{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# SymPy Examples"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This example is related to the article [*Computer Algebra with Jupyter and SymPy*](https://jaantollander.com/2019-11-16-computer-algebra.html). It contains a showcase of the essential features of SymPy. The Sympy webpage contains the full list of [features](https://www.sympy.org/en/features.html) and [documentation](https://docs.sympy.org/latest/index.html)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Imports and Printing"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Import SymPy and initialize the printing of SymPy expressions with MathJax."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "# init_printing(\"mathjax\") # sympy expression printing"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Import SymEngine and initialize the printing of SymEngine expressions with MathJax. SymEngine is a fast symbolic manipulation library written in C++ instead of Python. SymEngine has Python wrappers that integrate with Sympy, although Symengine does not yet support all of SymPy's functionality. We can override some of the SymPy functions with SymEngine variants for improved performance.  It is wise to import only computationally critical functions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "#from symengine import *  # overrides some sympy functions\n",
    "#init_printing(\"mathjax\") # symengine expression printing"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Define Symbols and Functions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`symbols`, `Symbol`, `Function`, `sympify`, `S`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = symbols(\"x\")\n",
    "a,b,c = symbols(\"a,b,c\", real=True)\n",
    "i,j,n = symbols(\"i,j,n\", integer=True)\n",
    "f = symbols(\"f\", cls=Function)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1, 1)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympify(1), S(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle i$"
      ],
      "text/plain": [
       "I"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I  # imaginary unit"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Notations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`Rational`, `Sum`, `Product`, `I`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{1}{2}$"
      ],
      "text/plain": [
       "1/2"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Rational(1, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sum_{i=1}^{n} x^{i}$"
      ],
      "text/plain": [
       "Sum(x**i, (i, 1, n))"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Sum(x**i, (i, 1, n))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\prod_{i=1}^{n} \\left(i + x\\right)$"
      ],
      "text/plain": [
       "Product(i + x, (i, 1, n))"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Product(x+i, (i, 1, n))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Evaluate Expressions Numerically"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`N`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle e^{5}$"
      ],
      "text/plain": [
       "exp(5)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "exp(5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 148.41$"
      ],
      "text/plain": [
       "148.41"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N(_, 5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Manipulate Expressions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`expand`, `collect`, `factor`, `simplify`, `.subs`, `.doit`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(- a + x\\right) \\left(- b + x\\right)$"
      ],
      "text/plain": [
       "(-a + x)*(-b + x)"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(x-a)*(x-b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a b - a x - b x + x^{2}$"
      ],
      "text/plain": [
       "a*b - a*x - b*x + x**2"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand(_)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a \\left(b - x\\right) - b x + x^{2}$"
      ],
      "text/plain": [
       "a*(b - x) - b*x + x**2"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "collect(_, a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(- a + x\\right) \\left(- b + x\\right)$"
      ],
      "text/plain": [
       "(-a + x)*(-b + x)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor(_)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(- a + c\\right) \\left(- b + c\\right)$"
      ],
      "text/plain": [
       "(-a + c)*(-b + c)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_.subs(x, c)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(a - c\\right) \\left(b - c\\right)$"
      ],
      "text/plain": [
       "(a - c)*(b - c)"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(_)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sum_{i=1}^{n} x^{i}$"
