{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "4fe0795e",
   "metadata": {},
   "source": [
    "# Leibniz Formula from a Fourier Series\n",
    "\n",
    "The Leibniz series\n",
    "\n",
    "$$\n",
    "\\frac{\\pi}{4}=1-\\frac13+\\frac15-\\frac17+\\cdots\n",
    "$$\n",
    "\n",
    "follows by evaluating the Fourier series of the sawtooth\n",
    "$f(x)=x$ at $x=\\pi/2$.  This notebook derives the coefficients\n",
    "symbolically before making that substitution.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7a46b4a7",
   "metadata": {},
   "source": [
    "## The sawtooth and its coefficients\n",
    "\n",
    "Since $f$ is odd, only sine terms occur:\n",
    "\n",
    "$$\n",
    "x=\\sum_{k=1}^{\\infty}b_k\\sin(kx),\\qquad\n",
    "b_k=\\frac1\\pi\\int_{-\\pi}^{\\pi}x\\sin(kx)\\,dx.\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "8fe91f34",
   "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": "ea1343e8",
   "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": 3,
   "id": "81fe69e4",
   "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": "92a62bbc",
   "metadata": {},
   "source": [
    "## Fourier terms\n",
    "\n",
    "Egison now combines each coefficient with its basis function.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "d71c6139",
   "metadata": {},
   "outputs": [],
   "source": [
    "def fourierTerms : [MathValue] :=\n",
    "  map (\\(k, b) -> b * sin (k * x)) (zip nats sineCoefficients)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "64cced3c",
   "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": "08076199",
   "metadata": {},
   "source": [
    "## Evaluate at $x=\\pi/2$\n",
    "\n",
    "Even harmonics vanish, while successive odd harmonics alternate in\n",
    "sign.  We encode the exact four-step pattern\n",
    "\n",
    "$$\n",
    "\\sin(k\\pi/2)=1,0,-1,0,\\ldots\n",
    "$$\n",
    "\n",
    "before dividing the Fourier terms by two.  Thus the identity\n",
    "$\\pi/2=2(1-1/3+1/5-\\cdots)$ gives the desired series.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "fb0deb25",
   "metadata": {},
   "outputs": [],
   "source": [
    "def sinAtHalfPi (k : Integer) : MathValue :=\n",
    "  if isEven k\n",
    "    then 0\n",
    "    else (-1) ^ (i.quotient (k - 1) 2)\n",
    "\n",
    "def leibnizTerms : [MathValue] :=\n",
    "  map\n",
    "    (\\(k, b) -> b * sinAtHalfPi k / 2)\n",
    "    (zip nats sineCoefficients)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "a08d47cd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\{1, 0, \\frac{-1}{3}, 0, \\frac{1}{5}, 0, \\frac{-1}{7}, 0, \\frac{1}{9}, 0\\}$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "take 10 leibnizTerms\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7ad715af",
   "metadata": {},
   "source": [
    "Reading the nonzero entries yields\n",
    "$1,-1/3,1/5,-1/7,\\ldots$.  Their infinite sum is $\\pi/4$;\n",
    "the zeros record the even Fourier modes that vanish at $\\pi/2$.\n"
   ]
  }
 ],
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   "display_name": "Egison",
   "language": "egison",
   "name": "egison"
  },
  "language_info": {
   "codemirror_mode": "egison",
   "file_extension": ".egi",
   "mimetype": "text/x-egison",
   "name": "egison"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}