Chapter-03.ipynb

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   "source": [
    "# 3 - Functions"
   ]
  },
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   "source": [
    "%matplotlib widget\n",
    "from sympy import *\n",
    "init_printing(use_latex=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3.1 - Undefined Functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = Function(\"f\")\n",
    "display(f)\n",
    "print(type(f))"
   ]
  },
  {
   "cell_type": "code",
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   "source": [
    "# argument of the function\n",
    "x = Symbol(\"x\")\n",
    "# fa represent the function application\n",
    "fa = f(x)\n",
    "display(fa)\n",
    "type(fa)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "isinstance(f, Function), issubclass(f, Function)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "isinstance(fa, f), isinstance(fa, Function)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy.core.function import UndefinedFunction, AppliedUndef\n",
    "print(isinstance(f, UndefinedFunction), isinstance(f, AppliedUndef))\n",
    "print(isinstance(fa, UndefinedFunction), isinstance(fa, AppliedUndef))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "isinstance(f, Expr), isinstance(fa, Expr)"
   ]
  },
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   "cell_type": "code",
   "execution_count": null,
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   "source": [
    "fa.args"
   ]
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   "execution_count": null,
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   "outputs": [],
   "source": [
    "y = Symbol(\"y\", integer=True)\n",
    "# note that g is an applied function!\n",
    "g = Function(\"g\", real=True)(x, y)\n",
    "display(g)\n",
    "print(g.is_real, g.is_integer)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "g.args"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "g.is_commutative"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "i = Function(\"i\", commutative=True)\n",
    "i.is_commutative"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "f, g = symbols(\"f, g\", cls=Function)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "symbols(\"f:3\", cls=Function)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3.2 - Elementary functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = Symbol(\"x\")\n",
    "y = Symbol(\"y\", positive=True)\n",
    "Max(-1, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Max(-1, x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Max(-1, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "g = Max(x, y)\n",
    "g.args"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = sin(pi * x)\n",
    "f.args"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(type(f))\n",
    "print(isinstance(f, sin), isinstance(f, Function))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3.3 - Lambda functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = symbols(\"x\")\n",
    "f = Lambda(x, x**2 + 3)\n",
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "f(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "y = symbols(\"y\")\n",
    "f(sin(y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "s, t, u = symbols(\"s, t, u\")\n",
    "f = Lambda((s, t, u), s * t + u)\n",
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "g = lambda z: z**2 + 3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "h = Lambda(z, z**2 + 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x, y, z = symbols(\"x:z\")\n",
    "f = Lambda((x, y, z), x + y * z)\n",
    "f.args"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "g(sin(y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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