ch03-symbolic-math-with-sympy.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Symbolic computing with [SymPy](https://www.sympy.org/en/index.html)\n",
    "\n",
    "Symbolic analysis can make a difference in determining __how__ to attack a problem before throwing numerical firepower at it. (Complexity or size reduction, for example.)\n",
    "\n",
    "- __Symbols__\n",
    "- __Numbers__ (Integer, Float, Rational)\n",
    "- __Constants__\n",
    "- __Functions__\n",
    "- __Expression__\n",
    "- __Manipulating expressions__\n",
    "- __Simplification__\n",
    "- __Expansion__\n",
    "- __Factor, Collect, Combine__\n",
    "- __Apart, Together, Cancel__\n",
    "- __Substitutions__\n",
    "- __Numerical evaluation__\n",
    "- __Calculus__ (Derivatives, Integrals, Series, Limits)\n",
    "- __Sums & Products__\n",
    "- __Equations__\n",
    "- __Linear Algebra__"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy\n",
    "from sympy import I, pi, oo # frequently used symbols"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "sympy.init_printing() # enables MathJax to render SymPy expressions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Symbols\n",
    "\n",
    "Symbols have a name & a set of attributes. They are not especially useful by themselves, but are used as nodes in algebraic expression trees.\n",
    "\n",
    "![symbol arguments](pics/symbol-arguments.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Symbols are used as nodes in algebraic expression trees.\n",
    "Normally, symbols are associated with Python variables with similar names."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "None\n",
      "True\n",
      "False\n"
     ]
    }
   ],
   "source": [
    "print(sympy.Symbol(\"x\"               ).is_real)\n",
    "print(sympy.Symbol('y',     real=True).is_real)\n",
    "print(sympy.Symbol(\"z\",imaginary=True).is_real)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Below: Sympy recognizes $\\sqrt{y^2} = y$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left( \\sqrt{x^{2}}, \\  y\\right)$"
      ],
      "text/plain": [
       "⎛   ____   ⎞\n",
       "⎜  ╱  2    ⎟\n",
       "⎝╲╱  x  , y⎠"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = sympy.Symbol(\"x\")\n",
    "y = sympy.Symbol(\"y\", positive=True)\n",
    "sympy.sqrt(x**2), sympy.sqrt(y**2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Integer representation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left( \\cos{\\left(\\pi n \\right)}, \\  \\left(-1\\right)^{n}, \\  -1\\right)$"
      ],
      "text/plain": [
       "⎛              n    ⎞\n",
       "⎝cos(π⋅n), (-1) , -1⎠"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n1 = sympy.Symbol(\"n\")\n",
    "n2 = sympy.Symbol(\"n\", integer=True)\n",
    "n3 = sympy.Symbol(\"n\", odd=True)\n",
    "\n",
    "sympy.cos(n1*pi), sympy.cos(n2*pi), sympy.cos(n3*pi)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Creating multiple symbols in one function call"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "a, b, c = sympy.symbols(\"a, b, c\", negative=True)\n",
    "d, e, f = sympy.symbols(\"d, e, f\", positive=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Numbers\n",
    "Can't directly use Python objects for integers, floats.\n",
    "\n",
    "Instead use __Sympy.Integer__, __Sympy.Float__ (Not often needed, because Sympy auto-promotes numbers to class instances when needed.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(True, True, True)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "i = sympy.Integer(19)\n",
    "i.is_Integer, i.is_real, i.is_odd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(False, True, False)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = sympy.Float(2.3)\n",
    "f.is_Integer, f.is_real, f.is_odd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(sympy.core.numbers.Integer, sympy.core.numbers.Float)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "i, f = sympy.sympify(19), sympy.sympify(2.3)\n",
    "type(i), type(f)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Integers\n",
    "\n",
    "There is a difference between a `Symbol instance` with `integer=True` and an instance of Integer. `Symbol` with `integer=True` represents some integer; an Integer instance represents a specific integer. \n",
    "\n",
    "\n",
    "`is_integer` is True for both. `is_Integer` (note the capital I) is only True for Integer instances. \n",
    "\n",
    "SymPy integers & floats have arbitrary precision, no upper/lower bounds. They are much easier to use with very large numbers."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = sympy.Symbol(\"n\", integer=True)\n",
    "i = sympy.Integer(19)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True False None True\n",
      "True True True False\n"
     ]
    }
   ],
   "source": [
    "print(n.is_integer, n.is_Integer, n.is_positive, n.is_Symbol)\n",
    "print(i.is_integer, i.is_Integer, i.is_positive, i.is_Symbol)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "8663234049605954426644038200675212212900743262211018069459689001\n"
     ]
    }
   ],
   "source": [
    "print(i**50)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000\n"
     ]
    }
   ],
   "source": [
    "print(sympy.factorial(100))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Float\n",
    "\n",
    "Also arbitrary precision. SymPy Float can represent the real number 0.3 without the limitations of floating-point format."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'0.2999999999999999888977698'"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\"%.25f\" % 0.3  # create a string represention with 25 decimals"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.2999999999999999888977698\n",
      "0.3000000000000000000000000\n"
     ]
    }
   ],
   "source": [
    "print(sympy.Float( 0.3, 25))\n",
    "print(sympy.Float('0.3',25))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Rationals\n",
    "\n",
    "Fractions of two integers."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\frac{11}{13}$"
      ],
      "text/plain": [
       "11\n",
       "──\n",
       "13"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.Rational(11, 13)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left( \\frac{8}{15}, \\  \\frac{5}{6}\\right)$"
      ],
      "text/plain": [
       "(8/15, 5/6)"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r1 = sympy.Rational(2, 3)\n",
    "r2 = sympy.Rational(4, 5)\n",
    "r1*r2, r1/r2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Constants & Special Symbols\n",
    "![constants-specials](pics/constants-specials.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left( \\pi, \\  e, \\  \\gamma, \\  i, \\  \\infty\\right)$"
      ],
      "text/plain": [
       "(π, ℯ, γ, ⅈ, ∞)"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.pi, sympy.E, sympy.EulerGamma, sympy.I, sympy.oo"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Functions\n",
    "\n",
    "Sympy understands __defined__ vs __undefined__ functions, and __applied__ vs __unapplied__ functions.\n",
    "\n",
    "Creating a function with __Function__ returns an undefined, unapplied function. It has a name but cannot be evaluated."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "<class 'sympy.core.function.UndefinedFunction'>\n"
     ]
    }
   ],
   "source": [
    "f = sympy.Function(\"f\"); print(type(f))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle f{\\left(x \\right)}$"
      ],
      "text/plain": [
       "f(x)"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left( g{\\left(x,y,z \\right)}, \\  \\left\\{x, y, z\\right\\}\\right)$"
      ],
      "text/plain": [
       "(g(x, y, z), {x, y, z})"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x,y,z = sympy.symbols(\"x, y, z\")\n",
    "g     = sympy.Function(\"g\")(x,y,z)\n",
    "g, g.free_symbols"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(sin, sin(x), -1)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# defined functions have a specific implementation.\n",
    "# they can be numerically evaluated.\n",
    "sympy.sin, sympy.sin(x), sympy.sin(pi*1.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAA0AAAAQCAYAAADNo/U5AAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjUuMiwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy8qNh9FAAAACXBIWXMAABJ0AAASdAHeZh94AAABE0lEQVR4nJXSv0rcURDF8c8uVop/QLBKQBT1DYxglQQkL2EfFARri2EEWwtFH8An0NYmVSRvoK4I6VKZICJ2bgrvyt0fu6CnOXBnvjPD5bS63a73aqT5kJkfsItvmMYfnCIj4h+06k2ZOY8LzOAMV1jGZ1xjNSLumpuOC7AVEYfVsH1sYw/f21VhDmv4jaPGsMAj1jNzrF0VvhQ/j4jnPiLiAT8xipUaWireaX5O0U3xxRqaLH4/BOq9T7WHNAxSq3i3hnqTJg3WRK+vhq6LLw6BFop3auhH8bXM7Ds7M8exiif8ei1GxC3OMYvNxpbEGE4i4rGZiA0vMTrIzK+4xCcvMepgh0b2yikfDQ/s34HQW/Qfnq1S3HXUdK4AAAAASUVORK5CYII=",
