{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# 10 Gauss積分, ガンマ函数, ベータ函数\n", "\n", "黒木玄\n", "\n", "2018-06-21～2019-04-03, 2020-04-25, 2023-06-22, 2023-11-05\n", "\n", "* Copyright 2018,2019,2020,2023 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 Julia Box で利用できる.\n", "\n", "自分のパソコンにJulia言語をインストールしたい場合には\n", "\n", "* [WindowsへのJulia言語のインストール](http://nbviewer.jupyter.org/gist/genkuroki/81de23edcae631a995e19a2ecf946a4f)\n", "\n", "* [Julia v1.1.0 の Windows 8.1 へのインストール](https://nbviewer.jupyter.org/github/genkuroki/msfd28/blob/master/install.ipynb)\n", "\n", "を参照せよ. 前者は古く, 後者の方が新しい.\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": [ "

目次

\n", "
" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "scrolled": false }, "outputs": [], "source": [ "using Base.MathConstants\n", "using Base64\n", "using Printf\n", "using Statistics\n", "const e = ℯ\n", "endof(a) = lastindex(a)\n", "linspace(start, stop, length) = range(start, stop, length=length)\n", "\n", "using Plots\n", "#gr(); ENV[\"PLOTS_TEST\"] = \"true\"\n", "#pyplot(fmt=:svg)\n", "pyplot()\n", "#clibrary(:colorcet)\n", "#clibrary(:misc)\n", "default(fmt=:png)\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\", \"\"\"\"\"\")\n", "end\n", "\n", "using SymPy\n", "#sympy.init_printing(order=\"lex\") # default\n", "#sympy.init_printing(order=\"rev-lex\")\n", "\n", "const latex = sympy.latex\n", "using LaTeXStrings\n", "\n", "using SpecialFunctions\n", "SpecialFunctions.lgamma(x::Real) = logabsgamma(x)[1]\n", "\n", "using QuadGK" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "# Override the Base.show definition of SymPy.jl:\n", "# https://github.com/JuliaPy/SymPy.jl/blob/29c5bfd1d10ac53014fa7fef468bc8deccadc2fc/src/types.jl#L87-L105\n", "\n", "@eval SymPy function Base.show(io::IO, ::MIME\"text/latex\", x::SymbolicObject)\n", " print(io, as_markdown(\"\\\\displaystyle \" * sympy.latex(x, mode=\"plain\", fold_short_frac=false)))\n", "end\n", "@eval SymPy function Base.show(io::IO, ::MIME\"text/latex\", x::AbstractArray{Sym})\n", " function toeqnarray(x::Vector{Sym})\n", " a = join([\"\\\\displaystyle \" * sympy.latex(x[i]) for i in 1:length(x)], \"\\\\\\\\\")\n", " \"\"\"\\\\left[ \\\\begin{array}{r}$a\\\\end{array} \\\\right]\"\"\"\n", " end\n", " function toeqnarray(x::AbstractArray{Sym,2})\n", " sz = size(x)\n", " a = join([join(\"\\\\displaystyle \" .* map(sympy.latex, x[i,:]), \"&\") for i in 1:sz[1]], \"\\\\\\\\\")\n", " \"\\\\left[ \\\\begin{array}{\" * repeat(\"r\",sz[2]) * \"}\" * a * \"\\\\end{array}\\\\right]\"\n", " end\n", " print(io, as_markdown(toeqnarray(x)))\n", "end" ] }, { "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": [ "#### Gauss積分\n", "\n", "**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": [ "#### 正規分布\n", "\n", "**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": [ "#### Lebesgueの収束定理が成立しない場合2\n", "\n", "**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": 3, "metadata": {}, "outputs": [ { "data": { "image/png": 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"#cycls = [:solid, :dash, :dashdot, :dashdotdot]\n", "cycls = [:solid, :dash, :dashdot, :dot]\n", "ns = [1,2,3,4,5,10, 30, 100]\n", "P = plot(size=(420,250))\n", "for k in 1:lastindex(ns)\n", " plot!(x, f.(ns[k],x), label=\"n = $(ns[k])\", ls=cycls[mod1(k, lastindex(cycls))], lw=1.3)\n", "end\n", "P" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### 正規分布のモーメント\n", "\n", "**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, ガンマ分布の中心極限定理とStirlingの公式\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, ガンマ分布の中心極限定理とStirlingの公式\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", "の第3章§35の例5,6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 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", "$$\n", "\n", "$\\Gamma(s)$ をガンマ函数と, $B(p,q)$ をベータ函数と呼ぶ." