Old_Ver_for_Julia_v0.6/02 series.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 02 級数\n",
    "\n",
    "黒木玄\n",
    "\n",
    "2018-04-18\n",
    "\n",
    "* Copyright 2018 Gen Kuroki\n",
    "* License: MIT https://opensource.org/licenses/MIT\n",
    "* Repository: https://github.com/genkuroki/Calculus\n",
    "\n",
    "このファイルは次の場所できれいに閲覧できる:\n",
    "\n",
    "* http://nbviewer.jupyter.org/github/genkuroki/Calculus/blob/master/02%20series.ipynb\n",
    "\n",
    "* https://genkuroki.github.io/documents/Calculus/02%20series.pdf\n",
    "\n",
    "このファイルは <a href=\"https://juliabox.com\">Julia Box</a> で利用できる.\n",
    "\n",
    "自分のパソコンに<a href=\"https://julialang.org/\">Julia言語</a>をインストールしたい場合には\n",
    "\n",
    "* <a href=\"http://nbviewer.jupyter.org/gist/genkuroki/81de23edcae631a995e19a2ecf946a4f\">WindowsへのJulia言語のインストール</a>\n",
    "\n",
    "を参照せよ.\n",
    "\n",
    "論理的に完璧な説明をするつもりはない. 細部のいい加減な部分は自分で訂正・修正せよ.\n",
    "\n",
    "$\n",
    "\\newcommand\\eps{\\varepsilon}\n",
    "\\newcommand\\ds{\\displaystyle}\n",
    "\\newcommand\\Z{{\\mathbb Z}}\n",
    "\\newcommand\\R{{\\mathbb R}}\n",
    "\\newcommand\\C{{\\mathbb C}}\n",
    "\\newcommand\\QED{\\text{□}}\n",
    "\\newcommand\\root{\\sqrt}\n",
    "\\newcommand\\bra{\\langle}\n",
    "\\newcommand\\ket{\\rangle}\n",
    "\\newcommand\\d{\\partial}\n",
    "\\newcommand\\sech{\\operatorname{sech}}\n",
    "\\newcommand\\cosec{\\operatorname{cosec}}\n",
    "\\newcommand\\sign{\\operatorname{sign}}\n",
    "\\newcommand\\sinc{\\operatorname{sinc}}\n",
    "\\newcommand\\real{\\operatorname{Re}}\n",
    "\\newcommand\\imag{\\operatorname{Im}}\n",
    "\\newcommand\\Li{\\operatorname{Li}}\n",
    "\\newcommand\\np[1]{:\\!#1\\!:}\n",
    "\\newcommand\\PROD{\\mathop{\\coprod\\kern-1.35em\\prod}}\n",
    "$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "toc": true
   },
   "source": [
    "<h1>目次<span class=\"tocSkip\"></span></h1>\n",
    "<div class=\"toc\"><ul class=\"toc-item\"><li><span><a href=\"#級数\" data-toc-modified-id=\"級数-1\"><span class=\"toc-item-num\">1&nbsp;&nbsp;</span>級数</a></span><ul class=\"toc-item\"><li><span><a href=\"#絶対収束と条件収束\" data-toc-modified-id=\"絶対収束と条件収束-1.1\"><span class=\"toc-item-num\">1.1&nbsp;&nbsp;</span>絶対収束と条件収束</a></span><ul class=\"toc-item\"><li><span><a href=\"#絶対収束と条件収束の定義\" data-toc-modified-id=\"絶対収束と条件収束の定義-1.1.1\"><span class=\"toc-item-num\">1.1.1&nbsp;&nbsp;</span>絶対収束と条件収束の定義</a></span></li><li><span><a href=\"#絶対収束級数の収束先は和を取る順番によらないこと\" data-toc-modified-id=\"絶対収束級数の収束先は和を取る順番によらないこと-1.1.2\"><span class=\"toc-item-num\">1.1.2&nbsp;&nbsp;</span>絶対収束級数の収束先は和を取る順番によらないこと</a></span></li></ul></li><li><span><a href=\"#絶対収束級数の例\" data-toc-modified-id=\"絶対収束級数の例-1.2\"><span class=\"toc-item-num\">1.2&nbsp;&nbsp;</span>絶対収束級数の例</a></span><ul class=\"toc-item\"><li><span><a href=\"#絶対収束級数の例1-(Riemannのゼータ函数)\" data-toc-modified-id=\"絶対収束級数の例1-(Riemannのゼータ函数)-1.2.1\"><span class=\"toc-item-num\">1.2.1&nbsp;&nbsp;</span>絶対収束級数の例1 (Riemannのゼータ函数)</a></span></li><li><span><a href=\"#絶対収束級数の例2\" data-toc-modified-id=\"絶対収束級数の例2-1.2.2\"><span class=\"toc-item-num\">1.2.2&nbsp;&nbsp;</span>絶対収束級数の例2</a></span></li></ul></li><li><span><a href=\"#条件収束級数の例-(交代級数)\" data-toc-modified-id=\"条件収束級数の例-(交代級数)-1.3\"><span class=\"toc-item-num\">1.3&nbsp;&nbsp;</span>条件収束級数の例 (交代級数)</a></span><ul class=\"toc-item\"><li><span><a href=\"#条件収束交代級数の例1-(log-2)\" data-toc-modified-id=\"条件収束交代級数の例1-(log-2)-1.3.1\"><span class=\"toc-item-num\">1.3.1&nbsp;&nbsp;</span>条件収束交代級数の例1 (log 2)</a></span></li><li><span><a href=\"#Euler変換による交代級数の数値計算\" data-toc-modified-id=\"Euler変換による交代級数の数値計算-1.3.2\"><span class=\"toc-item-num\">1.3.2&nbsp;&nbsp;</span>Euler変換による交代級数の数値計算</a></span></li><li><span><a href=\"#無限和と積分の交換に関する注意\" data-toc-modified-id=\"無限和と積分の交換に関する注意-1.3.3\"><span class=\"toc-item-num\">1.3.3&nbsp;&nbsp;</span>無限和と積分の交換に関する注意</a></span></li><li><span><a href=\"#条件収束交代級数の例2-(arctan-1-=-π/4)\" data-toc-modified-id=\"条件収束交代級数の例2-(arctan-1-=-π/4)-1.3.4\"><span class=\"toc-item-num\">1.3.4&nbsp;&nbsp;</span>条件収束交代級数の例2 (arctan 1 = π/4)</a></span></li></ul></li></ul></li><li><span><a href=\"#複素数の級数\" data-toc-modified-id=\"複素数の級数-2\"><span class=\"toc-item-num\">2&nbsp;&nbsp;</span>複素数の級数</a></span><ul class=\"toc-item\"><li><span><a href=\"#複素数の絶対収束級数\" data-toc-modified-id=\"複素数の絶対収束級数-2.1\"><span class=\"toc-item-num\">2.1&nbsp;&nbsp;</span>複素数の絶対収束級数</a></span></li><li><span><a href=\"#条件収束する複素級数の例\" data-toc-modified-id=\"条件収束する複素級数の例-2.2\"><span class=\"toc-item-num\">2.2&nbsp;&nbsp;</span>条件収束する複素級数の例</a></span></li><li><span><a href=\"#多重対数函数\" data-toc-modified-id=\"多重対数函数-2.3\"><span class=\"toc-item-num\">2.3&nbsp;&nbsp;</span>多重対数函数</a></span></li><li><span><a href=\"#二重対数函数のζ(2)の近似計算への応用\" data-toc-modified-id=\"二重対数函数のζ(2)の近似計算への応用-2.4\"><span class=\"toc-item-num\">2.4&nbsp;&nbsp;</span>二重対数函数のζ(2)の近似計算への応用</a></span></li><li><span><a href=\"#二重対数函数のζ(2)の正確な計算への応用\" data-toc-modified-id=\"二重対数函数のζ(2)の正確な計算への応用-2.5\"><span class=\"toc-item-num\">2.5&nbsp;&nbsp;</span>二重対数函数のζ(2)の正確な計算への応用</a></span></li><li><span><a href=\"#二重対数函数の高木貞治『解析概論』§35の例2(p.113)への応用\" data-toc-modified-id=\"二重対数函数の高木貞治『解析概論』§35の例2(p.113)への応用-2.6\"><span class=\"toc-item-num\">2.6&nbsp;&nbsp;</span>二重対数函数の高木貞治『解析概論』§35の例2(p.113)への応用</a></span></li><li><span><a href=\"#二重対数函数の五項関係式\" data-toc-modified-id=\"二重対数函数の五項関係式-2.7\"><span class=\"toc-item-num\">2.7&nbsp;&nbsp;</span>二重対数函数の五項関係式</a></span></li></ul></li><li><span><a href=\"#無限積\" data-toc-modified-id=\"無限積-3\"><span class=\"toc-item-num\">3&nbsp;&nbsp;</span>無限積</a></span><ul class=\"toc-item\"><li><span><a href=\"#無限積の絶対収束\" data-toc-modified-id=\"無限積の絶対収束-3.1\"><span class=\"toc-item-num\">3.1&nbsp;&nbsp;</span>無限積の絶対収束</a></span></li><li><span><a href=\"#絶対収束する無限積の収束先が積の順序によらないこと\" data-toc-modified-id=\"絶対収束する無限積の収束先が積の順序によらないこと-3.2\"><span class=\"toc-item-num\">3.2&nbsp;&nbsp;</span>絶対収束する無限積の収束先が積の順序によらないこと</a></span></li><li><span><a href=\"#量子二重対数函数\" data-toc-modified-id=\"量子二重対数函数-3.3\"><span class=\"toc-item-num\">3.3&nbsp;&nbsp;</span>量子二重対数函数</a></span><ul class=\"toc-item\"><li><span><a href=\"#量子二重対数函数と量子指数函数との関係\" data-toc-modified-id=\"量子二重対数函数と量子指数函数との関係-3.3.1\"><span class=\"toc-item-num\">3.3.1&nbsp;&nbsp;</span>量子二重対数函数と量子指数函数との関係</a></span></li><li><span><a href=\"#q二項定理\" data-toc-modified-id=\"q二項定理-3.3.2\"><span class=\"toc-item-num\">3.3.2&nbsp;&nbsp;</span>q二項定理</a></span></li><li><span><a href=\"#q指数函数の加法公式\" data-toc-modified-id=\"q指数函数の加法公式-3.3.3\"><span class=\"toc-item-num\">3.3.3&nbsp;&nbsp;</span>q指数函数の加法公式</a></span></li><li><span><a href=\"#量子二重対数函数の五角形恒等式\" data-toc-modified-id=\"量子二重対数函数の五角形恒等式-3.3.4\"><span class=\"toc-item-num\">3.3.4&nbsp;&nbsp;</span>量子二重対数函数の五角形恒等式</a></span></li></ul></li></ul></li></ul></div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Plots\n",
    "gr(); ENV[\"PLOTS_TEST\"] = \"true\"\n",
    "#clibrary(:colorcet)\n",
    "clibrary(:misc)\n",
    "\n",
    "function pngplot(P...; kwargs...)\n",
    "    sleep(0.1)\n",
    "    pngfile = tempname() * \".png\"\n",
    "    savefig(plot(P...; kwargs...), pngfile)\n",
    "    showimg(\"image/png\", pngfile)\n",
    "end\n",
    "pngplot(; kwargs...) = pngplot(plot!(; kwargs...))\n",
    "\n",
    "showimg(mime, fn) = open(fn) do f\n",
    "    base64 = base64encode(f)\n",
    "    display(\"text/html\", \"\"\"<img src=\"data:$mime;base64,$base64\">\"\"\")\n",
    "end\n",
    "\n",
    "using SymPy\n",
    "#sympy[:init_printing](order=\"lex\") # default\n",
    "#sympy[:init_printing](order=\"rev-lex\")\n",
    "\n",
    "using SpecialFunctions\n",
    "using QuadGK\n",
    "import PyPlot"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 級数"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 絶対収束と条件収束"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 絶対収束と条件収束の定義\n",
    "\n",
    "**定理:** $a_n\\geqq 0$ であると仮定する. そのとき, ある正の定数 $M$ が存在して, すべての番号 $n$ について $\\ds\\sum_{k=1}^n a_k\\leqq M$ が成立しているならば, 無限級数 $\\ds\\sum_{n=1}^\\infty a_n$ は有限の値に収束する. \n",
    "\n",
    "**証明:** $\\ds s_n=\\sum_{k=1}^n a_n$ とおくと, $s_n$ は上に有界な単調増加実数列になる. ゆえに実数の連続性より, $s_n$ は有限の値に収束する. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**定理:** $\\ds\\sum_{n=1}^\\infty |a_n| < \\infty$ ならば級数 $\\ds\\sum_{n=1}^\\infty a_n$ は収束する. そのとき, 級数 $\\ds\\sum_{n=1}^\\infty a_n$ は**絶対収束する**という.\n",
    "\n",
    "**証明:** $\\ds s_n = \\sum_{k=1}^n a_k$ とおく. 任意に $\\eps>0$ を取る. $\\ds\\sum_{n=1}^\\infty |a_n| < \\infty$ より, ある番号 $N$ が存在して, $n\\geqq N$ ならば\n",
    "\n",
    "$$\n",
    "\\sum_{k=n+1}^\\infty |a_k| < \\eps\n",
    "$$\n",
    "\n",
    "となる. ゆえに $m\\geqq n\\geqq N$ のとき, \n",
    "\n",
    "$$\n",
    "\\left|s_m - s_n\\right| = \\left|\\sum_{k=n+1}^m a_k\\right| \\leqq \\sum_{k=n+1}^m|a_k| \\leqq\n",
    "\\sum_{k=n+1}^\\infty |a_k| < \\eps.\n",
    "$$\n",
    "\n",
    "ゆえに $s_n$ はCauchy列になる. ゆえに実数の完備性より, $s_n$ は収束する. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**注意:** 上の定理の逆は成立しないことに注意に注意せよ. 級数 $\\ds\\sum_{n=1}^\\infty a_n$ が収束していても $\\ds\\sum_{n=1}^\\infty |a_n| = \\infty$ となる場合がある. そのとき, 級数 $\\ds\\sum_{n=1}^\\infty a_n$ は**条件収束する**という. $\\QED$\n",
    "\n",
    "**注意:** 上の定理は $a_n$ が実数列でも複素数列でも成立している. さらに $a_n$ が $\\R^n$ や $\\C^n$ の点列でかつ絶対値をノルムで置き換えた場合にも成立している. 直観的には「長さが有限の紐の先端に対応する点が収束先として存在する」というような発想で考えればよい.\n",
    "\n",
    "より一般にはノルム空間においては上の定理が成立することと完備性の条件は同値になっている. 実数列の場合には**実数の連続性**と同値になっている.\n",
    "\n",
    "**実数の連続性**について理解していなくても, 絶対収束の概念を理解していれば足りることは多い. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 絶対収束級数の収束先は和を取る順番によらないこと\n",
    "\n",
    "**定理:** 絶対収束する無限級数の収束先は足し上げる順序をどのように変えても同じになる. $\\QED$\n",
    "\n",
    "このノートでは, 絶対収束する無限級数 $\\ds\\sum_{n=1}^\\infty a_n$ に関する以下の2つの結果を証明しておくことにしよう.\n",
    "\n",
    "(1) $k\\mapsto n_k$ は正の整数全体と正の整数全体を一対一に対応させる写像であるならば, $\\ds\\sum_{k=1}^\\infty a_{n_k}$ も絶対収束して, $\\ds\\sum_{k=1}^\\infty a_{n_k}=\\sum_{n=1}^\\infty a_n$ が成立する.\n",
    "\n",
    "(2) $(k,l)\\mapsto n_{k,l}$ が正の整数の対の全体と正の整数全体を一対一に対応させる写像であるならば, すべての $k$ について $\\ds\\sum_{l=1}^\\infty a_{n_{k,l}}$ は絶対収束し, さらに $\\ds\\sum_{k=1}^\\infty\\sum_{l=1}^\\infty a_{n_{k,l}}$ も絶対収束して, $\\ds\\sum_{k=1}^\\infty\\sum_{l=1}^\\infty a_{n_{k,l}}=\\sum_{n=1}^\\infty a_n$ が成立する.\n",
    "\n",
    "**証明:**  $\\ds M = \\sum_{n=1}^\\infty |a_n|$ とおく. \n",
    "\n",
    "(1) $n_1,n_2,\\ldots,n_K$ の最大値を $N$ と書くと, \n",
    "\n",
    "$$\n",
    "\\sum_{k=1}^K |a_{n_k}| \\leqq \\sum_{n=1}^N |a_n| \\leqq M < \\infty\n",
    "$$\n",
    "\n",
    "なので $\\ds\\sum_{k=1}^\\infty |a_{n_k}|$ は収束する. つまり $\\ds\\sum_{k=1}^\\infty a_{n_k}$ は絶対収束する.  \n",
    "\n",
    "$n_k=1,2,\\ldots,N$ となる $k$ の最大値を $K_N$ 書くと, $n_1,n_2,\\ldots,n_{K_N}$ は $1,2,\\ldots,N$ をすべて含み, $N\\to\\infty$ のとき $K_N\\to\\infty$ となり, \n",
    "\n",
    "$$\n",
    "\\left|\\sum_{k=1}^{K_N}a_{n_k}-\\sum_{n=1}^N a_n\\right| \\leqq\n",
    "\\sum_{n=N+1}^\\infty|a_n|=M-\\sum_{n=1}^N|a_n|\\to 0 \\quad(N\\to\\infty)\n",
