Introduccion-a-Sympy.ipynb

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   "cells": [
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     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "# SymPy - C\u00e1lculo Simb\u00f3lico en Python\n",
      "\n",
      "Base del texto original por  J.R. Johansson (robert@riken.jp) http://dml.riken.jp/~rob/\n",
      "\n",
      "Traducido y modificado por F.J. Navarro para [CAChemE.org](http://cacheme.org) - ([@cachemeorg](https://twitter.com/CAChemEorg)).\n",
      "\n",
      "Otros Notebooks de aprendizaje en Python se pueden encontrar en la web de su autor:\n",
      "[http://jrjohansson.github.com](http://jrjohansson.github.com)."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# En Python los comentarios (l\u00edneas de c\u00f3digo ignoradas) van con almohadillas\n",
      "# Las siguiente l\u00ednea har\u00e1 que las respuestas del c\u00f3digo se muestren en este mismo notebook\n",
      "\n",
      "%pylab inline"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Populating the interactive namespace from numpy and matplotlib\n"
       ]
      }
     ],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Introducci\u00f3n\n",
      "\n",
      "Con este documento vas a aprender a calcular derivadas, integrales, l\u00edmites... y resolver ecuaciones con Python de forma libre (\u00a1y gratuita!) tal y como trabajan Mathematica, Maple o Mupad.\n",
      "\n",
      "Hay dos sistemas de \u00e1lgebra computacional (CAS, computer algebraic system) a destacar para Python:\n",
      "\n",
      "* [SymPy](http://sympy.org/en/index.html) - Biblioteca o librer\u00eda de Python que se puede utilizar en cualquier programa de Python, o en una sesi\u00f3n de IPython y que proporciona poderosas funciones de CAS.\n",
      "* [Sage](http://www.sagemath.org/) - Sage es un entorno CAS completo y muy potente que tiene como objetivo proporcionar un sistema de c\u00f3digo abierto compitiendo con Mathematica y Maple. Sage no es una librer\u00eda de Python regular, sino m\u00e1s bien un entorno de CAS que utiliza Python como lenguaje de programaci\u00f3n.\n",
      "\n",
      "Sage es, en algunos aspectos, m\u00e1s poderoso que SymPy, pero ambos ofrecen una funcionalidad CAS muy completa. La ventaja de SymPy es que es una librer\u00eda m\u00e1s de Python y se puede usar directamente con c\u00f3digo o integrar en con el bloc de notas IPython.\n",
      "\n",
      "En esta lecci\u00f3n, por lo tanto, vamos a ver c\u00f3mo utilizar SymPy con Notebooks de IPython. Si est\u00e1s interesado en un entorno de c\u00f3digo abierto CAS te recomendamos leer m\u00e1s acerca de Sage.\n",
      "\n",
      "SymPy se puede usar directamente online sin registro alguno en [SymPy Live](http://live.sympy.org/) o mediante Notebooks con registro gratuito en [Cloud Sage](https://cloud.sagemath.com/). Existen otras alternativas gratutias online como [Wakari](https://www.wakari.io/) pero esta son las no comerciales. Si lo que quieres instal\u00e1rtelo en tu ordenador: [Python para ingenieros](http://www.cacheme.org/python-para-ingenieros/)\n",
      "\n",
      "Para empezar a utilizar SymPy en un programa de Python o Notebook de IPython, importamos `sympy` directamente:\n"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Importamos todo SymPy y accedemos a sus funciones de forma directa (de ah\u00ed el asterisco).\n",
      "\n",
      "from sympy import *"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Para obtener las respuestas con estilo de ecuaciones a las que estamos acostumbrados ($\\LaTeX$):"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# El siguiente comando es opcional pero har\u00e1 que la scoluci\u00f3n sea vea con Latex\n",
      "init_printing()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Variables simb\u00f3licas\n",
      "\n",
      "En SymPy necesitamos crear s\u00edmbolos para las variables que queremos trabajar. Podemos crear un nuevo s\u00edmbolo con la instancia de clase `Symbol`:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = Symbol('x')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$x$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAAJBAMAAAAWSsseAAAAKlBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADmU0mKAAAADXRSTlMAEHarIkSJZt3NVLsyNZ302gAA\nAEJJREFUCB1jYBBSdlVTYGBMYG/imMDAJsB2kcuBgZGBcwMDCPAqgKmzBxgEGLgEdBmYBBhiF+xl\ncGdgEFYSSi9gAAAZ4Ql2LUcmxgAAAABJRU5ErkJggg==\n",
       "prompt_number": 5,
       "text": [
        "x"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "\u00bfQu\u00e9 ha pasado aqu\u00ed arriba? En primer lugar creamos el s\u00edmbolo con la variable 'x' y despu\u00e9s vimos su valor. Si te has fijado, a diferencia de c\u00f3digo normal en Python (o MATLAB), $x$ no devuelve ning\u00fan valor (n\u00famero, texto, verdadero o falso) simplemente devuelve esa inc\u00f3ginta en el que a priori no sabemos que contiene. "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "As\u00ed, escribiendo una expresi\u00f3n con esta variable no habr\u00e1 evaluaci\u00f3n num\u00e9rica:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(pi + x)**2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left(x + \\pi\\right)^{2}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEgAAAAbBAMAAAAt2dQtAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMmYiu80QdonvRN2Z\nVKvu110NAAABZElEQVQoFY2Sv0vDQBTHv0na/CBE+w+oWcQ1qJOTKDgJdomDWAiIdBPFQSctguBQ\npKM4qZPgYDdxEIqzQzcHcXARR0HI4BLf5d21HJamb7j3vt/78O5dckBRNFeTIgRuNFYthPzI+CmE\nvCvrtxACgnQEyN8dAWoSY4S0DA6zQr7ZoWV+MJC7H7Te4hpYHwKV6aDa+wqchgZtaYo2vSz7hhdp\n9qum7BbLO82FDqHGu+KC9aOFQ/kxJORkFB1A7FIsAXbDfPCrLCV0NrE/M0fONLtPgJu4aSBHY8jo\nutGe2H9j6JI6obzIAr2ZSvgU1hT7BAHjPNFpHD/G8aZwJtEWSUJ0HNUVJCL3Oz0j//fy7jR4kBzD\nSXJGHWenyF+RHPwcuO+uYZYZBTltXAjnhW1quHNQPwlZKagU4ks4G2z78u46JJXV4iJ/LtKjdNMv\nqXL51sCyZutiW8lhj67XwAoV/i+bct4/Xs5GamR386YAAAAASUVORK5CYII=\n",
       "prompt_number": 6,
       "text": [
        "       2\n",
        "(x + \u03c0) "
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Se pueden definir simbolos de forma alternativa y m\u00e1s compacta as\u00ed\n",
      "a, b, c = symbols(\"a, b, c\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 7
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Para ver que tipo de variable es, usamos la funci\u00f3n type()\n",
      "type(a)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 8,
       "text": [
        "sympy.core.symbol.Symbol"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Podemos a\u00f1adir suposiciones a los s\u00edmbolos cuando los creamos:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Si quiero definir x como una variable real\n",
      "\n",
      "x = Symbol('x', real=True)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Para comprobar si x es imaginaria\n",
      "\n",
      "x.is_imaginary"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 10,
       "text": [
        "False"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = Symbol('x', positive=True)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 11
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x > 0"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 12,
       "text": [
        "True"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### N\u00fameros complejos\n",
      "\n",
      "La unidad imaginaria en SymPy se denota con `I` (`i` may\u00fascula). "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "1+1*I"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$1 + i$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACoAAAAQBAMAAACSDPCjAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAzRAiu5mrdu/dZlSJRDIL\nYZgAAAB2SURBVBgZY2BgEGJAA2wODAwmj9AEGdgWMKiEwUSZDJCk2WGi3AWki+qEA/VgmHDgBhZR\n5g2BWESZGJ7DRXkEBSUbBQUDgALcIEdhmMuYoIBFVG/DASyi95gXMLBmvMwCmgMEML+xlEP4EBIm\niizGwMC7AM4HAA07Hr1Hv8R3AAAAAElFTkSuQmCC\n",
       "prompt_number": 13,
       "text": [
        "1 + \u2148"
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# En Python la exponencial se indica con ** (en MATLAB con ^)\n",
      "\n",
      "I**2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$-1$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAABgAAAAPBAMAAAAMihLoAAAAJ1BMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAilU6eAAAADHRSTlMAEM3dMiK7mat272a9noTYAAAALklE\nQVQIHWNggANFOIuBIewggiNWgcRh4CSTI2QMAioMZBsAcxB71+FuGBtMAwDSnw2AS5zPhQAAAABJ\nRU5ErkJggg==\n",
       "prompt_number": 14,
       "text": [
        "-1"
       ]
      }
     ],
     "prompt_number": 14
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(x * I + 1)**2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left(i x + 1\\right)^{2}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEwAAAAbBAMAAAAkMnRXAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 15,
       "text": [
        "         2\n",
        "(\u2148\u22c5x + 1) "
       ]
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### N\u00fameros racionales\n",
      "\n",
      "Hay tres tipos num\u00e9ricos diferentes en SymPy: `Real` (real), `Rational` (racional), `Integer` (entero): "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "r1 = Rational(4,5)\n",
      "r2 = Rational(5,4)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 16
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(r1, r2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{pmatrix}\\frac{4}{5}, & \\frac{5}{4}\\end{pmatrix}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEIAAAAcBAMAAAAn/nRcAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n/olfG4RccEcInd0AAAAASUVORK5CYII=\n",
       "prompt_number": 17,
       "text": [
        "(4/5, 5/4)"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "r1+r2 "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{41}{20}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAABUAAAAqBAMAAACuFQ3dAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMpndu3bvImbNiRBU\nq0Qb3U6NAAAA6klEQVQYGWNgYGAQAmLODUACCEwYGNj9IWzWFAZup/sQdsUUoFw8hL0AweYoQLC3\nMiDYCxBsHgEEexcDgn327t1vV+FmMqQgzGf4gmAv+X+DQTfftgEoQmPwHw4+UGZTkYojA8OmpYpA\nU7gvMKx/wDCHYXMBAwPvBga2BI4EBq4JDAxsBgy8H1kuMPD8Bgb8byCb/wID3x+wtcy/9ysw8H0F\ns98vqF/AwPgLzDZhgLOZHBj2L4CqiWVg4Fdg4AHp5VFgKGd2YOAAmgmK0XDWBAYmoF3sVqt0DRhs\nGLYJAO0C+tGAYVfoRQYGAGg1Ulux6DimAAAAAElFTkSuQmCC\n",
       "prompt_number": 18,
       "text": [
        "41\n",
        "\u2500\u2500\n",
        "20"
       ]
      }
     ],
     "prompt_number": 18