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### ガンマ函数のGauss積分型表示\n", "\n", "**問題(ガンマ函数のGauss積分型の表示):** 次を示せ:\n", "\n", "$$\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", "$$\n", "\n", "**注意:** この公式より, ガンマ函数は本質的にGauss積分の一般化になっていることがわかる. $\\QED$" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\ds 2\\int_0^\\infty e^{-y^2} y^{2s-1} \\,dy =\\Gamma\\left(s\\right)$" ], "text/plain": [ "L\"$\\ds 2\\int_0^\\infty e^{-y^2} y^{2s-1} \\,dy =\\Gamma\\left(s\\right)$\"" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "y = symbols(\"y\")\n", "s = symbols(\"s\", positive=true)\n", "sol = 2*integrate(e^(-y^2)*y^(2s-1), (y,0,oo))\n", "latexstring(raw\"\\ds 2\\int_0^\\infty e^{-y^2} y^{2s-1} \\,dy =\", latex(sol))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**問題:** 次を示せ: $r>0$ について\n", "\n", "$$\n", "\\int_0^\\infty e^{-x^r}\\,dx = \n", "\\frac{1}{r}\\Gamma\\left(\\frac{1}{r}\\right).\n", "$$\n", "\n", "**略解:** $x=t^{1/r}$ と置換すればただちに得られる. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**注意:** ガンマ函数の函数等式(下の方で示す)もしくは部分積分によって $\\ds \\frac{1}{r}\\Gamma\\left(\\frac{1}{r}\\right)=\\Gamma\\left(1+\\frac{1}{r}\\right)$ が成立することもわかる. $\\QED$" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\ds \\int_0^\\infty e^{-x^r} \\,dx =\\Gamma\\left(1 + \\frac{1}{r}\\right)$" ], "text/plain": [ "L\"$\\ds \\int_0^\\infty e^{-x^r} \\,dx =\\Gamma\\left(1 + \\frac{1}{r}\\right)$\"" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "x = symbols(\"x\")\n", "r = symbols(\"r\", positive=true)\n", "sol = integrate(e^(-x^r), (x,0,oo))\n", "latexstring(raw\"\\ds \\int_0^\\infty e^{-x^r} \\,dx =\", latex(sol))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### ガンマ函数のスケール変換\n", "\n", "**問題(ガンマ函数のスケール変換):** 次を示せ:\n", "\n", "$$\n", "\\int_0^\\infty e^{-x/\\theta}x^{s-1}\\,dx = \\theta^s\\Gamma(s) \\quad (\\theta>0,\\ s>0).\n", "$$\n", "\n", "ガンマ函数はこの形式でも非常によく使われる.\n", "\n", "**解答例:** $x=\\theta y$ と置換すると, $x^{s-1}\\,dx = \\theta^s y^{s-1}\\,dy$ なので示したい公式が得られる. $\\QED$." ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\ds \\int_0^\\infty e^{-x/t} x^{s-1} \\,dx =t^{s} \\Gamma\\left(s\\right)$" ], "text/plain": [ "L\"$\\ds \\int_0^\\infty e^{-x/t} x^{s-1} \\,dx =t^{s} \\Gamma\\left(s\\right)$\"" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "x = symbols(\"x\")\n", "s = symbols(\"s\", positive=true)\n", "t = symbols(\"t\", positive=true)\n", "sol = simplify(integrate(e^(-x/t)*x^(s-1), (x,0,oo)))\n", "latexstring(raw\"\\ds \\int_0^\\infty e^{-x/t} x^{s-1} \\,dx =\", latex(sol))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### ベータ函数の様々な表示\n", "\n", "**問題(ベータ函数の様々な表示):** 次を示せ:\n", "\n", "$$\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", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**問題:** 次を示せ. $a0$, $q>0$ のとき,\n", "\n", "$$\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": [ "#### ガンマ函数を定義する積分の被積分函数のグラフ\n", "\n", "**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": 7, "metadata": {}, "outputs": [], "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" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "image/png": 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lw=2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### ベータ函数を定義する積分の被積分函数のグラフ\n", "\n", "**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": 12, "metadata": { "scrolled": false }, "outputs": [], "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" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "image/png": 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ICAjA6tWrq/Zhax0RuZfly5dj+fLlta7bvHlz1e89evTA3r17a93O1jp7/LSbgiGhAj/dqWH4BxpeHCixqL8C1RWHFImaEBbuRG5Ek44/DOYOH+JE1Px0CRDYPU3Fcwd1PL1fx/azEm/fqyKsJYt3osYyZKgMC3ci+7jy5FQiIkfzUgQWD1KxfbKKY0US/dMs+PAMh84QNRYLdyI34qqTU4mInGl0RwWHk0wY1k7gvq0aHvtcQ5mFxQFRQxlTuBvRCFETwB53Imqqgn0FNo5X8fdhCv55XEf8RguOFrJ4J2oIY2aVYV4S2UUHe9yJqOkSQmB+HxVf3W+CRQcGbrDgjWO6XdeGICKDCnfO405kH01n4U5ETV//tgIHZpjw8+4K5u3R8MB2DQVlLBaI6sMedyI3ooNDZYioefAzCawYoSJtrIod5yQGrLdg13kOriWyhT3uRG6EPe5E1NwkdlZwKMmEzv4CCZs0PH9Qg4WzWhDVqtbCvaysDNOnT0dUVBSio6MxceJE5OTkVK3Pzs7GsGHDEBUVhfj4eBw9etRmI0w/Ivuwx52ImqOIVgI7p6h4LlbBf2fquPcjDWdusHogulOdPe7Jyck4fvw4srKyMHXqVCQnJ1etmzdvHpKTk3HixAksWrQIc+fOtdmIxrEyRHZhjzsRNVeqIvBsrIr0qSryiiWi11uw7hSHzhDdrtbC3dfXF5MnT4YQ1gpiyJAhOHXqFADg0qVLyMjIwKxZswAASUlJOH36dLUe+TuxbieyD3vciai5G95eQVaiCWM7CvzoUw3Juy0ormAhQQQAJns2ev311zFt2jQAQF5eHsLCwmAyWR8qhEBERARyc3NhNptrffzaf6/DoX89i8TERCQlJTkmcgcqLCx0dQj1YoyO4e4xlt9qAS+ThJ2pSUTUJLXxEXh/jIqVxyUWfKFh9wUL3httwoC27Nmg5q3e6mDx4sXIzs7GihUrqpZV9sRXqm/+1cSkB/Dqgy83MkRjBAUFuTqEejFGx3DnGBWTBb4+ro6CiMj1hBD4VU+Be9oJPLTDgvj/WPDKYAWP9VFq1CFEzYXNWWWWLl2K9evX4+OPP4afnx8AoFOnTsjPz4fFYgFgLdrz8vIQERFR5354gIvIPrxyKhFRdb3aCOy734SHeyn4zV4d0z7RcLmUlQU1T3UW7q+99hpSU1Oxbds2tG7dump5aGgoYmJisGbNGgBAWloazGZzncNkAE4HSWQvTfLkVCKiO/maBP5nmIqPJqj48rJ1zvdPz/LEVWp+ai3c8/Pz8cQTT6CoqAgJCQmIjo7G4MGDq9anpKQgJSUFUVFReOmll7By5UqbjfDkVCL7sMediKhuUyIUHE4yoU8bgXGbNTz1lYYKzvlOzUitY9zDw8Ntjlvv0aMH9u7da3cj/E5MZB+dPe5ERDZ18BP4ZJKKpYd1PL1fx45zEqmjVXQN4JsnNX2GXDlVZ+VOZBdNGpSUREQeTBECiwao+Pw+FQXlEjHrLXjnJIsNavqMKdyNaISoCdAloLJyJyKyS3yogowZJtwfKTBrp4af77Tgxi0OnaGmi4U7kRux9rjzcC8Rkb0CvAXeTjDh7XtVbDgjEbPBgv2XWXlQ02RM4c4vv0R20aVkjzsRUSPM6q4gc4YJQT4CwzZqeOWQBp2zY1ATw8KdyI1wjDsRUeN1CxTYM03Fb/spWPSVjokfazhfwiKEmg4W7kRuhGPciYjujrcq8PJgFdsmq/i6QKJ/mgWbczl0hpoGQ0oEHqkiso/OHnciIocY29E653t8iMCUTzT88ZAPynlFSPJwhtQImhGNEDUBvHIqEZHjhLQQ+GiCir8NVbDqlBeGbLTg2yIW7+S52ONO5EZ45VQiIscSQuA3fVVsTShBmQbEbbBg5be6zQtNErkrY3rcmRtEdmGPOxGRc/RrrePAdBN+2lXgV7s1PPiphqJyFijkWdjjTuRG2ONOROQ8Lb0E/jnShPfHqNh6ViJ6vQWfX+CJq+Q52ONO5EbY405E5Hw/6qLgUKIJ4S0FRn6k4b8zNGicAo88AHvcidyIDucU7tnZ2Rg2bBiioqIQHx+Po0ePVltfVFSE6OjoqltUVBRMJhMKCgoAAGazGT179qxav3btWscHSUQOxby3LdJf4LOpKp6JUfB8ho7RmzTk3WTBQu7NZEQjOpgIRPZw1lCZefPmITk5GXPmzMG6deswd+5c7N27t2p969atkZWVVXV/6dKlSE9PR1BQUNWydevWoW/fvo4PjoicgnlfP5Mi8EKcijFhAjN3ahiw3oKVI1TM6MyJeck9cagMkRtxxlCZS5cuISMjA7NmzQIAJCUl4fTp08jJyanzMatWrcLcuXMdGwgRGYZ53zAjO1iHziR0EEjcruHh3RpKLCxeyP0Y0uPOoTJE9nFGj3teXh7CwsJgMlnTXQiBiIgI5Obmwmw219h+7969uHr1KqZOnVpt+cyZM6HrOgYPHowlS5YgJCTEZrvz589HYGAgEhMTkZSU5LC/x1EKCwtdHUK9GKNjeEKMt/dyO4Ir8r4p5PwbscA9bbzw9CEffHa2Av83uAy9A407edUTXqueECPgGXE2Ju8NKdzZ405kH2ednCpE9Z3amr/4X//6F37+859XfeADwK5duxAREYGKigo888wzmD17NjZv3myzzWXLliE2NvbuAncyRxdLzsAYHcMTYnQ0o/O+qeT8b9sCE7pIPLTTgrE7W+LVwQoe6a3UeD6dxRNeq54QI+A5cTYEe9yJ3Igzetw7deqE/Px8WCwWmEwmSCmRl5eHiIiIGtsWFxdj7dq1+Oqrr6otr9zWy8sLCxcuRFRUlGODJCKHYt7fnT5BAl/db8LvvtQx/wsdW89KrBypItiX036Raxkyxp0zpBLZR3dCj3toaChiYmKwZs0aAEBaWhrMZnOth8v//e9/o3///ujZs2fVsuLiYhQVFVXdT01NRUxMjGODJCKHYt7fPV+TwN/vUbFxvIrPL0gMSLNg5zlWNORaxhTu7HEnsouzhsqkpKQgJSUFUVFReOmll7By5cqqdZMnT8aBAwcAACtXrqxxctrFixeRkJCA/v37o1+/fkhPT8dbb73l+CCJyKGY945xX6SCQ0km9AgUGLNJw9P7NVSwsCEXMWg6SCKyh7Omg+zRo0e1aeBud/uY1d27d9dY36VLF2RmZjo+KCJyKua943RsKbBtsor/d1jHnw7o+PScRGqCis4BHDpDxjKkx93Cyp3ILhbdOYU7ERHdHVUR+EO0ij3TVFwqlYheb0HqSRY4ZCxjCnceUSKql5QSFgl4O2OsDBEROcSQdgoyE02YEiHw050afpFuwc0KFjpkDPa4E7mJiu/zxIsX7CMicmuB3gLvJKh4c5SKf5+SiF1vwcHLLN7J+eosERYsWACz2QwhBI4cOVJtndlsRs+ePREdHY3o6GisXbvWZiMVLNyJ6nXr+zzxVl0bBxER1U8IgdlRCjJmmODvDQz9wIJXD2vQOQc2OVGdJ6c+8MADWLRoEYYPH17r+nXr1qFv3752NcIed6L6scediMjzRLUW2HufCX/cr+PJL3VsOyuxepSKdn4c9kiOV2eJMHLkSISHhzukERbuRPWrLNy9WbgTEXkUb1Vg6RAVWyaqyLoq0X+9BVvyWPyQ4zV6OsiZM2dC13UMHjwYS5YsQUhISJ3bnsk7i0mTJiExMRFJSUmNbdJpCgsLXR1CvRijY7hzjBdLBIBWKCu+ASDQ1eEQEVEDTeik4FCiwJx0DZO2aPhtP4nFgxT4cLowcpBGFe67du1CREQEKioq8Mwzz2D27NnV5oS9U2hYR3z8Px83OkgjBAUFuTqEejFGx3DXGK+ZJAAL2gb6uzoUIiJqpHZ+ApsmqvifIzp+/5WOz87rSE0wIao1i3e6e406KB8REQEA8PLywsKFC2u9eMPtOB0kUf1ucYw7EVGToAiBx/up2He/CTcrgNgNFrx5Qofkiat0lxpcIhQXF6OoqKjqfmpqKmJiYmw+hmPciepXwVlliIialNhggYMzTPhxF4FfpGv46U4N126xeKfGq7Nwf/TRRxEeHo78/HyMHTsW3bp1AwBcvHgRCQkJ6N+/P/r164f09HS89dZbNhvhdJBE9fuhx52HU4mImopWXgL/GmVC6mgVm3MlotMs2HeRhRE1Tp1j3JcvX47ly5fXWN6lSxdkZmY2qBH2uBPVr0K39sJwVhkioqbnJ10VDA6xXm11+IcaXoiTeGqAApWdNdQAhpQI7HEnqt8tzfqTY9yJiJqmzgECu6apeGqAgj8d0DF2s4azxRw6Q/YzpETgyalE9eMFmIiImj4vReDPg1R8OkVF9nWJ/mkWbMxhDyfZhz3uRG7iFi/ARETUbCSEKTiUaMKI9gLTt2l49HMNpZqroyJ3Z0iJICWg6ex2J7KFPe5ERM1LW1+BDeNULL9HwcrjOsbt8MM3BayXqG6GlQjsdSey7RangyQianaEEHikt4r9002QAAb+x4IVRzXO+U61YuFO5CbY405E1Hz1CxLYProEv4hS8F+f60jcpqGgjMU7VWdYiXCLhTuRTZWzynCMOxFR89RCBf53uIoN41SkX5AYsN6C9PMsoOgHLNyJ3MQt9rgTERGA6WYFhxNN6BogMHqThmcPaLDwXEGCgYV7qcWolog8U6lFwle1jnckIqLmLbyVwKeTVbwQq2Bxlo5RH2nIucHivblj4U7kJko1oEWd1zImIqLmRlUEnolVsWuairPFEtHrLXj/Ow5haM4MK9xLNH5LJLKlxGId30hERHS7Ye0UZCWaMDFc4MEdGn61y4LiCtZVzRF73IncRKkF8GOPOxER1aK1j0DqaBUrR6pI/U4iboMFmVdYvDc3LNyJ3ASHyhARkS1CCPyyh4KMGSa0MAFDNlrwt68553tzYlzhzsv4EtlUapFoofLEVCIisq1Ha4F995