use the function `lambdify` to \"compile\" a Sympy expression into a function that is much more efficient to evaluate numerically:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "f = lambdify([x], (x + pi)**2, 'numpy')  # the first argument is a list of variables that\n",
    "                                         # f will be a function of: in this case only x -> f(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "y_vec = f(x_vec)  # now we can directly pass a numpy array and f(x) is efficiently evaluated"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The speedup when using \"lambdified\" functions instead of direct numerical evaluation can be significant, often several orders of magnitude. Even in this simple example we get a significant speed up:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "16.6 ms ± 752 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)\n"
     ]
    }
   ],
   "source": [
    "%%timeit\n",
    "\n",
    "y_vec = numpy.array([N(((x + pi)**2).subs(x, xx)) for xx in x_vec])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.52 µs ± 58.8 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)\n"
     ]
    }
   ],
   "source": [
    "%%timeit\n",
    "\n",
    "y_vec = f(x_vec)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Algebraic manipulations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "One of the main uses of an CAS is to perform algebraic manipulations of expressions. For example, we might want to expand a product, factor an expression, or simply an expression. The functions for doing these basic operations in SymPy are demonstrated in this section."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Expand and factor"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The first steps in an algebraic manipulation "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMEAAAAUBAMAAADGn0QzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrkJggg==\n",
      "text/latex": [
       "$$\\left(x + 1\\right) \\left(x + 2\\right) \\left(x + 3\\right)$$"
      ],
      "text/plain": [
       "(x + 1)⋅(x + 2)⋅(x + 3)"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(x+1)*(x+2)*(x+3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKsAAAAWBAMAAABNknGBAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nESpbZVPGoMA/vluPpCTgHFcAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$x^{3} + 6 x^{2} + 11 x + 6$$"
      ],
      "text/plain": [
       " 3      2           \n",
       "x  + 6⋅x  + 11⋅x + 6"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand((x+1)*(x+2)*(x+3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The `expand` function takes a number of keywords arguments which we can tell the functions what kind of expansions we want to have performed. For example, to expand trigonometric expressions, use the `trig=True` keyword argument:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFgAAAAUBAMAAAD7IecQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ngg==\n",
      "text/latex": [
       "$$\\sin{\\left (a + b \\right )}$$"
      ],
      "text/plain": [
       "sin(a + b)"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sin(a+b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAP8AAAAUBAMAAACwpfa4AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ndIV4Paqf5pWV3KYDa0RE5S/Z/+i1KXyz9uD7AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\sin{\\left (a \\right )} \\cos{\\left (b \\right )} + \\sin{\\left (b \\right )} \\cos{\\left (a \\right )}$$"
      ],
      "text/plain": [
       "sin(a)⋅cos(b) + sin(b)⋅cos(a)"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand(sin(a+b), trig=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "See `help(expand)` for a detailed explanation of the various types of expansions the `expand` functions can perform."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The opposite a product expansion is of course factoring. The factor an expression in SymPy use the `factor` function: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMEAAAAUBAMAAADGn0QzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrkJggg==\n",
      "text/latex": [
       "$$\\left(x + 1\\right) \\left(x + 2\\right) \\left(x + 3\\right)$$"
      ],
      "text/plain": [
       "(x + 1)⋅(x + 2)⋅(x + 3)"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor(x**3 + 6 * x**2 + 11*x + 6)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Simplify"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The `simplify` tries to simplify an expression into a nice looking expression, using various techniques. More specific alternatives to the `simplify` functions also exists: `trigsimp`, `powsimp`, `logcombine`, etc. \n",
