Old_Ver_for_Julia_v0.6/10 Gauss, Gamma, Beta.ipynb

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   "source": [
    "# 10 Gauss積分, ガンマ函数, ベータ函数\n",
    "\n",
    "黒木玄\n",
    "\n",
    "2018-06-21\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/10%20Gauss%2C%20Gamma%2C%20Beta.ipynb\n",
    "\n",
    "* https://genkuroki.github.io/documents/Calculus/10%20Gauss%2C%20Gamma%2C%20Beta.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",
    "$\n",
    "\\newcommand\\eps{\\varepsilon}\n",
    "\\newcommand\\ds{\\displaystyle}\n",
    "\\newcommand\\Z{{\\mathbb Z}}\n",
    "\\newcommand\\R{{\\mathbb R}}\n",
    "\\newcommand\\C{{\\mathbb C}}\n",
    "\\newcommand\\QED{\\text{□}}\n",
    "\\newcommand\\root{\\sqrt}\n",
    "\\newcommand\\bra{\\langle}\n",
    "\\newcommand\\ket{\\rangle}\n",
    "\\newcommand\\d{\\partial}\n",
    "\\newcommand\\sech{\\operatorname{sech}}\n",
    "\\newcommand\\cosec{\\operatorname{cosec}}\n",
    "\\newcommand\\sign{\\operatorname{sign}}\n",
    "\\newcommand\\sinc{\\operatorname{sinc}}\n",
    "\\newcommand\\real{\\operatorname{Re}}\n",
    "\\newcommand\\imag{\\operatorname{Im}}\n",
    "\\newcommand\\Li{\\operatorname{Li}}\n",
    "\\newcommand\\PROD{\\mathop{\\coprod\\kern-1.35em\\prod}}\n",
    "$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "toc": true
   },
   "source": [
    "<h1>Table of Contents<span class=\"tocSkip\"></span></h1>\n",
    "<div class=\"toc\"><ul class=\"toc-item\"><li><span><a href=\"#Gauss積分\" data-toc-modified-id=\"Gauss積分-1\"><span class=\"toc-item-num\">1&nbsp;&nbsp;</span>Gauss積分</a></span><ul class=\"toc-item\"><li><span><a href=\"#Gauss積分の公式\" data-toc-modified-id=\"Gauss積分の公式-1.1\"><span class=\"toc-item-num\">1.1&nbsp;&nbsp;</span>Gauss積分の公式</a></span></li><li><span><a href=\"#Gauss積分を使う簡単な計算例\" data-toc-modified-id=\"Gauss積分を使う簡単な計算例-1.2\"><span class=\"toc-item-num\">1.2&nbsp;&nbsp;</span>Gauss積分を使う簡単な計算例</a></span></li><li><span><a href=\"#Gauss分布のFourier変換\" data-toc-modified-id=\"Gauss分布のFourier変換-1.3\"><span class=\"toc-item-num\">1.3&nbsp;&nbsp;</span>Gauss分布のFourier変換</a></span></li><li><span><a href=\"#Gauss積分の公式の導出\" data-toc-modified-id=\"Gauss積分の公式の導出-1.4\"><span class=\"toc-item-num\">1.4&nbsp;&nbsp;</span>Gauss積分の公式の導出</a></span><ul class=\"toc-item\"><li><span><a href=\"#方法1:-高さ-$z$-で輪切りにする方法\" data-toc-modified-id=\"方法1:-高さ-$z$-で輪切りにする方法-1.4.1\"><span class=\"toc-item-num\">1.4.1&nbsp;&nbsp;</span>方法1: 高さ $z$ で輪切りにする方法</a></span></li><li><span><a href=\"#方法2:-極座標を使う方法\" data-toc-modified-id=\"方法2:-極座標を使う方法-1.4.2\"><span class=\"toc-item-num\">1.4.2&nbsp;&nbsp;</span>方法2: 極座標を使う方法</a></span></li><li><span><a href=\"#方法3:-$y=x-\\tan\\theta$-と変数変換する方法\" data-toc-modified-id=\"方法3:-$y=x-\\tan\\theta$-と変数変換する方法-1.4.3\"><span class=\"toc-item-num\">1.4.3&nbsp;&nbsp;</span>方法3: $y=x \\tan\\theta$ と変数変換する方法</a></span></li></ul></li></ul></li><li><span><a href=\"#ガンマ函数とベータ函数\" data-toc-modified-id=\"ガンマ函数とベータ函数-2\"><span class=\"toc-item-num\">2&nbsp;&nbsp;</span>ガンマ函数とベータ函数</a></span><ul class=\"toc-item\"><li><span><a href=\"#ガンマ函数とベータ函数の定義\" data-toc-modified-id=\"ガンマ函数とベータ函数の定義-2.1\"><span class=\"toc-item-num\">2.1&nbsp;&nbsp;</span>ガンマ函数とベータ函数の定義</a></span></li><li><span><a href=\"#ガンマ函数の特殊値と函数等式\" data-toc-modified-id=\"ガンマ函数の特殊値と函数等式-2.2\"><span class=\"toc-item-num\">2.2&nbsp;&nbsp;</span>ガンマ函数の特殊値と函数等式</a></span></li><li><span><a href=\"#Riemannのゼータ函数の積分表示と函数等式と負の整数と正の偶数における特殊値\" data-toc-modified-id=\"Riemannのゼータ函数の積分表示と函数等式と負の整数と正の偶数における特殊値-2.3\"><span class=\"toc-item-num\">2.3&nbsp;&nbsp;</span>Riemannのゼータ函数の積分表示と函数等式と負の整数と正の偶数における特殊値</a></span><ul class=\"toc-item\"><li><span><a href=\"#Riemannのゼータ函数の積分表示1\" data-toc-modified-id=\"Riemannのゼータ函数の積分表示1-2.3.1\"><span class=\"toc-item-num\">2.3.1&nbsp;&nbsp;</span>Riemannのゼータ函数の積分表示1</a></span></li><li><span><a href=\"#Hurwitzのゼータ函数の積分表示1\" data-toc-modified-id=\"Hurwitzのゼータ函数の積分表示1-2.3.2\"><span class=\"toc-item-num\">2.3.2&nbsp;&nbsp;</span>Hurwitzのゼータ函数の積分表示1</a></span></li><li><span><a href=\"#Riemannのゼータ函数の積分表示2と函数等式\" data-toc-modified-id=\"Riemannのゼータ函数の積分表示2と函数等式-2.3.3\"><span class=\"toc-item-num\">2.3.3&nbsp;&nbsp;</span>Riemannのゼータ函数の積分表示2と函数等式</a></span></li></ul></li><li><span><a href=\"#ベータ函数とガンマ函数の関係\" data-toc-modified-id=\"ベータ函数とガンマ函数の関係-2.4\"><span class=\"toc-item-num\">2.4&nbsp;&nbsp;</span>ベータ函数とガンマ函数の関係</a></span><ul class=\"toc-item\"><li><span><a href=\"#方法1:-置換積分と積分の順序交換のみを使う方法\" data-toc-modified-id=\"方法1:-置換積分と積分の順序交換のみを使う方法-2.4.1\"><span class=\"toc-item-num\">2.4.1&nbsp;&nbsp;</span>方法1: 置換積分と積分の順序交換のみを使う方法</a></span></li><li><span><a href=\"#方法2:-極座標変換を使う方法\" data-toc-modified-id=\"方法2:-極座標変換を使う方法-2.4.2\"><span class=\"toc-item-num\">2.4.2&nbsp;&nbsp;</span>方法2: 極座標変換を使う方法</a></span></li><li><span><a href=\"#方法3:-y-=-tx-と変数変換する方法\" data-toc-modified-id=\"方法3:-y-=-tx-と変数変換する方法-2.4.3\"><span class=\"toc-item-num\">2.4.3&nbsp;&nbsp;</span>方法3: y = tx と変数変換する方法</a></span></li><li><span><a href=\"#ベータ函数とガンマ函数の関係の簡単な計算問題への応用\" data-toc-modified-id=\"ベータ函数とガンマ函数の関係の簡単な計算問題への応用-2.4.4\"><span class=\"toc-item-num\">2.4.4&nbsp;&nbsp;</span>ベータ函数とガンマ函数の関係の簡単な計算問題への応用</a></span></li><li><span><a href=\"#B(s,-1/2)の級数展開\" data-toc-modified-id=\"B(s,-1/2)の級数展開-2.4.5\"><span class=\"toc-item-num\">2.4.5&nbsp;&nbsp;</span>B(s, 1/2)の級数展開</a></span></li></ul></li><li><span><a href=\"#ガンマ函数の無限積表示\" data-toc-modified-id=\"ガンマ函数の無限積表示-2.5\"><span class=\"toc-item-num\">2.5&nbsp;&nbsp;</span>ガンマ函数の無限積表示</a></span></li><li><span><a href=\"#sinとガンマ函数の関係\" data-toc-modified-id=\"sinとガンマ函数の関係-2.6\"><span class=\"toc-item-num\">2.6&nbsp;&nbsp;</span>sinとガンマ函数の関係</a></span><ul class=\"toc-item\"><li><span><a href=\"#sinの無限積表示\" data-toc-modified-id=\"sinの無限積表示-2.6.1\"><span class=\"toc-item-num\">2.6.1&nbsp;&nbsp;</span>sinの無限積表示</a></span></li><li><span><a href=\"#Euler's-reflection-formula\" data-toc-modified-id=\"Euler's-reflection-formula-2.6.2\"><span class=\"toc-item-num\">2.6.2&nbsp;&nbsp;</span>Euler's reflection formula</a></span></li></ul></li><li><span><a href=\"#Wallisの公式\" data-toc-modified-id=\"Wallisの公式-2.7\"><span class=\"toc-item-num\">2.7&nbsp;&nbsp;</span>Wallisの公式</a></span></li><li><span><a href=\"#Legendre's-duplication-formula\" data-toc-modified-id=\"Legendre's-duplication-formula-2.8\"><span class=\"toc-item-num\">2.8&nbsp;&nbsp;</span>Legendre's duplication formula</a></span></li><li><span><a href=\"#sinとガンマ函数の関係の再証明\" data-toc-modified-id=\"sinとガンマ函数の関係の再証明-2.9\"><span class=\"toc-item-num\">2.9&nbsp;&nbsp;</span>sinとガンマ函数の関係の再証明</a></span><ul class=\"toc-item\"><li><span><a href=\"#Euler's-reflection-formula-の再証明\" data-toc-modified-id=\"Euler's-reflection-formula-の再証明-2.9.1\"><span class=\"toc-item-num\">2.9.1&nbsp;&nbsp;</span>Euler's reflection formula の再証明</a></span></li><li><span><a href=\"#sinの無限積表示の再証明\" data-toc-modified-id=\"sinの無限積表示の再証明-2.9.2\"><span class=\"toc-item-num\">2.9.2&nbsp;&nbsp;</span>sinの無限積表示の再証明</a></span></li></ul></li><li><span><a href=\"#Lerchの定理とゼータ正規化積\" data-toc-modified-id=\"Lerchの定理とゼータ正規化積-2.10\"><span class=\"toc-item-num\">2.10&nbsp;&nbsp;</span>Lerchの定理とゼータ正規化積</a></span><ul class=\"toc-item\"><li><span><a href=\"#Lerchの定理-(Hurwitzのゼータ函数とガンマ函数の関係)\" data-toc-modified-id=\"Lerchの定理-(Hurwitzのゼータ函数とガンマ函数の関係)-2.10.1\"><span class=\"toc-item-num\">2.10.1&nbsp;&nbsp;</span>Lerchの定理 (Hurwitzのゼータ函数とガンマ函数の関係)</a></span></li><li><span><a href=\"#ゼータ正規化積\" data-toc-modified-id=\"ゼータ正規化積-2.10.2\"><span class=\"toc-item-num\">2.10.2&nbsp;&nbsp;</span>ゼータ正規化積</a></span></li></ul></li></ul></li><li><span><a href=\"#Stirlingの公式とLaplaceの方法\" data-toc-modified-id=\"Stirlingの公式とLaplaceの方法-3\"><span class=\"toc-item-num\">3&nbsp;&nbsp;</span>Stirlingの公式とLaplaceの方法</a></span><ul class=\"toc-item\"><li><span><a href=\"#Stirlingの公式\" data-toc-modified-id=\"Stirlingの公式-3.1\"><span class=\"toc-item-num\">3.1&nbsp;&nbsp;</span>Stirlingの公式</a></span></li><li><span><a href=\"#Wallisの公式のStirlingの公式を使った証明\" data-toc-modified-id=\"Wallisの公式のStirlingの公式を使った証明-3.2\"><span class=\"toc-item-num\">3.2&nbsp;&nbsp;</span>Wallisの公式のStirlingの公式を使った証明</a></span></li><li><span><a href=\"#Gauss's-multiplication-formula\" data-toc-modified-id=\"Gauss's-multiplication-formula-3.3\"><span class=\"toc-item-num\">3.3&nbsp;&nbsp;</span>Gauss's multiplication formula</a></span></li><li><span><a href=\"#Laplaceの方法\" data-toc-modified-id=\"Laplaceの方法-3.4\"><span class=\"toc-item-num\">3.4&nbsp;&nbsp;</span>Laplaceの方法</a></span></li><li><span><a href=\"#Laplaceの方法の弱形\" data-toc-modified-id=\"Laplaceの方法の弱形-3.5\"><span class=\"toc-item-num\">3.5&nbsp;&nbsp;</span>Laplaceの方法の弱形</a></span></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"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Gauss積分"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Gauss積分の公式\n",
    "\n",
    "$$\n",
    "\\int_{-\\infty}^\\infty e^{-x^2}\\,dx = \\sqrt{\\pi}\n",
    "$$\n",
    "\n",
    "を**Gauss積分の公式**と呼ぶことにする. 証明は後で行う.\n",
    "\n",
    "このノートの筆者は大学新入生が習う積分の公式の中でこれが**最も重要**であると考えている. ガウス積分が重要だと考える理由は以下の通り.\n",
    "\n",
    "(1) この公式自体が非常に面白い形をしている. 左辺を見てもどこにも円周率は見えないが, 右辺には円周率が出て来る. しかも円周率がそのまま出て来るのではなく, その平方根が出て来る.\n",
    "\n",
    "(2) 様々な方法を使ってGauss積分の公式を証明できる.\n",
    "\n",
    "(3) Gauss積分の公式は確率論や統計学で正規分布を扱うときには必須である.  正規分布は中心極限定理によって特別に重要な役目を果たす確率分布である. \n",
    "\n",
    "(4) Gauss積分はガンマ函数に一般化される. \n",
    "\n",
    "(5) Gauss積分はLaplaceの方法の基礎である.  Laplaceの方法はある種の積分の漸近挙動を調べるための最も基本的な方法であり, 解析学の応用において基本的かつ重要である.\n",
    "\n",
    "(6) 特にGauss積分で階乗に等しい積分を近似することによって, Stirlingの公式が得られる.  (Stirlingの公式 $n!\\sim n^n e^{-n}\\sqrt{2\\pi n}$ の平行根の因子はGauss積分を経由して得られる.)\n",
    "\n",
    "以上のようにGauss積分は純粋数学的にも応用数学的にも基本的かつ重要である."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Gauss積分を使う簡単な計算例"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題:** 上の公式を使って, $a>0$ のとき,\n",
    "\n",
    "$$\n",
    "\\int_{-\\infty}^\\infty e^{-y^2/a}\\,dy = \\sqrt{a\\pi}\n",
    "$$\n",
    "\n",
    "となることを示せ. \n",
    "\n",
    "**注意:** $a$ を $1/a$ で置き換えれば\n",
    "\n",
    "$$\n",
    "\\int_{-\\infty}^\\infty e^{-ay^2}\\,dy = \\sqrt{\\frac{\\pi}{a}}\n",
    "$$\n",
    "\n",
    "も得られる.\n",
    "\n",
    "**解答例:** Gauss積分の公式で $\\ds x=\\frac{y}{\\sqrt{a}}$ と置換積分すると\n",
    "\n",
    "$$\n",
    "\\sqrt{\\pi} = \\int_{-\\infty}^\\infty e^{-x^2}\\,dx  =\n",
    "\\frac{1}{\\sqrt{a}}\\int_{-\\infty}^\\infty e^{-y^2/a}\\,dy\n",
    "$$\n",
    "\n",
    "なので, 両辺に $\\sqrt{a}$ をかければ示したい公式が得られる. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題:** 分散 $\\sigma^2>0$, 平均 $\\mu$ の正規分布の確率密度函数 $p(x)$ が\n",
    "\n",
    "$$\n",
    "p(x) = \\frac{1}{\\sqrt{2\\pi\\sigma^2}} e^{-(x-\\mu)^2/(2\\sigma^2)}\n",
    "$$\n",
    "\n",
    "で定義される. このとき\n",
    "\n",
    "$$\n",
    "\\int_{-\\infty}^\\infty p(x)\\,dx = 1\n",
    "$$\n",
    "\n",
    "となることを示せ. (この問題より, 確率統計学においてGauss積分の公式は必須であることがわかる.)\n",
    "\n",
    "**解答例:** $x=y+\\mu$ と置換し, 上の問題の結果を使うと, \n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\int_{-\\infty}^\\infty p(x)\\,dx &=\n",
    "\\frac{1}{\\sqrt{2\\pi\\sigma^2}} \\int_{-\\infty}^\\infty e^{-(x-\\mu)^2/(2\\sigma^2)}\\,dx =\n",
    "\\frac{1}{\\sqrt{2\\pi\\sigma^2}} \\int_{-\\infty}^\\infty e^{-y^2/(2\\sigma^2)}\\,dy \n",
    "\\\\ &=\n",
    "\\frac{1}{\\sqrt{2\\pi\\sigma^2}} \\sqrt{2\\sigma^2\\pi} = 1.\n",
    "\\qquad \\QED\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題(Lebesgueの収束定理の結論が成立しない場合2):** 函数列 $f_n(x)$ を\n",
    "\n",
    "$$\n",
    "f_n(x)=\\frac{1}{\\sqrt{n\\pi}}e^{-x^2/n}\n",
    "$$\n",
    "\n",
    "と定める. 以下を示せ.\n",
    "\n",
    "(1) $\\ds\\int_{-\\infty}^\\infty f_n(x)\\,dx = 1$.\n",
    "\n",
    "(2) 各 $x\\in\\R$ ごとに $\\ds\\lim_{n\\to\\infty}f_n(x)= 0$.\n",
    "\n",
    "(3) したがって $\\ds\\lim_{n\\to\\infty}\\int_{-\\infty}^\\infty f_n(x)\\,dx \\ne \\int_{-\\infty}^\\infty \\lim_{n\\to\\infty}f_n(x)\\,dx$.\n",
    "\n",
    "**解答例:** (1)はGauss積分の公式から得られる(詳細は自分で計算して確認せよ). (3)は(1)と(2)からただちに得られるので, あとは(2)のみを示せば十分である. $x\\in\\R$ を任意に取って固定する. このとき $n\\to\\infty$ で $\\dfrac{x^2}{n}\\to 0$, $\\dfrac{1}{\\sqrt{n\\pi}}\\to 0$ となるので, $f_n(x)\\to 0$ となることもわかる. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題:** すぐ上の問題の函数 $f_n(x)$ のグラフを描いてみよ. \n",
    "\n",
    "**解答例:** 以下のセルのようになる. \n",
    "\n",
    "$n$ が大きくなると, $f_n(x)$ の「分布」は広く拡がる. $\\QED$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
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       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:start;\" transform=\"rotate(0, 319.835, 85.671)\" x=\"319.835\" y=\"85.671\">n = 4</text>\n",
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       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:start;\" transform=\"rotate(0, 319.835, 100.791)\" x=\"319.835\" y=\"100.791\">n = 5</text>\n",
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       "<g clip-path=\"url(#clip8100)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:start;\" transform=\"rotate(0, 319.835, 115.911)\" x=\"319.835\" y=\"115.911\">n = 10</text>\n",
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       "<g clip-path=\"url(#clip8100)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:start;\" transform=\"rotate(0, 319.835, 131.031)\" x=\"319.835\" y=\"131.031\">n = 30</text>\n",
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       "  \"/>\n",
       "<g clip-path=\"url(#clip8100)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:start;\" transform=\"rotate(0, 319.835, 146.151)\" x=\"319.835\" y=\"146.151\">n = 100</text>\n",
       "</g>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(n,x) = exp(-x^2/n)/√(n*π)\n",
    "x = -10.0:0.05:10.0\n",
    "P = plot(size=(400,250))\n",
    "for n in [1,2,3,4,5,10, 30, 100]\n",
    "    plot!(x, f.(n,x), label=\"n = $n\")\n",
    "end\n",
    "P"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題:** 次を示せ: $a>0$ と $k=0,1,2,\\ldots$ について\n",
    "\n",
    "$$\n",
    "\\int_{-\\infty}^\\infty e^{-ax^2}x^{2k}\\,dx = \n",
    "\\sqrt{\\pi}\\; \\frac{1\\cdot3\\cdots(2k-1)}{2^k} a^{-(2k+1)/2} =\n",
    "\\sqrt{\\pi}\\; \\frac{(2k)!}{2^{2k}k!} a^{-(2k+1)/2}.\n",
    "\\tag{1}\n",
    "$$\n",
    "\n",
    "**注意:** $a$ を $1/a$ で置き換えれば次も得られる:\n",
    "\n",
    "$$\n",
    "\\int_{-\\infty}^\\infty e^{-x^2/a}x^{2k}\\,dx = \n",
    "\\frac{1\\cdot3\\cdots(2k-1)}{2^k} \\sqrt{a^{2k+1}\\pi} =\n",
    "\\frac{(2k)!}{2^{2k}k!} \\sqrt{a^{2k+1}\\pi}.\n",
    "\\tag{2}\n",
    "$$\n",
    "\n",
    "**解答例:** Gauss積分の公式から得られる公式\n",
    "\n",
    "$$\n",
    "\\int_{-\\infty}^\\infty e^{-ax^2}\\,dx = \\sqrt{\\pi}\\;a^{-1/2}\n",
    "$$\n",
    "\n",
    "の両辺を $a$ で微分して $-1$ 倍する操作を繰り返すと((K)を使う),\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "&\n",
    "\\int_{-\\infty}^\\infty e^{-ax^2}x^2\\,dx = \\sqrt{\\pi}\\;\\frac{1}{2}a^{-3/2},\n",
    "\\\\ &\n",
    "\\int_{-\\infty}^\\infty e^{-ax^2}x^4\\,dx = \\sqrt{\\pi}\\;\\frac{1}{2}\\frac{3}{2}a^{-5/2},\n",
    "\\\\ &\n",
    "\\int_{-\\infty}^\\infty e^{-ax^2}x^6\\,dx = \\sqrt{\\pi}\\;\\frac{1}{2}\\frac{3}{2}\\frac{5}{2}a^{-7/2}.\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "$k$ 回その操作を繰り返すと, \n",
    "\n",
    "$$\n",
    "\\int_{-\\infty}^\\infty e^{-ax^2}x^{2k}\\,dx = \n",
    "\\sqrt{\\pi}\\;\\frac{1}{2}\\frac{3}{2}\\cdots\\frac{2k-1}{2}a^{-(2k+1)/2}.\n",
    "$$\n",
    "\n",
    "これより, (1)の前半が成立することがわかる. 後半の成立は\n",
    "\n",
    "$$\n",
    "\\frac{1\\cdot3\\cdots(2k-1)}{2^k} =\n",
    "\\frac{1\\cdot3\\cdots(2k-1)}{2^k} \\frac{2\\cdot4\\cdots(2k)}{2^k k!} =\n",
    "\\frac{(2k)!}{2^{2k}k!}\n",
    "\\tag{3}\n",
    "$$\n",
    "\n",
    "によって確認できる. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**注意:** 奇数の積 $1\\cdot3\\cdots(2k-1)$ について(3)の計算法はよく使われる:\n",
    "\n",
    "$$\n",
    "1\\cdot3\\cdots(2k-1) = \n",
    "1\\cdot3\\cdots(2k-1) \\frac{2\\cdot4\\cdots(2k)}{2^k k!} =\n",
    "\\frac{(2k)!}{2^k k!}.\n",
    "$$\n",
    "\n",
    "例えば二項係数に関する\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "(-1)^k\\binom{-1/2}{k} &=\n",
    "(-1)^k\\frac{(-1/2)(-3/2)\\cdots(-(2k-1)/2)}{k!} \\\\ &=\n",
    "\\frac{1\\cdot3\\cdots(2k-1)}{2^k k!} =\n",
    "\\frac{(2k)!}{2^{2k}k!k!} =\n",
    "\\frac{1}{2^{2k}}\\binom{2k}{k}\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "もよく出て来る. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Gauss分布のFourier変換"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$a>0$ であるとする.  $e^{-x^2/a}$ 型の函数を**Gauss分布函数**と呼ぶことがある.\n",
    "\n",
    "一般に函数 $f(x)$ に対して,\n",
    "\n",
    "$$\n",
    "\\hat{f}(p) = \\int_{-\\infty}^\\infty e^{-ipx} f(x)\\,dx\n",
    "$$\n",
    "\n",
    "を $f$ の**Fourier変換**(フーリエ変換)と呼ぶ. もしも実数値函数 $f(x)$ が偶函数であれば, \n",
    "\n",
    "$$\n",
    "e^{-ipx} f(x) = f(x)\\cos(px) - i f(x)\\sin(px)\n",
    "$$\n",
    "\n",
    "の虚部は奇函数になり, その積分は消えるので\n",
    "\n",
    "$$\n",
    "\\hat{f}(p) = \\int_{-\\infty}^\\infty f(x)\\cos(px)\\,dx\n",
    "$$\n",
    "\n",
    "となる."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題:** $a>0$ とする. $f(x)=e^{-x^2/a}$ のFourier変換を求めよ.\n",
    "\n",
    "**解答例1:** $\\ds\\cos(px)=\\sum_{k=0}^\\infty\\frac{(-p^2)^k x^{2k}}{(2k)!}$ より,\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "\\hat{f}(p) &=\n",
    "\\int_{-\\infty}^\\infty e^{-x^2/a} \\cos(px)\\,dx =\n",
    "\\sum_{k=0}^\\infty\\frac{(-p^2)^k}{(2k)!}\\int_{-\\infty}^\\infty e^{-x^2/a}x^{2k}\\,dx\n",
    "\\\\ &=\n",
    "\\sum_{k=0}^\\infty\\frac{(-p^2)^k}{(2k)!}\\frac{(2k)!}{2^{2k}k!} \\sqrt{a^{2k+1}\\pi} =\n",
    "\\sqrt{a\\pi}\\sum_{k=0}^\\infty\\frac{(-ap^2/4)^k}{k!} = \\sqrt{a\\pi}\\;e^{-ap^2/4}.\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "3つ目の等号で上の方の問題の結果を用いた. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**解答例2:** 複素解析を用いる. 複素解析さえ認めて使えば, 形式的によりわかり易く計算できる.\n",
    "\n",
    "$$\n",
    "-\\frac{x^2}{a}-ipx = \n",
    "-\\frac{1}{a}\\left(x^2 + iapx\\right) =\n",
    "-\\frac{1}{a}\\left(\\left(x+\\frac{iap}{2}\\right)^2-\\frac{-a^2p^2}{4}\\right) =\n",
    "-\\frac{1}{a}\\left(x+\\frac{iap}{2}\\right)^2 - \\frac{ap^2}{4}\n",
    "$$\n",
    "\n",
    "と平方完成し, $\\ds x=y-\\frac{iap}{2}$ と置換すると,\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\hat{f}(p) &= \\int_{-\\infty}^\\infty e^{-x^2/a}e^{-ipx}\\,dx =\n",
    "\\int_{-\\infty}^\\infty e^{-(x^2/a+ipx)}\\,dx \n",
    "\\\\ &=\n",
    "\\int_{-\\infty}^\\infty \\exp\\left(-\\frac{1}{a}\\left(x+\\frac{iap}{2}\\right)^2 - \\frac{ap^2}{4}\\right)\\,dx =\n",
    "e^{-ap^2/4} \\int_{-\\infty+iap/2}^{\\infty+iap/2} e^{-y^2/a}\\,dy.\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "Cauchyの積分定理より,\n",
    "\n",
    "$$\n",
    "\\int_{-\\infty+iap/2}^{\\infty+iap/2} e^{-y^2/a}\\,dy =\n",
    "\\int_{-\\infty}^{\\infty} e^{-y^2/a}\\,dy = \\sqrt{a\\pi}.\n",
    "$$\n",
    "\n",
    "したがって, \n",
    "\n",
    "$$\n",
    "\\hat{f}(p) = \\int_{-\\infty}^\\infty e^{-x^2/a}e^{-ipx}\\,dx = \\sqrt{a\\pi}\\;e^{-ap^2/4}.\n",
    "\\qquad \\QED\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**補足:** 複素平面上の経路 $C$ を次のように定める: まず $-R$ から $R$ に直線的に移動する. 次に $R$ から $R+iap/2$ に直線的に移動する. その次に $R+iap/2$ から $-R+iap/2$ に直線的に移動する. 最後に $-R+iap/2$ から $-R$ に直線的に移動する. これによって得られる長方形型の経路が $C$ である. 上の解答例2の中のCauchyの積分定理をこの経路 $C_R$ に適用した場合を使っている. $R\\to\\infty$ とすると, 左右の縦方向に移動する経路上での積分が $0$ に収束することを使う. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$e^{-ax^2}$ のFourier変換については\n",
    "\n",
    "* 黒木玄, <a href=\"https://genkuroki.github.io/documents/20160501StirlingFormula.pdf\">ガンマ分布の中心極限定理とStirlingの公式</a>\n",
    "\n",
    "の第6節も参照せよ."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Gauss積分の公式の導出"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Gauss積分の計算の仕方については\n",
    "\n",
    "* 黒木玄, <a href=\"https://genkuroki.github.io/documents/20160501StirlingFormula.pdf\">ガンマ分布の中心極限定理とStirlingの公式</a>\n",
    "\n",
    "の第7節および\n",
    "\n",
    "* 高木貞治, 解析概論, 岩波書店 (1983)\n",
    "\n",
    "の<a href=\"https://twitter.com/genkuroki/status/1018654177164083201\">第3章§35の例5,6</a>を参照せよ."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\ds I = \\int_{-\\infty}^\\infty e^{-x^2}\\,dx$ とおく. $I=\\sqrt{\\pi}$ であることを示したい. そのためには\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "I^2 &= \\int_{-\\infty}^\\infty e^{-x^2}\\,dx\\cdot \\int_{-\\infty}^\\infty e^{-y^2}\\,dy\n",
    "\\\\ &=\n",
    "\\int_{-\\infty}^\\infty \\left(\\int_{-\\infty}^\\infty e^{-x^2}\\,dx\\right)e^{-y^2}\\,dy =\n",
    "\\int_{-\\infty}^\\infty\\left(\\int_{-\\infty}^\\infty e^{-(x^2+y^2)}\\,dx\\right)\\,dy\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "が $\\pi$ に等しいことを証明すればよい. 上の計算の2つ目と3つ目の等号で積分の線形性(A)を用いた."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 方法1: 高さ $z$ で輪切りにする方法\n",
    "\n",
    "$\\ds I^2 = \\int_{-\\infty}^\\infty\\left(\\int_{-\\infty}^\\infty e^{-(x^2+y^2)}\\,dx\\right)\\,dy$ は2変数函数 $z=e^{-(x^2+y^2)}$ の $xyz$ 空間内のグラフと $xy$ 平面 $z=0$ のあいだに挟まれた山型の領域の体積を意味する. \n",
    "\n",
    "なぜならば, $S(y) = \\ds \\int_{-\\infty}^\\infty e^{-(x^2+y^2)}\\,dx$ はその領域の $y$ を固定したときの切断面の面積に等しく, $\\int_{-\\infty}^\\infty S(y)\\,dy$ はその切断面の面積の積分なので領域全体の体積に等しいからである.  一般に, 長さを積分すれば面積になり、面積を積分すれば体積になる.\n",
    "\n",
    "その山型の領域の体積は高さ $z$ での切断面の面積の $z=0$ から $z=1$ までの積分に等しい.  高さ $z$ での切断面は半径 $\\sqrt{x^2+y^2}=\\sqrt{-\\log z}$ の円盤になり, その面積は $-\\pi\\log z$ になる. ゆえに\n",
    "\n",
    "$$\n",
    "I^2 = \\int_0^1 (-\\pi\\log z)\\,dz = -\\pi\\,[z\\log z - z]_0^1 = \\pi.\n",
    "$$\n",
    "\n",
    "これより $I=\\int_{-\\infty}^\\infty e^{-x^2}\\,dx = \\sqrt{\\pi}$ であることがわかる."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 方法2: 極座標を使う方法\n",
    "\n",
    "以下の方法は2重積分の積分変数の変換の仕方(Jacobianが出て来る)を知っておかなければ使えない. \n",
    "\n",
    "$x=r\\cos\\theta$, $y=r\\sin\\theta$ とおくと,\n",
    "\n",
    "$$\n",
    "I^2 = \\int_0^{2\\pi}d\\theta\\int_0^\\infty r e^{-r^2}\\,dr =\n",
    "2\\pi \\left[\\frac{e^{-r^2}}{-2}\\right]_0^\\infty = 2\\pi\\frac{1}{2}=\\pi.\n",
    "$$\n",
    "\n",
    "ゆえに $I=\\sqrt{\\pi}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 方法3: $y=x \\tan\\theta$ と変数変換する方法 \n",
    "\n",
    "$I^2$ は次のようにも表せる:\n",
    "\n",
    "$$\n",
    "I^2 = 2\\int_0^\\infty\\left(\\int_{-\\infty}^\\infty e^{-(x^2+y^2)}\\,dy\\right)\\,dx.\n",
    "$$\n",
    "\n",
    "この積分内で $x$ は $x>0$ を動くと考える.\n",
    "\n",
    "内側の積分で積分変数を $-\\infty<y<\\infty$ から $-\\pi/2<\\theta<\\pi/2$ に $y=x\\tan\\theta$ によって変換すると,\n",
    "\n",
    "$$\n",
    "\\cos\\theta > 0, \\quad\n",
    "dy = \\frac{x}{\\cos^2\\theta}\\,d\\theta, \\quad\n",
    "x^2+y^2 = x^2(1+\\tan^2\\theta) = \\frac{x^2}{\\cos^2\\theta}\n",
    "$$\n",
    "\n",
    "なので\n",
    "\n",
    "$$\n",
    "I^2 = 2 \\int_0^\\infty\\left(\\int_{-\\pi/2}^{\\pi/2} \\exp\\left(-\\frac{x^2}{\\cos^2\\theta}\\right)\\frac{x}{\\cos^2\\theta}\\,d\\theta\\right)\\,dx.\n",
    "$$\n",
    "\n",
    "ゆえに積分の順序を交換すると((J)を使う),\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "I^2 &= 2 \\int_{-\\pi/2}^{\\pi/2}\\left(\\int_0^\\infty \\exp\\left(-\\frac{x^2}{\\cos^2\\theta}\\right)\\frac{x}{\\cos^2\\theta}\\,dx\\right)\\,d\\theta\n",
    "\\\\ &=\n",
    "2 \\int_{-\\pi/2}^{\\pi/2}\\left[\\frac{1}{-2}\\exp\\left(-\\frac{x^2}{\\cos^2\\theta}\\right)\\right]_{x=0}^{x=\\infty}\\,d\\theta =\n",
    "2 \\int_{-\\pi/2}^{\\pi/2}\\frac{1}{2}\\,d\\theta = 2\\frac{\\pi}{2} = \\pi.\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "したがって $I=\\sqrt{\\pi}$.\n",
    "\n",
    "**注意:** 極座標変換 $(x,y)=(r\\cos\\theta, r\\sin\\theta)$ が有効な場面では, $y=x\\tan\\theta$ という変数変換も有効なことが多い. $\\tan\\theta$ の幾何的な意味は「原点を通る直線の傾き」であった. その意味でも $y=x\\tan\\theta$ は自然な変数変換だと言える. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## ガンマ函数とベータ函数"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### ガンマ函数とベータ函数の定義\n",
    "\n",
    "$s>0$, $p>0$, $q>0$ と仮定する. $\\Gamma(s)$ と $B(p,q)$ を次の積分で定義する:\n",
    "\n",
    "$$\n",
    "\\Gamma(s) = \\int_0^\\infty e^{-x}x^{s-1}\\,dx, \\quad\n",
    "B(p,q) = \\int_0^1 x^{p-1}(1-x)^{q-1}\\,dx.\n",
    "$$\n",
    "\n",
    "$\\Gamma(s)$ をガンマ函数と, $B(p,q)$ をベータ函数と呼ぶ."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題(ガンマ函数のGauss積分型の表示):** 次を示せ:\n",
    "\n",
    "$$\n",
    "\\Gamma(s) = 2\\int_0^\\infty e^{-y^2} y^{2s-1}\\,dy.\n",
    "$$\n",
    "\n",
    "**解答例:** ガンマ函数の積分による定義式において $x=y^2$ と置換すると, \n",
    "\n",
    "$$\n",
    "\\Gamma(s) = \\int_0^\\infty e^{-y^2} y^{2s-2}\\,2y\\,dy = 2\\int_0^\\infty e^{-y^2} y^{2s-1}\\,dy.\n",
    "\\qquad\\QED\n",
    "$$\n",
    "\n",
    "**注意:** この公式より, ガンマ函数は本質的にGauss積分の一般化になっていることがわかる. $\\QED$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\Gamma\\left(s\\right)$$"
      ],
      "text/plain": [
       "Gamma(s)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = symbols(\"y\")\n",
    "s = symbols(\"s\", positive=true)\n",
    "2*integrate(e^(-y^2)*y^(2s-1), (y,0,oo))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題:** 次を示せ: $r>0$ について\n",
    "\n",
    "$$\n",
    "\\int_0^\\infty e^{-x^r}\\,dx = \n",
    "\\frac{1}{r}\\Gamma\\left(\\frac{1}{r}\\right).\n",
    "$$\n",
    "\n",
    "**略解:** $x=t^{1/r}$ と置換すればただちに得られる. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**注意:** ガンマ函数の函数等式(下の方で示す)もしくは部分積分によって $\\ds \\frac{1}{r}\\Gamma\\left(\\frac{1}{r}\\right)=\\Gamma\\left(1+\\frac{1}{r}\\right)$ が成立することもわかる. $\\QED$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\Gamma\\left(1 + \\frac{1}{r}\\right)$$"
      ],
      "text/plain": [
       "     /    1\\\n",
       "Gamma|1 + -|\n",
       "     \\    r/"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = symbols(\"x\")\n",
    "r = symbols(\"r\", positive=true)\n",
    "integrate(e^(-x^r), (x,0,oo))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題(ガンマ函数のスケール変換):** 次を示せ:\n",
    "\n",
    "$$\n",
    "\\int_0^\\infty e^{-x/\\theta}x^{s-1}\\,dx = \\theta^s\\Gamma(s) \\quad (\\theta>0,\\ s>0).\n",
    "$$\n",
    "\n",
    "ガンマ函数はこの形式でも非常によく使われる.\n",
    "\n",
    "**解答例:** $x=\\theta y$ と置換すると, $x^{s-1}\\,dx = \\theta^s y^{s-1}\\,dy$ なので示したい公式が得られる. $\\QED$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$t^{s} \\Gamma\\left(s\\right)$$"
      ],
      "text/plain": [
       " s         \n",
       "t *Gamma(s)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = symbols(\"x\")\n",
    "s = symbols(\"s\", positive=true)\n",
    "t = symbols(\"t\", positive=true)\n",