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "r1/r2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{16}{25}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAABUAAAAqBAMAAACuFQ3dAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAzRAiu5mrdu/dZoky\nRFRd1xrCAAABAklEQVQYGWNgYGBUYGBg2F19AEgy+wPZihdYBRgYmFzzgWxpBhYHoDhDvQID12cQ\nAwiAbJYGIA0CQDa/zPZYGFtfioHjAVRc/xMDaxOUzT+BgesvlM0hwMD1C8pmaUCI8wHVN0DFGbwZ\nNB4A2WH90xYwcIfmgIRpDP7DwQfKbDIN8WFgEFc4egEYhgkM+y8w9P//DjSRTYGBo4FBOt0AyOaY\nwMD2hyEBbA/jJwQbKMD5iSFreQxYhuH+BgYnBnugOUAwBUSwTACRrA5gEhwdZUCDBRi4QWHLF8Bg\nDAxb1k9A9lEGhmJWoGECwCidvTtsAnMAwwqgxYxAP05gMN0WxMAAAJ3jRrgsItfHAAAAAElFTkSu\nQmCC\n",
       "prompt_number": 19,
       "text": [
        "16\n",
        "\u2500\u2500\n",
        "25"
       ]
      }
     ],
     "prompt_number": 19
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Evaluaci\u00f3n num\u00e9rica\n",
      "\n",
      "SymPy utiliza una biblioteca de precisi\u00f3n arbitraria como backend num\u00e9rico, y tiene predefinidas expresiones SymPy para una serie de constantes matem\u00e1ticas, tales como: `pi`, `e`, `oo` para el infinito.\n",
      "\n",
      "Para evaluar una expresi\u00f3n num\u00e9ricamente podemos utilizar la funci\u00f3n `evalf` (la funci\u00f3n `N`). Es necesario especificar un argumento `n` que especifica el n\u00famero d\u00edgitos."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Eval\u00faa num\u00e9ricamente pi hasta 50 d\u00edgitos\n",
      "pi.evalf(n=50)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$3.1415926535897932384626433832795028841971693993751$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAgsAAAAPBAMAAACYf5HCAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nH/8ADtF52HEjSsAAAAAASUVORK5CYII=\n",
       "prompt_number": 115,
       "text": [
        "3.1415926535897932384626433832795028841971693993751"
       ]
      }
     ],
     "prompt_number": 115
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Creamos una expresi\u00f3n que llamaremos y\n",
      "y = (x + pi)**2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 116
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "N(y, 5) # Equivalente a evalf()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left(x + 3.1416\\right)^{2}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAHUAAAAbBAMAAACw1N2lAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nSsPWAAAAAElFTkSuQmCC\n",
       "prompt_number": 117,
       "text": [
        "            2\n",
        "(x + 3.1416) "
       ]
      }
     ],
     "prompt_number": 117
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Cuando evaluamos num\u00e9ricamente expresiones algebraicas muchas veces queremos sustituir un s\u00edmbolo con un valor num\u00e9rico. Para ello, en SymPy hacemos uso de la funci\u00f3n `subs`:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Sustituye el valor de la variable 'x' en 'y' por un n\u00famero\n",
      "y.subs(x, 1.5)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left(1.5 + \\pi\\right)^{2}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFYAAAAbBAMAAAAUvmV2AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nXM5wR5BIiif9AI4FWSBAw5hOAAAAAElFTkSuQmCC\n",
       "prompt_number": 118,
       "text": [
        "         2\n",
        "(1.5 + \u03c0) "
       ]
      }
     ],
     "prompt_number": 118
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Sustituye con \".subs\" y eval\u00faa la expresi\u00f3n con \"N()\"\n",
      "N(y.subs(x, 1.5),10)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$21.54438236$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAGsAAAAPBAMAAAAR9CxzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 119,
       "text": [
        "21.54438236"
       ]
      }
     ],
     "prompt_number": 119
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "La funci\u00f3n `subs`  tambi\u00e9n puede utilizarse para sustituir los s\u00edmbolos y expresiones:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "y.subs(x, a+b+pi)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left(\\alpha + \\beta + 2 \\pi\\right)^{2}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAHoAAAAbBAMAAABB34YoAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 120,
       "text": [
        "             2\n",
        "(\u03b1 + \u03b2 + 2\u22c5\u03c0) "
       ]
      }
     ],
     "prompt_number": 120
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Si queremos representar estas expresiones podemos hacerlo con la funci\u00f3n `plot`. Para ello, cargamos las librer\u00edas necesarias mediante el siguiente c\u00f3digo:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sympy.plotting import plot\n",
      "\n",
      "# Representa las figuras en l\u00ednea con el documento (no en una ventana emergente)\n",
      "%matplotlib inline"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 134
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Representamos $f(x) = x^2$ $\\forall x \\in [-5,5]$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot(x**2, (x, -5, 5))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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o55dfbKcKv/R0s3d106ZmffLKlVCtmu1U4gcqzFJounaFL7+E336DevXMjRV+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\nRK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0xc8276a0>"
       ]
      },
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 135,
       "text": [
        "<sympy.plotting.plot.Plot at 0xc51aa90>"
       ]
      }
     ],
     "prompt_number": 135
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Si creamos otra variable para trabajar con ella:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "y = Symbol('y')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 138
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "podemos adem\u00e1s representar funciones en 3D de forma an\u00e1loga."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sympy.plotting import plot3d\n",
      "\n",
      "plot3d(x*y, (x, -5, 5), (y, -5, 5))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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L9mHlRumNYuCktTX6ih0hXkmECoXCkkIXh0xuhqNnP0c69SCFhTydbR5UK0+5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qb+hkGZAmowS2NQCnE47QLJjkaGVIWkSkzUuz5Hkn0ArW0/SOMYkkTQ5YwgwthFhmSQaHkHk5pc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\nPwjIOo7DfLssAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0xd071860>"
       ]
      },
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 139,
       "text": [
        "<sympy.plotting.plot.Plot at 0xd071940>"
       ]
      }
     ],
     "prompt_number": 139
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Tambi\u00e9n podemos combinar la evoluci\u00f3n num\u00e9rica de las expresiones con los arrays NumPy:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Importamos numpy\n",
      "import numpy"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 26
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Creamos un vector de \n",
      "x_vec = numpy.arange(0, 10, 0.1)\n",
      "x_vec"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 27,
       "text": [
        "array([ 0. ,  0.1,  0.2,  0.3,  0.4,  0.5,  0.6,  0.7,  0.8,  0.9,  1. ,\n",
        "        1.1,  1.2,  1.3,  1.4,  1.5,  1.6,  1.7,  1.8,  1.9,  2. ,  2.1,\n",
        "        2.2,  2.3,  2.4,  2.5,  2.6,  2.7,  2.8,  2.9,  3. ,  3.1,  3.2,\n",
        "        3.3,  3.4,  3.5,  3.6,  3.7,  3.8,  3.9,  4. ,  4.1,  4.2,  4.3,\n",
        "        4.4,  4.5,  4.6,  4.7,  4.8,  4.9,  5. ,  5.1,  5.2,  5.3,  5.4,\n",
        "        5.5,  5.6,  5.7,  5.8,  5.9,  6. ,  6.1,  6.2,  6.3,  6.4,  6.5,\n",
        "        6.6,  6.7,  6.8,  6.9,  7. ,  7.1,  7.2,  7.3,  7.4,  7.5,  7.6,\n",
        "        7.7,  7.8,  7.9,  8. ,  8.1,  8.2,  8.3,  8.4,  8.5,  8.6,  8.7,\n",
        "        8.8,  8.9,  9. ,  9.1,  9.2,  9.3,  9.4,  9.5,  9.6,  9.7,  9.8,\n",
        "        9.9])"
       ]
      }
     ],
     "prompt_number": 27
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "y_vec = numpy.array([N(((x + pi)**2).subs(x, xx)) for xx in x_vec])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 28
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "fig, ax = subplots()\n",
      "ax.plot(x_vec, y_vec);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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hKSmJ6upqvxYbiHbvhtmz4eKL4Sc/MX3155+HTz6BP/xBwS4i9vCpLVNRUUFF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\naFlJKJFyhJ8AAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0xb089128>"
       ]
      }
     ],
     "prompt_number": 29
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Sin embargo, este tipo de evoluci\u00f3n num\u00e9rica puede ser muy lenta, y hay una manera mucho m\u00e1s eficiente de hacerlo. La funci\u00f3n `lambdify` para \"compilar\" una expresi\u00f3n Sympy a una funci\u00f3n es mucho m\u00e1s eficiente para evaluar num\u00e9ricamente una expresi\u00f3n:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Creamos la funci\u00f3n y(x) con lambify\n",
      "# El primer argumento es una lista de variables \"[]\" siendo f una funci\u00f3n de, en este caso, s\u00f3lo x -> f (x)\n",
      "f = lambdify([x], (x + pi)**2, 'numpy')  #"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 30
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Ahora podemos pasar directamente 'y' a una array numpy y f(x) se eval\u00faa de manera eficiente\n",
      "y_vec = f(x_vec)  "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 31
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "El aumento de velocidad al utilizar las funciones de \"lambdify\" en lugar de la evaluaci\u00f3n num\u00e9rica directa puede ser significativa, \u00a1a menudo de varios \u00f3rdenes de magnitud!. Incluso en este sencillo ejemplo que acabamos de ver:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%timeit \n",
      "\n",
      "# 'timeit' calcula el tiempo medio ejecuci\u00f3n de un c\u00f3digo tras varias iteraciones.\n",
      "# NOTA: El c\u00f3digo escrito despu\u00e9s de 'timeit' se ejecutar\u00e1 varias veces.\n",
      "\n",
      "y_vec = numpy.array([N(((x + pi)**2).subs(x, xx)) for xx in x_vec])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "10 loops, best of 3: 37.1 ms per loop\n"
       ]
      }
     ],
     "prompt_number": 32
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%%timeit\n",
      "\n",
      "y_vec = f(x_vec)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "100000 loops, best of 3: 5.66 \u00b5s per loop\n"