      "text/latex": [
       "$\\displaystyle 0$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n = sympy.Symbol(\"n\", integer=True)\n",
    "sympy.sin(n*pi)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left( \\left( x \\mapsto x^{2} \\right), \\  25, \\  \\left(x + 1\\right)^{2}\\right)$"
      ],
      "text/plain": [
       "⎛     2             2⎞\n",
       "⎝x ↦ x , 25, (x + 1) ⎠"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# lambda functions: no name, but an executable body.\n",
    "h = sympy.Lambda(x, x**2)\n",
    "h, h(5), h(1+x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Expressions\n",
    "Sympy expressions = represented as tree structures.\n",
    "\n",
    "Symbols = leaves; nodes = math op class instances."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left( 3 x^{3} + 2 x^{2} + 1, \\  \\left( 1, \\  2 x^{2}, \\  3 x^{3}\\right), \\  2 x^{2}\\right)$"
      ],
      "text/plain": [
       "⎛   3      2      ⎛      2     3⎞     2⎞\n",
       "⎝3⋅x  + 2⋅x  + 1, ⎝1, 2⋅x , 3⋅x ⎠, 2⋅x ⎠"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = sympy.Symbol(\"x\")\n",
    "e = 1 + 2*x**2 + 3*x**3\n",
    "e, e.args, e.args[1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![expression tree](pics/expression-tree.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left( x^{2}, \\  2, \\  x^{3}\\right)$"
      ],
      "text/plain": [
       "⎛ 2      3⎞\n",
       "⎝x , 2, x ⎠"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# explore the expression tree using the args attribute\n",
    "# args is a tuple of sub-expressions\n",
    "e.args[1].args[1], e.args[1].args[0], e.args[2].args[1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Manipulating Expressions\n",
    "\n",
    "SymPy's main job to provide different transforms to expression trees. These transforms create new expressions & do not change the originals. In other words, *expression trees are immutable*."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Simplifications\n",
    "\n",
    "Simplification is a _good thing_, but often very ambiguous. (It's often non-trivial to determine if an expression appears simplier to a human being.)\n",
    "\n",
    "![simplifications](pics/simplifications.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle 2 x^{2} - x \\left(x + 1\\right) - 2 x$"
      ],
      "text/plain": [
       "   2                  \n",
       "2⋅x  - x⋅(x + 1) - 2⋅x"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = 2*(x**2-x) - x*(x+1); expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left( x \\left(x - 3\\right), \\  x \\left(x - 3\\right)\\right)$"
      ],
      "text/plain": [
       "(x⋅(x - 3), x⋅(x - 3))"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.simplify(expr), expr.simplify()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Both `sympy.simplify(expr)` and `expr.simplify()` return new\n",
    "expression trees and leave the expression expr untouched. \n",
    "\n",
    "In this example, `expr` can be simplified by expanding the products, canceling terms, then refactoring. `sympy.simplify` will attempt\n",
    "different strategies and will simplify trigonometric and\n",
    "power expressions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left( 2 \\sin{\\left(x \\right)} \\cos{\\left(x \\right)}, \\  \\sin{\\left(2 x \\right)}\\right)$"
      ],
      "text/plain": [
       "(2⋅sin(x)⋅cos(x), sin(2⋅x))"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = 2 * sympy.cos(x) * sympy.sin(x)\n",
    "expr, sympy.simplify(expr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left( e^{x} e^{y}, \\  e^{x + y}\\right)$"
      ],
      "text/plain": [
       "⎛ x  y   x + y⎞\n",
       "⎝ℯ ⋅ℯ , ℯ     ⎠"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = sympy.exp(x) * sympy.exp(y)\n",
    "expr, sympy.simplify(expr)\n",
    "\n",
    "# can also use sympy.trigsimp and sympy.powsimp \n",
    "# to perform only their specified simplifications\n",
    "# leaving the rest of the expression untouched."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Expansion\n",
    "\n",
    "When __sympy.simplify__ does not provide acceptable results. Expression expansion can help with designing a more manual approach."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle x^{2} + 3 x + 2$"
      ],
      "text/plain": [
       " 2          \n",
       "x  + 3⋅x + 2"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = (x+1)*(x+2)\n",
    "sympy.expand(expr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAREAAAAVCAYAAABhctosAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjUuMiwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy8qNh9FAAAACXBIWXMAABJ0AAASdAHeZh94AAAHvUlEQVR4nO2cf7BVVRXHPzy0iXJGJsgsSZ9NE+E40RAhNuLIBGpZWjk1Y5MGhlRmmamJjbr4Fr0ohRgzichCnBxxIH1F1qsm+gH0AwwziaYfgwmGmiHlqKQI/bH2tcN+58d9555737uX9515s9/be+111vqetX/v80YcOHCAYQxjGMMoi67BNmAYwxhGe+OwgVaQ1A1sB241s1lVGyRpJXAmcLyZPVVSx5uAzcAcM7ulSvtagUY5aCf/mx1P4RmHDJ9paHabGlIzEUmTgQ8AC8s6C2Bm9wJ3AwskHVGReS1BFRy0s/9V41DnsxVtqkwn8jAwAbi6rEE56AH+AyytQNcXgKOBT1Sgq5WoioN28b+Z8QSHHp8xmt6mRgyVjVVJrwP+BHzDzOZWpHMb8BLgNWb2fBU6m4mqOWjUf0mzgG8B083sZ43a02oMNT5bjVa1qcMigbOBS4ETgJcB/wL+Aqwys5uDTDfRGjaZB8wHFgIzgCOAB4D5Zra2wL4LgRHAqgzjfwTMBM41s+8k8kfggf5B4ItmNi9R7Y5gzwygL0XnFOBy4BRgLLAb+ANO+p2R7PuAS4CJwIuAvwK3A4vN7L+RbCGPA+VA0uXADcAVZrYopXw8cD/wGzM7tR7/m42y8RTnU3FMleQSOiiegt5K2lRXouJcoDcY+j1gEXAPMAqYnWNoEscBvwW6gduC8ScCvZKmF9SdATwP/Dqj/EpgP74mG5nIvwF3dnnkLMCGkM6MlUm6CNgIvCuki4DvA0cBF0eyPcGXCfiLvgl/OT1An6TDE7KN8JjHwfqQTs2o+xVgJB6YNWT632xUFE/QnJgqwyV0VjxBRW0qORP5MPAsMNHMHkvWkjS2wNgaTsNHCCXq3g78MBi8Lq2SpJcCbwS2ZW3+mNnvJd2GO3c+sELSZ4BPAXcCH0mptimkydEESScAN+NrxWlmtjUqH5f4/WR8vb4DmGJmj4T8q4G7gHcE33pClVI81sHB74BngJNS6r4Xf6k3mtn9iaJU/1uEKuIJmhNTZbiEzoqnytpUvLG6D3gu5WGPZxkb4e/AgqhuH/AQMCWn3jF4z7+rQP81wF5gvqRLgM/jU6rzzWx/it3/DvLHRkUfxTvQz8UvPNTbmfjzwpAuqL3wILMPn7ruB+ZEKsrwmMuBmT2Hv8BXS3pVLT8Ey2LgMeC6qE6W/61Co/EETYipMlyGeh0TTwk03KaSM5Fv41OlrZJWAT8HNpjZPwuMSOK+jA2nHcDJOfXGhPSJPOVmtlPSEmAePuXcCLzHzJ7NqbYbeEWUV5vG/iDveQGTQvrTFHv+LGkncLyk0Wa2h/I81sPBBnwEmArU1rDXAeOA2eEFx0jzvx8kPYgvHdKwTlKcV3Svo4p4gubFVBkuobPiqZI29cJMxMwW49Oah/AjnLuARyWtC2fN9WBPRv4+8o+Tnwnpi+t4RpK8D5nZ0wXyoxL6axgd0ofreN6RIc3q0Xcl5RrgsR4OauvRkwAkvR64DPgVvgGZhjT/07AEUPTTG8puTSm7O09ZRfEEzYupMlxCZ8VTDQ21qYNOZ8xsJbBS0mjgLcC78elXn6QJ8ZqsQtT0jskTknQevunzCH5efSk+lcyS78Jf8PaoaE9Ij8GPwPJQG5GOBv6WUv7KSK4sj/VwsBE4wP9HvpvwKevHzKzfWX2O//1gZktS6s8CzgFWlDniHcR4gmI+B8QldGQ8VdKmUntyM9tjZveY2UXACvxYaVqeMQ1iF94bjs8SkPR2fITYCrwBf1lzwgiShfH4rvd9UX5tt/ptddi2JaSnpdj0Wnz6uz1MPQ/CAHks5MDMngC2AZMlvR94K7DMzLZkVMnyv6UYhHiCAj5LcAkdFk9VtankEe+ZktK+pTkqpEVTnNIIPf8vgLGByIMg6RRgNbATOD2sB6/FZ1ILc1TXRpl4B38pPh2+Nuysx88bl/jzmyG9RtLLEzIj8R68C7glkV+KxyIOEliPX/ZZBjyOb4xlIcv/pmMw4wnq5nMgXEIHxVOVbSpp3B3AXknrgQfx3mYa8GbgXuAnOYqrwBrgXOAM/OINAJImAmvx6d1MM9sFYGarJW0GzpE0zcx+maLzdPycvDeZaWZ/lHQx8DVgi6Re/PLOGGAy8CQwPchulPQl4NPAA5JWA0/ho86JeCBen1DfCI+pHETYAMzFL11dZma7c/Sl+t8iDHY8QTGfA+ESOiSeqm5TyeXMPHxTaRJ+OWY2cDhwFX7tud8RU8VYAzwKXFDLCD1oH752PcPM4jVk7XuL66N8JB2JX/xZa2Y74nIzW47fLFyLTy2vBM7GR6SvRrJXAefhgXEBvsHVhY9cM6Od7EZ47MdBCmpr0U0kRqwYRf63AIMdT1DMZ11cQufEUzPa1JD5dgZeuHDTA0wqWJvWo+vjwI3AqRk96pBEEQeSvgucBUw1s01xeUKuLf2vGnl81stlkG1LPlvRpobUvwIAvowfZX22ESWSRuE96pp2euEBmRyEDcB3AksLOpB29r9qpPJZL5dBtp35bHqbGlKdiJntxa/fbg63B8uiG/g6cEUVdrUSMQeSjpU0T9Jy/KOorfh6Og/dtKn/VSPi87gSXEIb89mKNjWkljPD6A/5B1jL8LsIPwY+aWb/GFSj2hTDXDYH/wOoKYZ7n2bm/QAAAABJRU5ErkJggg==",
      "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": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# trig expansions\n",
    "sympy.sin(x+y).expand(trig=True) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\log{\\left(a \\right)} + \\log{\\left(b \\right)}$"
      ],
      "text/plain": [
       "log(a) + log(b)"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# logarithmic expansions\n",
    "a, b = sympy.symbols(\"a, b\", positive=True)\n",