    "simplify(integrate(e^(-x/t)*x^(s-1), (x,0,oo)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題(ベータ函数の別の表示):** 次を示せ:\n",
    "\n",
    "$$\n",
    "B(p,q) = \n",
    "2\\int_0^{\\pi/2} (\\cos\\theta)^{2p-1}(\\sin\\theta)^{2q-1}\\,d\\theta =\n",
    "\\int_0^\\infty \\frac{t^{p-1}}{(1+t)^{p+q}}\\,dt =\n",
    "\\frac{1}{p}\\int_0^\\infty \\frac{du}{(1+u^{1/p})^{p+q}}.\n",
    "$$\n",
    "\n",
    "ベータ函数のこれらの表示もよく使われる.\n",
    "\n",
    "**解答例:** $B(p,q)=\\int_0^1 x^{p-1}(1-x)^{q-1}\\,dx$ で $x=\\cos^2\\theta$ と置換すると,\n",
    "\n",
    "$$\n",
    "dx = -2\\cos\\theta\\;\\sin\\theta\\;d\\theta\n",
    "$$\n",
    "\n",
    "より, \n",
    "\n",
    "$$\n",
    "B(p,q) = 2\\int_0^{\\pi/2} (\\cos\\theta)^{2p-1}(\\sin\\theta)^{2q-1}\\,d\\theta.\n",
    "$$\n",
    "\n",
    "$B(p,q)=\\int_0^1 x^{p-1}(1-x)^{q-1}\\,dx$ で $\\ds x=\\frac{t}{1+t}=1-\\frac{1}{1+t}$ と置換すると,\n",
    "\n",
    "$$\n",
    "dx = \\frac{dt}{1+t}\n",
    "$$\n",
    "\n",
    "より, \n",
    "\n",
    "$$\n",
    "B(p,q) = \n",
    "\\int_0^\\infty \\left(\\frac{t}{1+t}\\right)^{p-1} \\left(\\frac{1}{1+t}\\right)^{q-1}\\,\\frac{dt}{(1+t)^2} =\n",
    "\\int_0^\\infty \\frac{t^{p-1}}{(1+t)^{p+q}}\\,dt\n",
    "$$\n",
    "\n",
    "さらに $t=u^{1/p}$ と置換すると, \n",
    "\n",
    "$$\n",
    "t^{p-1}\\,dt = \\frac{1}{p} \\, du\n",
    "$$\n",
    "\n",
    "より, \n",
    "\n",
    "$$\n",
    "B(p,q) = \\int_0^\\infty \\frac{t^{p-1}}{(1+t)^{p+q}}\\,dt =\n",
    "\\frac{1}{p}\\int_0^\\infty \\frac{du}{(1+u^{1/p})^{p+q}}.\n",
    "\\qquad \\QED\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題:** 次を示せ. $a<b$, $p>0$, $q>0$ のとき,\n",
    "\n",
    "$$\n",
    "\\int_a^b (x-a)^{p-1}(b-x)^{q-1}\\,dx = (b-a)^{p+q-1} B(p,q).\n",
    "$$\n",
    "\n",
    "**証明:** $x=(1-t)a+tb=a+(b-a)t$ と積分変数を置換すると,\n",
    "\n",
    "$$\n",
    "\\int_a^b (x-a)^{p-1}(b-x)^{q-1}\\,dx =\n",
    "\\int_0^1 ((b-a)t)^{p-1}((b-a)(1-t))^{q-1}(b-a)\\,dt = (b-a)^{p+q-1}B(p,q).\n",
    "\\qquad\\QED\n",
    "$$\n",
    "\n",
    "**例:** $\\ds B(2,2)=\\int_0^1 x(1-x)\\,dx = \\frac{1}{2}-\\frac{1}{3}=\\frac{1}{6}$ なので\n",
    "\n",
    "$$\n",
    "\\int_a^b (x-a)(b-x)\\,dx = (b-a)^3 B(2,2) = \\frac{(b-a)^3}{6}.\n",
    "\\qquad \\QED\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題:** ガンマ函数を定義する積分の被積分函数のグラフを色々な $s>0$ について描いてみよ.\n",
    "\n",
    "**解答例:** 次のセルを見よ. $\\QED$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# ガンマ函数の積分の被積分函数のグラフ\n",
    "\n",
    "f(s,x) = e^(-x)*x^(s-1)\n",
    "x = 0.00:0.05:30.0\n",
    "PP = []\n",
    "for s in [1/2, 1, 2, 3, 6, 10]\n",
    "    P = plot(x, f.(s,x), title=\"s = $s\", titlefontsize=10)\n",
    "    push!(PP, P)\n",
    "end\n",
    "for s in [15, 20, 30]\n",
    "    x = 0:0.02:2.2s\n",
    "    P = plot(x, f.(s,x), title=\"s = $s\", titlefontsize=10)\n",
    "    push!(PP, P)\n",
    "end\n",
    "plot(PP[1:3]..., size=(750, 200), legend=false, layout=@layout([a b c]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
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       "<g clip-path=\"url(#clip8500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:middle;\" transform=\"rotate(0, 667.501, 195.449)\" x=\"667.501\" y=\"195.449\">20</text>\n",
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       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:middle;\" transform=\"rotate(0, 725.128, 195.449)\" x=\"725.128\" y=\"195.449\">30</text>\n",
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       "<g clip-path=\"url(#clip8500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:end;\" transform=\"rotate(0, 539.559, 181.17)\" x=\"539.559\" y=\"181.17\">0</text>\n",
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       "<g clip-path=\"url(#clip8500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:start;\" transform=\"rotate(0, 507.024, 153.123)\" x=\"507.024\" y=\"153.123\">1×10</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip8500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:10; text-anchor:start;\" transform=\"rotate(0, 534.123, 146.27)\" x=\"534.123\" y=\"146.27\">4</text>\n",
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       "<g clip-path=\"url(#clip8500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:10; text-anchor:start;\" transform=\"rotate(0, 534.123, 116.791)\" x=\"534.123\" y=\"116.791\">4</text>\n",
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       "  606.992,37.5998 607.28,37.9825 607.568,38.3983 607.856,38.8465 608.144,39.3265 608.433,39.8374 608.721,40.3788 609.009,40.9498 609.297,41.5497 609.585,42.1779 \n",
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     "execution_count": 7,
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   ],
   "source": [
    "plot(PP[4:6]..., size=(750, 200), legend=false, layout=@layout([a b c]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
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       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:start;\" transform=\"rotate(0, 24.4522, 152.094)\" x=\"24.4522\" y=\"152.094\">2×10</text>\n",
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       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:10; text-anchor:start;\" transform=\"rotate(0, 51.5515, 145.242)\" x=\"51.5515\" y=\"145.242\">9</text>\n",
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       "  78.9799,176.67 79.0708,176.67 79.1616,176.67 79.2525,176.67 79.3434,176.67 79.4342,176.67 79.5251,176.67 79.6159,176.67 79.7068,176.67 79.7977,176.67 \n",
       "  79.8885,176.67 79.9794,176.67 80.0703,176.67 80.1611,176.67 80.252,176.67 80.3428,176.67 80.4337,176.67 80.5246,176.67 80.6154,176.67 80.7063,176.67 \n",
       "  80.7971,176.67 80.888,176.669 80.9789,176.669 81.0697,176.669 81.1606,176.669 81.2514,176.669 81.3423,176.669 81.4332,176.669 81.524,176.669 81.6149,176.669 \n",
       "  81.7058,176.669 81.7966,176.668 81.8875,176.668 81.9783,176.668 82.0692,176.668 82.1601,176.668 82.2509,176.668 82.3418,176.667 82.4326,176.667 82.5235,176.667 \n",
       "  82.6144,176.667 82.7052,176.666 82.7961,176.666 82.887,176.666 82.9778,176.665 83.0687,176.665 83.1595,176.665 83.2504,176.664 83.3413,176.664 83.4321,176.663 \n",
       "  83.523,176.663 83.6138,176.662 83.7047,176.662 83.7956,176.661 83.8864,176.661 83.9773,176.66 84.0681,176.659 84.159,176.659 84.2499,176.658 84.3407,176.657 \n",
       "  84.4316,176.656 84.5225,176.655 84.6133,176.654 84.7042,176.653 84.795,176.652 84.8859,176.651 84.9768,176.65 85.0676,176.649 85.1585,176.647 85.2493,176.646 \n",
       "  85.3402,176.645 85.4311,176.643 85.5219,176.642 85.6128,176.64 85.7036,176.638 85.7945,176.636 85.8854,176.634 85.9762,176.632 86.0671,176.63 86.158,176.628 \n",
       "  86.2488,176.626 86.3397,176.623 86.4305,176.621 86.5214,176.618 86.6123,176.615 86.7031,176.612 86.794,176.609 86.8848,176.606 86.9757,176.602 87.0666,176.599 \n",
       "  87.1574,176.595 87.2483,176.591 87.3391,176.587 87.43,176.583 87.5209,176.579 87.6117,176.574 87.7026,176.57 87.7935,176.565 87.8843,176.56 87.9752,176.554 \n",
       "  88.066,176.549 88.1569,176.543 88.2478,176.537 88.3386,176.531 88.4295,176.524 88.5203,176.517 88.6112,176.51 88.7021,176.503 88.7929,176.495 88.8838,176.488 \n",
       "  88.9747,176.479 89.0655,176.471 89.1564,176.462 89.2472,176.453 89.3381,176.443 89.429,176.434 89.5198,176.423 89.6107,176.413 89.7015,176.402 89.7924,176.391 \n",
       "  89.8833,176.379 89.9741,176.367 90.065,176.354 90.1558,176.341 90.2467,176.328 90.3376,176.314 90.4284,176.3 90.5193,176.285 90.6102,176.27 90.701,176.254 \n",
       "  90.7919,176.238 90.8827,176.221 90.9736,176.203 91.0645,176.185 91.1553,176.167 91.2462,176.148 91.337,176.128 91.4279,176.108 91.5188,176.087 91.6096,176.065 \n",
       "  91.7005,176.043 91.7913,176.02 91.8822,175.996 91.9731,175.972 92.0639,175.947 92.1548,175.921 92.2457,175.895 92.3365,175.868 92.4274,175.839 92.5182,175.811 \n",
       "  92.6091,175.781 92.7,175.75 92.7908,175.719 92.8817,175.687 92.9725,175.654 93.0634,175.62 93.1543,175.585 93.2451,175.549 93.336,175.512 93.4268,175.474 \n",
       "  93.5177,175.435 93.6086,175.395 93.6994,175.354 93.7903,175.312 93.8812,175.269 93.972,175.225 94.0629,175.18 94.1537,175.134 94.2446,175.086 94.3355,175.038 \n",
       "  94.4263,174.988 94.5172,174.937 94.608,174.884 94.6989,174.831 94.7898,174.776 94.8806,174.72 94.9715,174.662 95.0623,174.603 95.1532,174.543 95.2441,174.482 \n",
       "  95.3349,174.419 95.4258,174.354 95.5167,174.289 95.6075,174.221 95.6984,174.153 95.7892,174.082 95.8801,174.01 95.971,173.937 96.0618,173.862 96.1527,173.785 \n",
       "  96.2435,173.707 96.3344,173.627 96.4253,173.546 96.5161,173.463 96.607,173.378 96.6979,173.291 96.7887,173.203 96.8796,173.113 96.9704,173.021 97.0613,172.927 \n",
       "  97.1522,172.831 97.243,172.734 97.3339,172.634 97.4247,172.533 97.5156,172.429 97.6065,172.324 97.6973,172.217 97.7882,172.108 97.879,171.996 97.9699,171.883 \n",
       "  98.0608,171.768 98.1516,171.65 98.2425,171.531 98.3334,171.409 98.4242,171.285 98.5151,171.159 98.6059,171.031 98.6968,170.9 98.7877,170.767 98.8785,170.632 \n",
       "  98.9694,170.495 99.0602,170.355 99.1511,170.213 99.242,170.069 99.3328,169.922 99.4237,169.773 99.5145,169.622 99.6054,169.468 99.6963,169.312 99.7871,169.153 \n",
       "  99.878,168.991 99.9689,168.827 100.06,168.661 100.151,168.492 100.241,168.321 100.332,168.146 100.423,167.97 100.514,167.79 100.605,167.608 100.696,167.424 \n",
       "  100.787,167.236 100.877,167.046 100.968,166.854 101.059,166.658 101.15,166.46 101.241,166.259 101.332,166.055 101.423,165.849 101.513,165.639 101.604,165.427 \n",
       "  101.695,165.212 101.786,164.994 101.877,164.773 101.968,164.55 102.059,164.323 102.15,164.094 102.24,163.862 102.331,163.626 102.422,163.388 102.513,163.147 \n",
       "  102.604,162.903 102.695,162.656 102.786,162.406 102.876,162.153 102.967,161.897 103.058,161.638 103.149,161.376 103.24,161.111 103.331,160.842 103.422,160.571 \n",
       "  103.512,160.297 103.603,160.02 103.694,159.739 103.785,159.456 103.876,159.169 103.967,158.88 104.058,158.587 104.148,158.291 104.239,157.993 104.33,157.691 \n",
       "  104.421,157.386 104.512,157.078 104.603,156.766 104.694,156.452 104.784,156.135 104.875,155.814 104.966,155.49 105.057,155.164 105.148,154.834 105.239,154.501 \n",
       "  105.33,154.165 105.421,153.826 105.511,153.484 105.602,153.138 105.693,152.79 105.784,152.438 105.875,152.084 105.966,151.726 106.057,151.366 106.147,151.002 \n",
       "  106.238,150.635 106.329,150.265 106.42,149.892 106.511,149.516 106.602,149.138 106.693,148.756 106.783,148.371 106.874,147.983 106.965,147.592 107.056,147.198 \n",
       "  107.147,146.801 107.238,146.402 107.329,145.999 107.419,145.593 107.51,145.185 107.601,144.774 107.692,144.359 107.783,143.942 107.874,143.522 107.965,143.1 \n",
       "  108.055,142.674 108.146,142.246 108.237,141.815 108.328,141.381 108.419,140.944 108.51,140.505 108.601,140.063 108.692,139.619 108.782,139.171 108.873,138.721 \n",
       "  108.964,138.269 109.055,137.814 109.146,137.356 109.237,136.896 109.328,136.433 109.418,135.968 109.509,135.5 109.6,135.03 109.691,134.557 109.782,134.082 \n",
       "  109.873,133.605 109.964,133.125 110.054,132.643 110.145,132.159 110.236,131.673 110.327,131.184 110.418,130.693 110.509,130.2 110.6,129.704 110.69,129.207 \n",
       "  110.781,128.708 110.872,128.206 110.963,127.703 111.054,127.197 111.145,126.69 111.236,126.18 111.326,125.669 111.417,125.156 111.508,124.641 111.599,124.124 \n",
       "  111.69,123.606 111.781,123.086 111.872,122.564 111.963,122.04 112.053,121.515 112.144,120.988 112.235,120.46 112.326,119.93 112.417,119.399 112.508,118.866 \n",
       "  112.599,118.332 112.689,117.797 112.78,117.26 112.871,116.722 112.962,116.183 113.053,115.642 113.144,115.1 113.235,114.558 113.325,114.014 113.416,113.469 \n",
       "  113.507,112.923 113.598,112.376 113.689,111.828 113.78,111.28 113.871,110.73 113.961,110.18 114.052,109.629 114.143,109.077 114.234,108.524 114.325,107.971 \n",
       "  114.416,107.417 114.507,106.863 114.598,106.308 114.688,105.753 114.779,105.197 114.87,104.641 114.961,104.085 115.052,103.528 115.143,102.971 115.234,102.414 \n",
       "  115.324,101.857 115.415,101.299 115.506,100.742 115.597,100.184 115.688,99.6263 115.779,99.0687 115.87,98.5112 115.96,97.9539 116.051,97.3967 116.142,96.8398 \n",
       "  116.233,96.2831 116.324,95.7268 116.415,95.1707 116.506,94.6151 116.596,94.06 116.687,93.5053 116.778,92.9511 116.869,92.3975 116.96,91.8445 117.051,91.2922 \n",
       "  117.142,90.7406 117.232,90.1897 117.323,89.6397 117.414,89.0904 117.505,88.5421 117.596,87.9946 117.687,87.4481 117.778,86.9027 117.869,86.3582 117.959,85.8149 \n",
       "  118.05,85.2727 118.141,84.7317 118.232,84.1919 118.323,83.6534 118.414,83.1161 118.505,82.5802 118.595,82.0458 118.686,81.5127 118.777,80.9811 118.868,80.4511 \n",
       "  118.959,79.9225 119.05,79.3956 119.141,78.8704 119.231,78.3468 119.322,77.8249 119.413,77.3048 119.504,76.7865 119.595,76.2701 119.686,75.7555 119.777,75.2428 \n",
       "  119.867,74.7322 119.958,74.2235 120.049,73.7168 120.14,73.2123 120.231,72.7098 120.322,72.2095 120.413,71.7115 120.503,71.2156 120.594,70.722 120.685,70.2307 \n",
       "  120.776,69.7418 120.867,69.2553 120.958,68.7711 121.049,68.2895 121.14,67.8103 121.23,67.3336 121.321,66.8595 121.412,66.3881 121.503,65.9192 121.594,65.453 \n",
       "  121.685,64.9895 121.776,64.5288 121.866,64.0708 121.957,63.6156 122.048,63.1632 122.139,62.7138 122.23,62.2672 122.321,61.8235 122.412,61.3829 122.502,60.9452 \n",
       "  122.593,60.5105 122.684,60.0789 122.775,59.6504 122.866,59.225 122.957,58.8027 123.048,58.3837 123.138,57.9678 123.229,57.5551 123.32,57.1458 123.411,56.7397 \n",
       "  123.502,56.3369 123.593,55.9374 123.684,55.5414 123.774,55.1487 123.865,54.7595 123.956,54.3737 124.047,53.9913 124.138,53.6125 124.229,53.2372 124.32,52.8654 \n",
       "  124.411,52.4972 124.501,52.1326 124.592,51.7716 124.683,51.4142 124.774,51.0605 124.865,50.7105 124.956,50.3642 125.047,50.0215 125.137,49.6827 125.228,49.3476 \n",
       "  125.319,49.0162 125.41,48.6887 125.501,48.365 125.592,48.0451 125.683,47.7291 125.773,47.417 125.864,47.1087 125.955,46.8044 126.046,46.5039 126.137,46.2074 \n",
       "  126.228,45.9149 126.319,45.6263 126.409,45.3417 126.5,45.0611 126.591,44.7845 126.682,44.5119 126.773,44.2434 126.864,43.9789 126.955,43.7184 127.045,43.4621 \n",
       "  127.136,43.2098 127.227,42.9616 127.318,42.7175 127.409,42.4775 127.5,42.2417 127.591,42.01 127.682,41.7824 127.772,41.5589 127.863,41.3397 127.954,41.1246 \n",
       "  128.045,40.9136 128.136,40.7068 128.227,40.5043 128.318,40.3059 128.408,40.1117 128.499,39.9217 128.59,39.7359 128.681,39.5543 128.772,39.3769 128.863,39.2037 \n",
       "  128.954,39.0348 129.044,38.8701 129.135,38.7096 129.226,38.5533 129.317,38.4013 129.408,38.2535 129.499,38.1099 129.59,37.9705 129.68,37.8354 129.771,37.7045 \n",
       "  129.862,37.5778 129.953,37.4554 130.044,37.3371 130.135,37.2231 130.226,37.1134 130.316,37.0078 130.407,36.9064 130.498,36.8093 130.589,36.7164 130.68,36.6277 \n",
       "  130.771,36.5431 130.862,36.4628 130.953,36.3867 131.043,36.3147 131.134,36.2469 131.225,36.1833 131.316,36.1239 131.407,36.0686 131.498,36.0174 131.589,35.9704 \n",
       "  131.679,35.9276 131.77,35.8888 131.861,35.8542 131.952,35.8237 132.043,35.7973 132.134,35.775 132.225,35.7568 132.315,35.7426 132.406,35.7325 132.497,35.7265 \n",
       "  132.588,35.7244 132.679,35.7264 132.77,35.7325 132.861,35.7425 132.951,35.7565 133.042,35.7745 133.133,35.7965 133.224,35.8224 133.315,35.8523 133.406,35.8861 \n",
       "  133.497,35.9238 133.587,35.9653 133.678,36.0108 133.769,36.0602 133.86,36.1134 133.951,36.1704 134.042,36.2313 134.133,36.2959 134.224,36.3644 134.314,36.4366 \n",
       "  134.405,36.5126 134.496,36.5923 134.587,36.6758 134.678,36.763 134.769,36.8538 134.86,36.9484 134.95,37.0465 135.041,37.1484 135.132,37.2538 135.223,37.3628 \n",
       "  135.314,37.4755 135.405,37.5916 135.496,37.7114 135.586,37.8346 135.677,37.9614 135.768,38.0916 135.859,38.2253 135.95,38.3625 136.041,38.5031 136.132,38.6471 \n",
       "  136.222,38.7944 136.313,38.9452 136.404,39.0992 136.495,39.2566 136.586,39.4173 136.677,39.5813 136.768,39.7486 136.858,39.919 136.949,40.0927 137.04,40.2696 \n",
       "  137.131,40.4496 137.222,40.6328 137.313,40.8192 137.404,41.0086 137.495,41.2011 137.585,41.3967 137.676,41.5953 137.767,41.7969 137.858,42.0015 137.949,42.209 \n",
       "  138.04,42.4196 138.131,42.633 138.221,42.8493 138.312,43.0685 138.403,43.2906 138.494,43.5154 138.585,43.7431 138.676,43.9736 138.767,44.2068 138.857,44.4427 \n",
       "  138.948,44.6813 139.039,44.9226 139.13,45.1666 139.221,45.4132 139.312,45.6624 139.403,45.9142 139.493,46.1686 139.584,46.4255 139.675,46.6849 139.766,46.9467 \n",
       "  139.857,47.2111 139.948,47.4778 140.039,47.747 140.129,48.0186 140.22,48.2925 140.311,48.5688 140.402,48.8473 140.493,49.1282 140.584,49.4113 140.675,49.6966 \n",
       "  140.766,49.9842 140.856,50.2739 140.947,50.5658 141.038,50.8599 141.129,51.156 141.22,51.4542 141.311,51.7545 141.402,52.0568 141.492,52.3612 141.583,52.6675 \n",
       "  141.674,52.9757 141.765,53.2859 141.856,53.598 141.947,53.912 142.038,54.2279 142.128,54.5455 142.219,54.865 142.31,55.1863 142.401,55.5093 142.492,55.8341 \n",
       "  142.583,56.1606 142.674,56.4887 142.764,56.8185 142.855,57.1499 142.946,57.483 143.037,57.8176 143.128,58.1538 143.219,58.4915 143.31,58.8308 143.4,59.1715 \n",
       "  143.491,59.5136 143.582,59.8573 143.673,60.2023 143.764,60.5487 143.855,60.8965 143.946,61.2456 144.037,61.596 144.127,61.9477 144.218,62.3007 144.309,62.6549 \n",
       "  144.4,63.0104 144.491,63.367 144.582,63.7248 144.673,64.0838 144.763,64.4438 144.854,64.805 144.945,65.1673 145.036,65.5306 145.127,65.8949 145.218,66.2603 \n",
       "  145.309,66.6266 145.399,66.9939 145.49,67.3621 145.581,67.7312 145.672,68.1013 145.763,68.4722 145.854,68.8439 145.945,69.2165 146.035,69.5899 146.126,69.964 \n",
       "  146.217,70.339 146.308,70.7146 146.399,71.091 146.49,71.468 146.581,71.8458 146.671,72.2241 146.762,72.6031 146.853,72.9827 146.944,73.3629 147.035,73.7437 \n",
       "  147.126,74.125 147.217,74.5068 147.308,74.8891 147.398,75.2719 147.489,75.6552 147.58,76.0389 147.671,76.423 147.762,76.8075 147.853,77.1924 147.944,77.5777 \n",
       "  148.034,77.9633 148.125,78.3492 148.216,78.7354 148.307,79.1219 148.398,79.5086 148.489,79.8956 148.58,80.2828 148.67,80.6702 148.761,81.0578 148.852,81.4456 \n",
       "  148.943,81.8335 149.034,82.2215 149.125,82.6096 149.216,82.9979 149.306,83.3862 149.397,83.7745 149.488,84.1629 149.579,84.5513 149.67,84.9397 149.761,85.3281 \n",
       "  149.852,85.7165 149.943,86.1048 150.033,86.4931 150.124,86.8812 150.215,87.2693 150.306,87.6572 150.397,88.045 150.488,88.4327 150.579,88.8202 150.669,89.2075 \n",
       "  150.76,89.5946 150.851,89.9815 150.942,90.3681 151.033,90.7546 151.124,91.1407 151.215,91.5266 151.305,91.9122 151.396,92.2975 151.487,92.6824 151.578,93.067 \n",
       "  151.669,93.4513 151.76,93.8352 151.851,94.2187 151.941,94.6019 152.032,94.9846 152.123,95.3669 152.214,95.7487 152.305,96.1301 152.396,96.5111 152.487,96.8916 \n",
       "  152.577,97.2715 152.668,97.651 152.759,98.03 152.85,98.4084 152.941,98.7863 153.032,99.1637 153.123,99.5404 153.214,99.9166 153.304,100.292 153.395,100.667 \n",
       "  153.486,101.042 153.577,101.415 153.668,101.789 153.759,102.161 153.85,102.533 153.94,102.904 154.031,103.274 154.122,103.644 154.213,104.013 154.304,104.382 \n",
       "  154.395,104.749 154.486,105.116 154.576,105.482 154.667,105.848 154.758,106.212 154.849,106.576 154.94,106.939 155.031,107.301 155.122,107.663 155.212,108.024 \n",
       "  155.303,108.383 155.394,108.742 155.485,109.1 155.576,109.458 155.667,109.814 155.758,110.17 155.848,110.524 155.939,110.878 156.03,111.231 156.121,111.583 \n",
       "  156.212,111.934 156.303,112.285 156.394,112.634 156.485,112.982 156.575,113.33 156.666,113.676 156.757,114.021 156.848,114.366 156.939,114.71 157.03,115.052 \n",
       "  157.121,115.394 157.211,115.734 157.302,116.074 157.393,116.413 157.484,116.75 157.575,117.087 157.666,117.423 157.757,117.757 157.847,118.091 157.938,118.423 \n",
       "  158.029,118.755 158.12,119.085 158.211,119.414 158.302,119.743 158.393,120.07 158.483,120.396 158.574,120.721 158.665,121.045 158.756,121.368 158.847,121.69 \n",
       "  158.938,122.011 159.029,122.331 159.119,122.649 159.21,122.967 159.301,123.283 159.392,123.598 159.483,123.912 159.574,124.225 159.665,124.537 159.756,124.848 \n",
       "  159.846,125.158 159.937,125.466 160.028,125.774 160.119,126.08 160.21,126.385 160.301,126.689 160.392,126.992 160.482,127.293 160.573,127.594 160.664,127.893 \n",
       "  160.755,128.191 160.846,128.488 160.937,128.784 161.028,129.079 161.118,129.372 161.209,129.665 161.3,129.956 161.391,130.246 161.482,130.535 161.573,130.822 \n",
       "  161.664,131.109 161.754,131.394 161.845,131.678 161.936,131.961 162.027,132.243 162.118,132.523 162.209,132.803 162.3,133.081 162.39,133.358 162.481,133.634 \n",
       "  162.572,133.908 162.663,134.182 162.754,134.454 162.845,134.725 162.936,134.995 163.027,135.263 163.117,135.531 163.208,135.797 163.299,136.062 163.39,136.326 \n",
       "  163.481,136.589 163.572,136.85 163.663,137.111 163.753,137.37 163.844,137.628 163.935,137.884 164.026,138.14 164.117,138.394 164.208,138.647 164.299,138.899 \n",
       "  164.389,139.15 164.48,139.399 164.571,139.648 164.662,139.895 164.753,140.141 164.844,140.386 164.935,140.629 165.025,140.872 165.116,141.113 165.207,141.353 \n",
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       "  390.813,38.146 390.881,38.0258 390.95,37.9086 391.018,37.7945 391.086,37.6835 391.154,37.5755 391.222,37.4707 391.29,37.3689 391.358,37.2702 391.427,37.1745 \n",
       "  391.495,37.0819 391.563,36.9924 391.631,36.906 391.699,36.8227 391.767,36.7424 391.835,36.6652 391.904,36.591 391.972,36.5199 392.04,36.4519 392.108,36.387 \n",
       "  392.176,36.3251 392.244,36.2662 392.312,36.2105 392.381,36.1577 392.449,36.108 392.517,36.0614 392.585,36.0178 392.653,35.9773 392.721,35.9397 392.789,35.9052 \n",
       "  392.858,35.8738 392.926,35.8453 392.994,35.8199 393.062,35.7975 393.13,35.7781 393.198,35.7617 393.266,35.7482 393.335,35.7378 393.403,35.7304 393.471,35.7259 \n",
       "  393.539,35.7244 393.607,35.7259 393.675,35.7304 393.743,35.7378 393.812,35.7481 393.88,35.7614 393.948,35.7776 394.016,35.7968 394.084,35.8188 394.152,35.8438 \n",
       "  394.221,35.8717 394.289,35.9025 394.357,35.9361 394.425,35.9727 394.493,36.0121 394.561,36.0544 394.629,36.0995 394.698,36.1475 394.766,36.1984 394.834,36.252 \n",
       "  394.902,36.3085 394.97,36.3678 395.038,36.4299 395.106,36.4947 395.175,36.5624 395.243,36.6328 395.311,36.706 395.379,36.782 395.447,36.8606 395.515,36.9421 \n",
       "  395.583,37.0262 395.652,37.1131 395.72,37.2026 395.788,37.2948 395.856,37.3898 395.924,37.4873 395.992,37.5876 396.06,37.6905 396.129,37.796 396.197,37.9041 \n",
       "  396.265,38.0149 396.333,38.1282 396.401,38.2441 396.469,38.3626 396.537,38.4837 396.606,38.6073 396.674,38.7334 396.742,38.8621 396.81,38.9933 396.878,39.127 \n",
       "  396.946,39.2631 397.014,39.4018 397.083,39.5429 397.151,39.6864 397.219,39.8324 397.287,39.9808 397.355,40.1316 397.423,40.2848 397.492,40.4404 397.56,40.5984 \n",
       "  397.628,40.7587 397.696,40.9213 397.764,41.0863 397.832,41.2536 397.9,41.4232 397.969,41.595 398.037,41.7692 398.105,41.9456 398.173,42.1242 398.241,42.3051 \n",
       "  398.309,42.4881 398.377,42.6734 398.446,42.8608 398.514,43.0505 398.582,43.2422 398.65,43.4362 398.718,43.6322 398.786,43.8303 398.854,44.0306 398.923,44.2329 \n",
       "  398.991,44.4373 399.059,44.6437 399.127,44.8522 399.195,45.0627 399.263,45.2751 399.331,45.4896 399.4,45.7061 399.468,45.9245 399.536,46.1448 399.604,46.3671 \n",
       "  399.672,46.5913 399.74,46.8174 399.808,47.0453 399.877,47.2752 399.945,47.5068 400.013,47.7403 400.081,47.9757 400.149,48.2128 400.217,48.4517 400.285,48.6924 \n",
       "  400.354,48.9348 400.422,49.179 400.49,49.4249 400.558,49.6725 400.626,49.9218 400.694,50.1727 400.763,50.4254 400.831,50.6796 400.899,50.9355 400.967,51.193 \n",
       "  401.035,51.4521 401.103,51.7128 401.171,51.975 401.24,52.2388 401.308,52.5041 401.376,52.7709 401.444,53.0393 401.512,53.3091 401.58,53.5803 401.648,53.853 \n",
       "  401.717,54.1272 401.785,54.4028 401.853,54.6797 401.921,54.9581 401.989,55.2378 402.057,55.5189 402.125,55.8013 402.194,56.085 402.262,56.3701 402.33,56.6564 \n",
       "  402.398,56.944 402.466,57.2329 402.534,57.523 402.602,57.8143 402.671,58.1068 402.739,58.4005 402.807,58.6954 402.875,58.9915 402.943,59.2887 403.011,59.587 \n",
       "  403.079,59.8865 403.148,60.187 403.216,60.4886 403.284,60.7913 403.352,61.095 403.42,61.3998 403.488,61.7056 403.556,62.0124 403.625,62.3202 403.693,62.6289 \n",
       "  403.761,62.9386 403.829,63.2493 403.897,63.5608 403.965,63.8733 404.034,64.1867 404.102,64.5009 404.17,64.8161 404.238,65.132 404.306,65.4488 404.374,65.7664 \n",
       "  404.442,66.0848 404.511,66.404 404.579,66.724 404.647,67.0447 404.715,67.3662 404.783,67.6884 404.851,68.0113 404.919,68.3349 404.988,68.6591 405.056,68.9841 \n",
       "  405.124,69.3097 405.192,69.6359 405.26,69.9628 405.328,70.2902 405.396,70.6183 405.465,70.9469 405.533,71.2761 405.601,71.6059 405.669,71.9361 405.737,72.2669 \n",
       "  405.805,72.5982 405.873,72.93 405.942,73.2623 406.01,73.5951 406.078,73.9283 406.146,74.2619 406.214,74.5959 406.282,74.9304 406.35,75.2653 406.419,75.6005 \n",
       "  406.487,75.9361 406.555,76.272 406.623,76.6083 406.691,76.945 406.759,77.2819 406.827,77.6191 406.896,77.9566 406.964,78.2944 407.032,78.6325 407.1,78.9708 \n",
       "  407.168,79.3093 407.236,79.6481 407.305,79.987 407.373,80.3262 407.441,80.6655 407.509,81.0051 407.577,81.3447 407.645,81.6845 407.713,82.0245 407.782,82.3645 \n",