       ]
      }
     ],
     "prompt_number": 33
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "*Nota*: Si es la primera vez que has visto NumPy no te preocupes si te has perdido un poco. Dale un vistazo a este [Curso intensivo de Python](http://nbviewer.ipython.org/urls/bitbucket.org/franktoffel/cfd-python-class-es/raw/master/lecciones/00%20-%20Introduccion%20rapida%20a%20Python.ipynb) y qued\u00e1te ahora con que existe la opci\u00f3n de evaluar con un vector de valores una funci\u00f3n de SymPy."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Manipulaciones algebraicas.\n",
      "\n",
      "Uno de los principales usos de un CAS es llevar a cabo manipulaciones algebraicas de las expresiones. Por ejemplo, podr\u00edamos querer expandir un producto, factorizar una expresi\u00f3n, o simplemente una trabajar con ella. Las funciones para realizar estas operaciones b\u00e1sicas en SymPy se muestran en esta secci\u00f3n."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Expand y factor (Expandir y factorizar)\n",
      "\n",
      "Los primeros pasos en la manipulaci\u00f3n algebraica"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Escribimos la expresi\u00f3n con la queremos trabajar\n",
      "\n",
      "(x+1)*(x+2)*(x+3)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left(x + 1\\right) \\left(x + 2\\right) \\left(x + 3\\right)$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAMEAAAAVBAMAAAANw5eWAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 34,
       "text": [
        "(x + 1)\u22c5(x + 2)\u22c5(x + 3)"
       ]
      }
     ],
     "prompt_number": 34
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expand((x+1)*(x+2)*(x+3))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$x^{3} + 6 x^{2} + 11 x + 6$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAKsAAAAWBAMAAABNknGBAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nBZqsk+Ep3JM5AAAAAElFTkSuQmCC\n",
       "prompt_number": 35,
       "text": [
        " 3      2           \n",
        "x  + 6\u22c5x  + 11\u22c5x + 6"
       ]
      }
     ],
     "prompt_number": 35
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "La funci\u00f3n `expand` (expandir) puede adem\u00e1s tomar una serie de argumentos (o palabras clave)  para especificar qu\u00e9 tipo de expansiones queremos que se realicen. Por ejemplo, para ampliar las expresiones trigonom\u00e9tricas, utilizaremos el `trig = True` como argumento y palabra clave:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "sin(a+b)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\sin{\\left (a + b \\right )}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFgAAAAVBAMAAAAwfTS1AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMnZUzRC73UTviSKZ\nZqszMyTAAAABn0lEQVQoFaWSzUsCURTFj+nk5CRUBC0CldpVSKuiWiSl1SJiFiEEUZsIwj6k/gF3\nUiuJqKAWEi0jXLkJwkW0CpIIglYqtBQqKawwu2/GmfcmbNXAe/fce35z530M8J/Hlv3jbXeLbth2\nOZDmkqvwAem8nts+eT3EpaBmSDuEXJcNSbEkpWpZiaLVYkaTYWuU3acFyG8UpYSWKK16jea4qZgw\nYFeUZeOwdXcF0t9oP27z+6hwxqqXJx0qiybsKPT1My8MRLEOTPiUL7LXaMhH2GAohxdDIPMe2yqS\nDF4BKmRv0ZjL4pqhHB5VQQc2COd7Lxi8BLySfUjjVsUmQzl8BYlgL6TTaobByxzeh/RCpCsYnNwL\nBjMkpyHT4XlxAXdZgGkZ1MT+LBHCO1fQTIcXxzBQFGDaoFRCY2DEApfZPmiDQyoibA21ZZwTNIaH\nXNICF7FK+Q127h5jnuqTp5of+Mjpl7LQ01mIWeB5PzXGlFYTJmdKSMxL0WpKQrSYdmfEiswaGo/L\nZygzRkz1W8z+LgB1f34Nq9NGEb8s9nKn8APn2mCn+s4IPwAAAABJRU5ErkJggg==\n",
       "prompt_number": 36,
       "text": [
        "sin(a + b)"
       ]
      }
     ],
     "prompt_number": 36
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expand(sin(a+b), trig=True)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\sin{\\left (a \\right )} \\cos{\\left (b \\right )} + \\sin{\\left (b \\right )} \\cos{\\left (a \\right )}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAP8AAAAVBAMAAAB7+SUdAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMnZUzRC73UTviSKZ\nZqszMyTAAAAEK0lEQVRIDc2WUWhbVRjH/9fcm6S5zRKnMIu4xuqDVB1hD4r60KCZqIwZQQabYwll\nCGNqg+KDDjTgg0yQVmUqdJNYB45Ox/WlUNQtTvFhCkYRBBXMCoKIZa2zo87Z+v/Ovefc03L37qE3\n93++33fu+d/vfKEB/nfjhtBRvpjsLMRXomaN01XyWxNYIyIKjZ3XY+w3In0ujlmKeOdbnCfRgVKc\nOANvfxPYkPwaM1Gixs6leKVbi7QbxywleAfnSXSwHSduY0qP00ocshRpOCpaxPdhLa9qa2XfBV/g\nZehjNjVacKHMqXmcQRRmcRKe1JnehFb2nTi7xIChW21qdF8AjLNY2GVClhAajgj7V+sAsk0j7zMq\nFoJzKkVTy8BHRZP4MtX7MnPLJhYLoeEgdm4aqsys4NojG7eUGMx1+HFqalMN+IAKe6bfgzc13MLG\nrTdzKtidu41PVZSRyACxP9ZzHvxsaI5BRXcMfR49zzs7XcbuoeNqs4j6v9/4rMI7gSaeAu4v+f+Q\npbus8STGKL/n5b+BgWBTgFGvgUyE69vATKEyQgMKp3twR3E6YPRJHtJftUKAfIWz2bIz6RxEuqQ2\nUxSHgvQCBB+qoS0G9gP/MjfDl9vVxZeUd/JyKxgpPg8MXncYuQiP1MBvjVAZoYGsYBpIT6DAV8Az\nQGoJhTZSTc6qSC1m2nAW1WaKekfR32BSE5mLt0IM7AP+ZG5fC/imhqcpB3nVO4DDri+0XxwNOBf8\nBTwaECojNADBYqCCQonRt4H+CurcdIG1uMzIYBm4pDYL6RIyTCT2jq12xMDjsYE34S3KEl7jTEot\n00Bj98PyGDGwHVlaUgaOVavPVavbCQSLgYYx4HYwzhajgay82QhXXlSbKQN9DdSLysDHyC9bBngE\nfL/UggdIr5oKfBXgFz6CmCfVzy+i7uSwAr5gywCPgEfBUsrXxlTgstpMHQGLMwKF7wbmLQNsQu8C\nK3lP2GbsAZSPsBIvtbGhG/bosjTJuiZMCbYMsAkLLZ5z2IRVlrGvg+yy2kw1Yb2MfSnVo3fVcEDq\nzz+pVK4H3Isfem3ga079o8gFA0U8lDqMdBDieTxBIlRGWAGFWX99BCf5b6Cb7rATeGG2ixPOa3Db\najOA1C3lX2BXE7/y3Y+tzau/bl49d8ffPZ5XE9g7fP1cC3iAS7H3zAn4Z0+2sh/efpxTwXu2dKkU\n5T00INg5v/LI+ZVrxt5hlAfknZmncFv88P4YDvDJ9G9qM86Fzr176pYIM2KNg1r7E1rZd40NDQ3Y\nKaIzQRSZXU/WUCTgn/WKXEkr+66xoa/a1Oh8J5JTJmQJQ5GA5aTVeNRaEUuNk2mcdyCUXiUOWSqi\nSMJ+I0rUSdY6So2TaZwb/eTQfmOglP5Bkog/DZPzuhLr1ob4StQk+10lfzKBNSKiIP4PvzMnUkUr\n/wUAAAAASUVORK5CYII=\n",
       "prompt_number": 37,
       "text": [
        "sin(a)\u22c5cos(b) + sin(b)\u22c5cos(a)"
       ]
      }
     ],
     "prompt_number": 37
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Escribe `help(expand)` para una explicaci\u00f3n detallada de los diversos tipos de expansiones que la funci\u00f3n `expand` puede realizar. "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Lo opuesto a la expansi\u00f3n del producto es, por supuesto, la factorizaci\u00f3n. Para ello en SymPy utilizaremos la funci\u00f3n `factor`:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Esta es la funci\u00f3n que queremos factorizar\n",
      "(x**3 + 6 * x**2 + 11*x + 6)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$x^{3} + 6 x^{2} + 11 x + 6$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAKsAAAAWBAMAAABNknGBAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nBZqsk+Ep3JM5AAAAAElFTkSuQmCC\n",
       "prompt_number": 38,
       "text": [
        " 3      2           \n",
        "x  + 6\u22c5x  + 11\u22c5x + 6"
       ]
      }
     ],
     "prompt_number": 38
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "factor(x**3 + 6 * x**2 + 11*x + 6)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left(x + 1\\right) \\left(x + 2\\right) \\left(x + 3\\right)$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAMEAAAAVBAMAAAANw5eWAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 39,
       "text": [
        "(x + 1)\u22c5(x + 2)\u22c5(x + 3)"
       ]
      }
     ],
     "prompt_number": 39
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Simplify (Simplificar)\n",
      "\n",
      "La funci\u00f3n `simplify` intenta simplificar una expresi\u00f3n en una expresi\u00f3n simple, utilizando diversas t\u00e9cnicas. En realidad, el termino \"simplificar\" es bastante arbitrario y subjetivo, por ello es mejor utilizar alternativas m\u00e1s espec\u00edficas para simplificar funciones como: `trigsimp`, `powsimp`, `logcombine`, etc."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# 'simplify()' reduce la expresi\u00f3n\n",
      "simplify((x+1)+(x+2)+(x+3))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$3 x + 6$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADkAAAAQBAMAAABelcpIAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIom7VJlmdt1E7xDN\nMqsI8sYEAAABAElEQVQYGWNgEFIyYcAGmNR1GRgYAxj8E7DIcp9kiGFgYP/CwN+ALLsRwuFRYDBm\nYOBcybD/ALKsO4STLwAVRDUZKmsF03GUgWF7nWRRAYQPlf1zqXQCUIBXI4CBW4GtneMAsiz3nwSG\nJWABrQtMG5h+s0Atgujl/r+BIWoDSJprNTcDjwFEJwMD1OQfDAz5ExgYBRiYvzAw8BeAZXXOnOk6\ncwboEIY1QNkEBv7fDMzfGBjuX2AAmwPXC3QzUC+XAQPHR5YN9QzsqLL+QHsZGNgSGPwL5jvYM4hB\nLYbay6zADXLz1FBLhn3l29USUGUZVOMuQEWQKaheZCEk9jQkNgBA0TjiNuqZVAAAAABJRU5ErkJg\ngg==\n",
       "prompt_number": 40,
       "text": [
        "3\u22c5x + 6"
       ]
      }
     ],
     "prompt_number": 40
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# 'simplify()' reduce expresiones a su identidad trigonom\u00e9trica\n",
      "simplify(sin(a)**2 + cos(a)**2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$1$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAPBAMAAAArJJMAAAAAJFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAADHJj5lAAAAC3RSTlMAzRAiu5mrdu/dZmiL4QAAAAAjSURBVAgd\nY2BgEGJgYDDZxMCgEgYkGNhJJVgzdmYB9TEwAACPpQrvlUCHcAAAAABJRU5ErkJggg==\n",
       "prompt_number": 41,
       "text": [
        "1"
       ]
      }
     ],
     "prompt_number": 41
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "simplify(cos(x)/sin(x))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{1}{\\tan{\\left (x \\right )}}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAD4AAAAvBAMAAABJZWRJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 42,
       "text": [
        "  1   \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "tan(x)"
       ]
      }
     ],
     "prompt_number": 42
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## apart and together (descomoner en fracciones simples y denominador com\u00fan)\n",
      "\n",
      "Para manipular expresiones simb\u00f3licas de las fracciones, se puede el usar las funciones `apart` y `together`:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f1 = 1/((a+1)*(a+2)) # escribimos la expresion y la llamamos 'f1'"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 43
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{1}{\\left(a + 1\\right) \\left(a + 2\\right)}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAIIAAAAvBAMAAADdrw/+AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAzRAiu5mrdu/dZjKJ\nRFRer8KoAAACC0lEQVRIDe2UO0wbQRCGf9/Z+G2d06BUviJCdL7CQtA5IjRpcEFog2ggUSTcxyhC\npEiQECiF6eCkVCiNpVR0boAiKVImnSVEjUWHIHFmZs+POy05S26Q4i1ud2a++XdndnVAyIjYIUBY\n2FwcUcFY2BhRAXg3VpB7GvdB2vAQ3sPyXt1Vhxl/xx0YqgOdUcdQu/wfUNq9r84vKhAKlPQCj4CJ\nhoS0gPniWaMLEKoZpWtyWhLQAlPI/PaAeHMw33CU9WSZFWps6IFV4IMHJLwclZmrqhlxVphmQw98\nBDYaCvjlpajJrxBpkVcPFGxWEOCUIPPN2086hUyFvAxg5vzE5hm9LYAjGwI8J/dPJ9qWeB+QKlIW\neRnIbmMxACB7BwhwCBg7iJUDgCjk9slLAKZdLAUAJMt0IgZ2gdg1ktUAIArptgLw0sZBABDJLpAo\no9ggIJPPP97K5yvM+hXew7gNAivkEAU6ZLGJK9jkGGiUKGT3ycVl3iHXNvzAJUwHAlCjii08zTl+\nQBRSFjkJMP4gal34gHQFcUd18gyIVFP1jMQDdxHjggjAJI5XmkJ0b3Pu++kr6iAD9GCMtW8zmxLv\nKcTWb16T9A8FYKr2dY3X/TILnc6NByQdiXif7hbKnOUpFEjxQXoj6/aWtDhnIxTA/GCOb21YYoYC\n2h+IpEZVgaGA6fo2HjA+q/W/gb9fh7sjP8fEcgAAAABJRU5ErkJggg==\n",