    "\n",
    "The basic usages of these functions are as follows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMEAAAAUBAMAAADGn0QzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrkJggg==\n",
      "text/latex": [
       "$$\\left(x + 1\\right) \\left(x + 2\\right) \\left(x + 3\\right)$$"
      ],
      "text/plain": [
       "(x + 1)⋅(x + 2)⋅(x + 3)"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# simplify expands a product\n",
    "simplify((x+1)*(x+2)*(x+3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAPBAMAAAArJJMAAAAAHlBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAACGjDitAAAACXRSTlMAVO8Qq5l2zWYZcMvdAAAACXBIWXMAAA7EAAAOxAGV\nKw4bAAAAHUlEQVQIHWNgAANGZQYGk5DJQDYbqQSr03QPsBkAJYgIYEZbtZEAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$1$$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# simplify uses trigonometric identities\n",
    "simplify(sin(a)**2 + cos(a)**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAD4AAAAvBAMAAABJZWRJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{1}{\\tan{\\left (x \\right )}}$$"
      ],
      "text/plain": [
       "  1   \n",
       "──────\n",
       "tan(x)"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(cos(x)/sin(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### apart and together"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To manipulate symbolic expressions of fractions, we can use the `apart` and `together` functions:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "f1 = 1/((a+1)*(a+2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIIAAAAvBAMAAADdrw/+AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\frac{1}{\\left(a + 1\\right) \\left(a + 2\\right)}$$"
      ],
      "text/plain": [
       "       1       \n",
       "───────────────\n",
       "(a + 1)⋅(a + 2)"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAI4AAAAsBAMAAABBB53eAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nUyuKBLIyAOpdmg617KB4AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$- \\frac{1}{a + 2} + \\frac{1}{a + 1}$$"
      ],
      "text/plain": [
       "    1       1  \n",
       "- ───── + ─────\n",
       "  a + 2   a + 1"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "apart(f1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "f2 = 1/(a+2) + 1/(a+3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHwAAAAsBAMAAABVvsF6AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nnOfh0q8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\frac{1}{a + 3} + \\frac{1}{a + 2}$$"
      ],
      "text/plain": [
       "  1       1  \n",
       "───── + ─────\n",
       "a + 3   a + 2"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIIAAAAvBAMAAADdrw/+AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrkJggg==\n",
      "text/latex": [
       "$$\\frac{2 a + 5}{\\left(a + 2\\right) \\left(a + 3\\right)}$$"
      ],
      "text/plain": [
       "    2⋅a + 5    \n",
       "───────────────\n",
       "(a + 2)⋅(a + 3)"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "together(f2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Simplify usually combines fractions but does not factor: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIIAAAAvBAMAAADdrw/+AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACtklEQVRIDe2WzWsTQRjGn02y+dpuSCMo9LQk\nYC9KK3pS0OhfsKdCvSQIXrRgiGIugrnWi5YiWAsSPNSDivVqhKQFUU9G/wHr1UPpwdDi1/rOzs42\n231NFkoPggPZd/eZ3zz77uzMvgHCrVA6FRYB403iOKeHNb2K282wjIzj1BiZkZI2MvcGdLMrL4yb\nawPqsNNMC8lvA4BySA5ow0+N/n4dyD/bhzZ57L68k5/D5npZKhGOd2uYzxtbklQOcVvbiTBWIivQ\nHyE2HXSgq9NRHWIWYn1ku8Sn2u1Xy+12VQ6dyss48rgOZKZRKUtQPcVV8OuEsTMtjFdq+Ox1KYdl\nYMpmcEa6AKxVejhjypyVgwU8ZWhG0p40jraMbm4lJTuVQwG5BQZnJMNxnJY+d+Pwu6CDNnkp4kPs\nNVU57NWjX+tRX2F0y//kAc0ALZ/9tQPK61+01Tb+lvUX2TESOMQZaMWTZSTlnuKBxvWeAl5yDjNI\n7UBfdbtYoADztwfkrEEH9TG4DCwBt0QXD3zo4bsHZAPbXzk8BK6VMSsceGC+qf+ABCYE5Tfl0LGF\ng9GkDh6AeAoJUHFgiiSJL2ykLIoCQKH+3hYR6hbAxRoVJIuks/QLF0kgvU1TIKZSAOkFdChQ8x2O\nPCZLF3gOvkhmq4S3aAwBmN1AiQI13wGpBx5AkS2SRcITW3QgQBSrZxSCVXTJ9gGmSGLMohG+wyJ0\n72+NymEcONeTACXJFEl8gpZHukU2BOjbMEU61JSDYwsHF6CJYopkwsJY3p9J/Rfiqx8DDov0sjyg\nTm81XCQnGvUrNEE1GkUAXuNrVZzv5rAJ86cH3KEkw0Wy4zi0aI0yjSEAM2/PzzUp7jqkiyUSXIBf\ntC4tl+NIILhxgkWS2VkMgBPyfszR290jAfYD4vrF824YCYz8iA0H/gDROfuPYC+5AgAAAABJRU5E\nrkJggg==\n",
      "text/latex": [
       "$$\\frac{2 a + 5}{\\left(a + 2\\right) \\left(a + 3\\right)}$$"
      ],
