lectures/L09_sympy.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Symbolic maths: Sympy\n",
    "\n",
    "*Davide Gerosa (Milano-Bicocca)*\n",
    "\n",
    "**Sources**: Michael Zingale at Stony Brook University: https://sbu-python-class.github.io"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "SymPy provides support for symbolic math to python, similar to what you would do with Mathematica or Maple.  The major difference is that it acts just like any other python module, so you can use the symbolic math together in your own python projects with the rest of python functionality. Like python is open source, while Mathematica is proprietary (that said, Mathematica remains more powerful IMO).\n",
    "\n",
    "The following import and function (`init_session()`) sets up a nice environment for us when working in Jupyter"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "IPython console for SymPy 1.12 (Python 3.11.4-64-bit) (ground types: python)\n",
      "\n",
      "These commands were executed:\n",
      ">>> from sympy import *\n",
      ">>> x, y, z, t = symbols('x y z t')\n",
      ">>> k, m, n = symbols('k m n', integer=True)\n",
      ">>> f, g, h = symbols('f g h', cls=Function)\n",
      ">>> init_printing()\n",
      "\n",
      "Documentation can be found at https://docs.sympy.org/1.12/\n",
      "\n"
     ]
    }
   ],
   "source": [
    "from sympy import init_session\n",
    "init_session(use_latex=\"mathjax\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import math"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Sympy types and basic symbolic manipulation\n",
    "\n",
    "Sympy defines its own types, you can convert them to python types, but you don't always want to (and will probably lose accuracy when you do).  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "sqrt(2)\n"
     ]
    }
   ],
   "source": [
    "print(sqrt(2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2*sqrt(2)\n"
     ]
    }
   ],
   "source": [
    "print(sqrt(8))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.4142135623730951\n"
     ]
    }
   ],
   "source": [
    "print(2**0.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "#help(sqrt(8))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can do symbolic math not just on numbers, but we can tell sympy what to treat as a symbol, using `symbols()`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import symbols\n",
    "x, y, z = symbols(\"x y z\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x + 2 y$"
      ],
      "text/plain": [
       "x + 2⋅y"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = x + 2*y\n",
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x + 2 y - 1$"
      ],
      "text/plain": [
       "x + 2⋅y - 1"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr - 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x + y$"
      ],
      "text/plain": [
       "x + y"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr - y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x \\left(x + 2 y\\right)$"
      ],
      "text/plain": [
       "x⋅(x + 2⋅y)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = x*expr\n",
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{2} + 2 x y$"
      ],
      "text/plain": [
       " 2        \n",
       "x  + 2⋅x⋅y"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "g = expand(f)\n",
    "g"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x \\left(x + 2 y\\right)$"
      ],
      "text/plain": [
       "x⋅(x + 2⋅y)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor(g)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Substitution\n",
    "\n",
    "Sympy provides methods to substitute values for symbols in symbolic expressions.  Note, the follow likely does not do what you expect:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sin{\\left(2 \\pi w \\right)}$"
      ],
      "text/plain": [
       "sin(2⋅π⋅w)"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "w = symbols(\"w\")\n",
    "expr = sin(w*2*pi)\n",
    "w = 0\n",
    "expr"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We've now redefined `z` to be a python type"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "int"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(w)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To do substitution, we use the `subs()` method (that's like doing `./` in Mathematica)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sin{\\left(2 \\pi x \\right)}$"
      ],
      "text/plain": [
       "sin(2⋅π⋅x)"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = sin(x*2*pi)\n",
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\sqrt{2}}{2}$"
      ],
      "text/plain": [
       "√2\n",
       "──\n",
       "2 "
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = expr.subs(x, 0.125)\n",
    "a"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that this is not a floating point number -- it is still a sympy object.  To make it floating point, we can use evalf()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.707106781186548 <class 'sympy.core.numbers.Float'>\n"
     ]
    }
   ],
   "source": [
    "b = a.evalf()\n",
