Topic 6d Sympy.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "14595680",
   "metadata": {},
   "source": [
    "## Symbolic Computing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "2dc3b78f",
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy\n",
    "sympy.init_printing()\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "759382c4",
   "metadata": {},
   "source": [
    "The emphasis in this course has been the use of computers for manipulting data or using numerical techniques to solve problems. These are central problems in scientific computing and are good windows into how computers work.\n",
    "\n",
    "Of course, computers can be designed to do much more complex tasks and we should use computers when they are available to simplify/speed up/check our work. Symbolic computing is an excellent example of a computer resource that can assist in solving problems that we might usually associate with a human and a piece of paper. Fundamentally, the computer can do a lot of our math for us, so we should use it to check our results at the very least."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "07ab8928",
   "metadata": {},
   "source": [
    "Sympy is a python library that can do a lot of math for us, in this way.  It is similar to mathematica in both function and design, but is freely available. One of the most basic sources of confusion is that these symbolic tools try to avoid making any assumptions:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "41c463d6",
   "metadata": {},
   "outputs": [],
   "source": [
    "x=sympy.Symbol('x')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "9afe4489",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACYAAAAVCAYAAAAq05ytAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACWUlEQVRIDbWW600bQRCAD4sCrJTgdGCSDqADpFQAdADil/0PmQ4gFURJB6QDIB2YEhAdON+33j3vrY7j7oCR5nZ2XjuPndXtbTabaiwsl8s5to9j7bvs9ruEPWQrdA4I8F8P3UEqk0HamXKsVvUZQXnM6MCwvQSt2KfAqFZSpRnRzFj/jokKuyl2JiboSziB/7Ilq2pUYBhfREx+hq4rgjhLRtA30A7R18Qb3MqY7TfWUdWKB59if5iCYPVK2AGnPMDgwLDSydXWfPTXaj10WvuO9cXFYjEF1331++rhc1X6HVqxD5/E2L5jqneQV7C+/CgofGHtujvHyOsLmjsaQ+PLiUyPdD2R+gqBoXAObTWewEbk7ANEHaenFZD7BJxG4XfWE9CDf0TePTp/Il1Bh6BYj+SxevEtjDFUEwgVbiPOo4KyEs6QXZfMbO8TcB117uH/BA/Z+7QYdP0Yw/NMk7yCTmc6EM9gAAN7Ai1jmjQr1wDkVqKrWsrrg6OxVyNV6Esh983yuXBN6BOya2c+OUzGDbgBZwV/DW+a83Ia2bzY38F7zHlD6QkR55CytvwByCJk3sgmCXc65d+F1fhVqA3aNgLjcC+eU2lZvReCrU1tDoyuD3YGJaQ2VvrK/G2lb3zDVBY6Vk3nBmegD6y73hfK8cDfsL38JuUlrqDDdEkDl+zrLmxZ3d9GxVTFgc5tjZUSU3shW8EkxGdsrbKTVScCT5lTOgj2X9G2dVbBic0zb1M3EZ8bA6jQ91m5A51iJ86A67aq0wf2nJY2wNka/hHrW4G1mb+b9x/WhJfYjdKcLQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle \\sqrt{x^{2}}$"
      ],
      "text/plain": [
       "   ____\n",
       "  ╱  2 \n",
       "╲╱  x  "
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.sqrt(x**2)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7074e2ec",
   "metadata": {},
   "source": [
    "Sympy does not know how to simplify this expression. What if $x$ is a complex number? We will get different answers depending on our assumptions:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "b03c514e",
   "metadata": {},
   "outputs": [],
   "source": [
    "y=sympy.Symbol('y',real=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "ef7af175",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left|{y}\\right|$"
      ],
      "text/plain": [
       "│y│"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.sqrt(y**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "3310ed13",
   "metadata": {},
   "outputs": [],
   "source": [
    "y=sympy.Symbol('y',positive=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "9a1247b3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAANCAYAAAB/9ZQ7AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA8klEQVQoFXWS4Q2CQAyFi3EAZ9ANnEE3UEdwBcMv+GdwA3ECoxvoBkY20A00bIDva+4AE23Sa/t4LY8eSdM0luf5yMzW8on8oXqn6BaeHRSXg4ClgXBRnQYshpWSBcVApLHijUI2lz896w6wKyWT32o4U8iYcvSsO2ZKeaNPrknUAIj2khoTNlUAayeDY0t5JYI3O2LGgFpYRR0/kBztf/VC6JO/iJrG1FYv5CFHsI0i+ywUX3J2jrlez7iUX55lWSG/95+5DE3by+/erUM5G+BGtxEjRs1o6+/3oLpUU9w/XEvCv8F1sg0MrScRO60Om30Aa7RlfSPkRy8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle y$"