    "sympy.log(a*b).expand(log=True) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMgAAAAVCAYAAAAD+KFvAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjUuMiwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy8qNh9FAAAACXBIWXMAABJ0AAASdAHeZh94AAAHoElEQVR4nO2bf5CWVRXHP8uPErMkQZMRDNBkYEi2VVEbNWnQbGwsc6YxDYeIdGKSGnVFZnC+HKJNMk0bI0dTCyaVcq2dkFqHAfEHmWFspeBghYpGCRGYP0hXtj/OfeXx8jzvu++zz/M6jPv95+x7f5x7n/u9P845925TT08PRcPM7gM2SrqicOX9KBT9XFXHgJL0NgNdJenuR7Fopp+rTDTFJ4iZXQZcB1wg6a56FZrZocCLwIXAV4CTgaeBSyQ9mkPfaGAz8FNJ0+ut38s2lgBnAWMkvZJTx3HAOmCmpNuK7F9ZKJqrdwpl8pd2ghwf5ON5GgI+FuQVwAKgBfgHsMzMBuXUWRrM7HjgS8A1eQcXQNLjwK+AhWZ2UEHdKxv7FVdpKJu/tAVyFTAe30nyoBl4HThX0hpJTwFzgCOB0Tn0vRD6Mzdnf2qhDXgJ+FEBur4DHA7MLkBXI9BMsVy9EyiVv31MrL7CzO4C9ki6MJF2FPBXYJykTYU22AeY2THAU8CPJV1ckM6NwIHAWElv1ll3OnAHMEXSA0X0p0Z7+w1XaWgEf4OizCnAKuBaSVdGeecBM4ETgA8Az+JkLoomQjMQ2+AtwC7g75HOc4BvABOAQ4B/4yfXMkmLQ5nRpPggyXRgPnANMBU4CHgCmC9peY3xmAE0AcvSMs3sAuDs8M0jgG588iyWdEeGzrtDf6YCnTXaLwVlcBX0TgYuB04BhgM7gL/gE/TnUdkvAF8HJgHvwcftTuB6Sf+LytacBxkonb/YxGoJcn2ikYFhp7kHOBr4BbAY2AN8G7g9UfZA4CPA4ERaU/j4pZK6E+kXAx34oPwaDwysAIYAX87ofBo+DDyGmwRL8cGaCHSEBV8NU4E3gX0cUjN7P774xgIPATcB9wJjgNvNbE6GzkeCPKOObygEZXEV8r4KrAU+F+R1wH3AYcCsqGwbzsN4fFHchE/kNqDTzJJt9mUelM5f7IhVFsgfE2k3AufjO/TVlYEzs1bgAeAiM1skaQNwLNADTDOzVfhOMB8YCZwTtXUJbv9OkvRi9HHDMzqfhtPx08IS9e8Efgu0AqvTKpnZ+/AddGOGc9cDjJL0z6jePGATTt6ilHp/CPK0Or6hKJTClZlNwBfaS8Cpkp6M8kcm/j4Z9xe3AJMr42dmc4FfAp/BeWkLVXLNg0bxl3aCvExw0M3sRHx36JA0N7mrSHoDX6EAJwbZDPwNmIfvXn/Gd4KTJO1I6Uw38EacKGl7StksPAssjOp3As8Bk6vUOwIYCGxNy5T0cjy4IX0rHuk5JKPeLmA37ug2DCVz9TV8M/1WvDiC/ucTP2cEuTA5fqE/l+On2cxIRZ550BD+3jpBwoo8BlgraU9IvhQ/Gl81s/kp+iYG2RSU3wzcHNLuzf42AH6GH6dPmtkyYA3wiKRtNerF6Mpwhrfgcf0sDAvyP2mZZvZB3IY+GxiH2/LJDWV9Wr2AHcCHquRjZs/g5mEaVptZnFbrHqhMrk4K8jc1ysFeK2RVnCFpk5k9D4wxs6GSdpJ/HjSEv6SJNSkoSJpXZwb5xRqdfa5G/j6QdL2Zbcd3vdnAN4EeM1sDtEpa10tVOzPSu6n+UuC1IA+IM8zsWOB+fJAewx23HfguNwa4CPhTFd1DEvqzcAMwNEprBj6L7/bPRHldNfSVxhV7+/lCL8oeHGTqzh7SjwzldvZhHjSEv+QCeZuDbmYHAIcCD0r6RBVluSFpCbDEzIYCHwfOxY/oTjMbH9ukBaOie1hK3lJ8UuwTbjWzBeHPVOLMbECou7la45JuSKk7HV8gP6knzNsArnYGeQQeVq2GXUEejptwMUZE5fLOg4bwl7ZAKidIU5D1OMy5EI7aFcCK0MEZwKlAe4nNbgW24cfvWzCzUbgD25kyuEPZG1nJemkwDh+7ruK6WhNlc/Uo/sLi09ReIOvxuXQ60QIxs6PxIMDmwPnbUOc8aAh/SROkBXdONoTOvoY7bhPM7PNpmszsFDMbmNFQVZjZWRnPGQ4L8tU8ensLST3Ag8DwQFwFu4McG4Ujh+Ghy5G4+daVobpir6dGz8pA2Vzht9TdwNUhohXrHpn4WQklzwtvvSplBgLfw+fcbYn0XPOgUfwNCpXfi8ehu6L4dyse6243s5U4CQPwo/Y4YLCkvNGau4HdZvYwbm834bvFCfjqXplTbz1oB84DPoVfICFpWwh7fhL4ffjuEfjuuRqPwmyQtDtdJWfisfmOkvseozSuJG0ws1m4U7/ezDrwSOcw/GT5LzAllF1rZt8FrgSeMLN7gFfw8ZsIPAxcm1Dfl3lQOn+VE2QifmGUdNCRdD9uE7YDH8WdqGn4BdBK6rvQi3EV8Dv85JoVdA3G3wJNCaHJstEO/At32pI4H7djR+EhzqOAy/DY/QCy7deD8Yu05ZK2lNPldJTMFZJuxW/Ql+PmUyt+X7Id+GFUdg4eLHgaH9vZ+LjNA86Q9HqieF/mQen8Ff4Wa39DuMBqA1okVQv99UbXpcAPgNMkPVRE//pRHWXzV9Y/TO1P+D4e+lxQq2A1mNkQ/Aa5vX9xNBSl8veuXyDBFp0GrAuXpXkxGrgF/9+KfjQIZfP3rjex+tGPavg/ktPoKjy74CsAAAAASUVORK5CYII=",
      "text/latex": [
       "$\\displaystyle i e^{b} \\sin{\\left(a \\right)} + e^{b} \\cos{\\left(a \\right)}$"
      ],
      "text/plain": [
       "   b           b       \n",
       "ⅈ⋅ℯ ⋅sin(a) + ℯ ⋅cos(a)"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# separating real & imag parts\n",
    "sympy.exp(I*a + b).expand(complex=True) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle a^{x} b^{x}$"
      ],
      "text/plain": [
       " x  x\n",
       "a ⋅b "
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# power expressions - expanding the base & exponent\n",
    "sympy.expand((a*b)**x, power_exp=True) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle e^{i a x} e^{- i b x}$"
      ],
      "text/plain": [
       " ⅈ⋅a⋅x  -ⅈ⋅b⋅x\n",
       "ℯ     ⋅ℯ      "
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.exp(I*(a-b)*x).expand(power_exp=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Factoring, Collecting, Combining\n",
    "\n",
    "__expand__ often used to expand a function, cancel some terms, then refactor or recombine the expression. __factor__ helps do this."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left(x - 1\\right) \\left(x + 1\\right)$"
      ],
      "text/plain": [
       "(x - 1)⋅(x + 1)"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.factor(x**2 - 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle x \\left(\\sin{\\left(z \\right)} + \\cos{\\left(y \\right)}\\right)$"
      ],
      "text/plain": [
       "x⋅(sin(z) + cos(y))"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.factor(x*sympy.cos(y) + x*sympy.sin(z))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\log{\\left(\\frac{a}{b} \\right)}$"
      ],
      "text/plain": [
       "   ⎛a⎞\n",
       "log⎜─⎟\n",
       "   ⎝b⎠"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.logcombine(sympy.log(a) - sympy.log(b))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left( x y z + x + y, \\  x \\left(y z + 1\\right) + y, \\  x + y \\left(x z + 1\\right)\\right)$"
      ],
      "text/plain": [
       "(x⋅y⋅z + x + y, x⋅(y⋅z + 1) + y, x + y⋅(x⋅z + 1))"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# use collect for fine-grained factor control\n",
    "# collect factors terms containing a given symbol or list of symbols.\n",
    "expr = x + y + x*y*z\n",
    "expr.factor(), expr.collect(x), expr.collect(y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left(\\sin{\\left(x \\right)} + \\cos{\\left(x \\right)}\\right) \\left(- \\sin{\\left(y \\right)} + \\cos{\\left(y \\right)}\\right)$"
      ],
      "text/plain": [
       "(sin(x) + cos(x))⋅(-sin(y) + cos(y))"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# collect also supports method chaining\n",
    "expr = sympy.cos(x+y) + sympy.sin(x-y)\n",
    "\n",
    "expr.expand(trig=True).collect(\n",
    "    [sympy.cos(x), \n",
    "     sympy.sin(x)]).collect(sympy.cos(y) - sympy.sin(y))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Together, apart, cancel - rewriting of fractions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHQAAAAcCAYAAACwPZsQAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjUuMiwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy8qNh9FAAAACXBIWXMAABJ0AAASdAHeZh94AAAC8ElEQVR4nO2ZP2hTURSHv4jgYAVx0sU6iI4tWBwqFdulVgVRKCpUyKo4VrBFPP6EIhKE2qG42YIuUlwU20GxIEi3+g8nC1WogkJAXFSEOLwnPELSJnn3vhtCPniEd3PfPeeXw3n3npxcqVSiTeuwOaRxSYeBy8ABYBcwbGZzIX3KAp+6N7lYJAVbgTfApcB+ZI033UEz1MzmgXkASSFdyRSfukNnaBvHtAPaYrQD2mIE3UObBUl54B7Qb2aLYb1JRztDW4zQdWgHsDcxtEdSN1A0s89hvPKPT92hX7k9wIvEfSH+nAXymXuTHd50h65DF4FcSB9C4FN3LvlfrqRVoLOO5x+Y2Yhrp3zSgMZZM8v78cY95Rm6Avyq4/kvDn3Jiklge9lYN3CS6JW3Wvbda8/+OCXnotsiqelaNmZW8yut0bKl2XSbWc7JHlrPj9dKNKPudh3aYoQuW+pG0hhwGtgP/AaWgDEzex/UMY/U0z9tigyVNCPpeo3TjwDTQC8wAPwFnkna4cc7f9Shu+b+qbMMlTQM3Af2mdmneOwOcAzoNbPvLuyY2WCZ3fPAD+AQ8LjBNWeAmUaezUJ3Pf1Tlxk6B7wDrsaGR4GzwFFXwazCNiIdRY821iOU7oo4C6iZlYBxIC/pCnANOG5mK65sVGGSqFZc8mynIgF1V8RJHZpE0ivgIHDCzBaqzBkn+hH+swUoAX8SY0Nm9nIDWwVgBOgzs4+pHE9JVrrj2rfqocjpKVfSANBFlPnf1pl6F3iYuL8FrAFTibG1DWzdJgpmfxMEMzPdG+HyUNQFPAIuAqeACWCo0lwzK5LY8yT9JGod1RQYSVPAGaJgfkjpeiqy1F0LTvZQSZ3AU6BgZrNE+8igpD4X65fZmiZqMZ0DipJ2xleHa1s1+JKJbkkdkrrjninE/VNJu8vnpg5oXP8tAE/MbALAzN4Snf5upl2/AheITrbPga+Ja9SDrapkrLsHWI4viPqny8CN8onOD0VtwvIPQ8BLAfSsuYoAAAAASUVORK5CYII=",
      "text/latex": [
       "$\\displaystyle - \\frac{1}{x + 2} + \\frac{1}{x + 1}$"
      ],
      "text/plain": [
       "    1       1  \n",
       "- ───── + ─────\n",
       "  x + 2   x + 1"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# rewrite fraction as a partial fraction\n",
    "sympy.apart(1/(x**2 + 3*x + 2), x) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\frac{y + 1}{y \\left(x + 1\\right)}$"
      ],
      "text/plain": [
       "  y + 1  \n",
       "─────────\n",
       "y⋅(x + 1)"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# combine partial into single fraction\n",
    "sympy.together(1/(y*x+y) + 1/(1+x)) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\frac{1}{x + 1}$"
      ],
      "text/plain": [
       "  1  \n",
       "─────\n",
       "x + 1"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# cancel shared factors btwn numerator, denominator\n",
    "sympy.cancel(y/(y*x+y)) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Substitutions\n",
    "\n",
    "Th most basic use case: method called on an expression. The 1st argument = symbol/expr to be replaced; the 2nd argument = new symbol/expr."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle 2 y$"
      ],
      "text/plain": [
       "2⋅y"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(x+y).subs(x,y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\sin{\\left(y e^{y} \\right)}$"
      ],
      "text/plain": [
       "   ⎛   y⎞\n",
       "sin⎝y⋅ℯ ⎠"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.sin(x*sympy.exp(x)).subs(x,y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Instead of chaining multiple __subs__ calls when multiple substitutions are required, we can pass a dictionary as the only argument. It maps old symbols or expressions to new ones."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\cos{\\left(y e^{y} \\right)}$"
      ],
      "text/plain": [
       "   ⎛   y⎞\n",
       "cos⎝y⋅ℯ ⎠"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.sin(x*z).subs({z: sympy.exp(y), x: y, sympy.sin: sympy.cos})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Typical use case: substitute numerical values for symbolic numbers. A convenient way of doing this is to define a dictionary that translates the symbols to numerical values, and passing this dictionary to the subs method."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle 13.3$"
