Chapter-14.ipynb

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  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib widget\n",
    "from sympy import *\n",
    "init_printing(use_latex=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 14 - Series Expansion"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 14.1 - Basic Usage"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = symbols(\"x\")\n",
    "s = series(sin(x))\n",
    "s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "s = log(x).series(x, 2, 4)\n",
    "s"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 14.1.1 - The Order class and the removeO() method"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "o = s.args[4]\n",
    "display(o)\n",
    "type(o)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "o.args"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "Order(x + x**2, (x, 0))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "O(x + x**2, (x, oo))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr1 = x + x**3 + x**5 + x**7\n",
    "expr2 = expr1 + Order(x**4)\n",
    "display(expr1, expr2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "s1 = cos(x).series(x, pi, 6)\n",
    "s2 = s1.removeO()\n",
    "display(s1, s2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 14.1.2 - Series expansion of multivariate expressions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "y = symbols(\"y\")\n",
    "expr = sin(x) * cos(y)\n",
    "expr.series(x, 0).series(y, 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr.series(x, 0).removeO().series(y, 0).removeO()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 14.1.3 - Series expansion of Undefined Functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = Function(\"f\")\n",
    "s = f(x).series(x, 0, 3)\n",
    "s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "t = s.args[0].args[1]\n",
    "display(t)\n",
    "print(type(t))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "s.args"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x0 = symbols(\"x_0\")\n",
    "s = f(x).series(x, x0, 3)\n",
    "s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "s.subs(f, sin).doit()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Fourier Expansion"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x, L = symbols(\"x, L\")\n",
    "expr = x / (2 * L)\n",
    "fs = fourier_series(expr, (x, 0, 2 * L))\n",
    "display(expr, fs)\n",
    "print(type(fs))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = 3\n",
    "display(fs.truncate(n))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from spb import plot\n",
    "plot((frac(x / 2), \"sawtooth\"),\n",
    "      (fs.truncate(2).subs(L, 1), \"n=2\"),\n",
    "      (fs.truncate(4).subs(L, 1), \"n=4\"),\n",
    "      (fs.truncate(10).subs(L, 1), \"n=10\"),\n",
    "      (x, -0.5, 2.5))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 14.2 - Example - Linearization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "t = symbols(\"t\")\n",
    "phi, psi, theta = (s(t) for s in symbols(\"phi, psi, theta\", cls=Function))\n",
    "expr = -cos(phi) * sin(psi) + sin(phi) * sin(theta) * cos(psi)\n",
    "expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "expr.series(phi, 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def linearize(expr, order=1, tup=None, n=3, apply_lin=True):\n",
    "    if n < order:\n",
    "        n = order\n",
    "    if not tup:\n",
    "        from sympy.core.function import AppliedUndef\n",
    "        tup = []\n",
    "        funcs = list(expr.find(AppliedUndef))\n",
    "        for f in funcs:\n",
    "            tup.append((f, 0))\n",
    "\n",
    "    subs_dict = dict()\n",
    "    for t in tup:\n",
    "        f, f0 = t\n",
    "        s = Dummy(f.func.name)\n",
    "        expr = expr.subs(f, s)\n",
    "        subs_dict[s] = f\n",
    "        expr = expr.series(s, f0, n).removeO()\n",
    "\n",
    "    if apply_lin:\n",
    "        expr = expr.expand()\n",
    "        get_degree = lambda expr, symbols: sum([degree(expr, gen=s) for s in symbols])\n",
    "        args = [a for a in expr.args if get_degree(a, list(subs_dict.keys())) <= order]\n",
    "        expr = expr.func(*args)\n",
    "    expr = expr.subs(subs_dict)\n",
    "    return expr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "linearize(expr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "linearize(expr, apply_lin=False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "linearize(expr, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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