      ],
      "text/plain": [
       "y"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.sqrt(y**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "fabcd059",
   "metadata": {},
   "outputs": [],
   "source": [
    "n1=sympy.Symbol('n1',integer=True)\n",
    "n2=sympy.Symbol('n2',odd=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "4d664f2f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(-1\\right)^{n_{1}}$"
      ],
      "text/plain": [
       "    n₁\n",
       "(-1)  "
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.cos(sympy.pi*n1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "424e5d1e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABgAAAAOCAYAAAA1+Nx+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAiElEQVQ4EWP8//8/A7VAQ0ODEdCs1UBsDGR/AJnLAiIoAUCDBID6ZwPxOyA2AWIlIIYDalgAcmkoyESgZWVACuQLOGCCs2jEGLWAYMDSPIhYoMlsL9ApoORGLAgF6jtHjGKQBaBkZkyMYnLU0DyIhr4FjNQo7IDxCCrgQIkEVBaB6HtADEoEpwEsISASVdEGngAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle -1$"
      ],
      "text/plain": [
       "-1"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.cos(sympy.pi*n2)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "008a103c",
   "metadata": {},
   "source": [
    "## Manipulating Expressions"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cf9fafbd",
   "metadata": {},
   "source": [
    "We can define experessions you might want to work with"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "abd1f510",
   "metadata": {},
   "outputs": [],
   "source": [
    "exp=x**2+x*(x*2+x*2)+x*(x+1)+3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "be4e2f0f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 5 x^{2} + x \\left(x + 1\\right) + 3$"
      ],
      "text/plain": [
       "   2                \n",
       "5⋅x  + x⋅(x + 1) + 3"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "exp"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "47d5a95f",
   "metadata": {},
   "source": [
    "We can simplify these expressions two ways:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "7ec7fbde",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGgAAAAVCAYAAACqoKu+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAADcElEQVRoBe2Z4VFbMQyAA8cAwAh0A6ATEDaAdgJgg3L8y1/YoDABhQ1gAwojsEFpNki/z9i+l+QlXF54xEejO0Wy7GfLkiXLsDIYDDptQa/XW2fuszj/VqRHyPttrfnZ5l1reUPnOOMkrQH/E/4R/JJkSzrdAqvTu+fuPcYp3cos5/BbyLYrsiU7xQJtO8jo+T1l/WXXGxZYafMOGl2byDGCDqDLFDdqnAntsTsoGrE6/BrZU1XQhGcO09oBuNPk+8/6DXaxeEr3tEWVbe/ue/ecHRQH3iE7SZ3QG9riXCc+zm307MAvKzgMIWALHXIKTQ5SdozsDnoI3lbvIB1xjzB4Dl5wgufANfxhvnQi9uH74HaUzTwj33VBN1AMzKmTe7GQMrMk+BWZ8DwJERQHmIL20igp8v1qe1ae73WOpbWnJFVunpbTWeeK4z0wYkkwj05eHf2IYU/YyUMceH9SitNbdjj4PcE3jxuQZmCdHNJZ+B8y2MFstVHdOrIUTR7s7CBP9xOd0u/gH9B75yZOAvsKtDV4SjNf4Y9AI8XvhAfG3MpAhxZXtkiYRfdF6Il+Xdb1rrYOuFSH1ai0vLBL23R0AXrKdVDyaBjAjxWG/RfwD+AV2KVt2tJ5LlAqFKk7tvNe/oHRtLlpL78dVyuWdFDwWkXmhXWFXMN3oEbOqAN0YIgY6GZNP6LFQ8m6o5vZy0N/iKWuwUf4EBjhoUrDv5g+Q4fKado6xFxoBWaFpxPzmwjesnwT+q5vG+ZzTcN9FDwAwssrGfp1k26wFuibS/c2dKpVFCFr/YUYFBupSLCU7oOTwDvGD7Nz4kCN2LQii1OME9apLSKQe6r8W57pdSaYV/eWdPLO79ToZorTtt2U4nRQSGPQOhh7CzFpOuEpvbnQulg3QUmyEnSPdrK6NZ1NtFlykPdKiJIRQ5q6LL9NbxrfF25yTDjltKvOO6M9LRJHpv+YJjoVp3u0k7a6rbHZbrTMfXAQA6zHdUIuAOD16jfQMloIIQd9iX3eA9kZyOy3qisRStXd68F7PAN2NI1re0vt/lrqoWEhYBkaHkjIvZD3aKd7Ryda5bnZDnInMKIcb6jquJzuHFMQFKk79roEu9GGyVxmslCUKfjQfzckDZpSNtK4SGi65lvfta1TSHFvKVFQvyk1p9VC9GpVp3/v95D988cbOgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle 6 x^{2} + x + 3$"
      ],
      "text/plain": [
       "   2        \n",
       "6⋅x  + x + 3"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.simplify(exp)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "1bbdc972",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 6 x^{2} + x + 3$"
      ],
      "text/plain": [
       "   2        \n",