      "text/plain": [
       "    2⋅a + 5    \n",
       "───────────────\n",
       "(a + 2)⋅(a + 3)"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(f2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Calculus"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In addition to algebraic manipulations, the other main use of CAS is to do calculus, like derivatives and integrals of algebraic expressions."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Differentiation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Differentiation is usually simple. Use the `diff` function. The first argument is the expression to take the derivative of, and the second argument is the symbol by which to take the derivative:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEgAAAAbBAMAAAAt2dQtAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRK5CYII=\n",
      "text/latex": [
       "$$\\left(x + \\pi\\right)^{2}$$"
      ],
      "text/plain": [
       "       2\n",
       "(x + π) "
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFgAAAAbBAMAAAAKd1XFAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYII=\n",
      "text/latex": [
       "$$4 \\left(x + \\pi\\right)^{3}$$"
      ],
      "text/plain": [
       "         3\n",
       "4⋅(x + π) "
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(y**2, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For higher order derivatives we can do:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGEAAAAbBAMAAACekfw3AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$12 \\left(x + \\pi\\right)^{2}$$"
      ],
      "text/plain": [
       "          2\n",
       "12⋅(x + π) "
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(y**2, x, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGEAAAAbBAMAAACekfw3AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$12 \\left(x + \\pi\\right)^{2}$$"
      ],
      "text/plain": [
       "          2\n",
       "12⋅(x + π) "
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(y**2, x, 2) # same as above"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To calculate the derivative of a multivariate expression, we can do:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x, y, z = symbols(\"x,y,z\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "f = sin(x*y) + cos(y*z)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\frac{d^3f}{dxdy^2}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAPgAAAAUBAMAAABSee3BAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$- x \\left(x y \\cos{\\left (x y \\right )} + 2 \\sin{\\left (x y \\right )}\\right)$$"
      ],
      "text/plain": [
       "-x⋅(x⋅y⋅cos(x⋅y) + 2⋅sin(x⋅y))"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(f, x, 1, y, 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Integration"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Integration is done in a similar fashion:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKAAAAAUBAMAAAD4uit9AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ndPqDmrj6L+AfOVyzbfsv4YgAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\sin{\\left (x y \\right )} + \\cos{\\left (y z \\right )}$$"
      ],
      "text/plain": [
       "sin(x⋅y) + cos(y⋅z)"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAVAAAAA/BAMAAABEE43RAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nZSa5leumKEui9XB43P4HnRvrjhXRBWkAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$x \\cos{\\left (y z \\right )} + \\begin{cases} 0 & \\text{for}\\: y = 0 \\\\- \\frac{1}{y} \\cos{\\left (x y \\right )} & \\text{otherwise} \\end{cases}$$"
      ],
      "text/plain": [
       "             ⎛⎧    0       for y = 0⎞\n",
       "             ⎜⎪                     ⎟\n",
       "x⋅cos(y⋅z) + ⎜⎨-cos(x⋅y)            ⎟\n",
       "             ⎜⎪──────────  otherwise⎟\n",
       "             ⎝⎩    y                ⎠"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(f, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By providing limits for the integration variable we can evaluate definite integrals:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFAAAAAUBAMAAADo9qfkAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nLu6LcvKsatbXARd/hb/komRAMlpwOAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$2 \\cos{\\left (y z \\right )}$$"
      ],
      "text/plain": [