    "$$\n",
    "\n",
    "となるので, $\\ds \\sum_{k=1}^\\infty a_{n_k}=\\sum_{n=1}^\\infty a_n$ となることがわかる.\n",
    "\n",
    "(2) 各 $n_{k,1},n_{k,2},\\ldots,n_{k,L}$ の最大値を $N$ と書くと, \n",
    "\n",
    "$$\n",
    "\\sum_{l=1}^L |a_{n_{k,l}}| \\leqq \\sum_{n=1}^N |a_n| \\leqq M < \\infty\n",
    "$$\n",
    "\n",
    "なので, $\\ds \\sum_{l=1}^\\infty a_{n_{k,l}}$ は絶対収束する.\n",
    "\n",
    "任意に $\\eps>0$ を取る. $k=1,2,\\ldots,K$ ごとに $L_k$ を十分に大きくすると, $\\ds \\sum_{l=1}^\\infty |a_{k,l}| \\leqq \\sum_{l=1}^{L_k} |a_{k,l}| + \\eps$ となるので, $n_{k,l}$ ($k=1,2,\\ldots,K$, $l=1,2,\\ldots,L_k$) の最大値を $N$ と書くと, \n",
    "\n",
    "$$\n",
    "\\sum_{k=1}^K \\left|\\sum_{l=1}^\\infty a_{k,l}\\right| \\leqq\n",
    "\\sum_{k=1}^K \\sum_{l=1}^\\infty |a_{n_{k,l}}| \\leqq\n",
    "\\sum_{k=1}^K \\sum_{l=1}^{L_k} |a_{n_{k,l}}| + K\\eps \\leqq\n",
    "\\sum_{n=1}^N |a_n| + K\\eps \\leqq \n",
    "M + K\\eps.\n",
    "$$\n",
    "\n",
    "$\\eps>0$ はいくらでも小さくできるので, \n",
    "\n",
    "$$\n",
    "\\sum_{k=1}^K \\left|\\sum_{l=1}^\\infty a_{k,l}\\right| \\leqq \n",
    "\\sum_{k=1}^K \\sum_{l=1}^\\infty |a_{n_{k,l}}| \\leqq\n",
    "M < \\infty.\n",
    "$$\n",
    "\n",
    "ゆえに $\\ds\\sum_{k=1}^\\infty\\sum_{l=1}^\\infty a_{n_{k,l}}$ も絶対収束する.\n",
    "\n",
    "任意に $\\eps>0$ を取る. $\\{\\,n_{k,l}\\mid k=1,2,\\ldots,K,\\; l=1,2,3,\\ldots\\,\\}\\supset \\{1,2,\\ldots,N\\}$ となるような最小の $K$ を $K_N$ と書くことにする. このとき, $N\\to\\infty$ ならば $K_N\\to\\infty$ となる. 各 $k=1,2,\\ldots,K_N$ に対して, ある $L_k$ が存在して, \n",
    "\n",
    "$$\n",
    "\\{\\,n_{k,l}\\mid k=1,2,\\ldots,K,\\; l=1,2,\\ldots,L_k\\,\\}\\supset \\{1,2,\\ldots,N\\}\n",
    "$$\n",
    "\n",
    "かつ\n",
    "\n",
    "$$\n",
    "\\left|\\sum_{l=1}^\\infty a_{n_{k,l}} - \\sum_{l=1}^{L_k} a_{n_{k,l}}\\right| < \\frac{\\eps}{K_N}\n",
    "$$\n",
    "\n",
    "となる. ゆえに,\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\left|\\sum_{k=1}^{K_N}\\sum_{l=1}^\\infty a_{n_{k,l}} - \\sum_{n=1}^N a_n\\right| &\\leqq\n",
    "\\left|\\sum_{k=1}^{K_N}\\sum_{l=1}^\\infty a_{n_{k,l}} - \\sum_{k=1}^{K_N}\\sum_{l=1}^{L_k} a_{n_{k,l}}\\right| +\n",
    "\\left|\\sum_{k=1}^{K_N}\\sum_{l=1}^{L_k} a_{n_{k,l}} - \\sum_{n=1}^N a_n\\right|\n",
    "\\\\ &\\leqq\n",
    "\\sum_{k=1}^{K_N}\\left|\\sum_{l=1}^\\infty a_{n_{k,l}} - \\sum_{l=1}^{L_k} a_{n_{k,l}}\\right| +\n",
    "\\sum_{n=N+1}^\\infty |a_n|\n",
    "\\\\ &<\n",
    "\\eps + \\sum_{n=N+1}^\\infty |a_n|.\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "$\\eps>0$ はいくらでも小さくできるので, \n",
    "\n",
    "$$\n",
    "\\left|\\sum_{k=1}^{K_N}\\sum_{l=1}^\\infty a_{n_{k,l}} - \\sum_{n=1}^N a_n\\right| \\leqq\n",
    "\\sum_{n=N+1}^\\infty |a_n|.\n",
    "$$\n",
    "\n",
    "右辺は $N\\to\\infty$ のとき $0$ に収束するので, $\\ds\\sum_{k=1}^\\infty\\sum_{l=1}^\\infty a_{n_{k,l}}=\\sum_{n=1}^\\infty a_n$ が成立することがわかる. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "上の定理から次の系(corollary, 当然の結果)がただちに導かれる.\n",
    "\n",
    "**系:** $a_n\\geqq 0$ のとき $a_1,a_2,\\ldots$ の無限和は足し上げる順序をどのように変えても同じになる. $\\QED$\n",
    "\n",
    "**注意:** $a_n\\geqq 0$ のとき $a_1,a_2,\\ldots$ の無限和の値として $\\infty$ も許せば, その無限和は常に値を持っているとみなせる.  非負の実数の無限和を扱うときには収束先として $\\infty$ も許したくなる場合が多いので, その場合には収束先が有限の値になるか否かを明瞭に述べておいた方が誤解が少なくなる. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題:** $\\ds\\sum_{m=1}^\\infty a_m$ と $\\ds\\sum_{n=1}^\\infty b_n$ が絶対収束しているならば, $\\ds\\sum_{m,n\\geqq 1,\\; m+n\\leqq N+1} |a_m b_n|$ も $N\\to\\infty$ で収束しており, \n",
    "\n",
    "$$\n",
    "\\lim_{N\\to\\infty}\\sum_{m,n\\geqq 1,\\; m+n\\leqq N+1} a_m b_n = \n",
    "\\sum_{m=1}^\\infty a_m \\sum_{n=1}^\\infty b_n\n",
    "\\tag{$*$}\n",
    "$$\n",
    "\n",
    "が成立していることを示せ.\n",
    "\n",
    "**解答例:** $\\ds L=\\sum_{m=1}^\\infty |a_m|$, $\\ds M=\\sum_{n=1}^\\infty |b_n|$ とおく. このとき,\n",
    "\n",
    "$$\n",
    "\\sum_{m,n\\geqq 1,\\; m+n\\leqq N+1} |a_m b_n|\n",
    "\\leqq \\sum_{m=1}^N |a_m| \\sum_{n=1}^N |b_n| \\leqq LM < \\infty.\n",
    "$$\n",
    "\n",
    "これより, $\\ds\\sum_{m,n\\geqq 1,\\; m+n\\leqq N+1} |a_m b_n|$ も $N\\to\\infty$ で収束することがわかる. これは $\\ds\\lim_{N\\to\\infty}\\sum_{m,n\\geqq 1,\\; m+n\\leqq N+1} a_m b_n$ が絶対収束することを意味している. したがって, 上の方の定理の(2)より($*$)が成立することもわかる. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**注意:** 条件収束級数の収束先は足し上げる順序によって変わるので扱いには注意が必要になる. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題:** 実数 $a_n$ の級数 $\\ds\\sum_{n=1}^\\infty a_n$ は条件収束していると仮定する. このとき, 任意の実数 $\\alpha$ に対して, その級数の足し上げる順序をうまく変えることによって, 収束先を $\\alpha$ にできることを示せ.\n",
    "\n",
    "**解答例:** $a_n$ うち非負のもの全体を $b_1\\geqq b_2\\geqq \\ldots$ と書き, 負のもの全体を $c_1\\geqq c_2\\geqq\\ldots$ と書く. このとき, $b_n\\to 0$, $c_n\\to 0$, $\\ds\\sum_{n=1}^\\infty b_n=\\infty$ かつ $\\sum_{n=1}^\\infty c_n=-\\infty$ が成立している. まず, $b_1+\\cdots+b_k$ が最初に $\\alpha$ 以上になる $k$ を $k_1$ と定める. 次に $b_1+\\cdots+b_{k_1}+c_1+\\cdots+c_l$ が最初に $\\alpha$ 以下になる $l$ を $l_1$ と定める. これを同様に繰り返すと, 級数 $\\ds\\sum_{n=1}^\\infty a_n$ の項の順序を並び換えた級数\n",
    "\n",
    "$$\n",
    "b_1+\\cdots+b_{k_1}+c_1+\\cdots+c_{l_1}+b_{k_1+1}+\\cdots+b_{k_2}+c_{l_1+1}+\\cdots+c_{l_2}+\\cdots\n",
    "$$\n",
    "\n",
    "ができ上がる. $b_{k_i}$ までの和は和の途中段階も含めてすべて $\\alpha+b_{k_i}$ 未満であり, $c_{l_j}$ までの和は和の途中段階も含めてすべて $\\alpha+c_{l_j}$ より大きくなる($c_l<0$ に注意せよ). したがって, $b_n\\to 0$ と $c_n\\to 0$ より, 上のように並び換えた級数が $\\alpha$ に収束することがわかる. $\\QED$\n",
    "\n",
    "**注意:** 上の解答例の最後の部分の議論は $a_n\\geqq 0$, $a_n\\geqq a_{n+1}$, $a_n\\to 0$ のとき, 交代級数 $\\ds\\sum_{n=1}^\\infty (-1)^{n-1}a_n$ が常に収束することを示すためにも使える. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 絶対収束級数の例"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 絶対収束級数の例1 (Riemannのゼータ函数)\n",
    "\n",
    "$s>1$ と仮定する. このとき $\\ds\\sum_{n=1}^\\infty \\frac{1}{n^s} < \\infty$ である. なぜならば $n\\to\\infty$ のとき\n",
    "\n",
    "$$\n",
    "\\sum_{k=1}^n \\frac{1}{k^s} \\leqq 1 + \\int_1^n \\frac{dx}{x^s} = \n",
    "1 + \\frac{1-n^{1-s}}{s-1} \\to 1 + \\frac{1}{s-1}.\n",
    "$$\n",
    "\n",
    "これより $s>1$ のとき\n",
    "\n",
    "$$\n",
    "\\sum_{n=1}^\\infty \\frac{1}{n^s} \\leqq 1 + \\frac{1}{s-1}\n",
    "$$\n",
    "\n",
    "となることがわかった. $s>1$ の函数 $\\zeta(s)$ を\n",
    "\n",
    "$$\n",
    "\\zeta(s) = \\sum_{n=1}^\\infty \\frac{1}{n^s}\n",
    "$$\n",
    "\n",
    "と定める.  これを**Riemannのゼータ函数**と呼ぶ. Riemannのゼータ函数の定義域は $s\\ne 1$ の複素数全体に自然に拡張される(解析接続される).  Riemannのゼータ函数は整数論における最も基本的で重要な函数である.\n",
    "\n",
    "$s > 1$ のとき, $\\ds\\sum_{n=1}^\\infty\\frac{(-1)^{n-1}}{n^s}$ は絶対収束級数である.  この級数は後で示すように $s=1$ でも収束している.  (さらに複素平面全体に解析接続されることも示せる.)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題:** $s>1$ と仮定する. この仮定のもとで次の公式を証明せよ:\n",
    "\n",
    "$$\n",
    "(1-2^{1-s})\\zeta(s) = \\sum_{n=1}^\\infty\\frac{(-1)^{n-1}}{n^s}.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**証明:** 絶対収束級数を扱っているので, 以下のように形式的に計算した結果がそのまま証明になる.\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "(1-2^{1-s})\\zeta(s) &=\n",
    "\\left(1-\\frac{2}{2^s}\\right)\\sum_{n=1}^\\infty\\frac{1}{n^s} =\n",
    "\\sum_{n=1}^\\infty\\frac{1}{n^s} - \\sum_{n=1}^\\infty\\frac{2}{(2n)^s}\n",
    "\\\\ &=\n",
    "\\left(\\frac{1}{1^s}+\\frac{1}{2^s}+\\frac{1}{3^s}+\\frac{1}{4^s}+\\cdots\\right)\n",
    "-\\left(\\frac{2}{2^s}+\\frac{2}{4^s}+\\cdots\\right)\n",
    "\\\\ &=\n",
    "\\frac{1}{1^s}-\\frac{1}{2^s}+\\frac{1}{3^s}-\\frac{1}{4^s}+\\cdots =\n",
    "\\sum_{n=1}^\\infty\\frac{(-1)^{n-1}}{n^s}.\n",
    "\\qquad\\QED\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題:** $\\zeta(2)$ を数値計算して, $\\pi^2/6$ に近いことを確認せよ."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "SimpleSum = 1.6449340568482278\n",
      "TrueValue = 1.6449340668482264\n",
      "SimpleSum - TrueValue = -9.99999860695766e-9\n"
     ]
    }
   ],
   "source": [
    "N = 10^8\n",
    "SimpleSum = sum(n->1/n^2, 1:N)\n",
    "TrueValue = π^2/6\n",
    "@show SimpleSum\n",
    "@show TrueValue\n",
    "@show SimpleSum - TrueValue;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題:** $\\zeta(2)=\\pi^2/6$ の証明について調べよ.\n",
    "\n",
    "**ヒント:** この問題は**Basel問題**などと呼ばれているようである. インターネットで検索してみよ. $\\QED$\n",
    "\n",
    "正の偶数 $2m$ について $\\zeta(2m)$ はすべて計算されている. しかし, 3以上の奇数 $2m+1$ に対する $\\zeta(2m+1)$ の数学的な素性はまだよくわかっていない."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 絶対収束級数の例2\n",
    "\n",
    "**問題:** $\\ds\\sum_{n=2}^\\infty(\\zeta(n)-1)=1$ を示せ."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**証明:** $k\\geqq 2$ のとき\n",
    "\n",
    "$$\n",
    "\\ds\\sum_{n=2}^\\infty\\frac{1}{k^n} = \\frac{1/k^2}{1-1/k} = \\frac{1}{k(k-1)} = \\frac{1}{k-1} - \\frac{1}{k}.\n",
    "$$\n",
    "\n",
    "そして\n",
    "\n",
    "$$\n",
    "\\zeta(n) - 1 = \\frac{1}{2^n}+\\frac{1}{3^n}+\\frac{1}{4^n}+\\cdots =\n",
    "\\sum_{k=2}^\\infty\\frac{1}{k^n}\n",
    "$$\n",
    "\n",
    "なので, \n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\sum_{n=2}^\\infty (\\zeta(n)-1) &=\n",
    "\\sum_{n=2}^\\infty\\sum_{k=2}^\\infty\\frac{1}{k^n} =\n",
    "\\sum_{k=2}^\\infty\\sum_{n=2}^\\infty\\frac{1}{k^n} =\n",
    "\\sum_{k=2}^\\infty\\left(\\frac{1}{k-1} - \\frac{1}{k}\\right) \n",
    "\\\\ &=\n",
    "\\left(\\frac{1}{1} - \\frac{1}{2}\\right) + \n",
    "\\left(\\frac{1}{2} - \\frac{1}{3}\\right) + \n",
    "\\left(\\frac{1}{3} - \\frac{1}{3}\\right) + \n",
    "\\cdots =\n",
    "1.\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "2つ目の等号で $1/k^n\\geqq 0$ なので足し上げる順序を変えても総和は等しいことを使った. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "以下はコンピューターによる計算."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0000000000000044"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N = 50\n",
    "sum(n->zeta(n)-1, 2:N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$1$$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "l, n = symbols(\"l n\", integer=true, positive=true) # l = k-1\n",
    "s = simplify(doit(Sum(1/(l+1)^n, (n,2,oo))))\n",
    "S = doit(Sum(s, (l,1,oo)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題:** 上の問題の結果より, $\\ds\\sum_{n=2}^\\infty(-1)^n(\\zeta(n)-1)$ は絶対収束する. その収束先を求めよ."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**証明:** $k\\geqq 2$ のとき\n",
    "\n",
    "$$\n",
    "\\ds\\sum_{n=2}^\\infty\\frac{(-1)^n}{k^n} = \\frac{1/k^2}{1+1/k} = \\frac{1}{k(k+1)} = \\frac{1}{k} - \\frac{1}{k+1}.\n",
    "$$\n",
    "\n",
    "そして\n",
    "\n",
    "$$\n",
    "(-1)^n(\\zeta(n) - 1) = \\frac{(-1)^n}{2^n}+\\frac{(-1)^n}{3^n}+\\frac{(-1)^n}{4^n}+\\cdots =\n",
    "\\sum_{k=2}^\\infty\\frac{(-1)^n}{k^n}\n",
    "$$\n",
    "\n",
    "なので, \n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\sum_{n=2}^\\infty (\\zeta(n)-1) &=\n",
    "\\sum_{n=2}^\\infty\\sum_{k=2}^\\infty\\frac{(-1)^n}{k^n} =\n",
    "\\sum_{k=2}^\\infty\\sum_{n=2}^\\infty\\frac{(-1)^n}{k^n} =\n",
    "\\sum_{k=2}^\\infty\\left(\\frac{1}{k} - \\frac{1}{k+1}\\right) \n",
    "\\\\ &=\n",
    "\\left(\\frac{1}{2} - \\frac{1}{3}\\right) + \n",
    "\\left(\\frac{1}{3} - \\frac{1}{4}\\right) + \n",
    "\\left(\\frac{1}{4} - \\frac{1}{5}\\right) + \n",
    "\\cdots =\n",