       "  407.85,82.7047 407.918,83.045 407.986,83.3853 408.054,83.7257 408.122,84.0662 408.19,84.4067 408.259,84.7473 408.327,85.0879 408.395,85.4285 408.463,85.7691 \n",
       "  408.531,86.1097 408.599,86.4502 408.667,86.7907 408.736,87.1312 408.804,87.4716 408.872,87.812 408.94,88.1522 409.008,88.4924 409.076,88.8325 409.144,89.1724 \n",
       "  409.213,89.5123 409.281,89.852 409.349,90.1915 409.417,90.5309 409.485,90.8702 409.553,91.2092 409.621,91.5481 409.69,91.8867 409.758,92.2252 409.826,92.5634 \n",
       "  409.894,92.9014 409.962,93.2392 410.03,93.5767 410.098,93.914 410.167,94.251 410.235,94.5877 410.303,94.9241 410.371,95.2603 410.439,95.5961 410.507,95.9316 \n",
       "  410.576,96.2668 410.644,96.6016 410.712,96.9361 410.78,97.2703 410.848,97.604 410.916,97.9375 410.984,98.2705 411.053,98.6031 411.121,98.9354 411.189,99.2672 \n",
       "  411.257,99.5987 411.325,99.9297 411.393,100.26 411.461,100.59 411.53,100.92 411.598,101.249 411.666,101.578 411.734,101.906 411.802,102.234 411.87,102.561 \n",
       "  411.938,102.888 412.007,103.215 412.075,103.54 412.143,103.866 412.211,104.19 412.279,104.515 412.347,104.838 412.415,105.162 412.484,105.484 412.552,105.806 \n",
       "  412.62,106.128 412.688,106.449 412.756,106.769 412.824,107.089 412.892,107.408 412.961,107.726 413.029,108.044 413.097,108.361 413.165,108.678 413.233,108.994 \n",
       "  413.301,109.309 413.369,109.624 413.438,109.938 413.506,110.252 413.574,110.564 413.642,110.877 413.71,111.188 413.778,111.499 413.847,111.809 413.915,112.118 \n",
       "  413.983,112.427 414.051,112.735 414.119,113.042 414.187,113.349 414.255,113.655 414.324,113.96 414.392,114.265 414.46,114.568 414.528,114.871 414.596,115.173 \n",
       "  414.664,115.475 414.732,115.776 414.801,116.076 414.869,116.375 414.937,116.673 415.005,116.971 415.073,117.268 415.141,117.564 415.209,117.86 415.278,118.154 \n",
       "  415.346,118.448 415.414,118.741 415.482,119.033 415.55,119.325 415.618,119.615 415.686,119.905 415.755,120.194 415.823,120.482 415.891,120.769 415.959,121.056 \n",
       "  416.027,121.342 416.095,121.627 416.163,121.911 416.232,122.194 416.3,122.476 416.368,122.758 416.436,123.038 416.504,123.318 416.572,123.597 416.64,123.875 \n",
       "  416.709,124.153 416.777,124.429 416.845,124.705 416.913,124.979 416.981,125.253 417.049,125.526 417.118,125.798 417.186,126.069 417.254,126.34 417.322,126.609 \n",
       "  417.39,126.878 417.458,127.145 417.526,127.412 417.595,127.678 417.663,127.943 417.731,128.207 417.799,128.47 417.867,128.733 417.935,128.994 418.003,129.255 \n",
       "  418.072,129.514 418.14,129.773 418.208,130.031 418.276,130.288 418.344,130.544 418.412,130.799 418.48,131.053 418.549,131.306 418.617,131.559 418.685,131.81 \n",
       "  418.753,132.061 418.821,132.31 418.889,132.559 418.957,132.807 419.026,133.054 419.094,133.3 419.162,133.545 419.23,133.789 419.298,134.032 419.366,134.274 \n",
       "  419.434,134.515 419.503,134.756 419.571,134.995 419.639,135.234 419.707,135.472 419.775,135.708 419.843,135.944 419.912,136.179 419.98,136.413 420.048,136.646 \n",
       "  420.116,136.878 420.184,137.109 420.252,137.339 420.32,137.569 420.389,137.797 420.457,138.025 420.525,138.251 420.593,138.477 420.661,138.701 420.729,138.925 \n",
       "  420.797,139.148 420.866,139.37 420.934,139.591 421.002,139.811 421.07,140.03 421.138,140.248 421.206,140.465 421.274,140.682 421.343,140.897 421.411,141.112 \n",
       "  421.479,141.325 421.547,141.538 421.615,141.75 421.683,141.96 421.751,142.17 421.82,142.379 421.888,142.587 421.956,142.794 422.024,143.001 422.092,143.206 \n",
       "  422.16,143.41 422.228,143.614 422.297,143.816 422.365,144.018 422.433,144.219 422.501,144.419 422.569,144.618 422.637,144.816 422.705,145.013 422.774,145.209 \n",
       "  422.842,145.404 422.91,145.599 422.978,145.792 423.046,145.985 423.114,146.176 423.183,146.367 423.251,146.557 423.319,146.746 423.387,146.934 423.455,147.121 \n",
       "  423.523,147.308 423.591,147.493 423.66,147.678 423.728,147.861 423.796,148.044 423.864,148.226 423.932,148.407 424,148.587 424.068,148.767 424.137,148.945 \n",
       "  424.205,149.123 424.273,149.299 424.341,149.475 424.409,149.65 424.477,149.824 424.545,149.997 424.614,150.169 424.682,150.341 424.75,150.512 424.818,150.681 \n",
       "  424.886,150.85 424.954,151.018 425.022,151.185 425.091,151.352 425.159,151.517 425.227,151.682 425.295,151.846 425.363,152.009 425.431,152.171 425.499,152.332 \n",
       "  425.568,152.493 425.636,152.652 425.704,152.811 425.772,152.969 425.84,153.126 425.908,153.283 425.976,153.438 426.045,153.593 426.113,153.747 426.181,153.9 \n",
       "  426.249,154.052 426.317,154.203 426.385,154.354 426.454,154.504 426.522,154.653 426.59,154.801 426.658,154.948 426.726,155.095 426.794,155.241 426.862,155.386 \n",
       "  426.931,155.53 426.999,155.674 427.067,155.816 427.135,155.958 427.203,156.099 427.271,156.24 427.339,156.379 427.408,156.518 427.476,156.656 427.544,156.794 \n",
       "  427.612,156.93 427.68,157.066 427.748,157.201 427.816,157.335 427.885,157.469 427.953,157.601 428.021,157.733 428.089,157.865 428.157,157.995 428.225,158.125 \n",
       "  428.293,158.254 428.362,158.382 428.43,158.51 428.498,158.637 428.566,158.763 428.634,158.888 428.702,159.013 428.77,159.137 428.839,159.26 428.907,159.382 \n",
       "  428.975,159.504 429.043,159.625 429.111,159.746 429.179,159.865 429.247,159.984 429.316,160.103 429.384,160.22 429.452,160.337 429.52,160.453 429.588,160.569 \n",
       "  429.656,160.684 429.725,160.798 429.793,160.911 429.861,161.024 429.929,161.136 429.997,161.248 430.065,161.359 430.133,161.469 430.202,161.578 430.27,161.687 \n",
       "  430.338,161.795 430.406,161.903 430.474,162.009 430.542,162.116 430.61,162.221 430.679,162.326 430.747,162.43 430.815,162.534 430.883,162.637 430.951,162.739 \n",
       "  431.019,162.841 431.087,162.942 431.156,163.043 431.224,163.142 431.292,163.242 431.36,163.34 431.428,163.438 431.496,163.536 431.564,163.633 431.633,163.729 \n",
       "  431.701,163.824 431.769,163.919 431.837,164.014 431.905,164.107 431.973,164.201 432.041,164.293 432.11,164.385 432.178,164.477 432.246,164.568 432.314,164.658 \n",
       "  432.382,164.748 432.45,164.837 432.518,164.925 432.587,165.013 432.655,165.101 432.723,165.188 432.791,165.274 432.859,165.36 432.927,165.445 432.996,165.529 \n",
       "  433.064,165.614 433.132,165.697 433.2,165.78 433.268,165.863 433.336,165.945 433.404,166.026 433.473,166.107 433.541,166.187 433.609,166.267 433.677,166.346 \n",
       "  433.745,166.425 433.813,166.503 433.881,166.581 433.95,166.658 434.018,166.735 434.086,166.811 434.154,166.887 434.222,166.962 434.29,167.037 434.358,167.111 \n",
       "  434.427,167.185 434.495,167.258 434.563,167.33 434.631,167.403 434.699,167.474 434.767,167.546 434.835,167.616 434.904,167.687 434.972,167.756 435.04,167.826 \n",
       "  435.108,167.895 435.176,167.963 435.244,168.031 435.312,168.098 435.381,168.165 435.449,168.232 435.517,168.298 435.585,168.364 435.653,168.429 435.721,168.494 \n",
       "  435.789,168.558 435.858,168.622 435.926,168.685 435.994,168.748 436.062,168.811 436.13,168.873 436.198,168.934 436.267,168.996 436.335,169.057 436.403,169.117 \n",
       "  436.471,169.177 436.539,169.236 436.607,169.296 436.675,169.354 436.744,169.413 436.812,169.47 436.88,169.528 436.948,169.585 437.016,169.642 437.084,169.698 \n",
       "  437.152,169.754 437.221,169.809 437.289,169.864 437.357,169.919 437.425,169.973 437.493,170.027 437.561,170.081 437.629,170.134 437.698,170.187 437.766,170.239 \n",
       "  437.834,170.291 437.902,170.343 437.97,170.394 438.038,170.445 438.106,170.496 438.175,170.546 438.243,170.596 438.311,170.645 438.379,170.694 438.447,170.743 \n",
       "  438.515,170.791 438.583,170.839 438.652,170.887 438.72,170.934 438.788,170.981 438.856,171.028 438.924,171.074 438.992,171.12 439.06,171.166 439.129,171.211 \n",
       "  439.197,171.256 439.265,171.301 439.333,171.345 439.401,171.389 439.469,171.432 439.538,171.476 439.606,171.519 439.674,171.561 439.742,171.604 439.81,171.646 \n",
       "  439.878,171.688 439.946,171.729 440.015,171.77 440.083,171.811 440.151,171.851 440.219,171.892 440.287,171.932 440.355,171.971 440.423,172.01 440.492,172.049 \n",
       "  440.56,172.088 440.628,172.127 440.696,172.165 440.764,172.202 440.832,172.24 440.9,172.277 440.969,172.314 441.037,172.351 441.105,172.387 441.173,172.424 \n",
       "  441.241,172.459 441.309,172.495 441.377,172.53 441.446,172.565 441.514,172.6 441.582,172.635 441.65,172.669 441.718,172.703 441.786,172.737 441.854,172.77 \n",
       "  441.923,172.803 441.991,172.836 442.059,172.869 442.127,172.901 442.195,172.934 442.263,172.965 442.331,172.997 442.4,173.029 442.468,173.06 442.536,173.091 \n",
       "  442.604,173.122 442.672,173.152 442.74,173.182 442.809,173.212 442.877,173.242 442.945,173.272 443.013,173.301 443.081,173.33 443.149,173.359 443.217,173.387 \n",
       "  443.286,173.416 443.354,173.444 443.422,173.472 443.49,173.5 443.558,173.527 443.626,173.554 443.694,173.582 443.763,173.608 443.831,173.635 443.899,173.661 \n",
       "  443.967,173.688 444.035,173.714 444.103,173.74 444.171,173.765 444.24,173.791 444.308,173.816 444.376,173.841 444.444,173.865 444.512,173.89 444.58,173.914 \n",
       "  444.648,173.939 444.717,173.963 444.785,173.987 444.853,174.01 444.921,174.034 444.989,174.057 445.057,174.08 445.125,174.103 445.194,174.125 445.262,174.148 \n",
       "  445.33,174.17 445.398,174.192 445.466,174.214 445.534,174.236 445.602,174.258 445.671,174.279 445.739,174.3 445.807,174.321 445.875,174.342 445.943,174.363 \n",
       "  446.011,174.384 446.08,174.404 446.148,174.424 446.216,174.444 446.284,174.464 446.352,174.484 446.42,174.503 446.488,174.523 446.557,174.542 446.625,174.561 \n",
       "  446.693,174.58 446.761,174.599 446.829,174.617 446.897,174.636 446.965,174.654 447.034,174.672 447.102,174.69 447.17,174.708 447.238,174.726 447.306,174.744 \n",
       "  447.374,174.761 447.442,174.778 447.511,174.795 447.579,174.812 447.647,174.829 447.715,174.846 447.783,174.862 447.851,174.879 447.919,174.895 447.988,174.911 \n",
       "  448.056,174.927 448.124,174.943 448.192,174.959 448.26,174.974 448.328,174.99 448.396,175.005 448.465,175.02 448.533,175.036 448.601,175.05 448.669,175.065 \n",
       "  448.737,175.08 448.805,175.095 448.874,175.109 448.942,175.123 449.01,175.138 449.078,175.152 449.146,175.166 449.214,175.179 449.282,175.193 449.351,175.207 \n",
       "  449.419,175.22 449.487,175.234 449.555,175.247 449.623,175.26 449.691,175.273 449.759,175.286 449.828,175.299 449.896,175.312 449.964,175.324 450.032,175.337 \n",
       "  450.1,175.349 450.168,175.361 450.236,175.373 450.305,175.385 450.373,175.397 450.441,175.409 450.509,175.421 450.577,175.433 450.645,175.444 450.713,175.456 \n",
       "  450.782,175.467 450.85,175.478 450.918,175.489 450.986,175.5 451.054,175.511 451.122,175.522 451.19,175.533 451.259,175.544 451.327,175.554 451.395,175.565 \n",
       "  451.463,175.575 451.531,175.585 451.599,175.596 451.667,175.606 451.736,175.616 451.804,175.626 451.872,175.636 451.94,175.645 452.008,175.655 452.076,175.665 \n",
       "  452.145,175.674 452.213,175.684 452.281,175.693 452.349,175.702 452.417,175.711 452.485,175.72 452.553,175.729 452.622,175.738 452.69,175.747 452.758,175.756 \n",
       "  452.826,175.765 452.894,175.773 452.962,175.782 453.03,175.79 453.099,175.799 453.167,175.807 453.235,175.815 453.303,175.823 453.371,175.832 453.439,175.84 \n",
       "  453.507,175.847 453.576,175.855 453.644,175.863 453.712,175.871 453.78,175.879 453.848,175.886 453.916,175.894 453.984,175.901 454.053,175.909 454.121,175.916 \n",
       "  454.189,175.923 454.257,175.93 454.325,175.937 454.393,175.945 454.461,175.952 454.53,175.958 454.598,175.965 454.666,175.972 454.734,175.979 454.802,175.986 \n",
       "  454.87,175.992 454.938,175.999 455.007,176.005 455.075,176.012 455.143,176.018 455.211,176.024 455.279,176.031 455.347,176.037 455.416,176.043 455.484,176.049 \n",
       "  455.552,176.055 455.62,176.061 455.688,176.067 455.756,176.073 455.824,176.079 455.893,176.085 455.961,176.09 456.029,176.096 456.097,176.102 456.165,176.107 \n",
       "  456.233,176.113 456.301,176.118 456.37,176.124 456.438,176.129 456.506,176.134 456.574,176.14 456.642,176.145 456.71,176.15 456.778,176.155 456.847,176.16 \n",
       "  456.915,176.165 456.983,176.17 457.051,176.175 457.119,176.18 457.187,176.185 457.255,176.19 457.324,176.194 457.392,176.199 457.46,176.204 457.528,176.208 \n",
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       "<g clip-path=\"url(#clip8700)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:middle;\" transform=\"rotate(0, 598.612, 195.449)\" x=\"598.612\" y=\"195.449\">10</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip8700)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:middle;\" transform=\"rotate(0, 621.327, 195.449)\" x=\"621.327\" y=\"195.449\">20</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip8700)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:middle;\" transform=\"rotate(0, 644.042, 195.449)\" x=\"644.042\" y=\"195.449\">30</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip8700)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:middle;\" transform=\"rotate(0, 666.758, 195.449)\" x=\"666.758\" y=\"195.449\">40</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip8700)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:middle;\" transform=\"rotate(0, 689.473, 195.449)\" x=\"689.473\" y=\"195.449\">50</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip8700)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:middle;\" transform=\"rotate(0, 712.188, 195.449)\" x=\"712.188\" y=\"195.449\">60</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip8700)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:end;\" transform=\"rotate(0, 563.899, 181.17)\" x=\"563.899\" y=\"181.17\">0</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip8700)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:start;\" transform=\"rotate(0, 525.927, 161.022)\" x=\"525.927\" y=\"161.022\">1×10</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip8700)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:10; text-anchor:start;\" transform=\"rotate(0, 553.027, 154.17)\" x=\"553.027\" y=\"154.17\">29</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip8700)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:start;\" transform=\"rotate(0, 525.927, 139.442)\" x=\"525.927\" y=\"139.442\">2×10</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip8700)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:10; text-anchor:start;\" transform=\"rotate(0, 553.027, 132.59)\" x=\"553.027\" y=\"132.59\">29</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip8700)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:start;\" transform=\"rotate(0, 525.927, 117.862)\" x=\"525.927\" y=\"117.862\">3×10</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip8700)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:10; text-anchor:start;\" transform=\"rotate(0, 553.027, 111.01)\" x=\"553.027\" y=\"111.01\">29</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip8700)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:start;\" transform=\"rotate(0, 525.927, 96.2824)\" x=\"525.927\" y=\"96.2824\">4×10</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip8700)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:10; text-anchor:start;\" transform=\"rotate(0, 553.027, 89.4298)\" x=\"553.027\" y=\"89.4298\">29</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip8700)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:start;\" transform=\"rotate(0, 525.927, 74.7025)\" x=\"525.927\" y=\"74.7025\">5×10</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip8700)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:10; text-anchor:start;\" transform=\"rotate(0, 553.027, 67.8499)\" x=\"553.027\" y=\"67.8499\">29</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip8700)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:start;\" transform=\"rotate(0, 525.927, 53.1225)\" x=\"525.927\" y=\"53.1225\">6×10</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip8700)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:10; text-anchor:start;\" transform=\"rotate(0, 553.027, 46.2699)\" x=\"553.027\" y=\"46.2699\">29</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip8700)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:15; text-anchor:middle;\" transform=\"rotate(0, 650.857, 13.2)\" x=\"650.857\" y=\"13.2\">s = 30</text>\n",
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       "  576.351,176.67 576.396,176.67 576.442,176.67 576.487,176.67 576.532,176.67 576.578,176.67 576.623,176.67 576.669,176.67 576.714,176.67 576.76,176.67 \n",
       "  576.805,176.67 576.85,176.67 576.896,176.67 576.941,176.67 576.987,176.67 577.032,176.67 577.078,176.67 577.123,176.67 577.168,176.67 577.214,176.67 \n",
       "  577.259,176.67 577.305,176.67 577.35,176.67 577.396,176.67 577.441,176.67 577.486,176.67 577.532,176.67 577.577,176.67 577.623,176.67 577.668,176.67 \n",
       "  577.714,176.67 577.759,176.67 577.804,176.67 577.85,176.67 577.895,176.67 577.941,176.67 577.986,176.67 578.032,176.67 578.077,176.67 578.122,176.67 \n",
       "  578.168,176.67 578.213,176.67 578.259,176.67 578.304,176.67 578.35,176.67 578.395,176.67 578.441,176.67 578.486,176.67 578.531,176.67 578.577,176.67 \n",
       "  578.622,176.67 578.668,176.67 578.713,176.67 578.759,176.67 578.804,176.67 578.849,176.67 578.895,176.67 578.94,176.67 578.986,176.67 579.031,176.67 \n",
       "  579.077,176.67 579.122,176.67 579.167,176.67 579.213,176.67 579.258,176.67 579.304,176.67 579.349,176.67 579.395,176.67 579.44,176.67 579.485,176.67 \n",
       "  579.531,176.67 579.576,176.67 579.622,176.67 579.667,176.67 579.713,176.67 579.758,176.67 579.803,176.67 579.849,176.67 579.894,176.67 579.94,176.67 \n",
       "  579.985,176.67 580.031,176.67 580.076,176.67 580.121,176.67 580.167,176.67 580.212,176.67 580.258,176.67 580.303,176.67 580.349,176.67 580.394,176.67 \n",
       "  580.439,176.67 580.485,176.67 580.53,176.67 580.576,176.67 580.621,176.67 580.667,176.67 580.712,176.67 580.757,176.67 580.803,176.67 580.848,176.67 \n",
       "  580.894,176.67 580.939,176.67 580.985,176.67 581.03,176.67 581.075,176.67 581.121,176.67 581.166,176.67 581.212,176.67 581.257,176.67 581.303,176.67 \n",
       "  581.348,176.67 581.393,176.67 581.439,176.67 581.484,176.67 581.53,176.67 581.575,176.67 581.621,176.67 581.666,176.67 581.712,176.67 581.757,176.67 \n",
       "  581.802,176.67 581.848,176.67 581.893,176.67 581.939,176.67 581.984,176.67 582.03,176.67 582.075,176.67 582.12,176.67 582.166,176.67 582.211,176.67 \n",
       "  582.257,176.67 582.302,176.67 582.348,176.67 582.393,176.67 582.438,176.67 582.484,176.67 582.529,176.67 582.575,176.67 582.62,176.67 582.666,176.67 \n",
       "  582.711,176.67 582.756,176.67 582.802,176.67 582.847,176.67 582.893,176.67 582.938,176.67 582.984,176.67 583.029,176.67 583.074,176.67 583.12,176.67 \n",
       "  583.165,176.67 583.211,176.67 583.256,176.67 583.302,176.67 583.347,176.67 583.392,176.67 583.438,176.67 583.483,176.67 583.529,176.67 583.574,176.67 \n",
       "  583.62,176.67 583.665,176.67 583.71,176.67 583.756,176.67 583.801,176.67 583.847,176.67 583.892,176.67 583.938,176.67 583.983,176.67 584.028,176.67 \n",
       "  584.074,176.67 584.119,176.67 584.165,176.67 584.21,176.67 584.256,176.67 584.301,176.67 584.346,176.67 584.392,176.67 584.437,176.67 584.483,176.67 \n",
       "  584.528,176.67 584.574,176.67 584.619,176.67 584.664,176.67 584.71,176.67 584.755,176.67 584.801,176.67 584.846,176.67 584.892,176.67 584.937,176.67 \n",
       "  584.983,176.67 585.028,176.67 585.073,176.67 585.119,176.67 585.164,176.67 585.21,176.67 585.255,176.67 585.301,176.67 585.346,176.67 585.391,176.67 \n",
       "  585.437,176.67 585.482,176.67 585.528,176.67 585.573,176.67 585.619,176.67 585.664,176.67 585.709,176.67 585.755,176.67 585.8,176.67 585.846,176.67 \n",
       "  585.891,176.67 585.937,176.67 585.982,176.67 586.027,176.67 586.073,176.67 586.118,176.67 586.164,176.67 586.209,176.67 586.255,176.67 586.3,176.67 \n",
       "  586.345,176.67 586.391,176.67 586.436,176.67 586.482,176.67 586.527,176.67 586.573,176.67 586.618,176.67 586.663,176.67 586.709,176.67 586.754,176.67 \n",
       "  586.8,176.67 586.845,176.67 586.891,176.67 586.936,176.67 586.981,176.67 587.027,176.67 587.072,176.67 587.118,176.67 587.163,176.67 587.209,176.67 \n",
       "  587.254,176.67 587.299,176.67 587.345,176.67 587.39,176.67 587.436,176.67 587.481,176.67 587.527,176.67 587.572,176.67 587.617,176.67 587.663,176.67 \n",
       "  587.708,176.67 587.754,176.67 587.799,176.67 587.845,176.67 587.89,176.67 587.935,176.67 587.981,176.67 588.026,176.67 588.072,176.67 588.117,176.67 \n",
       "  588.163,176.67 588.208,176.67 588.254,176.67 588.299,176.67 588.344,176.67 588.39,176.67 588.435,176.67 588.481,176.67 588.526,176.67 588.572,176.67 \n",
       "  588.617,176.67 588.662,176.67 588.708,176.67 588.753,176.67 588.799,176.67 588.844,176.67 588.89,176.67 588.935,176.67 588.98,176.67 589.026,176.67 \n",
       "  589.071,176.67 589.117,176.67 589.162,176.67 589.208,176.67 589.253,176.67 589.298,176.67 589.344,176.67 589.389,176.67 589.435,176.67 589.48,176.67 \n",
       "  589.526,176.67 589.571,176.67 589.616,176.67 589.662,176.67 589.707,176.67 589.753,176.67 589.798,176.67 589.844,176.67 589.889,176.67 589.934,176.67 \n",
       "  589.98,176.67 590.025,176.67 590.071,176.67 590.116,176.67 590.162,176.67 590.207,176.67 590.252,176.67 590.298,176.67 590.343,176.67 590.389,176.67 \n",
       "  590.434,176.67 590.48,176.67 590.525,176.67 590.57,176.67 590.616,176.67 590.661,176.67 590.707,176.67 590.752,176.67 590.798,176.67 590.843,176.67 \n",
       "  590.888,176.67 590.934,176.67 590.979,176.67 591.025,176.67 591.07,176.67 591.116,176.67 591.161,176.67 591.206,176.67 591.252,176.67 591.297,176.67 \n",
       "  591.343,176.67 591.388,176.67 591.434,176.67 591.479,176.67 591.525,176.67 591.57,176.67 591.615,176.67 591.661,176.67 591.706,176.67 591.752,176.67 \n",
       "  591.797,176.67 591.843,176.67 591.888,176.67 591.933,176.67 591.979,176.67 592.024,176.67 592.07,176.67 592.115,176.67 592.161,176.67 592.206,176.67 \n",
       "  592.251,176.67 592.297,176.67 592.342,176.67 592.388,176.67 592.433,176.67 592.479,176.67 592.524,176.67 592.569,176.67 592.615,176.67 592.66,176.67 \n",
       "  592.706,176.67 592.751,176.67 592.797,176.67 592.842,176.67 592.887,176.67 592.933,176.67 592.978,176.67 593.024,176.67 593.069,176.67 593.115,176.67 \n",
       "  593.16,176.67 593.205,176.67 593.251,176.67 593.296,176.67 593.342,176.67 593.387,176.67 593.433,176.67 593.478,176.67 593.523,176.67 593.569,176.67 \n",
       "  593.614,176.67 593.66,176.67 593.705,176.67 593.751,176.67 593.796,176.67 593.841,176.67 593.887,176.67 593.932,176.67 593.978,176.67 594.023,176.67 \n",
       "  594.069,176.67 594.114,176.67 594.159,176.67 594.205,176.67 594.25,176.67 594.296,176.67 594.341,176.67 594.387,176.67 594.432,176.67 594.477,176.67 \n",
       "  594.523,176.67 594.568,176.67 594.614,176.67 594.659,176.67 594.705,176.67 594.75,176.67 594.796,176.67 594.841,176.67 594.886,176.67 594.932,176.67 \n",
       "  594.977,176.67 595.023,176.67 595.068,176.67 595.114,176.67 595.159,176.67 595.204,176.67 595.25,176.67 595.295,176.67 595.341,176.67 595.386,176.67 \n",
       "  595.432,176.67 595.477,176.67 595.522,176.67 595.568,176.67 595.613,176.67 595.659,176.67 595.704,176.67 595.75,176.67 595.795,176.67 595.84,176.67 \n",
       "  595.886,176.67 595.931,176.67 595.977,176.67 596.022,176.67 596.068,176.67 596.113,176.67 596.158,176.67 596.204,176.67 596.249,176.67 596.295,176.67 \n",
       "  596.34,176.67 596.386,176.67 596.431,176.67 596.476,176.67 596.522,176.67 596.567,176.67 596.613,176.67 596.658,176.67 596.704,176.67 596.749,176.67 \n",
       "  596.794,176.67 596.84,176.67 596.885,176.67 596.931,176.67 596.976,176.67 597.022,176.67 597.067,176.67 597.112,176.67 597.158,176.67 597.203,176.67 \n",
       "  597.249,176.67 597.294,176.67 597.34,176.67 597.385,176.67 597.43,176.67 597.476,176.67 597.521,176.67 597.567,176.67 597.612,176.67 597.658,176.67 \n",
       "  597.703,176.67 597.749,176.67 597.794,176.67 597.839,176.67 597.885,176.67 597.93,176.67 597.976,176.67 598.021,176.67 598.067,176.67 598.112,176.67 \n",
       "  598.157,176.67 598.203,176.67 598.248,176.67 598.294,176.67 598.339,176.67 598.385,176.669 598.43,176.669 598.475,176.669 598.521,176.669 598.566,176.669 \n",
       "  598.612,176.669 598.657,176.669 598.703,176.669 598.748,176.669 598.793,176.669 598.839,176.669 598.884,176.669 598.93,176.669 598.975,176.669 599.021,176.669 \n",
       "  599.066,176.669 599.111,176.669 599.157,176.669 599.202,176.669 599.248,176.669 599.293,176.669 599.339,176.669 599.384,176.668 599.429,176.668 599.475,176.668 \n",
       "  599.52,176.668 599.566,176.668 599.611,176.668 599.657,176.668 599.702,176.668 599.747,176.668 599.793,176.668 599.838,176.668 599.884,176.668 599.929,176.667 \n",
       "  599.975,176.667 600.02,176.667 600.065,176.667 600.111,176.667 600.156,176.667 600.202,176.667 600.247,176.667 600.293,176.667 600.338,176.666 600.383,176.666 \n",
       "  600.429,176.666 600.474,176.666 600.52,176.666 600.565,176.666 600.611,176.666 600.656,176.665 600.701,176.665 600.747,176.665 600.792,176.665 600.838,176.665 \n",
       "  600.883,176.665 600.929,176.664 600.974,176.664 601.02,176.664 601.065,176.664 601.11,176.664 601.156,176.663 601.201,176.663 601.247,176.663 601.292,176.663 \n",
       "  601.338,176.662 601.383,176.662 601.428,176.662 601.474,176.662 601.519,176.661 601.565,176.661 601.61,176.661 601.656,176.66 601.701,176.66 601.746,176.66 \n",