       "2⋅cos(y⋅z)"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(f, (x, -1, 1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and also improper integrals"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAB0AAAAVBAMAAABI7vhRAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAInarRM2ZVBDdiWbv\nuzJCz3LGAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAr0lEQVQYGWNggADG/2DwAcplYHaAsSC0KCqX\nIR2VzzEBlc9WgMrvROUyrEblg7TrAx3wDSrMtIGBa12RtKIAkO8CxJwMDM8ZFjI9ADKZjgIJJSBm\nuMDcACQl9B0YGEC28xkAVQFB/wUG7gVAmm0DfwOQYmD7yMAJYvQ38DsAKQbGbwy7QLQmA88CEM1g\n3zADRN1mYF4AohneL08A0zCC9WgDjAmm2WGuhIkGwhhAGgDwdic2xV4k0wAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\sqrt{\\pi}$$"
      ],
      "text/plain": [
       "√π"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(exp(-x**2), (x, -oo, oo))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Remember, `oo` is the SymPy notation for inifinity."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Sums and products"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can evaluate sums and products using the functions: 'Sum'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "n = Symbol(\"n\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADsAAAA9BAMAAADhUgydAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMs2Zq91U7yJ2iWZE\nELuNX9C8AAAACXBIWXMAAA7EAAAOxAGVKw4bAAACFklEQVQ4Ec2Vz2sTQRTHv/nRpMnuklBPnhK8\niYdW2lMlpgjSBg+KFytEsgVbb017bEESUARBSaBKoafgqT25oCh4ykWEXsxB8OTFP8AmodpYLOPs\nzr7ZTXYz577Dznvfz763MzszPMC2KSB10HLckMeFm8AnPAwhQloFysh0xvFVGD0kzfE4MsCEpcBn\nmGgq8ABJBebfzrTHZytn/gjYGb/uJ5US0vMFf+3EY3806n+bOxuVhuLUecDTzLO/cnpyalWWdcQf\nL97XToI4zu6QqFVMcmU2KsekIZYn18M5ZpGIq+R5WGMDEpFyvdhi/xaJR7/JCx2/s5lQ3RV19keF\n8fako+JRlldho3aqwqiytorHmPJooeKrnvF2sOvWbDxQFcduXYX1XpCmrxdIzJnkeeM+briBseWp\n0ltGri6C6IIYh55lNNpC+Dmky6AqdiLtrSohGXfui+BjXYofpAekxSnwrUrf8OF3wm+YpBlFJ+Fw\n6fmrJvSWnuXA2Cb67La4ETO/msYAU2vrHU6i/UuO3eM74ZzJiHkXkz3UGLPzit4GMeeOGcY/aF0q\nGRwTXcTzQZkUfpUylv3dcEu2UO20whlX+b9+o9mLGjXeFbh9Bp76fx+9ZXcFlfGuoLIg1q5d3JEZ\nIfirlazH7d96BQjil699LSiIUcIXRXFsYjbiFL8cVnyyi/IKpfOuMGL6AvYsV7O7wrD9B0+roo4Z\nb2oXAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\sum_{n=1}^{10} \\frac{1}{n^{2}}$$"
      ],
      "text/plain": [
       "  10    \n",
       " ____   \n",
       " ╲      \n",
       "  ╲   1 \n",
       "   ╲  ──\n",
       "   ╱   2\n",
       "  ╱   n \n",
       " ╱      \n",
       " ‾‾‾‾   \n",
       "n = 1   "
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Sum(1/n**2, (n, 1, 10))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJ8AAAAPBAMAAAAIUwCQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nt0ciqgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$1.54976773116654$$"
      ],
      "text/plain": [
       "1.54976773116654"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Sum(1/n**2, (n,1, 10)).evalf()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJ8AAAAPBAMAAAAIUwCQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$1.64493406684823$$"
      ],
      "text/plain": [
       "1.64493406684823"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Sum(1/n**2, (n, 1, oo)).evalf()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Products work much the same way:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAC8AAAA9BAMAAADPFy0PAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\prod_{n=1}^{10} n$$"
      ],
      "text/plain": [
       "  10   \n",
       "┬───┬  \n",
       "│   │ n\n",
       "│   │  \n",
       "n = 1  "
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Product(n, (n, 1, 10)) # 10!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Limits"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Limits can be evaluated using the `limit` function. For example, "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAPBAMAAAArJJMAAAAAHlBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAACGjDitAAAACXRSTlMAVO8Qq5l2zWYZcMvdAAAACXBIWXMAAA7EAAAOxAGV\nKw4bAAAAHUlEQVQIHWNgAANGZQYGk5DJQDYbqQSr03QPsBkAJYgIYEZbtZEAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$1$$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit(sin(x)/x, x, 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can use 'limit' to check the result of derivation using the `diff` function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKAAAAAUBAMAAAD4uit9AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ndPqDmrj6L+AfOVyzbfsv4YgAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\sin{\\left (x y \\right )} + \\cos{\\left (y z \\right )}$$"
      ],
      "text/plain": [
       "sin(x⋅y) + cos(y⋅z)"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFMAAAAUBAMAAAADwRznAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$y \\cos{\\left (x y \\right )}$$"