    "\\frac{1}{2}.\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "2つ目の等号で絶対収束級数なので足し上げる順序を変えても総和は等しいことを使った. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "以下はコンピューターによる計算."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5000000000000018"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N = 50\n",
    "sum(n->(-1)^n*(zeta(n)-1), 2:N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\frac{1}{2}$$"
      ],
      "text/plain": [
       "1/2"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "l, n = symbols(\"l n\", integer=true, positive=true) # l = k-1\n",
    "s = simplify(doit(Sum((-1)^n/(l+1)^n, (n,2,oo))))\n",
    "S = doit(Sum(s, (l,1,oo)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題:** $\\ds\\sum_{n=2}^\\infty \\frac{1}{n\\log n}=\\infty$ を示せ.\n",
    "\n",
    "**解答例:** $N\\to\\infty$ のとき,\n",
    "\n",
    "$$\n",
    "\\sum_{n=2}^N \\frac{1}{n\\log n} \\geqq \\sum_{n=2}^N \\int_n^{n+1} \\frac{dx}{x\\log x} =\n",
    "\\int_2^{N+1}\\frac{dx}{x\\log x} = [\\log\\log x]_2^{N+1} \\geqq \\log\\log(N+1)\\to\\infty.\n",
    "\\qquad\\QED\n",
    "$$\n",
    "\n",
    "**問題:** $\\ds\\sum_{n=3}^\\infty \\frac{1}{n(\\log n)^2} < \\infty$ を示せ.\n",
    "\n",
    "**解答例:** 左辺の部分和が有界であることが次のようにして確認できる:\n",
    "\n",
    "$$\n",
    "\\sum_{n=3}^N \\frac{1}{n(\\log n)^2} \\leqq \\sum_{n=3}^N\\int_{n-1}^n \\frac{dx}{x(\\log x)^2} =\n",
    "\\int_2^N \\frac{dx}{x(\\log x)^2} = \\left[\\frac{-1}{\\log x}\\right]_2^N \\leqq \\frac{1}{\\log 2}.\n",
    "\\qquad \\QED\n",
    "$$\n",
    "\n",
    "**問題:** $a>1$ のとき $\\ds\\sum_{n=3}^\\infty \\frac{1}{n(\\log n)^a}<\\infty$ となることを示せ.\n",
    "\n",
    "**解答例:** 左辺の部分和が有界であることが次のようにして確認できる:\n",
    "\n",
    "$$\n",
    "\\sum_{n=3}^N \\frac{1}{n(\\log n)^a} \\leqq \\sum_{n=3}^N\\int_{n-1}^n \\frac{dx}{x(\\log x)^a} =\n",
    "\\int_2^N \\frac{dx}{x(\\log x)^a} = \\left[\\frac{\\log(x)^{1-a}}{1-a}\\right]_2^N \\leqq \\frac{(\\log 2)^{1-a}}{a-1}.\n",
    "\\qquad \\QED\n",
    "$$\n",
    "\n",
    "**注意:** $x = e^y$ と置換すると, $\\log x = y$, $\\ds\\frac{dx}{x}=dy$ なので, $0<r<R$ について, \n",
    "\n",
    "$$\n",
    "\\int_r^R \\frac{dx}{x(\\log x)^a} = \\int_{\\log r}^{\\log R} \\frac{dy}{y^a}.\n",
    "$$\n",
    "\n",
    "これは, $a\\leqq 1$ ならば $R\\to\\infty$ で無限大に発散し, $a>1$ ならば有限の値に収束する. $\\QED$\n",
    "\n",
    "**問題:** 以上の例以外の絶対収束級数の例について調べ, ノートにまとめよ."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 条件収束級数の例 (交代級数)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**定理:** 一般に $a_1\\geqq a_2\\geqq a_3\\geqq\\cdots$ かつ $a_n\\geqq 0$ かつ $a_n\\to 0$ のとき, **交代級数**\n",
    "\n",
    "$$\n",
    "\\sum_{n=1}^\\infty (-1)^{n-1} a_n = a_1 - a_2 + a_3 - a_4 + \\cdots\n",
    "$$\n",
    "\n",
    "は常に収束する. \n",
    "\n",
    "**注意:** この定理中の $a_n$ について $\\ds\\sum_{n=1}^\\infty a_n=\\infty$ が成立しているとき, 交代級数は条件収束級数になる.\n",
    "\n",
    "**証明:** $\\ds c_n = \\sum_{k=1}^n (-1)^{k-1}$ とおくと, $n$ が奇数のとき $c_n=1$ となり, $n$ が偶数のとき $c_n=0$ となる.  $c_n-c_{n-1}=(-1)^{n-1}$ なので,\n",
    "\n",
    "$$\n",
    "\\sum_{k=1}^n (-1)^{k-1} a_k = \\sum_{k=1}^n (c_k - c_{k-1}) a_k =\n",
    "\\sum_{k=1}^n c_k a_k - \\sum_{k=0}^{n-1} c_k a_{k+1} =\n",
    "c_n a_n - c_0 a_1 + \\sum_{k=1}^{n-1} c_k(a_k-a_{k+1})\n",
    "$$\n",
    "\n",
    "であり, $|c_k|\\leqq 1$ かつ $a_n$ が単調減少することより $a_k - a_{k+1}\\geqq 0$ であり, $a_n\\to 0$ より $a_n\\geqq 0$ なので, \n",
    "\n",
    "$$\n",
    "\\sum_{k=1}^{n-1}|c_k||a_k-a_{k+1}| \\leqq\n",
    "\\sum_{k=1}^{n-1} (a_k-a_{k+1}) = a_1-a_n \\leqq a_1.\n",
    "$$\n",
    "\n",
    "ゆえに $\\ds \\sum_{k=1}^\\infty c_k(a_k-a_{k+1})$ は絶対収束する. そして, $a_n\\to 0$ かつ $|c_n|\\leqq 1$ より $c_n a_n\\to 0$ となるので, $\\ds \\sum_{k=1}^n (-1)^{k-1} a_k$ が $n\\to\\infty$ で収束することがわかる. $\\QED$\n",
    "\n",
    "**注意:** 上の証明は部分積分と類似の計算を和の場合に行っている. 部分積分やそれと同様の和の変形は収束し難い積分や無限和を収束し易くするために役に立つ. $\\QED$\n",
    "\n",
    "**注意:** 上の証明の議論は $c_n$ が有界で $a_n$ が単調減少しながら $0$ に収束する場合に一般化される. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**別証明:** $a_n$ が0以上で単調減少であることより, $s_n = a_1-a_2+a_3-a_4+\\cdots+(-1)^{n-1}a_n$ は\n",
    "\n",
    "$$\n",
    "0\\leqq s_2 \\leqq s_4\\leqq s_6\\leqq \\cdots\\leqq s_5 \\leqq s_3 \\leqq s_1\n",
    "$$\n",
    "\n",
    "を満たしていることがわかる. $s_1,s_3,s_5,\\ldots$ は下に有界な単調減少数列なので収束し, $s_2,s_4,s_6,\\ldots$ は上に有界な単調増加数列なので収束し,  $s_n - s_{n-1} = a_n$ が $0$ に収束することからそれらの収束先は一致し, $s_n$ も収束することがわかる. $\\QED$\n",
    "\n",
    "**注意:** 交代級数の部分和の誤差の評価にはこの別証明の議論が有用であり, よく使われる. 上の議論より, $s_n$ の収束先を $\\alpha$ と書くとき, \n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "&\n",
    "\\text{$n$ is odd} \\implies\n",
    "-a_{n+1} \\leqq \\alpha - s_n \\leqq 0,\n",
    "\\\\ &\n",
    "\\text{$n$ is even} \\implies\n",
    "0\\leqq \\alpha - s_n \\leqq a_{n+1}\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "が成立していることがわかる. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 条件収束交代級数の例1 (log 2)\n",
    "\n",
    "$\\ds\\sum_{n=1}^\\infty\\frac{(-1)^{n-1}}{n}$ は収束するが, $\\ds\\sum_{n=1}^\\infty\\frac{1}{n}=\\infty$ となる.\n",
    "\n",
    "なぜならば, $\\ds\\sum_{k=1}^n \\frac{1}{k} \\geqq \\int_1^{n+1}\\frac{dx}{x} = \\log(n+1) \\to \\infty$ ($n\\to\\infty$) が成立するからである."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題:** $\\ds\\sum_{n=1}^\\infty\\frac{(-1)^{n-1}}{n}$ を求めよ. \n",
    "\n",
    "**解答:** $\\log 2$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**証明:** 一般に\n",
    "\n",
    "$$\n",
    "\\int_1^x \\frac{dt}{t} = \\log x.\n",
    "$$\n",
    "\n",
    "ゆえに, \n",
    "\n",
    "$$\n",
    "\\int_0^1\\frac{dx}{1+x} = \\int_1^2 \\frac{dy}{y} = \\log 2.\n",
    "$$\n",
    "\n",
    "一方, $|x|<1$ のとき\n",
    "\n",
    "$$\n",
    "\\frac{1}{1+x} = 1 - x + x^2 - x^3 + \\cdots\n",
    "$$\n",
    "\n",
    "なので\n",
    "\n",
    "$$\n",
    "\\int_0^1 \\frac{dx}{1+x} = \n",
    "\\int_0^1 (1 - x + x^2 - x^3 + \\cdots) =\n",
    "1 - \\frac{1}{2} + \\frac{1}{3} - \\frac{1}{4} + \\cdots =\n",
    "\\sum_{n=1}^\\infty \\frac{(-1)^{n-1}}{n}.\n",
    "$$\n",
    "\n",
    "したがって,  \n",
    "\n",
    "$$\n",
    "\\int_0^1 \\frac{dx}{1+x} = \\log 2 = \\sum_{n=1}^\\infty \\frac{(-1)^{n-1}}{n}.\n",
    "\\qquad \\QED\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "以下はコンピューターによる計算."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\log{\\left (2 \\right )}$$"
      ],
      "text/plain": [
       "log(2)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n = symbols(\"n\", integer=true)\n",
    "S = doit(Sum((-1)^(n-1)/n, (n, 1, oo)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "SimpleSum = 0.6926474305598223\n",
      "TrueValue = 0.6931471805599453\n",
      "SimpleSum - TrueValue = -0.0004997500001230337\n"
     ]
    }
   ],
   "source": [
    "N = 1000\n",
    "SimpleSum = sum(n->(-1)^(n-1)/n, 1:N)\n",
    "TrueValue = log(2)\n",
    "@show SimpleSum\n",
    "@show TrueValue\n",
    "@show SimpleSum - TrueValue;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Euler変換による交代級数の数値計算\n",
    "\n",
    "$\\ds\\sum_{n=1}^N\\frac{(-1)^{n-1}}{n}$ で $N=1000$ としても収束先の値を小数点以下第2桁までしか再現してくれない. 一般に条件収束する交代級数を定義通り足し上げると収束は遅い.\n",
    "\n",
    "収束を加速するための様々な方法が知られているが, 交代級数の場合には**Euler変換**が有名である. 詳しくは\n",
    "\n",
    "* <a href=\"http://nbviewer.jupyter.org/gist/genkuroki/4a8fea5e2ed1b4e3b737c99acf237042\">交代ゼータ函数のオイラー変換</a>\n",
    "\n",
    "を参照せよ. 交代級数 $\\ds\\sum_{k=0}^\\infty (-1)^k a_k$ のEuler変換の結果は\n",
    "\n",
    "$$\n",
    "\\lim_{L\\to\\infty} \\sum_{k=0}^{L-1} w^{(L)}_k (-1)^k a_k, \\qquad\n",
    "w^{(L)}_k = \\sum_{n=k}^{L-1} 2^{-n-1}\\binom{n}{k}.\n",
    "$$\n",
    "\n",
    "すなわち交代級数のEuler変換は交代級数の各項にウェイト $w^{(L)}_k$ をかけてから $L\\to\\infty$ の極限を取る形になる.\n",
    "\n",
    "以下のようにEuler変換を使えば数十項で精密な計算が可能になる."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "EulerWeight{Array{Float64,2}}([0.5 0.0 … 0.0 0.0; 0.75 0.25 … 0.0 0.0; … ; 1.0 1.0 … 1.0842e-19 0.0; 1.0 1.0 … 3.52366e-18 5.42101e-20])"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "binom(n,k) = exp(lgamma(n+1) - lgamma(k+1) - lgamma(n-k+1))\n",
    "\n",
    "struct EulerWeight{T} W::T end\n",
    "function EulerWeight(; Lmax=2^6)\n",
    "    W = zeros(Lmax,Lmax)\n",
    "    for L in 1:Lmax\n",
    "        for k in 0:L-1\n",
    "            W[L,k+1] = sum(n->2.0^(-n-1)*binom(n,k), k:L-1)\n",
    "        end\n",
    "    end\n",
    "    EulerWeight(W)\n",
    "end\n",
    "(eulerweight::EulerWeight)(L,k) = eulerweight.W[L,k+1]\n",
    "eulerweight = EulerWeight()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "EulerTran = 0.6931471805599451\n",
      "TrueValue = 0.6931471805599453\n",
      "EulerTran - TrueValue = -2.220446049250313e-16\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "-2.220446049250313e-16"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "L = 2^6\n",
    "EulerTran = sum(k->eulerweight(L,k)*(-1)^k/(k+1), 0:L-1)\n",
    "TrueValue = log(2)\n",
    "@show EulerTran\n",
    "@show TrueValue\n",
    "@show EulerTran - TrueValue"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Euler変換を使うと, 次のセルのように $\\zeta(s)$ の非自明な零点をプロットすることができる."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  2.099687 seconds (13.50 M allocations: 339.937 MiB, 1.97% gc time)\n"
     ]
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     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(s) = sum(k -> eulerweight(L,k)*(-1)^k/(1+k)^s, 0:L-1)/(1-2^(1-s))\n",
    "x = 0.0:0.005:1.0\n",
    "y = 13.75:0.005:14.75\n",
    "z = x' .+ im.*y\n",
    "@time w = f.(z)\n",
    "plot(size=(600, 500), title=\"first non-trivial zero of zeta(s)\")\n",
    "heatmap!(x, y, abs.(w), color=:rainbow, xlabel=\"Re s\", ylabel=\"Im s\")\n",
    "vline!([0.5], color=\"gray\", ls=:dot, label=\"\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "複素平面における実部が $0$ から $1$ の縦に無限に長い帯状の領域を $\\zeta(s)$ の critical strip と呼ぶ.  $\\zeta(s)$ の critical strip における零点を $\\zeta(s)$ の非自明な零点と呼ぶ.\n",
    "\n",
    "**Riemann予想:** $\\zeta(s)$ の非自明な零点の実部はすべて $1/2$ である. $\\QED$\n",
    "\n",
    "この予想は非常に有名であり, もしも証明されれば素数の分布に関する精密な評価が得られる.\n",
    "\n",
    "**問題:** $\\zeta(s)$ のcritical stripにおける様子をプロットせよ. $\\QED$\n",
    "\n",
    "次のセルを見よ. 虚部が12から45までのcritical stripでの $\\zeta(s)$ の様子がプロットされている. 少なくともその範囲内で $\\zeta(s)$ の非自明な零点の実部はどれも $1/2$ になっていそうなことがわかる."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " "
     ]
    },
    {
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",
      "text/plain": [
       "PyPlot.Figure(PyObject <Figure size 400x1000 with 4 Axes>)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " 4.125797 seconds (152.44 k allocations: 16.536 MiB, 0.19% gc time)\n"
     ]
    }
   ],
   "source": [
    "x = 0:0.01:1\n",
    "y = 12:0.005:45\n",
    "s = x' .+ y*im\n",
    "@time w = zeta.(s)\n",
    "\n",
    "PyPlot.figure(figsize=(4,10))\n",
    "\n",
    "PyPlot.subplot(121)\n",
    "PyPlot.pcolormesh(x, y, log.(abs.(w)), cmap=\"prism\")\n",
    "PyPlot.colorbar()\n",
    "PyPlot.grid(ls=\":\")\n",
    "PyPlot.title(\"log(abs(ζ(s)))\")\n",
    "\n",
    "PyPlot.subplot(122)\n",