      ],
      "text/plain": [
       "13.3000000000000"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = x*y + z**2*x\n",
    "values = {x: 1.25, y: 0.4, z: 3.2}\n",
    "expr.subs(values)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Numerical evaluation\n",
    "\n",
    "Sooner or later you'll need to evaluate symbolic expressions numerically (for plots or  numerical results). A SymPy expression can be evaluated using either __sympy.N__ or the __evalf__ method."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left( 4.14159265358979, \\  3.1415926535897932385\\right)$"
      ],
      "text/plain": [
       "(4.14159265358979, 3.1415926535897932385)"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 2nd argument = #significant digits\n",
    "sympy.N(1+pi), sympy.N(pi, 20)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle x + 0.3183098862$"
      ],
      "text/plain": [
       "x + 0.3183098862"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(x + 1/pi).evalf(10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "When we need to evaluate an expression for multiple inputs, we can loop the evaluation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left[ 0, \\  0.774, \\  0.642, \\  0.722, \\  0.944, \\  0.205, \\  0.974, \\  0.977, \\  -0.87, \\  -0.695\\right]$"
      ],
      "text/plain": [
       "[0, 0.774, 0.642, 0.722, 0.944, 0.205, 0.974, 0.977, -0.87, -0.695]"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = sympy.sin(pi*x*sympy.exp(x))\n",
    "[expr.subs(x, xx).evalf(3) for xx in range(0, 10)]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Looping is slow. SymPy provides __sympy.lambdify__ instead. It takes a set of free symbols and an expression, and generates a function that evaluates the value of the expression.\n",
    "\n",
    "The function takes the same number of arguments as the number of free symbols passed as first argument to sympy.lambdify."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle 0.773942685266709$"
      ],
      "text/plain": [
       "0.773942685266709"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr_func = sympy.lambdify(x, expr)\n",
    "expr_func(1.0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "__expr_func__ expects numerical (scalar) values, so we can't pass a symbol. SymPy can generate functions that are NumPy-aware. Below: How a SymPy expression is converted to NumPy-aware vectorized function for efficient evaluation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle 0.773942685266709$"
      ],
      "text/plain": [
       "0.773942685266709"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr_func = sympy.lambdify(x, expr, 'numpy')\n",
    "expr_func(1.0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 0.        ,  0.77394269,  0.64198244,  0.72163867,  0.94361635,\n",
       "        0.20523391,  0.97398794,  0.97734066, -0.87034418, -0.69512687])"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import numpy as np\n",
    "xvalues = np.arange(0, 10)\n",
    "expr_func(xvalues)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Calculus - Derivatives\n",
    "\n",
    "The derivative of a function describes a rate of change at a given point. We can use __sympy.diff__ or the __diff__ method of SymPy expressions. These functions accept a symbol or symbols for which the function or the expression is to be derived with respect to."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = sympy.Function('f')(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left( \\frac{d}{d x} f{\\left(x \\right)}, \\  \\frac{d^{2}}{d x^{2}} f{\\left(x \\right)}, \\  \\frac{d^{3}}{d x^{3}} f{\\left(x \\right)}\\right)$"
      ],
      "text/plain": [
       "⎛            2          3      ⎞\n",
       "⎜d          d          d       ⎟\n",
       "⎜──(f(x)), ───(f(x)), ───(f(x))⎟\n",
       "⎜dx          2          3      ⎟\n",
       "⎝          dx         dx       ⎠"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# first-order derivatives\n",
    "sympy.diff(f,x), sympy.diff(f,x,x), sympy.diff(f,x,3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left( \\frac{\\partial^{2}}{\\partial y\\partial x} g{\\left(x,y \\right)}, \\  \\frac{\\partial^{5}}{\\partial y^{2}\\partial x^{3}} g{\\left(x,y \\right)}\\right)$"
      ],
      "text/plain": [
       "⎛   2                5           ⎞\n",
       "⎜  ∂                ∂            ⎟\n",
       "⎜─────(g(x, y)), ───────(g(x, y))⎟\n",
       "⎜∂y ∂x             2   3         ⎟\n",
       "⎝                ∂y  ∂x          ⎠"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# multivariate function derivatives\n",
    "g = sympy.Function('g')(x,y)\n",
    "g.diff(x,y), g.diff(x,3,y,2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can evaluate derivatives of defined functions, which result in new expressions that correspond to the evaluated derivatives. Below: using sympy.diff to evaluate derivatives of arbitrary mathematical expressions, such as polynomials."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left( 4 x^{3} + 3 x^{2} + 2 x + 1, \\  2 \\cdot \\left(6 x^{2} + 3 x + 1\\right)\\right)$"
      ],
      "text/plain": [
       "⎛   3      2              ⎛   2          ⎞⎞\n",
       "⎝4⋅x  + 3⋅x  + 2⋅x + 1, 2⋅⎝6⋅x  + 3⋅x + 1⎠⎠"
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = x**4 + x**3 + x**2 + x+1\n",
    "expr.diff(x), expr.diff(x,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle 6 y \\left(x + 1\\right)^{2}$"
      ],
      "text/plain": [
       "           2\n",
       "6⋅y⋅(x + 1) "
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = (x+1)**3 * y**2 * (z-1)\n",
    "expr.diff(x, y, z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle y \\cos{\\left(\\frac{x}{2} \\right)} \\cos{\\left(x y \\right)} - \\frac{\\sin{\\left(\\frac{x}{2} \\right)} \\sin{\\left(x y \\right)}}{2}$"
      ],
      "text/plain": [
       "                       ⎛x⎞         \n",
       "                    sin⎜─⎟⋅sin(x⋅y)\n",
       "     ⎛x⎞               ⎝2⎠         \n",
       "y⋅cos⎜─⎟⋅cos(x⋅y) - ───────────────\n",
       "     ⎝2⎠                   2       "
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# trig function differentials\n",
    "expr = sympy.sin(x*y) * sympy.cos(x/2)\n",
    "expr.diff(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\frac{2^{x} \\sqrt{\\pi} \\operatorname{polygamma}{\\left(0,\\frac{1}{2} - \\frac{x}{2} \\right)}}{2 \\Gamma\\left(\\frac{1}{2} - \\frac{x}{2}\\right)} + \\frac{2^{x} \\sqrt{\\pi} \\log{\\left(2 \\right)}}{\\Gamma\\left(\\frac{1}{2} - \\frac{x}{2}\\right)}$"
      ],
      "text/plain": [
       " x             ⎛   1   x⎞               \n",
       "2 ⋅√π⋅polygamma⎜0, ─ - ─⎟    x          \n",
       "               ⎝   2   2⎠   2 ⋅√π⋅log(2)\n",
       "───────────────────────── + ────────────\n",
       "           ⎛1   x⎞             ⎛1   x⎞  \n",
       "        2⋅Γ⎜─ - ─⎟            Γ⎜─ - ─⎟  \n",
       "           ⎝2   2⎠             ⎝2   2⎠  "
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# special function differentials (updated: formerly \"sympy.special...\")\n",
    "expr = sympy.functions.special.polynomials.hermite(x,0)\n",
    "expr.diff(x).doit()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Calling sympy.diff results in a new expression.\n",
    "\n",
    "If instead we want to represent a derivative of a definite expression, create an instance of the Derivative class. 1st argument = the expression; 2nd argument = the derivative.\n",
    "\n",
    "This (formal) representation can be evaluated using __doit__."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left( \\frac{d}{d x} e^{\\cos{\\left(x \\right)}}, \\  - e^{\\cos{\\left(x \\right)}} \\sin{\\left(x \\right)}\\right)$"
      ],
      "text/plain": [
       "⎛d ⎛ cos(x)⎞    cos(x)       ⎞\n",
       "⎜──⎝ℯ      ⎠, -ℯ      ⋅sin(x)⎟\n",
       "⎝dx                          ⎠"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d = sympy.Derivative(sympy.exp(sympy.cos(x)), x)\n",
    "d, d.doit()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Integrals\n",
    "\n",
    "Integrals are evaluated using __sympy.integrate__.\n",
    "\n",
    "Formal integrals represented using __sympy.Integral__.\n",
    "\n",
    "Two forms: __definite__ (specified integration limits, can be interpreted as an area or volume) & __indefinite__ (no integration limits)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [],
   "source": [
    "a, b = sympy.symbols(\"a, b\")\n",
    "x, y = sympy.symbols('x, y')\n",
    "f    = sympy.Function('f')(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left( \\int f{\\left(x \\right)}\\, dx, \\  \\int\\limits_{a}^{b} f{\\left(x \\right)}\\, dx\\right)$"
