07-Sympy.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Introducci&oacute;n a la Computaci&oacute;n Cient&iacute;fica con Python\n",
    "\n",
    "Adaptado a partir de la versi&oacute;n original en ingl&eacute;s de [J.R. Johansson](http://jrjohansson.github.io/) (robert@riken.jp), por [G.F. Rubilar](https://gfrubi.github.io).\n",
    "\n",
    "La &uacute;ltima versi&oacute;n del original (en ingl&eacute;s) est&aacute; disponible en [http://github.com/jrjohansson/scientific-python-lectures](http://github.com/jrjohansson/scientific-python-lectures)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## SymPy"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Introducci&oacute;n\n",
    "\n",
    "[SymPy](http://sympy.org/en/index.html) es un m&oacute;dulo de matem&aacute;tica simb&oacute;lica para Pytho. Tiene el objetivo de ofrecer un \"sistema de &aacute;lgebra computacional\" (CAS) completo. SymPy est&aacute; escrito completamente en Python.\n",
    "\n",
    "Para comenzar a usar SymPy en un programa Python o en un notebook, importamos el m&oacute;dulo `sympy`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy as sp"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para que los resultados sean formateados en $\\LaTeX$ podemos usar:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Variables simb&oacute;licas\n",
    "\n",
    "En SymPy, los objetos fundamentales con los que trabajaremos son *s&iacute;mbolos* para las variables con las que deseamos realizar operaciones. Podemos crear un nuevo s&iacute;mbolo usando la clase `Symbol`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = sp.Symbol('x')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "(x + pi)**2"
      ],
      "text/latex": "$\\displaystyle \\left(x + \\pi\\right)^{2}$"
     },
     "metadata": {},
     "execution_count": 3
    }
   ],
   "source": [
    "(sp.pi + x)**2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Tambi&eacute;n podemos definir varios s&iacute;mbolos simult&aacute;neamente, usando la funci&oacute;n `symbols`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "a, b, c = sp.symbols(\"a, b, c\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "sympy.core.symbol.Symbol"
      ]
     },
     "metadata": {},
     "execution_count": 5
    }
   ],
   "source": [
    "type(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Podemos agregar algunas propiedades a los s&iacute;mbolos cuando son creados:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = sp.Symbol('x', real=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "metadata": {},
     "execution_count": 7
    }
   ],
   "source": [
    "x.is_imaginary"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = sp.Symbol('x', positive=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "True"
      ],
      "text/latex": "$\\displaystyle \\text{True}$"
     },
     "metadata": {},
     "execution_count": 9
    }
   ],
   "source": [
    "x > 0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### N&uacute;meros complejos\n",
    "\n",
    "La unidad imaginaria es denotada por `I` en SymPy. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "2 + 3*I"
      ],
      "text/latex": "$\\displaystyle 2 + 3 i$"
     },
     "metadata": {},
     "execution_count": 10
    }
   ],
   "source": [
    "2+3*sp.I"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "-1"
      ],
      "text/latex": "$\\displaystyle -1$"
     },
     "metadata": {},
     "execution_count": 11
    }
   ],
   "source": [
    "sp.I**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "(I*x + 1)**2"
      ],
      "text/latex": "$\\displaystyle \\left(i x + 1\\right)^{2}$"
     },
     "metadata": {},
     "execution_count": 12
    }
   ],
   "source": [
    "(x*sp.I + 1)**2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### N&uacute;meros racionales\n",
    "\n",
    "Existen tres tipos distintos de n&uacute;meros en SymPy: `Real`, `Rational`, `Integer`: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "r1 = sp.Rational(4,5)\n",
    "r2 = sp.Rational(5,4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "4/5"
      ],
      "text/latex": "$\\displaystyle \\frac{4}{5}$"
     },
     "metadata": {},
     "execution_count": 14
    }
   ],
   "source": [
    "r1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "41/20"
      ],
      "text/latex": "$\\displaystyle \\frac{41}{20}$"
     },
     "metadata": {},
     "execution_count": 15
    }
   ],
   "source": [
    "r1 + r2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "16/25"
      ],
      "text/latex": "$\\displaystyle \\frac{16}{25}$"
     },
     "metadata": {},
     "execution_count": 16
    }
   ],
   "source": [
    "r1/r2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Evaluaci&oacute;n num&eacute;rica\n",
    "\n",
    "SymPy usa una biblioteca para trabajar con n&uacute;meros con precisi&oacute;n arbitraria, y tiene expresiones SymPy  predefinidas para varias constantes matem&aacute;ticas, tales como: `pi`, `e` y `oo` para el infinito.\n",
    "\n",
    "Para evaluar num&eacute;ricamente una expresi&oacute;n podemos usar la funci&oacute;n `evalf` (o bien `N`). &Eacute;stas usan un argumento `n` que especif&iacute;ca el n&uacute;mero de cifras significativas."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "3.1415926535897932384626433832795028841971693993751"
      ],
      "text/latex": "$\\displaystyle 3.1415926535897932384626433832795028841971693993751$"
     },
     "metadata": {},
     "execution_count": 17
    }
   ],
   "source": [
    "sp.pi.evalf(n=50)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "(x + pi)**2"
      ],
      "text/latex": "$\\displaystyle \\left(x + \\pi\\right)^{2}$"
     },
     "metadata": {},
     "execution_count": 18
    }
   ],
   "source": [
    "y = (x + sp.pi)**2\n",
    "y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "9.8696*(0.31831*x + 1)**2"
      ],
      "text/latex": "$\\displaystyle 9.8696 \\left(0.31831 x + 1\\right)^{2}$"
     },
     "metadata": {},