    "PyPlot.pcolormesh(x, y, angle.(w), cmap=\"rainbow\")\n",
    "PyPlot.colorbar()\n",
    "PyPlot.grid(ls=\":\")\n",
    "PyPlot.title(\"angle(ζ(s))\")\n",
    "\n",
    "PyPlot.tight_layout()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 無限和と積分の交換に関する注意\n",
    "\n",
    "上では $\\int_0^1 x^n\\,dx = 1/(n+1)$ となることと, 無限和 $1-x+x^2-x^3+\\cdots$ と積分 $\\int_0^1 dx$ の順序を交換できることを用いた. \n",
    "\n",
    "無限和と積分の順序交換が可能なことは以下のようにして確認できる. まず $|x|<1$ で\n",
    "\n",
    "$$\n",
    "\\frac{1}{1+x} = 1 - x + x^2 - x^3 + \\cdots + (-1)^{N-1}x^{N-1}+(-1)^N\\frac{x^N}{1+x}\n",
    "$$\n",
    "\n",
    "となることを確認せよ. この公式を使うと, \n",
    "\n",
    "$$\n",
    "\\int_0^1\\frac{dx}{1+x} = \n",
    "1-\\frac{1}{2}+\\frac{1}{3}+(-1)^{N-1}\\frac{1}{N} + (-1)^N\\int_0^1 \\frac{x^N}{1+x}\\,dx\n",
    "$$\n",
    "\n",
    "さらに $0\\leqq x\\leqq 1$ のとき, $1 \\leqq 1+x$ なので\n",
    "\n",
    "$$\n",
    "0 \\leqq \\frac{x^N}{1+x} \\leqq x^N\n",
    "$$\n",
    "\n",
    "となるから, \n",
    "\n",
    "$$\n",
    "0\\leqq \\int_0^1 \\frac{x^N}{1+x}\\,dx \\leqq \\int_0^1 x^N\\,dx = \\frac{1}{N+1}\n",
    "$$\n",
    "\n",
    "なので, $N\\to\\infty$ で\n",
    "\n",
    "$$\n",
    "\\int_0^1 \\frac{x^N}{1+x}\\,dx \\to 0.\n",
    "$$\n",
    "\n",
    "以上を合わせると, \n",
    "\n",
    "$$\n",
    "\\int_0^1\\frac{dx}{1+x} = \n",
    "\\lim_{N\\to\\infty} \\sum_{n=1}^N \\frac{(-1)^{n-1}}{n} = \\sum_{n=1}^\\infty \\frac{(-1)^{n-1}}{n}\n",
    "$$\n",
    "\n",
    "となることがわかる.\n",
    "\n",
    "**注意:** 上の計算における $\\ds (-1)^N\\int_0^1 \\frac{x^N}{1+x}\\,dx$ のように, 極限で $0$ になって欲しい項が積分表示を持つことは多い. 極限で $0$ になって欲しい部分を「$+\\cdots$」のように書くと正体不明で不安になるのだが, 積分表示を持つ場合にはそのような不安は完全に解消してしまう. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 条件収束交代級数の例2 (arctan 1 = π/4)\n",
    "\n",
    "**問題:** $\\ds\\sum_{n=1}^\\infty\\frac{(-1)^{n-1}}{2n-1}$ を求めよ.\n",
    "\n",
    "**答え:** $\\pi/4$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**証明:** $a = \\tan\\theta$ の $-\\pi/2<\\theta<\\pi/2$ における逆函数を $\\theta = \\arctan a$ と書くと,\n",
    "\n",
    "$$\n",
    "\\int_0^a \\frac{dt}{1+t^2} = \\arctan a.\n",
    "$$\n",
    "\n",
    "この公式は左辺で $t=\\tan\\theta$ と置換すれば確認できる.\n",
    "\n",
    "ゆえに, \n",
    "\n",
    "$$\n",
    "\\int_0^1\\frac{dx}{1+x^2} = \\arctan 1 = \\frac{\\pi}{4}\n",
    "$$\n",
    "\n",
    "一方, $|x|<1$ のとき\n",
    "\n",
    "$$\n",
    "\\frac{1}{1+x^2} = 1 - x^2 + x^4 - x^6 + \\cdots\n",
    "$$\n",
    "\n",
    "なので\n",
    "\n",
    "$$\n",
    "\\int_0^1 \\frac{dx}{1+x^2} = \n",
    "\\int_0^1 (1 - x^2 + x^4 - x^6 + \\cdots) =\n",
    "1 - \\frac{1}{3} + \\frac{1}{5} - \\frac{1}{7} + \\cdots =\n",
    "\\sum_{n=1}^\\infty \\frac{(-1)^{n-1}}{2n-1}.\n",
    "$$\n",
    "\n",
    "したがって,  \n",
    "\n",
    "$$\n",
    "\\int_0^1 \\frac{dx}{1+x^2} = \\arctan 1 = \\frac{\\pi}{4} = \\sum_{n=1}^\\infty \\frac{(-1)^{n-1}}{2n-1}.\n",
    "\\qquad\\QED\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題:** 上の例の計算における無限和と積分の交換の正当化を実行せよ.\n",
    "\n",
    "**解答例:** $0<t<1$ のとき\n",
    "\n",
    "$$\n",
    "\\frac{1}{1+t^2} = \n",
    "\\sum_{k=0}^{n-1}(-t^2)^k + \\frac{(-t^2)^n}{1+t^2} =\n",
    "\\sum_{k=0}^{n-1}(-1)^k t^{2k} + (-1)^n\\frac{t^{2n}}{1+t^2}\n",
    "$$\n",
    "\n",
    "なので,\n",
    "\n",
    "$$\n",
    "\\int_0^1\\frac{dt}{1+t^2} = \n",
    "\\sum_{k=0}^{n-1}\\frac{(-1)^k}{2k+1} + (-1)^n\\int_0^1\\frac{t^{2n}\\,dt}{1+t^2}\n",
    "\\tag{$*$}\n",
    "$$\n",
    "\n",
    "でかつ\n",
    "\n",
    "$$\n",
    "0 \\leqq \\int_0^1\\frac{t^{2n}\\,dt}{1+t^2} \\leqq\n",
    "\\int_0^1 t^{2n}\\,dt = \\frac{1}{2n+1} \\to 0 \\quad(n\\to\\infty)\n",
    "$$\n",
    "\n",
    "であるから, $\\ds \\int_0^1\\frac{t^{2n}\\,dt}{1+t^2}\\to 0$ となる. したがって, ($*$)で $n\\to\\infty$ とすることによって, \n",
    "\n",
    "$$\n",
    "\\frac{\\pi}{4} = \\int_0^1\\frac{dt}{1+t^2} = \\sum_{k=0}^{\\infty}\\frac{(-1)^k}{2k+1}\n",
    "$$\n",
    "\n",
    "を得る. $\\QED$\n",
    "\n",
    "**注意:** 多くの解説で, この例の証明にAbelの連続性定理を使っているようだ. しかし, この場合には以上のように初等的かつ明瞭かつ単純な方法で証明可能である. 特別に簡単な場合について, Abelの連続性定理のような一般的な定理を適用して処理することを機械的に繰り返すと, 数学の内容を誤解してしまう危険性があるので注意した方がよい. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題:** 以上の例以外の条件収束級数の例について調べ, ノートにまとめよ. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "以下はコンピューターによる計算."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\frac{\\pi}{4}$$"
      ],
      "text/plain": [
       "pi\n",
       "--\n",
       "4 "
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n = symbols(\"n\", integer=true)\n",
    "S = doit(Sum((-1)^(n-1)/(2n-1), (n, 1, oo)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "SimpleSum = 0.7853981608974486\n",
      "TrueValue = 0.7853981633974483\n",
      "SimpleSum - TrueValue = -2.499999651739415e-9\n"
     ]
    }
   ],
   "source": [
    "N = 10^8\n",
    "SimpleSum = sum(n->(-1)^(n-1)/(2n-1), 1:N)\n",
    "TrueValue = π/4\n",
    "@show SimpleSum\n",
    "@show TrueValue\n",
    "@show SimpleSum - TrueValue;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "EulerTran = 0.7853981633974486\n",
      "TrueValue = 0.7853981633974483\n",
      "EulerTran - TrueValue = 3.3306690738754696e-16\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "3.3306690738754696e-16"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "L = 2^6\n",
    "EulerTran = sum(k->eulerweight(L,k)*(-1)^k/(2k+1), 0:L-1)\n",
    "TrueValue = π/4\n",
    "@show EulerTran\n",
    "@show TrueValue\n",
    "@show EulerTran - TrueValue"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "以下はコンピューターによる $\\ds\\int_0^1\\frac{dx}{1+x^3}$ や $\\ds\\int_0^1\\frac{dx}{1+x^4}$ などの計算."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\frac{\\log{\\left (2 \\right )}}{3} + \\frac{\\sqrt{3} \\pi}{9}$$"
      ],
      "text/plain": [
       "           ___   \n",
       "log(2)   \\/ 3 *pi\n",
       "------ + --------\n",
       "  3         9    "
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = symbols(\"x\", real=true, positive=true)\n",
    "integrate(1/(1+x^3), (x,0,1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$- \\frac{\\log{\\left (2 \\right )}}{3} + \\frac{\\sqrt{3} \\pi}{9}$$"
      ],
      "text/plain": [
       "             ___   \n",
       "  log(2)   \\/ 3 *pi\n",
       "- ------ + --------\n",
       "    3         9    "
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = symbols(\"x\", real=true, positive=true)\n",
    "integrate(x/(1+x^3), (x,0,1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\frac{\\log{\\left (2 \\right )}}{3}$$"
      ],
      "text/plain": [
       "log(2)\n",
       "------\n",
       "  3   "
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = symbols(\"x\", real=true, positive=true)\n",
    "integrate(x^2/(1+x^3), (x,0,1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$- \\frac{\\sqrt{2} \\log{\\left (- \\sqrt{2} + 2 \\right )}}{8} + \\frac{\\sqrt{2} \\log{\\left (\\sqrt{2} + 2 \\right )}}{8} + \\frac{\\sqrt{2} \\pi}{8}$$"
      ],
      "text/plain": [
       "    ___    /    ___    \\     ___    /  ___    \\     ___   \n",
       "  \\/ 2 *log\\- \\/ 2  + 2/   \\/ 2 *log\\\\/ 2  + 2/   \\/ 2 *pi\n",
       "- ---------------------- + -------------------- + --------\n",
       "            8                       8                8    "
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = symbols(\"x\", real=true, positive=true)\n",
    "integrate(1/(1+x^4), (x,0,1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\frac{\\pi}{8}$$"
      ],
      "text/plain": [
       "pi\n",
       "--\n",
       "8 "
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = symbols(\"x\", real=true, positive=true)\n",
    "integrate(x/(1+x^4), (x,0,1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$- \\frac{\\sqrt{2} \\log{\\left (\\sqrt{2} + 2 \\right )}}{8} + \\frac{\\sqrt{2} \\log{\\left (- \\sqrt{2} + 2 \\right )}}{8} + \\frac{\\sqrt{2} \\pi}{8}$$"
      ],
      "text/plain": [
       "    ___    /  ___    \\     ___    /    ___    \\     ___   \n",
       "  \\/ 2 *log\\\\/ 2  + 2/   \\/ 2 *log\\- \\/ 2  + 2/   \\/ 2 *pi\n",
       "- -------------------- + ---------------------- + --------\n",
       "           8                       8                 8    "
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = symbols(\"x\", real=true, positive=true)\n",
    "integrate(x^2/(1+x^4), (x,0,1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\frac{\\log{\\left (2 \\right )}}{4}$$"
      ],
      "text/plain": [
       "log(2)\n",
       "------\n",
       "  4   "
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = symbols(\"x\", real=true, positive=true)\n",
    "integrate(x^3/(1+x^4), (x,0,1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$s \\Phi\\left(e^{i \\pi}, 1, s\\right)$$"
      ],
      "text/plain": [
       "          / I*pi      \\\n",
       "s*lerchphi\\e    , 1, s/"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = symbols(\"x\", real=true, positive=true)\n",
    "s = symbols(\"s\", real=true, positive=true)\n",
    "J = simplify(integrate(1/(1+x^(1/s)), (x,0,1)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\Phi\\left(z, s, a\\right)$$"
      ],
      "text/plain": [
       "lerchphi(z, s, a)"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sympy[:lerchphi](Sym(:z), Sym(:s), Sym(:a))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Lerchの超越函数(レルヒの超越函数) $\\Phi(z,s,a)$ は次のように定義される:\n",
    "\n",
    "$$\n",
    "\\Phi(z, s, a) = \\sum_{n=0}^\\infty \\frac{z^n}{(n + a)^s}.\n",
    "$$\n",
    "\n",
    "Lerchの超越函数についてはノート「12 Fourier解析」のPoissonの和公式の応用の解説を参照せよ."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\begin{cases} \\frac{\\pi s}{\\sin{\\left (\\pi s \\right )}} & \\text{for}\\: - s + 2 \\gt 1 \\\\\\int_{0}^{\\infty} \\frac{1}{x^{\\frac{1}{s}} + 1}\\, dx & \\text{otherwise} \\end{cases}$$"
      ],
      "text/plain": [
       "/      pi*s                      \n",
       "|   ---------      for -s + 2 > 1\n",
       "|   sin(pi*s)                    \n",
       "|                                \n",
       "| oo                             \n",
       "|  /                             \n",
       "| |                              \n",
       "< |      1                       \n",
       "| |  --------- dx    otherwise   \n",
       "| |  s ___                       \n",
       "| |  \\/ x  + 1                   \n",
       "| |                              \n",
       "|/                               \n",
       "|0                               \n",
       "\\                                "
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = symbols(\"x\", real=true, positive=true)\n",
    "s = symbols(\"s\", real=true, positive=true)\n",
    "J = simplify(integrate(1/(1+x^(1/s)), (x,0,oo)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 複素数の級数\n",
    "\n",
    "この節では複素数 $a_n$ の級数 $\\ds\\sum_{n=1}^\\infty a_n$ を扱う."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 複素数の絶対収束級数\n",
    "\n",