      ],
      "text/plain": [
       "⎛           b        ⎞\n",
       "⎜⌠          ⌠        ⎟\n",
       "⎜⎮ f(x) dx, ⎮ f(x) dx⎟\n",
       "⎜⌡          ⌡        ⎟\n",
       "⎝           a        ⎠"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.integrate(f), sympy.integrate(f,(x,a,b))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle - \\cos{\\left(x \\right)}$"
      ],
      "text/plain": [
       "-cos(x)"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.integrate(sympy.sin(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\cos{\\left(a \\right)} - \\cos{\\left(b \\right)}$"
      ],
      "text/plain": [
       "cos(a) - cos(b)"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.integrate(sympy.sin(x), (x,a,b))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\frac{\\sqrt{\\pi}}{2}$"
      ],
      "text/plain": [
       "√π\n",
       "──\n",
       "2 "
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Definite integrals can use limits from negative infinity to positive infinity.\n",
    "sympy.integrate(sympy.exp(-x**2), (x,0,oo))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [],
   "source": [
    "a,b,c = sympy.symbols(\"a, b, c\", positive=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADUAAAAUCAYAAAAtFnXjAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjUuMiwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy8qNh9FAAAACXBIWXMAABJ0AAASdAHeZh94AAADNElEQVR4nN3XX4hVVRQG8N+YRZGJFRZKE2ShD4aNRUFEoL2USBERZhZKTxEFYYT6EKxWkFBQRIQ92R8TMqGwSIIQE5LAsBotfYgew4TILKxMxelh7yvX47njDN0h7INhn1l7r3W+b5+91tp3YGRkxP8Nk/5rAhOByeNZnJl34UNc0C8CETHQr1gdjEsUlmNlRKzrN5F+YszHLzOnYhE2TRyd/mA8ObUE2yPi0ESR6RfGI2oF3p4oIv3EmHIqM2dhDj7pMb8YH4/xnfMjYrjhvwyLcTNm4AR+wLqIeHMUXgvwGG7DdBzC3kmNRRdn5owW/+V4NyKO94j/G16vz/uQXX/vVftOBPY23nmJcgJm4XO8hg9wDd7IzNU9BL2Cz3AHtuHl+v9Nk+uC2ZXAPdiCh7qcB6qo+3sIEhE7M3O6smtbIuLZLv9VeABvRcT6FvcRDEbEwQbpZ/A9HsELjbm1eBLvY0VE/NE1N6XzpX7HS/gK92bmlK4Yt+PPiPi6l6iKoToON+w39rCDiDjSFFTtP+EALmsImo/V2I1l3YI68SbXh4M4mJkbq4j7sKGuG2uB6Ija0yLqBL5rc8rMS/GEklNzMNXpBeybhstTdX5NRBxri9ksFJvxKh7Ghsy8SBE4t7eWUxjCESXBO4Sn4Drsi4i/WwTNw6e4El8qPfAQjis5tdyZm3QnflXypxWniYqIw5m5VTmCM7EQuyLiwGhq6m5fjS8iovuGPB8Dehw9vINpWBgROxoxn6uPu7tsFypVbjgiTvbi09anNlb7g8pO/dujR4uozBzEPOxoETRNKRCUPO+gc0+8YjQybaK2Kp/3cdyiVMOzYaiOww379XX8tsXnaB1nZeb5HWNmXq60gauUXDwVMyL+UnJzZmYuaQbMzNmZed4ZzTcijmXmZjyK9TXQ2TBUx+aX6lSuuZm5PyJ+7HrPz5m5XekzuzJzm9J4Fyn5chL7I+JoI+YafIRNmblC6YvTKofBiJjR65q0sY4besw3cUMl0fwim/ELnsfSFr+lSl4NKj3uWqW6rVVO0e6mQ0RsxQLldnMrVuJuHMbTMND2y7c23BexqpH45wRaRZ3r+Ac+mhQRGroOxgAAAABJRU5ErkJggg==",
      "text/latex": [
       "$\\displaystyle \\sqrt{\\pi} a c$"
      ],
      "text/plain": [
       "√π⋅a⋅c"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.integrate(a * sympy.exp(-((x-b)/c)**2), (x, -oo, oo))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Computing integrals symbolically is a difficult problem. When SymPy fails to evaluate an integral, an instance of __sympy.Integral__, representing the formal integral, is returned instead."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\int \\sin{\\left(x \\cos{\\left(x \\right)} \\right)}\\, dx$"
      ],
      "text/plain": [
       "⌠                 \n",
       "⎮ sin(x⋅cos(x)) dx\n",
       "⌡                 "
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.integrate(sympy.sin(x * sympy.cos(x)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Multivariable expressions can be integrated. In the case of indefinite integral of a multivariable expression, the integration variable has to be specified explicitly."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle - e^{- y} \\cos{\\left(x e^{y} \\right)}$"
      ],
      "text/plain": [
       "  -y    ⎛   y⎞\n",
       "-ℯ  ⋅cos⎝x⋅ℯ ⎠"
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = sympy.sin(x*sympy.exp(y))\n",
    "sympy.integrate(expr, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\frac{x^{3}}{3} + x^{2} y + x y^{2}$"
      ],
      "text/plain": [
       " 3              \n",
       "x     2        2\n",
       "── + x ⋅y + x⋅y \n",
       "3               "
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = (x+y)**2\n",
    "sympy.integrate(expr,x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By passing more than one symbol, or more than one tuple that contain symbols and their integration limits, we can carry out multiple integration."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAANwAAAAXCAYAAABgdSH2AAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjUuMiwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy8qNh9FAAAACXBIWXMAABJ0AAASdAHeZh94AAAHBElEQVR4nO2bZ4wVZRSGn4WNMfaoUaJibz8sJBYIIraoKCIsRhONBWMvQVCMYsTjaywYLGtDEQ3F8sMo2LCLFSwoIgJWIkYjioiSYAN0/XG+K8Ps3L0z9869e132SchwZ86c+8275yvnfHMbWlpa6KSTTmpDYyU3S7oIOA/YMZyaD1xvZtMqbFeHRtJIYDCwB/AX8C4w0szm1aPfjkytY7hLhfd/B1wJ7AfsD0wHnpS0T6UN6+AcCowFegOHA6uBVyRtXqd+OzI1jeGGvJeUkpbho+q4XB13YCRtBCwHBpnZM/Xut6NTzRiuaEkZRVJX4CRgI2BmTj63Br4H7jGzoeHczfhItDuwJfAH8A3wJHC3mf2cx3fXmI3x1cayWvqN6gsIaAL6A3sD2wIrgU+ACcAEM/sn5/bVFWljOB6XWWKy0iUlkvaWtALPGcbio+knlfoNDMTbODVybjiwIfAycAfwCL50uhaYK6l7Tt9dS5qBOXjOVUu/UX1PBMYDPYH3wr1PAHsBDwCPSWrIuX11QRkxHI/L1DGZxwz3OdAD2Aw4AZgs6dCcEvUm4Gfgzci5Tczsz7ihpBuAq4CRwIVZvkTSEHwUP8zMXi+3seUgaQzQFzjYzP6usd+ovg3A8cC06Ewm6SrgffxvOxjvhFnbMoR20jclWWM4HpepY7IxZvAScCRwgplNiZxvwAU7A7jZzK4sXDOzlcBX4eMHkg4AhgFnh3svA24BRpjZrQmN2gOYC7xnZn0j5zfFE/9HowGT9GCBx8LD7VbketUoR7dw/VbgVDwQv4pdK0u3Un4jNnF9pyfZmdkPku4DbsCLMpk7XKVk0bcc3UrFcOz+VnGZJSbjS8rLgX+A68N6tsAt4aHGx4MmgS7A+pHPb4djryL2dwFdgYtj5/sD6wFTWt2RzIBwnJvSPk8y6ybpTtZ0igUJPsvSLYXfAln0XRWOq1PYVoMs+pYbb1HiMRwli26tYnKtGc7MPpb0EP4QpwETw5LiUry3nh+1lzQamAZ8iyfop+CjYP+I2Ww8iewZb42kE/GR604zi3eUJuA3fF3cCkkj8OR2Uzxh7RMebHSSfTUpQ7exeKcYBCyT1C1cWmFmK8L/M+uW0m+BNvWN+GwETg8fX2jLtlpk1DeTbiljOEpR3dLEZFLR5GrgT+BaSRfjS4kXgdMSqlTdgIfxNfCrwAHAMWb2fMHAzFYBs4DukraJNG5D4DZgCXBNrOHrA/2A59uYrkcAhk/9ffBgOMrMfipiX22y6HYB/sd9FVgc+TeiYFCObmn8Bh9p9C0wGi+cPGdmL5awrSap9C1Dt5IxHPFRSreSMdmqaGJm30lqxjcD78LLo4PDOjduOyThS5OYgSfwvVgzFV8DbAecaWbLY/ZH4SPFVIpgZt3gvxJtbzwwPpJ0nJnNLnafpEXADkUuvyYpfm5SmufMqFvaal8m3TL4LakvgKShwGXAZ/jMUpJ60JcMumWIYSihW5qYLFaljM4SZ5nZ7xkalcSMcOwJTJG0J15KfQeYlGDfhO8BlXy9xsx+BKZKmg18AUzGR+RiNOPVqCg98FLvJGBR7NqcUm2I0N66paWkvuGVpzuABcARZpZ2j7CZ9te33XSDtmOyVYeTdDKejP6AT7eX4EuVSpgJtLAmkb0bT1wvMrO1XnUJSfEAYHrCzFcUM/tG0gKgh6QtzWxpEbvm+LlQth4ITCy3bN3eumVoZ0l9JQ0Dbgfm4Z1tSVr/daJvu+gWJykm18rhJB2LjwDzgX3wpcTZYYQoGzP7BfgU2F/SKcARwDgz+yjBvC+wBSWWO0UorNlz289KQ53olpY29ZV0Bd7Z5uCVztSdrVpk1bc9dGuDtWLyvw4nqQ/wOP4yZyHRG4XPgnlU/t4GNgDGAUvxJDiJwXgJ+Kn4BUl7Ripv0fNdwibjVsDMIHhNqCPd0tKWvqPwNn+Iz2yJq4RaUoG+NdEta0w2hov7As/iL7oeaWaLAczscUkfAAMlHWxmb1XQ4BnAuXjSObyNnGBQaOCPCdf6AWMkvQksxHf7twYOAXbGlxvnVNDGTNSZbmkZRIK+ks4ArsNH4reAoQkFjkVmNrHC709NhfrWRDcyxmSjpF3x8moLcLSZLYw5HInvOYyh+GZiGr4Ox1nAg0kGYYd/O3xJk8QrwP3AQcC+eHL+G56YPoTvr+T9AnAi9aRbWkrou1M4dsXL2km8AUyspA1pyUHfWumWKSZz/3lOMSQ9jW8m9jKzWUVsbsSF3NnMvk6yWddIo1sGX+uMvvWqW8W/FkhDSFwHAPeWePgm4OOOHgxpyaBbWtYJfetZt6rNcJK2x1+T2QV/NehL4MAc9qY6NJ26lcf/RbfcfoCaQD/gJuBXvLIzrN4evk7p1K08/he61SyH66STTuBfU6p+2iDHJCYAAAAASUVORK5CYII=",
      "text/latex": [
       "$\\displaystyle \\frac{x^{3} y}{3} + \\frac{x^{2} y^{2}}{2} + \\frac{x y^{3}}{3}$"
      ],
      "text/plain": [
       " 3      2  2      3\n",
       "x ⋅y   x ⋅y    x⋅y \n",
       "──── + ───── + ────\n",
       " 3       2      3  "
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.integrate(expr,x,y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\frac{7}{6}$"
      ],
      "text/plain": [
       "7/6"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.integrate(expr,(x,0,1),(y,0,1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Series expansions\n",
    "\n",
    "With a series expansion, a function can be written as a polynomial with coefficients given by the derivatives of the function at the point where the expansion is made. By truncating the expansion at some order n, the nth order approximation of the function is obtained."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle f{\\left(0 \\right)} + x \\left. \\frac{d}{d \\xi} f{\\left(\\xi \\right)} \\right|_{\\substack{ \\xi=0 }} + \\frac{x^{2} \\left. \\frac{d^{2}}{d \\xi^{2}} f{\\left(\\xi \\right)} \\right|_{\\substack{ \\xi=0 }}}{2} + \\frac{x^{3} \\left. \\frac{d^{3}}{d \\xi^{3}} f{\\left(\\xi \\right)} \\right|_{\\substack{ \\xi=0 }}}{6} + \\frac{x^{4} \\left. \\frac{d^{4}}{d \\xi^{4}} f{\\left(\\xi \\right)} \\right|_{\\substack{ \\xi=0 }}}{24} + \\frac{x^{5} \\left. \\frac{d^{5}}{d \\xi^{5}} f{\\left(\\xi \\right)} \\right|_{\\substack{ \\xi=0 }}}{120} + O\\left(x^{6}\\right)$"
      ],
      "text/plain": [
       "                             ⎛  2      ⎞│         ⎛  3      ⎞│         ⎛  4      ⎞│         ⎛  5      ⎞│           \n",