     "execution_count": 19
    }
   ],
   "source": [
    "sp.N(y, 5) # equivalente a evalf"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Cuando evaluamos num&eacute;ricamente expresiones a menudo deseamos substituir un s&iacute;mbolo por un valor num&eacute;rico. En SymPy hacemos esto usando la funci&oacute;n `subs`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "(1.5 + pi)**2"
      ],
      "text/latex": "$\\displaystyle \\left(1.5 + \\pi\\right)^{2}$"
     },
     "metadata": {},
     "execution_count": 20
    }
   ],
   "source": [
    "y.subs(x, 1.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "La funci&oacute;n `subs` tambi&eacute;n puede ser usada para substituir s&iacute;mbolos y expresiones:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "(a + 2*pi)**2"
      ],
      "text/latex": "$\\displaystyle \\left(a + 2 \\pi\\right)^{2}$"
     },
     "metadata": {},
     "execution_count": 21
    }
   ],
   "source": [
    "y.subs(x, a+sp.pi)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Manipulaciones algebr&aacute;icas\n",
    "\n",
    "Uno de los usos principales de un sistema de c&aacute;lculo simb&oacute;lico es realizar manipulaciones algebr&aacute;icas de expresiones. Por ejemplo, si queremos expandir un producto, factorizar una expresi&oacute;n, o simplificar un resultado. En esta secci&oacute;n presentamos las funciones para realizar estas operaciones b&aacute;sicas en SymPy."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Expand y factor\n",
    "\n",
    "Primeros pasos en la manipulaci&oacute;n algebr&aacute;ica"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "(x + 1)*(x + 2)*(x + 3)"
      ],
      "text/latex": "$\\displaystyle \\left(x + 1\\right) \\left(x + 2\\right) \\left(x + 3\\right)$"
     },
     "metadata": {},
     "execution_count": 22
    }
   ],
   "source": [
    "(x+1)*(x+2)*(x+3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "x**3 + 6*x**2 + 11*x + 6"
      ],
      "text/latex": "$\\displaystyle x^{3} + 6 x^{2} + 11 x + 6$"
     },
     "metadata": {},
     "execution_count": 23
    }
   ],
   "source": [
    "sp.expand((x+1)*(x+2)*(x+3))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "La funci&oacute;n `expand` acepta varios argumentos con los que se puede indicar qu&eacute; tipo de expansi&oacute;n deseamos realizar. Por ejemplo, para expandir expresiones trigonom&eacute;tricas, usamos el argumento clave `trig=True`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "sin(a + b)"
      ],
      "text/latex": "$\\displaystyle \\sin{\\left(a + b \\right)}$"
     },
     "metadata": {},
     "execution_count": 24
    }
   ],
   "source": [
    "sp.sin(a+b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "sin(a)*cos(b) + sin(b)*cos(a)"
      ],
      "text/latex": "$\\displaystyle \\sin{\\left(a \\right)} \\cos{\\left(b \\right)} + \\sin{\\left(b \\right)} \\cos{\\left(a \\right)}$"
     },
     "metadata": {},
     "execution_count": 25
    }
   ],
   "source": [
    "sp.expand(sp.sin(a+b), trig=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Ver `help(expand)` para una descripci&oacute;n detallada de los distintos tipos de expansiones que la funci&oacute;n `expand` puede realizar."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Tambi&eacute;n podemos factorizar expresiones, usando la funci&oacute;n `factor` de SymPy: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "(x + 1)*(x + 2)*(x + 3)"
      ],
      "text/latex": "$\\displaystyle \\left(x + 1\\right) \\left(x + 2\\right) \\left(x + 3\\right)$"
     },
     "metadata": {},
     "execution_count": 26
    }
   ],
   "source": [
    "sp.factor(x**3 + 6*x**2 + 11*x + 6)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Simplify\n",
    "\n",
    "La funci&oacute;n `simplify` intenta simplificar una expresi&oacute;n usando distinta t&eacute;cnicas. Existen tambi&eacute;n alternativas m&aacute;s espec&iacute;ficas a la funci&oacute;n `simplify`: `trigsimp`, `powsimp`, `logcombine`, etc. \n",
    "\n",
    "El uso b&aacute;sico de estas funciones en el siguiente:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "x**2 - 1"
      ],
      "text/latex": "$\\displaystyle x^{2} - 1$"
     },
     "metadata": {},
     "execution_count": 27
    }
   ],
   "source": [
    "# simplify expande un producto\n",
    "sp.simplify((x+1)*(x-1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "1"
      ],
      "text/latex": "$\\displaystyle 1$"
     },
     "metadata": {},
     "execution_count": 28
    }
   ],
   "source": [
    "# simplify usa identidades trigonometricas\n",
    "sp.simplify(sp.sin(a)**2 + sp.cos(a)**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "tan(x)"
      ],
      "text/latex": "$\\displaystyle \\tan{\\left(x \\right)}$"
     },
     "metadata": {},
     "execution_count": 29
    }
   ],
   "source": [
    "sp.simplify(sp.sin(x)/sp.cos(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## apart and together\n",
    "\n",
    "Podemos tambi&eacute;n manipular expresiones simb&oacute;licas que involucran fracciones usando las funciones `apart` y `together`. La primera de estas funciones separa una fracci&oacute;n en sus correspondientes fracciones parciales; la segunda hace todo lo contrario."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "f1 = 1/((a+1)*(a+2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "1/((a + 1)*(a + 2))"
      ],
      "text/latex": "$\\displaystyle \\frac{1}{\\left(a + 1\\right) \\left(a + 2\\right)}$"
     },
     "metadata": {},
     "execution_count": 31
    }
   ],
   "source": [
    "f1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "-1/(a + 2) + 1/(a + 1)"
      ],
      "text/latex": "$\\displaystyle - \\frac{1}{a + 2} + \\frac{1}{a + 1}$"
     },
     "metadata": {},
     "execution_count": 32
    }
   ],
   "source": [
    "sp.apart(f1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [],
   "source": [
    "f2 = 1/(a+2) + 1/(a+3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "1/(a + 3) + 1/(a + 2)"
      ],
