tutorial2.2_sympy_examples.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# SymPy Examples"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This example is related to the article [*Computer Algebra with Jupyter and SymPy*](https://jaantollander.com/post/computer-algebra-with-sympy-and-jupyter/). It contains a showcase of the essential features of SymPy. The Sympy webpage contains the full list of [features](https://www.sympy.org/en/features.html) and [documentation](https://docs.sympy.org/latest/index.html)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Imports and Printing"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Import SymPy and initialize the printing of SymPy expressions with MathJax."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy as sym\n",
    "# init_printing(\"mathjax\") # sympy expression printing"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Import SymEngine and initialize the printing of SymEngine expressions with MathJax. SymEngine is a fast symbolic manipulation library written in C++ instead of Python. SymEngine has Python wrappers that integrate with Sympy, although Symengine does not yet support all of SymPy's functionality. We can override some SymPy functions with SymEngine variants for improved performance.  It is wise to import only computationally critical functions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [],
   "source": [
    "# from symengine import *  # overrides some sympy functions\n",
    "# init_printing(\"mathjax\") # symengine expression printing"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Define Symbols and Functions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`symbols`, `Symbol`, `Function`, `sympify`, `S`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = sym.symbols(\"x\")\n",
    "a, b, c = sym.symbols(\"a,b,c\", real=True)\n",
    "i, j, n = sym.symbols(\"i,j,n\", integer=True)\n",
    "f = sym.symbols(\"f\", cls=sym.Function)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": "(1, 1)",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle \\left( 1, \\  1\\right)$"
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.sympify(1), sym.S(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": "ⅈ",
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAATCAYAAACtHkzTAAAAjElEQVR4nIXRMQoCMRAF0LeiWHgPQREE8QIew2tYWW7t3XYVBBs7z2BhIWoTUWOSnWbI5DH5kKqua6XqJWYbPLHOgWXoTQ5sMcEZ+glwKWVYhfd3ObAIfd8F2hK4vgPGYIQxDnikwDyc26/ZD/gLmAPFDTecUmCIKY64p8AMg3g9n79oUMWXcYZkdYIX6pcXaw1qyPoAAAAASUVORK5CYII=\n",
      "text/latex": "$\\displaystyle i$"
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.I  # imaginary unit"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Notations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`Rational`, `Sum`, `Product`, `I`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": "1/2",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle \\frac{1}{2}$"
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.Rational(1, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": "  n     \n ___    \n ╲      \n  ╲    i\n  ╱   x \n ╱      \n ‾‾‾    \ni = 1   ",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle \\sum_{i=1}^{n} x^{i}$"
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.Sum(x ** i, (i, 1, n))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": "  n        \n─┬─┬─      \n │ │  i + x\n │ │       \ni = 1      ",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle \\prod_{i=1}^{n} \\left(i + x\\right)$"
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.Product(x + i, (i, 1, n))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Evaluate Expressions Numerically"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`N`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": " 5\nℯ ",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle e^{5}$"
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.exp(5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": "148.41",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle 148.41$"