      ],
      "text/plain": [
       "y⋅cos(x⋅y)"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(f, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\displaystyle \\frac{\\mathrm{d}f(x,y)}{\\mathrm{d}x} = \\frac{f(x+h,y)-f(x,y)}{h}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "h = Symbol(\"h\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFMAAAAUBAMAAAADwRznAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$y \\cos{\\left (x y \\right )}$$"
      ],
      "text/plain": [
       "y⋅cos(x⋅y)"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit((f.subs(x, x+h) - f)/h, h, 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "OK!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can change the direction from which we approach the limiting point using the `dir` keywork argument:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAALBAMAAABv+6sJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKvvZom7mXYyzVQi\n3UQ6SGZXAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAaklEQVQIHWNgYBBgAAIQwaj82YGBIayogYGB\nbQLHLwapDQxTGRg8GRj2J6xkYGA5wACUYP0LJBgcQEyGfBDRAGYm/wNqd2BwZGDgiDE+wMBxgIGd\ngSGcYb4dgytQolxtAwNjvXEAUDncNgBJUBUwaYAbUgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\infty$$"
      ],
      "text/plain": [
       "∞"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit(1/x, x, 0, dir=\"+\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACMAAAALBAMAAAAHCCkxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMqvvZom7mXZU\nIkRJD0iWAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAfklEQVQIHWNggAMBEAtMQIUYw74VMDB0Lt0A\nV8LA6cD9iUHoAIMHQqiEgeH8BBUGBvYLDELGIKDCANTA8RmkC6gdCkC8+SAChCEAxJr2j4GBsQAm\nwlDIwMDdm3aBgfsCXIiLgaGLwT+PoQIuwsC4KvIAA+P6tAaEENThAgwMAMSLGqu/gFQwAAAAAElF\nTkSuQmCC\n",
      "text/latex": [
       "$$-\\infty$$"
      ],
      "text/plain": [
       "-∞"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit(1/x, x, 0, dir=\"-\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Series"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Series expansion is also one of the most useful features of a CAS. In SymPy we can perform a series expansion of an expression using the `series` function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "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": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "series(exp(x), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By default it expands the expression around $x=0$, but we can expand around any value of $x$ by explicitly include a value in the function call:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAvwAAAAoBAMAAABwRjOsAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$e + e \\left(x - 1\\right) + \\frac{e}{2} \\left(x - 1\\right)^{2} + \\frac{e}{6} \\left(x - 1\\right)^{3} + \\frac{e}{24} \\left(x - 1\\right)^{4} + \\frac{e}{120} \\left(x - 1\\right)^{5} + \\mathcal{O}\\left(\\left(x - 1\\right)^{6}; x\\rightarrow1\\right)$$"
      ],
      "text/plain": [
       "                         2            3            4            5             \n",
       "                ℯ⋅(x - 1)    ℯ⋅(x - 1)    ℯ⋅(x - 1)    ℯ⋅(x - 1)     ⎛       6\n",
       "ℯ + ℯ⋅(x - 1) + ────────── + ────────── + ────────── + ────────── + O⎝(x - 1) \n",
       "                    2            6            24          120                 \n",
       "\n",
       "        \n",
       "       ⎞\n",
       "; x → 1⎠\n",
       "        "
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "series(exp(x), x, 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And we can explicitly define to which order the series expansion should be carried out:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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hb1TtaJxuFT5vf/bFP/gFvDPsY44WFvBn7+nm6DiHuB02N1LNUUT/TfimANKqqNEclQJO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\nAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$e + e \\left(x - 1\\right) + \\frac{e}{2} \\left(x - 1\\right)^{2} + \\frac{e}{6} \\left(x - 1\\right)^{3} + \\frac{e}{24} \\left(x - 1\\right)^{4} + \\frac{e}{120} \\left(x - 1\\right)^{5} + \\frac{e}{720} \\left(x - 1\\right)^{6} + \\frac{e}{5040} \\left(x - 1\\right)^{7} + \\frac{e}{40320} \\left(x - 1\\right)^{8} + \\frac{e}{362880} \\left(x - 1\\right)^{9} + \\mathcal{O}\\left(\\left(x - 1\\right)^{10}; x\\rightarrow1\\right)$$"
      ],
      "text/plain": [
       "                         2            3            4            5            6\n",