       "                           2 ⎜ d       ⎟│       3 ⎜ d       ⎟│       4 ⎜ d       ⎟│       5 ⎜ d       ⎟│           \n",
       "                          x ⋅⎜───(f(ξ))⎟│      x ⋅⎜───(f(ξ))⎟│      x ⋅⎜───(f(ξ))⎟│      x ⋅⎜───(f(ξ))⎟│           \n",
       "                             ⎜  2      ⎟│         ⎜  3      ⎟│         ⎜  4      ⎟│         ⎜  5      ⎟│           \n",
       "         ⎛d       ⎞│         ⎝dξ       ⎠│ξ=0      ⎝dξ       ⎠│ξ=0      ⎝dξ       ⎠│ξ=0      ⎝dξ       ⎠│ξ=0    ⎛ 6⎞\n",
       "f(0) + x⋅⎜──(f(ξ))⎟│    + ────────────────── + ────────────────── + ────────────────── + ────────────────── + O⎝x ⎠\n",
       "         ⎝dξ      ⎠│ξ=0           2                    6                    24                  120                "
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# undefined functions: expansion is computed up to 6th order around x0=0.\n",
    "x = sympy.Symbol(\"x\")\n",
    "f = sympy.Function(\"f\")(x)\n",
    "sympy.series(f,x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle f{\\left({x_0} \\right)} + \\left(x - {x_0}\\right) \\left. \\frac{d}{d \\xi_{1}} f{\\left(\\xi_{1} \\right)} \\right|_{\\substack{ \\xi_{1}={x_0} }} + O\\left(\\left(x - {x_0}\\right)^{2}; x\\rightarrow {x_0}\\right)$"
      ],
      "text/plain": [
       "                       ⎛ d        ⎞│            ⎛           2           ⎞\n",
       "f({x_0}) + (x - {x_0})⋅⎜───(f(ξ₁))⎟│         + O⎝(x - {x_0}) ; x → {x_0}⎠\n",
       "                       ⎝dξ₁       ⎠│ξ₁={x_0}                             "
      ]
     },
     "execution_count": 79,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# to specify point around which to expand, provide x0 keyword.\n",
    "# n=2 tells function to return expansion up to 2nd order term.\n",
    "x0 = sympy.Symbol(\"{x_0}\")\n",
    "f.series(x, x0, n=2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(x - {x_0}\\right) \\left. \\frac{d}{d \\xi_{1}} f{\\left(\\xi_{1} \\right)} \\right|_{\\substack{ \\xi_{1}={x_0} }} + f{\\left({x_0} \\right)}$"
      ],
      "text/plain": [
       "            ⎛ d        ⎞│                   \n",
       "(x - {x_0})⋅⎜───(f(ξ₁))⎟│         + f({x_0})\n",
       "            ⎝dξ₁       ⎠│ξ₁={x_0}           "
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# use remove0() to remove order term\n",
    "f.series(x, x0, n=2).removeO()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle 1 - \\frac{x^{2}}{2} + \\frac{x^{4}}{24} + O\\left(x^{6}\\right)$"
      ],
      "text/plain": [
       "     2    4        \n",
       "    x    x     ⎛ 6⎞\n",
       "1 - ── + ── + O⎝x ⎠\n",
       "    2    24        "
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# some well-known series expansions:\n",
    "sympy.cos(x).series()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle x - \\frac{x^{3}}{6} + \\frac{x^{5}}{120} + O\\left(x^{6}\\right)$"
      ],
      "text/plain": [
       "     3     5        \n",
       "    x     x     ⎛ 6⎞\n",
       "x - ── + ─── + O⎝x ⎠\n",
       "    6    120        "
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.sin(x).series()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle 1 + x + \\frac{x^{2}}{2} + \\frac{x^{3}}{6} + \\frac{x^{4}}{24} + \\frac{x^{5}}{120} + O\\left(x^{6}\\right)$"
      ],
      "text/plain": [
       "         2    3    4     5        \n",
       "        x    x    x     x     ⎛ 6⎞\n",
       "1 + x + ── + ── + ── + ─── + O⎝x ⎠\n",
       "        2    6    24   120        "
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.exp(x).series()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle 1 - x + x^{2} - x^{3} + x^{4} - x^{5} + O\\left(x^{6}\\right)$"
      ],
      "text/plain": [
       "         2    3    4    5    ⎛ 6⎞\n",
       "1 - x + x  - x  + x  - x  + O⎝x ⎠"
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(1/(1+x)).series()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle \\left( 1 - x y + x^{2} \\left(y^{2} - \\frac{1}{2}\\right) + x^{3} \\left(- \\frac{5 y^{3}}{6} + \\frac{y}{2}\\right) + O\\left(x^{4}\\right), \\  \\cos{\\left(x \\right)} - x y \\cos{\\left(x \\right)} + x^{2} y^{2} \\cos{\\left(x \\right)} - \\frac{5 x^{3} y^{3} \\cos{\\left(x \\right)}}{6} + O\\left(y^{4}\\right)\\right)$"
      ],
      "text/plain": [
       "⎛                           ⎛     3    ⎞                                                  3  3               ⎞\n",
       "⎜           2 ⎛ 2   1⎞    3 ⎜  5⋅y    y⎟    ⎛ 4⎞                         2  2          5⋅x ⋅y ⋅cos(x)    ⎛ 4⎞⎟\n",
       "⎜1 - x⋅y + x ⋅⎜y  - ─⎟ + x ⋅⎜- ──── + ─⎟ + O⎝x ⎠, cos(x) - x⋅y⋅cos(x) + x ⋅y ⋅cos(x) - ────────────── + O⎝y ⎠⎟\n",
       "⎝             ⎝     2⎠      ⎝   6     2⎠                                                     6               ⎠"
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = sympy.cos(x) / (1 + sympy.sin(x * y))\n",
    "\n",
    "expr.series(x,n=4), expr.series(y,n=4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle - \\frac{61 x^{5} y^{5}}{120} + \\frac{5 x^{5} y^{3}}{12} - \\frac{x^{5} y}{24} + \\frac{2 x^{4} y^{4}}{3} - \\frac{x^{4} y^{2}}{2} + \\frac{x^{4}}{24} - \\frac{5 x^{3} y^{3}}{6} + \\frac{x^{3} y}{2} + x^{2} y^{2} - \\frac{x^{2}}{2} - x y + 1$"
      ],
      "text/plain": [
       "      5  5      5  3    5        4  4    4  2    4      3  3    3              2          \n",
       "  61⋅x ⋅y    5⋅x ⋅y    x ⋅y   2⋅x ⋅y    x ⋅y    x    5⋅x ⋅y    x ⋅y    2  2   x           \n",
       "- ──────── + ─────── - ──── + ─────── - ───── + ── - ─────── + ──── + x ⋅y  - ── - x⋅y + 1\n",
       "    120         12      24       3        2     24      6       2             2           "
      ]
     },
     "execution_count": 86,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.series(y).removeO().series(x).removeO().expand()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Limits\n",
    "\n",
    "Value of a function as a dependent variable approaches a specific value (or approaches pos/neg infinity)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAA0AAAAPCAYAAAA/I0V3AAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjUuMiwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy8qNh9FAAAACXBIWXMAABJ0AAASdAHeZh94AAAAi0lEQVR4nO3SIQoCYRQE4E+xajYbxObewGjdYBTMRmGj8PgP4zkM3kMQjEb7Wv6wrC6sxeTACzO8YSbMoK5r32LUJCmlDVYosMQYp4jYdppwzM9P3LH4lDRs8QPmmGDfq15EnBtVuzxvSb3wN/3cNGgONqVUosx0ijWuuGTtERFVe3sFdi1tlg9uqF5xyRu/uhi7owAAAABJRU5ErkJggg==",
      "text/latex": [
       "$\\displaystyle 1$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#To find the limit of sin(x)/x as x goes to zero:\n",
    "sympy.limit(sympy.sin(x) / x, x, 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can also use __sympy.limit__ to compute symbolic limits, which can be illustrated by computing derivatives using the previous definition."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = sympy.Function('f')\n",
    "x, h = sympy.symbols(\"x, h\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [],
   "source": [
    "diff_limit = (f(x+h)-f(x))/h"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle - \\sin{\\left(x \\right)}$"
      ],
      "text/plain": [
       "-sin(x)"
      ]
     },
     "execution_count": 91,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.limit(diff_limit.subs(f, sympy.cos), h, 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\cos{\\left(x \\right)}$"
      ],
      "text/plain": [
       "cos(x)"
      ]
     },
     "execution_count": 92,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.limit(diff_limit.subs(f, sympy.sin), h, 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Another example: find the asymptotic behavior as a function, for example\n",
    "as its dependent variable approaches infinity."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left( \\frac{1}{2}, \\  -1\\right)$"
      ],
      "text/plain": [
       "(1/2, -1)"
      ]
     },
     "execution_count": 93,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = (x**2-3*x)/(2*x-2)\n",
    "\n",
    "p = sympy.limit(expr/x,   x, oo)\n",
    "q = sympy.limit(expr-p*x, x, oo)\n",
    "\n",
    "p,q # result: asymptotic behavior of f(x) as x becomes large:\n",
    "# f(x) --> x/2-1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Sums and products\n",
    "\n",
    "1st argument is an expression; 2nd argument is a tuple (n,n1,n2) - n is a symbol; n1 & n2 are upper & lower limits for symbol n."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = sympy.symbols(\"n\", integer=True)\n",