    "**定理:** 複素数 $a_1,a_2,\\ldots$ について, 級数 $\\ds\\sum_{n=1}^\\infty |a_n|$ が有限の値に収束するならば, 級数 $\\ds\\sum_{n=1}^\\infty a_n$ も収束する. このとき, 級数 $\\ds\\sum_{n=1}^\\infty a_n$ は**絶対収束**するという. $\\QED$\n",
    "\n",
    "絶対収束していないが収束する級数は**条件収束**するという.\n",
    "\n",
    "上の定理の証明は略す. しかし, 下の方で, 級数 $\\ds\\sum_{n=1}^\\infty |a_n|$ が有限の値に収束するならば, 級数 $\\ds\\sum_{n=1}^\\infty a_n$ も収束する直観的な理由を説明する."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**例:** $\\theta$ は実数であるとする. このとき, 複素級数 $\\ds\\sum_{n=1}^\\infty\\frac{e^{in\\theta}}{n^{3/2}}$ は絶対収束する. 以下のセルにおける計算とプロットを参照せよ."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "θ = 2π/10\n",
    "a(n) = exp(im*n*θ)/n^1.5\n",
    "sum_a(n) = iszero(n) ? Complex(0.0) : sum(k->a(k), 1:n)\n",
    "N = 70\n",
    "n = 0:N\n",
    "z = sum_a.(n)\n",
    "plot(size=(400, 400), xlims=(0,1.0), ylim=(0,1.3))\n",
    "plot!(real.(z), imag.(z), legend=false)#, line=:arrow)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  3.386838 seconds (19.49 M allocations: 502.626 MiB, 3.46% gc time)\n",
      "  1.076321 seconds (91.73 k allocations: 4.912 MiB)\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "\u001b[1m\u001b[36mINFO: \u001b[39m\u001b[22m\u001b[36mSaved animation to C:\\Users\\genkuroki\\OneDrive\\Calculus\\sum_a.gif\n",
      "\u001b[39m"
     ]
    },
    {
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\">"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "θ = 2π/10\n",
    "a(n) = exp(im*n*θ)/n^1.5\n",
    "sum_a(n) = iszero(n) ? Complex(0.0) : sum(k->a(k), 1:n)\n",
    "@time anim = @animate for N in 0:70\n",
    "    n = 0:N\n",
    "    z = sum_a.(n)\n",
    "    plot(size=(400, 400), xlims=(0,1.0), ylim=(0,1.3))\n",
    "    plot!(real.(z), imag.(z), legend=false, line=:arrow)\n",
    "end\n",
    "\n",
    "gifname = \"sum_a.gif\"\n",
    "@time gif(anim, gifname; fps=10)\n",
    "sleep(0.1)\n",
    "showimg(\"image/gif\", gifname)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**絶対収束の概念の直観的な説明:**\n",
    "\n",
    "各複素数 $a_n$ は平面上の点と同一視できる(複素平面の考え方).\n",
    "\n",
    "平面上の点と原点からその点への矢線(ベクトル)は同一視できる(ベクトルの考え方).\n",
    "\n",
    "複素数の和はベクトルの和に対応している.\n",
    "\n",
    "ベクトルの和は矢線としてのベクトルを連結することを意味する(ベクトルの和の幾何学的意味).\n",
    "\n",
    "ゆえに, $\\ds s_N = \\sum_{n=1}^N a_n$ は各 $a_1,a_2,\\ldots,a_N$ の各々に対応する矢線を連結して得られる折れ線の先端の点に対応する複素数に等しい.\n",
    "\n",
    "$\\ds L = \\sum_{n=1}^\\infty|a_n|$ は各 $a_n$ に対応する矢線の長さの総和を意味する. \n",
    "\n",
    "ゆえに, $L$ が有限の値になることは, 各 $a_n$ に対応する矢線をすべて連結して得られる折れ線の長さが有限になることを意味する.\n",
    "\n",
    "そのとき, $a_n$ に対応する矢線をすべて連結して得られる折れ線の「先端」の点に対応する複素数が級数 $\\ds \\sum_{n=1}^\\infty a_n$ の収束先になる.\n",
    "\n",
    "1つ前のセルのアニメーションはそのような様子の一例になっている."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**注意:** 上の説明を理解できれば, 級数 $\\ds\\sum_{n=1}^\\infty |a_n|$ が有限の値に収束するならば, 級数 $\\ds\\sum_{n=1}^\\infty a_n$ も収束することも十分に納得できると思う. その直観は数学的には<a href=\"https://www.google.co.jp/search?q=%E5%AE%8C%E5%82%99%E6%80%A7+%E7%B5%B6%E5%AF%BE%E5%8F%8E%E6%9D%9F\">**完備性**</a>の概念によって正当化される. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 条件収束する複素級数の例"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題:** $\\theta$ は実数であり, $e^{i\\theta}\\ne 1$ であると仮定する. このとき $\\ds\\sum_{n=1}^\\infty\\frac{e^{in\\theta}}{n}$ は条件収束することを示せ. $\\QED$\n",
    "\n",
    "**注意:** $-\\log(1-z)$ の収束半径 $1$ のMaclaurin展開は\n",
    "\n",
    "$$\n",
    "-\\log(1-z) = \\sum_{n=1}^\\infty\\frac{z^n}{n}\n",
    "$$\n",
    "\n",
    "の形をしている. $\\theta\\in\\R$, $e^{i\\theta}\\ne 1$ のとき, $\\ds\\sum_{n=1}^\\infty\\frac{e^{in\\theta}}{n}$ の収束先は $-\\log(1-e^{i\\theta})$ になる. $\\QED$\n",
    "\n",
    "**解答例1:** $z=e^{i\\theta}$, $\\theta\\in\\R$ とおき,  $\\ds s_n(z) = \\sum_{k=1}^n \\frac{z^k}{k}$ とおく. $z\\ne 1$ のとき, $s_n(z)$ が $n\\to\\infty$ で収束することを示せばよい.\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "(z-1)s_n(z) &=\n",
    "\\left(\\frac{z^2}{1}+\\frac{z^3}{2}+\\cdots+\\frac{z^n}{n-1}+\\frac{z^{n+1}}{n}\\right)\n",
    "\\\\ &\\;-\n",
    "\\left(\\frac{z}{1}+\\frac{z^2}{2}+\\frac{z^3}{3}+\\cdots+\\frac{z^n}{n}\\right)\n",
    "\\\\ &=\n",
    "\\frac{z^{n+1}}{n}-\\frac{z}{1} + S_n(z).\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "ここで,\n",
    "\n",
    "$$\n",
    "S_n(z) = \n",
    "\\left(\\frac{1}{1}-\\frac{1}{2}\\right)z^2+\n",
    "\\left(\\frac{1}{2}-\\frac{1}{3}\\right)z^3+\n",
    "\\cdots+\n",
    "\\left(\\frac{1}{n-1}-\\frac{1}{n}\\right)z^n.\n",
    "$$\n",
    "\n",
    "$|z|=1$ なので, $n\\to\\infty$ のとき $\\ds\\frac{z^{n+1}}{n}\\to 0$ となるので, $(z-1)s_n(z)$ が $n\\to\\infty$ で収束することを示すためには, $S_n(z)$ が絶対収束することを示せば十分であり, $|z|=1$ より,\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "&\n",
    "\\left|\\left(\\frac{1}{1}-\\frac{1}{2}\\right)z^2\\right|+\n",
    "\\left|\\left(\\frac{1}{2}-\\frac{1}{3}\\right)z^3\\right|+\n",
    "\\cdots+\n",
    "\\left|\\left(\\frac{1}{n-1}-\\frac{1}{n}\\right)z^n\\right|\n",
    "\\\\ &=\n",
    "\\left(\\frac{1}{1}-\\frac{1}{2}\\right)+\n",
    "\\left(\\frac{1}{2}-\\frac{1}{3}\\right)+\n",
    "\\cdots+\n",
    "\\left(\\frac{1}{n-1}-\\frac{1}{n}\\right) =\n",
    "\\frac{1}{1}-\\frac{1}{n}\\to 1 \\quad(n\\to\\infty)\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "なので, $S_n(z)$ は絶対収束する. これで $n\\to\\infty$ で $(z-1)s_n(z)$ が収束することがわかった. $z\\ne 1$ なので, $s_n(z)$ も $n\\to\\infty$ で収束する. $\\QED$\n",
    "\n",
    "**解答例1の一般化:** $a_n\\geqq 0$ であり, $a_n$ は単調減少しながら $0$ に収束していると仮定する. このとき, $\\theta$ は実数であり, $e^{i\\theta}\\ne 1$ ならば, $\\ds \\sum_{n=1}^\\infty a_n e^{in\\theta}$ は収束している. 例えば, $\\ds a_n = \\frac{1}{\\sqrt{n}}$ や $\\ds a_n = \\frac{1}{\\log n}$ の場合にも $\\ds \\sum_{n=1}^\\infty a_n e^{in\\theta}$ は収束している.\n",
    "\n",
    "**証明:** $z=e^{i\\theta}$, $\\ds s_n(z) = \\sum_{k=0}^n a_k z^k$ とおく. $s_n(z)$ が $n\\to\\infty$ で収束していることを示せばよい. 上の解答例と同様にして,\n",
    "\n",
    "$$\n",
    "(z-1)s_n(z) = a_n z^{n+1} - a_1 z + S_n(z), \\quad\n",
    "S_n(z) = \\sum_{k=2}^n (a_{k-1}-a_k)z^k.\n",
    "$$\n",
    "\n",
    "そして, $|z|=1$ でかつ $a_n\\to 0$ より, $a_n z^{n+1}\\to 0$ となる. さらに,\n",
    "\n",
    "$$\n",
    "\\sum_{k=2}^n \\left|(a_{k-1}-a_k)z^k\\right| =\n",
    "\\sum_{k=2}^n(a_{k-1}-a_k) = a_1 - a_n \\to a_1 \\quad (n\\to\\infty)\n",
    "$$\n",
    "\n",
    "なので, $S_n(z)$ は $n\\to\\infty$ で絶対収束する. $z\\ne 1$ なので $s_n(z)$ も $n\\to\\infty$ で収束する. $\\QED$\n",
    "\n",
    "**さらなる一般化:** 一般に $\\ds S_n = \\sum_{k=1}^n a_k b_k$ のとき, $\\ds A_k = \\sum_{j=1}^k a_j$, $A_0=0$ とおくと, $a_k = A_k - A_{k-1}$ なので\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "S_n &= \\sum_{k=1}^n (A_k - A_{k-1}) b_k =\n",
    "\\sum_{k=1}^n A_k b_k - \\sum_{k=0}^{n-1} A_k b_{k+1}\n",
    "\\\\ &=\n",
    "A_n b_n - A_0 b_1 + \\sum_{k=1}^{n-1} A_k(b_k - b_{k+1}).\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "この公式は部分積分の公式の和に関するバージョンであることに注意せよ. (「部分積分」のアイデアは普遍的によく現われる.)\n",
    "\n",
    "仮定の追加1: 数列 $b_k$ は単調減少実数列でかつ $0$ に収束すると仮定する.\n",
    "\n",
    "このとき, $b_k-b_{k+1}\\geqq 0$ でかつ\n",
    "\n",
    "$$\n",
    "\\sum_{k=1}^{n-1} (b_k - b_{k+1}) = b_1 - b_n \\to b_1 \\quad (n\\to\\infty).\n",
    "$$\n",
    "\n",
    "仮定の追加2: $A_k$ は有界数列であると仮定する. すなわちある正の実数で $|A_k|\\leqq M$ ($k=1,2,\\ldots$) を満たすものが存在すると仮定する.\n",
    "\n",
    "このとき, \n",
    "\n",
    "$$\n",
    "\\sum_{k=1}^\\infty|A_k||b_k-b_{k+1}|\\leqq\n",
    "M\\sum_{k=1}^\\infty(b_k-b_{k+1}) = M b_1 < \\infty\n",
    "$$\n",
    "\n",
    "なので 級数 $\\ds\\sum_{k=1}^\\infty A_k(b_{k+1} - b_k)$ は絶対収束する. さらに, $n\\to\\infty$ で $b_n\\to 0$ なので $A_n b_n\\to 0$ となる. ゆえに級数 $\\ds\\sum_{k=1}^\\infty a_k b_k$ は収束する. (積分の場合にも, 絶対収束しない積分を部分積分によって絶対収束する積分に変形できる場合がある. 各自自分で研究してみよ.) $\\QED$\n",
    "\n",
    "**解答例2:** 以上の一般論を上の問題に適用しよう. すなわち $a_n = e^{in\\theta}$, $b_n=1/n$ の場合を考える. このとき\n",
    "\n",
    "$$\n",
    "A_k = \\sum_{j=1}^k a_j = \\frac{e^{i\\theta}-e^{i(n+1)\\theta}}{1-e^{i\\theta}}\n",
    "$$\n",
    "\n",
    "であるから, $|A_k|\\leqq 2/|1-e^{i\\theta}|$ なので, $A_k$ は有界である. 実数列 $b_k$ は単調減少しながら $0$ に収束する. したがって上の一般論より, 級数 $\\ds\\sum_{n=1}^\\infty\\frac{e^{in\\theta}}{n}$ は条件収束する. $\\QED$"
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     "execution_count": 28,
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   "source": [
    "f(z) = -log(1-z)\n",
    "g(z; N=1000) = sum(n->z^n/n, 1:N)\n",
    "\n",
    "θ = 0.1:0.01:π\n",
    "v = @.(f(exp(im*θ)))\n",
    "w = @.(g(exp(im*θ)))\n",
    "\n",
    "P1 = plot(xlims=(0,π), xlabel=\"t\")\n",
    "plot!(θ, real.(w), label=\"Re(sum exp(int)/n)\",  lw=2)\n",
    "plot!(θ, real.(v), label=\"Re(-log(1-exp(it)))\", lw=2, ls=:dash)\n",
    "\n",
    "P2 = plot(xlims=(0,π), xlabel=\"t\")\n",
    "plot!(θ, imag.(w), label=\"Im(sum exp(int)/n)\",  lw=2)\n",
    "plot!(θ, imag.(v), label=\"Im(-log(1-exp(it)))\", lw=2, ls=:dash)\n",
    "\n",
    "plot(P1, P2, size=(700,300))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
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       "</svg>\n"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "θ = 2π/10\n",
    "a(n) = exp(im*n*θ)/n\n",
    "sum_a(n) = iszero(n) ? Complex(0.0) : sum(k->a(k), 1:n)\n",
    "N = 400\n",
    "n = 0:N\n",
    "z = sum_a.(n)\n",
    "plot(size=(400, 400), xlims=(0,1.0), ylim=(0,1.6))\n",
    "plot!(real.(z), imag.(z), legend=false)#, line=:arrow)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  5.355350 seconds (40.30 M allocations: 1.076 GiB, 4.53% gc time)\n",
      "  0.748441 seconds (1.05 k allocations: 69.156 KiB)\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "\u001b[1m\u001b[36mINFO: \u001b[39m\u001b[22m\u001b[36mSaved animation to C:\\Users\\genkuroki\\OneDrive\\Calculus\\sum_b.gif\n",
      "\u001b[39m"
     ]
    },
    {
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\">"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "θ = 2π/10\n",
    "a(n) = exp(im*n*θ)/n\n",
    "sum_a(n) = iszero(n) ? Complex(0.0) : sum(k->a(k), 1:n)\n",
    "@time anim = @animate for N in 0:4:400\n",
    "    n = 0:N\n",
    "    z = sum_a.(n)\n",
    "    plot(size=(400, 400), xlims=(0,1.0), ylim=(0,1.6))\n",
    "    plot!(real.(z), imag.(z), legend=false, line=:arrow)\n",
    "end\n",
    "\n",
    "gifname = \"sum_b.gif\"\n",
    "@time gif(anim, gifname; fps=10)\n",
    "sleep(0.1)\n",
    "showimg(\"image/gif\", gifname)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "上のセルのアニメーションの折れ線の長さは無限大である. 無限に長くなる折れ線の先端が無限に回転しながら収束している."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 多重対数函数"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "複素数 $z$ に関するべき級数\n",
    "\n",
    "$$\n",
    "\\Li_r(z) = \\sum_{n=1}^\\infty \\frac{z^n}{n^r}\n",
    "$$\n",
    "\n",
    "は $r>1$, $|z|\\leqq 1$ および $r=1$, $|z|<1$ のとき絶対収束する. $r=1$, $|z|=1$ の場合には $z\\ne 1$ なら条件収束し, $z=1$ ならば $\\infty$ に発散する.  このように定義される函数 $\\Li_r(z)$ を $r$ 重の**多重対数函数**(polylogarithm)と呼ぶ. $r=2$ の場合はdilogarithmと呼ばれ, $r=3$ の場合にはtrilogarithmと呼ばれる. このとき, \n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "&\n",
    "\\Li_1(z)=\\sum_{n=1}^\\infty\\frac{z^n}{n} = -\\log(1-z) = \\int_0^z \\frac{dt}{1-t},\n",