       "prompt_number": 44,
       "text": [
        "       1       \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "(a + 1)\u22c5(a + 2)"
       ]
      }
     ],
     "prompt_number": 44
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "apart(f1) # Desomponer en fracciones simples"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- \\frac{1}{a + 2} + \\frac{1}{a + 1}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAI4AAAAsBAMAAABBB53eAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nym7j1IoigawMAGt/qLcv8MeSAAAAAElFTkSuQmCC\n",
       "prompt_number": 45,
       "text": [
        "    1       1  \n",
        "- \u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\n",
        "  a + 2   a + 1"
       ]
      }
     ],
     "prompt_number": 45
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f2 = 1/(a+2) + 1/(a+3)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 46
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{1}{a + 3} + \\frac{1}{a + 2}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAHwAAAAsBAMAAABVvsF6AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAAAElFTkSuQmCC\n",
       "prompt_number": 47,
       "text": [
        "  1       1  \n",
        "\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\n",
        "a + 3   a + 2"
       ]
      }
     ],
     "prompt_number": 47
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "together(f2) # Denominador com\u00fan"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{2 a + 5}{\\left(a + 2\\right) \\left(a + 3\\right)}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAIIAAAAvBAMAAADdrw/+AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nTkSuQmCC\n",
       "prompt_number": 48,
       "text": [
        "    2\u22c5a + 5    \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "(a + 2)\u22c5(a + 3)"
       ]
      }
     ],
     "prompt_number": 48
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## C\u00e1lculo\n",
      "\n",
      "Adem\u00e1s de las manipulaciones algebraicas, el otro uso principal de CAS es hacer el c\u00e1lculo, como derivadas e integrales de expresiones algebraicas."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Diferenciaci\u00f3n\n",
      "\n",
      "La diferenciaci\u00f3n es generalmente simple y se utiliza la funci\u00f3n `diff`. El primer argumento es la expresi\u00f3n de la derivada  y el segundo argumento es el s\u00edmbolo para el cual derivamos:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "y"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left(x + \\pi\\right)^{2}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEgAAAAbBAMAAAAt2dQtAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMmYiu80QdonvRN2Z\nVKvu110NAAABZElEQVQoFY2Sv0vDQBTHv0na/CBE+w+oWcQ1qJOTKDgJdomDWAiIdBPFQSctguBQ\npKM4qZPgYDdxEIqzQzcHcXARR0HI4BLf5d21HJamb7j3vt/78O5dckBRNFeTIgRuNFYthPzI+CmE\nvCvrtxACgnQEyN8dAWoSY4S0DA6zQr7ZoWV+MJC7H7Te4hpYHwKV6aDa+wqchgZtaYo2vSz7hhdp\n9qum7BbLO82FDqHGu+KC9aOFQ/kxJORkFB1A7FIsAXbDfPCrLCV0NrE/M0fONLtPgJu4aSBHY8jo\nutGe2H9j6JI6obzIAr2ZSvgU1hT7BAHjPNFpHD/G8aZwJtEWSUJ0HNUVJCL3Oz0j//fy7jR4kBzD\nSXJGHWenyF+RHPwcuO+uYZYZBTltXAjnhW1quHNQPwlZKagU4ks4G2z78u46JJXV4iJ/LtKjdNMv\nqXL51sCyZutiW8lhj67XwAoV/i+bct4/Xs5GamR386YAAAAASUVORK5CYII=\n",
       "prompt_number": 49,
       "text": [
        "       2\n",
        "(x + \u03c0) "
       ]
      }
     ],
     "prompt_number": 49
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "diff(y**2, x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$4 \\left(x + \\pi\\right)^{3}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFgAAAAbBAMAAAAKd1XFAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 50,
       "text": [
        "         3\n",
        "4\u22c5(x + \u03c0) "
       ]
      }
     ],
     "prompt_number": 50
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Para derivadas de orden superior que podemos hacer:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "diff(y**2, x, x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$12 \\left(x + \\pi\\right)^{2}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAGEAAAAbBAMAAACekfw3AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nIULXrxZWopLz3R9V5WldlZ6jtHVtfQDvsmc+fdUw/AAAAABJRU5ErkJggg==\n",
       "prompt_number": 51,
       "text": [
        "          2\n",
        "12\u22c5(x + \u03c0) "
       ]
      }
     ],
     "prompt_number": 51
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "diff(y**2, x, 2) # same as above"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$12 \\left(x + \\pi\\right)^{2}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAGEAAAAbBAMAAACekfw3AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nIULXrxZWopLz3R9V5WldlZ6jtHVtfQDvsmc+fdUw/AAAAABJRU5ErkJggg==\n",
       "prompt_number": 52,
       "text": [
        "          2\n",
        "12\u22c5(x + \u03c0) "
       ]
      }
     ],
     "prompt_number": 52
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Para calcular la derivada de una expresi\u00f3n con varias variables:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x, y, z = symbols(\"x,y,z\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 53
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f = sin(x*y) + cos(y*z)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 54
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$\\frac{d^3f}{dxdy^2}$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "diff(f, x, 1, y, 2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- x \\left(x y \\cos{\\left (x y \\right )} + 2 \\sin{\\left (x y \\right )}\\right)$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAPgAAAAVBAMAAACZJT5kAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 55,
       "text": [
        "-x\u22c5(x\u22c5y\u22c5cos(x\u22c5y) + 2\u22c5sin(x\u22c5y))"
       ]
      }
     ],
     "prompt_number": 55
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Integrales\n",
      "\n",
      "La integraci\u00f3n se realiza de una manera similar:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\sin{\\left (x y \\right )} + \\cos{\\left (y z \\right )}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAKAAAAAVBAMAAAAz5vjYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nSUVORK5CYII=\n",
       "prompt_number": 56,
       "text": [
        "sin(x\u22c5y) + cos(y\u22c5z)"
       ]
      }
     ],
     "prompt_number": 56
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "integrate(f, x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$x \\cos{\\left (y z \\right )} + \\begin{cases} 0 & \\text{for}\\: y = 0 \\\\- \\frac{1}{y} \\cos{\\left (x y \\right )} & \\text{otherwise} \\end{cases}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAVAAAAA/BAMAAABEE43RAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nWL/nQWgAAAAASUVORK5CYII=\n",
       "prompt_number": 57,
       "text": [
        "             \u239b\u23a7    0       for y = 0\u239e\n",
        "             \u239c\u23aa                     \u239f\n",
        "x\u22c5cos(y\u22c5z) + \u239c\u23a8-cos(x\u22c5y)            \u239f\n",
        "             \u239c\u23aa\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500  otherwise\u239f\n",
        "             \u239d\u23a9    y                \u23a0"
       ]
      }
     ],
     "prompt_number": 57
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Al proporcionar l\u00edmites para la variable de integraci\u00f3n podemos evaluar integrales definidas:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "integrate(f, (x, -1, 1))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$2 \\cos{\\left (y z \\right )}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFAAAAAVBAMAAAAjqnRBAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAAAElFTkSuQmCC\n",
       "prompt_number": 58,
       "text": [
        "2\u22c5cos(y\u22c5z)"
       ]
      }
     ],
     "prompt_number": 58
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "y tambi\u00e9n integrales impropias"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "integrate(exp(-x**2), (x, -oo, oo))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\sqrt{\\pi}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAB0AAAAVBAMAAABI7vhRAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAInarRM2ZVBDdiWbv\nuzJCz3LGAAAAr0lEQVQYGWNggADG/2DwAcplYHaAsSC0KCqXIR2VzzEBlc9WgMrvROUyrEblg7Tr\nAx3wDSrMtIGBa12RtKIAkO8CxJwMDM8ZFjI9ADKZjgIJJSBmuMDcACQl9B0YGEC28xkAVQFB/wUG\n7gVAmm0DfwOQYmD7yMAJYvQ38DsAKQbGbwy7QLQmA88CEM1g3zADRN1mYF4AohneL08A0zCC9WgD\njAmm2WGuhIkGwhhAGgDwdic2xV4k0wAAAABJRU5ErkJggg==\n",
       "prompt_number": 59,
       "text": [
        "  ___\n",
        "\u2572\u2571 \u03c0 "
       ]
      }
     ],
     "prompt_number": 59
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Recuerde, `oo` es la notaci\u00f3n SymPy para infinito."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Sumpatorios y productorios\n",
      "Podemos evaluar sumas y productos mediante las funciones: 'Sum'"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "n = Symbol(\"n\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 60
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Sum(1/n**2, (n, 1, 10))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\sum_{n=1}^{10} \\frac{1}{n^{2}}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADsAAAA9BAMAAADhUgydAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nUPOWOmbnWTm9FTyhDGFTsz3jVvg3/gK+T7KE+MaJIAAAAABJRU5ErkJggg==\n",
       "prompt_number": 61,
       "text": [
        "  10    \n",
        " ____   \n",
        " \u2572      \n",
        "  \u2572   1 \n",
        "   \u2572  \u2500\u2500\n",
        "   \u2571   2\n",
        "  \u2571   n \n",
        " \u2571      \n",
        " \u203e\u203e\u203e\u203e   \n",
        "n = 1   "
       ]
      }
     ],
     "prompt_number": 61
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Sum(1/n**2, (n,1, 10)).evalf()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$1.54976773116654$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJ8AAAAPBAMAAAAIUwCQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nTkSuQmCC\n",
       "prompt_number": 62,
       "text": [
        "1.54976773116654"
       ]
      }
     ],
     "prompt_number": 62