      "text/latex": "$\\displaystyle \\frac{1}{a + 3} + \\frac{1}{a + 2}$"
     },
     "metadata": {},
     "execution_count": 34
    }
   ],
   "source": [
    "f2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "(2*a + 5)/((a + 2)*(a + 3))"
      ],
      "text/latex": "$\\displaystyle \\frac{2 a + 5}{\\left(a + 2\\right) \\left(a + 3\\right)}$"
     },
     "metadata": {},
     "execution_count": 35
    }
   ],
   "source": [
    "sp.together(f2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Sumas y productos\n",
    "\n",
    "Podemos evaluar sumatorias y productos usando las funciones `Sum` y `Product`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = sp.Symbol(\"n\", integer=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "Sum(n**(-2), (n, 1, 10))"
      ],
      "text/latex": "$\\displaystyle \\sum_{n=1}^{10} \\frac{1}{n^{2}}$"
     },
     "metadata": {},
     "execution_count": 37
    }
   ],
   "source": [
    "suma = sp.Sum(1/n**2, (n, 1, 10))\n",
    "suma"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "1968329/1270080"
      ],
      "text/latex": "$\\displaystyle \\frac{1968329}{1270080}$"
     },
     "metadata": {},
     "execution_count": 38
    }
   ],
   "source": [
    "suma.doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "1.54976773116654"
      ],
      "text/latex": "$\\displaystyle 1.54976773116654$"
     },
     "metadata": {},
     "execution_count": 39
    }
   ],
   "source": [
    "suma.evalf()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "Sum(n**(-2), (n, 1, oo))"
      ],
      "text/latex": "$\\displaystyle \\sum_{n=1}^{\\infty} \\frac{1}{n^{2}}$"
     },
     "metadata": {},
     "execution_count": 40
    }
   ],
   "source": [
    "suma2 = sp.Sum(1/n**2, (n, 1, sp.oo))\n",
    "suma2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "1.64493406684823"
      ],
      "text/latex": "$\\displaystyle 1.64493406684823$"
     },
     "metadata": {},
     "execution_count": 41
    }
   ],
   "source": [
    "suma2.evalf()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "Sum(x**n/factorial(n), (n, 0, oo))"
      ],
      "text/latex": "$\\displaystyle \\sum_{n=0}^{\\infty} \\frac{x^{n}}{n!}$"
     },
     "metadata": {},
     "execution_count": 42
    }
   ],
   "source": [
    "suma3 = sp.Sum(x**n/(sp.factorial(n)),(n,0,sp.oo))\n",
    "suma3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "exp(x)"
      ],
      "text/latex": "$\\displaystyle e^{x}$"
     },
     "metadata": {},
     "execution_count": 43
    }
   ],
   "source": [
    "suma3.doit()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Productos funcionan de manera similar:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "Product(n, (n, 1, 10))"
      ],
      "text/latex": "$\\displaystyle \\prod_{n=1}^{10} n$"
     },
     "metadata": {},
     "execution_count": 44
    }
   ],
   "source": [
    "producto = sp.Product(n, (n, 1, 10)) # 10!\n",
    "producto"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "3628800"
      ],
      "text/latex": "$\\displaystyle 3628800$"
     },
     "metadata": {},
     "execution_count": 45
    }
   ],
   "source": [
    "producto.doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "3628800.00000000"
      ],
      "text/latex": "$\\displaystyle 3628800.0$"
     },
     "metadata": {},
     "execution_count": 46
    }
   ],
   "source": [
    "producto.evalf()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### L&iacute;mites\n",
    "\n",
    "SymPy permite evaluar l&iacute;mites usando la funci&oacute;n `limit`. Por ejemplo, "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "1"
      ],
      "text/latex": "$\\displaystyle 1$"
     },
     "metadata": {},
     "execution_count": 47
    }
   ],
   "source": [
    "sp.limit(sp.sin(x)/x, x, 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Tambi&eacute;n es posible calcular l&iacute;mites \"por la derecha\" y \"por la izquierda\" usando el argumento opcional `dir`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "oo"
      ],
      "text/latex": "$\\displaystyle \\infty$"
     },
     "metadata": {},
     "execution_count": 48
    }
   ],
   "source": [
    "sp.limit(1/x, x, 0, dir=\"+\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "-oo"
      ],
      "text/latex": "$\\displaystyle -\\infty$"
     },
     "metadata": {},
     "execution_count": 49
    }
   ],
   "source": [
    "sp.limit(1/x, x, 0, dir=\"-\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## C&aacute;lculo\n",
    "\n",
    "Adem&aacute;s de realizar manipulaciones algebr&aacute;icas, SimPy puede realizar operaciones de c&aacute;lculo, tales como derivar y derivar expresiones."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Derivaci&oacute;n\n",
    "\n",
    "Derviar es usualmente algo simple. Usamos la funci&oacute;n `diff`. El primer argumento es una expresi&oacute;n que ser&aacute; derivada, y el segundo argumento es el s&iacute;mbolo respecto al cual se realizar&aacute; la derivada:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "(x + pi)**4"
      ],
      "text/latex": "$\\displaystyle \\left(x + \\pi\\right)^{4}$"
     },
     "metadata": {},
     "execution_count": 50
    }
   ],
   "source": [
    "y**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "4*(x + pi)**3"
      ],
      "text/latex": "$\\displaystyle 4 \\left(x + \\pi\\right)^{3}$"
     },
     "metadata": {},
     "execution_count": 51
    }
   ],
   "source": [
    "sp.diff(y**2, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Para calcular derivadas de orden superior podemos usar:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "12*(x + pi)**2"
      ],
      "text/latex": "$\\displaystyle 12 \\left(x + \\pi\\right)^{2}$"
     },
     "metadata": {},
     "execution_count": 52
    }
   ],
   "source": [
    "sp.diff(y**2, x, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "o bien"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "12*(x + pi)**2"
      ],
      "text/latex": "$\\displaystyle 12 \\left(x + \\pi\\right)^{2}$"