    "print(b, type(b))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is still a sympy object, because sympy can do arbitrary precision "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 0.70710678118654752440084436210484903928483593768847$"
      ],
      "text/plain": [
       "0.70710678118654752440084436210484903928483593768847"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.evalf(50)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Want regular python types?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.7071067811865476 <class 'float'>\n"
     ]
    }
   ],
   "source": [
    "c = float(b)\n",
    "print(c, type(c))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Python and sympy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "x, y, z, t = symbols('x y z t')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sympy symbols are just objects and when you do operations on two sympy objects the result is a sympy object.  \n",
    "\n",
    "When you combine a sympy and python object, the result is also a sympy object.  \n",
    "\n",
    "But we need to be careful when doing fractions.  For instance doing `x + 1/3` will first compute `1/3` in python (giving `0.333...`) and then add it to the sympy `x` symbol.  The `Rational()` function makes this all happen in sympy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sin{\\left(2 \\pi x \\right)} + \\frac{1}{3}$"
      ],
      "text/plain": [
       "sin(2⋅π⋅x) + 1/3"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = expr + Rational(1,3)\n",
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sin{\\left(2 \\pi x \\right)} + 0.333333333333333$"
      ],
      "text/plain": [
       "sin(2⋅π⋅x) + 0.333333333333333"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr + 1/3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Equality"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`=` is still the assignment operator of python (it does not mean symbolic equality), and `==` is still the logical test (exact structural equality).  There is a separate object, `Eq()` to specify symbolic equality.\n",
    "\n",
    "And testing for _algebraic_ equality is not always accomplished using `==`, since that tests for _structural equality_."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x + 1 == 4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x + 1 = 4$"
      ],
      "text/plain": [
       "x + 1 = 4"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq(x + 1, 4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = (x + 1)**2\n",
    "b = x**2 + 2*x + 1    # these are algebraically equal"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a == b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(x + 1\\right)^{2} = x^{2} + 2 x + 1$"
      ],
      "text/plain": [
       "       2    2          \n",
       "(x + 1)  = x  + 2⋅x + 1"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq(a,b) # That's not enough..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can use `simplify()` to test for algebraic equality"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 0$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(a - b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle i \\sin{\\left(x \\right)} + \\cos{\\left(x \\right)}$"
      ],
      "text/plain": [
       "ⅈ⋅sin(x) + cos(x)"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = cos(x) + I*sin(x)\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle e^{i x}$"
      ],
      "text/plain": [
       " ⅈ⋅x\n",
       "ℯ   "
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## More substitution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that substitution returns a new expression, that's because sympy expressions are immutable"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = cos(x)\n",
    "expr.subs(x, 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\cos{\\left(x \\right)}$"
      ],
      "text/plain": [
       "cos(x)"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x$"
      ],
      "text/plain": [
       "x"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For multiple substitutions, pass a list of tuples"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{3} + 4 x y - z$"
      ],
      "text/plain": [
       " 3            \n",
       "x  + 4⋅x⋅y - z"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = x**3 + 4*x*y - z\n",
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 40$"
      ],
      "text/plain": [
       "40"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.subs([(x, 2), (y, 4), (z, 0)])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Simplifying"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There is not unique definition of what the simplest form of an expression is.\n",
    "\n",
    "`simplify()` tries lots of methods for simplification (check the docs for details)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(sin(x)**2 + cos(x)**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x - 1$"
      ],
      "text/plain": [
       "x - 1"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify( (x**3 + x**2 - x - 1)/(x**2 + 2*x + 1) )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(x - 2\\right) \\left(x - 1\\right)$"