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.N(_, 5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Manipulate Expressions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`expand`, `collect`, `factor`, `simplify`, `.subs`, `.doit`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": "(-a + x)⋅(-b + x)",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle \\left(- a + x\\right) \\left(- b + x\\right)$"
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(x - a) * (x - b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": "                   2\na⋅b - a⋅x - b⋅x + x ",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle a b - a x - b x + x^{2}$"
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.expand(_)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": "                   2\na⋅(b - x) - b⋅x + x ",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle a \\left(b - x\\right) - b x + x^{2}$"
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.collect(_, a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": "(-a + x)⋅(-b + x)",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle \\left(- a + x\\right) \\left(- b + x\\right)$"
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.factor(_)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": "(-a + c)⋅(-b + c)",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle \\left(- a + c\\right) \\left(- b + c\\right)$"
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_.subs(x, c)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": "(a - c)⋅(b - c)",
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIAAAAAVCAYAAACOleY7AAAETElEQVR4nO3aW6gWVRQH8N8xTU0jy25WR8tCH8zIDLoQYWZFD0EZREVp2kPUQzeiDAok4lQPlUURBAVZgQWGlVFJpCRB5am0O11IK7SUsouhWXl6WPNx5syZ+ea7nQ87+odh8+29Z+31n7VmrbX3Nx0LFy60F3suhhT0L8ZmjGqjLgOJ6ejB1TXMrcb9ukTO/Nap1hYU8s9zgJNxBe7FnwOrV9vwPpbhboyuMq+M+0kpef8nFPLPc4Au/I7HBlyt9uIeHI7rq8wp4z4dO/Bpa1VrC3L5Zx1gEmbheWxvj15tw3v4Atdgn5zxMu7DMQUf4Z8B0nEgkcs/6wDz0YHnCoRcjmfxJf7AVqzBvBYrWy9mCJ1/wF/YhNdxYWbeEowXhs6ijPtUDBPh9EQsxc8iVazA8Y2r3zRmaJB/1gFm4V+8k7PI/ngKE7Eaj+AFHIMncVtTFBrHIqzETLyBB5Lf03F6Zu7bSXtOjpxq3CXyYEIipwdP4INE3psY04D+zWKRJvgPTQ2OEp79ufwCqAed+DHTf4eICPNwX/36N4Uu3CDexrn66j0aYzPz1yTtmZn+Mu70OsApOAMfpsYW40qxS+iqWfvm0TT/dAQ4UuSGTQWLbdPf+JL5G3FQrVq3CNNE1OkWqSlruG3YkOn7TRRx4zP9Zdzp3QHcqq/x4dGknVqqdevQEv5pB6h4y9aCBQ/EnSJEbhXhsie5Jon8Uw3rU/NruZ4pkXdzov8C7CyZm8YvODjTV8Z9mDDuBvG2Z1F5MUZUWXe93ZB/OgVUKt88EieIQucwUU0uSQT9LWqAOVhXsvA3wvtqxcaS8fOEwVbWIRNG6l/lV+NOGH9fvCx/BzAhabNvXBq7Jf+0A2xO2mzegKdFgXMWVmXG7kra7pKFz65VwxowAodgLXbVcd8QwePbTH817vSG//UF4xcl7Yoqa++W/NMpYBO2YHLmpk4RAVbpb/wxereA7Twd60jaQ+u8b3Jy79pMfxH3CioFYF6dM07srb9S3QFaiZbxTztAD94S+eG4VH8lbE0UubCCsWLveZQIi2u1D9vxCY7AJTnjk+Qf9pyatNmwWcS9gooDXKbvfwSjRU0wXJywteuAqGX8h2YmLMXFIr98nfRtEXvcmXhX7DXH4fxE0C58pr781goswEuiHpkrjmfHiO1cZ6JjFueK4vXFnLE87sQzmioq//2Eoy8TRp8tjHAjXmuCSyNoCf/sQdBS/CSKujQuFXVAJ67FsaIK7UpklOX/gcAr4gTsVZyGm3ABfsUtOfMPECdjy/F9zngR9yki53aLPP6xCPlXiYc+Ew83TqNhtIR/NgLsxEPCsNP07ne36P9gKugo6G8HVidXLZgjDHl/wXgR93X6cpxdv5oDhqb55/0b+CC+01vdDwaMxO3iLa/2wAYjd6rwz3OAHeJYs9vg+SDkaDwuPzSmMRi5U4V/x95PwvZsFH0Sthd7CP4DOYY2JRBulzgAAAAASUVORK5CYII=\n",
      "text/latex": "$\\displaystyle \\left(a - c\\right) \\left(b - c\\right)$"
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.simplify(_)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": "  n     \n ___    \n ╲      \n  ╲    i\n  ╱   x \n ╱      \n ‾‾‾    \ni = 1   ",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle \\sum_{i=1}^{n} x^{i}$"
     },
     "execution_count": 86,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.Sum(x ** i, (i, 1, n))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": "⎧    n       for x = 1\n⎪                     \n⎪     n + 1           \n⎨x - x                \n⎪──────────  otherwise\n⎪  1 - x              \n⎩                     ",