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Sum(1/n**2, (n, 1, oo)).evalf()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$1.64493406684823$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJ8AAAAPBAMAAAAIUwCQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n//n5B+ab/k/pD/tGAAAAAElFTkSuQmCC\n",
       "prompt_number": 63,
       "text": [
        "1.64493406684823"
       ]
      }
     ],
     "prompt_number": 63
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Los productos funcionan de la misma manera:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Product(n, (n, 1, 10)) # 10!"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\prod_{n=1}^{10} n$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAC8AAAA9BAMAAADPFy0PAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMruZq91U7yJ2iWYQ\nRM2bSrnQAAABW0lEQVQ4EdWUP0vDQBjGn2g10UQidtLRyUUquKmDi0NHP0C1gsWx+A2CLgqChaqD\nU1YXm9nFfIQOSqGLnQoiSCoKDsp5d+mfu+vr6p9nCHfvL+/DXfLwAkIzwMRVKJfaI7sB3GBXq6Wb\nErAJPx4mJXjvsIsUsD4wFn0HKiTgViTwOvDrVAcK5Kn2gCp1j8PtPCZX1oad/mDFZgN11PM5d+zl\nSOiWtadVAORe5b7MYr2OWhe8GfV/DpoLx2dpYIwLLs1XeC6FdGAVt+CkX1MHnvcJNyE6MJ4g06LA\naAt+FAuiW8EOUY5DApzWce3KX2Z03AMHF6LBtJK13we5NDc1I1dOlrHFAN4cY+1gcFRAZjfBiEiw\nll31pZ9Y80lHSkw6WnzS0VKBuzpb7b+lgYfIDjLPXE+ACk7OlUGlAuTRIK2wj2VLWj3qVk6Cwk6v\nhU+6vqbWcRl1d2LS9fQFXLmvHr1wylwAAAAASUVORK5CYII=\n",
       "prompt_number": 64,
       "text": [
        "  10   \n",
        "\u252c\u2500\u2500\u2500\u252c  \n",
        "\u2502   \u2502 n\n",
        "\u2502   \u2502  \n",
        "n = 1  "
       ]
      }
     ],
     "prompt_number": 64
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### L\u00edmites\n",
      "\n",
      "Los l\u00edmites pueden ser evaluados mediante la funci\u00f3n `limit`'. Por ejemplo:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expresion = (sin(x)/x)\n",
      "expresion"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{1}{x} \\sin{\\left (x \\right )}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEgAAAAqBAMAAADi/wAlAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAzRAiu5mrdu/dZkSJ\nVDL+6OvUAAABk0lEQVQ4Ee2SsUvDUBDGvyTF2jRpi5MgYlx0EcngpKAFkYJLXHSUbm5WFwsitHRw\ncXDTSYx/ga5OZtGhLm7i1k0XKQh21XtJLi+YJ92cPEjuy+++3Ht5FwCag6FheMNN+lpjuAlo/Zv+\n9gi2Ts98KKIUU7P3o/i2J8ECy10Wce4HEoyx1Fhkcz5gJhWTJI+6LPU2qygv95Ln20RhUsrX533D\nq5fGt5s7At7TNXuwdESfImQUehUF5OrQalgRK60DemCeFyrAVWwBih1YwpRro+wTvQAsx/qw6YUX\nfIVB9LLmhqYKyrQETqgTtCop9MUtirmpz9BUTUyIVMpkuLi+EcvFJlqOevTgAPJD7QAjfspEG7ed\nFvJkkhu3O8i51IY7PQB3Tx4Wqd8mXVEUu4ePpcZgvjF4944J0RozzdmuT3IicijuBXFYYRjtWGST\nucHMEifyS6wyn2ahyMlPl7gVJsOPoCk2F49aYZOIRy2JQvGoFSWJeNSSKFX4ZygrKRiNOgWykked\nraQIjzqFspJHna1I8g0WTlrMmHy/3AAAAABJRU5ErkJggg==\n",
       "prompt_number": 65,
       "text": [
        "sin(x)\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "  x   "
       ]
      }
     ],
     "prompt_number": 65
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "limit(expresion, x, 0) # equivalente a limit((sin(x)/x), x, 0)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$1$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAPBAMAAAArJJMAAAAAJFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAADHJj5lAAAAC3RSTlMAzRAiu5mrdu/dZmiL4QAAAAAjSURBVAgd\nY2BgEGJgYDDZxMCgEgYkGNhJJVgzdmYB9TEwAACPpQrvlUCHcAAAAABJRU5ErkJggg==\n",
       "prompt_number": 66,
       "text": [
        "1"
       ]
      }
     ],
     "prompt_number": 66
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Podemos utilizar \"limit\" para comprobar el resultado de la derivaci\u00f3n mediante la funci\u00f3n `diff`:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "f"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\sin{\\left (x y \\right )} + \\cos{\\left (y z \\right )}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAKAAAAAVBAMAAAAz5vjYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nSUVORK5CYII=\n",
       "prompt_number": 67,
       "text": [
        "sin(x\u22c5y) + cos(y\u22c5z)"
       ]
      }
     ],
     "prompt_number": 67
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "diff(f, x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$y \\cos{\\left (x y \\right )}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFMAAAAVBAMAAADInc9CAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 68,
       "text": [
        "y\u22c5cos(x\u22c5y)"
       ]
      }
     ],
     "prompt_number": 68
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$\\displaystyle \\frac{\\mathrm{d}f(x,y)}{\\mathrm{d}x} = \\frac{f(x+h,y)-f(x,y)}{h}$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "h = Symbol(\"h\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 69
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "limit((f.subs(x, x+h) - f)/h, h, 0)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$y \\cos{\\left (x y \\right )}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFMAAAAVBAMAAADInc9CAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 70,
       "text": [
        "y\u22c5cos(x\u22c5y)"
       ]
      }
     ],
     "prompt_number": 70
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "\u00a1Correcto!"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Podemos cambiar la direcci\u00f3n de la que nos acercamos al punto del l\u00edmite con el uso de la palabra clave y argumento `dir`:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "limit(1/x, x, 0, dir=\"+\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\infty$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAALBAMAAABv+6sJAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAEKvvZom7mXYyzVQiRHuO\nwuMAAABqSURBVAgdY2BgEGAAAhDBqPzYgYEhrKiBgYFtAscrBqkNDFMZGDwZGPYlrGRgYDnAAJRg\nBSlhcAAxGfJAzAYwM/klULsDgyMDA0eM8QEGjgMM7AwM4QzzbBlcgRLlahsYGOuMA4DK4bYBAA/G\nFFwDPj79AAAAAElFTkSuQmCC\n",
       "prompt_number": 71,
       "text": [
        "\u221e"
       ]
      }
     ],
     "prompt_number": 71
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "limit(1/x, x, 0, dir=\"-\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$-\\infty$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAACMAAAALBAMAAAAHCCkxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMqvvZom7mXZU\nIkRJD0iWAAAAf0lEQVQIHWNggAMBEAtMQIUYw74VMDB0Lt0AV8LA6cD9iUHoAIMHQqiEgeH8BBUG\nBvYLDELGIKDCANTAAdLKAMIQADJjPogLYkAAiDXtCwMDI0gYAgoZGLh70y4wcF+AiTBwMTB0Mfjn\nMVTARRgYV0UeYGBcn9aAEII6XICBAQCMCRoksfUeRwAAAABJRU5ErkJggg==\n",
       "prompt_number": 72,
       "text": [
        "-\u221e"
       ]
      }
     ],
     "prompt_number": 72
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Series\n",
      "\n",
      "La expansi\u00f3n de la serie es tambi\u00e9n una de las caracter\u00edsticas m\u00e1s \u00fatiles de un CAS. En SymPy podemos realizar un desarrollo en serie de una expresi\u00f3n mediante la funci\u00f3n `series`:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "series(exp(x), x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$1 + x + \\frac{x^{2}}{2} + \\frac{x^{3}}{6} + \\frac{x^{4}}{24} + \\frac{x^{5}}{120} + \\mathcal{O}\\left(x^{6}\\right)$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAVoAAAAwBAMAAACiZ6/NAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n11uvr4GoTFNSL/V/PfoX0LjCXF+s8nwAAAAASUVORK5CYII=\n",
       "prompt_number": 73,
       "text": [
        "         2    3    4     5        \n",
        "        x    x    x     x     \u239b 6\u239e\n",
        "1 + x + \u2500\u2500 + \u2500\u2500 + \u2500\u2500 + \u2500\u2500\u2500 + O\u239dx \u23a0\n",
        "        2    6    24   120        "
       ]
      }
     ],
     "prompt_number": 73
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Por defecto se expande la expresi\u00f3n de alrededor de $ x = 0 $, pero podemos expandir en torno a cualquier valor de $ x $ incluyendo expl\u00edcitamente un valor en la llamada a la funci\u00f3n:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "series(exp(x), x, 1)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$e + e x + \\frac{e x^{2}}{2} + \\frac{e x^{3}}{6} + \\frac{e x^{4}}{24} + \\frac{e x^{5}}{120} + \\mathcal{O}\\left(x^{6}\\right)$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAYEAAAAwBAMAAAAFlttzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nKlZAV16roOD9H1z/ANP52lDicFqVAAAAAElFTkSuQmCC\n",
       "prompt_number": 74,
       "text": [
        "             2      3      4      5        \n",
        "          \u212f\u22c5x    \u212f\u22c5x    \u212f\u22c5x    \u212f\u22c5x     \u239b 6\u239e\n",
        "\u212f + \u212f\u22c5x + \u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500 + O\u239dx \u23a0\n",
        "           2      6      24    120         "
       ]
      }
     ],
     "prompt_number": 74
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Y podemos definir de forma expl\u00edcita que orden en la expansi\u00f3n de la serie se debe realizar:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "series(exp(x), x, 1, 10)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$e + e x + \\frac{e x^{2}}{2} + \\frac{e x^{3}}{6} + \\frac{e x^{4}}{24} + \\frac{e x^{5}}{120} + \\frac{e x^{6}}{720} + \\frac{e x^{7}}{5040} + \\frac{e x^{8}}{40320} + \\frac{e x^{9}}{362880} + \\mathcal{O}\\left(x^{10}\\right)$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAr0AAAAwBAMAAADuu9/EAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 75,
       "text": [
        "             2      3      4      5      6      7       8       9          \n",
        "          \u212f\u22c5x    \u212f\u22c5x    \u212f\u22c5x    \u212f\u22c5x    \u212f\u22c5x    \u212f\u22c5x     \u212f\u22c5x     \u212f\u22c5x      \u239b 10\u239e\n",
        "\u212f + \u212f\u22c5x + \u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500 + O\u239dx  \u23a0\n",
        "           2      6      24    120    720    5040   40320   362880         "
       ]
      }
     ],
     "prompt_number": 75
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "La expansi\u00f3n de la serie incluye el orden de la aproximaci\u00f3n, que es muy \u00fatil para hacer el seguimiento de la orden de validez cuando hacemos c\u00e1lculos con desarrollos en serie de distinto orden:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "s1 = cos(x).series(x, 0, 5)\n",
      "s1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$1 - \\frac{x^{2}}{2} + \\frac{x^{4}}{24} + \\mathcal{O}\\left(x^{5}\\right)$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAMQAAAAwBAMAAAC8i8hXAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 76,
       "text": [
        "     2    4        \n",
        "    x    x     \u239b 5\u239e\n",
        "1 - \u2500\u2500 + \u2500\u2500 + O\u239dx \u23a0\n",