      ],
      "text/plain": [
       "(x - 2)⋅(x - 1)"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(gamma(x)/gamma(x - 2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But sometimes it doesn't have your idea of what the simplest form is"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{2} + 2 x + 1$"
      ],
      "text/plain": [
       " 2          \n",
       "x  + 2⋅x + 1"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(x**2 + 2*x + 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Instead factor may be what you want (the very same thing happens in Mathematica)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(x + 1\\right)^{2}$"
      ],
      "text/plain": [
       "       2\n",
       "(x + 1) "
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor(x**2 + 2*x + 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Polynomial simplification"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{2} + 2 x + 1$"
      ],
      "text/plain": [
       " 2          \n",
       "x  + 2⋅x + 1"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand((x + 1)**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{2} - x - 6$"
      ],
      "text/plain": [
       " 2        \n",
       "x  - x - 6"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand((x + 2)*(x - 3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle -2$"
      ],
      "text/plain": [
       "-2"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand( (x + 1)*(x - 2) - (x - 1)*x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{2} z + 4 x y z + 4 y^{2} z$"
      ],
      "text/plain": [
       " 2                  2  \n",
       "x ⋅z + 4⋅x⋅y⋅z + 4⋅y ⋅z"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = x**2*z + 4*x*y*z + 4*y**2*z\n",
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle z \\left(x + 2 y\\right)^{2}$"
      ],
      "text/plain": [
       "           2\n",
       "z⋅(x + 2⋅y) "
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor(expr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left( 1, \\  \\left[ \\left( z, \\  1\\right), \\  \\left( x + 2 y, \\  2\\right)\\right]\\right)$"
      ],
      "text/plain": [
       "(1, [(z, 1), (x + 2⋅y, 2)])"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "factor_list(expr)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "collect collects common powers"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{3} - x^{2} z + 2 x^{2} + x y + x - 3$"
      ],
      "text/plain": [
       " 3    2        2              \n",
       "x  - x ⋅z + 2⋅x  + x⋅y + x - 3"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = x*y + x - 3 + 2*x**2 - z*x**2 + x**3\n",
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{3} + x^{2} \\cdot \\left(2 - z\\right) + x \\left(y + 1\\right) - 3$"
      ],
      "text/plain": [
       " 3    2                        \n",
       "x  + x ⋅(2 - z) + x⋅(y + 1) - 3"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "collected_expr = collect(expr, x)\n",
    "collected_expr"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Cancel cancels"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{x^{2} + 2 x + 1}{x^{2} + x}$"
      ],
      "text/plain": [
       " 2          \n",
       "x  + 2⋅x + 1\n",
       "────────────\n",
       "    2       \n",
       "   x  + x   "
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = (x**2 + 2*x + 1)/(x**2 + x)\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{x + 1}{x}$"
      ],
      "text/plain": [
       "x + 1\n",
       "─────\n",
       "  x  "
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cancel(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "trigsimp simplifies trigonometric identities"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sin^{4}{\\left(x \\right)} - 2 \\sin^{2}{\\left(x \\right)} \\cos^{2}{\\left(x \\right)} + \\cos^{4}{\\left(x \\right)}$"
      ],
      "text/plain": [
       "   4           2       2         4   \n",
       "sin (x) - 2⋅sin (x)⋅cos (x) + cos (x)"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = sin(x)**4 - 2*cos(x)**2*sin(x)**2 + cos(x)**4\n",
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\cos{\\left(4 x \\right)}}{2} + \\frac{1}{2}$"
      ],
      "text/plain": [
       "cos(4⋅x)   1\n",
       "──────── + ─\n",
       "   2       2"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "trigsimp(expr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sin^{2}{\\left(x \\right)}$"
      ],
      "text/plain": [
       "   2   \n",
       "sin (x)"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "trigsimp(sin(x)*tan(x)/sec(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The tutorial discusses some of the nuances of simplification of powers and special functions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Calculus"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Calculus operations are simple in sympy\n",
    "\n",
    "### Derivatives"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\sin{\\left(x \\right)}$"
      ],
      "text/plain": [
       "-sin(x)"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(cos(x), x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2 x e^{x^{2}}$"