       "6⋅x  + x + 3"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "exp.simplify()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "435a9fd5",
   "metadata": {},
   "source": [
    "Similarly we can factor expressions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "d75e4810",
   "metadata": {},
   "outputs": [],
   "source": [
    "exp2=x**2+2*x+1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "c89144d1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle x^{2} + 2 x + 1$"
      ],
      "text/plain": [
       " 2          \n",
       "x  + 2⋅x + 1"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "exp2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "e92a0fe3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(x + 1\\right)^{2}$"
      ],
      "text/plain": [
       "       2\n",
       "(x + 1) "
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.factor(exp2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "696f7b4e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEYAAAAaCAYAAAAKYioIAAAACXBIWXMAAA7EAAAOxAGVKw4bAAADAklEQVRYCeWY21EbMRSGDZMCICU4HZhQQUgHQCoAd4CHJ/uVdJCkhNCBSQVcOjAlEHfgfN9G2pHXK+xlxsaXM3Ms6ej6H52L1nuTyaS1qzQYDA7Afh3wt0N5gXz8YVeVEnDfoIRu1AH1H9Qf4U/7Ubij5SXKOEmw31BvI+vsumK0lodEMWV1b5djTKmFUMFStJhTyllXQhiDUHXeRrWb4mB8R6XARwKdciU6r5A5YBvIWCGeuRSUqLUcUR87oXQlBGrrmLJnxyYQZ/USf8MloPTc9KuYZ8rbVJ7W6dNDzE5nyildc1wohob5/A9lYUYOWFcKZ/3F+V7gz7BADpEXN019ipCbfr/U9SNTKabo1BgMyL34jtGMHLAy4lCmSc39Z5NNA8B4u4u4vrjEV75Xkv1UmkZhWRJ7dKNizm2UPaupeCB5qaTi4b9wD56yKtqHuc336fTmnnMDtkQuvvMmWLQYzfIuNwnFeauXof+Y8gLWN78F2T1jssEtjHnvQnxf4YXd1nRtABvBOTJif5cZcA8b+E5oG7BUmv677iQ+L3NhUjGCM8LPEOC1lCpw03q0kI81/TPrrIFAfOJcmHQlwY0zMx5QThp/NMenKKMsskNmbiFmjFnBOFYl923RXxf03WPu2tUFX2mLoZHFqJgscbinSqcA05xf6Z5tZoC3kGt5pmtddNn02uXX7q0rqc25ZgaAeOvRjQR3INeuvF5Cz1gbLnLH1GKcMGNmAbDPbYOvUb0weeqpa13TbmRBrPMepMWk5557Bi1GdzENV0kLkV+CklRgGYuQ2WeW2gTyU2fY5KBajBPqPge0EvO+CmihiC48hB3rE1qFlW7lmFUR+2rJuodPDekRmRfsm6ouZomhml2dlyf/qOr3+yO4Y31VzH6n8NWy92OPNjxquo+uJOU+sv73LudXtyxdczlbFKsaA5tZi9OiJtHqUO3G9jaUwVqGb8ESLUYd+aCqizX2bSqJ500PxfIfPJETuEzbfgct/LHlvHUkMPg5c0fZKE1HLP8ALhqH+fWPhJ0AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle \\left(x + 1\\right)^{2}$"
      ],
      "text/plain": [
       "       2\n",
       "(x + 1) "
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "exp2.factor()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "50be5c20",
   "metadata": {},
   "source": [
    "or expand"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "b290e757",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle x^{5} + 10 x^{4} + 40 x^{3} + 80 x^{2} + 80 x + 32$"
      ],
      "text/plain": [
       " 5       4       3       2            \n",
       "x  + 10⋅x  + 40⋅x  + 80⋅x  + 80⋅x + 32"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "((x+2)**5).expand()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "9a17e932",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\sin{\\left(x \\right)} \\cos{\\left(y \\right)} + \\sin{\\left(y \\right)} \\cos{\\left(x \\right)}$"
      ],
      "text/plain": [
       "sin(x)⋅cos(y) + sin(y)⋅cos(x)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.sin(x+y).expand(trig=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dc05c0d3",
   "metadata": {},
   "source": [
    "### Evaluating Expressions "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "07bec3aa",
   "metadata": {},
   "source": [
    "In a lot of situations, we need to combine our symbolic work with something numerical.  Sometimes, this means we just want to force the computer to return a demical instead of a symbolic expression for the number:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "97edba54",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 3.14159265358979$"
      ],
      "text/plain": [
       "3.14159265358979"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.N(sympy.pi)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "b7f84458",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 3.1415926535897932384626433832795028841971693993751$"
      ],
      "text/plain": [