    "\\\\ &\n",
    "\\Li_2(z)=\\sum_{n=1}^\\infty\\frac{z^n}{n^2} = \\int_0^z \\frac{\\Li_1(t)}{t}\\,dt =\n",
    "\\int_0^z dt_2 \\int_0^{t_2} dt_1\\,\\frac{1}{(1-t_1)t_2},\n",
    "\\\\ &\n",
    "\\Li_3(z)=\\sum_{n=1}^\\infty\\frac{z^n}{n^3} = \\int_0^z \\frac{\\Li_2(t)}{t}\\,dt =\n",
    "\\int_0^z dt_3 \\int_0^{t_3} dt_2 \\int_0^{t_2} dt_1\\, \\frac{1}{(1-t_1)t_2t_3},\n",
    "\\\\ &\\ldots\\ldots\\ldots\\ldots\\ldots\n",
    "\\\\ &\n",
    "\\Li_r(z)=\\sum_{n=1}^\\infty\\frac{z^n}{n^r} = \\int_0^z \\frac{\\Li_2(t)}{t}\\,dt =\n",
    "\\int_0^z dt_r \\int_0^{t_r} dt_{r-1}\\cdots\\int_0^{t_2} dt_1\\,\\frac{1}{(1-t_1)t_2\\cdots t_r},\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "$r>1$ のとき,\n",
    "\n",
    "$$\n",
    "\\Li_r(1) = \\sum_{n=1}^\\infty \\frac{1}{n^r} = \\zeta(r).\n",
    "$$\n",
    "\n",
    "したがって, 2以上の整数 $m$ に対して,\n",
    "\n",
    "$$\n",
    "\\zeta(m) = \\Li_m(1) = \n",
    "\\int_0^1 dt_m\\int_0^{t_m}dt_{m-1}\\cdots\\int_0^{t_2}dt_1\\frac{1}{(1-t_1)t_2\\cdots t_m}.\n",
    "$$\n",
    "\n",
    "$t_1=x_1 x_2\\cdots x_m$, $t_2=x_2\\cdots x_m$, $\\ldots$, $t_m=x_m$ と置換すると,\n",
    "\n",
    "$$\n",
    "\\zeta(m) = \\Li_m(1) = \n",
    "\\int_0^1 dx_m\\int_0^1 dx_{m-1}\\cdots\\int_0^1 dx_1\\frac{1}{1-x_1\\cdots x_m}.\n",
    "$$\n",
    "\n",
    "これの右辺が $\\zeta(m)$ に一致することは\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "&\n",
    "\\frac{1}{1-x_1\\cdots x_m} = \\sum_{n=1}^\\infty x_1^{n-1}\\cdots x_m^{n-1}, \n",
    "\\\\ &\n",
    "\\int_0^1 dx_m\\int_0^1 dx_{m-1}\\cdots\\int_0^1 dx_1\\,x_1^n\\cdots x_m^n =\n",
    "\\left(\\int_0^1 x^{n-1}\\,dx\\right)^m = \\frac{1}{n^m}\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "に注意すれば明らかであろう."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 二重対数函数のζ(2)の近似計算への応用\n",
    "\n",
    "$\\ds\\zeta(2)=\\sum_{n=1}^\\infty \\frac{1}{n^2}$ を計算せよという問題は**Basel問題**と呼ばれているらしい. Basel問題はEulerによって1743年ころに解かれたらしい. Eulerがどのように考えたかについては次の文献を参照せよ.\n",
    "\n",
    "* 杉本敏夫, <a href=\"http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1583-12.pdf\">バーゼル問題とオイラー</a>, 2007年8月23日, 数理解析研究所講究録, 第1583巻, 2008年, pp.159-167\n",
    "\n",
    "その第6節によれば, Eulerは, 積分 $\\ds\\int\\frac{-\\log(1-x)}{x}\\,dx$ を2通りに計算して比較することによって, $\\zeta(2)$ を定義に基いて直接計算するよりもずっと少ない項数の和で同精度で $\\zeta(2)$ の計算する方法を発見している.\n",
    "\n",
    "現代的には二重対数函数(dilogarithm, 以下ダイログと呼ぶ)を用いてその近似計算法を見付けたことになる. ダイログ $\\Li_2(z)$ の定義は\n",
    "\n",
    "$$\n",
    "\\Li_2(z) = \\int_0^z \\frac{-\\log(1-x)}{x}\\,dx = \\sum_{n=1}^\\infty \\frac{z^n}{n^2}\n",
    "$$\n",
    "\n",
    "であった. $x=1-y$ と置換すると,\n",
    "\n",
    "$$\n",
    "\\Li(z) = \\int_1^{1-z} \\frac{\\log y}{1-y}\\,dy\n",
    "$$\n",
    "\n",
    "となり, \n",
    "\n",
    "$$\n",
    "\\frac{d}{dy}\\left(-\\Li_2(y) -\\log(1-y)\\log y\\right) = \\frac{\\log y}{1-y}\n",
    "$$\n",
    "\n",
    "なので, $\\Li_2(1)=\\zeta(2)$ と $y\\to 1$ で $\\log(1-y)\\log y\\to 0$ となることより, \n",
    "\n",
    "$$\n",
    "\\Li_2(z) = \\left[-\\Li_2(y)-\\log(1-y)\\log y\\right]_1^{1-z} = -\\Li_2(1-z)+\\zeta(2)-\\log z\\,\\log(1-z).\n",
    "$$\n",
    "\n",
    "すなわち,\n",
    "\n",
    "$$\n",
    "\\zeta(2) = \\Li_2(z) + \\Li_2(1-z) + \\log z\\,\\log(1-z).\n",
    "$$\n",
    "\n",
    "特に $z=1/2$ のとき,\n",
    "\n",
    "$$\n",
    "\\zeta(2) = 2\\Li_2(1/2) + (\\log 2)^2 =\n",
    "\\sum_{n=1}^\\infty \\frac{1}{2^{n-1} n^2} + (\\log 2)^2.\n",
    "$$\n",
    "\n",
    "$\\zeta(2)$ の定義に基いて最初の10項を足した結果\n",
    "\n",
    "$$\n",
    "\\sum_{n=1}^{10} \\frac{1}{n^2} = 1.54976\\;77311\\;665\\cdots\n",
    "$$\n",
    "\n",
    "は正確な値 $\\ds\\zeta(2)=\\frac{\\pi^2}{6}=1.64493\\;40668\\;482\\cdots$ と整数部分しか一致しないが,\n",
    "\n",
    "$$\n",
    "\\sum_{n=1}^{10} \\frac{1}{2^{n-1} n^2} + (\\log 2)^2 = 1.64492\\;00516\\;736\\cdots\n",
    "$$\n",
    "\n",
    "は正確な値に小数点以下第4桁まで一致している. 最初の30項の和では, 前者は $1.61215\\;01176\\;015\\cdots$ で正確な値と小数点以下第1桁までしか一致しないが, 後者の $1.64493\\;40668\\;463\\cdots$ は小数点以下第11桁まで一致している."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(Q(10), Q(10) - π ^ 2 / 6, R(10), R(10) - π ^ 2 / 6) = (1.5497677311665408, -0.09516633568168564, 1.6449200516736968, -1.4015174529591334e-5)\n",
      "(Q(30), Q(30) - π ^ 2 / 6, R(30), R(30) - π ^ 2 / 6) = (1.6121501176015975, -0.03278394924662886, 1.6449340668463974, -1.828981410767483e-12)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "5-element Array{Tuple{Int64,Float64,Float64,Float64,Float64},1}:\n",
       " (10, 1.54977, -0.0951663, 1.64492, -1.40152e-5) \n",
       " (20, 1.59616, -0.0487708, 1.64493, -3.98311e-9) \n",
       " (30, 1.61215, -0.0327839, 1.64493, -1.82898e-12)\n",
       " (40, 1.62024, -0.0246901, 1.64493, -1.11022e-15)\n",
       " (50, 1.62513, -0.0198013, 1.64493, -2.22045e-16)"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Q(N) = sum(n->1/n^2, 1:N)\n",
    "R(N) = sum(n->1/(2.0^(n-1)*n^2), 1:N) + log(2)^2\n",
    "@show Q(10), Q(10)-π^2/6, R(10), R(10)-π^2/6\n",
    "@show Q(30), Q(30)-π^2/6, R(30), R(30)-π^2/6\n",
    "[(N, Q(N), Q(N)-π^2/6, R(N), R(N)-π^2/6) for N in 10:10:50]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 二重対数函数のζ(2)の正確な計算への応用\n",
    "\n",
    "以下の議論は次のリンク先の方法の引き写しである:\n",
    "\n",
    "* https://www.theoremoftheday.org/Analysis/Basel/TotDBasel.pdf\n",
    "\n",
    "複素函数としての $\\log z = \\log|z| + i\\arg z$, $-\\pi<\\arg z<\\pi$ を使う. \n",
    "\n",
    "$|z|<1$, $\\ds -\\pi<\\arg z<\\pi$ と仮定する. このとき, $\\ds 1+z^{-1}=\\frac{1+z}{z}$ より, 図を描いて確認すれば\n",
    "\n",
    "$$\n",
    "\\arg(1+z^{-1}) = \\arg(1+z) - \\arg z\n",
    "$$\n",
    "\n",
    "となっていることがわかる. ゆえに\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\log(1+z^{-1}) - \\log(1+z) &=\n",
    "\\log\\left|\\frac{1+z^{-1}}{1+z}\\right| + i(\\arg(1+z^{-1})-\\arg(1+z)) \n",
    "\\\\ &=\n",
    "-\\log|z|-i\\arg z = -\\log z.\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "そして, \n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "&\n",
    "(\\Li_2(-z^{-1}))' = \\frac{-\\log(1-(-z^{-1}))}{-z^{-1}}z^{-2} = \\frac{\\log(1+z^{-1})}{z},\n",
    "\\\\ &\n",
    "(\\Li_2(-z))' = \\frac{-\\log(1-(-z))}{-z}(-1) = \\frac{-\\log(1+z)}{z}, \n",
    "\\\\ &\\therefore\\quad\n",
    "(\\Li_2(-z^{-1})+\\Li_2(-z))' = \\frac{\\log(1+z^{-1})-\\log(1+z)}{z} = \\frac{-\\log z}{z}\n",
    "\\\\ &\n",
    "\\left(-\\frac{1}{2}(\\log z)^2\\right)' = \\frac{-\\log z}{z}\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "なので, ある定数 $C$ が存在して,\n",
    "\n",
    "$$\n",
    "\\Li_2(-z^{-1})+\\Li_2(-z) = -\\frac{1}{2}(\\log z)^2 + C.\n",
    "$$\n",
    "\n",
    "$-\\pi<\\arg z<\\pi$ と仮定していたことに注意しながら, $z \\to 1$ とすると,\n",
    "\n",
    "$$\n",
    "2\\Li_2(-1) = C.\n",
    "$$\n",
    "\n",
    "$-\\pi<\\arg z<\\pi$ と仮定していたことに注意しながら, $z\\to -1$ とすると,\n",
    "\n",
    "$$\n",
    "2\\Li_2(1) = \n",
    "-\\frac{1}{2}(\\pm\\pi i)^2 + C =\n",
    "\\frac{\\pi^2}{2} + 2\\Li_2(-1).\n",
    "$$\n",
    "\n",
    "$\\ds \\Li_2(1)+\\Li_2(-1)=\\sum_{n=1}^\\infty \\frac{1}{n^2}+\\sum_{n=1}^\\infty \\frac{(-1)^n}{n^2}=\\sum_{k=1}^\\infty\\frac{2}{(2k)^2} = \\frac{1}{2}\\Li_2(1)$ より, $2\\Li_2(-1) = -\\Li(1)$ なので, 上で示した $\\ds 2\\Li_2(1)=\\frac{\\pi^2}{2}+2\\Li_2(-1)$ と合わせて, $\\ds \\zeta(2)=\\Li_2(1)=\\frac{\\pi^2}{6}$ となることがわかる.\n",
    "\n",
    "要するに, $\\zeta(2)$ は二重対数函数の $\\Li_2(-z^{-1})$ と $\\Li_2(-z)$ のあいだの接続公式のモノドロミー(不定性)を注意深く扱えば自然に計算できてしまうということである. (注意: $\\Li_2(z^{-1})$ と $\\Li_2(z)$ を使っても同様の方法で示せる.)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 二重対数函数の高木貞治『解析概論』§35の例2(p.113)への応用\n",
    "\n",
    "**問題:** 次を示せ:\n",
    "\n",
    "$$\n",
    "I := \\int_0^{\\pi/2} \\log(\\sin x)\\,dx = -\\frac{\\pi}{2}\\log 2.\n",
    "$$\n",
    "\n",
    "**解答例(高木貞治『解析概論』§35の例2(p.113)):** $\\sin(\\pi/2-x)=\\cos x$ より, $\\ds I = \\int_0^{\\pi/2}\\log(\\cos x))\\,dx$. $\\sin(\\pi-x)=\\sin x$ より, $\\ds I=\\int_{\\pi/2}^\\pi\\log(\\sin x)\\,dx$ なので $\\ds\\int_0^\\pi\\log(\\sin x)\\,dx=2I$. そこで, $x=2t$ と置換して, $\\sin(2t)=2\\cos t\\sin t$ を使うと,\n",
    "\n",
    "$$\n",
    "I = \\frac{1}{2}\\int_0^\\pi\\log(\\sin x)\\,dx =\n",
    "\\int_0^{\\pi/2}\\log(2\\cos t\\sin t)\\,dt =\n",
    "\\frac{\\pi}{2}\\log 2 + I + I.\n",
    "$$\n",
    "\n",
    "ゆえに $\\ds I = -\\frac{\\pi}{2}\\log 2$. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**注意:** 上の問題の定積分は, 高木貞治『解析概論』§35の例2(p.113)だけではなく, 同書§62の例3(p.225)でも扱われている. そこでは\n",
    "\n",
    "$$\n",
    "\\frac{\\log(1-z)}{z} = -1 -z -\\frac{z^2}{2} - \\cdots\n",
    "$$\n",
    "\n",
    "の原点を中心とする周回積分を用いて,\n",
    "\n",
    "$$\n",
    "\\int_0^\\pi \\log(1-2r\\cos\\theta+r^2)\\,\\theta = 2\\pi\\log r \\qquad(r>0)\n",
    "$$\n",
    "\n",
    "を得ている. そしてこれの $r=1$ の場合から上の問題の結果を得ている. 二重対数函数の定義は $\\ds\\frac{-\\log(1-z)}{z}$ の不定積分だったので, 上の問題は二重対数函数と関係していることになる.  以下の2つの問題も見よ. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題(WolframAlpha):** 次の $F(x)$ について $F'(x)=\\log(\\sin x)$ となることを示せ.\n",
    "\n",
    "$$\n",
    "F(x) = \\frac{i}{2}x^2+\\frac{i}{2}\\Li_2(e^{2ix}) - x\\log(1-e^{2ix}) + x\\log(\\sin x). \n",
    "$$\n",
    "\n",
    "**解答例:** $\\ds\\Li_2'(x)=-\\frac{\\log(1-x)}{x}$, $\\ds(-\\log(1-x))'=\\frac{1}{1-x}$ より,\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "F'(x) &=\n",
    "ix + \\frac{i}{2}2i e^{2ix}\\frac{-\\log(1-e^{2ix})}{e^{2ix}} - \\log(1-e^{2ix}) + x \\frac{2ie^{2ix}}{1-e^{2ix}} + x\\frac{\\cos x}{\\sin x} + \\log(\\sin x).\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "第2項と第3項がキャンセルして消える. さらに, \n",
    "\n",
    "$$\n",
    "x\\frac{\\cos x}{\\sin x}=ix\\frac{e^{ix}+e^{-ix}}{e^{ix}-e^{-ix}} = \n",
    "ix\\frac{e^{2ix}+1}{e^{2ix}-1}\n",
    "$$\n",
    "\n",
    "を代入すると, 第1,4,5項の和を $ix$ で割ったものは\n",
    "\n",
    "$$\n",
    "1 + \\frac{-2e^{2ix}}{e^{2ix}-1} + \\frac{e^{2ix}+1}{e^{2ix}-1} = 0\n",
    "$$\n",
    "\n",
    "となるので, $F'(x)=\\log(\\sin x)$ となることがわかる. $\\QED$\n",
    "\n",
    "筆者はこの問題の $F(x)$ の式は<a href=\"http://www.wolframalpha.com/input/?i=%5Cint+log(sin+x)+dx\">WolframAlphaによる計算</a>で知った."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題:** すぐ上の問題の結果を用いて, 次を示せ:\n",
    "\n",
    "$$\n",
    "\\int_0^{\\pi/2}\\log(\\sin x)\\,dx = -\\frac{\\pi}{2}\\log 2, \\quad\n",
    "\\zeta(2) = \\Li_2(1) = \\frac{\\pi^2}{6}.\n",
    "$$\n",
    "\n",
    "**解答例:** $\\Li_2(1)$ と $\\Li(-1)$ は\n",
    "\n",
    "$$\n",
    "\\Li_2(1) = \\sum_{n=1}^\\infty \\frac{1}{n^2}, \\quad\n",
    "\\Li_2(-1) = \\sum_{n=1}^\\infty \\frac{(-1)^n}{n^2}\n",
    "$$\n",
    "\n",
    "と表わされるので, \n",
    "\n",
    "$$\n",
    "\\Li_2(1)+\\Li_2(-1)=\\sum_{k=1}^\\infty\\frac{2}{(2k)^2} = \\frac{1}{2}\\Li_2(1).\n",
    "$$\n",
    "\n",
    "ゆえに\n",
    "\n",
    "$$\n",
    "\\Li_2(-1) = -\\frac{1}{2}\\Li(1).\n",
    "\\tag{1}\n",
    "$$\n",
    "\n",
    "そして, 上の問題の $F(x)$ について\n",
    "\n",
    "$$\n",
    "F\\left(\\frac{\\pi}{2}\\right) = \\frac{\\pi^2}{8}i + \\frac{i}{2}\\Li_2(-1) -\\frac{\\pi}{2}\\log 2, \\quad\n",
    "F(0) = \\frac{i}{2}\\Li_2(1)\n",
    "$$\n",
    "\n",
    "なので, 上の問題の結論より, \n",
    "\n",
    "$$\n",