       "                ℯ⋅(x - 1)    ℯ⋅(x - 1)    ℯ⋅(x - 1)    ℯ⋅(x - 1)    ℯ⋅(x - 1) \n",
       "ℯ + ℯ⋅(x - 1) + ────────── + ────────── + ────────── + ────────── + ──────────\n",
       "                    2            6            24          120          720    \n",
       "\n",
       "            7            8            9                      \n",
       "   ℯ⋅(x - 1)    ℯ⋅(x - 1)    ℯ⋅(x - 1)     ⎛       10       ⎞\n",
       " + ────────── + ────────── + ────────── + O⎝(x - 1)  ; x → 1⎠\n",
       "      5040        40320        362880                        "
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "series(exp(x), x, 1, 10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The series expansion includes the order of the approximation, which is very useful for keeping track of the order of validity when we do calculations with series expansions of different order:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMQAAAAwBAMAAAC8i8hXAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRK5CYII=\n",
      "text/latex": [
       "$$1 - \\frac{x^{2}}{2} + \\frac{x^{4}}{24} + \\mathcal{O}\\left(x^{5}\\right)$$"
      ],
      "text/plain": [
       "     2    4        \n",
       "    x    x     ⎛ 5⎞\n",
       "1 - ── + ── + O⎝x ⎠\n",
       "    2    24        "
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s1 = cos(x).series(x, 0, 5)\n",
    "s1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAF8AAAAcBAMAAAD1rn4EAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYII=\n",
      "text/latex": [
       "$$x + \\mathcal{O}\\left(x^{2}\\right)$$"
      ],
      "text/plain": [
       "     ⎛ 2⎞\n",
       "x + O⎝x ⎠"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s2 = sin(x).series(x, 0, 2)\n",
    "s2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAF8AAAAcBAMAAAD1rn4EAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYII=\n",
      "text/latex": [
       "$$x + \\mathcal{O}\\left(x^{2}\\right)$$"
      ],
      "text/plain": [
       "     ⎛ 2⎞\n",
       "x + O⎝x ⎠"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand(s1 * s2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we want to get rid of the order information we can use the `removeO` method:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAG8AAAAwBAMAAADtMzlxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRK5CYII=\n",
      "text/latex": [
       "$$\\frac{x^{5}}{24} - \\frac{x^{3}}{2} + x$$"
      ],
      "text/plain": [
       " 5    3    \n",
       "x    x     \n",
       "── - ── + x\n",
       "24   2     "
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand(s1.removeO() * s2.removeO())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But note that this is not the correct expansion of $\\cos(x)\\sin(x)$ to $5$th order:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAANsAAAAwBAMAAABqLhIyAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$x - \\frac{2 x^{3}}{3} + \\frac{2 x^{5}}{15} + \\mathcal{O}\\left(x^{6}\\right)$$"
      ],
      "text/plain": [
       "       3      5        \n",
       "    2⋅x    2⋅x     ⎛ 6⎞\n",
       "x - ──── + ──── + O⎝x ⎠\n",
       "     3      15         "
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(cos(x)*sin(x)).series(x, 0, 6)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Linear algebra"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Matrices"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Matrices are defined using the `Matrix` class:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "m11, m12, m21, m22 = symbols(\"m11, m12, m21, m22\")\n",
    "b1, b2 = symbols(\"b1, b2\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGgAAAAyBAMAAABCJ4MDAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nt/sAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}m_{11} & m_{12}\\\\m_{21} & m_{22}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡m₁₁  m₁₂⎤\n",
       "⎢        ⎥\n",
       "⎣m₂₁  m₂₂⎦"
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = Matrix([[m11, m12],[m21, m22]])\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "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": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = Matrix([[b1], [b2]])\n",
    "b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With `Matrix` class instances we can do the usual matrix algebra operations:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXIAAAAzBAMAAAB4eZ5HAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nN3P+AQTjoqnEhDFNAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}m_{11}^{2} + m_{12} m_{21} & m_{11} m_{12} + m_{12} m_{22}\\\\m_{11} m_{21} + m_{21} m_{22} & m_{12} m_{21} + m_{22}^{2}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡    2                               ⎤\n",