    "x = sympy.Sum(1/(n**2), (n, 1, oo))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAI8AAAAzCAYAAACua1udAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjUuMiwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy8qNh9FAAAACXBIWXMAABJ0AAASdAHeZh94AAAKv0lEQVR4nO2ceZBcVRWHvwCBSECFoCwSCEFlJxMWETAEIYQdCRJBNokKElEQEGQrfvy00MIQJKCIlmiAMsgaLQgIQpQQ2bQ0YVFTyCoEQQxEwp4Q/zi3mTc9vbzX3TOTCf1VTd3q++42M6fPPfecc9+ApUuXUhTbewG7Szq1cOfoPwY4GPgmcBywN7AWcDjwOvBzSaMbGbtNcWzvAoyVdHaRfis0MNE44DuAi/bNMBK4QtIiSRcAqwAXS3pY0mPAPNsfamL8NgWQNAsYbvu0Iv0KCY/tEcCPgc9JWlSkbxnzgH3TmLsRwnOK7Y/YXh3YFPhvE+O3Kc5XgC/bPiBvh9zCY3tNYDogSU80sLgsNwFr2J4PfJ/Yws4B7gXmAhdIeqfJOdoUQNIrwARgqu1hefoMyGvz2J4GDAN2llTcUGrTL7B9BaH5d5K0pFbbXJonbS2fB85oC85yz6nAZsDEeg3rah7bA4EHgacl7dmS5bVZprF9PnAs8HFJ/6nWLo/mOZxQYz9o0draLPtcCAwCvl2rUU3hsT2AUGOPAre1bGltlmkkPQ9cCUywvW61dvU0z37A5sBlbVvnPcdUwoVycrUGNW0e2zOAvYChkuY3sgLb2wDTGulbxlhJT7dgnDY5sT0PWA9YR9Kr5c9XqtFxTWAP4IFGBSexKrBJE/1LrNyCMdoU42pAwIHAL8sf1tq2DgYGAr9tcgFvNtm/Td9xeyoPr/SwquYBPpvKu5qZXdKfbE8BTkxVS4EjJLViK6tLCvqdCmwLrAuMl3R9b8zdCLbPAA4itPWbwH2Ef+3hPljOA8D/gD1sD5HUJWRUUfPYXgnYGViSBmiWk+hUewMIF3hv+YwGEyGPr/XSfM2yK3ApsBOwG7AYuCOZEb2KpMXALELJdMtyqKZ5RhJ/9IclvdaCRSy1PQEYQhjgA4EbbO8u6f5mx68z963ArQB2M4kAvUO5I9b2kcBC4st8Ux8s6a/EqXsUcGP2QTWbZ1QqH2rVCiS9TWyF96WqwcAM25u2ao7llNWJ/9OCPpp/bipHlT+opXkAmo2ed0HSa7b3Be4m/EdDgNtt7yTpmVbOtRxxETCHzi9dN2yvDcwHfiTphEz9KOAbxBa4JiGADwEXSbol5/wl4emwPTApAaC65tk4lU/lnCA3khYAY4GSz2YocFtf7OnLOrYnAbsQRn6tCPdniP/l9Ezfswl7ZRfixDyZlApD2FV5eYo45KwIbJh9UE94niwwSW4kPUsI0IupanNiC1u1J+brj9ieDBxFpPv+s07zcUTy3KzUdzyR7XkHMFzSBElnSjpW0vbAWXnXkTRNKTi6cfZZt23L9mDgw+njv/JOUhRJ82zvA8wEVgM+CVxv+4Bk5fd70hZ9c87mIyXNSf0uBg4BPi3pb3Xm+ABxKpsmaYntFYDzgdeAw1KSVxeyW09O5hMyMTxbWcnmyeYON5NqWpfkAxoHzCA8yHsDv7B9VKtiabZXAz6aqRpmuwNY0AvhjoVE2u5E4BEg61/alBCQ2cDviLQXbF8KHEF4dRfYXie1X1Ql9Xdf4m9XOgntBGyU5nopCfCWwBtEtODeBn6PF1LZJa+8kvBkt46mj+n1kHRHOo5eTWyjRxBqsmpAriDbAb/PfJ6UyiuAo1s0R0UkzU6J/BOBX0s6t/QsJZsfAkyVdHmmWykJ686y4QycS3fGAa8SAgiwfSqfB/4CbNVlEHsWcHCtPJ0KvJ7KwdnKSsKTbdDjwpO4lbCvSmpxjO1Bkt5odmBJfyAck31FRyrnlNVvU6leUu612h5E+M1uzfytSibHccRpeQxwP2HsTgb2BK6jmNFcGruL8FQymEsNlkp6vcLzlpK82dfRKTjPAPu0QnCWETpSObesfhvCe9xM2GEsYS9Oz9StmMoBhIa5M11xeoTQUs8Ao23vWGCeipqnkvCUjoQDUjJYT/MT4tsAYSPsvZz5fDoI2/HdE1PGDvuHpGYCx+OAtwibscRLqXxcUheBTcqglNT3iQLzlHaoLu6CSsKTzdsYVGCCwtg+F/hi+vgWcGAfBQB7BNtrABsAD5YdAEYSmmFOE2OvCOwPzJS0MPNoXipfrtK1JFzvKzBdyQ7uktNTyebJ2jmD6FRZLSXFupQ+LgWOTvZJM2O25IRWxO6oQ0cqK21Z0ITwEM6/IXTdsiB8PYuBj9leWdJbZc+3TOWTBebKLTzZ4+BgOiW1ZdgeC/w0U/UtSVc3O26ef7rtocBVhGH5NmBJN1ZpOxX4AjBB0tQGltSRyjll9aV/YDOxw4OAd4DfZCslvWj7GiIH5xzg3fvntvcgTISFFMvTqig8lbatf9O5t1VNfm6U5GO5nk7BvUTSpOo9Ws5i4ERJmxOZklNqeLZXyPRphI5UlmueUihmC9vrNzj2gcA9KVm9nJMJG+ss27NsX2D7OuJUuwQ4RtLLBeZaI5VdbNFuwpO8uyXnWUuFx/YGhHG3eqqaTgTueg1Jz5UMSUkvEJp1rSrNtwJeoatBWoQRhHYo1zDXEuGE84BDiw5qe3tgfbpvWcC7v9cOxHWpocAJhBd6BjBK0nUFpyzFtB7LVlaLqj9GeCnXKzhJVWx/ELglM+Y9hPu8z+6k296OyC3qFoZJ690amCypoa1bUkeV+muAaxoZMzEulRWFJ82xgNBATTlbU8D6/enj49ln1QKjJYu9FYnr2F6Z+EW3yIx/QF/6cmwPIe4mfalKKGQUYRNd2KsLy8c4YG4LXjiRh2GpfBV4Lvugmua5FzieTsOuYZKvaCqdHs3nCV9Oj7xCJb3h4QniDzwR+BRhxx0jaWZqswohzN+TdE+lcSTdRA+7KhpF0ma9OF0pkn5/+ZesmvDMTmXTwgOcTrwkocTawOMNpIQ6GxuqwYhUnkSkJRxPaI/JwMiMMM+UdFXRRbwH6Ujl3eUPKm5bkp4ijOZ1bG9YqU0BRtRv0lJGEEbuoZLuSLkwN9AZEd6ZCEgeaHtO+tmqylht4tYJdCqUd6l19WYGofZHE7ZBozxLpw3VDC/WbwKE8NwsKbs/DyeFByTNpoHX6b0XSblBOxKO427bey3hmUYIz240ITySTgFOabR/A4wAppTVjSRuAbQpxrbESetXlW7R1PoG/pHIX90/Rb6XeVLAcTjdBaUtPI2xTyorev+rCk+yrKcS3tAxLV9Wz7A1ESebU6pIR/L1aQtPI4wnblxUDGXU2/t/SOx3E1q8qJ5iBPBoWbrmSOLa7t/7Zkn9E9sjCb/cZRWCq0C+18pNAb4KbFBmhPY6/e3eeX/G9s+I4OqwKvGzXKeOSYSn9estXFuj9Ld75/2SlId0GPGi9YqCAzlfpZsukJ0GbFTJM5zHq9tqUu5OW/P0ALZFnJA3qbXb5D1FTSIuoJ1ObBvl1PTqli3sTODMOvPtLambR7NNz2N7LUJwVM9MyeUsS3m2xwDH264ULK3n1c1yGeHyrvXz5zzratMjnENEzy+p1zC3/0bSXba/C1xue3TZ3emaXt2ycRbQd298aFMD2zsTRvIOeW7tFnX+nUfEhkRIaIncXt32trVsYvv9wOXAkTnuxgMFYzzJcTge2N326DRpUa9ue9taxkg3MaYBkwq8eqWw5kHSItv7Eb6fuyjo1W1m2+rje+fLM7sCV0q6tkinhmJWKS3zvPSxN726fXbvfHlGUvm9+Fz8H/QzyNHZsiH4AAAAAElFTkSuQmCC",
      "text/latex": [
       "$\\displaystyle \\left( \\sum_{n=1}^{\\infty} \\frac{1}{n^{2}}, \\  \\frac{\\pi^{2}}{6}\\right)$"
      ],
      "text/plain": [
       "⎛  ∞         ⎞\n",
       "⎜ ____       ⎟\n",
       "⎜ ╲          ⎟\n",
       "⎜  ╲   1    2⎟\n",
       "⎜   ╲  ──  π ⎟\n",
       "⎜   ╱   2, ──⎟\n",
       "⎜  ╱   n   6 ⎟\n",
       "⎜ ╱          ⎟\n",
       "⎜ ‾‾‾‾       ⎟\n",
       "⎝n = 1       ⎠"
      ]
     },
     "execution_count": 95,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x, x.doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left( \\prod_{n=1}^{7} n, \\  5040\\right)$"
      ],
      "text/plain": [
       "⎛  7          ⎞\n",
       "⎜─┬─┬─        ⎟\n",
       "⎜ │ │  n, 5040⎟\n",
       "⎜ │ │         ⎟\n",
       "⎝n = 1        ⎠"
      ]
     },
     "execution_count": 96,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = sympy.Product(n,(n,1,7))\n",
    "x, x.doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = sympy.Symbol(\"x\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle e^{x} - 1$"
      ],
      "text/plain": [
       " x    \n",
       "ℯ  - 1"
      ]
     },
     "execution_count": 98,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.Sum((x)**n/(sympy.factorial(n)), (n, 1, oo)).doit().simplify()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Equation Solving\n",
    "\n",
    "If an equation can be solved analytically, there is a good chance that SymPy is able to find the solution. If not, numerical methods might be the only option.\n",
    "\n",
    "In its simplest form, equation solving involves a single equation with a single unknown variable, and no additional parameters: for example, finding the value of x that satisfy the second-degree polynomial\n",
    "equation x 2 + 2 x – 3 = 0 ."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left[ -3, \\  1\\right]$"
      ],
      "text/plain": [
       "[-3, 1]"
      ]
     },
     "execution_count": 99,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = sympy.symbols(\"x\")\n",
    "sympy.solve(x**2 + 2*x - 3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That is, the solutions are x = -3 and x = 1. \n",
    "\n",
    "The argument to sympy.solve is an expression that will be solved under the assumption that it equals zero. When this expression contains more than one symbol, the variable that is to be solved for must be given as a second argument."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left[ \\frac{- b - \\sqrt{- 4 a c + b^{2}}}{2 a}, \\  \\frac{- b + \\sqrt{- 4 a c + b^{2}}}{2 a}\\right]$"
      ],
      "text/plain": [
       "⎡        _____________          _____________⎤\n",
       "⎢       ╱           2          ╱           2 ⎥\n",
       "⎢-b - ╲╱  -4⋅a⋅c + b    -b + ╲╱  -4⋅a⋅c + b  ⎥\n",
       "⎢─────────────────────, ─────────────────────⎥\n",
       "⎣         2⋅a                    2⋅a         ⎦"
      ]
     },
     "execution_count": 100,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a,b,c = sympy.symbols(\"a, b, c\")\n",
    "sympy.solve(a*x**2 + b*x + c, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "sympy.solve can also solve trigonometric expressions and other special functions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left[ \\frac{\\pi}{4}\\right]$"
      ],
      "text/plain": [
       "⎡π⎤\n",
       "⎢─⎥\n",
       "⎣4⎦"
      ]
     },
     "execution_count": 101,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.solve(sympy.sin(x) - sympy.cos(x), x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left[ - W\\left(\\frac{1}{2}\\right)\\right]$"
      ],
      "text/plain": [
       "[-W(1/2)]"
      ]
     },
     "execution_count": 102,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.solve(sympy.exp(x) + 2*x, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is not uncommon to encounter equations that are not solvable algebraically, or that SymPy is unable to solve. \n",
    "\n",
    "In these cases SymPy will return a formal solution, which can be evaluated numerically, or raise an error if no method is available."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left[ \\operatorname{CRootOf} {\\left(x^{5} - x^{2} + 1, 0\\right)}, \\  \\operatorname{CRootOf} {\\left(x^{5} - x^{2} + 1, 1\\right)}, \\  \\operatorname{CRootOf} {\\left(x^{5} - x^{2} + 1, 2\\right)}, \\  \\operatorname{CRootOf} {\\left(x^{5} - x^{2} + 1, 3\\right)}, \\  \\operatorname{CRootOf} {\\left(x^{5} - x^{2} + 1, 4\\right)}\\right]$"
      ],
      "text/plain": [
       "⎡       ⎛ 5    2       ⎞         ⎛ 5    2       ⎞         ⎛ 5    2       ⎞         ⎛ 5    2       ⎞         ⎛ 5    2       ⎞⎤\n",
       "⎣CRootOf⎝x  - x  + 1, 0⎠, CRootOf⎝x  - x  + 1, 1⎠, CRootOf⎝x  - x  + 1, 2⎠, CRootOf⎝x  - x  + 1, 3⎠, CRootOf⎝x  - x  + 1, 4⎠⎦"
      ]
     },
     "execution_count": 103,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.solve(x**5 - x**2 + 1, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {},
