{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "62bac615",
   "metadata": {},
   "source": [
    "# Sympy pour le calcul formel"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d199cfd8",
   "metadata": {},
   "source": [
    "> **Importez les bibliothèques utiles**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "d8696c37",
   "metadata": {},
   "outputs": [],
   "source": [
    "import math\n",
    "import sympy"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5c170b7c",
   "metadata": {},
   "source": [
    "## Premiers essais\n",
    "> **Testez les commandes suivantes**"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ad76441b",
   "metadata": {},
   "source": [
    "Avec la bibliothèque math \n",
    "Ces instructions fonctionnent de la même manière avec la bibliothèque Numpy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "dc98c6bc",
   "metadata": {},
   "outputs": [],
   "source": [
    "math.sqrt(9)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "85f541b7",
   "metadata": {},
   "outputs": [],
   "source": [
    "math.sqrt(8)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5ecbd569",
   "metadata": {},
   "source": [
    "Avec la bibliothèque Sympy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "6d933928",
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "sympy.sqrt(9)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "0bd354eb",
   "metadata": {},
   "outputs": [],
   "source": [
    "sympy.sqrt(8)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "3222c90a",
   "metadata": {},
   "outputs": [],
   "source": [
    "m = 3/2\n",
    "print(m)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "57173084",
   "metadata": {},
   "outputs": [],
   "source": [
    "k = sympy.Rational(3,2)\n",
    "print(k)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "70060fa7",
   "metadata": {},
   "source": [
    "## Expressions"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "80fc7f06",
   "metadata": {},
   "source": [
    "Il est possible de mettre en forme des équations avec Sympy.\n",
    "> **Testez l'exemple suivant**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "55ede9e3",
   "metadata": {},
   "outputs": [],
   "source": [
    "x, y = sympy.symbols('x y')\n",
    "expr = x**2 - 4 * x + 5\n",
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "baf3c438",
   "metadata": {},
   "outputs": [],
   "source": [
    "expr.subs(x, 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ea30ab0a",
   "metadata": {},
   "source": [
    "> **A quoi sert la fonction *subs* ?**"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "07ca4ddb",
   "metadata": {},
   "source": [
    "Réponse : "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d67ec2e7",
   "metadata": {},
   "source": [
    "### Développer et factoriser"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "2ba58ca8",
   "metadata": {},
   "outputs": [],
   "source": [
    "x*expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "9d6f8f67",
   "metadata": {},
   "outputs": [],
   "source": [
    "expand_exp = sympy.expand(x*expr)\n",
    "expand_exp"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "8dd656a8",
   "metadata": {},
   "outputs": [],
   "source": [
    "factor_exp = sympy.factor(expand_exp)\n",
    "factor_exp"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "14528e6a",
   "metadata": {},
   "source": [
    "## Fonctions"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f4995b2f",
   "metadata": {},
   "source": [
    "Soit la fonction $f(x,y) = x^2 + y$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "f2438c85",
   "metadata": {},
   "outputs": [],
   "source": [
    "f = sympy.Function('f')\n",
    "f = x**2 + y\n",
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "9a195faa",
   "metadata": {},
   "outputs": [],
   "source": [
    "f.subs(x, 4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "8117f3ce",
   "metadata": {},
   "outputs": [],
   "source": [
    "f.subs(y, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "f081c66b",
   "metadata": {},
   "outputs": [],
   "source": [
    "f.subs({x:1, y: 2})"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1e4ce429",
   "metadata": {},
   "source": [
    "### Vectorisation et affichage"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6cf71007",
   "metadata": {},
   "source": [
    "Soit la fonction $g(x) = \\sin(\\frac{x}{2} + \\sin(x))$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "39411d59",
   "metadata": {},
   "outputs": [],
   "source": [
    "g = sympy.Function('g')\n",
    "g = sympy.sin(x/2 + sympy.sin(x))\n",
    "g"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "58bfb293",
   "metadata": {},
   "source": [
    "On souhaite afficher cette fonction sur l'intervalle $[-\\pi, \\pi]$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "e76793e6",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "xlin = np.linspace(-np.pi, np.pi, 21)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "62e6aa1d",
   "metadata": {},
   "source": [
    "> Que donnent les instructions suivantes ?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "5b57a106",
   "metadata": {},
   "outputs": [],
   "source": [
    "result = g.subs(x, xlin)\n",
    "result"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "a00b04c3",
   "metadata": {},
   "outputs": [],
   "source": [
    "type(g)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5dc7437a",
   "metadata": {},
   "source": [
    "> **Testez les instructions suivantes**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "80c38e77",