       "3.1415926535897932384626433832795028841971693993751"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.N(sympy.pi,50) # tell it you want 50 decimal places"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "df89d314",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\cos{\\left(1 \\right)}$"
      ],
      "text/plain": [
       "cos(1)"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.cos(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "fae3f0fa",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 0.54030230586814$"
      ],
      "text/plain": [
       "0.540302305868140"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.N(sympy.cos(1))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0be3d03d",
   "metadata": {},
   "source": [
    "In other situations, we have found a function that we want to be able to evaluate numerically:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "c0bfe01c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 5 x^{2} + x \\left(x + 1\\right) + 3$"
      ],
      "text/plain": [
       "   2                \n",
       "5⋅x  + x⋅(x + 1) + 3"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "exp"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f4f6b9ca",
   "metadata": {},
   "source": [
    "We can just substitute $x$ with a number (or another variable) if we like"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "fcae2cb1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 613$"
      ],
      "text/plain": [
       "613"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "exp.subs(x,10)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "84d5618d",
   "metadata": {},
   "source": [
    "But we can also define an actual function we can use as normal with lambdify"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "bf27be1e",
   "metadata": {},
   "outputs": [],
   "source": [
    "exp_fnc=sympy.lambdify(x,exp)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "514f4b4f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 613$"
      ],
      "text/plain": [
       "613"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "exp_fnc(10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "710f4d2e",
   "metadata": {},
   "outputs": [],
   "source": [
    "x_array=np.linspace(1,10,10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "ad07e1e9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 10.,  29.,  60., 103., 158., 225., 304., 395., 498., 613.])"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "exp_fnc(x_array)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "a7fb53af",
   "metadata": {},
   "outputs": [],
   "source": [
    "exp3=sympy.sin(x**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "066830bd",
   "metadata": {},
   "outputs": [],
   "source": [
    "exp_fnc3=sympy.lambdify(x,exp3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "56b49d07",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.84147098, -0.7568025 ,  0.41211849, -0.28790332, -0.13235175,\n",
       "       -0.99177885, -0.95375265,  0.92002604, -0.62988799, -0.50636564])"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "exp_fnc3(x_array)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "48eb953d",
   "metadata": {},
   "source": [
    "### Solving Equations"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "759d2054",
   "metadata": {},
   "source": [
    "Not suprisingly, sympy is pretty good at solving equations."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "abdd7135",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ -1, \\  1\\right]$"
      ],
      "text/plain": [
       "[-1, 1]"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.solve(x**2-1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "e9f513c7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ 0, \\  \\pi\\right]$"
      ],
      "text/plain": [
       "[0, π]"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.solve(sympy.sin(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "id": "9b1c0533",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left\\{ x : -2, \\  y : 1\\right\\}$"