     },
     "metadata": {},
     "execution_count": 53
    }
   ],
   "source": [
    "sp.diff(y**2, x, 2) # hace lo mismo"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Podemos calcular la derivada de una expresi&oacute;n que depende de m&aacute;s de una variable:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [],
   "source": [
    "x, y, z = sp.symbols(\"x,y,z\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = sp.sin(x*y) + sp.cos(y*z)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Entonces, la dervidada \n",
    "\n",
    "$$\n",
    "\\frac{\\partial^3f}{\\partial x\\partial y^2},\n",
    "$$\n",
    "puede calcularse como"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "-x*(x*y*cos(x*y) + 2*sin(x*y))"
      ],
      "text/latex": "$\\displaystyle - x \\left(x y \\cos{\\left(x y \\right)} + 2 \\sin{\\left(x y \\right)}\\right)$"
     },
     "metadata": {},
     "execution_count": 56
    }
   ],
   "source": [
    "sp.diff(f, x, 1, y, 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Integraci&oacute;n\n",
    "\n",
    "La integraci&oacute;n se realiza de forma similar:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = x*sp.sin(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "-x*cos(x) + sin(x)"
      ],
      "text/latex": "$\\displaystyle - x \\cos{\\left(x \\right)} + \\sin{\\left(x \\right)}$"
     },
     "metadata": {},
     "execution_count": 58
    }
   ],
   "source": [
    "sp.integrate(f, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Tambi&eacute;n podemos evaluar integrales definidas, agregando la informaci&oacute;n de los l&iacute;mites de integraci&oacute;n:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "2*pi"
      ],
      "text/latex": "$\\displaystyle 2 \\pi$"
     },
     "metadata": {},
     "execution_count": 59
    }
   ],
   "source": [
    "sp.integrate(f, (x, -sp.pi, sp.pi))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Esto incluye el caso en que los l&iacute;mites son infinitos"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "sqrt(pi)"
      ],
      "text/latex": "$\\displaystyle \\sqrt{\\pi}$"
     },
     "metadata": {},
     "execution_count": 60
    }
   ],
   "source": [
    "sp.integrate(sp.exp(-x**2), (x, -sp.oo, sp.oo))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "Piecewise((x**(n + 1)/(n + 1), Ne(n, -1)), (log(x), True))"
      ],
      "text/latex": "$\\displaystyle \\begin{cases} \\frac{x^{n + 1}}{n + 1} & \\text{for}\\: n \\neq -1 \\\\\\log{\\left(x \\right)} & \\text{otherwise} \\end{cases}$"
     },
     "metadata": {},
     "execution_count": 61
    }
   ],
   "source": [
    "n = sp.Symbol('n', int=True)\n",
    "resultado = sp.integrate(x**n,x)\n",
    "resultado"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Series\n",
    "\n",
    "SymPy puede realizar expansiones en series de una expresi&oacute;n, usando la funci&oacute;n `series`:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Series de Fourier"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "FourierSeries(Heaviside(x), (x, -pi, pi), (1/2, SeqFormula(Piecewise((pi, Eq(_n, 0)), (sin(_n*pi)/_n, True))*cos(_n*x)/pi, (_n, 1, oo)), SeqFormula((-Piecewise((0, Eq(_n, 0)), (-1/_n, True)) + Piecewise((0, Eq(_n, 0)), (-cos(_n*pi)/_n, True)))*sin(_n*x)/pi, (_n, 1, oo))))"
      ],
      "text/latex": "$\\displaystyle \\frac{2 \\sin{\\left(x \\right)}}{\\pi} + \\frac{2 \\sin{\\left(3 x \\right)}}{3 \\pi} + \\frac{1}{2} + \\ldots$"
     },
     "metadata": {},
     "execution_count": 62
    }
   ],
   "source": [
    "sp.fourier_series(sp.Heaviside(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "2*sin(x)/pi + 2*sin(3*x)/(3*pi) + 2*sin(5*x)/(5*pi) + 2*sin(7*x)/(7*pi) + 2*sin(9*x)/(9*pi) + 2*sin(11*x)/(11*pi) + 2*sin(13*x)/(13*pi) + 2*sin(15*x)/(15*pi) + 2*sin(17*x)/(17*pi) + 1/2"
      ],
      "text/latex": "$\\displaystyle \\frac{2 \\sin{\\left(x \\right)}}{\\pi} + \\frac{2 \\sin{\\left(3 x \\right)}}{3 \\pi} + \\frac{2 \\sin{\\left(5 x \\right)}}{5 \\pi} + \\frac{2 \\sin{\\left(7 x \\right)}}{7 \\pi} + \\frac{2 \\sin{\\left(9 x \\right)}}{9 \\pi} + \\frac{2 \\sin{\\left(11 x \\right)}}{11 \\pi} + \\frac{2 \\sin{\\left(13 x \\right)}}{13 \\pi} + \\frac{2 \\sin{\\left(15 x \\right)}}{15 \\pi} + \\frac{2 \\sin{\\left(17 x \\right)}}{17 \\pi} + \\frac{1}{2}$"
     },
     "metadata": {},
     "execution_count": 63
    }
   ],
   "source": [
    "sp.fourier_series(sp.Heaviside(x)).truncate(10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Transformada de Fourier"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "DiracDelta(-a + x)"
      ],
      "text/latex": "$\\displaystyle \\delta\\left(- a + x\\right)$"
     },
     "metadata": {},
     "execution_count": 64
    }
   ],
   "source": [
    "f = sp.DiracDelta(x-a)\n",
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "exp(-2*I*pi*a*k)"
      ],
      "text/latex": "$\\displaystyle e^{- 2 i \\pi a k}$"
     },
     "metadata": {},
     "execution_count": 65
    }
   ],
   "source": [
    "k = sp.Symbol('k', real=True)\n",
    "sp.fourier_transform(f,x,k)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "1 + x + x**2/2 + x**3/6 + x**4/24 + x**5/120 + O(x**6)"
      ],
      "text/latex": "$\\displaystyle 1 + x + \\frac{x^{2}}{2} + \\frac{x^{3}}{6} + \\frac{x^{4}}{24} + \\frac{x^{5}}{120} + O\\left(x^{6}\\right)$"
     },
     "metadata": {},
     "execution_count": 66
    }
   ],
   "source": [
    "sp.series(sp.exp(x), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Por defecto, la expansi&oacute;n se realiza en torno a cero ($x=0$), pero podemos tambi&eacute;n expandir en torno a otro valor, incluy&eacute;ndolo en los argumentos de la funci&oacute;n:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "E + E*(x - 1) + E*(x - 1)**2/2 + E*(x - 1)**3/6 + E*(x - 1)**4/24 + E*(x - 1)**5/120 + O((x - 1)**6, (x, 1))"
      ],
      "text/latex": "$\\displaystyle e + e \\left(x - 1\\right) + \\frac{e \\left(x - 1\\right)^{2}}{2} + \\frac{e \\left(x - 1\\right)^{3}}{6} + \\frac{e \\left(x - 1\\right)^{4}}{24} + \\frac{e \\left(x - 1\\right)^{5}}{120} + O\\left(\\left(x - 1\\right)^{6}; x\\rightarrow 1\\right)$"