    "\\R\\ni \\int_0^{\\pi/2}\\log(\\sin x)\\,dx = [F(x)]_0^{\\pi/2} =\n",
    "\\frac{\\pi^2}{8}i + \\frac{i}{2}\\Li_2(-1) -\\frac{\\pi}{2}\\log 2 - \\frac{i}{2}\\Li_2(1).\n",
    "$$\n",
    "\n",
    "両辺の実部と虚部(の2倍)を比較して, $\\ds \\int_0^{\\pi/2}\\log(\\sin x)\\,dx = -\\frac{\\pi}{2}\\log 2$ と\n",
    "\n",
    "$$\n",
    "0 = \\frac{\\pi^2}{4}+\\Li_2(-1)-\\Li_2(1)\n",
    "\\tag{2}\n",
    "$$\n",
    "\n",
    "を得る. 以上の(1),(2)より, $\\ds\\zeta(2)=\\Li_2(1)=\\frac{\\pi^2}{6}$ が得られる. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**注意:** 以上の計算に関連する話題については, <a href=\"https://twitter.com/genkuroki/status/830016144933167104\">筆者によるツイッターにおける一連の発言</a>も参照せよ. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 二重対数函数の五項関係式\n",
    "\n",
    "<a href=\"https://www.google.co.jp/search?q=Rogers+dilogarithm\">Rogersの二重対数函数</a> $L(z)$ を次のように定義する:\n",
    "\n",
    "$$\n",
    "L(z) = \n",
    "\\frac{1}{2}\\int_0^z\\left(\\frac{-\\log(1-x)}{x}+\\frac{-\\log x}{1-x}\\right)\\,dx =\n",
    "\\Li_2(z) + \\frac{1}{2}\\log z\\;\\log(1-z).\n",
    "$$\n",
    "\n",
    "1つ目の等号は $L(z)$ の定義であり, 2つ目の等号は次を使えば得られる:\n",
    "\n",
    "$$\n",
    "\\frac{d}{dx}(\\Li_2(x) +\\log x\\,\\log(1-x)) = \\frac{-\\log x}{1-x}.\n",
    "$$\n",
    "\n",
    "$L(z)$ と $\\Li_2(z)$ の違いは初等函数 $\\ds\\frac{1}{2}\\log z\\;\\log(1-z)$ の分だけである.\n",
    "\n",
    "$\\ds \\Li_2(0)=0$, $\\ds \\Li_2(1)=\\zeta(2)=\\frac{\\pi^2}{6}$ より,\n",
    "\n",
    "$$\n",
    "L(0) = 0, \\quad L(1) = \\frac{\\pi^2}{6}.\n",
    "$$\n",
    "\n",
    "この節では次の**五項関係式** (five-term relation)\n",
    "\n",
    "$$\n",
    "L(x) + L(1-xy) + L(y) + L\\left(\\frac{1-y}{1-xy}\\right) + L\\left(\\frac{1-x}{1-xy}\\right) = \\frac{\\pi^2}{2}.\n",
    "\\tag{FTR}\n",
    "$$\n",
    "\n",
    "を示そう. 二重対数函数の五項関係式については19世紀にすでに先行する研究結果があったが, この形式に整理された五項関係式は\n",
    "\n",
    "* R. Rogers, On Function Sum Theorems Connected with the Series $\\ds\\sum_{n=1}^\\infty\\frac{z^n}{n^2}$,  Proc. London Math. Soc. 4 (1907), 169-189, <a href=\"https://doi.org/10.1112/plms/s2-4.1.169\">DOI</a>\n",
    "\n",
    "で証明された. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**補題:** 函数 $f_i(x)$ 達について\n",
    "\n",
    "$$\n",
    "F(x, X) = \\sum_{i=1}^N \\log(f_i(x))\\log(1-f_i(X))\n",
    "$$\n",
    "\n",
    "は $F(x,X)=F(X,x)$ を満たしていると仮定する. このとき, \n",
    "\n",
    "$$\n",
    "f(x) = \\sum_{i=1}^N L(f_i(x))\n",
    "$$\n",
    "\n",
    "は定数函数になる.\n",
    "\n",
    "**証明:** $df=0$ を示せばよい.\n",
    "\n",
    "$$\n",
    "-2\\,dL(x) = \n",
    "\\frac{\\log(1-x)\\,dx}{x}+\\frac{\\log(x)\\,dx}{1-x} =\n",
    "d\\log(x)\\,\\log(1-x) - \\log(x)\\,d\\log(1-x) \n",
    "$$\n",
    "\n",
    "なので, \n",
    "\n",
    "$$\n",
    "-2\\,df(x) = \n",
    "\\sum_{i=1}^N d\\log(f_i(x))\\,\\log(1-f_i(x)) - \n",
    "\\sum_{i=1}^N \\log(f_i(x))\\,d\\log(1-f_i(x)).\n",
    "$$\n",
    "\n",
    "$F(x,X)$ と $F(X,x)$ の $x$ に関する外微分 $d$ を取ると, \n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "&\n",
    "dF(x,X) = \\sum_{i=1}^N d\\log(f_i(x))\\,\\log(1-f_i(X)),\n",
    "\\\\ &\n",
    "dF(X,x) = \\sum_{i=1}^N \\log(f_i(X))\\,d\\log(1-f_i(x)).\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "$F(x,X)=F(X,x)$ より, これらは等しい:\n",
    "\n",
    "$$\n",
    "\\sum_{i=1}^N d\\log(f_i(x))\\,\\log(1-f_i(X)) =\n",
    "\\sum_{i=1}^N \\log(f_i(X))\\,d\\log(1-f_i(x))\n",
    "$$\n",
    "\n",
    "$X=x$ とおくと, \n",
    "\n",
    "$$\n",
    "\\sum_{i=1}^N d\\log(f_i(x))\\,\\log(1-f_i(x)) =\n",
    "\\sum_{i=1}^N \\log(f_i(x))\\,d\\log(1-f_i(x)).\n",
    "$$\n",
    "\n",
    "以上を合わせると, $-2df=0$ が得られる. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**五項関係式の証明:** $f_i(x,y)$ ($i=1,2,3,4,5$) をそれぞれ $x$, $1-xy$, $y$, $\\ds\\frac{1-y}{1-xy}$, $\\ds\\frac{1-x}{1-xy}$ と定め, $\\ds f(x,y)=\\sum_{i=1}^5 L(f_i(x,y))$ とおく.  $\\ds\n",
    "\n",
    "$f(x,y)$ が定数函数になることを示そう. 上の補題より, そのためには\n",
    "\n",
    "$$\n",
    "F(x,y;X,Y) = \\sum_{i=1}\\log(f_i(x,y))\\log(1-f_i(X,Y))\n",
    "$$\n",
    "\n",
    "が $F(x,y;X,Y)=F(X,Y;x,y)$ を満たしていることを示せば十分である. 具体的に計算すると,\n",
    "\n",
    "$$\n",
    "1-(1-XY)=XY, \\quad\n",
    "1-\\frac{1-Y}{1-XY} = \\frac{Y(1-X)}{1-XY}, \\quad\n",
    "1-\\frac{1-X}{1-XY} = \\frac{X(1-Y)}{1-XY}\n",
    "$$\n",
    "\n",
    "より\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "F(x,y;X,Y) &=\n",
    "\\log(x)\\log(1-X) +\n",
    "\\log(1-xy)\\log(1-XY) +\n",
    "\\log(y)\\log(1-Y)\n",
    "\\\\ &\\,+\n",
    "(\\log(1-y)-\\log(1-xy))(\\log(Y)+\\log(1-X)-\\log(1-XY)) \n",
    "\\\\ &\\,+\n",
    "(\\log(1-x)-\\log(1-xy))(\\log(X)+\\log(1-Y)-\\log(1-XY))\n",
    "\\\\ &=\n",
    "\\log(x)\\log(1-X) + \\log(1-x)\\log(X)\n",
    "\\\\ &\\,+\n",
    "\\log(y)\\log(1-Y) + \\log(1-y)\\log(Y)\n",
    "\\\\ &\\,+\n",
    "\\log(1-x)\\log(1-Y) + \\log(1-y)\\log(1-X)\n",
    "\\\\ &\\,-\n",
    "\\log(1-x)\\log(1-XY) - \\log(1-xy)\\log(1-X) \n",
    "\\\\ &\\,-\n",
    "\\log(1-y)\\log(1-XY) - \\log(1-xy)\\log(1-Y)\n",
    "\\\\ &\\,+\n",
    "2\\log(1-xy)\\log(1-XY) \n",
    "\\\\ &\\,+\n",
    "\\log(1-xy)\\log(XY) - \\log(1-xy)\\log(X)-\\log(1-xy)\\log(Y).\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "最下段の3つの項がキャンセルして消えるので, $F(x,y;X,Y)=F(X,Y;x,y)$ となることがわかる. これで $f(x,y)$ が定数函数であることがわかった.\n",
    "\n",
    "そして, $\\ds L(0)=0$, $\\ds L(1)=\\frac{\\pi^2}{6}$ より,\n",
    "\n",
    "$$\n",
    "f(0,0) = L(0)+L(0)+L(1)+L(1)+L(1) = \\frac{\\pi^2}{2}.\n",
    "$$\n",
    "\n",
    "以上によって $\\ds f(x,y)=\\frac{\\pi^2}{2}$ となることが示された. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**注意:** 上の補題を使えば, $L(x)+L(1-x)$ も定数函数になることがわかり, $L(0)=0$, $\\ds L(1)=\\frac{\\pi^2}{6}$ より,\n",
    "\n",
    "$$\n",
    "L(x) + L(1-x) = \\frac{\\pi^2}{6}\n",
    "$$\n",
    "\n",
    "となることも示される. これを使って, 五項関係式は次のように書き直される:\n",
    "\n",
    "$$\n",
    "L(x) + L(y) = L(xy) + L\\left(\\frac{x(1-y)}{1-xy}\\right) + L\\left(\\frac{y(1-x)}{1-xy}\\right).\n",
    "$$\n",
    "\n",
    "この形の公式も各種文献でよく見る. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**注意:** $x_1=x$, $x_2=1-xy$ とおき, 帰納的に $x_n$ を \n",
    "\n",
    "$$\n",
    "x_{n-1}x_{n+1} = 1 - x_n\n",
    "$$\n",
    "\n",
    "という関係式で定めると,\n",
    "\n",
    "$$\n",
    "x_3 = y, \\quad\n",
    "x_4 = \\frac{1-y}{1-xy}, \\quad\n",
    "x_5 = \\frac{1-x}{1-xy}, \\quad\n",
    "x_6 = x, \\quad\n",
    "x_7 = 1-xy, \\quad, \\ldots\\ldots\n",
    "$$\n",
    "\n",
    "$x_{n+5}=x_n$ が成立しており, 出て来た $x,y$ の式がちょうど五項関係式の5つの $f_i(x,y)$ に一致していることがわかる. この事実は<a href=\"https://www.google.co.jp/search?q=%E3%82%AF%E3%83%A9%E3%82%B9%E3%82%BF%E3%83%BC%E4%BB%A3%E6%95%B0\">クラスター代数</a>と <a href=\"https://www.google.co.jp/search?q=dilogarithm+identities\">dilogarithm identities</a> の理論として大幅に一般化されている. クラスター代数の入門的解説については\n",
    "\n",
    "* 黒木玄, <a href=\"https://genkuroki.github.io/documents/cluster_algebra_rank2.pdf\">ランク2のクラスター代数</a>, 2010年\n",
    "\n",
    "* 黒木玄, <a href=\"https://genkuroki.github.io/documents/20120810FriezePattern.pdf\">フリーズパターン ―― 数の繰返し模様の不思議</a>, 2013年\n",
    "\n",
    "* 黒木玄, <a href=\"https://genkuroki.github.io/documents/20121115Akita.pdf\">フリーズパターンの不思議</a>, 2013年\n",
    "\n",
    "\n",
    "などを見よ. 検索すれば他にもたくさん見付かる. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**注意:** このノートの下の方で二重対数函数の量子版と五項関係式の量子化とみあされる五角形恒等式について説明する. $\\QED$ "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 無限積\n",
    "\n",
    "多項式の計算で因数分解が極めて有用であるのと同様に, より一般の函数の無限積への因数分解も極めて有用である. そのためには無限積を正しく扱う方法を知っておかなければいけない.\n",
    "\n",
    "数列 $a_n$ から作られる無限積 $\\ds\\prod_{n=1}^\\infty a_n$ が収束するとは, 数列 $\\ds p_n=\\prod_{k=1}^n a_k=a_1 a_2\\cdots a_n$ が収束することであると定める. そのとき, その無限積の収束先が数列 $p_n$ の収束先で定義される.\n",
    "\n",
    "無限積については高木貞治『解析概論』の§51(pp.178-181)を参照せよ."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**例:** 以下では $p_i$ は $i$ 番目の素数であるとする. $p_1=2$, $p_2=3$, $p_3=5$, $\\ldots$ である. さらに $s>1$ であると仮定する. このとき, 素数全体に関する無限積 $\\ds\\prod_{i=1}^\\infty\\frac{1}{1-p_i^{-s}}$ がどうなるかを調べてみよう. $\\ds Z_N = \\prod_{i=1}^N\\frac{1}{1-p_i^{-s}}$ とおく. このとき,\n",
    "\n",
    "$$\n",
    "Z_N = \\prod_{i=1}^N (1 + p_i^{-s} + p_i^{-2s} + \\cdots)\n",
    "$$\n",
    "\n",
    "なので, $Z_N$ は非負の整数たち $k_1,\\ldots,k_N$ に対する\n",
    "\n",
    "$$\n",
    "p_1^{-k_1 s}\\cdots p_N^{-k_N s} = \\frac{1}{\\left(p_1^{k_1}\\cdots p_N^{k_N}\\right)^s}\n",
    "$$\n",
    "\n",
    "達すべての和に等しい. 正の整数は一意的に素因数分解できるので, $N\\to\\infty$ で $Z_N$ は正の整数 $n$ に対する $\\ds\\frac{1}{n^s}$ 達全体の和に等しくなる:\n",
    "\n",
    "$$\n",
    "\\prod_{i=1}^\\infty\\frac{1}{1-p^{-s}} = \\sum_{n=1}^\\infty \\frac{1}{n^s} = \\zeta(s).\n",
    "$$\n",
    "\n",
    "左辺をゼータ函数 $\\zeta(s)$ の**Euler積**と呼ぶ. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 無限積の絶対収束\n",
    "\n",
    "**定理:** 級数 $\\ds\\sum_{n=1}^\\infty a_n$ が絶対収束していれば, 無限積 $\\ds\\prod_{n=1}^\\infty(1 + a_n)$ も収束する. このとき, その無限積は**絶対収束**するという.\n",
    "\n",
    "**証明:** 級数 $\\ds\\sum_{n=1}^\\infty a_n$ は絶対収束していると仮定し, \n",
    "\n",
    "$$\n",
    "M = \\sum_{n=1}^\\infty |a_n|, \\quad\n",
    "p_n = \\prod_{k=1}^n(1+a_k), \\quad\n",
    "b_n = p_n - p_{n-1} = p_{n-1}a_n\n",
    "$$\n",
    "\n",
    "とおく. このとき, $\\ds p_n = \\sum_{k=1}^n b_k$ であり, $x\\geqq 0$ のとき $1+x\\leqq e^x$ であることより, \n",
    "\n",
    "$$\n",
    "|p_n| \\leqq \\prod_{k=1}^n(1+|a_k|) \\leqq \\prod_{k=1}^n e^{|a_n|} = e^M\n",
    "$$\n",
    "\n",
    "となり, $|b_k|=|p_{k-1}||a_k|\\leqq e^M |a_k|$ となるので, \n",
    "\n",
    "$$\n",
    "\\sum_{n=1}^\\infty |b_n| \\leqq e^M\\sum_{n=1}^\\infty |a_n| = e^M M <\\infty.\n",
    "$$\n",
    "\n",
    "よって, 級数 $\\ds\\sum_{n=1}^\\infty b_n$ は絶対収束する. $\\ds p_n=\\sum_{k=1}^n b_k$ なので無限積 $\\ds\\prod_{n=1}^\\infty (1+a_n)=\\lim_{n\\to\\infty} p_n$ が収束することがわかった. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**例:** $n\\to\\infty$ のとき $a_n = O(1/n^2)$ であるとする. このとき, ある定数 $C\\geqq 0$ が存在して, $|a_n|\\leqq C/n^2$ となる. このとき,\n",
    "\n",
    "$$\n",
    "\\sum_{n=1}^\\infty |a_n|\\leqq C\\sum_{n=1}^\\infty \\frac{1}{n^2} = C\\zeta(2) < \\infty\n",
    "$$\n",
    "\n",
    "より $\\ds\\sum_{n=1}^\\infty a_n$ は絶対収束するので, 無限積 $\\ds\\prod_{k=1}^\\infty(1+a_n)$ は絶対収束する. 例えば, \n",
    "\n",
    "$$\n",
    "f(s) = s\\prod_{n=1}^\\infty\\left(1-\\frac{s^2}{n^2}\\right)\n",
    "$$\n",
    "\n",
    "の右辺の無限積は絶対収束している. このとき, すべての整数 $n\\in\\Z$ について $f(n)=0$ となる. 実はこの $f(s)$ は $\\ds f(s)=\\frac{\\sin(\\pi s)}{\\pi}$ となっていることを示せる. この一連のノート群の後の方で複数の方法で証明することになるだろう. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**注意:** $\\ds\\prod_{n=1}^\\infty (1+a_n)$ は絶対収束していると仮定する. そのとき, $\\ds\\prod_{n=1}^\\infty\\frac{1}{1-a_n}=\\prod_{n=1}^\\infty\\left(1+\\frac{a_n}{1-a_n}\\right)$ も絶対収束している. なぜならば, $a_n$ は $0$ に収束しているので, ある番号 $N$ が存在して, $n\\geqq N$ ならば $\\ds |1-a_n|\\geqq\\frac{1}{2}$ となる. 絶対収束性を論じる場合には $N=1$ の場合だけを考えれば十分なので, $N=1$ であると仮定する. そのとき,\n",
    "\n",
    "$$\n",
    "\\sum_{n=1}^\\infty\\left|\\frac{a_n}{1-a_n}\\right|\\leqq 2\\sum_{n=1}^\\infty|a_n|<\\infty.\n",
    "$$\n",
    "\n",