      ],
      "text/plain": [
       "{x: -2, y: 1}"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.solve([x+y+1,x+2*y],[x,y])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "20138e06",
   "metadata": {},
   "source": [
    "## Calculus"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6ac3bba5",
   "metadata": {},
   "source": [
    "Calculus is one of the symbolic tools that tends to be very powerful and reliable. In some cases it would be enormously difficult to do it yourself yet the computer seems to find a nice answer quickly."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2aaa01a8",
   "metadata": {},
   "source": [
    "#### Derivatives"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "340880c2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\sin{\\left(x \\cos{\\left(x^{2} + x + 2 \\right)} \\right)}$"
      ],
      "text/plain": [
       "   ⎛     ⎛ 2        ⎞⎞\n",
       "sin⎝x⋅cos⎝x  + x + 2⎠⎠"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "exp4=sympy.sin(x*sympy.cos(x**2 +x+2))\n",
    "exp4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "098a77f9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(- x \\left(2 x + 1\\right) \\sin{\\left(x^{2} + x + 2 \\right)} + \\cos{\\left(x^{2} + x + 2 \\right)}\\right) \\cos{\\left(x \\cos{\\left(x^{2} + x + 2 \\right)} \\right)}$"
      ],
      "text/plain": [
       "⎛                 ⎛ 2        ⎞      ⎛ 2        ⎞⎞    ⎛     ⎛ 2        ⎞⎞\n",
       "⎝- x⋅(2⋅x + 1)⋅sin⎝x  + x + 2⎠ + cos⎝x  + x + 2⎠⎠⋅cos⎝x⋅cos⎝x  + x + 2⎠⎠"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.diff(exp4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "35ffe14f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(- x \\left(2 x + 1\\right) \\sin{\\left(x^{2} + x + 2 \\right)} + \\cos{\\left(x^{2} + x + 2 \\right)}\\right) \\cos{\\left(x \\cos{\\left(x^{2} + x + 2 \\right)} \\right)}$"
      ],
      "text/plain": [
       "⎛                 ⎛ 2        ⎞      ⎛ 2        ⎞⎞    ⎛     ⎛ 2        ⎞⎞\n",
       "⎝- x⋅(2⋅x + 1)⋅sin⎝x  + x + 2⎠ + cos⎝x  + x + 2⎠⎠⋅cos⎝x⋅cos⎝x  + x + 2⎠⎠"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "exp4.diff(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "id": "e31a3015",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/latex": [
       "$\\displaystyle - \\left(x \\left(2 x + 1\\right) \\sin{\\left(x^{2} + x + 2 \\right)} - \\cos{\\left(x^{2} + x + 2 \\right)}\\right)^{2} \\sin{\\left(x \\cos{\\left(x^{2} + x + 2 \\right)} \\right)} - \\left(x \\left(2 x + 1\\right)^{2} \\cos{\\left(x^{2} + x + 2 \\right)} + 2 x \\sin{\\left(x^{2} + x + 2 \\right)} + \\left(4 x + 2\\right) \\sin{\\left(x^{2} + x + 2 \\right)}\\right) \\cos{\\left(x \\cos{\\left(x^{2} + x + 2 \\right)} \\right)}$"
      ],
      "text/plain": [
       "                                                 2                            \n",
       "  ⎛               ⎛ 2        ⎞      ⎛ 2        ⎞⎞     ⎛     ⎛ 2        ⎞⎞   ⎛ \n",
       "- ⎝x⋅(2⋅x + 1)⋅sin⎝x  + x + 2⎠ - cos⎝x  + x + 2⎠⎠ ⋅sin⎝x⋅cos⎝x  + x + 2⎠⎠ - ⎝x\n",
       "\n",
       "                                                                              \n",
       "          2    ⎛ 2        ⎞          ⎛ 2        ⎞                ⎛ 2        ⎞⎞\n",
       "⋅(2⋅x + 1) ⋅cos⎝x  + x + 2⎠ + 2⋅x⋅sin⎝x  + x + 2⎠ + (4⋅x + 2)⋅sin⎝x  + x + 2⎠⎠\n",
       "\n",
       "                       \n",
       "    ⎛     ⎛ 2        ⎞⎞\n",
       "⋅cos⎝x⋅cos⎝x  + x + 2⎠⎠"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "exp4.diff(x,x).simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "id": "b682da9f",
   "metadata": {},
   "outputs": [],
   "source": [
    "exp5=x**2*y+y**3*x+x+3*y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "id": "93135a0a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle x^{2} + 3 x y^{2} + 3$"
      ],
      "text/plain": [
       " 2        2    \n",