        "    2    24        "
       ]
      }
     ],
     "prompt_number": 76
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "s2 = sin(x).series(x, 0, 2)\n",
      "s2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$x + \\mathcal{O}\\left(x^{2}\\right)$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAF8AAAAcBAMAAAD1rn4EAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nst0fI6fB3bFCQ4yRovAvviFnm72g1k0AAAAASUVORK5CYII=\n",
       "prompt_number": 77,
       "text": [
        "     \u239b 2\u239e\n",
        "x + O\u239dx \u23a0"
       ]
      }
     ],
     "prompt_number": 77
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expand(s1 * s2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$x + \\mathcal{O}\\left(x^{2}\\right)$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAF8AAAAcBAMAAAD1rn4EAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nst0fI6fB3bFCQ4yRovAvviFnm72g1k0AAAAASUVORK5CYII=\n",
       "prompt_number": 78,
       "text": [
        "     \u239b 2\u239e\n",
        "x + O\u239dx \u23a0"
       ]
      }
     ],
     "prompt_number": 78
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Si queremos deshacernos de la informaci\u00f3n del orden, podemos utilizar el m\u00e9todo `removeO`:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expand(s1.removeO() * s2.removeO())"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{x^{5}}{24} - \\frac{x^{3}}{2} + x$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAG8AAAAwBAMAAADtMzlxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nQmCC\n",
       "prompt_number": 79,
       "text": [
        " 5    3    \n",
        "x    x     \n",
        "\u2500\u2500 - \u2500\u2500 + x\n",
        "24   2     "
       ]
      }
     ],
     "prompt_number": 79
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Pero ten en cuenta que esta no es la correcta expansi\u00f3n de $ \\ cos (x) \\ sin (x) $ hasta el $ 5 $ \u00ba orden:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "(cos(x)*sin(x)).series(x, 0, 6)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$x - \\frac{2 x^{3}}{3} + \\frac{2 x^{5}}{15} + \\mathcal{O}\\left(x^{6}\\right)$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAANsAAAAwBAMAAABqLhIyAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nXUuVWH7i1sNgXn78rq2lqsj2sf8BdRcRYxcHw0UAAAAASUVORK5CYII=\n",
       "prompt_number": 80,
       "text": [
        "       3      5        \n",
        "    2\u22c5x    2\u22c5x     \u239b 6\u239e\n",
        "x - \u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500 + O\u239dx \u23a0\n",
        "     3      15         "
       ]
      }
     ],
     "prompt_number": 80
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Algebra lineal"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Matrices\n",
      "\n",
      "Matrices se definen con la clase `Matrix`:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "m11, m12, m21, m22 = symbols(\"m11, m12, m21, m22\")\n",
      "b1, b2 = symbols(\"b1, b2\")"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 81
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A = Matrix([[m11, m12],[m21, m22]])\n",
      "A"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}m_{11} & m_{12}\\\\m_{21} & m_{22}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAGgAAAAyBAMAAABCJ4MDAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 82,
       "text": [
        "\u23a1m\u2081\u2081  m\u2081\u2082\u23a4\n",
        "\u23a2        \u23a5\n",
        "\u23a3m\u2082\u2081  m\u2082\u2082\u23a6"
       ]
      }
     ],
     "prompt_number": 82
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "b = Matrix([[b1], [b2]])\n",
      "b"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}b_{1}\\\\b_{2}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAB4AAAAyBAMAAAC5cHbcAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMA74lUMhC73c2rRHaZ\nImaqCQggAAAA9klEQVQoFWOQ//+JAQb0/39kEHZxhXEZQlwMGUTgPBDDEcRnTSyACYL5DBwTUPn8\nBqj89QGofHUYF2IeQ+fuw1ARiPk/AvgvMLiBhMB85q8M/Ao+0+F8ngaGeAUGczif4wDDegMkPv8D\nhvMByPwFrHIMSHzOB2wHkPmsp+cBzUKYB+Rg4RuCBCH+BbHu1M9G4YPEkOXpyKdLfDAdOwrzHzg+\ndBlmwPjg+Khl8L8A9T84PqQY7i+A8iHxwWAPTAXg8OMHxwdDNUw/JD6YDsD4kPh4A+QihTeXApcD\nMt/D2AxmHkghg/z//0ASET9gQSL46OkdLT8AABDBTMmz6EtEAAAAAElFTkSuQmCC\n",
       "prompt_number": 83,
       "text": [
        "\u23a1b\u2081\u23a4\n",
        "\u23a2  \u23a5\n",
        "\u23a3b\u2082\u23a6"
       ]
      }
     ],
     "prompt_number": 83
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Con las instancias de clase `Matrix` podemos hacer las operaciones habituales de \u00e1lgebra matricial:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A**2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}m_{11}^{2} + m_{12} m_{21} & m_{11} m_{12} + m_{12} m_{22}\\\\m_{11} m_{21} + m_{21} m_{22} & m_{12} m_{21} + m_{22}^{2}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAXIAAAAzBAMAAAB4eZ5HAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\npW+JFL+Z007cClX4Zs5fZLuVy5E1BG0AAAAASUVORK5CYII=\n",
       "prompt_number": 84,
       "text": [
        "\u23a1    2                               \u23a4\n",
        "\u23a2 m\u2081\u2081  + m\u2081\u2082\u22c5m\u2082\u2081    m\u2081\u2081\u22c5m\u2081\u2082 + m\u2081\u2082\u22c5m\u2082\u2082\u23a5\n",
        "\u23a2                                    \u23a5\n",
        "\u23a2                                 2  \u23a5\n",
        "\u23a3m\u2081\u2081\u22c5m\u2082\u2081 + m\u2082\u2081\u22c5m\u2082\u2082   m\u2081\u2082\u22c5m\u2082\u2081 + m\u2082\u2082   \u23a6"
       ]
      }
     ],
     "prompt_number": 84
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A * b"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}b_{1} m_{11} + b_{2} m_{12}\\\\b_{1} m_{21} + b_{2} m_{22}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJEAAAAyBAMAAACufiRQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYII=\n",
       "prompt_number": 85,
       "text": [
        "\u23a1b\u2081\u22c5m\u2081\u2081 + b\u2082\u22c5m\u2081\u2082\u23a4\n",
        "\u23a2               \u23a5\n",
        "\u23a3b\u2081\u22c5m\u2082\u2081 + b\u2082\u22c5m\u2082\u2082\u23a6"
       ]
      }
     ],
     "prompt_number": 85
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Y el c\u00e1lculo de los determinantes e inversas:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A.det()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$m_{11} m_{22} - m_{12} m_{21}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAKYAAAAMBAMAAAAaIdvMAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYII=\n",
       "prompt_number": 86,
       "text": [
        "m\u2081\u2081\u22c5m\u2082\u2082 - m\u2081\u2082\u22c5m\u2082\u2081"
       ]
      }
     ],
     "prompt_number": 86
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A.inv()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}\\frac{1}{m_{11}} + \\frac{m_{12} m_{21}}{m_{11}^{2} \\left(m_{22} - \\frac{m_{12} m_{21}}{m_{11}}\\right)} & - \\frac{m_{12}}{m_{11} \\left(m_{22} - \\frac{m_{12} m_{21}}{m_{11}}\\right)}\\\\- \\frac{m_{21}}{m_{11} \\left(m_{22} - \\frac{m_{12} m_{21}}{m_{11}}\\right)} & \\frac{1}{m_{22} - \\frac{m_{12} m_{21}}{m_{11}}}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAa4AAABSCAMAAAA7DMB+AAAAPFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAo1xBWAAAAE3RSTlMA\nMquZdlQQQOkwRLvd7yJmic1sqjZpngAACPhJREFUeAHtXeu64ioMRdt6ZrxVj+//rgOUYJNAuRRo\n9ZMfuxoCZLFaGsNlC+FJh4sn4yfeXw8cuuN1f1b9LPL2wPCjy9s3O8z40bVDUoQ4vHQ6UuPq0jV2\n46HvO9H3lwNtWYjlXK6/C0lVo68TS4qurpdpoJAr03V+jULcjifRTzeK/CRteTwud3kdcS6IhVai\nhu7lOzFaTNZa29eZOSqOji9Fl+P2lnVXpet+6tWr8XUWonvID+fHTdEkndGzlNNcIzZKUm+PiRqN\nIRWxeMym66R6d0069vJOuckartNzfVUVPoU4vdRjhnOtWCutabVqWWy0RIYgFWg6n65B9vaqdJPP\ndC8fm/tL6KcbmJgeapprnnVQWtVytcLUaLC22Di1SJe69b1pLV2SpekZGi9CjYbTrSgfrqe6JVnu\nJAYlpb+/xIw2dBnbCxjsp+veXV/q4faltXSdlYOhqLkfH2r0M0ycJmeC5hrxvumiRmNIvo5Mkfvp\nCtWyli5Wv74VT939pJ81nG3FMLzg7L1+W4KUZfNu6Dr0r066HOrXhSOYYsRGKQtp+0LLkLLs2Q1d\nWdavK3R6Xm16un/LrGugfGlM1/TD2fcXWn88Vbre9KUDqa/U9nKwMHzd3lafBcZ2TFcY0Fsj7901\neRXvWup9OtkbqXQbzTBwCI3p6hqOObWm6xpiYBDa0tXJKGG7NFZ5vppioBCa0sVulsrUXVjUen2D\njTEQCPXocgSic9yvsRvGsTsPj2kYnYftUZZtbhaz1xHJ9QyhGjIwIDtJmB7lOTAQCISuhBmbc2Bg\nM0H0GdZBipLT0Mvw7m0QAw/boyxnzP4ZsDHZGpGDAdlJwvQoz4UBQ6B04WkmcyvYSwo8G0S3hY5Z\nnXc8iJP84fxwhO3nWba5edxjkNKyKQvD3E4app/nuTBgCJguz4xN7jQTCUTriZHk3pNUDdJnuArN\nF5Chq8ZZjpj9NBeT3ORCgSwM2E4EQeKaw+MYMARM1xQiZ5NQEH5dgOHKIoHo82KA31WBkp1kJLiX\nc5jHXgXqIcSrqyZZrpj9VZYsmbIwEDsNXZO5JM+BAUGgdHlmbOCOSIEOQXQo88h5dUFhuE6G0KpV\nLsiQqUdHuBiqyrmWwOCH4MSAIBC6fDM2qA88OImbYoPooK7n+OUXogfZcVdtCKtalrUyZGpq94Zs\nK4HBC8GNAUEgdHlmbGAMWuxSsrCExdYvZvKM6Bnv1TqxS20sxLjdMXvlVqYkYpv1seDXwXoMCxBg\nOgLPOyAIhC4Gzdyr6JZlSlpA3RSmZVxSqme8V3NhpdYJRvkmT0jUNuNjvV2tDTAgCMt0GZ4x3V74\ndGEJUbyCH4/1jPcKTuxwgaSnmH0R6pDctD2kujfYNuvawO2agyFkqi/fBWGZLtLhga/UTTEjCAwk\nT1hJQPVghRzx+wOtRWWPiYOhoLaR0WUDDAhCQbqIm8IGkovx0ogeLKYxa2qiWIhWUiutUhKzjdC1\nAQYEoSBdITelMwFyqmc8cHDEU3o3qAttBhWNArUNfCwYDKE+qlcRAzSpLSxIF+sScmd6xjrjgVtH\nnFUzE6CAKHPbZorw0Q5eIEi9bo8BQWhI10ktLOTJeODM7+eaMlzjiPe+3TZHiZcOhTgy4kTM1doA\nA4LQkC5xWx8RmgdEmdvGKTik+fG8AiZpjgFDaEnXg21JYr0REuCAKBmpeGEIQvCcXElzDBhCPbrY\nQGL2KuR2lCpHAqJBum76t9uaFmlZHCCnuTHfEzFgCPXocpj+KBxwDdHVwy89hy25osYYCISmdOkl\n8bn95CgXoEvNapZPesdFuWqXMVAIbem6pwYZlrqFj7ZE+7jOLSS1wdemGCiEtnSJw3pvA7oteK21\nHnAZQ9GVUgyCouu/P3+D2EspFH/3+w2r1tRCxYUPI2Et/f/HvzfZ3xO/HG8PeGI3Xv20jMaDYZpx\nn6j9dXSFJtg/kaS3zd9HF17LaGf9YV7sDf0TP30bXXSC3cz6L0ZqP4i3b6OLrmWEWX+YWvogalym\nfh9ddIKdL1119cOHyL6OLjbB7li6+iHcOMxMXszjqMMv2sCR90ycf8VgGDqMxE9EXM4GdBHD7Ky/\nCXaS7N/XeQ9sT5d7+e3cxt9n2wMRdLHjJ3zrGLeXW1ilPrSCFGtvBF2xVf306vdAPbqanT4hN27U\n76eoFlzr6iBog/JAmHz4aTW62FRNFOA8paJzTHkm6FKudXWwVQPlZUdyatHV9PQJuWVsRScXLOpY\nV2eDNvM8K0z1hivR1fp+J8dPFGQgqSrnujoI2gQ2IUc1VImuxqdPTGf6RgGuqeRekxa7CTnGsjp0\nNT99QuDjJ2KQx+msmZybRjrnVg0Q7mMwbH76hMDHT8RREaPFdr/GFDI6mgsbtJkXtMJ90NX89IkC\nK4Tn3Wk/08k544FbR9wq8g9mXZ1zq0Z2JKfKYLjB6RMCHT/BOy9bgne/Gg/ceuewMfei1izlhkAS\nbKtCFzpbIMGYuWrqwI+On5hXtPIznpwzHrh1xFdWnlw8i67Q+xd2TYT0lqxNHfhL3CIOe9jkXMWN\n1I7mqSiPLrw4hm1j3OD0CYGOn6Ao87/TyTnYQW2mVPMrziuZQxd9/7JN4xucPiHQ8RN5fRFTKmET\nMooSxtQdoZNDF10cw7YxbnD6hKg76Q49aTxw64iD3HVFUUI2ArlKhGV5dOH3L6PL7n6mehUHfnT8\nRBh3pobxwJ3eOatyHiVkIxDTjhJk0cXev+bXHvzo2+D0Cf2/iqIQt1NyRhChj/LMyKKLvX8JXXAU\nBNVLGPiT0UCbyQWrFXBHEDegiyEkdHnW2qUM/OQ9Tc4rYgbI/1VUYWOro5l8EemjvIqyni7SFNvG\nuMHpEwIdP0EM3MXX3dDFe6P56RMCHz/BLdpesmO6mp8+Mf270e058VrARiCv5mJGicGQN9D89AmB\nj5/gFn2JZKJLx5JL7vJufPqE/F/ZX8KHH4b6rSeTPAtG/QPlXp3+XS41PX1C/1+Acrbvsyb9b65r\n3ZZNT58Q9PiJfXb4nq1aPn2irOUt1zSWtXw/tbEzIeqZ1rCpeiBia/4H6cC42oziP8UAAAAASUVO\nRK5CYII=\n",