      ],
      "text/plain": [
       "Sum(x**i, (i, 1, n))"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Sum(x**i, (i, 1, n))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\begin{cases} n & \\text{for}\\: x = 1 \\\\\\frac{x - x^{n + 1}}{1 - x} & \\text{otherwise} \\end{cases}$"
      ],
      "text/plain": [
       "Piecewise((n, Eq(x, 1)), ((x - x**(n + 1))/(1 - x), True))"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_.doit()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Define Equations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`Eq`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a x^{2} + b x = c$"
      ],
      "text/plain": [
       "Eq(a*x**2 + b*x, c)"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq(a*x**2+b*x, c)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solve Equations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`solve`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a x^{2} + b x + c$"
      ],
      "text/plain": [
       "a*x**2 + b*x + c"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a*x**2 + b*x + c "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(-b + sqrt(-4*a*c + b**2))/(2*a), -(b + sqrt(-4*a*c + b**2))/(2*a)]"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve(_, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solve Systems of Equations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "z = symbols(\"z_{0:2}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[{z_{0}: -a/2 - sqrt(a**2 + 4*b)/2, z_{1}: -a/2 + sqrt(a**2 + 4*b)/2},\n",
       " {z_{0}: -a/2 + sqrt(a**2 + 4*b)/2, z_{1}: -a/2 - sqrt(a**2 + 4*b)/2}]"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve([z[0] + z[1] + a, z[0] * z[1] + b], z, dict=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solve Recurrence Relations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`rsolve`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\begin{aligned}\n",
    "y_0&=1, \\\\\n",
    "y_1&=4, \\\\\n",
    "y_n &= 2y_{n-1} + 5y_{n-2}\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(\\frac{1}{2} - \\frac{\\sqrt{6}}{4}\\right) \\left(1 - \\sqrt{6}\\right)^{n} + \\left(\\frac{1}{2} + \\frac{\\sqrt{6}}{4}\\right) \\left(1 + \\sqrt{6}\\right)^{n}$"
      ],
      "text/plain": [
       "(1/2 - sqrt(6)/4)*(1 - sqrt(6))**n + (1/2 + sqrt(6)/4)*(1 + sqrt(6))**n"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = Function('y')\n",
    "f = y(n) - 2*y(n-1) - 5*y(n-2)\n",
    "rsolve(f, y(n), [1, 4])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Differential Calculus"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`limit`, `series`, `diff`, `integrate`, `Derivative`, `Integral`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{1}{x}$"
      ],
      "text/plain": [
       "1/x"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(log(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\log{\\left(x^{2} + x + 1 \\right)}}{2} + \\frac{\\sqrt{3} \\operatorname{atan}{\\left(\\frac{2 \\sqrt{3} x}{3} + \\frac{\\sqrt{3}}{3} \\right)}}{3}$"
      ],
      "text/plain": [
       "log(x**2 + x + 1)/2 + sqrt(3)*atan(2*sqrt(3)*x/3 + sqrt(3)/3)/3"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate((x**2-1)/(x**3-1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\sqrt{3} \\pi}{18} + \\frac{\\log{\\left(3 \\right)}}{2}$"
      ],
      "text/plain": [
       "sqrt(3)*pi/18 + log(3)/2"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate((x**2-1)/(x**3-1), (x, 0, 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x - \\frac{x^{3}}{6} + \\frac{x^{5}}{120} + O\\left(x^{6}\\right)$"
      ],
      "text/plain": [
       "x - x**3/6 + x**5/120 + O(x**6)"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "series(sin(x), x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit(sin(x)/x, x, 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solve Differential Equations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`dsolve`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = Function(\"f\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a \\frac{d}{d x} f{\\left(x \\right)} + b f{\\left(x \\right)} + c + \\frac{d^{2}}{d x^{2}} f{\\left(x \\right)}$"
      ],
      "text/plain": [
       "a*Derivative(f(x), x) + b*f(x) + c + Derivative(f(x), (x, 2))"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(f(x), x, 2) + a*diff(f(x), x) + b*f(x) + c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle f{\\left(x \\right)} = C_{1} e^{\\frac{x \\left(- a - \\sqrt{a^{2} - 4 b}\\right)}{2}} + C_{2} e^{\\frac{x \\left(- a + \\sqrt{a^{2} - 4 b}\\right)}{2}} - \\frac{c}{b}$"
      ],
      "text/plain": [
       "Eq(f(x), C1*exp(x*(-a - sqrt(a**2 - 4*b))/2) + C2*exp(x*(-a + sqrt(a**2 - 4*b))/2) - c/b)"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dsolve(_)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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