     },
     "metadata": {},
     "execution_count": 67
    }
   ],
   "source": [
    "sp.series(sp.exp(x), x, 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Adem&aacute;s podemos definir expl&iacute;citamente el orden hasta el que se realiza la expansi&oacute;n:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "1 + x + x**2/2 + x**3/6 + x**4/24 + x**5/120 + x**6/720 + x**7/5040 + x**8/40320 + x**9/362880 + O(x**10)"
      ],
      "text/latex": "$\\displaystyle 1 + x + \\frac{x^{2}}{2} + \\frac{x^{3}}{6} + \\frac{x^{4}}{24} + \\frac{x^{5}}{120} + \\frac{x^{6}}{720} + \\frac{x^{7}}{5040} + \\frac{x^{8}}{40320} + \\frac{x^{9}}{362880} + O\\left(x^{10}\\right)$"
     },
     "metadata": {},
     "execution_count": 68
    }
   ],
   "source": [
    "sp.series(sp.exp(x), x, 0, 10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "La funci&oacute;n `series` define un tipo especial de objeto en SymPy (sympy.core.add.Add) que incluye la informaci&oacute;n del orden hasta el que se ha realizado la expansi&oacute;n. Esto es &uacute;til para llevar el registro de estos &oacute;rdenes en un c&aacute;lculo:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "1 - x**2/2 + x**4/24 + O(x**5)"
      ],
      "text/latex": "$\\displaystyle 1 - \\frac{x^{2}}{2} + \\frac{x^{4}}{24} + O\\left(x^{5}\\right)$"
     },
     "metadata": {},
     "execution_count": 69
    }
   ],
   "source": [
    "s1 = sp.cos(x).series(x, 0, 5)\n",
    "s1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "x + O(x**2)"
      ],
      "text/latex": "$\\displaystyle x + O\\left(x^{2}\\right)$"
     },
     "metadata": {},
     "execution_count": 70
    }
   ],
   "source": [
    "s2 = sp.sin(x).series(x, 0, 2)\n",
    "s2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "x + O(x**2)"
      ],
      "text/latex": "$\\displaystyle x + O\\left(x^{2}\\right)$"
     },
     "metadata": {},
     "execution_count": 71
    }
   ],
   "source": [
    "sp.expand(s1 * s2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Si queremos eliminar la informaci&oacute;n del orden de expansi&oacute;n y s&oacute;lo quedarnos con los t&eacute;rminos calculados, podemos usar el m&eacute;todo `removeO`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "x**4/24 - x**2/2 + 1"
      ],
      "text/latex": "$\\displaystyle \\frac{x^{4}}{24} - \\frac{x^{2}}{2} + 1$"
     },
     "metadata": {},
     "execution_count": 72
    }
   ],
   "source": [
    "s1.removeO()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Funciones especiales\n",
    "\n",
    "Ver [documentaci&oacute;n](https://docs.sympy.org/latest/modules/functions/special.html)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Polinomios de Legendre"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "x"
      ],
      "text/latex": "$\\displaystyle x$"
     },
     "metadata": {},
     "execution_count": 73
    }
   ],
   "source": [
    "sp.legendre(1,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "1",
      "text/latex": "$\\displaystyle 1$"
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "x",
      "text/latex": "$\\displaystyle x$"
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "3*x**2/2 - 1/2",
      "text/latex": "$\\displaystyle \\frac{3 x^{2}}{2} - \\frac{1}{2}$"
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "5*x**3/2 - 3*x/2",
      "text/latex": "$\\displaystyle \\frac{5 x^{3}}{2} - \\frac{3 x}{2}$"
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "35*x**4/8 - 15*x**2/4 + 3/8",
      "text/latex": "$\\displaystyle \\frac{35 x^{4}}{8} - \\frac{15 x^{2}}{4} + \\frac{3}{8}$"
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "63*x**5/8 - 35*x**3/4 + 15*x/8",
      "text/latex": "$\\displaystyle \\frac{63 x^{5}}{8} - \\frac{35 x^{3}}{4} + \\frac{15 x}{8}$"
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "231*x**6/16 - 315*x**4/16 + 105*x**2/16 - 5/16",
      "text/latex": "$\\displaystyle \\frac{231 x^{6}}{16} - \\frac{315 x^{4}}{16} + \\frac{105 x^{2}}{16} - \\frac{5}{16}$"
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "429*x**7/16 - 693*x**5/16 + 315*x**3/16 - 35*x/16",
      "text/latex": "$\\displaystyle \\frac{429 x^{7}}{16} - \\frac{693 x^{5}}{16} + \\frac{315 x^{3}}{16} - \\frac{35 x}{16}$"
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "6435*x**8/128 - 3003*x**6/32 + 3465*x**4/64 - 315*x**2/32 + 35/128",
      "text/latex": "$\\displaystyle \\frac{6435 x^{8}}{128} - \\frac{3003 x^{6}}{32} + \\frac{3465 x^{4}}{64} - \\frac{315 x^{2}}{32} + \\frac{35}{128}$"
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": "12155*x**9/128 - 6435*x**7/32 + 9009*x**5/64 - 1155*x**3/32 + 315*x/128",
      "text/latex": "$\\displaystyle \\frac{12155 x^{9}}{128} - \\frac{6435 x^{7}}{32} + \\frac{9009 x^{5}}{64} - \\frac{1155 x^{3}}{32} + \\frac{315 x}{128}$"
     },
     "metadata": {}
    }
   ],
   "source": [
    "for n in range(10):\n",
    "    display(sp.legendre(n,x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "besselj(1, x)"
      ],
      "text/latex": "$\\displaystyle J_{1}\\left(x\\right)$"
     },
     "metadata": {},
     "execution_count": 75
    }
   ],
   "source": [
    "J1 = sp.besselj(1,x)\n",
    "J1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "x/2 - x**3/16 + x**5/384 - x**7/18432 + x**9/1474560 + O(x**10)"
      ],
      "text/latex": "$\\displaystyle \\frac{x}{2} - \\frac{x^{3}}{16} + \\frac{x^{5}}{384} - \\frac{x^{7}}{18432} + \\frac{x^{9}}{1474560} + O\\left(x^{10}\\right)$"
     },
     "metadata": {},
     "execution_count": 76
    }
   ],
   "source": [
    "J1.series(x,0,10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "besselj(0, x)/2 - besselj(2, x)/2"
      ],
      "text/latex": "$\\displaystyle \\frac{J_{0}\\left(x\\right)}{2} - \\frac{J_{2}\\left(x\\right)}{2}$"
     },
     "metadata": {},
     "execution_count": 77
    }
   ],
   "source": [
    "J1.diff(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [
    {
     "output_type": "display_data",
     "data": {
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\n"
     },
     "metadata": {
      "needs_background": "light"
     }
    },
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x7f4d8b770af0>"
      ]
     },
     "metadata": {},
     "execution_count": 78
    }
   ],
   "source": [
    "sp.plot(J1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [],
   "source": [
    "th, ph = sp.symbols('theta, varphi', real=True, positive=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "Ynm(1, 1, theta, varphi)"
      ],
      "text/latex": "$\\displaystyle Y_{1}^{1}\\left(\\theta,\\varphi\\right)$"
     },
     "metadata": {},
     "execution_count": 80
    }
   ],
   "source": [
    "sp.Ynm(1,1,th,ph)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "exp(-2*I*varphi)*Ynm(1, 1, theta, varphi)"
      ],
      "text/latex": "$\\displaystyle e^{- 2 i \\varphi} Y_{1}^{1}\\left(\\theta,\\varphi\\right)$"
     },
     "metadata": {},
     "execution_count": 81
    }
   ],
   "source": [
    "sp.Ynm(1,1,th,-ph)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "-sqrt(6)*exp(I*varphi)*sin(theta)/(4*sqrt(pi))"
      ],
      "text/latex": "$\\displaystyle - \\frac{\\sqrt{6} e^{i \\varphi} \\sin{\\left(\\theta \\right)}}{4 \\sqrt{\\pi}}$"
     },
     "metadata": {},
     "execution_count": 82
    }
   ],
   "source": [
    "sp.Ynm(1,1,th,ph).expand(func=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "Heaviside(x)"
      ],
      "text/latex": "$\\displaystyle \\theta\\left(x\\right)$"
     },
     "metadata": {},
     "execution_count": 83
    }
   ],
   "source": [
    "H = sp.Heaviside(x)\n",
    "H"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "output_type": "display_data",
     "data": {
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qapSeni6/36+VK1f2Wm+t1T333CO/36/MzEy99dZbHqQEIFFWiFOhUEilpaWqrq5WU1OTNm7cqKamph5jqqurFQgEFAgEVFFRobvvvtujtAAoK8Sluro6+f1+paWlKSkpSYWFhaqqquoxpqqqSosXL5YxRrNmzdLJkyd16NAhjxID8c1Ya/tb3+/K83HHr3ZEalPA13b06FEdP3Fc6VPSJUmHDx/Wqc9OabJ/cteYd997VxNSJ2j06NGSpMbGRqWlpWnUqFFht/nRJ39S86/v0eH9TWHXAwjLDGpQf2WVl5dnjx07FpE0+w6e1EUXXRSRbUXTmTNnYjK3RPb+tLefUXsopBHDR3TcXvsZfRUKaXjnsiSdPv25kpKGKyEhQZL0+enPNXz4cCUMS+iV9cyZto7tnjiorKysIcs9lI4ePaorrrjC6xjnLVZzS7GbPZK5Gxoatlhr8wYcaK3t7yNiZs6cGcnNRU2s5raW7P1544037Lx587qWH330Ufvoo4/2GFNcXGw3bNjQtTxlyhR78ODBfrc7cuTIyAaNolj9eYnV3NbGbvYI5x6oh2St5TkrxKecnBwFAgE1Nzerra1NlZWVKigo6DGmoKBAa9eulbVWb775pkaPHq3x48d7lBiIb4leBwC8kJiYqLKyMuXm5ioUCqmoqEgZGRkqLy+XJJWUlCg/P1+bN2+W3+/XyJEjtWbNGo9TA/EramVVXFwcrZuKqFjNLZF9IPn5+crPz+9xWUlJSdfXxhitXr36vLY5duzYiGTzQqz+vMRqbil2s3uRO2qvBgTiQXZ2turr672OAcSSQb0akOesAADOi2hZ/eY3v1FGRoaGDRvW66/LFStWyO/3Kz09XVu2bAl7/ePHj2vu3LmaPHmy5s6dqxMnTkQy3qDccccdysrKUlZWliZOnNjny5AnTpyo6dOnKysrS9nZ2dEN2Yef//znuuqqq7ryb968Oey4gU4zFG333XefvvnNbyozM1MLFizQyZMnw45zac5j8VRNLS0tmjNnjqZOnaqMjAw98cQTvcbU1tZq9OjRXT9DDz/8sAdJwxvo/ndxziVp3759XfOZlZWlSy+9VI8//niPMa7Me1FRkZKTk3Xttdd2XTbYx+Uhf1wZ4OWC56Wpqcnu3bvX3njjjXbXrl1dl+/Zs8dmZmbaL774wu7fv9+mpaXZ9vb2Xte/77777IoVK6y11q5YscLef//95xshon70ox/Zhx56KOy6q6++2h49ejTKifr3s5/9zD722GP9jmlvb7dpaWn2ww8/tF9++aXNzMy0e/bsiVLC8LZs2WLPnDljrbX2/vvv7/N+d2XO+5vDsy/pffnll21eXp796quv7I4dO+z111/vZWRrrbUHDx60DQ0N1lprT506ZSdPntzrvt++fbv97ne/60W8AQ10/7s45+dqb2+348aNswcOHOhxuSvz/uqrr9qGhgabkZHRddlgHpe/5uNK9F+6PnXqVKWnp/e6vKqqSoWFhRo+fLgmTZokv9+vurq6sOOWLFkiSVqyZIlefPHFSMY7L9ZaPffcc1q0aJFnGYbCYE4zFG3z5s1TYmLHa31mzZql1tZWT/MMJFZP1TR+/HjNmDFDkjRq1ChNnTpVwWDQ00yR5OKcn2vbtm265pprdPXVV3sdJazZs2drzJgxPS4bzONyNB5XovKcVTAYVGpqateyz+cL+0ty+PDhrvexjB8/XkeOHIlGvLBee+01jRs3TpMnTw673hijefPmaebMmaqoqIhyur6VlZUpMzNTRUVFYXfXB3tfeOWZZ57R/Pnzw65zZc4HM4euz/OBAwe0e/dufetb3+q1bseOHbruuus0f/587dmzx4N04Q10/7s+55JUWVnZ5x/Ars77YB6XozL3g90FO/sh6feS3gvzcVu3MbWSsrstr5Z0Z7flpyXdHmbbJ89ZPnG++SL4PTwl6d5+tpHS+TlZUqOk2UOR9XyySxonKUEdf4T8s6Rnwlz/ryT9utvy30p60svc3cb8RNIL6nyVqitzfj5zKKmm8/PLkv6i25htkmZ6kTdM/m9IapC0MMy6SyV9o/PrfEkBr/MO9v53ec478yRJOiZpnMvzLmmipPe6LZ88Z/2JMNcZ8seV836flbX25vO9jqRWSandln2SDoYZd9gYM95ae8gYM17SkOxaDfQ9GGMSJS2UNLOfbRzs/HzEGPOCpOsl/U8kc/Zxu4Oaf2PMv0t6Kcyqwd4XETWIOV8i6RZJ37GdP+1htuHJnIfR5xza/z/HmSfzPBBjzEWSnpe03lr73+eut9ae6vb1ZmPMvxljxlprI3OS0K9hEPe/k3PezXxJb1lrD5+7wuV51+Ael4d87qP10vXfSio0xgw3xkySNFlS7yetOsYt6fx6iSSvnky5WdJea23YJ0+MMZcYY0ad/VrSPHXsJXiq8wfprAUKn2mXpMnGmEnGmCRJheqYd88YY/IkPSCpwFr7eR9jXJrzwczhbyUtNh1mSfrUWuvpEyjGGKOOoxrvW2v/tY8xV3aOkzHmenU8RnwSvZThDfL+d27Oz7FI0sZwK1yd906DeVwe+seVCO8+LlBHw34p6bCkLd3W/UTSh5L2SZrf7fJfq/OQoaTL1bHrHuj8PMaj3eBnJZWcc1mKpM2dX6ep4zBEo6Q9kn7iRc4wuddJelfSO50/KOPPzd65nC/pg877w/Pskv5XUouktzs/yl2f83BzKKnk7M+NOt7ouLpz/bvqdljcw8x/oY43+r/Tba7zz8m9vHN+GyW9KekGr3P3d/+7Pufd8o9UR/mM7naZc/OujjI9JOlM52P53/X1uBztx5WBzmABAIDnOIMFAMB5lBUAwHmUFQDAeZQVAMB5lBUAwHmUFQDAeZQVAMB5lBUAIKqMMTnGmHeMMSM6z06yxxhzbb/X4U3BAIBoM8Y8ImmEpIsltVprV/Q7nrICAERb5zkEd0n6Qh2nlwr1N57DgAAAL4xRx7+rGaWOPax+sWcFAIg6Y8xvJVVKmqSOk24v72/8ef8/KwAAvg5jzGJJ7dbaDcaYBElvGGO+ba19pc/rsGcFAHAdz1kBAJxHWQEAnEdZAQCcR1kBAJxHWQEAnEdZAQCcR1kBAJxHWQEAnPd/5gCDo17roWcAAAAASUVORK5CYII=\n"
     },
     "metadata": {
      "needs_background": "light"
     }
    },
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x7f4d8b72cdf0>"
      ]
     },
     "metadata": {},
     "execution_count": 84
    }
   ],
   "source": [
    "sp.plot(H)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "DiracDelta(x)"
      ],
      "text/latex": "$\\displaystyle \\delta\\left(x\\right)$"
     },
     "metadata": {},
     "execution_count": 85
    }
   ],
   "source": [
    "sp.diff(H,x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## &Aacute;lgebra lineal"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Matrices\n",
    "\n",
    "En SymPy podemos usar la clase `Matrix` para definir matrices:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [],
   "source": [
    "v1, v2 = sp.symbols(\"v_1, v_2\")\n",
    "alpha, beta =  sp.symbols('alpha, beta', real=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "Matrix([\n",
       "[I*alpha,    beta],\n",
       "[  -beta, I*alpha]])"
      ],