    "これで, $\\ds\\prod_{n=1}^\\infty\\frac{1}{1-a_n}$ も絶対収束していることが示された. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**例:** $|q|<1$ のとき, $\\ds\\sum_{n=1}^\\infty q^n$ は絶対収束しているので, 無限積 $f(q)=\\ds\\prod_{n=1}^\\infty (1-q^n)$ も絶対収束しており, $\\ds \\frac{1}{f(q)}=\\ds\\prod_{n=1}^\\infty \\frac{1}{1-q^n}$ も絶対収束している. そして, \n",
    "\n",
    "$$\n",
    "\\frac{1}{f(q)} = \\prod_{m=1}^\\infty \\sum_{k=0}^\\infty q^{mk} =\n",
    "\\sum_{k_1,k_2,\\ldots=0}^\\infty q^{k_1+2k_2+3k_3+\\cdots} =\n",
    "\\sum_{n=0}^\\infty p(n) q^n.\n",
    "$$\n",
    "\n",
    "ただし, 有限個を除いて $0$ であるような非負の整数たちの組全体を $(k_1,k_2,\\ldots)$ は動くとする. その約束のもとで\n",
    "\n",
    "$$\n",
    "p(n) = \\sharp\\{\\, (k_1,k_2,k_3,\\ldots)\\mid k_1+2k_2+3k_3+\\cdots = n\\,\\}.\n",
    "$$\n",
    "\n",
    "ここで $\\sharp X$ は集合 $X$ の元の個数を表わす. すなわち, $p(n)$ は $n$ を正の整数の和に分解する方法の個数に等しい. $p(n)$ は $n$ の**分割数**と呼ばれている. 例えば, $p(1)=1$, $p(2)=2$, $p(3)=3$, $p(4)=5$ であり, \n",
    "\n",
    "$$\n",
    "5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1\n",
    "$$\n",
    "\n",
    "なので $p(5)=7$ である.  $\\eta(q)=q^{1/24}f(q)$ は**Dedekindoのη函数**と呼ばれるモジュラー変換性を持つ重要な特殊函数である. 分割数のような組み合わせ論的な数とモジュラ₀変換性を持つ特殊函数が関係しており, 豊かな数学の世界を作っている. $\\QED$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10-element Array{Tuple{Int64,SymPy.Sym},1}:\n",
       " (1, 1)  \n",
       " (2, 2)  \n",
       " (3, 3)  \n",
       " (4, 5)  \n",
       " (5, 7)  \n",
       " (6, 11) \n",
       " (7, 15) \n",
       " (8, 22) \n",
       " (9, 30) \n",
       " (10, 42)"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "partition_number(n) = sympy[:functions][:combinatorial][:numbers][:nT](n)\n",
    "[(n, partition_number(n)) for n in 1:10]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$1 + q + 2 q^{2} + 3 q^{3} + 5 q^{4} + 7 q^{5} + 11 q^{6} + 15 q^{7} + 22 q^{8} + 30 q^{9} + 42 q^{10} + O\\left(q^{11}\\right)$$"
      ],
      "text/plain": [
       "           2      3      4      5       6       7       8       9       10    \n",
       "1 + q + 2*q  + 3*q  + 5*q  + 7*q  + 11*q  + 15*q  + 22*q  + 30*q  + 42*q   + O\n",
       "\n",
       "/ 11\\\n",
       "\\q  /"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q = symbols(\"q\")\n",
    "N = 10\n",
    "series(prod(1/(1-q^n) for n in 1:N), n=N+1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**注意:** $\\ds\\prod_{n=1}^\\infty (1-q^n)$ について\n",
    "\n",
    "$$\n",
    "\\prod_{n=1}^\\infty (1-q^n) = \n",
    "\\sum_{n=-\\infty}^\\infty (-1)^n q^{n(3n-1)/2} =\n",
    "\\sum_{n=-\\infty}^\\infty (-1)^n q^{((6n-1)^2-1)/24}\n",
    "$$\n",
    "\n",
    "となることが知られている. 次のセルを見よ. $\\QED$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$1 - q - q^{2} + q^{5} + q^{7} - q^{12} - q^{15} + q^{22} + q^{26} - q^{35} + O\\left(q^{36}\\right)$$"
      ],
      "text/plain": [
       "         2    5    7    12    15    22    26    35    / 36\\\n",
       "1 - q - q  + q  + q  - q   - q   + q   + q   - q   + O\\q  /"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " "
     ]
    },
    {
     "data": {
      "text/latex": [
       "$$1 - q - q^{2} + q^{5} + q^{7} - q^{12} - q^{15} + q^{22} + q^{26} - q^{35} + O\\left(q^{36}\\right)$$"
      ],
      "text/plain": [
       "         2    5    7    12    15    22    26    35    / 36\\\n",
       "1 - q - q  + q  + q  - q   - q   + q   + q   - q   + O\\q  /"
      ]
     },
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     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " 0.514364 seconds (69.04 k allocations: 3.235 MiB)\n",
      "  7.514937 seconds (26.06 k allocations: 1.200 MiB)\n"
     ]
    }
   ],
   "source": [
    "q = symbols(\"q\")\n",
    "N = 5\n",
    "M = N*(3N-1)÷2\n",
    "@time series(sum((-1)^n*q^(n*(3n-1)÷2) for n in -M:M), q, n=M+1) |>display\n",
    "@time series(prod(1-q^n for n in 1:M), q, n=M+1) |> display"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 絶対収束する無限積の収束先が積の順序によらないこと\n",
    "\n",
    "**定理:** 絶対収束する無限積の収束先は積の順序によらない.\n",
    "\n",
    "**証明:** 絶対収束する級数の収束先が和の順序によらないことに帰着する. 無限積 $\\ds\\prod_{n=1}^\\infty(1+a_n)$ は絶対収束していると仮定し,  $\\ds M=\\sum_{n=1}^\\infty |a_n|$ とおく. その無限積の収束先は\n",
    "\n",
    "$$\n",
    "p_n=\\prod_{k=1}^n(1+a_k) = \\sum_{I\\subset\\{1,2,\\ldots,n\\}} \\prod_{i\\in I}a_i\n",
    "$$\n",
    "\n",
    "の収束先に等しい. さらに, \n",
    "\n",
    "$$\n",
    "\\sum_{I\\subset\\{1,2,\\ldots,n\\}} \\left|\\prod_{i\\in I}a_i\\right| =\n",
    "\\prod_{k=1}^n (1+|a_k|) \\leqq \\prod_{k=1}^n e^{|a_k|} \\leqq e^M < \\infty.\n",
    "$$\n",
    "\n",
    "ゆえに, 正の整数全体の集合の有限部分集合 $I$ に対する積 $\\ds\\prod_{i\\in I}a_i$ 達すべての無限和は絶対収束している. したがって, その無限和の収束先は和の順序によらない.  無限積の積の順序を変えることは, その無限和の和の順序を変える場合の特別な場合になっているので, 無限積の収束先も積の順序によらない. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 量子二重対数函数\n",
    "\n",
    "以下, $|q|<1$ であると仮定する. \n",
    "\n",
    "このノートでは**量子二重対数函数**(quantum dilogarithm) $E_q(x)$ を\n",
    "\n",
    "$$\n",
    "E_q(x) = \\left(\\prod_{k=0}^\\infty\\left(1 + q^{k+1/2}x\\right)\\right)^{-1} =\n",
    "\\frac{1}{(1+q^{1/2}x)(1+q^{1+1/2}x)(1+q^{2+1/2}x)\\cdots}.\n",
    "$$\n",
    "\n",
    "と定める. (注意: $E_q(-x)$ の方を量子二重対数と呼んでいる文献が多い.) この $E_q(x)$ は\n",
    "\n",
    "$$\n",
    "E_q(qx) = (1+q^{1/2}x) E_q(x), \\quad E_q(0) = 1\n",
    "$$\n",
    "\n",
    "という条件で一意に特徴付けられる.\n",
    "\n",
    "上の $E_q(x)$ が量子二重対数函数と呼ばれる理由はその対数が次の表示を持つからである:\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\log E_q(-x) &= -\\sum_{k=0}^\\infty \\log(1 - q^{k+1/2}x) =\n",
    "\\sum_{k=0}^\\infty \\sum_{n=1}^\\infty \\frac{\\left(q^{k+1/2}x\\right)^n}{n} \n",
    "\\\\ &=\n",
    "\\sum_{n=1}^\\infty\\left(\\sum_{k=0}^\\infty q^{nk}\\right)(q^{1/2}x)^n =\n",
    "\\sum_{n=1}^\\infty\\frac{(q^{1/2}x)^n}{n(1-q^n)}.\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "ゆえに, \n",
    "\n",
    "$$\n",
    "(1-q)\\log E_q(-x) = \\sum_{n=1}^\\infty \\frac{(q^{1/2}x)^n}{n (n)_q}, \n",
    "\\qquad\n",
    "(n)_q = \\frac{1-q^n}{1-q}.\n",
    "$$\n",
    "\n",
    "ここで, $(n)_q$ は **q数** (q-number)と呼ばれ, $q\\to 1$ で $n$ に収束する. ゆえに, $q\\to 1$ のとき,\n",
    "\n",
    "$$\n",
    "(1-q)\\log E_q(-x) \\to \\sum_{n=1}^\\infty\\frac{x^n}{n^2} = \\Li_2(x).\n",
    "$$\n",
    "\n",
    "$E_q(x)$ の $q\\to 1$ での漸近挙動は(古典)二重対数函数 $\\Li_2(x)$ を含んでいる."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 量子二重対数函数と量子指数函数との関係\n",
    "\n",
    "**量子指数函数** $e_q(x)$ を\n",
    "\n",
    "$$\n",
    "e_q(x) = \\sum_{n=0}^\\infty \\frac{x^n}{(n)_q!},\n",
    "\\qquad\n",
    "(n)_q! = (1)_q(2)_q\\cdots(n)_q\n",
    "$$\n",
    "\n",
    "と定める. $(n)_q!$ は**q階乗 (q-factorial)** と呼ばれている. このとき,\n",
    "\n",
    "$$\n",
    "\\frac{e_q(x) - e_q(qx)}{(1-q)x} = \\sum_{n=1}^\\infty\\frac{(n)_q x^{n-1}}{(n)_q!} =\n",
    "\\sum_{m=0}^\\infty \\frac{x^m}{(m)_q!} = e_q(x).\n",
    "$$\n",
    "\n",
    "ゆえに, $n\\to\\infty$ のとき, $e_q(q^n x)\\to e_q(0)=1$ より, \n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "e_q(x) &= \\frac{e_q(qx)}{1-(1-q)x} = \n",
    "\\frac{e_q(q^2 x)}{(1-(1-q)x)(1-(1-q)xq)} \n",
    "\\\\ &= \n",
    "\\frac{e_q(q^3 x)}{(1-(1-q)x)(1-(1-q)xq)(1-(1-q)xq^2)}\n",
    "\\\\ &= \n",
    "\\cdots\\cdots\\cdots\\cdots\\cdots\n",
    "\\\\ &=\n",
    "\\frac{1}{(1-(1-q)x)(1-(1-q)xq)(1-(1-q)xq^2)\\cdots}\n",
    "\\\\ &=\n",
    "\\left(\\prod_{k=0}^\\infty(1-(1-q)xq^k)\\right)^{-1}.\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "これより,\n",
    "\n",
    "$$\n",
    "E_q(x) = e_q\\left(-\\frac{q^{1/2}x}{1-q}\\right) =\n",
    "\\sum_{n=0}^\\infty \\frac{q^{n/2}(-x)^n}{(1-q)(1-q^2)\\cdots(1-q^n)}.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### q二項定理\n",
    "\n",
    "**q二項係数** ($q$-binomial coefficients)$ を\n",
    "\n",
    "$$\n",
    "\\binom{n}{k}_{\\!q} = \n",
    "\\frac{(n)_q!}{(k)_q!(n-k)_q!} = \n",
    "\\frac{(n)_q(n-1)_q\\cdots(n-k+1)_q}{(k)_q!}\n",
    "$$\n",
    "\n",
    "と定める. このとき, \n",
    "\n",
    "$$\n",
    "(k)_q + q^k(n-k+1)_q =\n",
    "\\frac{1-q^k+q^k(1-q^{n-k+1})}{1-q} =\n",
    "\\frac{1-q^{n+1}}{1-q} =\n",
    "(n+1)_q\n",
    "$$\n",
    "\n",
    "より, \n",
    "\n",
    "$$\n",
    "\\binom{n}{k-1}_{\\!q} + q^k \\binom{n}{k}_{\\!q} =\n",
    "\\binom{n}{k-1}_{\\!q}\\frac{(k)_q+q^k(n-k+1)_q}{(k)_q} =\n",
    "\\binom{n+1}{k}_{\\!q}.\n",
    "\\tag{1}\n",
    "$$\n",
    "\n",
    "文字 $x$, $y$ は関係式 $yx=qxy$ を満たしていると仮定する. このとき, (1)を使うと, 通常の二項定理の場合と同様に $n$ に関して帰納的に次の公式が成立することを示せる(実際に示してみよ):\n",
    "\n",
    "$$\n",
    "(x+y)^n = \\sum_{k=0}^n\\binom{n}{k}_{\\!q}\\; x^k y^{n-k}\n",
    "\\qquad (yx=qxy).\n",
    "$$\n",
    "\n",
    "これを **q二項定理** と呼ぶ."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### q指数函数の加法公式\n",
    "\n",
    "$yx=qxy$ のとき, \n",
    "\n",
    "$$\n",
    "e_q(x+y) = e_q(x)e_q(y)\n",
    "$$\n",
    "\n",
    "となることを示そう. \n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "e_q(x+y) &= \\sum_{n=0}^\\infty \\frac{(x+y)^n}{(n)_q!}=\n",
    "\\sum_{n=0}^\\infty\\frac{1}{(n)_q!}\\sum_{k=0}^n\\binom{n}{k}_{\\!q}\\; x^k y^{n-k}\n",
    "\\\\ &=\n",
    "\\sum_{n=0}^\\infty\\sum_{k=0}^n\\frac{x^k}{(k)_q!}\\frac{y^{n-k}}{(n-k)_q!} =\n",
    "\\sum_{k,l=0}^\\infty \\frac{x^k}{(k)_q!}\\frac{y^l}{(l)_q!} =\n",
    "e_q(x)e_q(y).\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "$\\ds E_q(x) = e_q\\left(-\\frac{q^{1/2}x}{1-q}\\right)$ より, \n",
    "\n",
    "$$\n",
    "E_q(x+y) = E_q(x)E_q(y) \\qquad (yx=qxy).\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 量子二重対数函数の五角形恒等式\n",
    "\n",
    "$yx=qxy$ と仮定し, \n",
    "\n",
    "$$\n",
    "\\np{xy}=q^{1/2}xy=q^{-1/2}yx\n",
    "$$ \n",
    "\n",
    "とおく. そのとき, $y E_q(x)= E_q(qx) y = E_q(x)(1+q^{1/2}x)y = E_q(x)(\\np{xy} + y)$ より,\n",
    "\n",
    "$$\n",
    "y^n E_q(x) = E_q(x)(\\np{xy} + y)^n.\n",
    "$$\n",
    "\n",
    "この等式の両辺に適当な係数をかけて $n$ について足し上げると,\n",
    "\n",
    "$$\n",
    "E_q(y) E_q(x) = E_q(x)E_q(\\np{xy}+y).\n",
    "$$\n",
    "\n",
    "$y\\np{xy} = q\\np{xy}y$ なので, 前節の結果より,\n",
    "\n",
    "$$\n",
    "E_q(\\np{xy}+y) = E_q(\\np{xy}) E_q(y).\n",
    "$$\n",
    "\n",
    "したがって,\n",
    "\n",
    "$$\n",
    "E_q(y)E_q(x) = E_q(x)E_q(\\np{xy})E_q(y).\n",
    "$$\n",
    "\n",
    "これを量子二重対数函数の**五角形恒等式** (pentagon identity)と呼ぶ.\n",
    "\n",
    "五角形恒等式は量子二重対数函数 $E_q(x)$ の $0$ でない形式べき級数として一意に特徴付ける. なぜならば, 五角形恒等式の $x,y$ に $0$ を代入すると $E_q(0)=1$ であることがわかり, 五角形恒等式の両辺の $y$ に関する1次の項を比較すると, \n",
    "\n",
    "$$\n",
    "E_q(qx)y = yE_q(x) = E_q(x)(\\np{xy}+y) = E_q(x)(1+q^{1/2}x)y\n",
    "$$\n",
    "\n",
    "となるので $E_q(qx)=E_q(1+q^{1/2}x)$ が得られ, 以上で得られた2つの条件で $E_q(x)$ が一意に特徴付けられるからである."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**注意:** 量子二重対数函数の五角形恒等式は $A_2$ 型の量子クラスター代数から得られる quantum dilogarithm identity とみなされる. さらにアフィン型の場合の無限積版の quantum dilogarithm identitities も存在する. それらの恒等式の背景には量子クラスター代数や量子群の普遍 $R$ 行列の理論が隠れている. 有限積の量子二重対数恒等式の作り方については\n",
    "\n",
    "* 黒木玄, <a href=\"https://genkuroki.github.io/documents/20101228HowToMakeQDIs.pdf\">量子 dilogarithm 恒等式の作り方</a>, 2010年12月\n",
    "\n",
    "にも解説がある. $\\QED$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
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