       "prompt_number": 87,
       "text": [
        "\u23a1 1          m\u2081\u2082\u22c5m\u2082\u2081                -m\u2081\u2082        \u23a4\n",
        "\u23a2\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500  \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5\n",
        "\u23a2m\u2081\u2081      2 \u239b      m\u2081\u2082\u22c5m\u2082\u2081\u239e      \u239b      m\u2081\u2082\u22c5m\u2082\u2081\u239e\u23a5\n",
        "\u23a2      m\u2081\u2081 \u22c5\u239cm\u2082\u2082 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f  m\u2081\u2081\u22c5\u239cm\u2082\u2082 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u23a5\n",
        "\u23a2           \u239d        m\u2081\u2081  \u23a0      \u239d        m\u2081\u2081  \u23a0\u23a5\n",
        "\u23a2                                               \u23a5\n",
        "\u23a2          -m\u2082\u2081                       1         \u23a5\n",
        "\u23a2   \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500         \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500   \u23a5\n",
        "\u23a2       \u239b      m\u2081\u2082\u22c5m\u2082\u2081\u239e               m\u2081\u2082\u22c5m\u2082\u2081   \u23a5\n",
        "\u23a2   m\u2081\u2081\u22c5\u239cm\u2082\u2082 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f         m\u2082\u2082 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500   \u23a5\n",
        "\u23a3       \u239d        m\u2081\u2081  \u23a0                 m\u2081\u2081     \u23a6"
       ]
      }
     ],
     "prompt_number": 87
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Resolviendo ecuaciones\n",
      "\n",
      "Para la resoluci\u00f3n de ecuaciones y sistemas de ecuaciones podemos usar la funci\u00f3n `solve`:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "solve(x**2 - 1, x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{bmatrix}-1, & 1\\end{bmatrix}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAEkAAAAZBAMAAACP0x4YAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAu90iEM0ymat272ZEieIx\nuvsAAAB6SURBVCgVYxBSMmDADw4pXWBQwK8EJMvcQI6qEAyDWR0wzDJ8gq6KWQ5DlWMFuioW8T4M\nVQyc6KoYGOYNelWuSiAQxjCoXe8XgBJLmKHK3vWom4EJJfwr1mlswJpyDqOYBeJgS18biFHFQpQq\nbgyjsNpIc1VKROTHBgCD+TIQNJw1owAAAABJRU5ErkJggg==\n",
       "prompt_number": 88,
       "text": [
        "[-1, 1]"
       ]
      }
     ],
     "prompt_number": 88
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "solve(x**4 - x**2 - 1, x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{bmatrix}- i \\sqrt{- \\frac{1}{2} + \\frac{\\sqrt{5}}{2}}, & i \\sqrt{- \\frac{1}{2} + \\frac{\\sqrt{5}}{2}}, & - \\sqrt{\\frac{1}{2} + \\frac{\\sqrt{5}}{2}}, & \\sqrt{\\frac{1}{2} + \\frac{\\sqrt{5}}{2}}\\end{bmatrix}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAgMAAAAmBAMAAACrC4XIAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nwX/DG7SInEemHQAAAABJRU5ErkJggg==\n",
       "prompt_number": 89,
       "text": [
        "\u23a1        _____________         _____________        ___________       ________\n",
        "\u23a2       \u2571         ___         \u2571         ___        \u2571       ___       \u2571       _\n",
        "\u23a2      \u2571    1   \u2572\u2571 5         \u2571    1   \u2572\u2571 5        \u2571  1   \u2572\u2571 5       \u2571  1   \u2572\u2571 \n",
        "\u23a2-\u2148\u22c5  \u2571   - \u2500 + \u2500\u2500\u2500\u2500\u2500 , \u2148\u22c5  \u2571   - \u2500 + \u2500\u2500\u2500\u2500\u2500 , -  \u2571   \u2500 + \u2500\u2500\u2500\u2500\u2500 ,   \u2571   \u2500 + \u2500\u2500\u2500\n",
        "\u23a3   \u2572\u2571      2     2       \u2572\u2571      2     2      \u2572\u2571    2     2     \u2572\u2571    2     2\n",
        "\n",
        "___\u23a4\n",
        "__ \u23a5\n",
        "5  \u23a5\n",
        "\u2500\u2500 \u23a5\n",
        "   \u23a6"
       ]
      }
     ],
     "prompt_number": 89
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Sistema de ecuaciones:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "solve([x + y - 1, x - y - 1], [x,y])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{Bmatrix}x : 1, & y : 0\\end{Bmatrix}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAH0AAAAaBAMAAABoX070AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nPpnIgxLNAx8vgBQ/CRD6L/ID1aNrABrAsl8AAAAASUVORK5CYII=\n",
       "prompt_number": 90,
       "text": [
        "{x: 1, y: 0}"
       ]
      }
     ],
     "prompt_number": 90
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "En t\u00e9rminos de otras expresiones simb\u00f3licas:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "solve([x + y - a, x - y - c], [x,y])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{Bmatrix}x : \\frac{a}{2} + \\frac{c}{2}, & y : \\frac{a}{2} - \\frac{c}{2}\\end{Bmatrix}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAANEAAAAaBAMAAADbO6SrAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n4m2lL7zgb0KdmHlCV4CcJVO6pfV/rMy8P+mJlX0AAAAASUVORK5CYII=\n",
       "prompt_number": 91,
       "text": [
        "\u23a7   a   c     a   c\u23ab\n",
        "\u23a8x: \u2500 + \u2500, y: \u2500 - \u2500\u23ac\n",
        "\u23a9   2   2     2   2\u23ad"
       ]
      }
     ],
     "prompt_number": 91
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Quantum mechanics: noncommuting variables\n",
      "\n",
      "How about non-commuting symbols? In quantum mechanics we need to work with noncommuting operators, and SymPy has a nice support for noncommuting symbols and even a subpackage for quantum mechanics related calculations!"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sympy.physics.quantum import *"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 92
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### States\n",
      "\n",
      "We can define symbol states, kets and bras:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Ket('psi')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$${\\left|\\psi\\right\\rangle }$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAABgAAAAWBAMAAAAoU0G7AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVCKJRHYyEJndZu+7\nq837RU9zAAAAvElEQVQYGU3PIQ7CQBAF0N9uYUupaIpBkNAjrNgEW4lEQcAAByBpQjAIFAhcHRYB\nnoQLNJwAheYMnIC/7DZhxO68zGQygxQuIv66RpjXGAHNZw3F+slBsoKZQ7Agdg6COeKM09YHCEwB\nTxGqjy3mBVBC+8kRKdrs20NL3KEQEiu2BR9iQ3QIUcozBsSDCKuo8HLAr0zbrYELCyIh0O1dM6Jl\nN1gyBYYWkx9eFuYAyLeFOQAR37FJTMTc7j++zlkbOXLgJM8AAAAASUVORK5CYII=\n",
       "prompt_number": 93,
       "text": [
        "\u2758\u03c8\u27e9"
       ]
      }
     ],
     "prompt_number": 93
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Bra('psi')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$${\\left\\langle \\psi\\right|}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAABgAAAAWBAMAAAAoU0G7AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZjKrmUTdVIm7Is12\nEO9EeINyAAAAvElEQVQYGTXPIQ7CMBQG4H/LQgu0ZBfgADiOgCEERwgadgQMCUFVDDwWVYfdEQoS\nNTQIjsAFSPj3tjVpX7++vpcWQIpmHIB43OIIJIsKhpPoSELX2AveNS6CpaBXCJxAp7B5gDMZG7Be\nq69NzY9YAdckUx4fYgqUA8fOE+LGyo3XsIF4ESc8oUui64E5htiBUA5YI0R8Id+WAWp2fvA2wXaI\nt1wqjBhM2aDPTfUByUSh/prAFnLM5d5uJP4BptQn4uAwXfEAAAAASUVORK5CYII=\n",
       "prompt_number": 94,
       "text": [
        "\u27e8\u03c8\u2758"
       ]
      }
     ],
     "prompt_number": 94
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "u = Ket('0')\n",
      "d = Ket('1')\n",
      "\n",
      "a, b = symbols('alpha beta', complex=True)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 95
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "phi = a * u + sqrt(1-abs(a)**2) * d; phi"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\alpha {\\left|0\\right\\rangle } + \\sqrt{- \\left\\lvert{\\alpha}\\right\\rvert^{2} + 1} {\\left|1\\right\\rangle }$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAMMAAAAmBAMAAACL2DKoAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 96,
       "text": [
        "           ____________    \n",
        "          \u2571      2         \n",
        "\u03b1\u22c5\u27580\u27e9 + \u2572\u2571  - \u2502\u03b1\u2502  + 1 \u22c5\u27581\u27e9"
       ]
      }
     ],
     "prompt_number": 96
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Dagger(phi)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\overline{\\alpha} {\\left\\langle 0\\right|} + \\overline{\\sqrt{- \\left\\lvert{\\alpha}\\right\\rvert^{2} + 1}} {\\left\\langle 1\\right|}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAMMAAAArBAMAAAA3RiF2AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nVAAAAABJRU5ErkJggg==\n",
       "prompt_number": 97,
       "text": [
        "        _______________    \n",
        "           ____________    \n",
        "_         \u2571      2         \n",
        "\u03b1\u22c5\u27e80\u2758 + \u2572\u2571  - \u2502\u03b1\u2502  + 1 \u22c5\u27e81\u2758"
       ]
      }
     ],
     "prompt_number": 97
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Dagger(phi) * d"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left(\\overline{\\alpha} {\\left\\langle 0\\right|} + \\overline{\\sqrt{- \\left\\lvert{\\alpha}\\right\\rvert^{2} + 1}} {\\left\\langle 1\\right|}\\right) {\\left|1\\right\\rangle }$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAPsAAAA/BAMAAADUJaM3AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nr7mo/gWQKOAzEXRvqAAAAABJRU5ErkJggg==\n",
       "prompt_number": 98,
       "text": [
        "\u239b        _______________    \u239e    \n",
        "\u239c           ____________    \u239f    \n",
        "\u239c_         \u2571      2         \u239f    \n",
        "\u239d\u03b1\u22c5\u27e80\u2758 + \u2572\u2571  - \u2502\u03b1\u2502  + 1 \u22c5\u27e81\u2758\u23a0\u22c5\u27581\u27e9"
       ]
      }
     ],
     "prompt_number": 98
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Use `qapply` to distribute a mutiplication:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "qapply(Dagger(phi) * d)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\overline{\\alpha} \\left\\langle 0 \\right. {\\left|1\\right\\rangle } + \\overline{\\sqrt{- \\left\\lvert{\\alpha}\\right\\rvert^{2} + 1}} \\left\\langle 1 \\right. {\\left|1\\right\\rangle }$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAPsAAAArBAMAAABMYuO6AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\naf7Z/X/WdRD/Rdnby5TsuQPIZOspivwDf1xTay7EXNUAAAAASUVORK5CYII=\n",
       "prompt_number": 99,
       "text": [
        "          _______________      \n",