      "text/latex": "$\\displaystyle \\begin{cases} n & \\text{for}\\: x = 1 \\\\\\frac{x - x^{n + 1}}{1 - x} & \\text{otherwise} \\end{cases}$"
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_.doit()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Define Equations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`Eq`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": "   2          \na⋅x  + b⋅x = c",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle a x^{2} + b x = c$"
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.Eq(a * x ** 2 + b * x, c)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solve Equations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`solve`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": "   2          \na⋅x  + b⋅x + c",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle a x^{2} + b x + c$"
     },
     "execution_count": 89,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a * x ** 2 + b * x + c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": "⎡        _____________          _____________⎤\n⎢       ╱           2          ╱           2 ⎥\n⎢-b - ╲╱  -4⋅a⋅c + b    -b + ╲╱  -4⋅a⋅c + b  ⎥\n⎢─────────────────────, ─────────────────────⎥\n⎣         2⋅a                    2⋅a         ⎦",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle \\left[ \\frac{- b - \\sqrt{- 4 a c + b^{2}}}{2 a}, \\  \\frac{- b + \\sqrt{- 4 a c + b^{2}}}{2 a}\\right]$"
     },
     "execution_count": 90,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.solve(_, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solve Systems of Equations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [],
   "source": [
    "z = sym.symbols(\"z_{0:2}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": "⎡⎧                __________                  __________⎫  ⎧                __\n⎢⎪               ╱  2                        ╱  2       ⎪  ⎪               ╱  \n⎢⎨         a   ╲╱  a  + 4⋅b            a   ╲╱  a  + 4⋅b ⎬  ⎨         a   ╲╱  a\n⎢⎪z_{0}: - ─ - ─────────────, z_{1}: - ─ + ─────────────⎪, ⎪z_{0}: - ─ + ─────\n⎣⎩         2         2                 2         2      ⎭  ⎩         2        \n\n________                  __________⎫⎤\n2                        ╱  2       ⎪⎥\n  + 4⋅b            a   ╲╱  a  + 4⋅b ⎬⎥\n────────, z_{1}: - ─ - ─────────────⎪⎥\n 2                 2         2      ⎭⎦",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle \\left[ \\left\\{ z_{0} : - \\frac{a}{2} - \\frac{\\sqrt{a^{2} + 4 b}}{2}, \\  z_{1} : - \\frac{a}{2} + \\frac{\\sqrt{a^{2} + 4 b}}{2}\\right\\}, \\  \\left\\{ z_{0} : - \\frac{a}{2} + \\frac{\\sqrt{a^{2} + 4 b}}{2}, \\  z_{1} : - \\frac{a}{2} - \\frac{\\sqrt{a^{2} + 4 b}}{2}\\right\\}\\right]$"
     },
     "execution_count": 92,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.solve([z[0] + z[1] + a, z[0] * z[1] + b], z, dict=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solve Recurrence Relations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`rsolve`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\begin{aligned}\n",
    "y_0&=1, \\\\\n",
    "y_1&=4, \\\\\n",
    "y_n &= 2y_{n-1} + 5y_{n-2}\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": "⎛1   √6⎞         n   ⎛1   √6⎞         n\n⎜─ - ──⎟⋅(1 - √6)  + ⎜─ + ──⎟⋅(1 + √6) \n⎝2   4 ⎠             ⎝2   4 ⎠          ",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle \\left(\\frac{1}{2} - \\frac{\\sqrt{6}}{4}\\right) \\left(1 - \\sqrt{6}\\right)^{n} + \\left(\\frac{1}{2} + \\frac{\\sqrt{6}}{4}\\right) \\left(1 + \\sqrt{6}\\right)^{n}$"
     },
     "execution_count": 93,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = sym.Function('y')\n",
    "f = y(n) - 2 * y(n - 1) - 5 * y(n - 2)\n",
    "sym.rsolve(f, y(n), [1, 4])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Differential Calculus"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`limit`, `series`, `diff`, `integrate`, `Derivative`, `Integral`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": "1\n─\nx",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle \\frac{1}{x}$"
