{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "3afbe59d",
   "metadata": {},
   "source": [
    "# Fourier Series of a Sawtooth Wave\n",
    "\n",
    "Consider the $2\\pi$-periodic extension of $f(x)=x$ on\n",
    "$-\\pi<x<\\pi$.  Its Fourier series is\n",
    "\n",
    "$$\n",
    "f(x)\\sim \\frac{a_0}{2}\n",
    "  +\\sum_{k=1}^{\\infty}a_k\\cos(kx)\n",
    "  +\\sum_{k=1}^{\\infty}b_k\\sin(kx).\n",
    "$$\n",
    "\n",
    "Egison symbolically evaluates the coefficient integrals and builds\n",
    "the first terms of the series.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bef7916d",
   "metadata": {},
   "source": [
    "## Symbolic antiderivatives\n",
    "\n",
    "The coefficients use\n",
    "\n",
    "$$\n",
    "a_k=\\frac1\\pi\\int_{-\\pi}^{\\pi}x\\cos(kx)\\,dx,\n",
    "\\qquad\n",
    "b_k=\\frac1\\pi\\int_{-\\pi}^{\\pi}x\\sin(kx)\\,dx.\n",
    "$$\n",
    "\n",
    "Integration by parts gives explicit primitives for\n",
    "$x\\cos(kx)$ and $x\\sin(kx)$.  We define those formulas directly,\n",
    "keeping the notebook independent of optional integration libraries.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "11e361ce",
   "metadata": {},
   "outputs": [],
   "source": [
    "declare symbol x, n : MathValue\n",
    "\n",
    "def f (x : MathValue) : MathValue := x\n",
    "\n",
    "def cosinePrimitive (k : MathValue) : MathValue :=\n",
    "  x * sin (k * x) / k + cos (k * x) / k^2\n",
    "\n",
    "def sinePrimitive (k : MathValue) : MathValue :=\n",
    "  (- x) * cos (k * x) / k + sin (k * x) / k^2\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "cfed2a88",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$-\\cos(n x) n^{-1} x + \\sin(n x) n^{-2}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sinePrimitive n\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "28ee912d",
   "metadata": {},
   "source": [
    "## Fourier coefficients\n",
    "\n",
    "The sawtooth is odd, so every cosine coefficient is zero.  The sine\n",
    "coefficients are $b_k=2(-1)^{k+1}/k$.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "f05ac1f2",
   "metadata": {},
   "outputs": [],
   "source": [
    "def cosineCoefficients : [MathValue] :=\n",
    "  map\n",
    "    (\\k ->\n",
    "      let primitive := cosinePrimitive k\n",
    "       in (substitute [(x, π)] primitive - substitute [(x, - π)] primitive) / π)\n",
    "    nats\n",
    "\n",
    "def sineCoefficients : [MathValue] :=\n",
    "  map\n",
    "    (\\k ->\n",
    "      let primitive := sinePrimitive k\n",
    "       in (substitute [(x, π)] primitive - substitute [(x, - π)] primitive) / π)\n",
    "    nats\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "974066ad",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\{0, 0, 0, 0, 0, 0, 0, 0, 0, 0\\}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "take 10 cosineCoefficients\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "3d203a5c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\{2, -1, \\frac{2}{3}, \\frac{-1}{2}, \\frac{2}{5}, \\frac{-1}{3}, \\frac{2}{7}, \\frac{-1}{4}, \\frac{2}{9}, \\frac{-1}{5}\\}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "take 10 sineCoefficients\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a3ef0cbd",
   "metadata": {},
   "source": [
    "## Reconstructing the series\n",
    "\n",
    "Multiplying each $b_k$ by $\\sin(kx)$ gives the successive Fourier\n",
    "terms.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "aeef0488",
   "metadata": {},
   "outputs": [],
   "source": [
    "def fourierTerms : [MathValue] :=\n",
    "  map (\\(k, b) -> b * sin (k * x)) (zip nats sineCoefficients)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "197ca621",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\{2 \\sin(x), -\\sin(2 x), \\frac{2}{3} \\sin(3 x), \\frac{-1}{2} \\sin(4 x), \\frac{2}{5} \\sin(5 x), \\frac{-1}{3} \\sin(6 x), \\frac{2}{7} \\sin(7 x), \\frac{-1}{4} \\sin(8 x), \\frac{2}{9} \\sin(9 x), \\frac{-1}{5} \\sin(10 x)\\}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "take 10 fourierTerms\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8f1ad57d",
   "metadata": {},
   "source": [
    "Thus\n",
    "\n",
    "$$\n",
    "x=2\\sum_{k=1}^{\\infty}\\frac{(-1)^{k+1}}{k}\\sin(kx)\n",
    "\\qquad(-\\pi<x<\\pi).\n",
    "$$\n",
    "\n",
    "The separate parity and coefficient cells make clear why no cosine\n",
    "terms survive.\n"
   ]
  }
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