       "x  + 3⋅x⋅y  + 3"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "exp5.diff(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "id": "62c3e092",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 2 x y + y^{3} + 1$"
      ],
      "text/plain": [
       "         3    \n",
       "2⋅x⋅y + y  + 1"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "exp5.diff(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "id": "f33fda49",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 2 x + 3 y^{2}$"
      ],
      "text/plain": [
       "         2\n",
       "2⋅x + 3⋅y "
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "exp5.diff(x,y)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e2628711",
   "metadata": {},
   "source": [
    "#### Series"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "280cab00",
   "metadata": {},
   "source": [
    "An extremely useful tool when trying to understand functions is the Taylor series around a given point"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "id": "9d508ea4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQUAAAAaCAYAAAC+YUlrAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAJmElEQVR4Ae2b6ZXUOBDHm3kTwAARABlwRABkwBEBkMHy5hvfeJABQwQcGcBEwJHBksEOnQH7/2lVQpZlW7bbjdvrek8jWUepbsnlnku/fv3arHCYEnjx4sWJKD/11F/39RP1bw+To5XqXUlANoA9PFDBFrCTOyov1f9ddSsctY6ug3OXwCsp+bkvD0Xshcq3uRO90lcmAen1gcq9stm1Wa/Uc1Xrz1Req/1F5TyepX7m1GANCjWRHFTH08RoUPJ19d08KC5WYmsSkA455e+r/lwbLOt4qWnvoqlX1f4RPdN8J/yfkr7NIoOCGOXq1AuGrOm1wTSTnwnt12lQT4t1iLyHrJmWi2mwi0/3Wqga/Q4Crf1OiRYTZN5Ezxs//kH1X3H/4oKCZ3DISckJWxFOLKg5tkUvV8M4f4AR/VBfbAyzI93L+X+ho4HC/6B1FQceiGeDrFV4pXyj+izF4/ueqSYQObi0pESjGCMa3lH93PPnKj1zc7C+22rz7s27eMV59ExQwKk+qj4oEM04GcZ0S+04UPTiQ2uRFUaEfGpG1AtZZrJwLlJH4guneqpyQ8Xkjyw/9ZGj5jr5q76stTsBTxu2AS3kFyqgPuwevyEvtVlMUPCMn6u+FXOsZ4RMQs4xzJjavHsjiNo7m8ZwiLuqTbF6nDeIVnjkZHk4lm6tJ7HFeya3kMHXV62vgfDhOIvTkfjCnpAbX37SgwZn5BZasUv1ZUHz0ONWtR1i2Xl9O4WPQwPb5tBIaUQvP1Uua2x7pMZSAMXkrlz0P4mZ9ALH6VFYCuBgzUGAeLGgR4DDmG76vkH0ay2JLYxjpwHBEzMbHYm/eyqc7INB609UcDRkXnM2EKufw4igUGpT0FRL/oGrFLQXdP1UiV/R7JDjplwBzWOMQPGIgSUFhUdiLnfdJYIjIKJhDBg/wsOpAngc4ErnhzlzaXjaCWJ8f8YwMQKc+WIMjcJjBjQGTW7tnHSEfsfqmE982ND9HLNRH7bGzbQVvP6YMyp57PWHDcRfG/AD9PpeJQfs6W7Tx7nRQ+uTEGA4FkDMAgrBYZoMPWcY4CJq5oJMjPtPtzmloJ86gHid4pQP+Ic0lqYj8cPJTxDuCgiIy9mm1mCHlas7gxE4O9acJluNpnY2cfBT4frHz7yjui3f9LfG3U0hBAUtxrjsOgUCrtycoo9VgC+ak03A+bUIKWamMj+aw+YACRk+h+C0DtRGyAgGPNADPFZ/1/sYAgh43Cr/R2sZy4G7Wmk8pyRwoey9BwXRU6wHze2djNKaRhlrDH27d2DVn/XsZOfX0H9FhZMGXZMwBLAVdF1LYLnR33+WpCPkxMnPZ7+s3f1m27XMlpFfGyBLbD8L2quPbWDXOdvO4vZzufWchKCgTpJx7oRRDcNvVVA2WWh7z64FBY0hIE4qkixuXDWOjcO7z2OqMUSMivfecKKrTTaU6PlaBYahoRJ59Vxy6vGelMsnqLsOwgk90N2UzCFwwcOfgEF6KCG0S8YaRzecJha4HVo9Y1w3VJOMwrD5pZ0LAqqR5TfVH1WCbtWXwpJ0ZDb5LmWy4dlsqU0+LMUHLhpw0D2ZbUT7Xj9iJymTGwLRPwZOAgsCGEI6bnNx9vQzHswB2/8qFxByRoNTwiiGhdHcVtvW+qWBBnvO1V3CTNdAs/38Mx3jGcWkdOTm7bRvpB5KaCmVcZPxftUmJOjCrUBtO43M8JvoWISOPHP4BtB5S/D2zAG0VbtJrg6Z/uBnW3uIa60d46Mxqqa27Xvl2M/4mhDMac3VyDGh2l0jU2zqR9E4dDAS5qifYGK3BsYRyheVCmgee9DHKwI3EpyRpCDCJgNLIGk6zTUcoFGYYYZvCB83Cq7GFu3TKTzDNzS3gnDA/7kKdSnw2dAcKV0zSA8pkqZn7QvfQ2VsaAkMOeiSwZ/SEfrOBSzo2UgeOTvALrM27xk32+hycqbbK3nJqygyRD85mNQ2tGHY1wUFCSA1UoRY4owmHEtm5JixOdvcoO8jcADkDk5V2J+bCbeInX0vFy6nINU5Q9B2AYoMWHjgqSvfEZB2NYRvqB66UMfjk8o43mhIewIdZXWtfTjt+VRYOdAKaXaHhta22fRG4zg59swtocSfwOeCVUqH1k9tG7bvxVFmc4uq7qRnHOYo6Vw9W6QkadgENie33tbw+uGCh2puDBguSTQU+lTPFjRsflqzRxv+jXBgBLwXByNRG6OwoBXjBFeInPHAvtqiq48eisgyXlUPkXHRHi2TlqQj56CSY6vNSRbk5Zhzt0UuvYemsA1PJ7Rsj2BMhYSfGaFzGj2bMzORTxtbGjH4PgTEu2oNNE5CinHWVhKITNaY7ckVD8e3qxZjRFeuXASnLH71G+DAOed248ID7trPn9VHoGBtCkTNmP90fOfPonGwHnoQM0bGPbbJTj14HUVcvfRt9wkv6g9N6ZObLvZFch0fKIFs4NyTbWDzgLsp4JiUCzanVgkBQH2M1fIB6jPg3YsTNzg0A3pGKCYMIiU/XMEoY2AOXx5sHsEHGmLguSuhw3o+59RA+AgWJBZxOv4pxAp9/CNI4DVazE1l1K/KIlylzbF6KN2nVMapHsBvhlO6VzxvCTpy/Mhm4CVOkgc+NeaCuzrQJzfTLtsNa9XAz3IH4D5sAz/hIN4eqwHRnMhsvFEnjsLNgdObT40Ei/AqwZwYNMbV/5r63qrGmczJcD532qomccMY11YbhwieTWj0Q0f8a8Kreiav0HVq48DQmwOcn70qQctPtGCUrkMWBKx9wig9FBLaKmPJmaAN384w9YxcXdD3/YxjOPTjFMyzhBx2E/6pRv0pLEFHgSfxymGGzgiyF6qR7YmfgM2aXfuuogp7JKhQwGewD9vAPx3NS/qHKL6tt2X2TcCttZRBACEotuVJWnGsg3kJSKaz0ZFo4Wo/NNGYZ3AHvaKL34Jgx0OCymAKtB8XAA7ys6PBWOa3kBMuJBFHkOeuhSPWr0ubJTAnHXESx6dxM9X7HXmv7Wr5tylJUCDghsMtkL2X8w9RRDjx0/Q1AV47QTi4JYADXCvsWAJz0pFo4Tcbc9Qzr8G5V90da6OCjoQpvwlyQXJJNwW45P22KbfAeBew1t6Ru+au48MksOqoRW5yTPIKBCxeb/YF3I4pDhYVFHykI+HVO9L6NU1fI0xeaz1SAquOigT4RLNOi2aOnOTtPnwUAN1iEo0jZbMuXyUwKwnIWbkp8BuHXeTJsrwJN6/LBIRKDmNRN4Us52vnKoEDlIAclZ8BxD8qnIILAk7tdflfQZasaQFvkS0AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle x \\cos{\\left(2 \\right)} - x^{2} \\sin{\\left(2 \\right)} + O\\left(x^{3}\\right)$"