       "  601.792,176.659 601.837,176.659 601.883,176.659 601.928,176.658 601.974,176.658 602.019,176.658 602.064,176.657 602.11,176.657 602.155,176.656 602.201,176.656 \n",
       "  602.246,176.656 602.292,176.655 602.337,176.655 602.382,176.654 602.428,176.654 602.473,176.653 602.519,176.653 602.564,176.652 602.61,176.652 602.655,176.651 \n",
       "  602.7,176.651 602.746,176.65 602.791,176.649 602.837,176.649 602.882,176.648 602.928,176.648 602.973,176.647 603.018,176.646 603.064,176.645 603.109,176.645 \n",
       "  603.155,176.644 603.2,176.643 603.246,176.643 603.291,176.642 603.336,176.641 603.382,176.64 603.427,176.639 603.473,176.638 603.518,176.637 603.564,176.637 \n",
       "  603.609,176.636 603.654,176.635 603.7,176.634 603.745,176.633 603.791,176.632 603.836,176.631 603.882,176.629 603.927,176.628 603.972,176.627 604.018,176.626 \n",
       "  604.063,176.625 604.109,176.624 604.154,176.622 604.2,176.621 604.245,176.62 604.291,176.618 604.336,176.617 604.381,176.615 604.427,176.614 604.472,176.613 \n",
       "  604.518,176.611 604.563,176.609 604.609,176.608 604.654,176.606 604.699,176.605 604.745,176.603 604.79,176.601 604.836,176.599 604.881,176.597 604.927,176.596 \n",
       "  604.972,176.594 605.017,176.592 605.063,176.59 605.108,176.588 605.154,176.586 605.199,176.583 605.245,176.581 605.29,176.579 605.335,176.577 605.381,176.574 \n",
       "  605.426,176.572 605.472,176.57 605.517,176.567 605.563,176.564 605.608,176.562 605.653,176.559 605.699,176.556 605.744,176.554 605.79,176.551 605.835,176.548 \n",
       "  605.881,176.545 605.926,176.542 605.971,176.539 606.017,176.536 606.062,176.532 606.108,176.529 606.153,176.526 606.199,176.522 606.244,176.519 606.289,176.515 \n",
       "  606.335,176.512 606.38,176.508 606.426,176.504 606.471,176.5 606.517,176.496 606.562,176.492 606.607,176.488 606.653,176.484 606.698,176.479 606.744,176.475 \n",
       "  606.789,176.471 606.835,176.466 606.88,176.461 606.925,176.457 606.971,176.452 607.016,176.447 607.062,176.442 607.107,176.437 607.153,176.431 607.198,176.426 \n",
       "  607.243,176.421 607.289,176.415 607.334,176.409 607.38,176.404 607.425,176.398 607.471,176.392 607.516,176.386 607.562,176.379 607.607,176.373 607.652,176.367 \n",
       "  607.698,176.36 607.743,176.353 607.789,176.347 607.834,176.34 607.88,176.332 607.925,176.325 607.97,176.318 608.016,176.31 608.061,176.303 608.107,176.295 \n",
       "  608.152,176.287 608.198,176.279 608.243,176.271 608.288,176.262 608.334,176.254 608.379,176.245 608.425,176.236 608.47,176.227 608.516,176.218 608.561,176.209 \n",
       "  608.606,176.2 608.652,176.19 608.697,176.18 608.743,176.17 608.788,176.16 608.834,176.15 608.879,176.139 608.924,176.129 608.97,176.118 609.015,176.107 \n",
       "  609.061,176.095 609.106,176.084 609.152,176.072 609.197,176.06 609.242,176.048 609.288,176.036 609.333,176.024 609.379,176.011 609.424,175.998 609.47,175.985 \n",
       "  609.515,175.972 609.56,175.958 609.606,175.945 609.651,175.931 609.697,175.917 609.742,175.902 609.788,175.887 609.833,175.873 609.878,175.857 609.924,175.842 \n",
       "  609.969,175.826 610.015,175.811 610.06,175.794 610.106,175.778 610.151,175.761 610.196,175.744 610.242,175.727 610.287,175.71 610.333,175.692 610.378,175.674 \n",
       "  610.424,175.656 610.469,175.637 610.514,175.618 610.56,175.599 610.605,175.58 610.651,175.56 610.696,175.54 610.742,175.52 610.787,175.499 610.833,175.478 \n",
       "  610.878,175.457 610.923,175.435 610.969,175.413 611.014,175.391 611.06,175.369 611.105,175.346 611.151,175.322 611.196,175.299 611.241,175.275 611.287,175.251 \n",
       "  611.332,175.226 611.378,175.201 611.423,175.176 611.469,175.15 611.514,175.124 611.559,175.097 611.605,175.07 611.65,175.043 611.696,175.016 611.741,174.988 \n",
       "  611.787,174.959 611.832,174.93 611.877,174.901 611.923,174.872 611.968,174.842 612.014,174.811 612.059,174.78 612.105,174.749 612.15,174.717 612.195,174.685 \n",
       "  612.241,174.653 612.286,174.62 612.332,174.586 612.377,174.552 612.423,174.518 612.468,174.483 612.513,174.448 612.559,174.412 612.604,174.376 612.65,174.339 \n",
       "  612.695,174.302 612.741,174.264 612.786,174.226 612.831,174.188 612.877,174.148 612.922,174.109 612.968,174.069 613.013,174.028 613.059,173.987 613.104,173.945 \n",
       "  613.149,173.903 613.195,173.86 613.24,173.817 613.286,173.773 613.331,173.728 613.377,173.683 613.422,173.638 613.467,173.592 613.513,173.545 613.558,173.498 \n",
       "  613.604,173.45 613.649,173.402 613.695,173.353 613.74,173.303 613.785,173.253 613.831,173.202 613.876,173.151 613.922,173.099 613.967,173.046 614.013,172.993 \n",
       "  614.058,172.939 614.104,172.884 614.149,172.829 614.194,172.773 614.24,172.717 614.285,172.66 614.331,172.602 614.376,172.544 614.422,172.485 614.467,172.425 \n",
       "  614.512,172.364 614.558,172.303 614.603,172.241 614.649,172.179 614.694,172.116 614.74,172.052 614.785,171.987 614.83,171.922 614.876,171.856 614.921,171.789 \n",
       "  614.967,171.721 615.012,171.653 615.058,171.584 615.103,171.514 615.148,171.444 615.194,171.373 615.239,171.3 615.285,171.228 615.33,171.154 615.376,171.08 \n",
       "  615.421,171.005 615.466,170.929 615.512,170.852 615.557,170.774 615.603,170.696 615.648,170.617 615.694,170.537 615.739,170.456 615.784,170.374 615.83,170.292 \n",
       "  615.875,170.209 615.921,170.125 615.966,170.04 616.012,169.954 616.057,169.867 616.102,169.78 616.148,169.691 616.193,169.602 616.239,169.512 616.284,169.421 \n",
       "  616.33,169.329 616.375,169.236 616.42,169.142 616.466,169.048 616.511,168.952 616.557,168.855 616.602,168.758 616.648,168.66 616.693,168.561 616.738,168.46 \n",
       "  616.784,168.359 616.829,168.257 616.875,168.154 616.92,168.05 616.966,167.945 617.011,167.84 617.057,167.733 617.102,167.625 617.147,167.516 617.193,167.406 \n",
       "  617.238,167.296 617.284,167.184 617.329,167.071 617.375,166.957 617.42,166.842 617.465,166.727 617.511,166.61 617.556,166.492 617.602,166.373 617.647,166.253 \n",
       "  617.693,166.133 617.738,166.011 617.783,165.888 617.829,165.764 617.874,165.639 617.92,165.512 617.965,165.385 618.011,165.257 618.056,165.128 618.101,164.997 \n",
       "  618.147,164.866 618.192,164.733 618.238,164.6 618.283,164.465 618.329,164.329 618.374,164.192 618.419,164.054 618.465,163.915 618.51,163.775 618.556,163.633 \n",
       "  618.601,163.491 618.647,163.347 618.692,163.203 618.737,163.057 618.783,162.91 618.828,162.762 618.874,162.613 618.919,162.462 618.965,162.311 619.01,162.158 \n",
       "  619.055,162.004 619.101,161.849 619.146,161.693 619.192,161.536 619.237,161.378 619.283,161.218 619.328,161.057 619.373,160.895 619.419,160.732 619.464,160.568 \n",
       "  619.51,160.402 619.555,160.236 619.601,160.068 619.646,159.899 619.691,159.729 619.737,159.557 619.782,159.385 619.828,159.211 619.873,159.036 619.919,158.86 \n",
       "  619.964,158.682 620.009,158.503 620.055,158.324 620.1,158.143 620.146,157.96 620.191,157.777 620.237,157.592 620.282,157.406 620.328,157.219 620.373,157.031 \n",
       "  620.418,156.841 620.464,156.65 620.509,156.458 620.555,156.265 620.6,156.07 620.646,155.875 620.691,155.678 620.736,155.48 620.782,155.28 620.827,155.08 \n",
       "  620.873,154.878 620.918,154.674 620.964,154.47 621.009,154.265 621.054,154.058 621.1,153.85 621.145,153.64 621.191,153.43 621.236,153.218 621.282,153.005 \n",
       "  621.327,152.79 621.372,152.575 621.418,152.358 621.463,152.14 621.509,151.921 621.554,151.7 621.6,151.479 621.645,151.256 621.69,151.031 621.736,150.806 \n",
       "  621.781,150.579 621.827,150.351 621.872,150.122 621.918,149.892 621.963,149.66 622.008,149.427 622.054,149.193 622.099,148.958 622.145,148.721 622.19,148.483 \n",
       "  622.236,148.244 622.281,148.004 622.326,147.762 622.372,147.52 622.417,147.276 622.463,147.03 622.508,146.784 622.554,146.536 622.599,146.288 622.644,146.038 \n",
       "  622.69,145.786 622.735,145.534 622.781,145.28 622.826,145.025 622.872,144.769 622.917,144.512 622.962,144.253 623.008,143.994 623.053,143.733 623.099,143.471 \n",
       "  623.144,143.208 623.19,142.943 623.235,142.677 623.28,142.411 623.326,142.143 623.371,141.874 623.417,141.603 623.462,141.332 623.508,141.059 623.553,140.785 \n",
       "  623.599,140.511 623.644,140.234 623.689,139.957 623.735,139.679 623.78,139.399 623.826,139.119 623.871,138.837 623.917,138.554 623.962,138.27 624.007,137.985 \n",
       "  624.053,137.699 624.098,137.411 624.144,137.123 624.189,136.834 624.235,136.543 624.28,136.251 624.325,135.958 624.371,135.665 624.416,135.37 624.462,135.074 \n",
       "  624.507,134.777 624.553,134.479 624.598,134.179 624.643,133.879 624.689,133.578 624.734,133.276 624.78,132.973 624.825,132.668 624.871,132.363 624.916,132.057 \n",
       "  624.961,131.749 625.007,131.441 625.052,131.132 625.098,130.822 625.143,130.51 625.189,130.198 625.234,129.885 625.279,129.571 625.325,129.256 625.37,128.94 \n",
       "  625.416,128.623 625.461,128.305 625.507,127.986 625.552,127.667 625.597,127.346 625.643,127.025 625.688,126.702 625.734,126.379 625.779,126.055 625.825,125.73 \n",
       "  625.87,125.404 625.915,125.078 625.961,124.75 626.006,124.422 626.052,124.093 626.097,123.763 626.143,123.432 626.188,123.1 626.233,122.768 626.279,122.435 \n",
       "  626.324,122.101 626.37,121.766 626.415,121.431 626.461,121.095 626.506,120.758 626.551,120.42 626.597,120.082 626.642,119.743 626.688,119.403 626.733,119.062 \n",
       "  626.779,118.721 626.824,118.379 626.87,118.037 626.915,117.694 626.96,117.35 627.006,117.006 627.051,116.661 627.097,116.315 627.142,115.969 627.188,115.622 \n",
       "  627.233,115.274 627.278,114.926 627.324,114.578 627.369,114.229 627.415,113.879 627.46,113.529 627.506,113.178 627.551,112.827 627.596,112.475 627.642,112.123 \n",
       "  627.687,111.77 627.733,111.417 627.778,111.063 627.824,110.709 627.869,110.354 627.914,109.999 627.96,109.644 628.005,109.288 628.051,108.932 628.096,108.576 \n",
       "  628.142,108.219 628.187,107.861 628.232,107.504 628.278,107.146 628.323,106.787 628.369,106.429 628.414,106.07 628.46,105.71 628.505,105.351 628.55,104.991 \n",
       "  628.596,104.631 628.641,104.271 628.687,103.91 628.732,103.549 628.778,103.188 628.823,102.827 628.868,102.466 628.914,102.104 628.959,101.742 629.005,101.381 \n",
       "  629.05,101.018 629.096,100.656 629.141,100.294 629.186,99.9317 629.232,99.5692 629.277,99.2066 629.323,98.844 629.368,98.4813 629.414,98.1186 629.459,97.7558 \n",
       "  629.504,97.393 629.55,97.0302 629.595,96.6674 629.641,96.3047 629.686,95.9419 629.732,95.5792 629.777,95.2165 629.822,94.8539 629.868,94.4914 629.913,94.129 \n",
       "  629.959,93.7667 630.004,93.4045 630.05,93.0424 630.095,92.6804 630.141,92.3187 630.186,91.957 630.231,91.5956 630.277,91.2343 630.322,90.8733 630.368,90.5124 \n",
       "  630.413,90.1518 630.459,89.7915 630.504,89.4313 630.549,89.0715 630.595,88.7119 630.64,88.3526 630.686,87.9936 630.731,87.635 630.777,87.2766 630.822,86.9187 \n",
       "  630.867,86.561 630.913,86.2037 630.958,85.8468 631.004,85.4903 631.049,85.1342 631.095,84.7786 631.14,84.4233 631.185,84.0685 631.231,83.7142 631.276,83.3603 \n",
       "  631.322,83.0069 631.367,82.654 631.413,82.3016 631.458,81.9497 631.503,81.5983 631.549,81.2475 631.594,80.8973 631.64,80.5476 631.685,80.1985 631.731,79.85 \n",
       "  631.776,79.5021 631.821,79.1548 631.867,78.8081 631.912,78.4621 631.958,78.1168 632.003,77.7721 632.049,77.4281 632.094,77.0848 632.139,76.7422 632.185,76.4003 \n",
       "  632.23,76.0592 632.276,75.7188 632.321,75.3791 632.367,75.0402 632.412,74.7021 632.457,74.3648 632.503,74.0283 632.548,73.6926 632.594,73.3578 632.639,73.0238 \n",
       "  632.685,72.6906 632.73,72.3583 632.775,72.0269 632.821,71.6963 632.866,71.3667 632.912,71.038 632.957,70.7102 633.003,70.3833 633.048,70.0574 633.094,69.7325 \n",
       "  633.139,69.4085 633.184,69.0855 633.23,68.7635 633.275,68.4425 633.321,68.1225 633.366,67.8036 633.412,67.4857 633.457,67.1688 633.502,66.8531 633.548,66.5383 \n",
       "  633.593,66.2247 633.639,65.9122 633.684,65.6008 633.73,65.2905 633.775,64.9813 633.82,64.6733 633.866,64.3665 633.911,64.0608 633.957,63.7562 634.002,63.4529 \n",
       "  634.048,63.1508 634.093,62.8498 634.138,62.5501 634.184,62.2516 634.229,61.9544 634.275,61.6584 634.32,61.3637 634.366,61.0702 634.411,60.778 634.456,60.4872 \n",
       "  634.502,60.1976 634.547,59.9093 634.593,59.6224 634.638,59.3368 634.684,59.0525 634.729,58.7696 634.774,58.488 634.82,58.2079 634.865,57.9291 634.911,57.6517 \n",
       "  634.956,57.3757 635.002,57.1011 635.047,56.8279 635.092,56.5562 635.138,56.2859 635.183,56.017 635.229,55.7496 635.274,55.4837 635.32,55.2193 635.365,54.9563 \n",
       "  635.41,54.6949 635.456,54.4349 635.501,54.1765 635.547,53.9195 635.592,53.6641 635.638,53.4103 635.683,53.158 635.728,52.9072 635.774,52.658 635.819,52.4104 \n",
       "  635.865,52.1644 635.91,51.92 635.956,51.6771 636.001,51.4359 636.046,51.1962 636.092,50.9582 636.137,50.7219 636.183,50.4871 636.228,50.254 636.274,50.0226 \n",
       "  636.319,49.7928 636.365,49.5647 636.41,49.3382 636.455,49.1135 636.501,48.8904 636.546,48.669 636.592,48.4494 636.637,48.2314 636.683,48.0151 636.728,47.8006 \n",
       "  636.773,47.5878 636.819,47.3768 636.864,47.1675 636.91,46.9599 636.955,46.7541 637.001,46.55 637.046,46.3478 637.091,46.1473 637.137,45.9485 637.182,45.7516 \n",
       "  637.228,45.5565 637.273,45.3631 637.319,45.1716 637.364,44.9819 637.409,44.7939 637.455,44.6079 637.5,44.4236 637.546,44.2412 637.591,44.0606 637.637,43.8818 \n",
       "  637.682,43.7049 637.727,43.5298 637.773,43.3566 637.818,43.1853 637.864,43.0158 637.909,42.8482 637.955,42.6825 638,42.5186 638.045,42.3567 638.091,42.1966 \n",
       "  638.136,42.0384 638.182,41.8821 638.227,41.7278 638.273,41.5753 638.318,41.4247 638.363,41.276 638.409,41.1293 638.454,40.9845 638.5,40.8416 638.545,40.7006 \n",
       "  638.591,40.5615 638.636,40.4244 638.681,40.2893 638.727,40.156 638.772,40.0247 638.818,39.8954 638.863,39.768 638.909,39.6425 638.954,39.519 638.999,39.3974 \n",
       "  639.045,39.2779 639.09,39.1602 639.136,39.0446 639.181,38.9308 639.227,38.8191 639.272,38.7093 639.317,38.6015 639.363,38.4957 639.408,38.3919 639.454,38.29 \n",
       "  639.499,38.1901 639.545,38.0921 639.59,37.9962 639.636,37.9022 639.681,37.8103 639.726,37.7203 639.772,37.6322 639.817,37.5462 639.863,37.4622 639.908,37.3801 \n",
       "  639.954,37.3 639.999,37.222 640.044,37.1459 640.09,37.0718 640.135,36.9996 640.181,36.9295 640.226,36.8614 640.272,36.7952 640.317,36.7311 640.362,36.6689 \n",
       "  640.408,36.6087 640.453,36.5506 640.499,36.4944 640.544,36.4402 640.59,36.3879 640.635,36.3377 640.68,36.2895 640.726,36.2432 640.771,36.1989 640.817,36.1566 \n",
       "  640.862,36.1163 640.908,36.078 640.953,36.0417 640.998,36.0073 641.044,35.9749 641.089,35.9445 641.135,35.9161 641.18,35.8896 641.226,35.8651 641.271,35.8426 \n",
       "  641.316,35.8221 641.362,35.8035 641.407,35.7869 641.453,35.7722 641.498,35.7595 641.544,35.7488 641.589,35.74 641.634,35.7332 641.68,35.7283 641.725,35.7254 \n",
       "  641.771,35.7244 641.816,35.7254 641.862,35.7283 641.907,35.7332 641.952,35.74 641.998,35.7487 642.043,35.7593 642.089,35.7719 642.134,35.7864 642.18,35.8028 \n",
       "  642.225,35.8212 642.27,35.8414 642.316,35.8636 642.361,35.8876 642.407,35.9136 642.452,35.9415 642.498,35.9712 642.543,36.0029 642.588,36.0365 642.634,36.0719 \n",
       "  642.679,36.1092 642.725,36.1484 642.77,36.1894 642.816,36.2324 642.861,36.2771 642.907,36.3238 642.952,36.3723 642.997,36.4226 643.043,36.4748 643.088,36.5289 \n",
       "  643.134,36.5848 643.179,36.6425 643.225,36.702 643.27,36.7634 643.315,36.8265 643.361,36.8915 643.406,36.9583 643.452,37.0269 643.497,37.0973 643.543,37.1695 \n",
       "  643.588,37.2434 643.633,37.3192 643.679,37.3967 643.724,37.476 643.77,37.5571 643.815,37.6399 643.861,37.7245 643.906,37.8108 643.951,37.8988 643.997,37.9886 \n",
       "  644.042,38.0802 644.088,38.1734 644.133,38.2684 644.179,38.3651 644.224,38.4635 644.269,38.5636 644.315,38.6654 644.36,38.7688 644.406,38.874 644.451,38.9808 \n",
       "  644.497,39.0893 644.542,39.1995 644.587,39.3113 644.633,39.4248 644.678,39.5399 644.724,39.6567 644.769,39.775 644.815,39.8951 644.86,40.0167 644.905,40.1399 \n",
       "  644.951,40.2647 644.996,40.3912 645.042,40.5192 645.087,40.6488 645.133,40.78 645.178,40.9127 645.223,41.0471 645.269,41.1829 645.314,41.3203 645.36,41.4593 \n",
       "  645.405,41.5998 645.451,41.7418 645.496,41.8853 645.541,42.0304 645.587,42.1769 645.632,42.3249 645.678,42.4745 645.723,42.6255 645.769,42.778 645.814,42.9319 \n",
       "  645.859,43.0873 645.905,43.2442 645.95,43.4025 645.996,43.5622 646.041,43.7234 646.087,43.886 646.132,44.05 646.178,44.2154 646.223,44.3822 646.268,44.5504 \n",
       "  646.314,44.7199 646.359,44.8909 646.405,45.0632 646.45,45.2369 646.496,45.4119 646.541,45.5882 646.586,45.7659 646.632,45.9449 646.677,46.1253 646.723,46.3069 \n",
       "  646.768,46.4899 646.814,46.6741 646.859,46.8596 646.904,47.0464 646.95,47.2345 646.995,47.4238 647.041,47.6143 647.086,47.8062 647.132,47.9992 647.177,48.1935 \n",
       "  647.222,48.389 647.268,48.5857 647.313,48.7836 647.359,48.9827 647.404,49.183 647.45,49.3844 647.495,49.587 647.54,49.7908 647.586,49.9958 647.631,50.2018 \n",
       "  647.677,50.409 647.722,50.6174 647.768,50.8268 647.813,51.0374 647.858,51.249 647.904,51.4618 647.949,51.6756 647.995,51.8905 648.04,52.1065 648.086,52.3235 \n",
       "  648.131,52.5416 648.176,52.7607 648.222,52.9808 648.267,53.202 648.313,53.4242 648.358,53.6473 648.404,53.8715 648.449,54.0967 648.494,54.3228 648.54,54.5499 \n",
       "  648.585,54.778 648.631,55.007 648.676,55.237 648.722,55.4679 648.767,55.6997 648.812,55.9324 648.858,56.1661 648.903,56.4007 648.949,56.6361 648.994,56.8724 \n",
       "  649.04,57.1096 649.085,57.3477 649.13,57.5866 649.176,57.8264 649.221,58.067 649.267,58.3084 649.312,58.5507 649.358,58.7938 649.403,59.0377 649.449,59.2823 \n",
       "  649.494,59.5278 649.539,59.774 649.585,60.0211 649.63,60.2688 649.676,60.5174 649.721,60.7666 649.767,61.0166 649.812,61.2674 649.857,61.5188 649.903,61.771 \n",
       "  649.948,62.0238 649.994,62.2774 650.039,62.5316 650.085,62.7865 650.13,63.0421 650.175,63.2983 650.221,63.5552 650.266,63.8127 650.312,64.0709 650.357,64.3297 \n",
       "  650.403,64.5891 650.448,64.8491 650.493,65.1097 650.539,65.3709 650.584,65.6326 650.63,65.895 650.675,66.1579 650.721,66.4214 650.766,66.6854 650.811,66.9499 \n",
       "  650.857,67.215 650.902,67.4806 650.948,67.7467 650.993,68.0134 651.039,68.2805 651.084,68.5481 651.129,68.8162 651.175,69.0848 651.22,69.3538 651.266,69.6233 \n",
       "  651.311,69.8932 651.357,70.1636 651.402,70.4344 651.447,70.7056 651.493,70.9773 651.538,71.2494 651.584,71.5218 651.629,71.7947 651.675,72.0679 651.72,72.3415 \n",
       "  651.765,72.6155 651.811,72.8898 651.856,73.1645 651.902,73.4395 651.947,73.7149 651.993,73.9906 652.038,74.2666 652.083,74.5429 652.129,74.8196 652.174,75.0965 \n",
       "  652.22,75.3737 652.265,75.6512 652.311,75.929 652.356,76.207 652.402,76.4853 652.447,76.7639 652.492,77.0427 652.538,77.3217 652.583,77.6009 652.629,77.8804 \n",
       "  652.674,78.1601 652.72,78.44 652.765,78.7201 652.81,79.0004 652.856,79.2808 652.901,79.5615 652.947,79.8423 652.992,80.1233 653.038,80.4044 653.083,80.6857 \n",
       "  653.128,80.9671 653.174,81.2486 653.219,81.5303 653.265,81.8121 653.31,82.094 653.356,82.376 653.401,82.6581 653.446,82.9403 653.492,83.2225 653.537,83.5049 \n",
       "  653.583,83.7873 653.628,84.0698 653.674,84.3523 653.719,84.6349 653.764,84.9175 653.81,85.2002 653.855,85.4829 653.901,85.7656 653.946,86.0483 653.992,86.3311 \n",
       "  654.037,86.6138 654.082,86.8965 654.128,87.1793 654.173,87.462 654.219,87.7446 654.264,88.0273 654.31,88.3099 654.355,88.5925 654.4,88.875 654.446,89.1574 \n",
       "  654.491,89.4398 654.537,89.7222 654.582,90.0044 654.628,90.2866 654.673,90.5687 654.718,90.8507 654.764,91.1326 654.809,91.4143 654.855,91.696 654.9,91.9776 \n",
       "  654.946,92.259 654.991,92.5403 655.036,92.8215 655.082,93.1025 655.127,93.3834 655.173,93.6641 655.218,93.9447 655.264,94.2251 655.309,94.5053 655.354,94.7854 \n",
       "  655.4,95.0653 655.445,95.345 655.491,95.6245 655.536,95.9038 655.582,96.1829 655.627,96.4618 655.673,96.7405 655.718,97.0189 655.763,97.2972 655.809,97.5752 \n",
       "  655.854,97.8529 655.9,98.1305 655.945,98.4078 655.991,98.6848 656.036,98.9616 656.081,99.2381 656.127,99.5143 656.172,99.7903 656.218,100.066 656.263,100.341 \n",
       "  656.309,100.617 656.354,100.891 656.399,101.166 656.445,101.44 656.49,101.714 656.536,101.988 656.581,102.261 656.627,102.534 656.672,102.807 656.717,103.079 \n",
       "  656.763,103.351 656.808,103.623 656.854,103.894 656.899,104.165 656.945,104.436 656.99,104.706 657.035,104.976 657.081,105.246 657.126,105.515 657.172,105.784 \n",
       "  657.217,106.052 657.263,106.32 657.308,106.588 657.353,106.855 657.399,107.122 657.444,107.389 657.49,107.655 657.535,107.921 657.581,108.186 657.626,108.451 \n",
       "  657.671,108.715 657.717,108.979 657.762,109.243 657.808,109.506 657.853,109.769 657.899,110.031 657.944,110.293 657.989,110.555 658.035,110.816 658.08,111.076 \n",
       "  658.126,111.336 658.171,111.596 658.217,111.855 658.262,112.114 658.307,112.372 658.353,112.63 658.398,112.887 658.444,113.144 658.489,113.401 658.535,113.657 \n",
       "  658.58,113.912 658.625,114.167 658.671,114.421 658.716,114.675 658.762,114.929 658.807,115.182 658.853,115.434 658.898,115.686 658.944,115.937 658.989,116.188 \n",
       "  659.034,116.439 659.08,116.689 659.125,116.938 659.171,117.187 659.216,117.435 659.262,117.683 659.307,117.93 659.352,118.177 659.398,118.423 659.443,118.669 \n",
       "  659.489,118.914 659.534,119.158 659.58,119.402 659.625,119.646 659.67,119.889 659.716,120.131 659.761,120.373 659.807,120.614 659.852,120.855 659.898,121.095 \n",
       "  659.943,121.335 659.988,121.574 660.034,121.812 660.079,122.05 660.125,122.287 660.17,122.524 660.216,122.76 660.261,122.996 660.306,123.231 660.352,123.465 \n",
       "  660.397,123.699 660.443,123.932 660.488,124.165 660.534,124.397 660.579,124.628 660.624,124.859 660.67,125.09 660.715,125.319 660.761,125.548 660.806,125.777 \n",
       "  660.852,126.005 660.897,126.232 660.942,126.459 660.988,126.685 661.033,126.91 661.079,127.135 661.124,127.359 661.17,127.583 661.215,127.806 661.26,128.028 \n",
       "  661.306,128.25 661.351,128.471 661.397,128.692 661.442,128.912 661.488,129.131 661.533,129.35 661.578,129.568 661.624,129.786 661.669,130.002 661.715,130.219 \n",
       "  661.76,130.434 661.806,130.649 661.851,130.863 661.896,131.077 661.942,131.29 661.987,131.503 662.033,131.715 662.078,131.926 662.124,132.136 662.169,132.346 \n",
       "  662.215,132.555 662.26,132.764 662.305,132.972 662.351,133.18 662.396,133.386 662.442,133.592 662.487,133.798 662.533,134.003 662.578,134.207 662.623,134.41 \n",
       "  662.669,134.613 662.714,134.816 662.76,135.017 662.805,135.218 662.851,135.418 662.896,135.618 662.941,135.817 662.987,136.016 663.032,136.213 663.078,136.41 \n",
       "  663.123,136.607 663.169,136.803 663.214,136.998 663.259,137.192 663.305,137.386 663.35,137.58 663.396,137.772 663.441,137.964 663.487,138.155 663.532,138.346 \n",
       "  663.577,138.536 663.623,138.725 663.668,138.914 663.714,139.102 663.759,139.29 663.805,139.476 663.85,139.662 663.895,139.848 663.941,140.033 663.986,140.217 \n",
       "  664.032,140.401 664.077,140.583 664.123,140.766 664.168,140.947 664.213,141.128 664.259,141.309 664.304,141.488 664.35,141.667 664.395,141.846 664.441,142.023 \n",
       "  664.486,142.2 664.531,142.377 664.577,142.553 664.622,142.728 664.668,142.902 664.713,143.076 664.759,143.249 664.804,143.422 664.849,143.594 664.895,143.765 \n",
       "  664.94,143.936 664.986,144.106 665.031,144.275 665.077,144.444 665.122,144.612 665.167,144.78 665.213,144.947 665.258,145.113 665.304,145.278 665.349,145.443 \n",
       "  665.395,145.608 665.44,145.771 665.486,145.934 665.531,146.097 665.576,146.259 665.622,146.42 665.667,146.58 665.713,146.74 665.758,146.899 665.804,147.058 \n",
       "  665.849,147.216 665.894,147.373 665.94,147.53 665.985,147.686 666.031,147.842 666.076,147.997 666.122,148.151 666.167,148.304 666.212,148.457 666.258,148.61 \n",
       "  666.303,148.762 666.349,148.913 666.394,149.063 666.44,149.213 666.485,149.363 666.53,149.511 666.576,149.659 666.621,149.807 666.667,149.954 666.712,150.1 \n",
       "  666.758,150.246 666.803,150.391 666.848,150.535 666.894,150.679 666.939,150.822 666.985,150.965 667.03,151.107 667.076,151.248 667.121,151.389 667.166,151.529 \n",
       "  667.212,151.669 667.257,151.808 667.303,151.946 667.348,152.084 667.394,152.221 667.439,152.358 667.484,152.494 667.53,152.629 667.575,152.764 667.621,152.898 \n",
       "  667.666,153.032 667.712,153.165 667.757,153.298 667.802,153.43 667.848,153.561 667.893,153.692 667.939,153.822 667.984,153.952 668.03,154.081 668.075,154.209 \n",
       "  668.12,154.337 668.166,154.464 668.211,154.591 668.257,154.717 668.302,154.843 668.348,154.968 668.393,155.093 668.438,155.217 668.484,155.34 668.529,155.463 \n",
       "  668.575,155.585 668.62,155.707 668.666,155.828 668.711,155.948 668.757,156.068 668.802,156.188 668.847,156.307 668.893,156.425 668.938,156.543 668.984,156.66 \n",
       "  669.029,156.777 669.075,156.893 669.12,157.009 669.165,157.124 669.211,157.238 669.256,157.352 669.302,157.466 669.347,157.579 669.393,157.691 669.438,157.803 \n",
       "  669.483,157.914 669.529,158.025 669.574,158.136 669.62,158.245 669.665,158.355 669.711,158.463 669.756,158.571 669.801,158.679 669.847,158.786 669.892,158.893 \n",
       "  669.938,158.999 669.983,159.105 670.029,159.21 670.074,159.314 670.119,159.418 670.165,159.522 670.21,159.625 670.256,159.727 670.301,159.829 670.347,159.931 \n",
       "  670.392,160.032 670.437,160.133 670.483,160.233 670.528,160.332 670.574,160.431 670.619,160.53 670.665,160.628 670.71,160.725 670.755,160.823 670.801,160.919 \n",