      ],
      "text/plain": [
       "     ⎛ 2⎞\n",
       "     ⎝x ⎠\n",
       "2⋅x⋅ℯ    "
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(exp(x**2), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "...third derivative"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 24 x$"
      ],
      "text/plain": [
       "24⋅x"
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(x**4, x, 3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Differentiate different variables"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(x^{2} y^{2} z^{2} + 3 x y z + 1\\right) e^{x y z}$"
      ],
      "text/plain": [
       "⎛ 2  2  2              ⎞  x⋅y⋅z\n",
       "⎝x ⋅y ⋅z  + 3⋅x⋅y⋅z + 1⎠⋅ℯ     "
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = exp(x*y*z)\n",
    "diff(expr, x, y, z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle x^{2} y^{2} z^{2} e^{x y z} + 3 x y z e^{x y z} + e^{x y z}$"
      ],
      "text/plain": [
       " 2  2  2  x⋅y⋅z            x⋅y⋅z    x⋅y⋅z\n",
       "x ⋅y ⋅z ⋅ℯ      + 3⋅x⋅y⋅z⋅ℯ      + ℯ     "
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(diff(diff(expr, x),y),z)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Unevaluated derivatives can be useful for building up ODEs and PDEs (which you then integrate numerically with scipy)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\partial^{3}}{\\partial z\\partial y\\partial x} e^{x y z}$"
      ],
      "text/plain": [
       "    3           \n",
       "   ∂    ⎛ x⋅y⋅z⎞\n",
       "────────⎝ℯ     ⎠\n",
       "∂z ∂y ∂x        "
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "deriv = Derivative(expr, x, y, z)\n",
    "deriv"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But of course if you can do it analytically you should"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(x^{2} y^{2} z^{2} + 3 x y z + 1\\right) e^{x y z}$"
      ],
      "text/plain": [
       "⎛ 2  2  2              ⎞  x⋅y⋅z\n",
       "⎝x ⋅y ⋅z  + 3⋅x⋅y⋅z + 1⎠⋅ℯ     "
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "deriv.doit()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Integrals\n",
    "\n",
    "Both definite and indefinite integrals are supported"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sin{\\left(x \\right)}$"
      ],
      "text/plain": [
       "sin(x)"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(cos(x), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Definite integral -- note the construction of the infinity"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(exp(-x), (x, 0, oo))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Double integral"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\pi$"
      ],
      "text/plain": [
       "π"
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(exp(-x**2 - y**2), (x, -oo, oo), (y, -oo, oo))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "if it is unable to do the integral, it returns an Integral object"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Integral(x**x, x)\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\int x^{x}\\, dx$"
      ],
      "text/plain": [
       "⌠      \n",
       "⎮  x   \n",
       "⎮ x  dx\n",
       "⌡      "
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = integrate(x**x, x)\n",
    "print(expr)\n",
    "expr"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This integral has a [known solution](https://en.wikipedia.org/wiki/Risch_algorithm#:~:text=8%20External%20links-,Description,(%2B%20%E2%88%92%20%C3%97%20%C3%B7) but sympy fails to find it"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{x}{\\sqrt{x^{4} + 10 x^{2} - 96 x - 71}}$"
      ],
      "text/plain": [
       "             x             \n",
       "───────────────────────────\n",
       "   ________________________\n",
       "  ╱  4       2             \n",
       "╲╱  x  + 10⋅x  - 96⋅x - 71 "
      ]
     },
     "execution_count": 86,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = x / sqrt(x**4 + 10*x**2 - 96*x - 71)   # example from Wikipedia Risch algorithm page\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\int \\frac{x}{\\sqrt{x^{4} + 10 x^{2} - 96 x - 71}}\\, dx$"
      ],
      "text/plain": [
       "⌠                               \n",
       "⎮              x                \n",
       "⎮ ─────────────────────────── dx\n",
       "⎮    ________________________   \n",
       "⎮   ╱  4       2                \n",
       "⎮ ╲╱  x  + 10⋅x  - 96⋅x - 71    \n",
       "⌡                               "
      ]
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate(a, x)"
   ]
  },
  {
   "attachments": {
    "Screenshot%202023-08-24%20at%2009.14.04.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is a case where instead Mathematica can do it a quickly...\n",
    "![Screenshot%202023-08-24%20at%2009.14.04.png](attachment:Screenshot%202023-08-24%20at%2009.14.04.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Limits"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "limit(sin(x)/x, x, 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Series expansions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 1 + x + \\frac{x^{2}}{2} - \\frac{x^{4}}{8} - \\frac{x^{5}}{15} - \\frac{x^{6}}{240} + \\frac{x^{7}}{90} + \\frac{31 x^{8}}{5760} + \\frac{x^{9}}{5670} + O\\left(x^{10}\\right)$"