       "⎢ m₁₁  + m₁₂⋅m₂₁    m₁₁⋅m₁₂ + m₁₂⋅m₂₂⎥\n",
       "⎢                                    ⎥\n",
       "⎢                                 2  ⎥\n",
       "⎣m₁₁⋅m₂₁ + m₂₁⋅m₂₂   m₁₂⋅m₂₁ + m₂₂   ⎦"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJEAAAAyBAMAAACufiRQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nDbOYk3PHxJxH96vsyH4p/gHeHTH4ApAfnQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}b_{1} m_{11} + b_{2} m_{12}\\\\b_{1} m_{21} + b_{2} m_{22}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡b₁⋅m₁₁ + b₂⋅m₁₂⎤\n",
       "⎢               ⎥\n",
       "⎣b₁⋅m₂₁ + b₂⋅m₂₂⎦"
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A * b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And calculate determinants and inverses, and the like:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKYAAAAMBAMAAAAaIdvMAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n9AyNtKqM6qgiUQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$m_{11} m_{22} - m_{12} m_{21}$$"
      ],
      "text/plain": [
       "m₁₁⋅m₂₂ - m₁₂⋅m₂₁"
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.det()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAa4AAABSCAMAAAA7DMB+AAAAPFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAo1xBWAAAAE3RSTlMA\nMquZdlQQQOkwRM3d7yJmu4lsc/NKdgAAAAlwSFlzAAAOxAAADsQBlSsOGwAACQdJREFUeAHtXemi\nsygMpS6db7rZTt//XQeQcE3CIgho+11+3LYhQA5HMYblCuFJp6sn41d8vB449d3teFb9WuTtgfGX\nLm/fHDDjl64DkiLE6a1TR42rS9fUT6dh6MUwPE+0ZSHCuVz/EJKqRt9mlhRd/SDTSCFXpuvynoS4\nd2cxzBeK/CZteb2eD/k54VwQC61EDT3Kb2K0mK21tm8zc1IcdW9Fl+PylnVXpetxHtSj8X0Ron/J\nL5fXXdH0lN+knOYasVGSekdM1GgMqYjFUzZd47DRgE5WcLrLSm7zfX1TdMl3h/Nb3WY414q10saG\n6xXHRktkCFKBdnek6y7v6UHeNo+30Hc3MDHf1DTX3OugVAB6hSqo0WBtsXEqSJe69L1p690lWZrv\noekp1Gg4X4ry5rqqS5LlzmJQUvrHS8xoQ5exvYDBfroe/e2tbm5f2krXRTkYippH91Kjn2HiPDsT\nNNeIj00XNRpD8nVkitxPV6yWrXSx+vWleO4fZ32v4WwrhuEFZx/1VwhSls2Hoes0vHvpcqi3C0cw\nxYiNUhbS9oXCkLLsyaHrdVXpdtcf7OU6y4xdCp2vN5uu7neZXewKNIrpml+cfX9xNXQw9JXaX47t\nDv3a31afBcZqTFcICs2jdNF89+/Zq3DnFZb2heuz1bXDwCA0pqtvOOac5DtdjdQQA4PQlq5eRgnb\npZFdnCXaboqBQmhKF7tYSnRfoI4ni1oHlFdmNcZAINSjyxGIznG/pn6cpv4yvuZhdBm2R1m2uUXM\nvsbqhQwMyE4Spkd5DgwEAqErYcbmrGJFgWSC6AuNMedhMg4yvHsfxcjD9ijLNIdj9lcZ7y+bcjAg\nO0mYHuW5MGAIlC48zWQuBfuRAt0G0W2hLuvJ1Z3EWb4493PP6zgBVL3MAhkKU41SWjZlYVjaScP0\nyzwXBgwB0+WZscGX7Hr8JBCtJ0bWlzaakir1wL0J/SCCIJSuGmc5YvbzXExyk4ECWRiwnQiCxLWE\nxzFgCJiuOUTOJqEg/BqA4coigehLMMDvqkDJzjJsMsg7qxv04Guw6qpJlitmfys8GmZhIHYuIVB4\nDgwIAqXLM2MDV4SvU11yCKJD3ivn0QWF4XM2hFatckGGTO0c4WKoKuezBAY/BCcGBIHQ5ZuxQX3g\nwUncFBtEB3U9xy9/ED3IXvepDWFVy7JWhkxN7d6YbSUweCG4MSAIhC7PjA16fvv6lSwsYbH1p5k8\nI3rGe7VOrK96JQ/EuN0xe+VWpiRim/Wx4O1gO4YABJiOwPMOCAKhi0Ez1yq6ZJmSFlA3hWldZ8eQ\n6hnv1XywUtsEk3ySJyRqm/GxflytHTAgCGG6DM+Ybi98urCEKN7Aj8d6xnsFJ3Z8QtJTzL4IdUxu\n2h5T3Rtsm/Wx4HLNwRAz1ZfvghCmi3R45Cd1U8wIAgPJFVYSUD1YIUf8/khrq7KnxMFQUNvI6LID\nBgShIF3ETWEDydN4aUQPFtOYNTWrWFitpFZapSRmG6FrBwwIQkG6Ym5KbwLkVM944OCIp/RuVBfa\njCoaBWob+FgwGEJ9VK8iBmhSW1iQLtYl5Mr0jHXGA7eOOKtmIUABUea2LRThqx28QJD6uT8GBKEh\nXWe1sJAn44Ezv59rynCNI97747Y5SrwjcWhHkaWIuVo7YEAQGtIl7tsjQsuAKHPblh09fz+l+fG8\nAiZpjgFDaEmX2WnCuiBBgAOiZKTi9UAQgufkSppjwBDq0cUGEjltpV+lcntKlnMHS8EP4BVvbrBC\nlYkYMIR6dHGkYigccI3dXQO86TlsyRU1xkAgNKVLL4nP7SdHuQhdalazfNI7LspVG8ZAIbSl65Ea\nZAh1i2O0xerdNrcQV2Z/NcVAIbSlS5wartGutR4wjIGshbEsZ31hEBRd//z5N6uynEJbnY2ENqs1\nFai48GEkrKX//vj3Jid0za8q9IAndgPZGz8bD4Ybrf2A4l9HV2yC/QM4CZj4fXThtYx21h/mxQJ9\n8QFZ30YXnWA3s/7BSO0H0AQmfhtddC0jzPrD1BLg/tDP76OLTrDzpasfSpUy++voYhPsjqWrn8tX\n8mKeJKg7OPKeifOvGAxjh5EkceNQ3oEuYoWd9ffPg5ASf/HP/elyL7/9iykJQV9BFzt+wreOcX95\nCGlWXitIa41bQdfaqn716vdAPbranT4hquzwz+h717o6CNqgPBAmH35ajS42VZMBf22RxrvxvWa5\n1tXBVg2Ulx3JqUVX09Mn9HZRbyc2zHCsq7NBm2WeFaZ6w5Xoan29k+MnGjKEmnKuq4OgTWQTMqrI\n96MSXY1PnxBFp9x9fRWVu9ekzUEbkpcZyalDV/PTJwQ+fiLasasVtkzOzSOdc6sGCI8xGDY/fULg\n4ydWsxFVZLtfoyV+FDQXNmjzI/dtpF5qeL7Xubuanz5hTjP3gMwX08k544FbRzxQs1lX59yqkR3J\nqULXDqdPCHT8RKAXU7Pw7lfjgVvvHDbmPtWapdwQSIJJVehCZwskGLNUTR340fETy4o2fseTc8YD\nt474xsqTi2fRFXv+wq6JmF7I2tSBv8Ql4rCHTc7B/GPdaUiHJVqURxdeHMO2Me5w+oRAx0/40KbL\n6eQc7KA2jnh6hdtK5NBFn79s0/gOp08IdPzEtj4JlU7YhIyihKE6E/Jy6KKLY9g2xh1OnxB1J92h\nR1M2UqMoIRuBoMa0zzy68POX0WV3P1O9igM/On4irQ8StI0H7vTOWTXLKCEbgZj2KkEWXez5a17O\nzYfY4fQJ/b+KViFup+SMIEIf5ZmRRRd7/hK64CgIqpcw8CejgTaTC1YrQKKEpI/yms2iizVFTPE4\nuSkDP3lOk/OKmAHyfxVV2NjqaCZfRPoor6ISdLFtjDucPiHQ8RN5fVG31GHo4jCbnz4h8PET3KL9\nJQemq/npE/O/G92fE68FbATyagYzSgyGjgbwaRAOhZgo9Tm9ucGYQcfIn+nSseSSu7wbnz4h/1f2\nMbqzohXqXU8mOVmm/oHyoE7/Lpeanj6h/y9AOduPWZP+N9e1Lsump08IevzEMTv8yFaFT58oa3nL\nNY1lLT9ObexMiHqmNWyqHoi1Nf8PHzi4lSipunoAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{1}{m_{11}} + \\frac{m_{12} m_{21}}{m_{11}^{2} \\left(m_{22} - \\frac{m_{12} m_{21}}{m_{11}}\\right)} & - \\frac{m_{12}}{m_{11} \\left(m_{22} - \\frac{m_{12} m_{21}}{m_{11}}\\right)}\\\\- \\frac{m_{21}}{m_{11} \\left(m_{22} - \\frac{m_{12} m_{21}}{m_{11}}\\right)} & \\frac{1}{m_{22} - \\frac{m_{12} m_{21}}{m_{11}}}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡ 1          m₁₂⋅m₂₁                -m₁₂        ⎤\n",