        "             ____________      \n",
        "_           \u2571      2           \n",
        "\u03b1\u22c5\u27e80\u27581\u27e9 + \u2572\u2571  - \u2502\u03b1\u2502  + 1 \u22c5\u27e81\u27581\u27e9"
       ]
      }
     ],
     "prompt_number": 99
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "qapply(Dagger(phi) * u)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\overline{\\alpha} \\left\\langle 0 \\right. {\\left|0\\right\\rangle } + \\overline{\\sqrt{- \\left\\lvert{\\alpha}\\right\\rvert^{2} + 1}} \\left\\langle 1 \\right. {\\left|0\\right\\rangle }$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAPsAAAArBAMAAABMYuO6AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 100,
       "text": [
        "          _______________      \n",
        "             ____________      \n",
        "_           \u2571      2           \n",
        "\u03b1\u22c5\u27e80\u27580\u27e9 + \u2572\u2571  - \u2502\u03b1\u2502  + 1 \u22c5\u27e81\u27580\u27e9"
       ]
      }
     ],
     "prompt_number": 100
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Operators"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A = Operator('A')\n",
      "B = Operator('B')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 101
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Check if they are commuting!"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "A * B == B * A"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 102,
       "text": [
        "False"
       ]
      }
     ],
     "prompt_number": 102
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "expand((A+B)**3)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$A B A + A \\left(B\\right)^{2} + \\left(A\\right)^{2} B + \\left(A\\right)^{3} + B A B + B \\left(A\\right)^{2} + \\left(B\\right)^{2} A + \\left(B\\right)^{3}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAloAAAAaBAMAAACdsu+cAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRK5CYII=\n",
       "prompt_number": 103,
       "text": [
        "           2    2      3              2    2      3\n",
        "A\u22c5B\u22c5A + A\u22c5B  + A \u22c5B + A  + B\u22c5A\u22c5B + B\u22c5A  + B \u22c5A + B "
       ]
      }
     ],
     "prompt_number": 103
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "c = Commutator(A,B)\n",
      "c"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[A,B\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAADEAAAAUBAMAAADW08iqAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAdt3NMolEZhAiVJmr\nu+8o6X5NAAABA0lEQVQoFWMQMmHAAtiNEhjCIOLOMGn283/OnL3GwAqXsYfJMPBfYGDY7wCXYddv\ngEn5CzAw5G+Ay8zaPwEmIw9k1QvAZQTkE2Ay9xgYOB41wGTYGtYXwGT+MDA4nYC7oI0BZDoYsPwR\nlNVigMskMMzfAJVh/MnA0HcAJsNy5sz5C1AZ3gcMDIy/YDJsDAx8BlAZ/gNAma8wmQUMDExApWDg\nX8DAwPMNKsO5gIGB+QNUBuQd/wtQmSagIO9PBo7PIEk7Bga2tw0Qmd7/Fgxc9z9tYACF0OH/Z87c\nXYBwNUgxUC3QIBiAhQGEzw4TBtKoMu04ZYDuhQOgHiEVOA+JwaiUAADdR0Tn9afoOgAAAABJRU5E\nrkJggg==\n",
       "prompt_number": 104,
       "text": [
        "[A,B]"
       ]
      }
     ],
     "prompt_number": 104
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can use the `doit` method to evaluate the commutator:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "c.doit()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$A B - B A$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFkAAAAOBAMAAAC2rq3TAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMs1m3RAiVJl2RKu7\n74l6DlUiAAABUklEQVQoFV2Sv0rDUBTGv7RWTVupiHt0cJR2d3BRHBwM4iQWujnqG5g3aNeCkLyA\nWAQHiZAMiiCKGdx08BEUxX8R63fuTdLEMxy+79zfvfecmwASM5KWhr5/44kaRTmI/fNr7RUksqn8\nDzD5oley3AiB7rSyGgLKGza98QVUPzNOC6sHHDuiNUSx2+0wm+8scUch3A6wzB0pRNFzPeZ6BLRX\nKfJxC1Tu5eYE4vX2QYu2MYCxrhZyeAzMnopPIGAb0h2ss/7Ffg4UacT9wzVVSyDAw57DClusPKiV\nUZJZ2gPxCQTD94OQni3ClcM3VyTmqdQs5gdFCqEKTM2xwBbhPjHlgrPA/GUhhcDjxiLufmUx8HIo\npdUCat8UCYRxitKzfm6Dn7MQ8txWiAzCFpfrHKYUAUeXBRZYZA+PNjJoZ7iAibs3p9bk/3P1Dz6R\nH413p9Af6+RkvbEYuuwAAAAASUVORK5CYII=\n",
       "prompt_number": 105,
       "text": [
        "A\u22c5B - B\u22c5A"
       ]
      }
     ],
     "prompt_number": 105
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can mix quantum operators with C-numbers:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "c = Commutator(a * A, b * B)\n",
      "c"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\alpha \\beta \\left[A,B\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFEAAAAUBAMAAAAHNMzaAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYII=\n",
       "prompt_number": 106,
       "text": [
        "\u03b1\u22c5\u03b2\u22c5[A,B]"
       ]
      }
     ],
     "prompt_number": 106
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "To expand the commutator, use the `expand` method with the `commutator=True` keyword argument:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "c = Commutator(A+B, A*B)\n",
      "c.expand(commutator=True)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- \\left[A,B\\right] B + A \\left[A,B\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAALwAAAAUBAMAAADFKEobAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nJOk/GEPK4QFPhpoAAAAASUVORK5CYII=\n",
       "prompt_number": 107,
       "text": [
        "-[A,B]\u22c5B + A\u22c5[A,B]"
       ]
      }
     ],
     "prompt_number": 107
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Dagger(Commutator(A, B))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- \\left[A^{\\dagger},B^{\\dagger}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFsAAAAaBAMAAAAqHD1jAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 108,
       "text": [
        " \u23a1 \u2020  \u2020\u23a4\n",
        "-\u23a3A ,B \u23a6"
       ]
      }
     ],
     "prompt_number": 108
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "ac = AntiCommutator(A,B)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 109
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "ac.doit()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$A B + B A$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFkAAAAQBAMAAACPcs44AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 110,
       "text": [
        "A\u22c5B + B\u22c5A"
       ]
      }
     ],
     "prompt_number": 110
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "#### Example: Quadrature commutator"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's look at the commutator of the electromagnetic field quadatures $x$ and $p$. We can write the quadrature operators in terms of the creation and annihilation operators as:\n",
      "\n",
      "$\\displaystyle x = (a + a^\\dagger)/\\sqrt{2}$\n",
      "\n",
      "$\\displaystyle p = -i(a - a^\\dagger)/\\sqrt{2}$\n"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "X = (A + Dagger(A))/sqrt(2)\n",
      "X"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{\\sqrt{2}}{2} \\left(A^{\\dagger} + A\\right)$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAHQAAAAvBAMAAADA/fMgAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAInarRM2ZVBDdiWbv\nuzJCz3LGAAACh0lEQVRIDZ1VPWgUURCe3du7273gcp1YyG0lghaBCMZGD2wsrzlQEG4rFSGa6jpx\n0crKVGIh/jQhWB1YaZUiQQSLNGolCdgqqESNkeScN+9n3ktWs+8e3Mw3833fvf0fALU6Y7F05ZVP\nealtcdCzKy/cmPeS2+J7duGHr0Jw5eOGn0eq4wW4C63NSaytEcwCvJ3Eiqf6BuDcxgTeEwCDwtsa\ndXGrBdruSUGpcjiJD0OYCXm0XdlEwuAOXteEYLLmZ4V4B+AVeeY8nQA32/BQmMLpEuv/34pOPvVU\nmIYQtKU57cqM8V/WjBS1raRAkOYQKusX6lNYYuig81Sl20ORjzxfPKPYiyoDvHv/2mAH4OMn1mWS\nDsbj35JtZDJjLJYKg11Alwdu5263U1Cd9jCddSkmjhGsq1PUqkUJ6l8xf9BNKxPRsRoM1em93MKW\ntsbW3xORjNhgUCosAMHwG8bThAGamQKaaKyZBoOpZcJhKN58fV8tqyRCqSKpCY0ewQvNP5hvqLZl\nlUQkj824CByiQ4m76Xcsn+2zKiLddV1U1adFiiD4iek4texzVUSwCTQvTBDCJBfx08wMvlJQCIyL\nD1gRwS9JOFHu2gVY1e1Wv39ptd/PqVYE7rp/kbWJ/UcWZ3bVRCpu3d5VE5fpMP4GFmOsmii9wuKO\n4ScdYL3NXm01hLr7rBAI/y9e/1HArfEDJpSVCf00ufNmhR0G6V1No55J6M6b64ZnEI0YE7pfyMas\nM28+71GVli9U1503tflSsdt8rEp33sQ9V1VWyY+oZOx5Q5+5MgP3jjJ05k3z4CO+xlbvecPWOYae\nqHTeVPsPnjfV9KxKed5wsxqy5k01A6t43nDvQPQXXiaau7w+Ga4AAAAASUVORK5CYII=\n",
       "prompt_number": 111,
       "text": [
        "  ___ \u239b \u2020    \u239e\n",
        "\u2572\u2571 2 \u22c5\u239dA  + A\u23a0\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "      2       "
       ]
      }
     ],
     "prompt_number": 111
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "P = -I * (A - Dagger(A))/sqrt(2)\n",
      "P"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- \\frac{\\sqrt{2} i}{2} \\left(- A^{\\dagger} + A\\right)$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJwAAAAvBAMAAADkyL6lAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ndOHBcPuUbqIRut43IPcP+22rFtFy7pEAAAAASUVORK5CYII=\n",
       "prompt_number": 112,
       "text": [
        "   ___   \u239b   \u2020    \u239e \n",
        "-\u2572\u2571 2 \u22c5\u2148\u22c5\u239d- A  + A\u23a0 \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "         2          "
       ]
      }
     ],
     "prompt_number": 112
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's expand the commutator $[x,p]$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Commutator(X, P).expand(commutator=True).expand(commutator=True)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$- i \\left[A^{\\dagger},A\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFkAAAAaBAMAAAAu6e1eAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nXKzbrvcJrthN86f5xysAAAAASUVORK5CYII=\n",
       "prompt_number": 113,
       "text": [
        "   \u23a1 \u2020  \u23a4\n",
        "-\u2148\u22c5\u23a3A ,A\u23a6"
       ]
      }
     ],
     "prompt_number": 113
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Here we see directly that the well known commutation relation for the quadratures\n",
      "\n",
      "$[x,p]=i$\n",
      "\n",
      "is a directly related to\n",
      "\n",
      "$[A, A^\\dagger]=1$ \n",
      "\n",
      "(which SymPy does not know about, and does not simplify)."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "For more details on the quantum module in SymPy, see:\n",
      "\n",
      "* http://docs.sympy.org/0.7.2/modules/physics/quantum/index.html\n",
      "* http://nbviewer.ipython.org/urls/raw.github.com/ipython/ipython/master/docs/examples/notebooks/sympy_quantum_computing.ipynb"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "## Further reading\n",
      "\n",
      "* http://sympy.org/en/index.html - The SymPy projects web page.\n",
      "* https://github.com/sympy/sympy - The source code of SymPy.\n",
      "* http://live.sympy.org - Online version of SymPy for testing and demonstrations.\n"
     ]
    }
   ],
   "metadata": {}
  }
 ]
}