   "outputs": [],
   "source": [
    "#sympy.solve(sympy.tan(x)+x, x)\n",
    "# NotImplementedError: multiple generators. Debug TODO."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Solving equations for >1 unknown variable is a generalization of univariate solvers. Pass a list of expressions instead of a single one as first argument to sympy.solve. The second argument should be a list of symbols to solve for. \n",
    "\n",
    "Below: how to solve two systems that are linear and nonlinear equations in x and y, respectively."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left[ \\left\\{ x : - \\frac{1}{3}, \\  y : \\frac{2}{3}\\right\\}\\right]$"
      ],
      "text/plain": [
       "[{x: -1/3, y: 2/3}]"
      ]
     },
     "execution_count": 105,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq1 = x+2 * y-1\n",
    "eq2 = x   - y+1\n",
    "\n",
    "sympy.solve([eq1, eq2], [x, y], dict=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 106,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left[ \\left\\{ x : 0, \\  y : 0\\right\\}, \\  \\left\\{ x : 1, \\  y : 1\\right\\}, \\  \\left\\{ x : \\left(- \\frac{1}{2} - \\frac{\\sqrt{3} i}{2}\\right)^{2}, \\  y : - \\frac{1}{2} - \\frac{\\sqrt{3} i}{2}\\right\\}, \\  \\left\\{ x : \\left(- \\frac{1}{2} + \\frac{\\sqrt{3} i}{2}\\right)^{2}, \\  y : - \\frac{1}{2} + \\frac{\\sqrt{3} i}{2}\\right\\}\\right]$"
      ],
      "text/plain": [
       "⎡                            ⎧               2               ⎫  ⎧               2               ⎫⎤\n",
       "⎢                            ⎪   ⎛  1   √3⋅ⅈ⎞        1   √3⋅ⅈ⎪  ⎪   ⎛  1   √3⋅ⅈ⎞        1   √3⋅ⅈ⎪⎥\n",
       "⎢{x: 0, y: 0}, {x: 1, y: 1}, ⎨x: ⎜- ─ - ────⎟ , y: - ─ - ────⎬, ⎨x: ⎜- ─ + ────⎟ , y: - ─ + ────⎬⎥\n",
       "⎢                            ⎪   ⎝  2    2  ⎠        2    2  ⎪  ⎪   ⎝  2    2  ⎠        2    2  ⎪⎥\n",
       "⎣                            ⎩                               ⎭  ⎩                               ⎭⎦"
      ]
     },
     "execution_count": 106,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq1 = x**2 - y\n",
    "eq2 = y**2 - x\n",
    "\n",
    "sols = sympy.solve([eq1, eq2], [x, y], dict=True)\n",
    "sols"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "sympy.solve returns a list where each element represents a solution to the equation system. \n",
    "\n",
    "The optional keyword argument __dict = True__ requests each solution in dictionary format, which maps the symbols that have been solved for\n",
    "to their values. \n",
    "\n",
    "This dictionary can be used in calls to __subs__, which is used below (checks that each solution indeed satisfies the two equations)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 107,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[True, True, True, True]"
      ]
     },
     "execution_count": 107,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[eq1.subs(sol).simplify() == 0 and eq2.subs(sol).simplify() == 0 \n",
    " for sol in sols]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Linear algebra\n",
    "![sympy-matrix-ops](pics/sympy-matrix-ops.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1\\\\2\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1⎤\n",
       "⎢ ⎥\n",
       "⎣2⎦"
      ]
     },
     "execution_count": 108,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.Matrix([1,2])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 2\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "[1  2]"
      ]
     },
     "execution_count": 109,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.Matrix([[1,2]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 110,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 2\\\\3 & 4\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1  2⎤\n",
       "⎢    ⎥\n",
       "⎣3  4⎦"
      ]
     },
     "execution_count": 110,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.Matrix([[1, 2], [3, 4]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0 & 1 & 2 & 3\\\\10 & 11 & 12 & 13\\\\20 & 21 & 22 & 23\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡0   1   2   3 ⎤\n",
       "⎢              ⎥\n",
       "⎢10  11  12  13⎥\n",
       "⎢              ⎥\n",
       "⎣20  21  22  23⎦"
      ]
     },
     "execution_count": 111,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy.Matrix(3, 4, lambda m,n: 10 * m + n)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sympy matrix elements can be symbolic expressions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "metadata": {},
   "outputs": [],
   "source": [
    "a, b, c, d = sympy.symbols(\"a, b, c, d\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 113,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left( \\left[\\begin{matrix}a & b\\\\c & d\\end{matrix}\\right], \\  \\left[\\begin{matrix}a^{2} + b c & a b + b d\\\\a c + c d & b c + d^{2}\\end{matrix}\\right]\\right)$"
      ],
      "text/plain": [
       "⎛        ⎡ 2                  ⎤⎞\n",
       "⎜⎡a  b⎤  ⎢a  + b⋅c   a⋅b + b⋅d⎥⎟\n",
       "⎜⎢    ⎥, ⎢                    ⎥⎟\n",
       "⎜⎣c  d⎦  ⎢                  2 ⎥⎟\n",
       "⎝        ⎣a⋅c + c⋅d  b⋅c + d  ⎦⎠"
      ]
     },
     "execution_count": 113,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M = sympy.Matrix([[a, b], [c, d]])\n",
    "M, M*M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}a x_{1} + b x_{2}\\\\c x_{1} + d x_{2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡a⋅x₁ + b⋅x₂⎤\n",
       "⎢           ⎥\n",
       "⎣c⋅x₁ + d⋅x₂⎦"
      ]
     },
     "execution_count": 114,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = sympy.Matrix(sympy.symbols(\"x_1, x_2\"))\n",
    "M*x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With purely numerical methods, we would have to choose particular values of p and q before beginning to solve this problem, for example, using an LU factorization (or by computing the inverse) of the matrix on the left-hand side of the equation. \n",
    "\n",
    "With a symbolic computing approach we can proceed with computing the solution, as if we did the calculation analytically by hand. With SymPy, we can simply define symbols for the unknown variables and parameters,\n",
    "and setup the required matrix objects."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & p\\\\q & 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1  p⎤\n",
       "⎢    ⎥\n",
       "⎣q  1⎦"
      ]
     },
     "execution_count": 115,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p,q = sympy.symbols(\"p, q\")\n",
    "M = sympy.Matrix([[1,p],[q,1]])\n",
    "M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}b_{1}\\\\b_{2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡b₁⎤\n",
       "⎢  ⎥\n",
       "⎣b₂⎦"
      ]
     },
     "execution_count": 116,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = sympy.Matrix(sympy.symbols(\"b_1, b_2\"))\n",
    "b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{- b_{1} + b_{2} p}{p q - 1}\\\\\\frac{b_{1} q - b_{2}}{p q - 1}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡-b₁ + b₂⋅p⎤\n",
       "⎢──────────⎥\n",
       "⎢ p⋅q - 1  ⎥\n",
       "⎢          ⎥\n",
       "⎢b₁⋅q - b₂ ⎥\n",
       "⎢───────── ⎥\n",
       "⎣ p⋅q - 1  ⎦"
      ]
     },
     "execution_count": 117,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.solve(b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 118,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}b_{1} - \\frac{p \\left(- b_{1} q + b_{2}\\right)}{- p q + 1}\\\\\\frac{- b_{1} q + b_{2}}{- p q + 1}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡     p⋅(-b₁⋅q + b₂)⎤\n",
       "⎢b₁ - ──────────────⎥\n",
       "⎢        -p⋅q + 1   ⎥\n",
       "⎢                   ⎥\n",
       "⎢    -b₁⋅q + b₂     ⎥\n",
       "⎢    ──────────     ⎥\n",
       "⎣     -p⋅q + 1      ⎦"
      ]
     },
     "execution_count": 118,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.LUsolve(b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{b_{1}}{- p q + 1} - \\frac{b_{2} p}{- p q + 1}\\\\- \\frac{b_{1} q}{- p q + 1} + \\frac{b_{2}}{- p q + 1}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡    b₁        b₂⋅p   ⎤\n",
       "⎢ ──────── - ──────── ⎥\n",
       "⎢ -p⋅q + 1   -p⋅q + 1 ⎥\n",
       "⎢                     ⎥\n",
       "⎢    b₁⋅q        b₂   ⎥\n",
       "⎢- ──────── + ────────⎥\n",
       "⎣  -p⋅q + 1   -p⋅q + 1⎦"
      ]
     },
     "execution_count": 119,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.inv()*b # multiple inverse of M by vector b:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  }
 ],
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