      "text/latex": "$\\displaystyle \\left[\\begin{matrix}i \\alpha & \\beta\\\\- \\beta & i \\alpha\\end{matrix}\\right]$"
     },
     "metadata": {},
     "execution_count": 87
    }
   ],
   "source": [
    "A = sp.Matrix([[sp.I*alpha, beta],[-beta, sp.I*alpha]])\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "Matrix([\n",
       "[v_1],\n",
       "[v_2]])"
      ],
      "text/latex": "$\\displaystyle \\left[\\begin{matrix}v_{1}\\\\v_{2}\\end{matrix}\\right]$"
     },
     "metadata": {},
     "execution_count": 88
    }
   ],
   "source": [
    "v = sp.Matrix([[v1], [v2]])\n",
    "v"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "La clase `Matrix` nos permite realizar las usuales operaciones matriciales:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "Matrix([\n",
       "[-alpha**2 - beta**2,      2*I*alpha*beta],\n",
       "[    -2*I*alpha*beta, -alpha**2 - beta**2]])"
      ],
      "text/latex": "$\\displaystyle \\left[\\begin{matrix}- \\alpha^{2} - \\beta^{2} & 2 i \\alpha \\beta\\\\- 2 i \\alpha \\beta & - \\alpha^{2} - \\beta^{2}\\end{matrix}\\right]$"
     },
     "metadata": {},
     "execution_count": 89
    }
   ],
   "source": [
    "A**2 # note la diferencia con lo que hace si A fuese un arreglo de Numpy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "Matrix([\n",
       "[I*alpha*v_1 + beta*v_2],\n",
       "[I*alpha*v_2 - beta*v_1]])"
      ],
      "text/latex": "$\\displaystyle \\left[\\begin{matrix}i \\alpha v_{1} + \\beta v_{2}\\\\i \\alpha v_{2} - \\beta v_{1}\\end{matrix}\\right]$"
     },
     "metadata": {},
     "execution_count": 90
    }
   ],
   "source": [
    "A * v"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "-alpha**2 + beta**2"
      ],
      "text/latex": "$\\displaystyle - \\alpha^{2} + \\beta^{2}$"
     },
     "metadata": {},
     "execution_count": 91
    }
   ],
   "source": [
    "sp.det(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "-alpha**2 + beta**2"
      ],
      "text/latex": "$\\displaystyle - \\alpha^{2} + \\beta^{2}$"
     },
     "metadata": {},
     "execution_count": 92
    }
   ],
   "source": [
    "A.det()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "Matrix([\n",
       "[I*alpha/(-alpha**2 + beta**2),     beta/(alpha**2 - beta**2)],\n",
       "[   beta/(-alpha**2 + beta**2), I*alpha/(-alpha**2 + beta**2)]])"
      ],
      "text/latex": "$\\displaystyle \\left[\\begin{matrix}\\frac{i \\alpha}{- \\alpha^{2} + \\beta^{2}} & \\frac{\\beta}{\\alpha^{2} - \\beta^{2}}\\\\\\frac{\\beta}{- \\alpha^{2} + \\beta^{2}} & \\frac{i \\alpha}{- \\alpha^{2} + \\beta^{2}}\\end{matrix}\\right]$"
     },
     "metadata": {},
     "execution_count": 93
    }
   ],
   "source": [
    "A.inv()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "Matrix([\n",
       "[     exp(I*alpha - I*beta)/2 + exp(I*alpha + I*beta)/2, I*exp(I*alpha - I*beta)/2 - I*exp(I*alpha + I*beta)/2],\n",
       "[-I*exp(I*alpha - I*beta)/2 + I*exp(I*alpha + I*beta)/2,     exp(I*alpha - I*beta)/2 + exp(I*alpha + I*beta)/2]])"
      ],
      "text/latex": "$\\displaystyle \\left[\\begin{matrix}\\frac{e^{i \\alpha - i \\beta}}{2} + \\frac{e^{i \\alpha + i \\beta}}{2} & \\frac{i e^{i \\alpha - i \\beta}}{2} - \\frac{i e^{i \\alpha + i \\beta}}{2}\\\\- \\frac{i e^{i \\alpha - i \\beta}}{2} + \\frac{i e^{i \\alpha + i \\beta}}{2} & \\frac{e^{i \\alpha - i \\beta}}{2} + \\frac{e^{i \\alpha + i \\beta}}{2}\\end{matrix}\\right]$"
     },
     "metadata": {},
     "execution_count": 94
    }
   ],
   "source": [
    "sp.exp(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Soluci&oacute;n de (sistemas de) ecuaciones\n",
    "\n",
    "Podemos solucionar (sistemas de) ecuaciones por medio de la funci&oacute;n `solve`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "[-1, 1]"
      ]
     },
     "metadata": {},
     "execution_count": 95
    }
   ],
   "source": [
    "sp.solve(x**2 - 1, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "[-I*sqrt(-1/2 + sqrt(5)/2),\n",
       " I*sqrt(-1/2 + sqrt(5)/2),\n",
       " -sqrt(1/2 + sqrt(5)/2),\n",
       " sqrt(1/2 + sqrt(5)/2)]"
      ]
     },
     "metadata": {},
     "execution_count": 96
    }
   ],
   "source": [
    "sp.solve(x**4 - x**2 - 1, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Sistemas de ecuaciones:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "{x: 1, y: 0}"
      ]
     },
     "metadata": {},
     "execution_count": 97
    }
   ],
   "source": [
    "sp.solve([x + y - 1, x - y - 1], [x,y])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "{x: a/2 + c/2, y: a/2 - c/2}"
      ]
     },
     "metadata": {},
     "execution_count": 98
    }
   ],
   "source": [
    "sp.solve([x + y - a, x - y - c], [x,y])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "[pi/2, 3*pi/2]"
      ]
     },
     "metadata": {},
     "execution_count": 99
    }
   ],
   "source": [
    "sp.solve(sp.cos(x),x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Ecuaciones diferenciales"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {},
   "outputs": [],
   "source": [
    "y = sp.Function('y')(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "y(x) + Derivative(y(x), (x, 2))"
      ],
      "text/latex": "$\\displaystyle y{\\left(x \\right)} + \\frac{d^{2}}{d x^{2}} y{\\left(x \\right)}$"
     },
     "metadata": {},
     "execution_count": 101
    }
   ],
   "source": [
    "eq = y.diff(x,2)+y\n",
    "eq"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "Eq(y(x), C1*sin(x) + C2*cos(x))"
      ],
      "text/latex": "$\\displaystyle y{\\left(x \\right)} = C_{1} \\sin{\\left(x \\right)} + C_{2} \\cos{\\left(x \\right)}$"
     },
     "metadata": {},
     "execution_count": 102
    }
   ],
   "source": [
    "sp.dsolve(eq)"
   ]
  },
  {
   "source": [
    "Sympy cuenta con un subm&oacute;dulo de F&iacute;sica, donde podemos encontrar objetos y funciones &uacute;tiles en &eacute;sa &aacute;rea. Por ejemplo, las matrices de Pauli est&aacute;n contenidas en los objetos `msigma`"
   ],
   "cell_type": "markdown",