      ],
      "text/plain": [
       "         2    4    5     6    7       8     9          \n",
       "        x    x    x     x    x    31⋅x     x      ⎛ 10⎞\n",
       "1 + x + ── - ── - ── - ─── + ── + ───── + ──── + O⎝x  ⎠\n",
       "        2    8    15   240   90    5760   5670         "
      ]
     },
     "execution_count": 89,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = exp(sin(x))\n",
    "a = expr.series(x, 0, 10)\n",
    "a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle -1 - \\frac{\\left(x - 1\\right)^{2}}{2} + \\frac{\\left(x - 1\\right)^{3}}{3} - \\frac{\\left(x - 1\\right)^{4}}{4} + \\frac{\\left(x - 1\\right)^{5}}{5} + x + O\\left(\\left(x - 1\\right)^{6}; x\\rightarrow 1\\right)$"
      ],
      "text/plain": [
       "            2          3          4          5                         \n",
       "     (x - 1)    (x - 1)    (x - 1)    (x - 1)         ⎛       6       ⎞\n",
       "-1 - ──────── + ──────── - ──────── + ──────── + x + O⎝(x - 1) ; x → 1⎠\n",
       "        2          3          4          5                             "
      ]
     },
     "execution_count": 90,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c = log(x).series(x, x0=1, n=6)\n",
    "c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{x^{5}}{5} - \\frac{5 x^{4}}{4} + \\frac{10 x^{3}}{3} - 5 x^{2} + 5 x - \\frac{137}{60}$"
      ],
      "text/plain": [
       " 5      4       3                   \n",
       "x    5⋅x    10⋅x       2         137\n",
       "── - ──── + ───── - 5⋅x  + 5⋅x - ───\n",
       "5     4       3                   60"
      ]
     },
     "execution_count": 91,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(c.removeO())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solvers\n",
    "\n",
    "`solveset()` is the main interface to solvers in sympy.  Note that it used to be `solve()`, but this has been replaced (see http://docs.sympy.org/latest/modules/solvers/solveset.html)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 95,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left\\{\\frac{1}{2} - \\frac{\\sqrt{17}}{2}, \\frac{1}{2} + \\frac{\\sqrt{17}}{2}\\right\\}$"
      ],
      "text/plain": [
       "⎧1   √17  1   √17⎫\n",
       "⎨─ - ───, ─ + ───⎬\n",
       "⎩2    2   2    2 ⎭"
      ]
     },
     "execution_count": 95,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solveset(Eq(x**2 - x,4), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If no Eq() is done, then it is assumed to be equal to 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left\\{0, 1\\right\\}$"
      ],
      "text/plain": [
       "{0, 1}"
      ]
     },
     "execution_count": 96,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solveset(x**2 - x, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can restrict the domain of the solution (e.g. to reals).  Recall that Z is the set of integers"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left\\{2 n \\pi + \\frac{\\pi}{2}\\; \\middle|\\; n \\in \\mathbb{Z}\\right\\}$"
      ],
      "text/plain": [
       "⎧        π │      ⎫\n",
       "⎨2⋅n⋅π + ─ │ n ∊ ℤ⎬\n",
       "⎩        2 │      ⎭"
      ]
     },
     "execution_count": 97,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solveset(sin(x) - 1, x, domain=S.Reals)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Linear systems\n",
    "\n",
    "`linsolve()` is the interface to linear systems"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left\\{\\left( \\frac{1}{2}, \\  \\frac{5}{2}\\right)\\right\\}$"
      ],
      "text/plain": [
       "{(1/2, 5/2)}"
      ]
     },
     "execution_count": 102,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "linsolve([x - y + 2, x + y - 3], [x, y])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left\\{\\left( - y - 1, \\  y, \\  2\\right)\\right\\}$"
      ],
      "text/plain": [
       "{(-y - 1, y, 2)}"
      ]
     },
     "execution_count": 103,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "linsolve([x + y + z - 1, x + y + 2*z - 3 ], (x, y, z))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`roots()` will report if a solution is multiple"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left\\{0, 3\\right\\}$"
      ],
      "text/plain": [
       "{0, 3}"
      ]
     },
     "execution_count": 108,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solveset(x**3 - 6*x**2 + 9*x, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left\\{ 0 : 1, \\  3 : 2\\right\\}$"
      ],
      "text/plain": [
       "{0: 1, 3: 2}"
      ]
     },
     "execution_count": 109,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "roots(x**3 - 6*x**2 + 9*x, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "0 is a single root, and 3 corresponds to two coincident roots"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Differential equations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You need an undefined function (f and g already are by our init_session() above, but we've probably reset these"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 110,