      ],
      "text/plain": [
       "            2           ⎛ 3⎞\n",
       "x⋅cos(2) - x ⋅sin(2) + O⎝x ⎠"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "exp4.series(x,0,3) # this means: series wrt x, at x=0 to order x^3."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "57e88331",
   "metadata": {},
   "source": [
    "This becomes much more useful when you start dealing with functions you don't konw a lot about"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "id": "fcaab46b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{x^{2}}{8} + O\\left(x^{3}\\right)$"
      ],
      "text/plain": [
       " 2        \n",
       "x     ⎛ 3⎞\n",
       "── + O⎝x ⎠\n",
       "8         "
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.besselj(2,x).series(x,0,3)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0cc952fb",
   "metadata": {},
   "source": [
    "Importantly, series is not really a Taylor series.  In includes inverse powers as well:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "id": "3ac92ee8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{1}{x} - \\gamma + x \\left(\\frac{\\gamma^{2}}{2} + \\frac{\\pi^{2}}{12}\\right) + x^{2} \\left(- \\frac{\\gamma \\pi^{2}}{12} + \\frac{\\operatorname{polygamma}{\\left(2,1 \\right)}}{6} - \\frac{\\gamma^{3}}{6}\\right) + O\\left(x^{3}\\right)$"
      ],
      "text/plain": [
       "          ⎛ 2    2⎞      ⎛     2                      3⎞        \n",
       "1         ⎜γ    π ⎟    2 ⎜  γ⋅π    polygamma(2, 1)   γ ⎟    ⎛ 3⎞\n",
       "─ - γ + x⋅⎜── + ──⎟ + x ⋅⎜- ──── + ─────────────── - ──⎟ + O⎝x ⎠\n",
       "x         ⎝2    12⎠      ⎝   12           6          6 ⎠        "
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.gamma(x).series(x,0,3)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6e8c833f",
   "metadata": {},
   "source": [
    "To see why this is useful consider"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "id": "0ed94561",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\text{NaN}$"
      ],
      "text/plain": [
       "nan"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(sympy.gamma(x)*sympy.sin(x)).subs(x,0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "id": "02d5c453",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 1 - \\gamma x + O\\left(x^{2}\\right)$"
      ],
      "text/plain": [
       "           ⎛ 2⎞\n",
       "1 - γ⋅x + O⎝x ⎠"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(sympy.gamma(x)*sympy.sin(x)).series(x,0,2)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2f7fa61d",
   "metadata": {},
   "source": [
    "The function $\\Gamma[x]\\sin(x)$ is perfectly well behaved at $x=0$ but you need to look at the series as $x\\to 0$ to see that.  There is also a limit option but in my experience series is always better than limit."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8e6e7efb",
   "metadata": {},
   "source": [
    "#### Integration"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5fee2090",
   "metadata": {},
   "source": [
    "Integration is where it is really obvious how much better a computer is than a person.  Even just a minor change to an integral can make it faily challenging"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "id": "24a47d62",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - \\cos{\\left(x \\right)}$"
      ],
      "text/plain": [