       "⎢─── + ────────────────────  ───────────────────⎥\n",
       "⎢m₁₁      2 ⎛      m₁₂⋅m₂₁⎞      ⎛      m₁₂⋅m₂₁⎞⎥\n",
       "⎢      m₁₁ ⋅⎜m₂₂ - ───────⎟  m₁₁⋅⎜m₂₂ - ───────⎟⎥\n",
       "⎢           ⎝        m₁₁  ⎠      ⎝        m₁₁  ⎠⎥\n",
       "⎢                                               ⎥\n",
       "⎢          -m₂₁                       1         ⎥\n",
       "⎢   ───────────────────         ─────────────   ⎥\n",
       "⎢       ⎛      m₁₂⋅m₂₁⎞               m₁₂⋅m₂₁   ⎥\n",
       "⎢   m₁₁⋅⎜m₂₂ - ───────⎟         m₂₂ - ───────   ⎥\n",
       "⎣       ⎝        m₁₁  ⎠                 m₁₁     ⎦"
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.inv()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solving equations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For solving equations and systems of equations we can use the `solve` function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEoAAAAUBAMAAADYerbFAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAdt3NMolEEFTvq5lmIsfa\npuIAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAB3SURBVCgVYxAyYSAEDqsxhBFSA5QPQ1bFMR2Ljs4C\nVFVVu55jqGJfvQ5NFQMjpioGBrlBrqpYCQSAgTuYXV9ngBb+6KHKvfLxGgYmdVRVu+ZZHUCNbbB8\nL6oqMA8lTYBFAohRxS5AjCoeLIqA7hJSwSaOIiakBgD0ZimSClhTrgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left [ -1, \\quad 1\\right ]$$"
      ],
      "text/plain": [
       "[-1, 1]"
      ]
     },
     "execution_count": 86,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve(x**2 - 1, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAjEAAABLCAMAAAC2lyZIAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nUA9GUf8DYe6QJALMc1YAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left [ - i \\sqrt{- \\frac{1}{2} + \\frac{\\sqrt{5}}{2}}, \\quad i \\sqrt{- \\frac{1}{2} + \\frac{\\sqrt{5}}{2}}, \\quad - \\sqrt{\\frac{1}{2} + \\frac{\\sqrt{5}}{2}}, \\quad \\sqrt{\\frac{1}{2} + \\frac{\\sqrt{5}}{2}}\\right ]$$"
      ],
      "text/plain": [
       "⎡       __________        __________       ________      ________⎤\n",
       "⎢      ╱   1   √5        ╱   1   √5       ╱ 1   √5      ╱ 1   √5 ⎥\n",
       "⎢-ⅈ⋅  ╱  - ─ + ── , ⅈ⋅  ╱  - ─ + ── , -  ╱  ─ + ── ,   ╱  ─ + ── ⎥\n",
       "⎣   ╲╱     2   2      ╲╱     2   2     ╲╱   2   2    ╲╱   2   2  ⎦"
      ]
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve(x**4 - x**2 - 1, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "System of equations:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAH4AAAAVBAMAAAByPkciAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n/1sDfgDNr2XH9LJoSQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left \\{ x : 1, \\quad y : 0\\right \\}$$"
      ],
      "text/plain": [
       "{x: 1, y: 0}"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve([x + y - 1, x - y - 1], [x,y])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In terms of other symbolic expressions:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAOAAAAAmBAMAAAAvsop7AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nfmHZAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left \\{ x : \\frac{a}{2} + \\frac{c}{2}, \\quad y : \\frac{a}{2} - \\frac{c}{2}\\right \\}$$"
      ],
      "text/plain": [
       "⎧   a   c     a   c⎫\n",
       "⎨x: ─ + ─, y: ─ - ─⎬\n",
       "⎩   2   2     2   2⎭"
      ]
     },
     "execution_count": 89,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve([x + y - a, x - y - c], [x,y])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Further reading"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Sympy\n",
    "* http://sympy.org/en/index.html - The SymPy projects web page.\n",
    "* https://github.com/sympy/sympy - The source code of SymPy.\n",
    "* http://live.sympy.org - Online version of SymPy for testing and demonstrations."
   ]
  }
 ],
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