       "  670.846,161.015 670.892,161.111 670.937,161.206 670.983,161.301 671.028,161.395 671.073,161.489 671.119,161.582 671.164,161.675 671.21,161.767 671.255,161.859 \n",
       "  671.301,161.95 671.346,162.041 671.391,162.131 671.437,162.221 671.482,162.311 671.528,162.4 671.573,162.489 671.619,162.577 671.664,162.664 671.71,162.752 \n",
       "  671.755,162.838 671.8,162.925 671.846,163.01 671.891,163.096 671.937,163.181 671.982,163.265 672.028,163.349 672.073,163.433 672.118,163.516 672.164,163.599 \n",
       "  672.209,163.681 672.255,163.763 672.3,163.845 672.346,163.926 672.391,164.006 672.436,164.086 672.482,164.166 672.527,164.245 672.573,164.324 672.618,164.403 \n",
       "  672.664,164.481 672.709,164.558 672.754,164.636 672.8,164.712 672.845,164.789 672.891,164.865 672.936,164.94 672.982,165.015 673.027,165.09 673.072,165.165 \n",
       "  673.118,165.238 673.163,165.312 673.209,165.385 673.254,165.458 673.3,165.53 673.345,165.602 673.39,165.674 673.436,165.745 673.481,165.815 673.527,165.886 \n",
       "  673.572,165.956 673.618,166.025 673.663,166.095 673.708,166.163 673.754,166.232 673.799,166.3 673.845,166.367 673.89,166.435 673.936,166.502 673.981,166.568 \n",
       "  674.026,166.634 674.072,166.7 674.117,166.766 674.163,166.831 674.208,166.895 674.254,166.96 674.299,167.024 674.344,167.087 674.39,167.15 674.435,167.213 \n",
       "  674.481,167.276 674.526,167.338 674.572,167.4 674.617,167.461 674.662,167.522 674.708,167.583 674.753,167.644 674.799,167.704 674.844,167.763 674.89,167.823 \n",
       "  674.935,167.882 674.981,167.94 675.026,167.999 675.071,168.057 675.117,168.114 675.162,168.172 675.208,168.229 675.253,168.285 675.299,168.342 675.344,168.398 \n",
       "  675.389,168.454 675.435,168.509 675.48,168.564 675.526,168.619 675.571,168.673 675.617,168.727 675.662,168.781 675.707,168.834 675.753,168.887 675.798,168.94 \n",
       "  675.844,168.993 675.889,169.045 675.935,169.097 675.98,169.148 676.025,169.2 676.071,169.251 676.116,169.301 676.162,169.352 676.207,169.402 676.253,169.451 \n",
       "  676.298,169.501 676.343,169.55 676.389,169.599 676.434,169.648 676.48,169.696 676.525,169.744 676.571,169.791 676.616,169.839 676.661,169.886 676.707,169.933 \n",
       "  676.752,169.979 676.798,170.026 676.843,170.072 676.889,170.117 676.934,170.163 676.979,170.208 677.025,170.253 677.07,170.297 677.116,170.342 677.161,170.386 \n",
       "  677.207,170.43 677.252,170.473 677.297,170.516 677.343,170.559 677.388,170.602 677.434,170.645 677.479,170.687 677.525,170.729 677.57,170.77 677.615,170.812 \n",
       "  677.661,170.853 677.706,170.894 677.752,170.935 677.797,170.975 677.843,171.015 677.888,171.055 677.933,171.095 677.979,171.134 678.024,171.173 678.07,171.212 \n",
       "  678.115,171.251 678.161,171.289 678.206,171.327 678.252,171.365 678.297,171.403 678.342,171.44 678.388,171.478 678.433,171.515 678.479,171.551 678.524,171.588 \n",
       "  678.57,171.624 678.615,171.66 678.66,171.696 678.706,171.732 678.751,171.767 678.797,171.802 678.842,171.837 678.888,171.872 678.933,171.906 678.978,171.941 \n",
       "  679.024,171.975 679.069,172.008 679.115,172.042 679.16,172.075 679.206,172.109 679.251,172.141 679.296,172.174 679.342,172.207 679.387,172.239 679.433,172.271 \n",
       "  679.478,172.303 679.524,172.335 679.569,172.366 679.614,172.398 679.66,172.429 679.705,172.46 679.751,172.49 679.796,172.521 679.842,172.551 679.887,172.581 \n",
       "  679.932,172.611 679.978,172.641 680.023,172.67 680.069,172.699 680.114,172.728 680.16,172.757 680.205,172.786 680.25,172.815 680.296,172.843 680.341,172.871 \n",
       "  680.387,172.899 680.432,172.927 680.478,172.954 680.523,172.982 680.568,173.009 680.614,173.036 680.659,173.063 680.705,173.09 680.75,173.116 680.796,173.143 \n",
       "  680.841,173.169 680.886,173.195 680.932,173.221 680.977,173.246 681.023,173.272 681.068,173.297 681.114,173.322 681.159,173.347 681.204,173.372 681.25,173.396 \n",
       "  681.295,173.421 681.341,173.445 681.386,173.469 681.432,173.493 681.477,173.517 681.523,173.541 681.568,173.564 681.613,173.587 681.659,173.611 681.704,173.634 \n",
       "  681.75,173.656 681.795,173.679 681.841,173.702 681.886,173.724 681.931,173.746 681.977,173.768 682.022,173.79 682.068,173.812 682.113,173.834 682.159,173.855 \n",
       "  682.204,173.876 682.249,173.898 682.295,173.919 682.34,173.94 682.386,173.96 682.431,173.981 682.477,174.001 682.522,174.022 682.567,174.042 682.613,174.062 \n",
       "  682.658,174.082 682.704,174.102 682.749,174.121 682.795,174.141 682.84,174.16 682.885,174.179 682.931,174.198 682.976,174.217 683.022,174.236 683.067,174.255 \n",
       "  683.113,174.273 683.158,174.292 683.203,174.31 683.249,174.328 683.294,174.346 683.34,174.364 683.385,174.382 683.431,174.4 683.476,174.417 683.521,174.434 \n",
       "  683.567,174.452 683.612,174.469 683.658,174.486 683.703,174.503 683.749,174.52 683.794,174.536 683.839,174.553 683.885,174.569 683.93,174.586 683.976,174.602 \n",
       "  684.021,174.618 684.067,174.634 684.112,174.65 684.157,174.666 684.203,174.681 684.248,174.697 684.294,174.712 684.339,174.728 684.385,174.743 684.43,174.758 \n",
       "  684.475,174.773 684.521,174.788 684.566,174.802 684.612,174.817 684.657,174.832 684.703,174.846 684.748,174.86 684.794,174.875 684.839,174.889 684.884,174.903 \n",
       "  684.93,174.917 684.975,174.931 685.021,174.944 685.066,174.958 685.112,174.971 685.157,174.985 685.202,174.998 685.248,175.011 685.293,175.025 685.339,175.038 \n",
       "  685.384,175.051 685.43,175.063 685.475,175.076 685.52,175.089 685.566,175.101 685.611,175.114 685.657,175.126 685.702,175.139 685.748,175.151 685.793,175.163 \n",
       "  685.838,175.175 685.884,175.187 685.929,175.199 685.975,175.211 686.02,175.222 686.066,175.234 686.111,175.245 686.156,175.257 686.202,175.268 686.247,175.279 \n",
       "  686.293,175.291 686.338,175.302 686.384,175.313 686.429,175.324 686.474,175.334 686.52,175.345 686.565,175.356 686.611,175.366 686.656,175.377 686.702,175.387 \n",
       "  686.747,175.398 686.792,175.408 686.838,175.418 686.883,175.428 686.929,175.438 686.974,175.448 687.02,175.458 687.065,175.468 687.11,175.478 687.156,175.488 \n",
       "  687.201,175.497 687.247,175.507 687.292,175.516 687.338,175.526 687.383,175.535 687.428,175.544 687.474,175.553 687.519,175.563 687.565,175.572 687.61,175.581 \n",
       "  687.656,175.589 687.701,175.598 687.746,175.607 687.792,175.616 687.837,175.624 687.883,175.633 687.928,175.642 687.974,175.65 688.019,175.658 688.065,175.667 \n",
       "  688.11,175.675 688.155,175.683 688.201,175.691 688.246,175.699 688.292,175.707 688.337,175.715 688.383,175.723 688.428,175.731 688.473,175.739 688.519,175.746 \n",
       "  688.564,175.754 688.61,175.762 688.655,175.769 688.701,175.777 688.746,175.784 688.791,175.791 688.837,175.799 688.882,175.806 688.928,175.813 688.973,175.82 \n",
       "  689.019,175.827 689.064,175.834 689.109,175.841 689.155,175.848 689.2,175.855 689.246,175.862 689.291,175.869 689.337,175.875 689.382,175.882 689.427,175.888 \n",
       "  689.473,175.895 689.518,175.901 689.564,175.908 689.609,175.914 689.655,175.921 689.7,175.927 689.745,175.933 689.791,175.939 689.836,175.945 689.882,175.952 \n",
       "  689.927,175.958 689.973,175.964 690.018,175.97 690.063,175.975 690.109,175.981 690.154,175.987 690.2,175.993 690.245,175.999 690.291,176.004 690.336,176.01 \n",
       "  690.381,176.016 690.427,176.021 690.472,176.027 690.518,176.032 690.563,176.037 690.609,176.043 690.654,176.048 690.699,176.053 690.745,176.059 690.79,176.064 \n",
       "  690.836,176.069 690.881,176.074 690.927,176.079 690.972,176.084 691.018,176.089 691.063,176.094 691.108,176.099 691.154,176.104 691.199,176.109 691.245,176.114 \n",
       "  691.29,176.118 691.336,176.123 691.381,176.128 691.426,176.132 691.472,176.137 691.517,176.141 691.563,176.146 691.608,176.151 691.654,176.155 691.699,176.159 \n",
       "  691.744,176.164 691.79,176.168 691.835,176.172 691.881,176.177 691.926,176.181 691.972,176.185 692.017,176.189 692.062,176.194 692.108,176.198 692.153,176.202 \n",
       "  692.199,176.206 692.244,176.21 692.29,176.214 692.335,176.218 692.38,176.222 692.426,176.225 692.471,176.229 692.517,176.233 692.562,176.237 692.608,176.241 \n",
       "  692.653,176.244 692.698,176.248 692.744,176.252 692.789,176.255 692.835,176.259 692.88,176.263 692.926,176.266 692.971,176.27 693.016,176.273 693.062,176.277 \n",
       "  693.107,176.28 693.153,176.283 693.198,176.287 693.244,176.29 693.289,176.294 693.334,176.297 693.38,176.3 693.425,176.303 693.471,176.307 693.516,176.31 \n",
       "  693.562,176.313 693.607,176.316 693.652,176.319 693.698,176.322 693.743,176.325 693.789,176.328 693.834,176.331 693.88,176.334 693.925,176.337 693.97,176.34 \n",
       "  694.016,176.343 694.061,176.346 694.107,176.349 694.152,176.352 694.198,176.354 694.243,176.357 694.289,176.36 694.334,176.363 694.379,176.365 694.425,176.368 \n",
       "  694.47,176.371 694.516,176.373 694.561,176.376 694.607,176.379 694.652,176.381 694.697,176.384 694.743,176.386 694.788,176.389 694.834,176.391 694.879,176.394 \n",
       "  694.925,176.396 694.97,176.399 695.015,176.401 695.061,176.404 695.106,176.406 695.152,176.408 695.197,176.411 695.243,176.413 695.288,176.415 695.333,176.418 \n",
       "  695.379,176.42 695.424,176.422 695.47,176.424 695.515,176.426 695.561,176.429 695.606,176.431 695.651,176.433 695.697,176.435 695.742,176.437 695.788,176.439 \n",
       "  695.833,176.441 695.879,176.443 695.924,176.445 695.969,176.447 696.015,176.449 696.06,176.451 696.106,176.453 696.151,176.455 696.197,176.457 696.242,176.459 \n",
       "  696.287,176.461 696.333,176.463 696.378,176.465 696.424,176.467 696.469,176.469 696.515,176.47 696.56,176.472 696.605,176.474 696.651,176.476 696.696,176.478 \n",
       "  696.742,176.479 696.787,176.481 696.833,176.483 696.878,176.484 696.923,176.486 696.969,176.488 697.014,176.489 697.06,176.491 697.105,176.493 697.151,176.494 \n",
       "  697.196,176.496 697.241,176.498 697.287,176.499 697.332,176.501 697.378,176.502 697.423,176.504 697.469,176.505 697.514,176.507 697.56,176.508 697.605,176.51 \n",
       "  697.65,176.511 697.696,176.513 697.741,176.514 697.787,176.516 697.832,176.517 697.878,176.518 697.923,176.52 697.968,176.521 698.014,176.522 698.059,176.524 \n",
       "  698.105,176.525 698.15,176.527 698.196,176.528 698.241,176.529 698.286,176.53 698.332,176.532 698.377,176.533 698.423,176.534 698.468,176.536 698.514,176.537 \n",
       "  698.559,176.538 698.604,176.539 698.65,176.54 698.695,176.542 698.741,176.543 698.786,176.544 698.832,176.545 698.877,176.546 698.922,176.547 698.968,176.549 \n",
       "  699.013,176.55 699.059,176.551 699.104,176.552 699.15,176.553 699.195,176.554 699.24,176.555 699.286,176.556 699.331,176.557 699.377,176.558 699.422,176.559 \n",
       "  699.468,176.56 699.513,176.561 699.558,176.562 699.604,176.563 699.649,176.564 699.695,176.565 699.74,176.566 699.786,176.567 699.831,176.568 699.876,176.569 \n",
       "  699.922,176.57 699.967,176.571 700.013,176.572 700.058,176.573 700.104,176.574 700.149,176.575 700.194,176.576 700.24,176.577 700.285,176.577 700.331,176.578 \n",
       "  700.376,176.579 700.422,176.58 700.467,176.581 700.512,176.582 700.558,176.583 700.603,176.583 700.649,176.584 700.694,176.585 700.74,176.586 700.785,176.587 \n",
       "  700.831,176.587 700.876,176.588 700.921,176.589 700.967,176.59 701.012,176.59 701.058,176.591 701.103,176.592 701.149,176.593 701.194,176.593 701.239,176.594 \n",
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     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(PP[7:9]..., size=(750, 200), legend=false, layout=@layout([a b c]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$s$ を大きくすると, ガンマ函数の被積分函数(を函数で割ったもの)は正規分布の確率密度函数とほとんどぴったり一致するようになる. 次のセルを見よ."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# f(s,x) = e^{-x} x^{s-1} / Γ(s)\n",
    "# g(s,x) = e^{-(x-s)^2/(2s)} / √(2πs)\n",
    "\n",
    "f(s,x) = e^(-x+(s-1)*log(x)-lgamma(s))\n",
    "g(s,x) = e^(-(x-s)^2/(2s)) / √(2π*s)\n",
    "s = 100\n",
    "x = 0:0.5:2s\n",
    "plot(size=(400, 250))\n",
    "plot!(title=\"y = e^(-x) x^(s-1)/Gamma(s),  s = $s\", titlefontsize=11)\n",
    "plot!(x, f.(s,x), label=\"Gamma dist\", lw=2)\n",
    "plot!(x, g.(s,x), label=\"normal dist\", ls=:dash, lw=2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題:** ベータ函数を定義する積分の被積分函数のグラフを色々な $p,q>0$ について描いてみよ.\n",
    "\n",
    "**解答例:** 次のセルを見よ. $\\QED$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
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   "source": [
    "# ベータ函数の積分の被積分函数のグラフ\n",
    "\n",
    "f(p,q,x) = x^(p-1)*(1-x)^(q-1)\n",
    "x = 0.002:0.002:0.998\n",
    "PP = []\n",
    "for (p,q) in [(1/2,1/2), (1,1), (1,2), (2,2), (2,3), (2,4), (4,6), (8, 12), (16, 24)]\n",
    "    y = f.(p,q,x)\n",
    "    P = plot(x, y, title=\"(p,q) = ($p,$q)\", titlefontsize=10, xlims=(0,1), ylims=(0,1.05*maximum(y)))\n",
    "    push!(PP, P)\n",
    "end\n",
    "plot(PP[1:3]..., size=(750, 200), legend=false, layout=@layout([a b c]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
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       "  175.304,171.339 175.623,171.662 175.942,171.977 176.261,172.285 176.58,172.585 176.899,172.878 177.218,173.164 177.537,173.443 177.856,173.714 178.176,173.979 \n",
       "  178.495,174.237 178.814,174.488 179.133,174.732 179.452,174.97 179.771,175.201 180.09,175.426 180.409,175.645 180.728,175.857 181.047,176.063 181.366,176.263 \n",
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       "<polyline clip-path=\"url(#clip9500)\" style=\"stroke:#000000; stroke-width:1; stroke-opacity:1; fill:none\" points=\"\n",
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       "<polyline clip-path=\"url(#clip9500)\" style=\"stroke:#000000; stroke-width:1; stroke-opacity:1; fill:none\" points=\"\n",
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       "<g clip-path=\"url(#clip9500)\">\n",
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       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
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       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:middle;\" transform=\"rotate(0, 380.171, 195.449)\" x=\"380.171\" y=\"195.449\">0.4</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:middle;\" transform=\"rotate(0, 412.08, 195.449)\" x=\"412.08\" y=\"195.449\">0.6</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:middle;\" transform=\"rotate(0, 443.99, 195.449)\" x=\"443.99\" y=\"195.449\">0.8</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:middle;\" transform=\"rotate(0, 475.899, 195.449)\" x=\"475.899\" y=\"195.449\">1.0</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:end;\" transform=\"rotate(0, 308.853, 185.399)\" x=\"308.853\" y=\"185.399\">0</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:start;\" transform=\"rotate(0, 270.608, 163.003)\" x=\"270.608\" y=\"163.003\">1×10</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:10; text-anchor:start;\" transform=\"rotate(0, 297.707, 156.15)\" x=\"297.707\" y=\"156.15\">-</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:10; text-anchor:start;\" transform=\"rotate(0, 303.417, 156.15)\" x=\"303.417\" y=\"156.15\">6</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:start;\" transform=\"rotate(0, 270.608, 139.175)\" x=\"270.608\" y=\"139.175\">2×10</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:10; text-anchor:start;\" transform=\"rotate(0, 297.707, 132.323)\" x=\"297.707\" y=\"132.323\">-</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:10; text-anchor:start;\" transform=\"rotate(0, 303.417, 132.323)\" x=\"303.417\" y=\"132.323\">6</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:start;\" transform=\"rotate(0, 270.608, 115.348)\" x=\"270.608\" y=\"115.348\">3×10</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:10; text-anchor:start;\" transform=\"rotate(0, 297.707, 108.495)\" x=\"297.707\" y=\"108.495\">-</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:10; text-anchor:start;\" transform=\"rotate(0, 303.417, 108.495)\" x=\"303.417\" y=\"108.495\">6</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:start;\" transform=\"rotate(0, 270.608, 91.52)\" x=\"270.608\" y=\"91.52\">4×10</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:10; text-anchor:start;\" transform=\"rotate(0, 297.707, 84.6674)\" x=\"297.707\" y=\"84.6674\">-</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:10; text-anchor:start;\" transform=\"rotate(0, 303.417, 84.6674)\" x=\"303.417\" y=\"84.6674\">6</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:start;\" transform=\"rotate(0, 270.608, 67.6923)\" x=\"270.608\" y=\"67.6923\">5×10</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:10; text-anchor:start;\" transform=\"rotate(0, 297.707, 60.8397)\" x=\"297.707\" y=\"60.8397\">-</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:10; text-anchor:start;\" transform=\"rotate(0, 303.417, 60.8397)\" x=\"303.417\" y=\"60.8397\">6</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:start;\" transform=\"rotate(0, 270.608, 43.8647)\" x=\"270.608\" y=\"43.8647\">6×10</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:10; text-anchor:start;\" transform=\"rotate(0, 297.707, 37.0121)\" x=\"297.707\" y=\"37.0121\">-</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:10; text-anchor:start;\" transform=\"rotate(0, 303.417, 37.0121)\" x=\"303.417\" y=\"37.0121\">6</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:15; text-anchor:middle;\" transform=\"rotate(0, 396.126, 13.2)\" x=\"396.126\" y=\"13.2\">(p,q) = (8,12)</text>\n",
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       "  319.863,180.899 320.182,180.899 320.501,180.899 320.82,180.898 321.139,180.898 321.458,180.898 321.777,180.898 322.096,180.897 322.415,180.897 322.735,180.896 \n",
       "  323.054,180.895 323.373,180.894 323.692,180.892 324.011,180.891 324.33,180.888 324.649,180.885 324.968,180.881 325.287,180.877 325.606,180.871 325.926,180.865 \n",
       "  326.245,180.857 326.564,180.848 326.883,180.837 327.202,180.825 327.521,180.81 327.84,180.794 328.159,180.774 328.478,180.752 328.797,180.727 329.116,180.699 \n",
       "  329.436,180.667 329.755,180.631 330.074,180.59 330.393,180.545 330.712,180.495 331.031,180.439 331.35,180.377 331.669,180.309 331.988,180.233 332.307,180.151 \n",
       "  332.626,180.061 332.946,179.962 333.265,179.854 333.584,179.737 333.903,179.61 334.222,179.473 334.541,179.324 334.86,179.164 335.179,178.991 335.498,178.806 \n",
       "  335.817,178.608 336.136,178.395 336.456,178.168 336.775,177.925 337.094,177.667 337.413,177.393 337.732,177.101 338.051,176.792 338.37,176.464 338.689,176.118 \n",
       "  339.008,175.753 339.327,175.367 339.646,174.961 339.966,174.534 340.285,174.086 340.604,173.615 340.923,173.122 341.242,172.605 341.561,172.066 341.88,171.502 \n",
       "  342.199,170.913 342.518,170.3 342.837,169.661 343.156,168.997 343.476,168.307 343.795,167.591 344.114,166.848 344.433,166.078 344.752,165.281 345.071,164.457 \n",
       "  345.39,163.606 345.709,162.727 346.028,161.82 346.347,160.886 346.666,159.924 346.986,158.934 347.305,157.917 347.624,156.872 347.943,155.8 348.262,154.7 \n",
       "  348.581,153.573 348.9,152.419 349.219,151.238 349.538,150.031 349.857,148.798 350.176,147.539 350.496,146.255 350.815,144.946 351.134,143.612 351.453,142.255 \n",
       "  351.772,140.874 352.091,139.47 352.41,138.045 352.729,136.597 353.048,135.129 353.367,133.641 353.686,132.134 354.006,130.607 354.325,129.063 354.644,127.503 \n",
       "  354.963,125.926 355.282,124.333 355.601,122.727 355.92,121.107 356.239,119.475 356.558,117.832 356.877,116.178 357.197,114.515 357.516,112.843 357.835,111.165 \n",
       "  358.154,109.48 358.473,107.791 358.792,106.097 359.111,104.401 359.43,102.704 359.749,101.006 360.068,99.3085 360.387,97.6132 360.707,95.9212 361.026,94.2334 \n",
       "  361.345,92.5513 361.664,90.8758 361.983,89.2083 362.302,87.5499 362.621,85.9019 362.94,84.2653 363.259,82.6415 363.578,81.0316 363.897,79.4367 364.217,77.8581 \n",
       "  364.536,76.2968 364.855,74.7541 365.174,73.231 365.493,71.7287 365.812,70.2483 366.131,68.7908 366.45,67.3573 366.769,65.949 367.088,64.5667 367.407,63.2115 \n",
       "  367.727,61.8844 368.046,60.5864 368.365,59.3184 368.684,58.0812 369.003,56.8759 369.322,55.7031 369.641,54.5639 369.96,53.4589 370.279,52.389 370.598,51.3548 \n",
       "  370.917,50.3571 371.237,49.3966 371.556,48.4739 371.875,47.5897 372.194,46.7444 372.513,45.9387 372.832,45.1731 373.151,44.448 373.47,43.7639 373.789,43.1211 \n",
       "  374.108,42.5202 374.427,41.9613 374.747,41.4448 375.066,40.971 375.385,40.54 375.704,40.1521 376.023,39.8074 376.342,39.5061 376.661,39.2482 376.98,39.0338 \n",
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       "  386.872,52.6617 387.191,53.6701 387.51,54.7064 387.829,55.7699 388.148,56.8596 388.468,57.9748 388.787,59.1144 389.106,60.2777 389.425,61.4637 389.744,62.6715 \n",
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       "<g clip-path=\"url(#clip9500)\">\n",
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       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:middle;\" transform=\"rotate(0, 634.587, 195.449)\" x=\"634.587\" y=\"195.449\">0.4</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:middle;\" transform=\"rotate(0, 666.497, 195.449)\" x=\"666.497\" y=\"195.449\">0.6</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:middle;\" transform=\"rotate(0, 698.406, 195.449)\" x=\"698.406\" y=\"195.449\">0.8</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:middle;\" transform=\"rotate(0, 730.315, 195.449)\" x=\"730.315\" y=\"195.449\">1.0</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:end;\" transform=\"rotate(0, 563.269, 185.399)\" x=\"563.269\" y=\"185.399\">0</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:start;\" transform=\"rotate(0, 519.588, 153.343)\" x=\"519.588\" y=\"153.343\">2×10</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:10; text-anchor:start;\" transform=\"rotate(0, 546.687, 146.49)\" x=\"546.687\" y=\"146.49\">-</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:10; text-anchor:start;\" transform=\"rotate(0, 552.397, 146.49)\" x=\"552.397\" y=\"146.49\">12</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:start;\" transform=\"rotate(0, 519.588, 119.855)\" x=\"519.588\" y=\"119.855\">4×10</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:10; text-anchor:start;\" transform=\"rotate(0, 546.687, 113.003)\" x=\"546.687\" y=\"113.003\">-</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:10; text-anchor:start;\" transform=\"rotate(0, 552.397, 113.003)\" x=\"552.397\" y=\"113.003\">12</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:start;\" transform=\"rotate(0, 519.588, 86.3675)\" x=\"519.588\" y=\"86.3675\">6×10</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:10; text-anchor:start;\" transform=\"rotate(0, 546.687, 79.5149)\" x=\"546.687\" y=\"79.5149\">-</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:10; text-anchor:start;\" transform=\"rotate(0, 552.397, 79.5149)\" x=\"552.397\" y=\"79.5149\">12</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:start;\" transform=\"rotate(0, 519.588, 52.8798)\" x=\"519.588\" y=\"52.8798\">8×10</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:10; text-anchor:start;\" transform=\"rotate(0, 546.687, 46.0272)\" x=\"546.687\" y=\"46.0272\">-</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:10; text-anchor:start;\" transform=\"rotate(0, 552.397, 46.0272)\" x=\"552.397\" y=\"46.0272\">12</text>\n",
       "</g>\n",
       "<g clip-path=\"url(#clip9500)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:15; text-anchor:middle;\" transform=\"rotate(0, 650.542, 13.2)\" x=\"650.542\" y=\"13.2\">(p,q) = (16,24)</text>\n",
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       "  580.661,180.899 580.98,180.899 581.299,180.899 581.618,180.899 581.937,180.899 582.256,180.899 582.575,180.899 582.895,180.899 583.214,180.899 583.533,180.899 \n",
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       "  587.043,180.897 587.362,180.896 587.681,180.896 588,180.895 588.319,180.894 588.638,180.893 588.957,180.891 589.276,180.89 589.595,180.887 589.915,180.885 \n",
       "  590.234,180.882 590.553,180.879 590.872,180.874 591.191,180.87 591.51,180.864 591.829,180.857 592.148,180.849 592.467,180.84 592.786,180.83 593.105,180.818 \n",
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     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "plot(PP[7:9]..., size=(750, 200), legend=false, layout=@layout([a b c]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$p,q$ がそれらの比を保ちながら大きくすると, ベータ函数の被積分函数をベータ函数で割ったものは正規分布の被積分函数にほとんどぴったり一致するようになる. 次のセルを見よ."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
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       "  \" fill=\"#ffffff\" fill-opacity=\"1\"/>\n",
       "<polyline clip-path=\"url(#clip9700)\" style=\"stroke:#000000; stroke-width:1; stroke-opacity:1; fill:none\" points=\"\n",
       "  269.472,85.7361 368.315,85.7361 368.315,40.3761 269.472,40.3761 269.472,85.7361 \n",
       "  \"/>\n",
       "<polyline clip-path=\"url(#clip9700)\" style=\"stroke:#009af9; stroke-width:2; stroke-opacity:1; fill:none\" points=\"\n",
       "  273.472,55.4961 297.472,55.4961 \n",
       "  \"/>\n",
       "<g clip-path=\"url(#clip9700)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:start;\" transform=\"rotate(0, 301.472, 59.9961)\" x=\"301.472\" y=\"59.9961\">Beta dist</text>\n",
       "</g>\n",
       "<polyline clip-path=\"url(#clip9700)\" style=\"stroke:#e26f46; stroke-width:2; stroke-opacity:1; fill:none\" stroke-dasharray=\"8, 5\" points=\"\n",
       "  273.472,70.6161 297.472,70.6161 \n",
       "  \"/>\n",
       "<g clip-path=\"url(#clip9700)\">\n",