       "-cos(x)"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.integrate(sympy.sin(x),x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "id": "6755151c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{3 \\sqrt{2} \\sqrt{\\pi} S\\left(\\frac{\\sqrt{2} x}{\\sqrt{\\pi}}\\right) \\Gamma\\left(\\frac{3}{4}\\right)}{8 \\Gamma\\left(\\frac{7}{4}\\right)}$"
      ],
      "text/plain": [
       "         ⎛√2⋅x⎞       \n",
       "3⋅√2⋅√π⋅S⎜────⎟⋅Γ(3/4)\n",
       "         ⎝ √π ⎠       \n",
       "──────────────────────\n",
       "       8⋅Γ(7/4)       "
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.integrate(sympy.sin(x**2),x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "id": "7dfe3e4f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 2 - \\frac{5}{e}$"
      ],
      "text/plain": [
       "       -1\n",
       "2 - 5⋅ℯ  "
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.integrate(x**2*sympy.exp(-x),(x,0,1))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a224059f",
   "metadata": {},
   "source": [
    "The notation for $\\infty$ is a bit unusual"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "id": "9f2737f4",
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{\\sqrt{\\pi}}{2}$"
      ],
      "text/plain": [
       "√π\n",
       "──\n",
       "2 "
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.integrate(sympy.exp(-x**2),(x,0,sympy.oo))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "818611da",
   "metadata": {},
   "source": [
    "We can also perform a integration that produces a function of some other variable:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "id": "62a85cb9",
   "metadata": {},
   "outputs": [],
   "source": [
    "a=sympy.Symbol('a')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "id": "63862dce",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\begin{cases} \\frac{\\sqrt{\\pi}}{2 \\sqrt{a}} & \\text{for}\\: \\left|{\\arg{\\left(a \\right)}}\\right| \\leq \\frac{\\pi}{2} \\\\\\int\\limits_{0}^{\\infty} e^{- a x^{2}}\\, dx & \\text{otherwise} \\end{cases}$"
      ],
      "text/plain": [
       "⎧    √π                      π\n",
       "⎪   ────      for │arg(a)│ ≤ ─\n",
       "⎪   2⋅√a                     2\n",
       "⎪                             \n",
       "⎪∞                            \n",
       "⎨⌠                            \n",
       "⎪⎮      2                     \n",
       "⎪⎮  -a⋅x                      \n",
       "⎪⎮ ℯ      dx     otherwise    \n",
       "⎪⌡                            \n",
       "⎩0                            "
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.integrate(sympy.exp(-a*x**2),(x,0,sympy.oo))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "97620f14",
   "metadata": {},
   "source": [
    "Notice again that it didn't know what to do with our variable $a$ (it could be real, imaginary, complex, negative, postive. Fortunately, we can tell it what to assume to avoid this problem (sometimes it can't figure it out on its own):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "id": "50894321",
   "metadata": {},
   "outputs": [],
   "source": [
    "c=sympy.Symbol('c',positive=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "id": "d8328597",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\frac{\\sqrt{\\pi}}{2 \\sqrt{c}}$"
      ],
      "text/plain": [
       " √π \n",
       "────\n",
       "2⋅√c"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.integrate(sympy.exp(-c*x**2),(x,0,sympy.oo))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e6ab8002",
   "metadata": {},
   "source": [
    "## Summary"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "aa89af77",
   "metadata": {},
   "source": [
    "Symbolic computation is extremely powerful and useful. The functionality is really quite endless.  However, one often does not appreciate all that it can do until you really need it.  At the very least, it will keep you from making stupid mistakes on your homework."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.9.7"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}