     },
     "execution_count": 94,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.diff(sym.log(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": "                         ⎛2⋅√3⋅x   √3⎞\n   ⎛ 2        ⎞   √3⋅atan⎜────── + ──⎟\nlog⎝x  + x + 1⎠          ⎝  3      3 ⎠\n─────────────── + ────────────────────\n       2                   3          ",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle \\frac{\\log{\\left(x^{2} + x + 1 \\right)}}{2} + \\frac{\\sqrt{3} \\operatorname{atan}{\\left(\\frac{2 \\sqrt{3} x}{3} + \\frac{\\sqrt{3}}{3} \\right)}}{3}$"
     },
     "execution_count": 95,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.integrate((x ** 2 - 1) / (x ** 3 - 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": "√3⋅π   log(3)\n──── + ──────\n 18      2   ",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle \\frac{\\sqrt{3} \\pi}{18} + \\frac{\\log{\\left(3 \\right)}}{2}$"
     },
     "execution_count": 96,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.integrate((x ** 2 - 1) / (x ** 3 - 1), (x, 0, 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": "     3     5        \n    x     x     ⎛ 6⎞\nx - ── + ─── + O⎝x ⎠\n    6    120        ",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle x - \\frac{x^{3}}{6} + \\frac{x^{5}}{120} + O\\left(x^{6}\\right)$"
     },
     "execution_count": 97,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.series(sym.sin(x), x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": "1",
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAA0AAAASCAYAAACAa1QyAAAAh0lEQVR4nO3RIQ7CYAwF4G8EPY0kCIIbllsg8NNIkknEfwcugOAeCE5CgkQR/GZ+1WTLMFM0eWlf05fXtEVKya8xC/yACx74oMUtiuaBn1Hhixc2Y5xOWKPEsW+96HTvGxxyGhV/0eSi+Nx9Bixy3uGa6zeaKNqiDr1VBjzRxPUSigEsmfIQHbJ7ECMDdcssAAAAAElFTkSuQmCC\n",
      "text/latex": "$\\displaystyle 1$"
     },
     "execution_count": 98,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.limit(sym.sin(x) / x, x, 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solve Differential Equations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`dsolve`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = sym.Function(\"f\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": "                            2      \n  d                        d       \na⋅──(f(x)) + b⋅f(x) + c + ───(f(x))\n  dx                        2      \n                          dx       ",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle a \\frac{d}{d x} f{\\left(x \\right)} + b f{\\left(x \\right)} + c + \\frac{d^{2}}{d x^{2}} f{\\left(x \\right)}$"
     },
     "execution_count": 100,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.diff(f(x), x, 2) + a * sym.diff(f(x), x) + b * f(x) + c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": "             ⎛        __________⎞          ⎛       __________⎞     \n             ⎜       ╱  2       ⎟          ⎜      ╱  2       ⎟     \n           x⋅⎝-a + ╲╱  a  - 4⋅b ⎠       -x⋅⎝a + ╲╱  a  - 4⋅b ⎠     \n           ──────────────────────       ───────────────────────    \n                     2                             2              c\nf(x) = C₁⋅ℯ                       + C₂⋅ℯ                        - ─\n                                                                  b",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle f{\\left(x \\right)} = C_{1} e^{\\frac{x \\left(- a + \\sqrt{a^{2} - 4 b}\\right)}{2}} + C_{2} e^{- \\frac{x \\left(a + \\sqrt{a^{2} - 4 b}\\right)}{2}} - \\frac{c}{b}$"
     },
     "execution_count": 101,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.dsolve(_)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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   "name": "python3"
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  "language_info": {
   "codemirror_mode": {
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