       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:12; text-anchor:start;\" transform=\"rotate(0, 301.472, 75.1161)\" x=\"301.472\" y=\"75.1161\">normal dist</text>\n",
       "</g>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# f(p,q,x) = x^{p-1} (1-x)^{q-1} / B(p,q)\n",
    "# μ = p/(p+q)\n",
    "# σ² = pq/((p+q)^2(p+q+1))\n",
    "# g(μ,σ²,x) = e^{-(x-μ)^2/(2σ²)} / √(2πσ²)\n",
    "\n",
    "f(p,q,x) = x^(p-1)*(1-x)^(q-1)/beta(p,q)\n",
    "g(μ,σ²,x) = e^(-(x-μ)^2/(2*σ²)) / √(2π*σ²)\n",
    "p, q = 45,55\n",
    "μ = p/(p+q)\n",
    "σ² = p*q/((p+q)^2*(p+q+1))\n",
    "x = 0.000:0.002:1.000\n",
    "plot(size=(400, 250))\n",
    "plot!(title=\"y = x^(p-1) (1-x)^(q-1) / B(p,q),  (p,q) = ($p,$q)\", titlefontsize=10)\n",
    "plot!(x, f.(p,q,x), label=\"Beta dist\", lw=2)\n",
    "plot!(x, g.(μ,σ²,x), label=\"normal dist\", lw=2, ls=:dash)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### ガンマ函数の特殊値と函数等式\n",
    "\n",
    "**問題(ガンマ函数の最も簡単な特殊値):** $\\Gamma(1)=1$ と $\\Gamma(1/2)=\\sqrt{\\pi}$ を示せ.\n",
    "\n",
    "**解答例:** 前者は\n",
    "\n",
    "$$\n",
    "\\Gamma(1)=\\int_0^\\infty e^{-x}\\,dx = [-e^{-x}]_0^\\infty = 1.\n",
    "$$\n",
    "\n",
    "と容易に示される. 後者を示すためには $\\Gamma(1/2)$ がGauss積分 $\\int_{-\\infty}^\\infty e^{-y^2}\\,dy=\\sqrt{\\pi}$ に等しいことを示せばよい.  $x=y^2$ で置換積分すると,\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\Gamma(1/2) &= \\int_0^\\infty e^{-x}x^{1/2-1}\\,dx =\n",
    "\\int_0^\\infty e^{-y^2} \\frac{1}{y} 2y\\,dy \n",
    "\\\\ &= \n",
    "2\\int_0^\\infty e^{-y^2}\\,dy =\n",
    "\\int_{-\\infty}^\\infty e^{-y^2}\\,dy = \\sqrt{\\pi}. \n",
    "\\qquad \\QED\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "**注意:** 上の問題の解答より, $\\Gamma(1/2)$ は本質的にGauss積分に等しい. その意味でガンマ函数はGauss積分の一般化になっていると言える. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題(ガンマ函数の函数等式):** $s>0$ のとき $\\Gamma(s+1)=s\\Gamma(s)$ となることを示せ.\n",
    "\n",
    "**解答例:** 部分積分を使う. $s>0$ と仮定する. このとき\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\Gamma(s+1) &=\n",
    "\\int_0^\\infty e^{-x}x^s\\,dx =\n",
    "\\int_0^\\infty (-e^{-x})'x^s\\,dx\n",
    "\\\\ &=\n",
    "\\int_0^\\infty e^{-x} (x^s)'\\,dx =\n",
    "\\int_0^\\infty e^{-x} sx^{s-1}\\,dx =\n",
    "s\\Gamma(s).\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "3つ目の等号で部分積分を行った. そのとき, $x\\searrow 0$ でも $x\\to\\infty$ でも $e^{-x}x^s\\to 0$ となることを使った($s>0$ と仮定したことに注意せよ).  (積分以外の項が消える.) $\\QED$\n",
    "\n",
    "**注意:** 上の問題の結果を使えば, $s< 0$, $s\\ne 0,-1,-2,\\ldots$ のとき $s+n>0$ となる整数 $n$ を取れば, \n",
    "\n",
    "$$\n",
    "\\Gamma(s) = \\frac{\\Gamma(s+n)}{s(s+1)\\cdots(s+n-1)}\n",
    "$$\n",
    "\n",
    "の右辺はwell-definedになるので, この公式によってガンマ函数を $s<0$, $s\\ne 0,-1,-2,\\ldots$ の場合に自然に拡張できる. $\\QED$ \n",
    "\n",
    "**注意(ガンマ函数は階乗の一般化):** 以上の問題の結果より, 非負の整数 $n$ について\n",
    "\n",
    "$$\n",
    "\\Gamma(n+1)=n\\Gamma(n)=n(n-1)\\Gamma(n-1)=\\cdots=n(n-1)\\cdots1\\,\\Gamma(1)=n!.\n",
    "$$\n",
    "\n",
    "すなわち, $\\Gamma(s+1)$ は階乗 $n!$ の連続変数 $s$ への拡張になっていることがわかる. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題(ガンマ函数の正の半整数での値):** 次を示せ: 非負の整数 $k$ に対して\n",
    "\n",
    "$$\n",
    "\\Gamma((2k+1)/2) = \\frac{1\\cdot3\\cdots(2k-1)}{2^k}\\sqrt{\\pi} =\n",
    "\\frac{(2k)!}{2^{2k}k!}\\sqrt{\\pi}\n",
    "$$\n",
    "\n",
    "**解答例1:** ガンマ函数の函数等式と $\\Gamma(1/2)=\\sqrt{\\pi}$ より\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\Gamma\\left(\\frac{2k+1}{2}\\right) &=\n",
    "\\frac{2k-1}{2}\\Gamma\\left(\\frac{2k-1}{2}\\right) =\n",
    "\\frac{2k-1}{2}\\frac{2k-3}{2}\\Gamma\\left(\\frac{2k-3}{2}\\right) = \\cdots \n",
    "\\\\ &=\n",
    "\\frac{2k-1}{2}\\frac{2k-3}{2}\\cdots\\frac{1}{2}\\Gamma\\left(\\frac{1}{2}\\right) = \n",
    "\\frac{1\\cdot3\\cdots(2k-1)}{2^k}\\sqrt{\\pi}.\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "これで示したい公式の1つ目の等号は示せた. 2つ目の等号は上の方のGauss積分の応用問題で使った方法を使えば同様に示される. $\\QED$\n",
    "\n",
    "**解答例2:** ガンマ函数の $\\Gamma(s)=2\\int_0^\\infty e^{-y^2}y^{2s-1}\\,dy$ という表示を使うと,\n",
    "\n",
    "$$\n",
    "\\Gamma((2k+1)/2) = 2\\int_0^\\infty e^{-y^2} y^{2k}\\,dy = \\int_{-\\infty}^\\infty e^{-y^2} y^{2k}\\,dy\n",
    "$$\n",
    "\n",
    "なので, 上の方のGauss積分の応用問題に関する結果から欲しい公式が得られる. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Riemannのゼータ函数の積分表示と函数等式と負の整数と正の偶数における特殊値\n",
    "\n",
    "この節はこのノートを最初に読むときには飛ばして読んでも構わない. ガンマ函数の理論がRiemannのゼータ函数の理論と密接に関係していることを認識しておけば問題ない. \n",
    "\n",
    "Bernoulli数やBernoulli多項式に関してはノート「<a href=\"http://nbviewer.jupyter.org/github/genkuroki/Calculus/blob/master/13%20Euler-Maclaurin%20summation%20formula.ipynb\">13 Euler-Maclaurinの和公式</a>」により詳しい解説がある."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Riemannのゼータ函数の積分表示1\n",
    "\n",
    "**問題(Riemannのゼータ函数の積分表示1):** 次をが成立することを示せ.\n",
    "\n",
    "$$\n",
    "\\zeta(s)=\\sum_{n=1}^\\infty \\frac{1}{n^s} = \n",
    "\\frac{1}{\\Gamma(s)}\\int_0^\\infty \\frac{x^{s-1}\\,dx}{e^x-1}\n",
    "\\quad (s>1).\n",
    "$$\n",
    "\n",
    "**注意:** $x\\to 0$ のとき $\\ds \\frac{e^x-1}{x}\\to 1$ となるので,  $\\ds \\frac{x}{e^x-1}$ は $x=0$ まで連続的に拡張され, この公式の積分は\n",
    "\n",
    "$$\n",
    "\\int_0^\\infty \\frac{x^{s-1}\\,dx}{e^x-1} = \n",
    "\\int_0^\\infty \\frac{x}{e^x-1} x^{s-2}\\,dx\n",
    "$$\n",
    "\n",
    "と書けるので, $s-2 > -1$ すなわち $s>1$ ならば収束している. $\\QED$\n",
    "\n",
    "**解答例:** 上の問題の結果より, $\\ds\\frac{1}{n^s} = \\frac{1}{\\Gamma(s)}\\int_0^\\infty e^{-nx}x^{s-1}\\,dx$ なので,\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\zeta(s) &=\\sum_{n=1}^\\infty \\frac{1}{n^s} =\n",
    "\\frac{1}{\\Gamma(s)}\\sum_{n=1}^\\infty\\int_0^\\infty e^{-nx}x^{s-1}\\,dx\n",
    "\\\\ &=\n",
    "\\frac{1}{\\Gamma(s)}\\int_0^\\infty\\sum_{n=1}^\\infty e^{-nx}x^{s-1}\\,dx =\n",
    "\\frac{1}{\\Gamma(s)}\\int_0^\\infty \\frac{x^{s-1}\\,dx}{e^x-1}.\n",
    "\\qquad \\QED\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**定義:** Bernoulli数 $B_n$ ($n=0,1,2,\\ldots$) を次の条件によって定める:\n",
    "\n",
    "$$\n",
    "\\frac{z}{e^z-1} = \\sum_{n=1}^\\infty \\frac{B_n}{n!}z^n.\n",
    "\\qquad\\QED\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題:** $B_0=1$, $\\ds B_1=-\\frac{1}{2}$ であり, $n$ が3以上の奇数のとき $B_n=0$ となることを示せ.\n",
    "\n",
    "**解答例:** $z\\to 0$ のとき, $\\ds \\frac{z}{e^z-1}\\to 1$ より $B_0=1$ となる.\n",
    "\n",
    "$$\n",
    "\\frac{z}{e^z-1}-\\frac{z}{2} = \\frac{z}{2}\\frac{e^z+1}{e^z-1} = \n",
    "\\frac{z}{2}\\frac{e^{z/2}+e^{-z/2}}{e^{z/2}-e^{-z/2}}\n",
    "$$\n",
    "\n",
    "であることと, これが偶函数であることから, $\\ds B_1=-\\frac{1}{2}$ で $n$ が3以上の奇数ならば $B_n=0$ となることがわかる. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題(Riemannのゼータ函数の積分表示1'):** 非負の整数 $N$ に対して, 次を示せ:\n",
    "\n",
    "$$\n",
    "\\zeta(s) = \n",
    "\\frac{1}{\\Gamma(s)}\\left[\n",
    "\\int_1^\\infty \\frac{x^{s-1}\\,dx}{e^x-1} +\n",
    "\\int_0^1 \\left(\\frac{x}{e^x-1} - \\sum_{k=0}^N \\frac{B_k}{k!}x^k\\right)x^{s-2}\\,dx +\n",
    "\\sum_{k=0}^N \\frac{B_k}{k!}\\frac{1}{s+k-1}\n",
    "\\right].\n",
    "$$\n",
    "\n",
    "さらに右辺の括弧の内側の2つ目の積分が $s>-N$ で絶対収束していることを示せ.\n",
    "\n",
    "**解答例:** Riemannのゼータ函数の積分表示1の公式で積分を $\\int_1^\\infty$ と $\\int_0^1$ に分けて, $k=0,1,\\ldots,N$ に対する\n",
    "\n",
    "$$\n",
    "\\ds\\int_0^1\\frac{B_k}{k!}x^{s+k-2}\\,dx = \\frac{B_k}{k!}\\frac{1}{s+k-1}\n",
    "$$\n",
    "\n",
    "を足して引けば示したい公式が得られる. \n",
    "\n",
    "$$\n",
    "\\frac{x}{e^x-1} - \\sum_{k=0}^N \\frac{B_k}{k!}x^k = O(x^{N+1})\n",
    "$$\n",
    "\n",
    "であり, $\\ds\\int_0^1 x^{N+1}x^{s-2}\\,dx=\\int_0^1 x^{s+N-1}\\,dx$ が $s>-N$ で絶対収束していることから, 右辺の括弧の内側の2つ目の積分もそこで収束している. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題:** Riemannのゼータ函数の積分表示1'の右辺で $\\zeta(s)$ を $s>-N$ まで拡張しておくとき, \n",
    "\n",
    "$$\n",
    "\\zeta(0) = -\\frac{1}{2}, \\quad \\zeta(-r) = -\\frac{B_{r+1}}{r+1} \\quad (r=1,2,3,\\ldots)\n",
    "$$\n",
    "\n",
    "となることを示せ. ($r$ が2以上の偶数のとき $B_{r+1}=0$ となることに注意せよ.)\n",
    "\n",
    "**解答例:** ガンマ函数の函数等式より,\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\frac{1}{\\Gamma(s)}\\frac{B_k}{k!}\\frac{1}{s+k-1} &=\n",
    "\\frac{s(s+1)\\cdots(s+k-2)(s+k-1)}{\\Gamma(s+k)}\\frac{B_k}{k!}\\frac{1}{s+k-1} \n",
    "\\\\ &=\n",
    "\\frac{s(s+1)\\cdots(s+k-2)}{\\Gamma(s+k)}\\frac{B_k}{k!}\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "なので, 非負の整数 $r$ に対して, $k=r+1$ とおいて $s\\to -r$ とすると,\n",
    "\n",
    "$$\n",
    "\\frac{1}{\\Gamma(s)}\\frac{B_k}{k!}\\frac{1}{s+k-1}\\to\n",
    "(-1)^r \\frac{B_{r+1}}{r+1} =\n",
    "\\begin{cases}\n",
    "-\\dfrac{1}{2} & (r=0) \\\\\n",
    "-\\dfrac{B_{r+1}}{r+1} & (r=1,2,3,\\ldots)\n",
    "\\end{cases}.\n",
    "$$\n",
    "\n",
    "ただし, 等号で, $\\ds B_1=-\\frac{1}{2}$ と $r+1$ が3以上の奇数のとき $B_{r+1}=0$ となることを使った. これをRiemannのゼータ函数の積分表示1'\n",
    "\n",
    "$$\n",
    "\\zeta(s) = \n",
    "\\frac{1}{\\Gamma(s)}\\left[\n",
    "\\int_1^\\infty \\frac{x^{s-1}\\,dx}{e^x-1} +\n",
    "\\int_0^1 \\left(\\frac{x}{e^x-1} - \\sum_{k=0}^N \\frac{B_k}{k!}x^k\\right)x^{s-2}\\,dx +\n",
    "\\sum_{k=0}^N \\frac{B_k}{k!}\\frac{1}{s+k-1}\n",
    "\\right]\n",
    "$$\n",
    "\n",
    "に適用すれば,\n",
    "\n",
    "$$\n",
    "\\zeta(0) = -\\frac{1}{2}, \\quad \\zeta(-r) = -\\frac{B_{r+1}}{r+1} \\quad (r=1,2,3,\\ldots)\n",
    "$$\n",
    "\n",
    "が得られる.  $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題:** 次を示せ.\n",
    "\n",
    "$$\n",
    "(1-2^{1-s})\\zeta(s)=\\sum_{n=1}^\\infty \\frac{(-1)^{n-1}}{n^s} = \\frac{1}{\\Gamma(s)}\\int_0^\\infty \\frac{x^{s-1}\\,dx}{e^x+1} \\quad (s>1).\n",
    "$$\n",
    "\n",
    "**注意:** この公式の積分は $s>0$ という条件を外して, $s$ が任意の複素数にしても絶対収束している. この公式は $(1-2^{1-s})\\zeta(s)$ の複素平面全体への解析接続を与える. $\\QED$\n",
    "\n",
    "**解答例:** 1つ目の等号を示そう:\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "&\n",
    "\\zeta(s) = \\frac{1}{1^s}+\\frac{1}{2^s}+\\frac{1}{3^s}+\\frac{1}{4^s}+\\cdots,\n",
    "\\\\ &\n",
    "2^{1-s}\\zeta(s) = \\frac{2}{2^s}+\\frac{2}{4^s}+\\frac{2}{6^s}+\\frac{2}{8^s}+\\cdots,\n",
    "\\\\ &\n",
    "(1-2^{1-s})\\zeta(s) = \\frac{1}{1^s}-\\frac{1}{2^s}+\\frac{1}{3^s}-\\frac{1}{4^s}+\\cdots =\n",
    "\\sum_{n=1}^\\infty \\frac{(-1)^{n-1}}{n^s}.\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "2つ目の等号を示そう. 上の問題の解答例と同様にして, $\\ds\\frac{1}{n^s} = \\frac{1}{\\Gamma(s)}\\int_0^\\infty e^{-nx}x^{s-1}\\,dx$ なので,\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\sum_{n=1}^\\infty \\frac{(-1)^{n-1}}{n^s} &=\n",
    "\\frac{1}{\\Gamma(s)}\\sum_{n=1}^\\infty(-1)^{n-1}\\int_0^\\infty e^{-nx}x^{s-1}\\,dx\n",
    "\\\\ &=\n",
    "\\frac{1}{\\Gamma(s)}\\int_0^\\infty\\sum_{n=1}^\\infty (-1)^{n-1}e^{-nx}x^{s-1}\\,dx =\n",
    "\\frac{1}{\\Gamma(s)}\\int_0^\\infty \\frac{x^{s-1}\\,dx}{e^x+1}\\,dx.\n",
    "\\qquad \\QED\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "**注意:** 以上の計算は統計力学における<a href=\"https://www.google.co.jp/search?q=Dirac-Fermi%E5%88%86%E5%B8%83+%CE%B6\">Fermi-Dirac統計</a>に関する議論に登場する. ゼータ函数は数論の基本であるだけではなく, 統計力学的にも意味を持っている. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Hurwitzのゼータ函数の積分表示1\n",
    "\n",
    "**問題:** **Hurwitzのゼータ函数** $\\zeta(s,x)$ と**Bernoulli多項式** $B_k(x)$ を\n",
    "\n",
    "$$\n",
    "\\zeta(s,x) = \\sum_{k=0}^\\infty \\frac{1}{(x+k)^s}\\quad (x>0,\\;\\; s>1), \\qquad\n",
    "\\frac{te^{xt}}{e^t-1} = \\sum_{k=0}^\\infty \\frac{B_k(x)}{k!}t^k\n",
    "$$\n",
    "\n",
    "と定める. $\\zeta(s)=\\zeta(s,1)$ なのでHurwitzのゼータ函数はRiemannのゼータ函数の拡張になっている. 以下を示せ:\n",
    "\n",
    "(1) $\\quad\\ds \\zeta(s,x) = \\frac{1}{\\Gamma(s)}\\int_0^\\infty \\frac{e^{(1-x)t}t^{s-1}}{e^t-1}\\,dt$.\n",
    "\n",
    "(2) $\\quad\\ds \\zeta(s,x) = \\frac{1}{\\Gamma(s)}\\left[\n",
    "\\int_1^\\infty \\frac{e^{(1-x)t}t^{s-1}}{e^t-1}\\,dt +\n",
    "\\int_1^\\infty\\left(\\frac{t e^{(1-x)t}}{e^t-1}-\\sum_{k=0}^N\\frac{B_k(1-x)}{k!}t^k\\right)t^{s-2}\\,dt +\n",
    "\\sum_{k=0}^N \\frac{B_k(1-x)}{k!}\\frac{1}{s+k-1}\n",
    "\\right].\n",
    "$\n",
    "\n",
    "(3) Hurwitzのゼータ函数を(2)によって $s<1$ に拡張すると, $0$ 以上の整数 $m$ について\n",
    "\n",
    "$$\n",
    "\\zeta(-m,x) = \\frac{(-1)^m B_{m+1}(1-x)}{m+1} = -\\frac{B_{m+1}(x)}{m+1}.\n",
    "$$\n",
    "\n",
    "**解答例:** (1) $x,s>0$, $k\\geqq 0$ に対して, $\\ds \\frac{1}{(x+k)^s}=\\frac{1}{\\Gamma(s)}\\int_0^\\infty e^{-(x+k)t}t^{s-1}\\,dt$ を使うと, \n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\zeta(s,x) &=\n",
    "\\sum_{k=0}^\\infty \\frac{1}{\\Gamma(s)}\\int_0^\\infty e^{-(x+k)t}t^{s-1}\\,dt =\n",
    "\\frac{1}{\\Gamma(s)}\\int_0^\\infty \\left(\\sum_{k=0}^\\infty e^{-kt}\\right)e^{-xt}t^{s-1}\\,dt \n",
    "\\\\ &=\n",
    "\\frac{1}{\\Gamma(s)}\\int_0^\\infty \\frac{e^{-xt}t^{s-1}}{1-e^{-t}}\\,dt =\n",
    "\\frac{1}{\\Gamma(s)}\\int_0^\\infty \\frac{e^{(1-x)t}t^{s-1}}{e^t-1}\\,dt.\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "(2) 上の(1)の結果の右辺の積分を $0$ から $1$ への積分と $1$ から $\\infty$ の積分に分けて, $0$ から $1$ への積分の被積分函数に $\\ds\\sum_{k=0}^N\\frac{B_k(1-x)}{k!}t^k$ を足して引き, 引いた方の積分を計算すれば, (2)の公式が得られる.\n",
    "\n",
    "(3) Bernoulli多項式の定義より, $\\ds \\left(\\frac{t e^{(1-x)t}}{e^t-1} - \\sum_{k=0}^N\\frac{B_k(1-x)}{k!}t^k\\right)t^{s-1} = O(t^{s+N-1})$ となるので, (2)の右辺の $0$ から $1$ への積分は $s>-N$ で絶対収束している. $N > m$ と仮定する. $s$ が $0$ 以下の整数に近付くと $\\ds\\frac{1}{\\Gamma(s)}\\to 0$ となり, \n",
    "\n",
    "$$\n",
    "\\frac{1}{\\Gamma(s)}\\frac{1}{s+(m+1)-1} = \\frac{s(s+1)\\cdots(s+m-1)}{\\Gamma(s+m+1)} \\to (-1)^m m! \\quad (s\\to -m)\n",
    "$$\n",
    "\n",
    "より, $s\\to -m$ のとき,\n",
    "\n",
    "$$\n",
    "\\frac{1}{\\Gamma(s)}\\frac{B_{m+1}(1-x)}{(m+1)!}\\frac{1}{s+(m+1)-1} \\to \n",
    "(-1)^m m! \\frac{B_{m+1}(1-x)}{(m+1)!} =\n",
    "\\frac{(-1)^m B_{m+1}(1-x)}{m+1}\n",
    "$$\n",
    "\n",
    "となることから, $\\ds\\zeta(s,x)=\\frac{(-1)^m B_{m+1}(1-x)}{m+1}$ が得られる. さらに, $\\ds\\frac{te^{(1-x)t}}{e^t-1} = \\frac{(-t)e^{a(-t)}}{e^{-t}-1}$ によって, $B_k(1-x)=(-1)^k B_k(x)$ となることがわかるので, $\\ds \\frac{(-1)^m B_{m+1}(1-x)}{m+1}=-\\frac{B_{m+1}(x)}{m+1}$ も得られる. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Riemannのゼータ函数の積分表示2と函数等式\n",
    "\n",
    "**問題(Riemannのゼータ函数の積分表示2):** $\\theta(t)$ を\n",
    "\n",
    "$$\n",
    "\\theta(t) = \\sum_{n=1}^\\infty e^{-\\pi n^2 t} \\quad (t>0)\n",
    "$$\n",
    "\n",
    "とおくと, 次が成立することを示せ:\n",
    "\n",
    "$$\n",
    "\\pi^{-s/2}\\Gamma(s/2)\\zeta(s) = \\int_0^\\infty \\theta(t) t^{s/2-1}\\,dt \\quad (s>2).\n",
    "$$\n",
    "\n",
    "**解答例:** \n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\pi^{-s/2}\\Gamma(s/2)\\zeta(s) &=\\sum_{n=1}^\\infty \\frac{\\Gamma(s/2)}{(\\pi n^2)^{s/2}} =\n",
    "\\sum_{n=1}^\\infty\\int_0^\\infty e^{-\\pi n^2 t} t^{s/2-1}\\,dx\n",
    "\\\\ &=\n",
    "\\int_0^\\infty\\sum_{n=1}^\\infty e^{-\\pi n^2 t} t^{s/2-1}\\,dx =\n",
    "\\int_0^\\infty \\theta(t) t^{s/2-1}\\,dt .\n",
    "\\qquad \\QED\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題(Riemannのゼータ函数の積分表示2'):** 上の問題の続き. $\\theta(t)$ が\n",
    "\n",
    "$$\n",
    "1+2\\theta(1/t)=t^{1/2}(1+2\\theta(t)) \\quad (t>0)\n",
    "$$\n",
    "\n",
    "すなわち\n",
    "\n",
    "$$\n",
    "\\theta(1/t) =-\\frac{1}{2} + \\frac{1}{2}t^{1/2} + t^{1/2}\\theta(t)\n",
    "$$\n",
    "\n",
    "を満たしていることを認めて, 次を示せ:\n",
    "\n",
    "$$\n",
    "\\pi^{-s/2}\\Gamma(s/2)\\zeta(s) = \n",
    "-\\frac{1}{s}-\\frac{1}{1-s} +\n",
    "\\int_1^\\infty \\theta(t) (t^{s/2}+t^{(1-s)/2})\\,\\frac{dt}{t}.\n",
    "$$\n",
    "\n",
    "$\\theta(t)$ に関する上の公式の証明についてはノート「<a href=\"http://nbviewer.jupyter.org/github/genkuroki/Calculus/blob/master/12%20Fourier%20analysis.ipynb\">12 Fourier解析</a>」におけるPoissonの和公式の解説を見よ.\n",
    "\n",
    "**注意:** 上の問題の公式の右辺の積分は $s$ が任意の複素数であってもしているので, 右辺は左辺の複素平面上への解析接続を与える. さらに, 右辺は $s$ を $1-s$ で置き換える操作で不変であるから,\n",
    "\n",
    "$$\n",
    "\\hat{\\zeta}(s) = \\pi^{-s/2}\\Gamma(s/2)\\zeta(s)\n",
    "$$\n",
    "\n",
    "とおくと, \n",
    "\n",
    "$$\n",
    "\\hat{\\zeta}(1-s) = \\hat{\\zeta}(s)\n",
    "$$\n",
    "\n",
    "が成立している. これを**ゼータ函数の函数等式**と呼ぶ.  $\\QED$\n",
    "\n",
    "**解答例:** 上の問題と以下の計算を合わせれば欲しい結果が得られる. 積分区間を $0$ から $1$ と $1$ から $\\infty$ に分けて, $t=1/u$ とおくと, $t^{s/2-1}\\,dt=-u^{-s/2+1}u^{-2}\\,du=-u^{-s/2-1}\\,du$ を使うと, \n",
    "$$\n",
    "\\begin{aligned}\n",
    "&\n",
    "\\int_0^\\infty \\theta(t) t^{s/2-1}\\,dt =\n",
    "\\int_0^1 \\theta(t) t^{s/2-1}\\,dt + \n",
    "\\int_1^\\infty \\theta(t) t^{s/2}\\,dt,\n",
    "\\\\ &\n",
    "\\int_0^1 \\theta(t) t^{s/2-1}\\,dt =\n",
    "\\int_1^\\infty \\left(-\\frac{1}{2} + \\frac{1}{2}t^{1/2} + t^{1/2}\\theta(t)\\right)t^{-s/2-1}\\,dt\n",
    "\\\\ &\\qquad =\n",
    "\\int_1^\\infty\\left(\n",
    "-\\frac{1}{2}t^{-s/2-1}+\\frac{1}{2}t^{(1-s)/2-1} + \\theta(t)t^{(1-s)/2-1}\n",
    "\\right)\\,dt\n",
    "\\\\ &\\qquad =\n",
    "-\\frac{1}{s}-\\frac{1}{s-1} + \\int_1^\\infty \\theta(t)t^{(1-s)/2-1}\\,dt.\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "上の問題の結果と以上の計算をまとめると, 欲しい結果が得られる. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題:** 上の問題の続き. ガンマ函数が Euler's reflection formula\n",
    "\n",
    "$$\n",
    "\\Gamma(s)\\Gamma(1-s) = \\frac{\\pi}{\\sin(\\pi s)}\n",
    "$$\n",
    "\n",
    "と Legendre's duplication formula\n",
    "\n",
    "$$\n",
    "\\Gamma(s)\\Gamma(s+1/2) = 2^{1-2s}\\pi^{1/2}\\Gamma(2s)\n",
    "$$\n",
    "\n",
    "を満たしていることを認めて, 上の問題の注意におけるゼータ函数の函数等式 $\\hat\\zeta(1-s)=\\hat\\zeta(s)$ が\n",
    "\n",
    "$$\n",
    "\\zeta(s) = 2^s \\pi^{s-1}\\sin\\frac{\\pi s}{2}\\,\\Gamma(1-s)\\,\\zeta(1-s)\n",
    "$$\n",
    "\n",
    "と書き直されることを示せ.\n",
    "\n",
    "Legendre's duplication formula と Euler's reflection formula はこのノートの下の方で初等的に証明される. Euler's reflection formulaの証明についてはノート「<a href=\"http://nbviewer.jupyter.org/github/genkuroki/Calculus/blob/master/12%20Fourier%20analysis.ipynb\">12 Fourier解析</a>」のガンマ函数とsinの関係の節も参照せよ.\n",
    "\n",
    "**解答例:** $\\hat\\zeta(s)=\\pi^{-s/2}\\Gamma(s/2)\\zeta(s)$, $\\hat\\zeta(s)=\\hat\\zeta(1-s)$ より,\n",
    "\n",
    "$$\n",
    "\\pi^{-s/2}\\Gamma(s/2)\\zeta(s) = \\pi^{-(1-s)/2}\\Gamma((1-s)/2)\\zeta(1-s).\n",
    "$$\n",
    "\n",
    "これは以下のように書き直される:\n",
    "\n",
    "$$\n",
    "\\zeta(s) = \\pi^{s-1/2}\\frac{\\Gamma((1-s)/2)}{\\Gamma(s/2)}\\zeta(s).\n",
    "$$\n",
    "\n",
    "一方, Euler's reflection formula の $s$ に $s/2$ を代入すると,\n",
    "\n",
    "$$\n",
    "\\Gamma(s/2)\\Gamma(1-s/2)=\\frac{\\pi}{\\sin(\\pi s/2)},\n",
    "\\quad\\text{i.e.}\\quad\n",
    "\\frac{1}{\\Gamma(s/2)} = \\pi^{-1}\\sin\\frac{\\pi s}{2}\\Gamma(1-s/2)\n",
    "$$\n",
    "\n",
    "となり, Legendre's duplication formula の $s$ に $(1-s)/2$ を代入すると,\n",
    "\n",
    "$$\n",
    "\\Gamma((1-s)/2)\\Gamma(1-s/2)=2^s \\pi^{1/2}\\,\\Gamma(1-s),\n",
    "$$\n",
    "\n",
    "となるので, それらを上の公式に代入すると,\n",
    "\n",
    "$$\n",
    "\\zeta(s) = 2^s \\pi^{s-1}\\sin\\frac{\\pi s}{2}\\,\\Gamma(1-s)\\,\\zeta(1-s)\n",
    "$$\n",
    "\n",
    "が得られる. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題:** $k$ が正の整数であるとき $\\ds\\zeta(-(2k-1)) = -\\frac{B_{2k}}{2k}$ であるという事実と上の問題の結果から\n",
    "\n",
    "$$\n",
    "\\zeta(2k) = \\frac{2^{2k-1}(-1)^{k-1}B_{2k}}{(2k)!}\\pi^{2k}\n",
    "$$\n",
    "\n",
    "が導かれることを示せ.\n",
    "\n",
    "**解答例:** $\\ds\\zeta(-(2k-1)) = -\\frac{B_{2k}}{2k}$ と上の問題の結果より, \n",
    "\n",
    "$$\n",
    "-\\frac{B_{2k}}{2k} = \\zeta(-(2k-1)) = 2^{-(2k-1)}\\pi^{-2k}(-1)^k(2k-1)!\\zeta(2k).\n",
    "$$\n",
    "\n",
    "これより示したい公式が得られる. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### ベータ函数とガンマ函数の関係\n",
    "\n",
    "ベータ函数はガンマ函数によって\n",
    "\n",
    "$$\n",
    "B(p,q) = \\frac{\\Gamma(p)\\Gamma(q)}{\\Gamma(p+q)}\n",
    "\\tag{$*$}\n",
    "$$\n",
    "\n",
    "と表わされる. これを証明したい. そのためには\n",
    "\n",
    "$$\n",
    "\\Gamma(p)\\Gamma(q)=\n",
    "\\int_0^\\infty\n",
    "\\left(\n",
    "\\int_0^\\infty e^{-(x+y)} x^{p-1} y^{q-1}\\,dy\n",
    "\\right)\\,dx\n",
    "$$\n",
    "\n",
    "が\n",
    "\n",
    "$$\n",
    "\\Gamma(p+q)B(p,q)=\\int_0^\\infty e^{-z}z^{p+q-1}\\,dz\n",
    "\\,\\int_0^1 t^{p-1}(1-t)^{q-1}\\,dt\n",
    "$$\n",
    "\n",
    "に等しいことを示せばよい. ガンマ函数とベータ函数の別の表示を使えば右辺も別の形になることに注意せよ."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 方法1: 置換積分と積分の順序交換のみを使う方法\n",
    "\n",
    "ガンマ函数とベータ函数のあいだの関係式は1変数の置換積分と積分の順序交換のみを使って証明可能である. 条件 $A$ に対して, $x,y$ が条件 $A$ をみたすとき値が $1$ になり, それ以外のときに値が $0$ になる $x,y$ の函数を $1_A(x,y)$ と書くことにすると,\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\Gamma(p)\\Gamma(q) &=\n",
    "\\int_0^\\infty\n",
    "\\left(\n",
    "\\int_0^\\infty e^{-(x+y)} x^{p-1} y^{q-1}\\,dy\n",
    "\\right)\\,dx\n",
    "\\\\ &=\n",
    "\\int_0^\\infty\n",
    "\\left(\n",
    "\\int_x^\\infty e^{-z} x^{p-1} (z-x)^{q-1}\\,dz\n",
    "\\right)\\,dx\n",
    "\\\\ &=\n",
    "\\int_0^\\infty\n",
    "\\left(\n",
    "\\int_0^\\infty 1_{x<z}(x,z) e^{-z} x^{p-1} (z-x)^{q-1}\\,dz\n",
    "\\right)\\,dx\n",
    "\\\\ &=\n",
    "\\int_0^\\infty\n",
    "\\left(\n",
    "\\int_0^\\infty 1_{x<z}(x,z) e^{-z} x^{p-1} (z-x)^{q-1}\\,dx\n",
    "\\right)\\,dz\n",
    "\\\\ &=\n",
    "\\int_0^\\infty\n",
    "\\left(\n",
    "\\int_0^z e^{-z} x^{p-1} (z-x)^{q-1}\\,dx\n",
    "\\right)\\,dz\n",
    "\\\\ &=\n",
    "\\int_0^\\infty\n",
    "\\left(\n",
    "\\int_0^1 e^{-z} (zt)^{p-1} (z-zt)^{q-1}z\\,dt\n",
    "\\right)\\,dz\n",
    "\\\\ &=\n",
    "\\int_0^\\infty e^{-z}z^{p+q-1}\\,dz\n",
    "\\,\\int_0^1 t^{p-1}(1-t)^{q-1}\\,dt =\n",
    "\\Gamma(p+q)B(p,q).\n",
    "\\end{aligned}\n",
    "$$\n",
    "2つ目の等号で $y=z-x$ と置換積分し, 4つ目の等号で積分の順序を交換し, 6つ目の等号で $x=zt$ と置換積分した.\n",
    "\n",
    "以上の計算は\n",
    "\n",
    "* 黒木玄, <a href=\"https://genkuroki.github.io/documents/20160501StirlingFormula.pdf\">ガンマ分布の中心極限定理とStirlingの公式</a>\n",
    "\n",
    "の第7.4節からの引き写しである."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 方法2: 極座標変換を使う方法\n",
    "\n",
    "この方法は2重積分に関する知識が必要になる. 2重積分について知らない人は次の節の別の方法を参照せよ.\n",
    "\n",
    "$x=X^2$, $y=Y^2$ と変数変換すると, \n",
    "\n",
    "$$\n",
    "\\Gamma(p)\\Gamma(q) = 4\\int_0^\\infty\\int_0^\\infty e^{-(X^2+Y^2)} X^{2p-1} Y^{2q-1}\\,dX\\,dY.\n",
    "$$\n",
    "\n",
    "さらに $X=r\\cos\\theta$, $Y=r\\sin\\theta$ と変数変換すると,\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\Gamma(p)\\Gamma(q) &= \n",
    "4\\int_0^{\\pi/2}d\\theta\\int_0^\\infty e^{-r^2} (r\\cos\\theta)^{2p-1} (r\\sin\\theta)^{2q-1} r\\,dr\n",
    "\\\\ &=\n",
    "4\\int_0^{\\pi/2}(\\cos\\theta)^{2p-1} (\\sin\\theta)^{2q-1}\\,d\\theta\n",
    "\\int_0^\\infty e^{-r^2} r^{2(p+q)-1}\\,dr =\n",
    "B(p,q)\\Gamma(p+q).\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "最後の等号でベータ函数の三角函数を用いた表示とガンマ函数のGauss積分に似た表示を用いた. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 方法3: y = tx と変数変換する方法\n",
    "\n",
    "$y=tx$ とおくと, $dy = x\\,dt$ より, \n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\Gamma(p)\\Gamma(q) &=\n",
    "\\int_0^\\infty\\left(\\int_0^\\infty e^{-(x+y)}x^{p-1}y^{q-1}\\,dy\\right)\\,dx =\n",
    "\\int_0^\\infty\\left(\\int_0^\\infty e^{-(1+t)x}x^{p+q-1}t^{q-1}\\,dt\\right)\\,dx\n",
    "\\\\ &=\n",
    "\\int_0^\\infty\\left(\\int_0^\\infty e^{-(1+t)x}x^{p+q-1}\\,dx\\right)t^{q-1}\\,dt =\n",
    "\\int_0^\\infty \\frac{\\Gamma(p+q)}{(1+t)^{p+q}} t^{q-1}\\,dt\n",
    "\\\\ &=\n",
    "\\Gamma(p+q)\\int_0^\\infty \\frac{t^{q-1}}{(1+t)^{p+q}}\\,dt =\n",
    "\\Gamma(p+q)B(p,q).\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "3つ目の等号で積分順序を交換し, 4つ目の等号で $s,c>0$ についてよく使われる公式($x=y/c$ と置けば得られる公式)\n",
    "\n",
    "$$\n",
    "\\int_0^\\infty e^{-cx}x^{s-1}\\,dx = \\frac{\\Gamma(s)}{c^s}\n",
    "$$\n",
    "\n",
    "を使い, 最後の等号でベータ函数の次の表示の仕方を用いた:\n",
    "\n",
    "$$\n",
    "B(p,q) = \\int_0^1 x^{p-1}(1-x)^{q-1}\\,dx =\n",
    "\\int_0^\\infty \\frac{t^{q-1}}{(1+t)^{p+q}}\\,dt\n",
    "$$\n",
    "\n",
    "この公式は積分変数を $\\ds x=\\frac{1}{1+t}$ と置換すれば得られる. $\\ds x = \\frac{t}{1+t}$ と置換すれば $p,q$ を交換した公式が得られる. \n",
    "\n",
    "ベータ函数に関するその公式を知っていれば, ガンマ函数とベータ函数の関係を導くにはこの方法が簡単かもしれない.\n",
    "\n",
    "$y=tx$ の $t$ は直線の傾きという意味を持っている. $xy$ 平面の第一象限の点を $(x,y)$ で指定していたのを, $(x,y)=(x,tx)$ と直線の傾き $t$ と $x$ で指定するようにしたことが, 上の計算で採用した方法である. この方法はJacobianが出て来る二重積分の積分変数の変換を避けたい場合に便利である."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### ベータ函数とガンマ函数の関係の簡単な計算問題への応用\n",
    "\n",
    "**問題:** ベータ函数とガンマ函数の関係を用いて $\\Gamma(1/2)=\\sqrt{\\pi}$ を証明せよ.\n",
    "\n",
    "**解答例:** \n",
    "$$\n",
    "\\Gamma(1/2)^2 = \\frac{\\Gamma(1/2)\\Gamma(1/2)}{\\Gamma(1)} = B(1/2,1/2) =\n",
    "2\\int_0^{\\pi/2}(\\cos\\theta)^{2\\cdot1/2-1}(\\sin\\theta)^{2\\cdot1/2-1}\\,d\\theta =\n",
    "2\\int_0^{\\pi/2}d\\theta = \\pi.\n",
    "$$\n",
    "\n",
    "1つ目の等号で $\\Gamma(1)=1$ を使い, 2つ目の等号でベータ函数とガンマ函数の関係を用い, 3つ目の等号でベータ函数の三角函数を用いた表示を使った. ゆえに $\\Gamma(1/2)=\\sqrt{\\pi}$. $\\QED$\n",
    "\n",
    "**注意:** この問題の解答例はGauss積分の公式の別証明 $\\int_{-\\infty}^\\infty e^{-x^2}\\,dx=\\Gamma(1/2)=\\sqrt{\\pi}$ を与える. $\\QED$\n",
    "\n",
    "**問題:** 次の積分を計算せよ:\n",
    "\n",
    "$$\n",
    "A = \\int_0^1 x^5(1-x^2)^{3/2}\\,dx.\n",
    "$$\n",
    "\n",
    "**解答例:** $x=t^{1/2}$ と置換すると $\\ds dx=\\frac{1}{2}t^{-1/2}\\,dt$ なので,\n",
    "\n",
    "$$\n",
    "A = \\int_0^1 t^{5/2}(1-t)^{3/2}\\,\\frac{1}{2}t^{-1/2}\\,dt = \n",
    "\\frac{1}{2}\\int_0^1 t^2(1-t)^{3/2}\\,dt = \\frac{1}{2}B(3, 5/2) =\n",
    "\\frac{\\Gamma(3)\\Gamma(5/2)}{2\\Gamma(3+5/2)}.\n",
    "$$\n",
    "\n",
    "3つ目の等号で $2=3-1$, $3/2=5/2-1$ とみなしてからベータ函数の表示を得ていることに注意せよ. このステップでよく間違う.\n",
    "\n",
    "一般に非負の整数 $n$ について\n",
    "\n",
    "$$\n",
    "\\Gamma(n+1) = n!, \\quad\n",
    "\\frac{\\Gamma(s)}{\\Gamma(s+n)} = \\frac{1}{s(s+1)\\cdots(s+n-1)}\n",
    "$$\n",
    "\n",
    "なので, \n",
    "\n",
    "$$\n",
    "\\Gamma(3) = 2! = 2, \\quad\n",
    "\\frac{\\Gamma(5/2)}{\\Gamma(3+5/2)} = \\frac{1}{(5/2)(7/2)(9/2)} = \\frac{2^3}{5\\cdot 7\\cdot 9}.\n",
    "$$\n",
    "\n",
    "したがって\n",
    "\n",
    "$$\n",
    "A = \\frac{2}{2}\\frac{2^3}{5\\cdot7\\cdot9} = \\frac{8}{315}.\n",
    "\\qquad \\QED\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\frac{8}{315}$$"
      ],
      "text/plain": [
       "8/315"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = symbols(\"x\", real=true)\n",
    "integrate(x^5*(1-x^2)^(Sym(3)/2), (x,0,1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### B(s, 1/2)の級数展開\n",
    "\n",
    "$\\ds\\binom{-1/2}{n}$ は次を満たしている:\n",
    "\n",
    "$$\n",
    "\\binom{-1/2}{n}(-x)^n =\n",
    "\\frac{(1/2)(3/2)\\cdots((2n-1)/2)}{n!}x^n =\n",
    "\\frac{1}{2^{2n}}\\binom{2n}{n}x^n.\n",