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import physics"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {},
   "outputs": [],
   "source": [
    "s1 = physics.matrices.msigma(1)\n",
    "s2 = physics.matrices.msigma(2)\n",
    "s3 = physics.matrices.msigma(3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "Matrix([\n",
       "[0, 1],\n",
       "[1, 0]])"
      ],
      "text/latex": "$\\displaystyle \\left[\\begin{matrix}0 & 1\\\\1 & 0\\end{matrix}\\right]$"
     },
     "metadata": {},
     "execution_count": 106
    }
   ],
   "source": [
    "s1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 107,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "Matrix([\n",
       "[0, -I],\n",
       "[I,  0]])"
      ],
      "text/latex": "$\\displaystyle \\left[\\begin{matrix}0 & - i\\\\i & 0\\end{matrix}\\right]$"
     },
     "metadata": {},
     "execution_count": 107
    }
   ],
   "source": [
    "s2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "Matrix([\n",
       "[1,  0],\n",
       "[0, -1]])"
      ],
      "text/latex": "$\\displaystyle \\left[\\begin{matrix}1 & 0\\\\0 & -1\\end{matrix}\\right]$"
     },
     "metadata": {},
     "execution_count": 108
    }
   ],
   "source": [
    "s3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 121,
   "metadata": {},
   "outputs": [],
   "source": [
    "a, b, c = sp.symbols('a,b,c', real=True)\n",
    "r = sp.Symbol('r', positive=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 127,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "Matrix([\n",
       "[-(-a - I*b)*(a - I*b)*exp(-sqrt(-a**2 - b**2 - c**2))/(2*(c + sqrt(a**2 + b**2 + c**2))*sqrt(a**2 + b**2 + c**2)) - (a - I*b)*(a + I*b)*exp(sqrt(-a**2 - b**2 - c**2))/(2*(c - sqrt(a**2 + b**2 + c**2))*sqrt(a**2 + b**2 + c**2)), -(a - I*b)*(a**2 + b**2)*exp(sqrt(-a**2 - b**2 - c**2))/((c - sqrt(a**2 + b**2 + c**2))*(2*a**2 + 2*b**2 + 2*c**2 + 2*c*sqrt(a**2 + b**2 + c**2))) - (a - I*b)*(a**2 + b**2 + 2*c**2 + 2*c*sqrt(a**2 + b**2 + c**2))*exp(-sqrt(-a**2 - b**2 - c**2))/((c + sqrt(a**2 + b**2 + c**2))*(2*a**2 + 2*b**2 + 2*c**2 + 2*c*sqrt(a**2 + b**2 + c**2)))],\n",
       "[                                                                                   (-a - I*b)*exp(-sqrt(-a**2 - b**2 - c**2))/(2*sqrt(a**2 + b**2 + c**2)) + (a + I*b)*exp(sqrt(-a**2 - b**2 - c**2))/(2*sqrt(a**2 + b**2 + c**2)),                                                                                        (a**2 + b**2)*exp(sqrt(-a**2 - b**2 - c**2))/(2*a**2 + 2*b**2 + 2*c**2 + 2*c*sqrt(a**2 + b**2 + c**2)) + (a**2 + b**2 + 2*c**2 + 2*c*sqrt(a**2 + b**2 + c**2))*exp(-sqrt(-a**2 - b**2 - c**2))/(2*a**2 + 2*b**2 + 2*c**2 + 2*c*sqrt(a**2 + b**2 + c**2))]])"
      ],
      "text/latex": "$\\displaystyle \\left[\\begin{matrix}- \\frac{\\left(- a - i b\\right) \\left(a - i b\\right) e^{- \\sqrt{- a^{2} - b^{2} - c^{2}}}}{2 \\left(c + \\sqrt{a^{2} + b^{2} + c^{2}}\\right) \\sqrt{a^{2} + b^{2} + c^{2}}} - \\frac{\\left(a - i b\\right) \\left(a + i b\\right) e^{\\sqrt{- a^{2} - b^{2} - c^{2}}}}{2 \\left(c - \\sqrt{a^{2} + b^{2} + c^{2}}\\right) \\sqrt{a^{2} + b^{2} + c^{2}}} & - \\frac{\\left(a - i b\\right) \\left(a^{2} + b^{2}\\right) e^{\\sqrt{- a^{2} - b^{2} - c^{2}}}}{\\left(c - \\sqrt{a^{2} + b^{2} + c^{2}}\\right) \\left(2 a^{2} + 2 b^{2} + 2 c^{2} + 2 c \\sqrt{a^{2} + b^{2} + c^{2}}\\right)} - \\frac{\\left(a - i b\\right) \\left(a^{2} + b^{2} + 2 c^{2} + 2 c \\sqrt{a^{2} + b^{2} + c^{2}}\\right) e^{- \\sqrt{- a^{2} - b^{2} - c^{2}}}}{\\left(c + \\sqrt{a^{2} + b^{2} + c^{2}}\\right) \\left(2 a^{2} + 2 b^{2} + 2 c^{2} + 2 c \\sqrt{a^{2} + b^{2} + c^{2}}\\right)}\\\\\\frac{\\left(- a - i b\\right) e^{- \\sqrt{- a^{2} - b^{2} - c^{2}}}}{2 \\sqrt{a^{2} + b^{2} + c^{2}}} + \\frac{\\left(a + i b\\right) e^{\\sqrt{- a^{2} - b^{2} - c^{2}}}}{2 \\sqrt{a^{2} + b^{2} + c^{2}}} & \\frac{\\left(a^{2} + b^{2}\\right) e^{\\sqrt{- a^{2} - b^{2} - c^{2}}}}{2 a^{2} + 2 b^{2} + 2 c^{2} + 2 c \\sqrt{a^{2} + b^{2} + c^{2}}} + \\frac{\\left(a^{2} + b^{2} + 2 c^{2} + 2 c \\sqrt{a^{2} + b^{2} + c^{2}}\\right) e^{- \\sqrt{- a^{2} - b^{2} - c^{2}}}}{2 a^{2} + 2 b^{2} + 2 c^{2} + 2 c \\sqrt{a^{2} + b^{2} + c^{2}}}\\end{matrix}\\right]$"
     },
     "metadata": {},
     "execution_count": 127
    }
   ],
   "source": [
    "M = sp.factor(sp.exp(sp.I*(a*s1+b*s2+c*s3)))\n",
    "M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 131,
   "metadata": {},
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "Matrix([\n",
       "[-(-a - I*b)*(a - I*b)*exp(-I*r)/(2*r*(c + r)) - (a - I*b)*(a + I*b)*exp(I*r)/(2*r*(c - r)), -(a - I*b)*(c**2 + 2*c*r + r**2)*exp(-I*r)/((c + r)*(2*c*r + 2*r**2)) - (a - I*b)*(-c**2 + r**2)*exp(I*r)/((c - r)*(2*c*r + 2*r**2))],\n",
       "[                                     (-a - I*b)*exp(-I*r)/(2*r) + (a + I*b)*exp(I*r)/(2*r),                                          (-c**2 + r**2)*exp(I*r)/(2*c*r + 2*r**2) + (c**2 + 2*c*r + r**2)*exp(-I*r)/(2*c*r + 2*r**2)]])"
      ],
      "text/latex": "$\\displaystyle \\left[\\begin{matrix}- \\frac{\\left(- a - i b\\right) \\left(a - i b\\right) e^{- i r}}{2 r \\left(c + r\\right)} - \\frac{\\left(a - i b\\right) \\left(a + i b\\right) e^{i r}}{2 r \\left(c - r\\right)} & - \\frac{\\left(a - i b\\right) \\left(c^{2} + 2 c r + r^{2}\\right) e^{- i r}}{\\left(c + r\\right) \\left(2 c r + 2 r^{2}\\right)} - \\frac{\\left(a - i b\\right) \\left(- c^{2} + r^{2}\\right) e^{i r}}{\\left(c - r\\right) \\left(2 c r + 2 r^{2}\\right)}\\\\\\frac{\\left(- a - i b\\right) e^{- i r}}{2 r} + \\frac{\\left(a + i b\\right) e^{i r}}{2 r} & \\frac{\\left(- c^{2} + r^{2}\\right) e^{i r}}{2 c r + 2 r^{2}} + \\frac{\\left(c^{2} + 2 c r + r^{2}\\right) e^{- i r}}{2 c r + 2 r^{2}}\\end{matrix}\\right]$"
     },
     "metadata": {},
     "execution_count": 131
    }
   ],
   "source": [
    "sp.factor(M.subs(a**2,r**2-b**2-c**2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Lectura adicional\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"
   ]
  }
 ],
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