   "metadata": {},
   "outputs": [],
   "source": [
    "f, g = symbols('f g', cls=Function)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle f{\\left(x \\right)}$"
      ],
      "text/plain": [
       "f(x)"
      ]
     },
     "execution_count": 111,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{d}{d x} f{\\left(x \\right)}$"
      ],
      "text/plain": [
       "d       \n",
       "──(f(x))\n",
       "dx      "
      ]
     },
     "execution_count": 112,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x).diff(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "metadata": {},
   "outputs": [],
   "source": [
    "diffeq = Eq(f(x).diff(x, 2) - 2*f(x).diff(x) + f(x), sin(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle f{\\left(x \\right)} - 2 \\frac{d}{d x} f{\\left(x \\right)} + \\frac{d^{2}}{d x^{2}} f{\\left(x \\right)} = \\sin{\\left(x \\right)}$"
      ],
      "text/plain": [
       "                      2               \n",
       "         d           d                \n",
       "f(x) - 2⋅──(f(x)) + ───(f(x)) = sin(x)\n",
       "         dx           2               \n",
       "                    dx                "
      ]
     },
     "execution_count": 115,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diffeq"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle f{\\left(x \\right)} = \\left(C_{1} + C_{2} x\\right) e^{x} + \\frac{\\cos{\\left(x \\right)}}{2}$"
      ],
      "text/plain": [
       "                    x   cos(x)\n",
       "f(x) = (C₁ + C₂⋅x)⋅ℯ  + ──────\n",
       "                          2   "
      ]
     },
     "execution_count": 116,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dsolve(diffeq, f(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Matrices"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider the Euler equations of fluid dynamics:\n",
    "\n",
    "$$q_t + A(q) q_x = 0$$\n",
    "\n",
    "where\n",
    "\n",
    "$$q = \\left ( \\begin{array}{c} \\rho \\\\ u \\\\ p \\end{array} \\right )\n",
    "\\qquad\n",
    "A(q) = \\left ( \\begin{array}{ccc} u  & \\rho     & 0 \\\\                          \n",
    "                                  0  &  u       & 1/\\rho \\\\                     \n",
    "                                  0  & c^2 \\rho & u \\end{array} \\right ) $$\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}u & \\rho & 0\\\\0 & u & \\frac{1}{\\rho}\\\\0 & c^{2} \\rho & u\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡u   ρ    0⎤\n",
       "⎢          ⎥\n",
       "⎢         1⎥\n",
       "⎢0   u    ─⎥\n",
       "⎢         ρ⎥\n",
       "⎢          ⎥\n",
       "⎢    2     ⎥\n",
       "⎣0  c ⋅ρ  u⎦"
      ]
     },
     "execution_count": 117,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy.abc import rho\n",
    "rho, u, c = symbols('rho u c')\n",
    "A = Matrix([[u, rho, 0], [0, u, rho**-1], [0, c**2 * rho, u]])\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 118,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}u & \\rho & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "[u  ρ  0]"
      ]
     },
     "execution_count": 118,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.row(0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The eigenvalues of the system are the speeds at which information propagates"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left\\{ u : 1, \\  - c + u : 1, \\  c + u : 1\\right\\}$"
      ],
      "text/plain": [
       "{u: 1, -c + u: 1, c + u: 1}"
      ]
     },
     "execution_count": 119,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.eigenvals()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can diagonalize it, such that\n",
    "$$ A = PDP^{-1}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "metadata": {},
   "outputs": [],
   "source": [
    "P, D = A.diagonalize()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$D$ will be a matrix of the eigenvalues"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 121,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}u & 0 & 0\\\\0 & - c + u & 0\\\\0 & 0 & c + u\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡u    0       0  ⎤\n",
       "⎢                ⎥\n",
       "⎢0  -c + u    0  ⎥\n",
       "⎢                ⎥\n",
       "⎣0    0     c + u⎦"
      ]
     },
     "execution_count": 121,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$P$ will be the matrix of right eigenvectors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 122,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & \\frac{1}{c^{2}} & \\frac{1}{c^{2}}\\\\0 & - \\frac{1}{c \\rho} & \\frac{1}{c \\rho}\\\\0 & 1 & 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡   1    1  ⎤\n",
       "⎢1  ──   ── ⎥\n",
       "⎢    2    2 ⎥\n",
       "⎢   c    c  ⎥\n",
       "⎢           ⎥\n",
       "⎢   -1    1 ⎥\n",
       "⎢0  ───  ───⎥\n",
       "⎢   c⋅ρ  c⋅ρ⎥\n",
       "⎢           ⎥\n",
       "⎣0   1    1 ⎦"
      ]
     },