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "from sympy.utilities.lambdify import lambdify\n",
    "func = lambdify([x], g)\n",
    "type(func)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "43fac308",
   "metadata": {},
   "outputs": [],
   "source": [
    "yres = func(xlin)\n",
    "type(yres), yres.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "72bb1bea",
   "metadata": {},
   "source": [
    "> **Affichez la fonction g(x) en fonction de xlin**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "e817b93c",
   "metadata": {},
   "outputs": [],
   "source": [
    "# TO DO"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f5d532b0",
   "metadata": {},
   "source": [
    "## Expressions avec des dérivées"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d5e9d586",
   "metadata": {},
   "source": [
    "La plupart des problèmes de physique font intervenir des équations différentielles nécessitant alors la possibilité de déclarer des expressions incluant des dérivées."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1f55c1bf",
   "metadata": {},
   "source": [
    "Dans la suite de cette exercice, nous nous intéresserons à l'équation différentielle régissant la décharge d'une capacité dans une résistance selon une constante de temps $\\tau$ : $$V_s(t) + \\tau \\cdot \\frac{d V_s(t)}{dt} = 0$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b2f027c2",
   "metadata": {},
   "source": [
    "On déclare au préalable les fonctions et les symboles dont nous aurons besoin. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "eedcc563",
   "metadata": {},
   "outputs": [],
   "source": [
    "t, tau = sympy.symbols('t tau')\n",
    "vs = sympy.Function('V_s')(t)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "69061b85",
   "metadata": {},
   "source": [
    "On définit ensuite la dérivée de $V_s(t)$ et l'expression précédente :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "77e08e40",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\tau \\frac{d}{d t} V_{s}{\\left(t \\right)} + V_{s}{\\left(t \\right)}$"
      ],
      "text/plain": [
       "tau*Derivative(V_s(t), t) + V_s(t)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dvs = sympy.Derivative(vs, t)\n",
    "exp = vs + tau * dvs\n",
    "exp"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "97ce005b",
   "metadata": {},
   "source": [
    "### Résolution d'une équation différentielle"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fc78d8cf",
   "metadata": {},
   "source": [
    "Lorsqu'une solution analytique peut être calculée, il est possible d'utiliser la fonction ***dsolve*** de Sympy pour obtenir cette expression."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "e36eb4e8",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "vs_t = C1*exp(-t/tau)\n"
     ]
    }
   ],
   "source": [
    "result = sympy.dsolve(exp, vs)\n",
    "vs_t = result.rhs\n",
    "print(f'vs_t = {vs_t}')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d1436101",
   "metadata": {},
   "source": [
    "> La solution donnée est-elle juste ? "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "21785300",
   "metadata": {},
   "source": [
    "Il est également possible d'ajouter des conditions initiales."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "72ef30b9",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "vs_t = 5*exp(-t/tau)\n"
     ]
    }
   ],
   "source": [
    "init_conds = {vs.subs(t,0): 5}\n",
    "result = sympy.dsolve(exp, vs, ics=init_conds)\n",
    "vs_t = result.rhs\n",
    "print(f'vs_t = {vs_t}')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ced0731e",
   "metadata": {},
   "source": [
    "> Quelle est la différence avec l'expression précédente ?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c4fd0734",
   "metadata": {},
   "source": [
    "## Calculs de limites\n",
    "Soit la fonction $g(x) = \\sin(\\frac{x}{2} + \\sin(x))$.\n",
    "\n",
    "On souhaite calculer la limite de cette fonction $g(x)$ en $\\pi$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "5c9b0710",
   "metadata": {},
   "outputs": [],
   "source": [
    "g = sympy.Function('g')\n",
    "g = sympy.sin(x/2 + sympy.sin(x))\n",
    "g"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "6739399f",
   "metadata": {},
   "outputs": [],
   "source": [
    "lg = sympy.limit(g, x, sympy.pi)\n",
    "lg"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a4e1ca24",
   "metadata": {},
   "source": [
    "## Dérivées"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "7bb1b568",
   "metadata": {},
   "outputs": [],
   "source": [
    "f = x**2 + y\n",
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "914889a3",
   "metadata": {},
   "outputs": [],
   "source": [
    "dfx = sympy.diff(f, x)\n",
    "dfx"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "e3649806",
   "metadata": {},
   "outputs": [],
   "source": [
    "dfy = sympy.diff(f, y)\n",
    "dfy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "bee61453",
   "metadata": {},
   "outputs": [],
   "source": [
    "k = sympy.Function('k')\n",
    "l = sympy.Function('l')\n",
    "k = k(x)\n",
    "l = l(x + k)\n",
    "k, l"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "59f1532d",
   "metadata": {},
   "outputs": [],
   "source": [
    "dfkl = sympy.diff(l, x)\n",
    "dfkl"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4d172a7d",
   "metadata": {},
   "source": [
    "## Intégrales"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "b3e23608",
   "metadata": {},
   "outputs": [],
   "source": [
    "inte_f = sympy.integrate(f, x)\n",
    "inte_f"
   ]
  }
 ],
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