    "$$\n",
    "\n",
    "ゆえに, $|x|<1$ のとき,\n",
    "\n",
    "$$\n",
    "(1-x)^{-1/2} = \\sum_{n=0}^\\infty \\frac{1}{2^{2n}}\\binom{2n}{n}x^n.\n",
    "$$\n",
    "\n",
    "したがって,\n",
    "\n",
    "$$\n",
    "B(s,1/2)=\\int_0^1 x^{s-1}(1-x)^{-1/2}\\,dx=\n",
    "\\sum_{n=0}^\\infty \\frac{1}{2^{2n}}\\binom{2n}{n}\\int_0^1 x^{s+n-1}\\,dx =\n",
    "\\sum_{n=0}^\\infty \\frac{1}{2^{2n}}\\binom{2n}{n}\\frac{1}{s+n}.\n",
    "$$\n",
    "\n",
    "例えば, $s=1/2$ のとき, $B(1/2,1/2)=\\Gamma(1/2)^2=\\pi$ なので, 両辺を2で割ると,\n",
    "\n",
    "$$\n",
    "\\sum_{n=0}^\\infty \\frac{1}{2^{2n}}\\binom{2n}{n}\\frac{1}{2n+1} =\n",
    "\\frac{1}{2}B(1/2,1/2) = \\frac{\\pi}{2}.\n",
    "$$\n",
    "\n",
    "このような公式はベータ函数について知らないと驚くべき公式に見えてしまうが, ベータ函数について知っていれば単に二項展開をベータ函数の被積分函数に適用しただけの公式に過ぎない."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### ガンマ函数の無限積表示\n",
    "\n",
    "**問題(Gaussの公式):** ベータ函数とガンマ函数の関係を用いて, 次の公式を示せ.\n",
    "\n",
    "$$\n",
    "\\Gamma(s) = \\lim_{n\\to\\infty}\\frac{n^s n!}{s(s+1)\\cdots(s+n)}.\n",
    "$$\n",
    "\n",
    "**解答例:** 右辺をベータ函数と表示することを考える. 以下では $n$ は正の整数であるとし, $s>0$ と仮定する. ベータ函数とガンマ函数の函数等式および $\\Gamma(n+1)=n!$ より,\n",
    "\n",
    "$$\n",
    "B(s,n+1) = \\frac{\\Gamma(s)\\Gamma(n+1)}{\\Gamma(s+n+1)} =\n",
    "\\frac{n!}{s(s+1)\\cdots(s+n)}.\n",
    "$$\n",
    "\n",
    "ゆえに\n",
    "\n",
    "$$\n",
    "n^s B(s,n+1) = \\frac{n^s n!}{s(s+1)\\cdots(s+n)}.\n",
    "$$\n",
    "\n",
    "左辺を $n\\to\\infty$ での極限を取り易い形に変形しよう. $x=t/n$ と置換することによって, $n\\to\\infty$ のとき\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "n^s B(s,n+1) &= n^s \\int_0^1 x^{s-1}(1-x)^n\\,dx\n",
    "\\\\ &=\n",
    "\\int_0^n t^{s-1}\\left(1-\\frac{t}{n}\\right)^n\\,dt \\to \\int_0^\\infty t^{s-1}e^{-t}\\,dt = \\Gamma(s).\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "以上をまとめると示したい結果が得られる. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題:** 上の解答例中で極限と積分の順序を交換した. その部分の議論を指数函数に関する不等式\n",
    "\n",
    "$$\n",
    "\\left(1+\\frac{t}{a}\\right)^a \\leqq e^t \\leqq \\left(1-\\frac{t}{b}\\right)^{-b}\\qquad(-a<t<b,\\;\\; a,b>0)\n",
    "\\tag{1}\n",
    "$$\n",
    "\n",
    "と $\\ds \\left(1+\\frac{t}{a}\\right)^a$, $\\ds \\left(1-\\frac{t}{b}\\right)^{-b}$ がそれぞれ $a,b$ について単調増加, 単調減少することを用いて正当化せよ.\n",
    "\n",
    "**解答例:** 問題文の中で与えられた不等式の全体の逆元を取り, $a=m$, $b=n$ とおくと, \n",
    "\n",
    "$$\n",
    "\\left(1-\\frac{t}{n}\\right)^n \\leqq e^{-t} \\leqq \\left(1+\\frac{t}{m}\\right)^{-m} \\qquad (-m<t<n)\n",
    "\\tag{2}\n",
    "$$\n",
    "\n",
    "となり, $\\ds \\left(1-\\frac{t}{n}\\right)^n$, $\\ds \\left(1+\\frac{t}{m}\\right)^{-m}$ はそれぞれ $n,m$ について単調増加, 単調減少する. ベータ函数の表式\n",
    "\n",
    "$$\n",
    "B(p,q) = \n",
    "\\int_0^1 x^{p-1}(1-x)^{q-1}\\,dx =\n",
    "\\int_0^\\infty \\frac{x^{p-1}}{(1+x)^{p+q}}\\,dx\n",
    "$$\n",
    "\n",
    "において, それぞれ $x=t/n$, $x=t/m$ とおくことによって, $s,n>0$, $m>s$ のとき, \n",
    "\n",
    "$$\n",
    "n^s B(s,n+1) = \\int_0^n t^{s-1}\\left(1-\\frac{t}{n}\\right)^n\\,dt, \\qquad \n",
    "m^s B(s,m-s) = \\int_0^n t^{s-1}\\left(1+\\frac{t}{m}\\right)^m\\,dt\n",
    "$$\n",
    "\n",
    "が得られ, それぞれ, $n$, $m$ について単調増加, 単調減少することがわかる.  これらと $\\ds\\Gamma(s)=\\int_0^\\infty t^{s-1}e^{-t}\\,dt$ を比較すると, \n",
    "\n",
    "$$\n",
    "n^s B(s,n+1) \\leqq \\Gamma(s) \\leqq m^s B(s,m-s).\n",
    "\\tag{$*$}\n",
    "$$\n",
    "\n",
    "$n^s B(s,n+1)$, $ m^s B(s,m-s)$ はそれぞれ $n$, $m$ について単調増加, 単調減少するので, どちらも $n,m\\to\\infty$ で収束する. そして, $m=n+s+1$ とおくと, \n",
    "\n",
    "$$\n",
    "\\frac{m^s B(s,m-s)}{n^s B(s,n+1)} = \\frac{(n+s+1)^s B(s,n+1)}{n^s B(s,n+1)} =\n",
    "\\left(1+\\frac{s+1}{n}\\right)^s \\to 1 \\quad(n\\to\\infty)\n",
    "$$\n",
    "\n",
    "なので, $n^s B(s,n+1)$, $ m^s B(s,m-s)$ は $n,m\\to\\infty$ で同じ値に収束する. これと不等式($*$)を合わせると,  $n^s B(s,n+1)$, $ m^s B(s,m-s)$ は $n,m\\to\\infty$ で $\\Gamma(s)$ に収束することがわかる. $\\QED$\n",
    "\n",
    "**注意:** 不等式(1),(2)と $a,b$, $m,n$ に関する単調性は極限で指数函数が現われる結果を初等的に正当化するために非常に便利である. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題(Weierstrassの公式):** 上の問題の結果を用いて, 次の公式を示せ.\n",
    "\n",
    "$$\n",
    "\\frac{1}{\\Gamma(s)} = \n",
    "e^{\\gamma s} s\\prod_{n=1}^\\infty\\left[\\left(1+\\frac{s}{n}\\right)e^{-s/n}\\right].\n",
    "\\tag{$*$}\n",
    "$$\n",
    "\n",
    "ここで $\\gamma$ はEuler定数である:\n",
    "\n",
    "$$\n",
    "\\gamma = \\lim_{n\\to\\infty}\\left(\\sum_{k=1}^n\\frac{1}{k}-\\log n\\right) =\n",
    "0.5772\\cdots\n",
    "$$\n",
    "\n",
    "**解答例:**\n",
    "$$\n",
    "\\begin{aligned}\n",
    "&\n",
    "\\frac{s(s+1)\\cdots(s+n)}{n^s n!}\n",
    "\\\\ &=\n",
    "s\\left(1+s\\right)\\left(1+\\frac{s}{2}\\right)\\cdots\\left(1+\\frac{s}{n}\\right) e^{-s\\log n}\n",
    "\\\\ &=\n",
    "s\\left(1+s\\right)e^{-s}\n",
    "\\left(1+\\frac{s}{2}\\right)e^{-s/2}\n",
    "\\cdots\n",
    "\\left(1+\\frac{s}{n}\\right)e^{-s/n}\n",
    "e^{s\\left(1+\\frac{1}{2}+\\cdots+\\frac{1}{n}-\\log n\\right)}\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "であるから, 公式($*$)を得る. $\\QED$\n",
    "\n",
    "**注意:** \n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\log\\left[\\left(1+\\frac{s}{n}\\right)e^{-s/n}\\right] &=\n",
    "\\log\\left(1+\\frac{s}{n}\\right) - \\frac{s}{n} \n",
    "\\\\ &=\n",
    "\\frac{s}{n} - \\frac{s^2}{2n^2} + O\\left(\\frac{1}{n^3}\\right) - \\frac{s}{n} \n",
    "\\\\ &= -\n",
    "\\frac{s^2}{2n^2} + O\\left(\\frac{1}{n^3}\\right)\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "なので\n",
    "\n",
    "$$\n",
    "\\prod_{n=1}^\\infty\\left[\\left(1+\\frac{s}{n}\\right)e^{-s/n}\\right] =\n",
    "\\prod_{n=1}^\\infty\\left[1 + O\\left(\\frac{1}{n^2}\\right)\\right]\n",
    "$$\n",
    "\n",
    "となり, この無限積は任意の複素数 $s$ について収束する. したがって, Weierstrassの公式は $1/\\Gamma(s)$ のすべての複素数 $s$ への自然な拡張を与える.  $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### sinとガンマ函数の関係\n",
    "\n",
    "sinの無限積表示と Euler's reflection formulaの証明についてはノート「<a href=\"http://nbviewer.jupyter.org/github/genkuroki/Calculus/blob/master/12%20Fourier%20analysis.ipynb\">12 Fourier解析</a>」のガンマ函数とsinの関係の節も参照せよ. 以下ではsinの奇数倍角の公式を用いた証明を紹介する."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### sinの無限積表示\n",
    "\n",
    "sinの無限積表示\n",
    "\n",
    "$$\n",
    "\\frac{\\sin(\\pi s)}{\\pi} =\n",
    "s \\prod_{n=1}^\\infty\\left(1-\\frac{s^2}{n^2}\\right)\n",
    "$$\n",
    "\n",
    "を導出したい. この公式は正弦函数の奇数倍角の公式の極限としても導出されることを以下で説明しよう. (他にも様々な経路での証明がある.)\n",
    "\n",
    "非負の整数 $n$ に関する $e^{inx} = (e^{ix})^n$ の右辺に $e^{ix} = \\cos x + i\\sin x$ を代入して二項定理を適用し, 両辺の虚部を取ると次が得られる:\n",
    "\n",
    "$$\n",
    "  \\sin(nx) = \\sum_{0\\leqq k<n/2}\n",
    "  (-1)^k\\binom{n}{2k+1} (\\cos x)^{n-2k-1}(\\sin x)^{2k+1}.\n",
    "$$\n",
    "\n",
    "$(\\cos x)^2=1-(\\sin x)^2$ より, $n$ が奇数ならば $\\sin(nx)$ は $\\sin x$ の $n$ 次多項式で表わされることがわかる.\n",
    "\n",
    "以下では, $m$ は非負の整数であるとし, $n=2m+1$ とおき, $\\sin(nx)=\\sin((2m+1)x)$ について考える. $\\sin((2m+1)x)$ は $\\sin x$ の $2m+1$ 次の多項式で表わされ, 周期 $\\frac{2\\pi}{2m+1}$ を持ち,\n",
    "\n",
    "$$\n",
    " x_k = \\frac{k\\pi}{2m+1}, \\quad k=0,\\pm1,\\ldots,\\pm m\n",
    "$$\n",
    "\n",
    "で $0$ になり,\n",
    "\n",
    "$$\n",
    "  -\\pi/2<x_{-m}<x_{-m+1}<\\cdots<x_{m-1}<x_m<\\pi/2\n",
    "$$\n",
    "\n",
    "なので, $\\sin x_k$ は互いに異なる. これで $\\sin((2m+1)x)$ の互いに異なる $(2m+1)$ 個の零点 $\\sin x_k$ が判明したことになる. $\\sin((2m+1)x)$ は $\\sin x$ の $2m+1$ 次の多項式で表われるので, $0$ でないある定数 $C$ が存在して,\n",
    "\n",
    "$$\n",
    "\\sin((2m+1)x) =\n",
    "  C \\sin x\n",
    "  \\prod_{k=1}^m\n",
    "  \\left[\n",
    "  \\left(\\sin x - \\sin\\frac{k\\pi}{2m+1}\\right)\n",
    "  \\left(\\sin x + \\sin\\frac{k\\pi}{2m+1}\\right)\n",
    "  \\right].\n",
    "$$\n",
    "\n",
    "両辺を $x$ で割って, $x\\to 0$ の極限を取ると,\n",
    "\n",
    "$$\n",
    "  2m+1 = C\n",
    "  \\prod_{k=1}^m\n",
    "  \\left[\n",
    "  \\left(-\\sin\\frac{k\\pi}{2m+1}\\right)\n",
    "  \\left(\\sin\\frac{k\\pi}{2m+1}\\right)\n",
    "  \\right].\n",
    "$$\n",
    "\n",
    "したがって,\n",
    "\n",
    "$$\n",
    "\\frac{\\sin((2m+1)x)}{2m+1} =\n",
    "  \\sin x\n",
    "  \\prod_{k=1}^m\n",
    "  \\left[\n",
    "  \\left(1-\\frac{\\sin x}{\\sin\\frac{k\\pi}{2m+1}}\\right)\n",
    "  \\left(1+\\frac{\\sin x}{\\sin\\frac{k\\pi}{2m+1}}\\right)\n",
    "  \\right].\n",
    "$$\n",
    "\n",
    "これで, $\\sin$ の奇数倍角の公式の積表示が得られた.  これに $x=\\frac{\\pi s}{2m+1}$ を代入し, 両辺に $\\frac{2m+1}{\\pi}$ をかけると\n",
    "\n",
    "$$\n",
    "  \\frac{\\sin(\\pi s)}{\\pi} =\n",
    "  \\frac{\\sin\\frac{\\pi s}{2m+1}}{\\frac{\\pi}{2m+1}}\n",
    "  \\prod_{k=1}^m\n",
    "  \\left[\n",
    "  \\left(1-\\frac{\\sin\\frac{\\pi s}{2m+1}}{\\sin\\frac{\\pi k}{2m+1}}\\right)\n",
    "  \\left(1+\\frac{\\sin\\frac{\\pi s}{2m+1}}{\\sin\\frac{\\pi k}{2m+1}}\\right)\n",
    "  \\right].\n",
    "$$\n",
    "\n",
    "$m\\to\\infty$ の極限を取ると正弦函数の無限積表示\n",
    "\n",
    "$$\n",
    "\\frac{\\sin(\\pi s)}{\\pi}\n",
    "=s \\prod_{k=1}^\\infty\\left[\n",
    "\\left(1-\\frac{s}{k}\\right)\\left(1+\\frac{s}{k}\\right)\n",
    "\\right]\n",
    "= s \\prod_{k=1}^\\infty\\left(1-\\frac{s^2}{k^2}\\right).\n",
    "$$\n",
    "\n",
    "が得られる.  ここで以下を使った: $t\\to 0$ のとき\n",
    "\n",
    "$$\n",
    "\\frac{\\sin(at)}{t}\\to a, \\qquad\n",
    "  \\frac{\\sin(at)}{\\sin(bt)} \\to \\frac{a}{b}.\n",
    "$$\n",
    "\n",
    "以上のようにsinの無限積表示はsinの奇数倍角の公式の極限として導出される."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Euler's reflection formula \n",
    "\n",
    "ガンマ函数の無限積表示より\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "&\n",
    "\\Gamma(s) = \n",
    "\\lim_{n\\to\\infty}\\frac{n^s n!}{s(1+s)(2+s)\\cdots(n+s)} =\n",
    "\\lim_{n\\to\\infty}\\frac{n^s}{s\\left(1+s\\right)\\left(1+\\frac{s}{2}\\right)\\cdots\\left(1+\\frac{s}{n}\\right)},\n",
    "\\\\ &\n",
    "\\Gamma(1-s) = \n",
    "\\lim_{n\\to\\infty}\\frac{n^{1-s} n!}{(1-s)(2-s)\\cdots(n+1-s)} =\n",
    "\\lim_{n\\to\\infty}\\frac{n^{-s}}{\\left(1-s\\right)\\left(1-\\frac{s}{2}\\right)\\cdots\\left(1-\\frac{s}{n}\\right)\\left(1+\\frac{1-s}{n}\\right)},\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "ゆえに, $\\sin$ の無限積表示も使うと, \n",
    "\n",
    "$$\n",
    "\\frac{1}{\\Gamma(s)\\Gamma(1-s)} =\n",
    "\\lim_{n\\to\\infty}s\\prod_{k=1}^n\\left(\\left(1+\\frac{s}{k}\\right)\\left(1-\\frac{s}{k}\\right)\\right) =\n",
    "s\\prod_{n=1}^\\infty\\left(1-\\frac{s^2}{n^2}\\right) =\n",
    "\\frac{\\sin(\\pi s)}{\\pi}.\n",
    "$$\n",
    "\n",
    "これで次の公式が得られた:\n",
    "\n",
    "$$\n",
    "\\Gamma(s)\\Gamma(1-s) = \\frac{\\pi}{\\sin(\\pi s)}\n",
    "$$\n",
    "\n",
    "これは **Euler's reflection formula** と呼ばれる. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Wallisの公式\n",
    "\n",
    "互いに同じ深さにある次の2つの公式の両方を**Wallisの公式**と呼ぶ.\n",
    "\n",
    "$$\n",
    "\\prod_{n=1}^\\infty\\frac{(2n)(2n)}{(2n-1)(2n+1)} = \\frac{\\pi}{2}, \\qquad\n",
    "\\frac{1}{2^{2n}}\\binom{2n}{n} = \\frac{(2n)!}{2^{2n}(n!)^2} \\sim \\frac{1}{\\sqrt{\\pi n}}.\n",
    "$$\n",
    "\n",
    "**問題:** 前者のWallisの公式から後者のWallisの公式を導け.\n",
    "\n",
    "**解答例:** 前者のWallisの公式と\n",
    "\n",
    "$$\n",
    "1\\cdot3\\cdots(2n-1) = \\frac{(2n)!}{2^n n!}\n",
    "$$\n",
    "\n",
    "より,\n",
    "\n",
    "$$\n",
    "\\frac{2}{\\pi}\\sim \n",
    "\\frac\n",
    "{1\\cdot3\\cdots(2n-1)\\times 3\\cdot5\\cdots(2n+1)}\n",
    "{2\\cdot4\\cdots(2n)\\times 2\\cdot4\\cdots(2n)} =\n",
    "\\left(\\frac{(2n)!}{2^{2n}(n!)^2}\\right)^2 (2n+1) =\n",
    "\\left(\\frac{1}{2^{2n}}\\binom{2n}{n}\\right)^2 (2n+1).\n",
    "$$\n",
    "\n",
    "ゆえに,\n",
    "\n",
    "$$\n",
    "\\left(\\frac{(2n)!}{2^{2n}(n!)^2}\\right)^2 = \\left(\\frac{1}{2^{2n}}\\binom{2n}{n}\\right)^2 \\sim \\frac{1}{\\pi n}\n",
    "$$\n",
    "\n",
    "全体の平方根を取れば, 後者のWallisの公式が得られる. $\\QED$\n",
    "\n",
    "**問題:** 後者のWallisの公式から前者のWallisの公式を導け.\n",
    "\n",
    "**解答例:** 前者のWallisの公式の無限積の最初の $N$ 個の因子の積は\n",
    "\n",
    "$$\n",
    "\\prod_{n=1}^N \\frac{(2n)(2n)}{(2n-1)(2n+1)} =\n",
    "\\frac{2^{2N} (N!)^2}{(1\\cdot3\\cdots(2N-1))}\\frac{1}{2N+1} =\n",
    "\\left(\\frac{2^{2N}(N!)^2}{(2N)!}\\right)\\frac{1}{2N+1}\n",
    "$$\n",
    "\n",
    "であり, 後者のWallisの公式より,  $N\\to\\infty$ において,\n",
    "\n",
    "$$\n",
    "\\left(\\frac{2^{2N}(N!)^2}{(2N)!}\\right)\\frac{1}{2N+1} \\sim\n",
    "\\frac{\\pi N}{2N+1} \\sim \\frac{\\pi}{2}\n",
    "$$\n",
    "\n",
    "となる. ゆえに \n",
    "\n",
    "$$\n",
    "\\lim_{N\\to\\infty} \\prod_{n=1}^N \\frac{(2n)(2n)}{(2n-1)(2n+1)} = \\frac{\\pi}{2}.\n",
    "$$\n",
    "\n",
    "これで後者のWallisの公式から前者のWallisの公式を導けた. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題(Wallisの公式の証明1):** $\\sin$ の無限積表示\n",
    "\n",
    "$$\n",
    "s \\prod_{n=1}^\\infty\\left(1-\\frac{s^2}{n^2}\\right) = \\frac{\\sin(\\pi s)}{\\pi s}\n",
    "\\tag{1}\n",
    "$$\n",
    "\n",
    "を用いて, 次の公式を証明せよ:\n",
    "\n",
    "$$\n",
    "\\prod_{n=1}^\\infty\\frac{(2n)(2n)}{(2n-1)(2n+1)} = \\frac{\\pi}{2}.\n",
    "\\tag{2}\n",
    "$$\n",
    "\n",
    "**解答例:** (1)に $\\ds s=\\frac{\\pi}{2}$ を代入すれば(2)がただちに得られる. $\\QED$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\frac{4 n^{2}}{\\left(2 n - 1\\right) \\left(2 n + 1\\right)}$$"
      ],
      "text/plain": [
       "           2       \n",
       "        4*n        \n",
       "-------------------\n",
       "(2*n - 1)*(2*n + 1)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n = symbols(\"n\")\n",
    "1/factor(1-1/(2n)^2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題(Wallisの公式の証明2):** 前節のガンマ函数のGaussの公式の問題の解答例の中で次を示した:\n",
    "\n",
    "$$\n",
    "\\lim_{n\\to\\infty}n^s B(s,n+1) = \\Gamma(s)\n",
    "\\tag{1}\n",
    "$$\n",
    "\n",
    "これは $n$ が整数以外の実数を動いても成立している. この公式を用いて次の公式を証明せよ.\n",
    "\n",
    "$$\n",
    "\\prod_{n=1}^\\infty\\frac{(2n)(2n)}{(2n-1)(2n+1)} = \\frac{\\pi}{2}.\n",
    "\\tag{2}\n",
    "$$\n",
    "\n",
    "**解答例:** ベータ函数とガンマ函数の関係とガンマ函数の特殊値に関する結果より,\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "&\n",
    "B(1/2, n+1/2) = \\frac{\\Gamma(1/2)\\Gamma(n+1/2)}{\\Gamma(n+1)}=\n",
    "\\pi\\frac{1\\cdot3\\cdots(2n-1)}{2\\cdot4\\cdots(2n)},\n",
    "\\\\ &\n",
    "B(1/2, n+1) = \\frac{\\Gamma(1/2)\\Gamma(n+1)}{\\Gamma(n+1+1/2)}=\n",
    "2\\frac{2\\cdot4\\cdots(2n)}{3\\cdot5\\cdots(2n+1)},\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "ゆえに,\n",
    "\n",
    "$$\n",
    "\\frac{B(1/2,n+1)}{B(1/2,n+1/2)} = \n",
    "\\frac{2}{\\pi}\\frac{2\\cdot4\\cdots(2n)\\times 2\\cdot4\\cdots(2n)}\n",
    "{1\\cdot3\\cdots(2n-1)\\times 3\\cdot5\\cdots(2n+1)} =\n",
    "\\frac{2}{\\pi}\\prod_{k=1}^n\\frac{(2k)(2k)}{(2k-1)(2k+1)}\n",
    "$$\n",
    "\n",
    "一方, \n",
    "\n",
    "$$\n",
    "\\lim_{n\\to\\infty}\\frac{(n+a-1)^s}{(n+b-1)^s} = 1\n",
    "$$\n",
    "\n",
    "と公式(1)より,\n",
    "\n",
    "$$\n",
    "\\lim_{n\\to\\infty}\\frac{B(s,n+a)}{B(s,n+b)} = \n",
    "\\lim_{n\\to\\infty}\\frac{(n+a-1)^s B(s,n+a)}{(n+b-1)^s B(s,n+b)} = \n",
    "\\frac{\\Gamma(s)}{\\Gamma(s)} = 1\n",
    "$$\n",
    "\n",
    "この等式の $s=1/2$, $a=1/2$, $b=1$ の場合と上の結果を合わせると, (2)が得られる. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**注意:** $\\ds B(p,q) = 2\\int_0^{\\pi/2} (\\cos\\theta)^{2p-1}(\\sin\\theta)^{2q-1}\\,d\\theta$ より,\n",
    "\n",
    "$$\n",
    "B(1/2,n+1/2) = 2\\int_0^{\\pi/2} \\sin^{2n}\\theta\\,d\\theta, \\quad\n",
    "B(1/2,n+1) = 2\\int_0^{\\pi/2} \\sin^{2n+1}\\theta\\,d\\theta\n",
    "$$\n",
    "\n",
    "であることに注意せよ. 巷でよく見るWallisの公式の証明は右辺のsinのべきの積分を部分積分によって計算することによって行われている. その正体はベータ函数の特殊値であり, 結局のところ $\\ds\\lim_{n\\to\\infty}\\frac{B(s,n+a+1)}{B(s,n+b+1)}=1$ の特別な場合として, Wallisの公式は得られているのである. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題(Wallisの公式からGauss積分の公式を出す方法):** 問題(Wallisの公式の証明2)の解答例を続けることによって, Gauss積分 $\\ds \\int_{-\\infty}^\\infty e^{-x^2}\\,dx=\\Gamma\\left(\\frac{1}{2}\\right)$ を計算してみよ.\n",
    "\n",
    "**解答例:** 上の解答例より, \n",
    "\n",
    "$$\n",
    "n^{1/2}B(1/2,n+1/2)\\cdot n^{1/2}B(1/2,n+1)=\\frac{2n\\pi}{2n+1}.\n",
    "$$\n",
    "\n",
    "左辺は $n\\to\\infty$ で $\\Gamma(1/2)^2$ に収束し, 右辺は $\\pi$ に収束する. このことから $\\Gamma(1/2)=\\sqrt{\\pi}$ であることがわかる. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Legendre's duplication formula\n",
    "\n",
    "**問題(Legendre's duplication formula):** 次を示せ:\n",
    "\n",
    "$$\n",
    "\\Gamma(s)\\Gamma(s+1/2) = 2^{1-2s}\\sqrt{\\pi}\\;\\Gamma(2s).\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**解答例1:** 次の公式を示そう:\n",
    "\n",
    "$$\n",
    "\\int_{-1}^1 (1-x^2)^{s-1}\\,dx = 2^{2s-1}B(s,s) = B(1/2,s).\n",
    "$$\n",
    "\n",
    "まず, $(1-x^2)^{s-1}=(1-x)^{s-1}(1+x)^{s-1}$ であることに注意し, $x=1-2t$ とおくと,\n",
    "\n",
    "$$\n",
    "\\int_{-1}^1 (1-x^2)^{s-1}\\,dx = 2^{2s-1}\\int_0^1 t^{s-1}(1-t)^{s-1}\\,dt = 2^{2s-1}B(s,s). \n",
    "$$\n",
    "\n",
    "以上とは別に, 偶函数の積分であることを使って, $x=\\sqrt{t}$ とおくと,\n",
    "\n",
    "$$\n",
    "\\int_{-1}^1 (1-x^2)^{s-1}\\,dx = 2\\int_0^1 (1-x^2)^{s-1}\\,dx =\n",
    "\\int_0^1 (1-t)^{s-1} t^{-1/2}\\,dt = B(1/2, s).\n",
    "$$\n",
    "\n",
    "これで上の公式が示された. $B(1/2,s) = 2^{2s-1}B(s,s)$ の両辺をガンマ函数を使って表すと,\n",
    "\n",
    "$$\n",
    "2^{2s-1}\\frac{\\Gamma(s)\\Gamma(s)}{\\Gamma(2s)} =\n",
    "\\frac{\\Gamma(s)\\Gamma(1/2)}{\\Gamma(s+1/2)}\n",
    "$$\n",
    "\n",
    "ゆえに $\\Gamma(1/2)=\\sqrt{\\pi}$ を使うと,\n",
    "\n",
    "$$\n",
    "\\Gamma(s)\\Gamma(s+1/2) = 2^{1-2s}\\Gamma(1/2)\\Gamma(2s) = 2^{1-2s}\\sqrt{\\pi}\\;\\Gamma(2s).\n",
    "\\QED\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**解答例2:** ガンマ函数の函数等式とこのノートの下の方で証明されているStirlingの近似公式を使っても証明できる. 実は Legendre's duplication formula を Gauss's multiplication formula に一般化しても証明の方針は完全に同様であるので, 一般化した場合の証明のみを書いておくことにする. このノートの下の方を見よ. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### sinとガンマ函数の関係の再証明\n",
    "\n",
    "#### Euler's reflection formula の再証明\n",
    "\n",
    "Legendre's reflection formula \n",
    "\n",
    "$$\n",
    "\\Gamma(s)\\Gamma(s+1/2) = 2^{1-2s} \\sqrt{\\pi}\\;\\Gamma(2s)\n",
    "$$\n",
    "\n",
    "で $s=t/2$ とおいて得られる\n",
    "\n",
    "$$\n",
    "\\Gamma(t) = \\frac{2^{t-1}}{\\sqrt{\\pi}}\\Gamma\\left(\\frac{t}{2}\\right)\\Gamma\\left(\\frac{t+1}{2}\\right)\n",
    "$$\n",
    "\n",
    "を用いた Euler's reflection formula の証明を紹介しよう. 以下の方法は\n",
    "\n",
    "* E. Artin, <a href=\"https://www.google.co.jp/search?q=Artin+The+Gamma+Function\">The Gamma Function</a>\n",
    "\n",
    "の第4節に書いてある.\n",
    "\n",
    "函数 $f(t)$ を\n",
    "\n",
    "$$\n",
    "f(t) = \\Gamma(t)\\Gamma(1-t)\\frac{\\sin(\\pi t)}{\\pi}\n",
    "$$\n",
    "\n",
    "と定める. $0<t<1$ において $f(t)$ が正値 $C^2$ 級(実際には解析的)であることがわかる. さらに $\\Gamma(t)=\\Gamma(1+t)/t$, $\\Gamma(1-t)=\\Gamma(2-t)/(1-t)$, $\\sin(\\pi t)=\\sin(\\pi(1-t))$ より,\n",
    "\n",
    "$$\n",
    "f(t) = \n",
    "\\Gamma(1+t)\\Gamma(1-t)\\frac{\\sin(\\pi t)}{\\pi t} =\n",
    "\\Gamma(t)\\Gamma(2-t)\\frac{\\sin(\\pi(1-t))}{\\pi(1-t)}\n",
    "$$\n",
    "\n",
    "なので, この前者の表示より $t=0$ の十分近くで, 後者の表示より $t=1$ の近くで, $f(t)$ は正値でかつ $C^2$ 級(実際には解析的)であることがわかる. 特に, 前者より $f(0)=1$ であることがわかり, 後者より $f(1)=1$ であることがわかる. \n",
    "\n",
    "Legendre's duplication formula と\n",
    "\n",
    "$$\n",
    "\\ds\\sin(\\pi t)=\n",
    "2\\sin\\frac{\\pi t}{2}\\,\\cos\\frac{\\pi t}{2}=\n",
    "2\\sin\\frac{\\pi t}{2}\\,\\sin\\frac{\\pi(t+1)}{2}\n",
    "$$\n",
    "\n",
    "を使うと, \n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "f(t) &= \n",
    "\\frac{2^{t-1}}{\\sqrt{\\pi}}\\Gamma\\left(\\frac{t}{2}\\right)\\Gamma\\left(\\frac{t+1}{2}\\right)\n",
    "\\frac{2^{-t}}{\\sqrt{\\pi}}\\Gamma\\left(\\frac{1-t}{2}\\right)\\Gamma\\left(\\frac{2-t}{2}\\right)\n",
    "\\frac{2}{\\pi}\\sin\\frac{\\pi t}{2}\\,\\sin\\frac{\\pi(t+1)}{2}\n",
    "\\\\ &= \n",
    "\\Gamma\\left(\\frac{t}{2}\\right)\\Gamma\\left(\\frac{2-t}{2}\\right)\\frac{\\sin(\\pi t/2)}{\\pi}\n",
    "\\Gamma\\left(\\frac{t+1}{2}\\right)\\Gamma\\left(\\frac{1-t}{2}\\right)\\frac{\\sin(\\pi (t+1)/2)}{\\pi}\n",
    "\\\\ &= \n",
    "\\Gamma\\left(\\frac{t}{2}\\right)\\Gamma\\left(1-\\frac{t}{2}\\right)\\frac{\\sin(\\pi t/2)}{\\pi}\n",
    "\\Gamma\\left(\\frac{t+1}{2}\\right)\\Gamma\\left(1-\\frac{t+1}{2}\\right)\\frac{\\sin(\\pi(t+1)/2)}{\\pi} \n",
    "\\\\ &=\n",
    "f\\left(\\frac{t}{2}\\right)f\\left(\\frac{t+1}{2}\\right).\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "これより, $g(t)=\\log f(t)$ とおくと, $g(0)=g(1)=\\log 1=0$ でかつ,\n",
    "\n",
    "$$\n",
    "g(t) = g\\left(\\frac{t}{2}\\right) + g\\left(\\frac{t+1}{2}\\right).\n",
    "$$\n",
    "\n",
    "ゆえに, $0\\leqq t\\leqq 1$ において, \n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "&\n",
    "g''(t) = \\frac{1}{4}\\left(g''\\left(\\frac{t}{2}\\right) + g''\\left(\\frac{t+1}{2}\\right)\\right), \n",
    "\\\\ &\n",
    "|g''(t)| \\leqq \\frac{1}{4}\\left(\\left|g''\\left(\\frac{t}{2}\\right)\\right| + \\left|g''\\left(\\frac{t+1}{2}\\right)\\right|\\right)\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "となるので, $\\ds M=\\max_{0\\leqq t\\leqq 1}|g''(t)|$ とおくと,\n",
    "\n",
    "$$\n",
    "M \\leqq \\frac{1}{4}(M+M)=\\frac{M}{2}\n",
    "$$\n",
    "\n",
    "となることがわかる. ゆえに $M=0$ である. これで $0\\leqq t\\leqq 1$ において $g''(t)=0$ となることがわかった. $g(0)=g(1)=0$ より, 恒等的に $g(t)=0$ となる. これで $f(t)=1$ すなわち \n",
    "\n",
    "$$\n",
    "\\Gamma(t)\\Gamma(1-t) = \\frac{\\pi}{\\sin(\\pi t)}\n",
    "$$\n",
    "\n",
    "が成立することが示された. この公式は **Euler's reflection formula** と呼ばれているのであった.\n",
    "\n",
    "#### sinの無限積表示の再証明\n",
    "\n",
    "Euler's reflection formulaとガンマ函数の無限積表示\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "&\n",
    "\\Gamma(s) = \n",
    "\\lim_{n\\to\\infty}\\frac{n^s n!}{s(1+s)(2+s)\\cdots(n+s)} =\n",
    "\\lim_{n\\to\\infty}\\frac{n^s}{s\\left(1+s\\right)\\left(1+\\frac{s}{2}\\right)\\cdots\\left(1+\\frac{s}{n}\\right)},\n",
    "\\\\ &\n",
    "\\Gamma(1-s) = \n",
    "\\lim_{n\\to\\infty}\\frac{n^{1-s} n!}{(1-s)(2-s)\\cdots(n+1-s)} =\n",
    "\\lim_{n\\to\\infty}\\frac{n^{-s}}{\\left(1-s\\right)\\left(1-\\frac{s}{2}\\right)\\cdots\\left(1-\\frac{s}{n}\\right)\\left(1+\\frac{1-s}{n}\\right)},\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "より,\n",
    "\n",
    "$$\n",
    "\\frac{\\sin(\\pi t)}{\\pi} =\n",
    "\\frac{1}{\\Gamma(t)\\Gamma(1-t)} =\n",
    "\\lim_{n\\to\\infty}s\\prod_{k=1}^n\\left(\\left(1+\\frac{s}{k}\\right)\\left(1-\\frac{s}{k}\\right)\\right) =\n",
    "s\\prod_{n=1}^\\infty\\left(1-\\frac{s^2}{n^2}\\right).\n",
    "$$\n",
    "\n",
    "**まとめ:** ガンマ函数とベータ函数をEuler型の積分表示式に基いて, ガンマ函数の函数等式やガンマ函数とベータ函数の関係を証明できる. $\\ds\\lim_{n\\to\\infty}n^s B(s,n+1)=\\Gamma(s)$ からガンマ函数の無限積表示が得られ, 同じ積分の2通りの計算 $\\ds \\int_{-1}^1 (1-x^2)^{s-1}\\,dx = 2^{2s-1}B(s,s) = B(1/2,s)$ から Legendre's duplication formula が得られる. これだけの準備のもとで Euler's reflection formula が得られ, sin の無限積表示も得られるというのが以上の筋道である. E. Artinの有名なガンマ函数の本ではガンマ函数は函数等式と対数凸性で定数倍を除いて一意に特徴付けられるという結果を全般的に用いてガンマ函数に関する様々な公式を証明している. そのような対数凸性に頼らなくても, Euler型積分表示式だけを使っても主要な公式を示せることが以上の議論を見ればわかる. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "ガンマ函数についてはさらに\n",
    "\n",
    "* 黒木玄, <a href=\"https://genkuroki.github.io/documents/20160501StirlingFormula.pdf\">ガンマ分布の中心極限定理とStirlingの公式</a>\n",
    "\n",
    "の第8節を参照せよ."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Lerchの定理とゼータ正規化積\n",
    "\n",
    "#### Lerchの定理 (Hurwitzのゼータ函数とガンマ函数の関係)\n",
    "\n",
    "**Lerchの定理:** Hurwitzのゼータ函数 $\\zeta(s,x)$ からガンマ函数が\n",
    "\n",
    "$$\n",
    "\\zeta_s(0,x) = \\log\\frac{\\Gamma(x)}{\\sqrt{2\\pi}}, \\qquad\n",
    "\\Gamma(x) = \\sqrt{2\\pi}\\;\\exp(\\zeta_s(0,x))\n",
    "$$\n",
    "\n",
    "によって得られる. ここで $\\zeta_s(s,x)$ は $\\zeta(s,x)$ の $s$ に関する偏導函数である.\n",
    "\n",
    "**証明:** $F(x)=\\zeta_s(0,x)-\\log\\Gamma(x)$ とおく. $F(x)=-\\log\\sqrt{2\\pi}$ であることを示せば十分である. \n",
    "\n",
    "(1) $(\\zeta_s(0,x))'' = (\\log\\Gamma(x))''$ を示そう. ここで $'$ は $x$ による微分を表わす. まずHurwitzのゼータ函数\n",
    "\n",
    "$$\n",
    "\\zeta(s,x) = \\sum_{k=0}^\\infty \\frac{1}{(x+k)^s}\n",
    "$$\n",
    "\n",
    "については\n",
    "\n",
    "$$\n",
    "\\zeta_x(s,x) = -s\\zeta(s+1,x)\n",
    "$$\n",
    "\n",
    "が成立しているので, \n",
    "\n",
    "$$\n",
    "\\zeta_{xx}(s,x) = s(s+1)\\zeta(s+2,x).\n",
    "$$\n",
    "\n",
    "これより, \n",
    "\n",
    "$$\n",
    "(\\zeta_s(0,x))'' = \\zeta_{xxs}(0,x) = \\zeta(2,x).\n",
    "$$\n",
    "\n",
    "一方, ガンマ函数\n",
    "\n",
    "$$\n",
    "\\Gamma(x) = \\lim_{\\to\\infty}\\frac{n!\\,n^x}{x(x+1)\\cdots(x+n)}\n",
    "$$\n",
    "\n",
    "については,\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "&\n",
    "\\log\\Gamma(x) =\n",
    "\\lim_{n\\to\\infty}\\left(\n",
    "\\log n! + n\\log x - \\log x - \\log(x+1) - \\cdots - \\log(x+n)\n",
    "\\right),\n",
    "\\\\ &\n",
    "(\\log\\Gamma(x))' =\n",
    "\\lim_{n\\to\\infty}\\left(\n",
    "\\log n - \\frac{1}{x} - \\frac{1}{x+1} - \\cdots - \\frac{1}{x+n}\n",
    "\\right),\n",
    "\\\\ &\n",
    "(\\log\\Gamma(x))'' =\n",
    "\\lim_{n\\to\\infty}\\left(\n",
    "\\frac{1}{x^2} + \\frac{1}{(x+1)^2} + \\cdots + \\frac{1}{(x+n)^2}\n",
    "\\right) = \\zeta(2,x).\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "これで $(\\zeta_s(0,x))'' = (\\log\\Gamma(x))''$ が示された.\n",
    "\n",
    "(2) 上の結果より, $F(x)=\\zeta_s(0,x)-\\log\\Gamma(x)$ は $x$ の一次函数である.\n",
    "\n",
    "(3) $\\zeta_s(0,x)$ と $\\log\\Gamma(x)$ がどちらも同一の函数等式 $f(x+1)=f(x)+\\log x$ を満たすことを示そう. \n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "&\n",
    "\\zeta(s,x+1) = \\zeta(s,x) - \\frac{1}{x^s},\n",
    "\\qquad\\therefore\\quad\n",
    "\\zeta_s(0,x+1) = \\zeta_s(0,x) + \\log x.\n",
    "\\\\ &\n",
    "\\log\\Gamma(x+1) = \\log(x\\Gamma(x)) = \\log\\Gamma(x) + \\log x.\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "(4) $F(x)=\\zeta_s(0,x)-\\log\\Gamma(x)$ は $x$ の一次函数だったので, 上の結果より $F(x)$ は定数になる.\n",
    "\n",
    "(5) $\\zeta_s(0,1/2)=-\\log\\sqrt{2}$ を示そう.\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\zeta(s) - 2^{-s}\\zeta(s) &=\n",
    "\\left(\\frac{1}{1^s}+\\frac{1}{2^s}+\\frac{1}{3^s}+\\frac{1}{4^s}+\\cdots\\right) -\n",
    "\\left(\\frac{1}{2^s}+\\frac{1}{4^s}+\\cdots\\right) \n",
    "\\\\ &=\n",
    "\\frac{1}{1^s}+\\frac{1}{3^s}+\\cdots =\n",
    "\\sum_{k=0}^\\infty\\frac{1}{(2k+1)^s}\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "なので\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "&\n",
    "\\zeta(s,1/2) = \\sum_{k=0}^\\infty\\frac{1}{(k+1/2)^s} =\n",