     "execution_count": 122,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Inverse"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 123,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{1}{u} & - \\frac{\\rho}{- c^{2} + u^{2}} & \\frac{1}{- c^{2} u + u^{3}}\\\\0 & \\frac{u}{- c^{2} + u^{2}} & - \\frac{1}{- c^{2} \\rho + \\rho u^{2}}\\\\0 & - \\frac{c^{2} \\rho}{- c^{2} + u^{2}} & \\frac{u}{- c^{2} + u^{2}}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1     -ρ            1      ⎤\n",
       "⎢─  ─────────   ─────────── ⎥\n",
       "⎢u     2    2      2      3 ⎥\n",
       "⎢   - c  + u    - c ⋅u + u  ⎥\n",
       "⎢                           ⎥\n",
       "⎢       u           -1      ⎥\n",
       "⎢0  ─────────  ─────────────⎥\n",
       "⎢      2    2     2        2⎥\n",
       "⎢   - c  + u   - c ⋅ρ + ρ⋅u ⎥\n",
       "⎢                           ⎥\n",
       "⎢       2                   ⎥\n",
       "⎢     -c ⋅ρ          u      ⎥\n",
       "⎢0  ─────────    ─────────  ⎥\n",
       "⎢      2    2       2    2  ⎥\n",
       "⎣   - c  + u     - c  + u   ⎦"
      ]
     },
     "execution_count": 123,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A**-1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Units\n",
    "\n",
    "Sympy can attach units to numbers and propagate them through (for the astronomers in the room, astropy also has a well-designed way of handling units)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 124,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy.physics.units import newton, kilogram, meter, second, convert_to"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 125,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 9.81 \\text{N}$"
      ],
      "text/plain": [
       "9.81⋅newton"
      ]
     },
     "execution_count": 125,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F = 1 * kilogram * 9.81 * meter / second**2\n",
    "convert_to(F, newton)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Exercises"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy as sym\n",
    "from sympy import init_session\n",
    "init_session()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Q1: Play with expressions\n",
    "\n",
    "Create the expression:\n",
    "\n",
    "$$f = x e^{-x} + x (1-x)$$\n",
    "\n",
    "Then evaluate it for \n",
    "\n",
    "$$x = 0, 0.1, 0.2, 0.4, 0.8$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Q2: Polynomials\n",
    "\n",
    "Factor this polynomial\n",
    "\n",
    "$$x^{4} - 6 x^{3} + x^{2} + 24 x + 16$$\n",
    "\n",
    "Then find its zeros."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Q3: There and back again \n",
    "Integrate the function:\n",
    "\n",
    "$$f = \\sin(x) e^{-x}$$\n",
    "\n",
    "Then differentiate the result to see if you get back the original function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Q4: Parsing an expression\n",
    "\n",
    "Write a program that reads in a mathematical expression as a string (e.g., `\"sin(2*pi*x)\"`), converts it to a sympy expression, and then evaluates it as needed.  \n",
    "\n",
    "Have your program either make a plot of the entered function, or use the input function as the function to fit a dataset to using curvefit.\n",
    "\n",
    "The following will be helpful:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`parse_expr()` will convert a string into a SymPy expression"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 133,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy.parsing.sympy_parser import parse_expr\n",
    "from sympy import lambdify"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 130,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\sin{\\left(2 \\pi x \\right)}$"
      ],
      "text/plain": [
       "sin(2⋅π⋅x)"
      ]
     },
     "execution_count": 130,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s = \"sin(2*pi*x)\"\n",
    "a = parse_expr(s)\n",
    "a"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`sympy.lambdify()` will convert a sympy expression into a function that is callable by python.  You can make it a numpy-compatible function too (this means, e.g., that any `sin()` in your SymPy expression will be evaluate using `np.sin()`)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 134,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = lambdify(x, a, \"numpy\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 135,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle -2.44929359829471 \\cdot 10^{-16}$"
      ],
      "text/plain": [
       "-2.4492935982947064e-16"
      ]
     },
     "execution_count": 135,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(1.0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 136,
   "metadata": {},
   "outputs": [],
   "source": [
    "#help(lambdify)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Q5: Let's go back to high school\n",
    "\n",
    "Newton's 2nd law is\n",
    "\n",
    "$$F = ma$$\n",
    "\n",
    "Create a mass of 1 kg and an acceleration of 10 m/s$^2$, and compute the force, $F$, and express the result in Newtons.\n",
    "\n",
    "Note: the `convert_to` function was added in SymPy 1.1, so if you are using an earlier version, you will need to divide by the target unit to do the conversion."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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