Jupyter notebook demonstrating the "change of variables" functionality in sympy

Integrals_change_of_variables.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "sp.init_printing(use_latex='mathjax')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sympy can perform \"change of variables\" substitution for integrals out of the box using the `Integral.transform()` method.\n",
    "\n",
    "For documentation see [here](https://docs.sympy.org/latest/modules/integrals/integrals.html#sympy.integrals.integrals.Integral.transform).\n",
    "\n",
    "Note that the goal here is not necessarily to evaluate the integral, but to transform a definite integral. Thus, the input and output are both integrals. Therefore we use `Integral` rather than `integrate`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "x, u = symbols('x, u')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Example 1**: This is the example from the documentation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\int\\limits_{0}^{1} x \\cos{\\left(x^{2} - 1 \\right)}\\, dx$"
      ],
      "text/plain": [
       "1                 \n",
       "⌠                 \n",
       "⎮      ⎛ 2    ⎞   \n",
       "⎮ x⋅cos⎝x  - 1⎠ dx\n",
       "⌡                 \n",
       "0                 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "I1 = Integral(x*cos(x**2 - 1), (x, 0, 1))\n",
    "display(I1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\int\\limits_{0}^{1} u \\cos{\\left(u^{2} - 1 \\right)}\\, du$"
      ],
      "text/plain": [
       "1                 \n",
       "⌠                 \n",
       "⎮      ⎛ 2    ⎞   \n",
       "⎮ u⋅cos⎝u  - 1⎠ du\n",
       "⌡                 \n",
       "0                 "
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# subs x = u, and rewrite the integral\n",
    "I1.transform(x, u)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\int\\limits_{-1}^{0} \\frac{\\cos{\\left(u \\right)}}{2}\\, du$"
      ],
      "text/plain": [
       "0           \n",
       "⌠           \n",
       "⎮  cos(u)   \n",
       "⎮  ────── du\n",
       "⎮    2      \n",
       "⌡           \n",
       "-1          "
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# more interesting subs x^2 - 1 = u, and rewrite the integral\n",
    "I1.transform(x**2 - 1, u)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\int\\limits_{0}^{1} \\frac{\\cos{\\left(u \\right)}}{2}\\, du$"
      ],
      "text/plain": [
       "1          \n",
       "⌠          \n",
       "⎮ cos(u)   \n",
       "⎮ ────── du\n",
       "⎮   2      \n",
       "⌡          \n",
       "0          "
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# this inspires another substitution\n",
    "I1.transform(1 - x**2, u)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "No ambiguity in this, even though $x = \\pm \\sqrt{1 - u}$, since the $-2 x dx = du$. The transform function complains in case of ambiguity. For example, consider the integral without the prefactor."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\int\\limits_{0}^{1} \\cos{\\left(x^{2} - 1 \\right)}\\, dx$"
      ],
      "text/plain": [
       "1               \n",
       "⌠               \n",
       "⎮    ⎛ 2    ⎞   \n",
       "⎮ cos⎝x  - 1⎠ dx\n",
       "⌡               \n",
       "0               "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "I2 = Integral(cos(x**2 - 1), (x, 0, 1))\n",
    "display(I2)\n",
    "#I2.transform(x**2 - 1, u)  # elicits a complaint"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can re-transform back to check."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\int\\limits_{-1}^{0} \\frac{\\cos{\\left(u \\right)}}{2}\\, du$"
      ],
      "text/plain": [
       "0           \n",
       "⌠           \n",
       "⎮  cos(u)   \n",
       "⎮  ────── du\n",
       "⎮    2      \n",
       "⌡           \n",
       "-1          "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "uI = I1.transform(x**2 - 1, u)\n",
    "display(uI)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "uI.transform(sqrt(u+1), x) == I1\n",
    "#or\n",
    "uI.transform(u, x**2 - 1) == I1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If the substitution involves additional variable, the actual variable of integration has to be made explicity by passing a tuple."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = symbols('a')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\int\\limits_{- a}^{1 - a} \\left(a + u\\right)\\, du$"
      ],
      "text/plain": [
       "1 - a           \n",
       "  ⌠             \n",
       "  ⎮   (a + u) du\n",
       "  ⌡             \n",
       "  -a            "
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Integral(x, (x, 0, 1)).transform(x, (u + a, u))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "*** Example 2**\n",
    "\n",
    "From Monte Carlo class: infinite domain integral."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\int\\limits_{0}^{\\infty} x^{2} e^{- x}\\, dx$"
      ],
      "text/plain": [
       "∞          \n",
       "⌠          \n",
       "⎮  2  -x   \n",
       "⎮ x ⋅ℯ   dx\n",
       "⌡          \n",
       "0          "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "I3 = Integral(x**2 * exp(-x), (x, 0, oo))\n",
    "display(I3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\int\\limits_{0}^{1} \\log{\\left(\\frac{1}{u} \\right)}^{2}\\, du$"
      ],
      "text/plain": [
       "1           \n",
       "⌠           \n",
       "⎮    2⎛1⎞   \n",
       "⎮ log ⎜─⎟ du\n",
       "⎮     ⎝u⎠   \n",
       "⌡           \n",
       "0           "
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I3.transform(exp(-x), u)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\log{\\left(x^{2} \\right)}$"
      ],
      "text/plain": [
       "   ⎛ 2⎞\n",
       "log⎝x ⎠"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\log{\\left(x \\right)}^{2}$"
      ],
      "text/plain": [
       "   2   \n",
       "log (x)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# note how sympy displays exponents of logs\n",
    "display(log(x**2), (log(x))**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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