    "2^s\\sum_{k=0}^\\infty\\frac{1}{(2k+1)^s} = \n",
    "2^s(\\zeta(s) - 2^{-s}\\zeta(s)) =\n",
    "(2^s-1)\\zeta(s),\n",
    "\\\\ &\\therefore\\quad\n",
    "\\zeta_s(0,1/2) = \\zeta(0)\\log 2 = -\\frac{1}{2}\\log 2 = -\\log\\sqrt{2}.\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "(6) $\\log\\Gamma(1/2)=\\log\\sqrt{\\pi}$ なので, 上の結果より, $F(x)=-\\log\\sqrt{2\\pi}$ であることがわかる. $\\QED$\n",
    "\n",
    "#### ゼータ正規化積 \n",
    "\n",
    "数列 $a_n$ に対して,\n",
    "\n",
    "$$\n",
    "f(s) = \\sum_{n=1}^N \\frac{1}{a_n^s}\n",
    "$$\n",
    "\n",
    "とおくとき, \n",
    "\n",
    "$$\n",
    "f'(0) = -\\sum_{n=1}^N \\log a_n\n",
    "$$\n",
    "\n",
    "なので, \n",
    "\n",
    "$$\n",
    "\\exp(-f'(0)) = \\prod_{n=1}^N a_n\n",
    "$$\n",
    "\n",
    "が成立している. もしも $N=\\infty$ のときの $\\ds\\prod_{n=1}^\\infty a_n$ が発散していても, $\\ds f(s)=\\sum_{n=1}^\\infty \\frac{1}{a_n^s}$ の解析接続によって, 左辺の $\\exp(-f'(0))$ はwell-definedになる可能性がある. そのとき, $\\exp(-f'(0))$ を\n",
    "\n",
    "$$\n",
    "\\exp(-f'(0)) = \\PROD_{n=1}^\\infty a_n\n",
    "$$\n",
    "\n",
    "と書き, $a_n$ 達の**ゼータ正規化積**と呼ぶ. \n",
    "\n",
    "例えば $x,x+1,x+2,x+3,\\ldots$ のゼータ正規化積はLerchの定理より, $\\ds\\frac{\\sqrt{2\\pi}}{\\Gamma(x)}$ になる. 特に $x=1$ のときの $1,2,3,4,\\ldots$ のゼータ正規化積は $\\sqrt{2\\pi}$ になる:\n",
    "\n",
    "$$\n",
    "\"\\! 1\\times 2\\times 3\\times 4\\times\\cdots \\!\" \\,= \n",
    "\\PROD_{n=1}^\\infty n = \\exp(-\\zeta'(0)) = \\sqrt{2\\pi}.\n",
    "$$\n",
    "\n",
    "これは\n",
    "\n",
    "$$\n",
    "\"\\! 1+2+3+4+\\cdots \\!\"\\, = \\zeta(-1) = -\\frac{1}{12}\n",
    "$$\n",
    "\n",
    "の積バージョンである. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Stirlingの公式とLaplaceの方法\n",
    "\n",
    "一般に数列 $a_n,b_n$ について\n",
    "\n",
    "$$\n",
    "\\lim_{n\\to\\infty}\\frac{a_n}{b_n} = 1\n",
    "$$\n",
    "\n",
    "が成立するとき,\n",
    "\n",
    "$$\n",
    "a_n\\sim b_n\n",
    "$$\n",
    "\n",
    "と書くことにする. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Stirlingの公式\n",
    "\n",
    "**Stirlingの(近似)公式:** $n\\to\\infty$ のとき,\n",
    "\n",
    "$$\n",
    "n!\\sim n^n e^{-n} \\sqrt{2\\pi n}.\n",
    "$$\n",
    "\n",
    "さらに, 両辺の対数を取ることによって, $n\\to\\infty$ のとき,\n",
    "\n",
    "$$\n",
    "\\log n! = n\\log n - n + \\frac{1}{2}\\log n + \\log\\sqrt{2\\pi} + o(1).\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Stirlingの公式の「物理学的」もしくは「情報理論的」な応用については\n",
    "\n",
    "* 黒木玄, <a href=\"https://genkuroki.github.io/documents/20160616KullbackLeibler.pdf\">Kullback-Leibler情報量とSanovの定理</a>\n",
    "\n",
    "の第1節を参照せよ."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Stirlingの公式の証明:**\n",
    "\n",
    "$$\n",
    "n! = \\Gamma(n+1) = \\int_0^\\infty e^{-x} x^n\\,dx\n",
    "$$\n",
    "\n",
    "で $x = n+\\sqrt{n}\\;y = n(1+y/\\sqrt{n})$ と置換すると, \n",
    "\n",
    "$$\n",
    "n! = \n",
    "n^n e^{-n} \\sqrt{n} \\int_{-\\sqrt{n}}^\\infty e^{-\\sqrt{n}\\;y}\\;\\left(1+\\frac{y}{\\sqrt{n}}\\right)^n\\,dy =\n",
    "n^n e^{-n} \\sqrt{n} \\int_{-\\sqrt{n}}^\\infty \\;f_n(y)\\,dy.\n",
    "$$\n",
    "\n",
    "ここで, 被積分函数を $f_n(y)$ と書いた. そのとき $n\\to\\infty$ で\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\log f_n(y) &= -\\sqrt{n}\\;y + n\\log\\left(1+\\frac{y}{\\sqrt{n}}\\right) =\n",
    "-\\sqrt{n}\\;y + n\\left(\\frac{y}{\\sqrt{n}} - \\frac{y^2}{2n} + O\\left(\\frac{1}{n\\sqrt{n}}\\right)\\right) \n",
    "\\\\ &=\n",
    "-\\frac{y^2}{2} + O\\left(\\frac{1}{\\sqrt{n}}\\right) \\to -\\frac{y^2}{2}.\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "すなわち $f_n(y)\\to e^{-y^2/2}$ となる. ゆえに\n",
    "\n",
    "$$\n",
    "\\frac{n!}{n^n e^{-n} \\sqrt{2\\pi n}} =\n",
    "\\frac{1}{\\sqrt{2\\pi}}\\int_{-\\sqrt{n}}^\\infty\\;f_n(y)\\,dy\n",
    "\\to \\frac{1}{\\sqrt{2\\pi}}\\int_{-\\infty}^\\infty e^{-y^2/2}\\,dy = 1.\n",
    "$$\n",
    "\n",
    "最後の等号でGauss積分の公式 $\\int_{-\\infty}^\\infty e^{-y^2/a}\\,dy=\\sqrt{a\\pi}$ を用いた. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Stirlingの公式の証明の解説:** 上の証明のポイントは $x=n+\\sqrt{n}\\;y$ という積分変数変換である. この変数変換の「正体」は $\\Gamma(n+1)=\\int_0^\\infty e^{-x} x^n\\,dx$ の被積分函数 $f(x)=e^{-x}x^n$ のグラフを描いてみれば見当がつく.\n",
    "\n",
    "$g(x)=\\log f(x)=n\\log x - x$ の導函数は $g'(x)=n/x-1$ は $x$ について単調減少であり, $x=n$ で $0$ になる. ゆえに $g(x)=\\log f(x)$ は $x=n$ で最大になる.  そこで $x=n$ における $g(x)=\\log f(x)$ のTaylor展開を求めてみよう. $g''(x)=-n/x^2$, $g'''(x)=2n/x^3$ なので, $g(n)=n\\log n - n$, $g'(n)=0$, $g''(n)=-1/n$, $g'''(n)=2/n^2$ なので,\n",
    "\n",
    "$$\n",
    "g(x) = n \\log n - n -\\frac{(x-n)^2}{2n} + \\frac{(x-n)^3}{3\\,n^2} + \\cdots\n",
    "$$\n",
    "\n",
    "これの2次の項が $-y^2/2$ になるような変数変換がちょうど $x=n+\\sqrt{n}\\;y$ になっている. これが上の証明で用いた変数変換の「正体」である. $\\QED$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
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     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "# y = f(x) = e^{-x} x^n / (n^n * e^{-n}) のグラフは n が大きなとき,\n",
    "# Gauss近似 y = e^{-(x-n)^2/(2n)} のグラフにほぼ一致する.\n",
    "\n",
    "f(n,x) = e^(-x + n*log(x) - (n*log(n) - n))\n",
    "g(n,x) = e^(-(x-n)^2/(2n))\n",
    "PP = []\n",
    "for n in [10, 30, 100, 300]\n",
    "    x = 0:2.5n/400:2.5n\n",
    "    n ≤ 20 && (x = 0:3n/400:3n)\n",
    "    P = plot()\n",
    "    plot!(title=\"n = $n\", titlefontsize=9)\n",
    "    plot!(x, f.(n,x), label=\"\")\n",
    "    plot!(x, g.(n,x), label=\"Gaussian\")\n",
    "    push!(PP, P)\n",
    "end\n",
    "\n",
    "plot(PP[1:2]..., size=(700, 200))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(PP[3:4]..., size=(700, 200))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**注意(ガンマ函数のStirlingの近似公式):** 上の証明で $n$ が整数であることは使っていない. ゆえに正の実数 $s$ について\n",
    "\n",
    "$$\n",
    "\\Gamma(s+1) \\sim s^s e^{-s} \\sqrt{2\\pi s} \\quad (s\\to\\infty)\n",
    "$$\n",
    "\n",
    "が証明されている. これの両辺を $s$ で割ると,\n",
    "\n",
    "$$\n",
    "\\Gamma(s) \\sim s^s e^{-s} s^{-1/2} \\sqrt{2\\pi} \\quad (s\\to\\infty)\n",
    "$$\n",
    "\n",
    "が得られる. これらをも**Stirlingの近似公式**と呼ぶ. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Stirlingの公式の重要な応用については\n",
    "\n",
    "* 黒木玄, <a href=\"http://nbviewer.jupyter.org/github/genkuroki/Calculus/blob/master/11%20Kullback-Leibler%20information.ipynb\">11 Kullback-Leibler情報量</a>\n",
    "\n",
    "も参照せよ. 「Stirlingの公式」とその応用としての「KL情報量に関するSanovの定理」についてはできるだけ早く理解しておいた方がよい. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題:** $n=1,2,\\ldots,10$ について Stirling の公式の相対誤差\n",
    "\n",
    "$$\n",
    "\\frac{n^n e^{-n} \\sqrt{2\\pi n}}{n!}-1\n",
    "$$\n",
    "\n",
    "を求めよ.\n",
    "\n",
    "**解答例:** 以下のセルを参照せよ. $n=5$ で相対誤差は2%を切っている. $\\QED$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10-element Array{Tuple{Int64,Int64,Float64,Float64},1}:\n",
       " (1, 1, 0.922137, -0.077863)         \n",
       " (2, 2, 1.919, -0.0404978)           \n",
       " (3, 6, 5.83621, -0.0272984)         \n",
       " (4, 24, 23.5062, -0.020576)         \n",
       " (5, 120, 118.019, -0.0165069)       \n",
       " (6, 720, 710.078, -0.0137803)       \n",
       " (7, 5040, 4980.4, -0.0118262)       \n",
       " (8, 40320, 39902.4, -0.0103573)     \n",
       " (9, 362880, 3.59537e5, -0.00921276) \n",
       " (10, 3628800, 3.5987e6, -0.00829596)"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(n) = factorial(n)\n",
    "g(n) = n^n * exp(-n) * √(2π*n)\n",
    "[(n, f(n), g(n), g(n)/f(n)-1) for n in 1:10]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**参考:** 上の計算を見れば, $n^n e^{-n} \\sqrt{2\\pi n}$ は $n!$ よりも微小に小さいことがわかる. その分を補正したより精密な近似式\n",
    "\n",
    "$$\n",
    "n! = n^n e^{-n} \\sqrt{2\\pi n}\\left(1+\\frac{1}{12n}+O\\left(\\frac{1}{n^2}\\right)\\right)\n",
    "$$\n",
    "\n",
    "が成立している. (実際には $O(1/n^2)$ の部分についてもっと詳しいことがわかる.)\n",
    "\n",
    "$1/(12n)$ で補正した近似式の相対誤差は $n=1$ ですでに0.1%程度と非常に小さくなる. 次のセルを見よ. $\\QED$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10-element Array{Tuple{Int64,Int64,Float64,Float64},1}:\n",
       " (1, 1, 0.998982, -0.00101824)        \n",
       " (2, 2, 1.99896, -0.000518567)        \n",
       " (3, 6, 5.99833, -0.000278913)        \n",
       " (4, 24, 23.9959, -0.00017137)        \n",
       " (5, 120, 119.986, -0.000115383)      \n",
       " (6, 720, 719.94, -8.28033e-5)        \n",
       " (7, 5040, 5039.69, -6.22504e-5)      \n",
       " (8, 40320, 40318.0, -4.84771e-5)     \n",
       " (9, 362880, 3.62866e5, -3.88063e-5)  \n",
       " (10, 3628800, 3.62868e6, -3.17601e-5)"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(n) = factorial(n)\n",
    "g1(n) = n^n * exp(-n) * √(2π*n) * (1+1/(12n))\n",
    "[(n, f(n), g1(n), g1(n)/f(n)-1) for n in 1:10]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Wallisの公式のStirlingの公式を使った証明\n",
    "\n",
    "**問題(Wallisの公式):** Stirlingの公式を用いて次を示せ:\n",
    "\n",
    "$$\n",
    "\\frac{1}{2^{2n}}\\binom{2n}{n} \\sim \\frac{1}{\\sqrt{\\pi n}}.\n",
    "$$\n",
    "\n",
    "**解答例:**\n",
    "$$\n",
    "\\frac{1}{2^{2n}}\\binom{2n}{n} = \\frac{(2n)!}{2^{2n}(n!)^2}\n",
    "\\sim \\frac{(2n)^{2n}e^{-2n}\\sqrt{4\\pi n}}{2^{2n}n^{2n}e^{-2n}2\\pi n} = \\frac{1}{\\sqrt{\\pi n}}.\n",
    "\\qquad \\QED\n",
    "$$\n",
    "\n",
    "**注意:** この形のWallisの公式は1次元の単純ランダムウォークの逆正弦法則に関係している.\n",
    "\n",
    "* 黒木玄, <a href=\"https://genkuroki.github.io/documents/#2016-11-02\">単純ランダムウォークの逆正弦法則</a> (<a href=\"https://genkuroki.github.io/documents/20161102ArcSineLawForSimpleRandomWalks.pdf\">手描きのノートのPDF</a>)\n",
    "\n",
    "を参照せよ. 特に手描きのノートのPDFファイルの12頁以降にまとまった解説がある. 1次元の単純ランダムウォークの場合には高校数学レベルの組み合わせ論的な議論とWallisの公式から逆正弦法則を導くことができる. 1次元の一般ランダムウォークの場合にはTauber型定理を使ってWallisの公式に対応する漸近挙動を証明することになる. $\\QED$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Wallis_pi = 3.14159265358979323846264338327950280112510935008936482449955348608333403219364\n",
      "Exact__pi = 3.141592653589793238462643383279502884197169399375105820974944592307816406286198\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "-8.307206004928574099647539110622448237409255782069327815244440488293374992724481e-35"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Wallisの公式より\n",
    "#\n",
    "# [ 2^{2n} (n!)^2 / ((2n)! √n) ]^2 ---→ π\n",
    "#\n",
    "# 以下はこれの数値的確認\n",
    "#\n",
    "# log n! を log lgamma(n+1) で計算している. ここで lgamma(x) = log(Γ(x)).\n",
    "# lgamma(x) は対数ガンマ函数を巨大な x についても計算してくれる.\n",
    "\n",
    "f(n) = exp((2n)*log(typeof(n)(2)) + 2lgamma(n+1) - lgamma(2n+1) - log(n)/2)^2\n",
    "Wallis_pi = f(big\"10.0\"^40)\n",
    "Exact__pi = big(π)\n",
    "@show Wallis_pi\n",
    "@show Exact__pi\n",
    "Wallis_pi - Exact__pi"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Gauss's multiplication formula\n",
    "\n",
    "**問題(Gauss's multiplication formula):** 次を示せ: 正の整数 $n$ に対して,\n",
    "\n",
    "$$\n",
    "\\Gamma(s)\\Gamma\\left(s+\\frac{1}{n}\\right)\\cdots\\Gamma\\left(s+\\frac{n-1}{n}\\right) =\n",
    "n^{1/2-ns}(2\\pi)^{(n-1)/2}\\Gamma(ns).\n",
    "$$\n",
    "\n",
    "**解答例:** 函数 $f(s)$ を次のように定める:\n",
    "\n",
    "$$\n",
    "f(s) = \\frac{\\Gamma(s)\\Gamma\\left(s+\\frac{1}{n}\\right)\\cdots\\Gamma\\left(s+\\frac{n-1}{n}\\right)}{n^{-ns}\\Gamma(ns)}\n",
    "$$\n",
    "\n",
    "$f(s)=n^{1/2}(2\\pi)^{(n-1)/2}$ を示せばよい.\n",
    "\n",
    "ガンマ函数の函数等式だけを使って, $f(s+1)=f(s)$ を示せる:\n",
    "\n",
    "$$\n",
    "f(s+1) = \n",
    "f(s)\\frac\n",
    "{s\\left(s+\\frac{1}{n}\\right)\\cdots\\left(s+\\frac{n-1}{n}\\right)}\n",
    "{n^{-n}(ns+n-1)\\cdots(ns+1)(ns)} = f(s).\n",
    "$$\n",
    "\n",
    "上で証明されているStirlingの近似公式\n",
    "\n",
    "$$\n",
    "\\Gamma(s) \\sim s^s e^{-s} s^{-1/2}\\sqrt{2\\pi} \\quad (s\\to\\infty)\n",
    "$$\n",
    "\n",
    "を使って, $s\\to\\infty$ のときの $f(s)$ の極限を求めよう. $s\\to\\infty$ のとき, $\\ds \\left(1+\\frac{a}{s}\\right)^s\\to e^a$ なので, $s\\to\\infty$ において, \n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\Gamma(s+a) &\\sim (s+a)^{s+a-1/2} e^{-s-a} \\sqrt{2\\pi} \n",
    "\\\\ &=\n",
    "s^{s+a-1/2}e^{-s}\\sqrt{2\\pi}\\;\\left(1+\\frac{a}{s}\\right)^{s+a-1/2} e^{-a} \n",
    "\\\\ &\\sim\n",
    "s^{s+a-1/2}e^{-s}\\sqrt{2\\pi}.\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "となる. ゆえに, $\\frac{1}{n}+\\frac{2}{n}+\\cdots+\\frac{n-1}{n}=\\frac{n-1}{2}$ なので, $s\\to\\infty$ において, \n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "&\n",
    "\\Gamma(s) \\sim\n",
    "s^{s-1/2} e^{-s}\\sqrt{2\\pi},\n",
    "\\\\ &\n",
    "\\Gamma\\left(s+\\frac{1}{n}\\right)\\sim\n",
    "s^{s+1/n-1/2} e^{-s} \\sqrt{2\\pi},\n",
    "\\\\ &\n",
    "\\qquad\\qquad\\cdots\\cdots\\cdots\\cdots\\cdots\n",
    "\\\\ &\n",
    "\\Gamma\\left(s+\\frac{n-1}{n}\\right)\\sim\n",
    "s^{s+(n-1)/n-1/2} e^{-s} \\sqrt{2\\pi}\n",
    "\\\\ &\n",
    "\\therefore\\quad\n",
    "\\Gamma(s)\\Gamma\\left(s+\\frac{1}{n}\\right)\\cdots\\Gamma\\left(s+\\frac{n-1}{n}\\right)\\sim\n",
    "s^{ns-1/2}e^{-ns}(2\\pi)^{n/2}\n",
    "\\\\ &\n",
    "n^{-ns}\\Gamma(ns)\\sim\n",
    "n^{-ns}(ns)^{ns-1/2}e^{-ns}\\sqrt{2\\pi} =\n",
    "n^{-1/2}s^{ns-1/2}e^{-ns}(2\\pi)^{1/2}\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "となり, \n",
    "\n",
    "$$\n",
    "f(s)\\sim\\frac{s^{ns-1/2}e^{-ns}(2\\pi)^{n/2}}{n^{-1/2}s^{ns-1/2}e^{-ns}(2\\pi)^{1/2}}=\n",
    "n^{1/2}(2\\pi)^{(n-1)/2}.\n",
    "$$\n",
    "\n",
    "ゆえに整数 $N$ について, $f(s+N)=f(s)$ なので, $N\\to\\infty$ のとき $f(s)=f(s+N)\\to n^{1/2}(2\\pi)^{(n-1)/2}$ となる. これで $f(s)=2^{1/2}(2\\pi)^{(n-1)/2}$ が示された. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題:** Gauss's multiplication formula の $n=2$ の場合である Legendre's duplication formula は定積分の計算だけで証明できるのであった. 上の解答例は本質的にStirlingの近似公式を使っている. Gauss's multiplication formula にも定積分の計算だけで証明する方法がないだろうか. 以下の方針で Gauss's multiplication formula を証明せよ. ただし, (3)の証明には Euler's reflection formula は使ってよいことにする. \n",
    "\n",
    "$t>0$ に対する $n-1$ 重積分 $I(t)$ を次のように定める:\n",
    "\n",
    "$$\n",
    "I(t) = \\int_0^\\infty\\cdots\\int_0^\\infty \n",
    "e^{-(t^n/(x_2\\cdots x_n)+x_2+\\cdots+x_n)}\n",
    "x_2^{-(n-1)/n}x_3^{-(n-2)/n}\\cdots x_n^{-1/n} \\,dx_2\\cdots dx_n.\n",
    "$$\n",
    "\n",
    "以下を示せ:\n",
    "\n",
    "(1) $\\ds I(t) = \\Gamma\\left(\\frac{1}{n}\\right)\\Gamma\\left(\\frac{2}{n}\\right)\\cdots\\Gamma\\left(\\frac{n-1}{n}\\right)e^{-nt}$.\n",
    "\n",
    "(2) $\\ds \\Gamma(s)\\Gamma\\left(s+\\frac{1}{n}\\right)\\cdots\\Gamma\\left(s+\\frac{n-1}{n}\\right) = \n",
    "n^{1-ns}I(0)\\Gamma(ns) =\n",
    "n^{1-ns}\\Gamma\\left(\\frac{1}{n}\\right)\\Gamma\\left(\\frac{2}{n}\\right)\\cdots\\Gamma\\left(\\frac{n-1}{n}\\right)\\Gamma(ns)$.\n",
    "\n",
    "(3) $\\ds I(0) = \n",
    "\\Gamma\\left(\\frac{1}{n}\\right)\\Gamma\\left(\\frac{2}{n}\\right)\\cdots\\Gamma\\left(\\frac{n-1}{n}\\right) =\n",
    "(2\\pi)^{(n-1)/2} n^{-1/2}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**注意:** (1), (2) の方針の証明は\n",
    "\n",
    "* Andrews, G.E., Askey,R., and Roy, R. Special functions. Encyclopedia of Mathematics and its Applications, Vol. 71, Cambridge University Press, 1999, 2000, 681 pages.\n",
    "\n",
    "のpp.24-25で解説されている. その方法は\n",
    "\n",
    "* Liouville, J. Sur un théorème relatif à l’intégrale eulérienne de seconde espèce. Journal de mathématiques pures et appliquées 1re série, tome 20 (1855), p. 157-160. <a href=\"http://sites.mathdoc.fr/JMPA/PDF/JMPA_1855_1_20_A15_0.pdf\">PDF</a>\n",
    "\n",
    "の方法の再構成ということらしい."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**解答例:** (1) $t=0$ のとき, $I(0)$ の積分は変数分離形になって, \n",
    "\n",
    "$$\n",
    "I(0) = \\Gamma\\left(\\frac{1}{n}\\right)\\Gamma\\left(\\frac{2}{n}\\right)\\cdots\\Gamma\\left(\\frac{n-1}{n}\\right)\n",
    "$$\n",
    "\n",
    "となることがすぐにわかる. $I(t)$ を $t$ で微分して, $\\ds x_2 = \\frac{t^n}{x_3\\cdots x_n x_1}$ によって積分変数 $x_2$ を積分変数 $x_1$ に変換すると, \n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "I'(t) &=\n",
    "\\int_0^\\infty\\cdots\\int_0^\\infty \n",
    "e^{-(t^n/(x_2\\cdots x_n)+x_2+\\cdots+x_n)}\n",
    "\\frac{-nt^{n-1}}{x_2\\cdots x_n}\n",
    "x_2^{-(n-1)/n}x_3^{-(n-2)/n}\\cdots x_n^{-1/n} \\,dx_2\\cdots dx_n\n",
    "\\\\ &=\n",
    "-n\\int_0^\\infty\\cdots\\int_0^\\infty \n",
    "e^{-(t^n/(x_2\\cdots x_n)+x_2+\\cdots+x_n)}\n",
    "t^{n-1}\n",
    "x_2^{-(n-1)/n-1}x_3^{-(n-2)/n-1}\\cdots x_n^{-1/n-1} \\,dx_2\\cdots dx_n\n",
    "\\\\ &=\n",
    "-n\\int_0^\\infty\\cdots\\int_0^\\infty \n",
    "e^{-(x_1+t^n/(x_3\\cdots x_n x_1)+x_3+\\cdots+x_n)}\n",
    "\\\\ & \\qquad\\qquad\\quad\\times\n",
    "t^{n-1}\n",
    "\\left(\\frac{t^n}{x_3\\cdots x_n x_1}\\right)^{-(2n-1)/n}\n",
    "x_3^{-(2n-2)/n}\\cdots x_n^{-(n+1)/n} \\frac{t^n}{x_3\\cdots x_n x_1^2}\\,dx_3\\cdots dx_n\\,dx_1\n",
    "\\\\ &=\n",
    "-n\\int_0^\\infty\\cdots\\int_0^\\infty \n",
    "e^{-(x_1+t^n/(x_3\\cdots x_n x_1)+x_3+\\cdots+x_n)}\n",
    "x_3^{-(n-1)/n}\\cdots x_n^{-2/n} x_1^{-1/n}\\,dx_3\\cdots dx_n\\,dx_1\n",
    "\\\\ &=\n",
    "-n I(t)\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "ゆえに $I(t)=I(0)e^{-nt}$.  これで(1)が示された."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(2) 左辺をLHSと書き, $\\ds x_1=\\frac{t^n}{x_2\\cdots x_n}$ とおくと, \n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\text{LHS} &=\n",
    "\\int_0^\\infty\\cdots\\int_0^\\infty e^{-(x_1+\\cdots+x_n)} x_1^{s-1}x_2^{s-(n-1)/n}\\cdots x_n^{s-1/n}\\,dx_1\\cdots dx_n\n",
    "\\\\ &=\n",
    "\\int_0^\\infty\\cdots\\int_0^\\infty e^{-(t^n/(x_2\\cdots x_n)+x_2\\cdots+x_n)}\n",
    "\\left(\\frac{t^n}{x_2\\cdots x_n}\\right)^{s-1}\n",
    "x_2^{s-(n-1)/n}\\cdots x_n^{s-1/n}\n",
    "\\frac{nt^{n-1}}{x_2\\cdots x_n}\n",
    "\\,dt\\,dx_2\\cdots dx_n\n",
    "\\\\ &=\n",
    "n\\int_0^\\infty\\cdots\\int_0^\\infty e^{-(t^n/(x_2\\cdots x_n)+x_2\\cdots+x_n)}\n",
    "x_2^{-(n-1)/n}\\cdots x_n^{-1/n} t^{ns-1}\n",
    "\\,dx_2\\cdots dx_n\\,dt\n",
    "\\\\ &=\n",
    "n\\int_0^\\infty I(t) t^{ns-1}\\,dt =\n",
    "nI(0)\\int_0^\\infty e^{-nt} t^{ns-1}\\,dt =\n",
    "n^{1-ns}I(0)\\Gamma(ns).\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "これで(2)が示された."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "(3) $I(0)=(2\\pi)^{(n-1)/2}n^{-1/2}$ を示したい. そのためには $I(0)^2 = (2\\pi)^{n-1} n^{-1}$ を示せばよい. Euler's reflection formula より, $\\ds \\Gamma\\left(\\frac{k}{n}\\right)\\Gamma\\left(\\frac{n-k}{n}\\right) = \\frac{\\pi}{\\sin(k\\pi/n)}$ なので\n",
    "\n",
    "$$\n",
    "I(0)^2 = \\prod_{k=1}^{n-1}\\frac{\\pi}{\\sin(k\\pi/n)} =\n",
    "\\frac{\\pi^{n-1}}{\\ds\\prod_{k=1}^{n-1}\\sin\\frac{k\\pi}{n}}.\n",
    "$$\n",
    "\n",
    "そして, \n",
    "\n",
    "$$\n",
    "\\prod_{k=1}^{n-1}\\sin\\frac{k\\pi}{n} = \n",
    "\\prod_{k=1}^{n-1}\\frac{e^{\\pi ik/n}-e^{-\\pi ik/n}}{2i} =\n",
    "\\frac{e^{\\pi i(1+2+\\cdots+(n-1))/n}}{2^{n-1}i^{n-1}}\\prod_{k=1}^{n-1}(1-e^{-2\\pi ik/n}) =\n",
    "\\frac{1}{2^{n-1}}\\prod_{k=1}^{n-1}(1-e^{-2\\pi ik/n})\n",
    "$$\n",
    "\n",
    "であり,  \n",
    "\n",
    "$$\n",
    "\\frac{x^n-1}{x-1} = \\prod_{k=1}^{n-1}(x-e^{-2\\pi ik/n})\n",
    "$$\n",
    "\n",
    "において $x\\to 1$ とすると $\\ds \\prod_{k=1}^{n-1}(1-e^{-2\\pi ik/n})=n$ が得られる. 以上を合わせると\n",
    "\n",
    "$$\n",
    "I(0)^2 = \\frac{(2\\pi)^{n-1}}{n}.\n",
    "$$\n",
    "\n",
    "両辺の平方根を取れば(3)が得られる. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Laplaceの方法\n",
    "\n",
    "**Laplaceの方法:** Stirlingの公式の証明の解説のようにして見付かる変数変換はより一般の場合に非常に有用である. 以下では $\\int_{-\\infty}^\\infty$ や $\\int_0^\\infty$ を単に $\\int$ と書くことにし, \n",
    "\n",
    "$$\n",
    "Z_n = \\int e^{-nf(x)}g(x)\\,dx\n",
    "$$\n",
    "\n",
    "とおく. ただし, $f(x)$ は実数値函数で唯一つの最小値 $f(x_0)$ を持ち, $x=x_0$ において, \n",
    "\n",
    "$$\n",
    "f(x) = f(x_0) + \\frac{a}{2}(x-x_0)^2 + O((x-x_0)^3), \\quad a=f''(x_0) > 0\n",
    "$$\n",
    "\n",
    "とTaylor展開されていると仮定するし, さらに, $0$ 以上の値を持つ実数値函数 $g(x)$ は積分 $Z_n$ がうまく定義されるような適当な条件を満たしていると仮定し, $x_0$ の近傍で $g(x)>0$ を満たしていると仮定する. (ここで, $x_0$ の近傍で $g(x)>0$ が成立しているとは, ある $\\delta>0$ が存在して, $|x-x_0|<\\delta$ ならば $g(x)>0$ となることである.) このとき, \n",
    "\n",
    "$$\n",
    "Z_n = e^{-nf(x_0)} \\int \\exp\\left(-n\\left(\\frac{a}{2}(x-x_0)^2+O((x-x_0)^3)\\right)\\right)\\;g(x)\\,dx.\n",
    "$$\n",
    "\n",
    "$x=x_0+y/\\sqrt{n}$ と変数変換すると\n",
    "\n",
    "$$\n",
    "Z_n = \\frac{e^{-nf(x_0)}}{\\sqrt{n}}\n",
    "\\int \\exp\\left(-\\frac{a}{2}y^2+O\\left(\\frac{1}{\\sqrt{n}}\\right)\\right)\\;\n",
    "g\\left(x_0+\\frac{y}{\\sqrt{n}}\\right)\\,dy.\n",
    "$$\n",
    "\n",
    "そして, $n\\to\\infty$ で\n",
    "\n",
    "$$\n",
    "\\int \\exp\\left(-\\frac{a}{2}y^2+O\\left(\\frac{1}{\\sqrt{n}}\\right)\\right)\\;\n",
    "g\\left(x_0+\\frac{y}{\\sqrt{n}}\\right)\\,dy \\to\n",
    "\\int \\exp\\left(-\\frac{a}{2}y^2\\right)g(x_0)\\,dy =\n",
    "\\sqrt{\\frac{2\\pi}{a}}\\;g(x_0).\n",
    "$$\n",
    "\n",
    "$a=f''(x_0)$ とおいたことを思い出しながら, 以上をまとめると, $n\\to\\infty$ で\n",
    "\n",
    "$$\n",
    "Z_n \\sim  \\frac{e^{-nf(x_0)}}{\\sqrt{n}} \\sqrt{\\frac{2\\pi}{f''(x_0)}}\\;g(x_0).\n",
    "$$\n",
    "\n",
    "すなわち, \n",
    "\n",
    "$$\n",
    "-\\log Z_n = nf(x_0) + \\frac{1}{2}\\log n - \\log\\left(\\sqrt{\\frac{2\\pi}{f''(x_0)}}\\;g(x_0)\\right) + o(1).\n",
    "$$\n",
    "\n",
    "$Z_n$ の $n\\to\\infty$ における漸近挙動を調べるための以上の方法を**Laplaceの方法**(Laplace's method)と呼ぶ. $\\QED$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**問題(Stirlingの公式):**  $\\ds n! = \\int_0^\\infty e^{-t}t^n\\,dt$ にLapalceの方法を適用して, Stirlingの公式を導出せよ.\n",
    "\n",
    "**解答例:** 積分変数を $t=nx$ で置換すると,\n",
    "\n",
    "$$\n",
    "n! = \\int_0^\\infty e^{-t+n\\log t}\\,dt = n^{n+1} \\int_0^\\infty e^{-n(x-\\log x)}\\,dx.\n",
    "$$\n",
    "\n",
    "$f(x)=x-\\log x$, $g(x)=1$ とおく. $f'(x)=1-1/x$, $f''(x)=1/x^2$ なので $f(x)$ は $x_0=1$ で最小になり, $f(1)=f''(1)=1$ となる. ゆえに, それらにLaplaceの方法を適用すると,\n",
    "\n",
    "$$\n",
    "n! \\sim n^{n+1}\\frac{e^{-n}}{\\sqrt{n}}\\sqrt{2\\pi} = n^n e^{-n}\\sqrt{2\\pi n}.\n",
    "\\qquad \\QED\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Laplaceの方法は本質的にGauss積分の応用である.\n",
    "\n",
    "Gauss積分をガンマ函数に置き換えることによって得られる一般化されたLaplaceの方法の素描については\n",
    "\n",
    "* 黒木玄, <a href=\"https://genkuroki.github.io/documents/20161014GeneralizedLaplace.pdf\">一般化されたLaplaceの方法</a>\n",
    "\n",
    "を参照せよ. 一般化されたLaplaceの方法は\n",
    "\n",
    "* 渡辺澄夫, <a href=\"https://www.amazon.co.jp/dp/4339024627\">ベイズ統計の理論と方法</a>, 2012\n",
    "\n",
    "の第4章の主結果であるベイズ統計における自由エネルギーの\n",
    "\n",
    "$$\n",
    "F_n = -\\log Z_n = nS + \\lambda \\log n - (m-1)\\log\\log n + O(1)\n",
    "$$\n",
    "\n",
    "の形の漸近挙動を導く議論を初等化するために役に立つ. 特異点解消は本質的に不可避だが, この形の漸近挙動だけが欲しいのであればゼータ函数を用いた精密な議論は必要ない."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Laplaceの方法の弱形\n",
    "\n",
    "**Laplaceの方法の弱形:** Laplaceの方法が使える状況では, \n",
    "\n",
    "$$\n",
    "Z_n = \\int e^{-nf(x)}g(x)\\,dx\n",
    "$$\n",
    "\n",
    "について, 特に, $n\\to\\infty$ のとき, \n",
    "\n",
    "$$\n",
    "-\\frac{1}{n}\\log Z_n \\to f(x_0) = \\min f(x), \\quad\\text{i.e.}\\quad\n",
    "Z_n = \\int e^{-nf(x)}g(x)\\,dx = \\exp\\left(-n\\min f(x)+o(n)\\right)\n",
    "$$\n",
    "\n",
    "が成立している. この結論を**Laplaceの方法の弱形**と呼ぶことにする. Laplaceの方法のような精密な形でなくても, こちらの弱形だけで用が足りることは結構多い. $\\QED$"
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    "**問題(Laplaceの方法の弱形が明瞭に成立する場合):** 閉区間 $[a,b]$ 上の実数値連続函数 $f(x)$ と $0$ 以上の値を持つ実数値函数 $g(x)$ は, $\\ds f(x_0) = \\min_{a\\leqq x\\leqq b} f(x)$ を満たすある $x_0\\in [a,b]$ の近傍で $g(x)>0$ を満たしていると仮定する. このとき, $n\\to\\infty$ において, \n",
    "\n",
    "$$\n",
    "\\int_a^b e^{-nf(x)}g(x)\\,dx = \\exp\\left(-n\\min_{a\\leqq x\\leqq b} f(x) + o(n)\\right)\n",
    "$$\n",
    "\n",
    "が成立していることを示せ. すなわち, $n\\to\\infty$ のとき, \n",
    "\n",
    "$$\n",
    "-\\frac{1}{n}\\log\\int_a^b e^{-nf(x)}g(x)\\,dx \\to \\min_{a\\leqq x\\leqq b} f(x)\n",
    "$$\n",
    "\n",
    "が成立していることを示せ."
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    "**解答例:** $\\ds f_0(x) = f(x)-\\min_{a\\leqq \\xi\\leqq b}f(\\xi)$ とおくと, $f_0(x)$ の最小値は $0$ になり, \n",
    "\n",
    "$$\n",
    "-\\frac{1}{n}\\log\\int_a^b e^{-nf(x)}g(x)\\,dx = \n",
    "\\min_{a\\leqq x\\leqq b}f(x) - \\frac{1}{n}\\log\\int_a^b e^{-nf_0(x)}g(x)\\,dx\n",
    "$$\n",
    "\n",
    "なので, $n\\to\\infty$ のとき\n",
    "\n",
    "$$\n",
    "-\\frac{1}{n}\\log\\int_a^b e^{-nf_0(x)}g(x)\\,dx \\to 0\n",
    "$$\n",
    "\n",
    "となることを示せばよい. \n",
    "\n",
    "$\\eps > 0$ を任意に取って固定し, $A = \\{\\, x\\in[a,b]\\mid f_0(x)\\leqq\\eps\\,\\}$ とおき, その $[a,b]$ での補集合を $A^c$ と書き, \n",
    "\n",
    "$$\n",
    "Z_{0,n} = \\int_a^b e^{-nf_0(x)}g(x)\\,dx = I_n + J_n, \\quad\n",
    "I_n = \\int_A e^{-nf_0(x)}g(x)\\,dx, \\quad\n",
    "J_n = \\int_{A^c} e^{-nf_0(x)}g(x)\\,dx.\n",
    "$$\n",
    "\n",
    "とおく. $n\\to\\infty$ のとき $-\\frac{1}{n}\\log Z_{0,n} \\to 0$ となることを示したい.\n",
    "\n",
    "$x\\in A$ について $\\eps\\geqq f_0(x)\\geqq 0$ なので, $e^{-n\\eps}\\leqq e^{-nf_0(x)}\\leqq 1$ となるので, \n",
    "\n",
    "$$\n",
    "e^{-n\\eps}\\int_A g(x)\\,dx\\leqq I_n = \\int_A e^{-nf_0(x)}g(x)\\,dx \\leqq \\int_A g(x)\\,dx.\n",
    "$$\n",
    "\n",
    "$\\ds f(x_0) = \\min_{a\\leqq x\\leqq b} f(x)$ を満たすある $x_0\\in [a,b]$ の近傍で $g(x)>0$ となっていると仮定したことより, $\\ds \\int_A g(x)\\,dx > 0$ となることにも注意せよ.\n",
    "\n",
    "$x\\in A^c$ について $f_0(x)>\\eps$ なので, $0 < e^{-nf_0(x)}<e^{-n\\eps}$ となるので,\n",
    "\n",
    "$$\n",
    "0\\leqq J_n = \\int_{A^c} e^{-nf_0(x)}g(x)\\,dx \\leqq e^{-n\\eps}\\int_{A^c} g(x)\\,dx.\n",
    "$$\n",
    "\n",
    "以上をまとめると,\n",
    "\n",
    "$$\n",
    "e^{-n\\eps}\\int_A g(x)\\,dx\\leqq Z_{0,n} \\leqq \\int_A g(x)\\,dx + e^{-n\\eps}\\int_{A^c} g(x)\\,dx.\n",
    "$$\n",
    "\n",
    "これの全体の対数を取って $-1/n$ 倍すると,\n",
    "\n",
    "$$\n",
    "\\eps - \\frac{1}{n}\\log\\int_A g(x)\\,dx \\geqq\n",
    "-\\frac{1}{n}\\log Z_{0,n}\\geqq\n",
    "-\\frac{1}{n}\\log\\left(\\int_A g(x)\\,dx + e^{-n\\eps}\\int_{A^c} g(x)\\,dx\\right).\n",
    "$$\n",
    "\n",
    "したがって, $n\\to 0$ とすると, \n",
    "\n",
    "$$\n",
    "\\eps\\geqq\n",
    "\\limsup_{n\\to\\infty}\\left(-\\frac{1}{n}\\log Z_{0,n}\\right)\\geqq\n",
    "\\liminf_{n\\to\\infty}\\left(-\\frac{1}{n}\\log Z_{0,n}\\right)\\geqq\n",
    "0.\n",
    "$$\n",
    "\n",
    "$\\eps>0$ は幾らでも小さくできるので, 下極限と上極限が等しくなることがわかり, \n",
    "\n",
    "$$\n",
    "\\lim_{n\\to\\infty}\\left(-\\frac{1}{n}\\log Z_{0,n}\\right) = 0\n",
    "$$\n",
    "\n",
    "が得られる. $\\QED$"
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