vwaG8Fj+/TMfUTDZdKWbw3B+xxJ3IT7HEnIiJ7+agCrw1VsXmiiv2XJQakWbAtnyeuNnXGnZzKwp3IJp6cSkREDTWpk4LDSSb0CxIY/7GGRV9quMUJQZosQwp3kwKU8kVEZFOpxTk97tnZ2Rg2bBiioqIQHx+Po0eP1rqd2WxGz549ER0djejoaKxdu7bB+yAi12PONz/t/QS2TFLxymAFfzui454PNJy8xrqrKTKkcPdVOVSGqD6lmnNmlZk3bx6Sk5Nx4sQJLFq0CHPnzq1z23Xr1iErKwtZWVl48MEHG7UPInIt5nzzpAiBJ/ur+OI+FUW3JGI2WPB2NofONDWGFe4cKkNkW6kThspcunQJGRkZmDVrFgAgKSkJp0+fRk5OjqH7ICJjMOdpYIh11plEs8DPP9Mwa6cF12+x972pMORUuJYm4EaFES0Rea4bFRKtvBz7XTovLw9hYWEwmaypLoRAREQEcnNzYTaba2w/c+ZM6LqOwYMHY8mSJQgJCWnwPirNnz8fgYGBSExMRFJSkkP/LkcoLCx0dQj1YoyO4QkxBgUFOWQ/zPm6ecLrwJEx/rU/cE9rE57I9MXn5zW8EV+KuKC774H3hOcR8Iw4G5P3hhTurbyAIn7bI7LpRgUQ4O34/QpRfYrJuub73bVrFyIiIlBRUYFnnnkGs2fPxubNmxu0j9stW7YMsbGxjYzaGI4qlpyJMTqGJ8ToKMz5unnC68CRMSYHAWO7SDy0Q8Pkz1rivwcqWDRAgSLubuphT3geAc+JsyEMGSrj7wVcu2VES0Se6/ota644UqdOnZCfnw+LxTpWTUqJvLw8RERE1Ni2cpmXlxcWLlyI3bt3N3gfRORazHm6U5cAgT33qXiyv4I/7tcxfrOGc8XsTPVUhhTu1h53I1oi8kxSSmuPu5djL8AUGhqKmJgYrFmzBgCQlpYGs9lc43B3cXExioqKqu6npqYiJiamQfsgItdjzlNtvBSBJfEqtk1WcbRIYsB6Cz46wxNXPZEhQ2X8vYB8Fu5EdSq2ABKO73EHgJSUFMyZMweLFy9GQEAAVq9eXbVu8uTJePHFFxEUFISkpCRomvXS2V26dMFbb71l1z6IyL0w56kuYzoqOJwk8It0DdO2anisj8T/i1fga+JVuz1FnYX7ggUL8MEHH+DMmTP4+uuv0bdv36p12dnZmD17Nq5cuYLWrVvjzTffRO/evetsxN9LoKich2WI6lJ58rYzCvcePXpg7969ta6rHM8KAJmZmY3aBxG5F+Y82RLsK/DBeBXLj+p48ksd6ed1vDfahF5tWLx7gjqHyjzwwAPYs2cPIiMja6xr6PyurbyAa5xVhqhO178/IuWMk1OJiIhuJ4TA/D4qvrrfhAodiNtgwT+/1e06CZlcq87CfeTIkQgPD6+xvDHzu/p7AUXl9p2VTtQcXa+w5kaAN3s8iIjIGP3bChyYYcLPuitI3q3hx59qKOQICbfW4DHujZnfNS31bdyaFI3//b9VeChp+t3E6xSeMNcnY3QMd40x94oKwA9KaREKCmSTnMKKiIjcj59JIGWEivHhAr/apWFAmgXvjlYxvL0h85dQAzXq5NSGzu/66Nyf4/FzQNLMXyDIzz17FD2hUGKMjuGOMd4q0AFo6NKuNVo5eGYZIiKi+iR1VjAoRGDWTg2jPtLwbIzE0zEKTAo/k9xJg79ONWZ+11bffz3glJBEtSsol/BSrFcZJiIicoWIVgI7pqh4NkbBi5k6Ej7SkHuTQ2fcSYML98bM7+r//bjda7x6KlGtCsqBtj41j2YREREZyaQIPBenIn2qitxiiQFpFqSd5pzv7qLOwv3RRx9FeHg48vPzMXbsWHTr1q1qXUpKClJSUhAVFYWXXnoJK1eutNlIq++nuCsqd0zQRE3N1XIgyMfVURAREVkNb68gK9GEsR0FHtiuYd5uDSUWdsC6Wp0H5pcvX47ly5fXuq6h87sGfj/F3VUW7kS1ulIm0daXve1EROQ+2vgIvD9GxcrjEgu+0LD7go4VcQpGut+pYs2GIacM+5kE/L2A8yX8pkZUm4ulQPsWro6CiIioOiEEftVTwcEZJngrwPidfvj7EY1TfLuIYXP9hPkB50qMao3Is1wokWjXgj3uRETknnq1Edh3vwmzO1dgwV4d92/VcKWMxbvRDCzcBc4W8x9MVJuLpUB7P1dHQUREVDdfk8CS6HJ8OF7F3ksS/dMs+PQsT1w1knGFe0v2uBPVpkKXuFIG9rgTEZFHmBqp4FCiCb1bC4zbrOEPX2mo0Nk5awTDCveOfgLnOMadqIazxYAEENHK1ZEQERHZJ6ylwNbJKpYMUrD0sI7hH2g4dZ11nrMZ2+NeXP9VVomam5wb1pwwt2KPOxEReQ5FCPw+WsXn96m4UiYRvd6Cd05y6IwzGTrGvVQDrvHqqUTV5Ny0/mSPOxEReaL4UAWZiSbcHykwa6eG2Z9ZcIMX3XQKwwr38JbWn2duGtUikWfIuSHRwc960g8REZEnCvAWeDvBhLfuVbE+RyJ2gwUHLrP33dEMK9y7B1qLkuxr/AZGdLucG5LDZIiIqEn4WXcFmTNMaO0tMOwDDa8c0qBzmLTDGFa4t/UB2vgAJ1i4E1WTcxMw+7s6CiIiIsfoFijw+X0qHu+rYNFXOiZ+rOECJyhxCMMKdyEEegQKHGfhTlRN9jWJLv7scScioqbDWxV4ebCKrZNUfF1gnfP94zwOnblbhhXuABAVCJy4ZmSLRO6toEziXAnQN4iFOxERNT3jwhUcSjJhUIjA5C0aHt+roVxjJ25jGVy4W3vcOSUkkdXXhdZc6NeGhTsRETVNoS0EPpqg4m9DFfzvUR1DNlpwvIi1YGMYWrj3DxIoLAdyObMMEQDg6wIJbwWIau3qSIiIiJxHCIHf9FXx5XQTSi1A7AYL/nVcZ2duAxlauA8MsfYqHrzCfxIRAHxdAPRqDXgp7HEnIqKmL7qtwMEZJjzUVWDuLg0/2aGhqJx1ob0MLdw7+Al0bAkcYOFOBAD46rKO2GAW7URE1Hy09BL4v5EmrB2t4pN86xVXv7jIE1ftYWjhDgADgwW+usTCnej6LYnDBcDw9oanIRERkcv9uKuCQ4kmdGwpMPJDDX/O0KDprBFtMbxiuKedwN5LErd4RjE1c/suSejSmhNERETNUaS/QPpUFU/HKHguQ8eYzRryb7JGrIvhhfuYjgpKLNaihag5Sz8vEexrnSaViIiouTIpAi/Eqdg5RcV31yUGrLfgPzkcOlMbwwv36LZAkA+w7SwLd2reNuXqmBAuIAR73ImIiEZ2sA6dGdVBYMY2DY/s0VBqYb14O8MLd0UITAwX/CZFzVr+TYlDBcCUTsakYHZ2NoYNG4aoqCjEx8fj6NGj1daXlZVh+vTpiIqKQnR0NCZOnIicnJxq25jNZvTs2RPR0dGIjo7G2rVrDYmdiBqOOU+eKshXIG2sihXDFaw6oWPQfyw4UsDivZJLzop7sKuCI4XAN/xHUDO1IUeHSQATwo3pbZ83bx6Sk5Nx4sQJLFq0CHPnzq2xTXJyMo4fP46srCxMnToVycnJNbZZt24dsrKykJWVhQcffNCI0ImoEZjz5MmEEJjXS8WB6SYoAhj0Hwv+96jGOd/hosJ9QrhAoDew9hR73al5WnNSYnKEQJCv8wv3S5cuISMjA7NmzQIAJCUl4fTp09V613x9fTF58uSqYTtDhgzBqVOnnB4bETkec56aij5BAl/eb8LcHgoe/VzHjG0arpY17+Ld5IpGfVSBGWaBd07qeC5WgcqLz1Az8k2BxFeXJd4foxrSXl5eHsLCwmAyWdNdCIGIiAjk5ubCbDbX+pjXX38d06ZNq7F85syZ0HUdgwcPxpIlSxASElJnu/Pnz0dgYCASExORlJTkkL/FkQoLC10dQr0Yo2N4QoxBQUEO2xdzvnae8DpgjLV7sRcwJNCEBQd90W+dhhWDyjA8RLP5GE94LhuT940q3M1mM3x9feHr6wsA+MMf/tDgQ2j/1UvBmyc0fJgrMd3Mwp2aj78d0RDmB9wfadzr/s4TYG0dbly8eDGys7OxYsWKast37dqFiIgIVFRU4JlnnsHs2bOxefPmOvezbNkyxMbG3l3gTubIYslZGKNjeEKMjsScr50nvA4YY+1mBQEJnSV+tlPD9F1++GO0gufiFJtXHveE57KhGt3jvm7dOvTt27fRDceHKhjeTserh3VMN/MCNNQ8XCyRePukxPOxCrxVYwr3Tp06IT8/HxaLBSaTCVJK5OXlISIiosa2S5cuxfr167F9+3b4+flVW1e5vZeXFxYuXIioqChD4ieihmHOU1PVsaXAtskqXj6k49mDOnack3h3tAqzf/PpAHZpxfxkfwV7LkpsP8ux7tQ8vHRIh5cCJPc0LvVCQ0MRExODNWvWAADS0tJgNptrHDJ/7bXXkJqaim3btqF169bV1hUXF6OoqKjqfmpqKmJiYpwcORE1BnOemjJVEfhjjIo901ScL5EYkGbBe981nzqy0T3ujhj3NtwfGNy2BX6zR8NnY0tgUAdkDZ4wDooxOoYrYzx5Q2DZNy3x+963gJIbKCipe1tHH95LSUnBnDlzsHjxYgQEBGD16tVV6yZPnow//vGPeOKJJ9ClSxckJCQAAHx8fPDll18CAC5evIikpCRomvWs/i5duuCtt95yaIxE5DjMeWrqhrRTkJUk8F97NDy0Q8PWfB2vD1PRyqtp9743qnB35Li314frGLxRwzvnArGwnzEn69XGE8ZBMUbHcEWMUko8s09DWEuJp+NbooWplaHt9+jRA3v37q11XWXu2hoD26VLF2RmZjolNiJyPOY8NQeB3gLvJKiYEC7x6OcaPr9oQepoE2KDm27x3qjj9XeOe9u9e3ejA4gPVbCgj4I/7NdxrLB5T/FDTdc/junYdlbijeEqWpia7hsKERGRkYQQmB2lIGOGCa28gCEbLXjtsAa9iZaUDS7cnTHubUm8gs7+wIxtFhSWN9FnmpqtLy/p+O0+HY/0VjDBoCulEhERNSdRrQW+uM+EBX0UPPGljp983gIXS5peTdngKuLixYtISEhA//790a9fP6Snp9/1uDc/k8DG8SZcLgN+tF3DLa3pPdHUPJ28JjF9q4a4YIHXhrBoJyIichYfVWDpEBVbJqo4XKRgwHoLtuY3rRNXGzzG3Vnj3roHCqwfp2L8Zg2J2zSsG6vCl0MKyIN9d10iYZMFgd7A+rEqfFx19jUREVEzMqGTgl1jS7DwkD8mfKzhiX4SiwcZNw2zM7lVF+CoDgo+nKBixzmJSVs0XC5lzzt5pi8v6bjnAwtaqMCOKSa08/P8NwsiIiJPEeorsXmiilcHK3j9Gx3DPtCQfc3z60q3KtwBYHy4gk8mqfimUCJugwX7LzetQxzUtEkpkXJMw6iPNHQNENhznwlhLVm0ExERGU0RAr/tr2Lf/SZcvyURs96C1Sd0mzMquTu3K9wBYEQH69nBHfwEhm3U8PR+DWUWz32SqXnIvykx9RMND+/RMae7gh1TVIS2YNFORETkSrHBAhmJJvy4i8CcdA0zd2q4dssz60q3LNwBILyVwO5pKv4Uq+CVwzr6pVnw/nc6dA/+lkRNU3GFxPMHNUS9b8HBKxIfTVCxYgTHtBMREbmLVl4C/xplQupoFZtyrb3v+y563qgOty3cAcBbFXg2VkVWoglRgQIP7tAwcIMF607psDTVCTrJYxSUSfwlU0PXtRYsydKxoK+CEz82YUqEW6cVERFRs/WTrgqyEk1o10Jg+IcalmRp0DyopmzUlVON1ruNwKaJJuw+r+PZgzp+9KmGTi2BR3ormB2loANP/CODSCmx/7LE6myJ1Sd0WCQwp7uC3w9Q0DmAr0MiIiJ31zlAYNc0FS8c1PH0fh3bz0q8fa/qEeekeUThXmlEBwU7pyrIuirx+hENz2fo+ON+HQlhAg91VTDdLBDs6/5POnkWKSW+KQQ2ntGx5qSOb4uAMD/g8X4K5vdWOGMMERGRh/FSBP48SMWYjgKzdmron2bBqlEqpkW691FzjyrcK0W3tY5Tem2IxPociXdP6pi3R0PybmBgiMCkTgITwgUGBosmMWcnGa+gTGLPRYmt+RIf5eo4cxNoaQKmmwX+Z6iCMWECqsLXFhERkSdLCFNwKElg7i4N923VML+3xCuDFbe9lpBHFu6VWvsI/LKHwC97KLhQIrElX2JLno6/f6PjxQzAVwUGhQgMa2e9xYcItGfvKN3BokucuAZkXJHYc0Fiz0Ud3xRa15lbAdMiFEyNEBjVQbhtIhMREVHjBPsK/Gecin8c0/HbfTrSL+h4b7QJvdu432e+Rxfut2vvJzAnSmBOlAJNlzh4ReLzixJfXJR4O1vHy4es24X4Av2DhPXWVqB3a6CtDgS5NHoygi4l8ksEskp1HC2SyLoqcegqcKRQokyzbtOrNTC8vcDvBygY3k7A7A8I4X6JS0RERI4jhMAjvVWMaK/goR0WDNxgwV+HKkjuqbhVHdBkCvfbqYpAfKhAfCjweD/rGOXcm9Ye1cMF1tuHuTr+dgSwnkfsjzY+FegWINAtAOgaINDFXyC8JdCxpfVngLf7/NOodrqUuFwK5BcD+cUSZ0skTl0Hsq9LnLwm8d0NoFxrBUCDtwL0bmMddjWzm4IBbQUGBAkE8RwJIiKiZqtfkMBX0014Yp+Oh/fo2Jov8c8RqtvUB02ycL+TEAKR/kCkv8CMzj8sv1khkX0NyDp3Axf0ljh5TeLkdSD9vI5zJdX30coL1kLeTyDMDwhpIRDsC4T4Vv60HmoJ8QVa+1iv1kV3r1yTuFoGXC4DrpRJXLnt5+Uy4HKZxNli4GyxxNkSoOK2KVlNAoj0B7oHCIwOUzAvEGgnbiI2PABmf+uJKURERES38zMJ/GO4ivEdBebu1jBgvQXvJKgY2cH1J642i8K9Lq28BGKCgUjFgqAgtdq6ck3iXDFwtkQi//vCsPLnqRvAV5d1XCkDCsore+1/IAAEeAOB3kCAFxDoLW67L6qW+3tbXxwtVKCFCfAzoer3Fqqw3v9+WalmHYutCvcbuiGlRIUOXK8AKkokSjWg1AKUaUCpJn/43WL9O0osEjcqgOu3rI+5fkt+/7P6/Wu3gJsVNdvzUiq/KFm/LHXxB0Z2UNDRDwhvKb6/ASEtan6BKijQEBToXs8fERERuZ8ZnRUMDLHOOpOwScMzMRJ/ilFgcmHHX7Mu3G3xUQU6B6DeubktukRh+Q89wpfLgKtlwLXbis/rtySu3QIulQLZ16T1fgVw45a1kLWPPwALAGtPspdivZmU236/bbmXAigCqKxbBX74Oyp/u72mrfxVfv83WSSg6YBFApbafuqAJq33K3TAeu2CH2K0RRGAv5f1y0vA919mAryB1t5ARCsgwEtBgLd1m+A7jmgE+1qXu9uXFyIiImp6OrUS2DFFxZIsHc9n6Pj0rMQ7CSoi/V1Th7Bwv0smRSCkhbV3F2j4P1FKiXINKPm+N/r2XunS25ZdvnYTPn6tYNGBCt3aw115s+hAhUSNZZqsbOO29qp+yttiqB6Tl6LA9P0XAfX7n6baft72ewtVQCu7ieDAVlVHCXxV8cPvVcsAH5WFNxEREXkGVRF4JlZFQpjAT3dah878c4SKH3UxfugMC3cXE0LA12QtbO9YU+1eQYEFQUGuH1tliyfESERERNQY97RXcChRYN4eDT/+VMOv8nX8baiKll7GdUayyiIiIiIiskNrH4H3RqtYOVLFu99JDPyPBYeu3nm2o/OwcCciIiIispMQ1ot/Hpxhgq8KxP/HgtePaJB3jj12AhbuREREREQN1LO1wL77TXikt4Lf7NUx7RMNl0udW7yzcCciIiIiagQfVeCvQ1VsmqDiq8sS/dMs2H5Wr/+BjcTCnYiIiIjoLkyOUHAoyYS+QQLjN2t46isNFbrje99ZuBMRERER3aUOfgKfTFLxcryCVw/ruOcDDd9dd2zxzsKdiIiIiMgBFCHwuwEqvrhfRUG5RPR6C9ZkO27oDAt3IiIiIiIHGhSiIHOGCYlmgZ99puHnOy24cevue98bVbhnZ2dj2LBhiIqKQnx8PI4ePXrXgbhSWlqaq0OoF2N0DE+IEQBSU1Mdti9789XWdk0t5wHPeC0wRsfwhBiZ887nCa8Dxug47hCnv7fA6ntNWJOg4j9nJGI2WLD/8g+9743Ke9kICQkJctWqVVJKKf/973/LIUOG1LrdwYMHJQB58ODBxjRjmIkTJ7o6hHoxRsfwhBillHLatGkO25e9+WprO3v3ISXz3pEYo2N4QozMeefzhNcBY3Qcd4vz5DVdxm+okKZ/3pIvZ1mkpuuNyntTQwv9S5cuISMjA1u3bgUAJCUlYf78+cjJyYHZbK62bWlpKQDg2LFjDf9GYaBr164hIyPD1WHYxBgdwxNiBICioiKUlJTAz8/vrvZjb77a2s7Pz8/unAeY947EGB3DE2JkzjufJ7wOGKPjuGOcfw+X+Md1Hb//t0TpKaVxed/QSv/AgQOyV69e1ZYNGjRIpqen19h2zZo1EgBvvPHWiJsjeq/szVdb2zUk55n3vPHW+Btznjfemt+toXnf4B53wHqp19vJOi7xOmHCBKxZswZmsxktWrRoTFNEzVbPnj0dsh9789XWdvbuA2DeEzUWc56o+Wlo3je4cO/UqRPy8/NhsVhgMpkgpUReXh4iIiJqbBscHIyZM2c2tAkichB789XWdn5+fnbnPMC8J3Il5jxR09bgWWVCQ0MRExODNWvWALCetWs2m2sd90ZErmVvvtrajjlP5DmY80RNm5C2jn/V4fjx45gzZw6uXr2KgIAArF69Gn369HFGfER0l2zl6+TJk/Hiiy9i4MCBNrdjzhN5DuY8UdPVqMKdiIiIiIiMxSunEhERERF5AIcU7o64Spuz1dd2WVkZpk+fjqioKERHR2PixInIyckxLD57YrzTCy+8ACEEjhw5YlCE9sdYXl6O+fPno3v37ujTpw9mzZrldjF+8skniIuLQ0xMDPr27YvVq1cbFuOCBQtgNpvr/f+569ULPSHn7Wnf1XnPnDc+TuZ943lC3rt7ztsT452Y93cXY5PL+QZNHlkHR1ylzdnqa7u0tFRu2rRJ6roupZTy73//uxw3bpxh8dkT4+0OHjwoJ06cKCMiIuTXX39tUIT2x7hw4UL52GOPVT2f586dMypEu2LUdV0GBQXJQ4cOSSmlPH36tPTx8ZHXr183JMb09HSZl5cnIyMjbf7/XJkztnhCztvTvqvznjnvOMx75/OEvHf3nLcnxtsx7+vWXHP+rgv3ixcvysDAQFlRUSGltD5J7dq1k6dPn27Uds7QmLb3798vu3bt6vTYKjUkxrKyMjlkyBB56tSpel8Mrojx5s2bMjAwUN64ccOQuBoTY2UyV15Q5NChQzIsLEyWl5cbGq+t/58rc8YWT8j5xrZvZN4z5x2Hee98npD37p7zUjLvjY6xKeb8XQ+VycvLQ1hYGEwm65TwQghEREQgNze3Uds5Q2Pafv311zFt2jSnx1apITE+++yzmDVrFjp37mxYfA2J8bvvvkPbtm3x5z//GQMHDsSIESPw6aefulWMQgi8//77SExMRGRkJIYPH47Vq1fD29vbkDjt4cqccURcro7f3fOeOW98nMx758fFz3rbmPfGxtgUc94hY9wdcZU2Z2tI24sXL0Z2djb+8pe/ODusauyJce/evdi/fz8eeeQRo8Kqxp4YKyoqcOrUKfTu3RsHDhzAsmXL8JOf/ASXL192mxgtFguWLFmCjRs34syZM/j0008xe/ZsFBQUGBKjvVyZM7Z4Qs43tH1X5D1z3nGY987nCXnv7jkPMO+NjLFJ5vzddv9fvHhRBgQE2HX4zJ7tnKEhbb/yyisyLi5OFhYWOj2uxsS4ZMkS2aFDBxkZGSkjIyOlqqoyLCxMbt682W1ivHz5slQURVoslqplgwYNkjt37nSbGPfv3y979epVbdnAgQPljh07nB7j7eo7fOaqnLHFE3K+oe27Iu+Z88bHybxvPE/Ie3fP+Yb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ee+81+/mablG98t+SJUt0ZQ4cOEA+Pj5mb3H8/vvvKSwsTO+C5p49ewzWP3fuXKP1WNJmtbW15O3tTQcPHjQZkznDhw9vteP85ptvqHfv3uTi4kKjR4+mjz76yKIEYuhzRkVF6ZWJjY2lzZs3G63j2WefpdmzZ5uN+8pOYr6+vjRmzJhWdxMa+puMGTOGHnzwQXrggQfI29ubfH196amnntK7qL5kyZJWsS9fvpz8/f3J09OT5s6dS0888USrBGLo80VFReltT5Zo7zbetD239OGHH1J4eDi5ubnR1KlTDd7GO3HiRHrppZfaFGN73HbbbSYTnqEE8t1331FSUhJ5enqSu7s79e/fn956661WPzLwT18RIu3dhFdffTUpFAqSyWTUq1cvWrBgAZWUlOjK//XXXzRs2DCSy+Xk6upKffr0oSVLlujdpEBEtGLFCpo0aVLHP7wVOEwCmTBhAv3f//2f2GG0280330wrVqzQvW76ctnCzJkzafny5WbLaTQaGjp0qNGOUta0evVquvbaaztcz86dOyk+Pt7knWpNSdDSu7CM+fbbbyk+Pt7k0cWAAQNo69atHVpPS4b+Jpb0D5g7d67JJE+kTTItE4ihz1dTU0Ourq4mb5M3pj3b+JIlS2jMmDEmy6xbt04vgaSnp1NgYGCrTna28Pfff1NgYGCr22k76ty5cySVSunUqVNWrbeuro4iIiLot99+s2q97SX6WFilpaXYsmULdu/ejYceekjscNrttdde07sbyFbq6+uRmJiod7eGMYIg4P3334darbZ5XM7Ozli1alWH67n++utx//33o6CgwApRmVZdXY1169ZBKjV8KbChoQEzZszAddddZ7V1tvdvsm/fPrzwwgttWsbQ59u3bx+uueYajBs3rk11Ae3bxn/44Qe8+uqrbVqmsLAQGzZsgFwub9Ny7TFgwAC8+uqrVh977fvvv8e8efMQExNj1XrPnz+PRYsWYdSoUVatt70EInEfKBUdHQ2lUonnnnsOjz32mJihWNVdd92FsrKyVheAWcft3bsX48aNg1KpbNNFS0c1duxYJCUldeh2cgBYunQpvvrqK7270mzJWtv4+vXr8cgjj6CsrMwqcTH7ET2BMMYY65xEP4XFGGOsc+IEwhhjrF04gTDGGGsXqySQmpoaHDlyxCHGZmGMMWYfVkkgmZmZGDRoEDIzM82WLS8vt8Yq2RW4XW2D29U2uF1tw97tavdTWI2NjfZeZbfA7Wob3K62we1qG/ZuV74GwhhjrF04gTDGGGsXTiCMMcbaxeTzQH744Qc888wz0Gg0UKlUePzxxzF37lx7xcYYcwC5ubkoLi62ap3l5eV2GeuquzHUrv7+/oiMjLTJ+owmECLC7bffjj179iAhIQE5OTmIi4tDSkpKhx6ExJit1Odk4MylaiypSMJJJSHJT8DigU6IPLMHbolXQXCWiR1ip5Obm4v4+Hi+Rb8Tc3d3R0ZGhk2SiNknEjYNcFZRUQE/P79Wj3pkzBGQugEFH6+Eh7II0tCHkOE7EVlKNQYcWIs7ir+B16V8yK+/03xFTE9xcTFqamrw6aefIj4+XuxwWBtlZGRgzpw5KC4utm8CEQQB27ZtQ0pKCjw8PKBUKvHFF19AJjP+K27+/PmQy+VISUnBjBkzDJZpy3OVmeW6e7sqD/wIF6V2CPhZyh+xzWcCIhsu4rbiXQCAit2fQd3/KgiePm2rt5u3a1O/gvj4eKPP7WaOr7y8HKWlpbrXhh5b3R5GE4harcZLL72EHTt2YNSoUTh06BCmT5+O9PR0oytfvXq1RRuZtYJn+rpru1JjI4p+34mmY+NdSf/GpRtlmP5jGNYrp+C+kh0Q1CpIj+2FfOrdba6/u7YrAL5O0UXI5XKbbMdG78I6evQoCgsLdQ8uGTJkCEJDQ3Hs2DGrB8FYR5w/nAp57WUAwF7vwXhhWh/4uwrYOM4J6wJT0CBofydV/vkziDuwMWY1RhNIREQE8vPzkZWVBQA4c+YMsrOzERsba7fgGLNEzr4fdNP1Q6Yi0E0AAER5Cbgl0R+/eA7RvllZirqsI2KEyFiXZPQUVlBQENauXYuZM2dCIpGAiPDuu+8iLCzMnvExZlJ9TTUiitIAABelCtw4frDe+4/0l+C+A9fgusqDAIDqQz/Bre8Qu8fJWFdk8i6sWbNmYdasWfaKhbE2O/zbn4giFQDgdNgIDHJ30ns/3FOAU5/BKCnwhl9jBWr+PghNfR0kLq5ihMtYl8I90VmnVnb0d9104OBRBsvM6uOCXd4jAQCSRhXqz/B1PMasgRMI67Sqa+vR+8JhAIDSyQujhiUYLDc9SsBB+SDd65qTh+wSH7Of6OhoxMXFISkpCfHx8bj99ttRXV1tdrm9e/fixx9/tFoMx48fBwBcf/31yM7ONll+6dKlaGhosMq6xcIJhHVavxeo8K7/TPzukYDsiFFwkRk+I+vhLMA3PhG1ggx/uvdFnlcPO0fK7OHzzz/H0aNHcfLkSVRUVGD9+vVml7FmAmlp165d6NWrl8kyzz//fKdPIGZ7ojPmqHZddsM7AbfinYBb8fkEJ5Nlx/fwRGLcRtRKXPGMnwRJ9gmRiaC+vh7V1dXw9fXVzVu5ciW2bdsGtVqN4OBgrF27FiUlJVizZg00Gg1+/vlnpKSk4JlnnsENN9yAkpIS1NbWIikpCR988AHc3d1brefXX3/Fgw8+CDc3NwwdOhREpHsvOjoa3377Lfr3748XX3wRGzdu1I3isWPHDrz00ksAgJEjR0IikeDHH39EYGCgjVvG+jiBsE7rh3wNAMBJAMaHCibLTgwXUCvRXjj/Pl+D5UNMJxzWNkU1hCIbDJcV4g6EuJv+2zaZOXMmXF1dce7cOQwaNAi33HILAGDTpk04deoUDh48CCcnJ3zyySeYP38+duzYgQceeABVVVVYuXIlAO0YgJs2bYKfnx+ICA8++CDeffddPPbYY3rrqq+vx2233YaNGzdi7Nix2LZtG/773/+2ikmpVGLlypUoKiqCm5sbampqIJFIsGbNGqxduxYHDhyAp6dnB1tJPJxAWKd0vpKQWaadHhYowMfF9E4m2F1Ash+QVgIcKQYu1hCCLNwxMfPWZmjw/BGN1etdMlCCpYMsS/aff/45+vfvD7Vajfvvvx9PPvkkXn/9dXz11Vc4fPgwBg3SXgdrbGyEk5PhOokIb775Jnbu3Am1Wo3y8nJcffXVrcplZWXB3d0dY8eOBQDccsstmDdvXqty3t7eiImJwZw5czBx4kTccMMNCA8Pt/DTOz5OIKxT+i2jCH1rK5HhGo1J4ZZtxpMjJEgr0QBE+O1EPlIGh0MQOIlYw/3xEkyLsv4l1ZDWZ47MkkqlmDFjBh5//HG8/vrrICI8++yzuPtu88PYbNq0Cfv27cP+/fvh5eWFd955B/v3729VruXpKlOcnJyQmpqKAwcOYO/evRg+fDg2b96M0aNHt/lzOSJOIKxT0hz+AT+c3YpSJy80DFwMYIDZZSaECqj5cQvuKv0W/ifL0dhzPaR+wbYPthsIcRfatbO3ld27d6NPnz4AgGnTpuHtt9/G9OnToVAooFKpcPz4cSQnJ8Pb2xsFBQW65ZRKJfz8/ODl5YXKykqsX78ePXv2bFV/XFwcamtrsX//flx99dX4/PPPdQNPtlRZWYnKykqMHj0ao0ePxokTJ5CWlobRo0fDy8sL5eXlfAqLMXvzupAJAFA0VkLRI8iiZYYFCvhWAPwbtV/0+rPHOYF0IU3XQFQqFaKjo7FmzRoAwB133IGSkhKMHTsWgiBArVbjnnvuQXJyMm666SZ88sknSEpKQkpKCh5++GHs2LEDffv2RVhYGEaPHq2XYJq4uLhg8+bNuovoY8eONThcenl5OWbOnInq6moIgoCYmBjdQ/kWLlyIa665Bm5ubnwRnTF7KapSo0/VKQBAiYsfwv0s++J5OAsoD+kHXNK+rjh9Ah5DJtgqTGZHOTk5Jt9/5JFH8Mgjj7Sa36NHD6SlpenN+/nnny1a5+jRo5Genq57/eqrrxqMJzU11eDyS5YswZIlSyxal6PifiCs00nLzIOXphYAUBbYp03L+sf0gfqfzb4q+6TVY2OsO+EEwjqdC1kZumnX6Lg2LTs8zA1ZrlEAAJfSPGga6qwaG2PdCScQ1ukI+Vm66Z592/aY1VHBAv527Q0AkJAGqoKzVo2Nse6EEwjrVBoaCeFKbQJphAR+vWLatHygm4BC3+Zl6s6fsmp8jHUnnEBYp3LicgN61+UBAC55RUAia/uw7JLw5oeiFZ89bbXYGOtuOIGwTuXUmVw4Q/tY2vrA1vfnWyKsZzTq/3nMrSqPj0AYay9OIKxTyS0sQZnEAwDgFWl6tFNjBgXLkOkSDQBwriwGqTr3iKjMsezduxeDBw82XxDAXXfdhdWrVwMAFi9ejK1bt5osv379epw65Tg/ergfCOtUtjoPxhNxmxGmuowTo9vX9Xmgn4CxIfejUuIO37Bw/OEss3KUrDNSq9WQSsXbJS5btsxsmfXr18Pf3x+xsbFmy9oDH4GwTqO+kfB3KQGCAK/AQMgVvuYXMsBLJqA2NA6nXSORppSgvtGycY2Y4xIEAa+88gqGDRuGHj16YN26dbr3Dh8+jBEjRiAhIQFDhw7F779rn2KZk5MDf39/LFu2DKNHj8aqVaswduxYPP7447j66qsRERGB1157DVu2bMHIkSMRFRWFLVu26OqdM2cOBg8ejISEBEyZMgWXLl0yG2dBQQHGjx+PhIQE3HjjjSguLta91/Jo5JtvvkFCQgKSkpLQv39/7NixA//73/9w+PBhLFiwAElJSdi1a5e1mq/d+AiEdRonlYDqnwFfB/l3bBDEwf4CMsoIKg1wQgkM9LdCgN1YY3kJGitK9eZJ3L0g9QsGqRqgunC+1TKyCO3dcKqLeaAr+uNIFcGQeHi1KQZXV1f88ccfyMjIwNChQ3HHHXdAo9EgJSUFH3zwASZNmoTffvsNM2fOxJkzZwAAJSUl6N27NxYvXgxA+6yO3Nxc7N27FxcuXECvXr2wcOFCHDhwAH/++SemT5+O2267DQDw1ltvwd9fu+G8/PLLWLZsmS4BGLNgwQJcffXVWLJkCc6ePYvExERMnjy5Vblnn30Wa9aswciRI6HRaFBRUQEfHx98+umneOyxxzBlypQ2tY2tcAJhncbfpc1HCkl+HUsgiX4CPjlDunoHdjAhdXdVB3ah8oeNevPcB42D4o4n0VhWjEuv/1+rZcLf+h4AoNz0OhrOZ+q95zvncXgMHt+mGGbPng0AiI+Ph1QqxYULF6BUKiGTyTBp0iQAwFVXXYXAwED8/fffCAkJgaurK2bNmqVXz8033wyJRILQ0FD4+/tj+vTpAIBBgwahqKgIdXV1cHV1xcaNG/HJJ5+gvr4etbW1CA42P67anj178M477wAAevbsifHjDX/G8ePH45FHHsHMmTMxceJEJCUltakt7IUTCOs0yk8cxqac7Tjp2gND1OMBtK0PSEsJCgFjK//CoNoMhH9bBHr0aR7avQM8R14Pt/7D9eZJ3LVHEE4+/ghcuMrosr63LzR4BNJWrq7Nt3Q7OTlBrVaDiAz+XZvmeXh4tHr/ynqaXjc9Q0StVuO3337D6tWrceDAAQQEBODrr7+26BqGpd544w2cOHECe/bswdy5czF79mw88cQTVqvfWjiBsE5Dkp+J0dXHMLr6GCSqvuhIAkn0E3BX6bcYX3UYANBYdi+kvp1vNFRH4ST3g5Pcz+B7grNMd7rKEOegCFuFhbi4ONTX12P37t245pprcODAAVy6dAkDBgzA5cuX212vUqmEt7c3FAoFGhoasHbtWouWu+aaa/DRRx/hueeeQ05ODn755RdMmNB6QM/MzEz069cP/fr1g1Qq1T233dvb2+Cw8WLhi+is05CXNA87EtCjfX1AmgS6Ccj1jNa9VhWc61B9zDHJZDJs374dixYtQkJCAh555BF89tln8PDw6FC91113HXr37o24uDhMmjTJ4lNMb7/9Nvbu3YuEhAQ89thjBpMHADz99NPo168fkpOT8cknn2Dp0qUAgHnz5mHZsmV8EZ2xtrhcS+hZrd3J1zm5Quof2uE6GwKigQva6dLccwjrP6zDdTJxXPmEwJZ3Nw0ZMgQHDx5stUx0dLReOUDbh6OlK4eJb7meK/tsLF++HAAwduxYHD582GCcYWFh+OWXXwy+t379et30l19+abDMlClTHOYCOsBHIKyTOH6hFlGqiwCAUnkkBEnHN133sB66aWUuH4Ew1lacQFinkHs+VzetCYiySp3hURFo+GdIE1zIsUqdjHUnnEBYp1Ce35xAvMJaPzq0PRICnXFGFq6tszwfpOYhTRhrC04grFOgS3m66bBo6ySQOB/glGs0AMCJGqG6mG+VehnrLjiBMIenIYK8rPkIxDPMOqewXJwEKH2ak1FdUa6J0oyxK3ECYQ7vbAXQ859ngNQ7ucDJiv01JIHNfRAu5uaZKMkYuxLfxssc3skywrqAWYirz8HV/mr0ssIdWE3kEdE4cGIAzriEY6BPnw50TWSs++EEwhzeqXLCVz5jAQADxjlZte7wyDDc1GMFAGCxXILWw9oxxozhU1jM4Z0qb+68FSu3bt3xPs3jIGUoeVh3xtqCj0CYw8sqa56OkVt3wMNe3oCzRDtM/MkyTiDGZGRkiB0Cawdb/904gTCHJyk8jQi1J1TegZDLrJtApBIBsXLtM0Eul5SjoUoGmae3VdfRmfn7+8Pd3R1z5swROxTWTu7u7rrnllgbJxDm0CoaCIuzX0dMQz4qpV6g2VutMoxJS9MbjmBT5kooGiuRv+8e9LzhZqvW35lFRkYiIyOj1ZhRHVVeXg653MrnI5nBdvX390dkpHX6Tl2JEwhzaKdL1YhuKAIAVHv4WT15AECgnxyKxkoAQEUB9wW5UmRkpNV3QKWlpVAoFFatk9m/XfkiOnNoeXlFcEYjAKDBN9wm6whqsXOky9wXhDFLmUwg9fX1mD9/PmJiYtCvXz8+D8rsrqSwQDftHBhmk3XEBboh3zkAAOCpzGs1NDhjzDCTp7CeeuopSCQSnDp1CoIgoKioyF5xMQYAaLjUPD6VT6htnlwXKwe2u0QgXHUZbupqaCqVcPLm0yuMmWM0gVRXV2PdunXIz8/XPTM4JCTEboExBgDOpc1HIKGRtjkCcZMKuOwZDlQdAQCoLuRyAmHMAkYTSHZ2Nvz8/PDiiy/i559/hpubG5YuXYrx48cbrWz+/PmQy+VISUnBjBkzDJZRKpUdj5q10hXblQjwqWxOIHUuHqgvLbXJuurkIbqnE54/fRoB/trrIl2xXR0Bt6ttWNqu1rrQbjSBqFQqnD17Fn379sXLL7+MY8eOYcKECTh58iQCAgIMLrN69WoMHDjQ7Er57gvb6GrtWlRDiKorBABUOnsj3ErPATHEIzQSyNJO15Yq9dqyq7Wro+B2tQ2HuAsrKioKEokEs2fPBgAkJiaiR48eOHHihN2CY93b6cs1CFGXAAAqvG1z+qqJf3jzHV4Nl/m5IIxZwmgC8ff3x/jx4/HDDz8AAM6fP49z586hT58+dguOdW/5hcWoE2QAgEY/2yaQnmF+qP1nXbLSQpuui7GuwuRdWGvWrMHdd9+NJ598Ek5OTnj//ff5QjqzmzRJOObEf4ZgdQk2j7XuECZXilc4YWrks7gkVSAsIgS/2HRtjHUNJhNIz549sXfvXjuFwpi+U+UEEiQocg5Az0jbDpoglwk4E5iMohqguMKmq2Ksy+Ce6MxhZf0zOq6bExDmYfv1xf0z0m9JPVBSx50JGTOHEwhzSGoNIfufI4EYOSARbHsKCwD6tHg2yOlyTiCMmcODKTKHlFNBeDPvNRRJ/eHuEgNgnM3XGe9ei6nlf6JHQyHK/wgEpk20+ToZ68w4gTCHlH2xHNPL9wMA8p2TYI8E0sddhZT8VwEAeSeTOIEwZgafwmIO6WJei74Y/ra9hbdJr2A5yiTaiy0e5XwrL2PmcAJhDqn6QvMQJp7B9kkg0d4Ccly06/KpvQxqqLfLehnrrDiBMMdU3JxAgiJsMwrvlaQSAcWeoQAACQgNxXwUwpgpnECYQ/Iob04gvmG2eZCUIXU+zUc7F3ILTJRkjHECYQ6nWkUIqdH++lcLUjgpAu22bqeA5gRSUsBjYjFmCicQ5nBOl2vQo0GbQEo9QyBInOy2bnlI89FO7UU+AmHMFE4gzOHk5F+GKzUAAOp97HMBvUlIRKhuWsKDKjJmEicQ5nBK8ptPHTnZ6DnoxsQEeuCi1BcAIK/gIxDGTOGOhMzhZDb6INfvRvSoL8CIHvZ9fECQG/CdRxy8VRW47BGKmzWNdl0/Y50JJxDmcH5FNP4IvhcCgOrh9t1EBUHAu0mLcLiYIACYRpV2XT9jnQmfwmIOhYh0o/BGegJuUtsPonilWPk/sQDIqeavCGPG8LeDOZTiOqBMe/0csXL7Jw9Af1TeM5X8FWHMGP52MIdySqlGkKoEIBItgbRcb3aFRpQYGOsM+BoIcygFuQU4fOoBVEtcke88FRh1j91jiPUmbMlZhN71+WjI9wVG/dfuMTDWGXACYQ5FWai9hddDUwdfN2dRYoj1kaCu4QKC1KWorqoDEUGwwwOtGOts+BQWcyiqS82d9xRh9u0D0sTTWUChm3bdHo010FSViRIHY46OEwhzKDJli1F4w8VJIABQ4d287lIeE4sxgziBMIfRqCEoqpqPQFyCxEsg5NdiVN487pHOmCGcQJjDyKsGouu1O+sqmTck7l6ixeIW1DwmVkUhJxDGDOEEwhzG6Us1CFaXAgAqvUPNlLatgPDmUXkbizmBMGYIJxDmMIrym3fUGj/7PUTKkOjwIDQI2psU3co4gTBmCCcQ5jAqipqvf7gHiXsEEu3jhFxZMABAUV0E0nCHQsauxAmEOYzGFs8gb3kKSQzOEgGX3LUX0l00DVCVXRY1HsYcEXckZA7jHUUK3uo9Cv01hfgyzr7DuBtyuPcUbCy6Gmdlodgl8UWE2AEx5mA4gTCHUKsmnKt2ArmEIjAgDFJv8TdNTa+B+LpGe+oqq9oJET7ixsOYo+FTWMwhZFdoh08HmodTF1vLUXlPlZOJkox1T5xAmENouYMWaxTeK7VMZKfKxYuDMUcl/nkCxgAU5hXg8Ys/4ZxLKBKoPwBxL6ID2kTWqz4PvevzEZReDoyYKnZIjDkUTiDMIahys7CgeBsAoLrwHmDQzSJHBAS7Ae/lv4b4unNQQwJqvA6CE39lGGvCp7CYY2jR2zsgXNw+IE0EQUCJZwgAQAoNai5dEDkixhwLJxDmEDwqmvuAeIWIf/qqSa08RDede557pDPWEicQJrrSOkJYjXbnrIEAqV+ImSXsR6JojqWkgBMIYy1xAmGiO12uQXRDEQCgwj0QgrNM5IiaeQcF66brLvJzQRhriRMIE93ZonLINdUAgDofx7j+0SQoNFA37aQsNFGSse6HEwgTXctTQ9IA8R4iZUiUvwfKnDwBAPIKPoXFWEsWJZDnn38egiDg+PHjto6HdUO1l5pPDXmHOFYCcZMKKPjn+eiB9Zehqa8TOSLGHIfZBHLkyBGkpqYiMjLSHvGwbsiptPnUUGCEYyUQAKjwaj6tVlRQJGIkjDkWkwmkvr4eDz30EN59910IgmMML8G6Fg0RjmmCsN8jCRdcAuEa5Di38DbR+IWhzMkTR9z64LyyXuxwGHMYJrvVLl68GHPmzEGPHj0sqmz+/PmQy+VISUnBjBkzDJZRKpVtj5KZ1VnbtaBGwAafSdjgMwmTQ9TYKKkFSkvFDktHqVSiIGEKbqNbAQArqQ5xDhRfZ9VZt1dHZ2m7KhQKq6zPaAI5ePAgDh06hJdfftniylavXo2BAweaLWet4Jm+ztiuR2o1ABoBAP38ZVAo3MQNyICBbt7AMW2MeSp3KBROIkfUNXTG7bUzsGe7Gj2FtW/fPmRmZqJHjx6Ijo5Gfn4+Jk2ahO+++85uwbGuL6useRTelsOnO5KWcbWMl7HuzmgCeeqpp1BYWIicnBzk5OQgPDwcP/zwA6677jp7xse6uKzSRoC0O+U+DvIckCsFuwHeztrpTE4gjOnw0KJMVCEnvkP6qQ3IdglHbOndQEii2CG1IggC/lP1NWKKDiG6oRDVU96Fh5en2GExJjqLE0hOTo4Nw2DdlasyDz6aagyqzYKPi+P2a01oOI8h1UcBADk5Beg3QPxntjMmNsf9xrIur1pFCKpq7kToHBwhYjSmCS16yF/M5x7pjAGcQJiITlcAvRq0CaTG2QsSDwe9CALAK7g5gdQU8aCKjAGcQJiITl+qQZiqGABQ5Rvu0J1VQyKbj46EkjwRI2HMcXACYaK5lN/8S17wd9zTVwDQIzoUKmj7f3iXcQJhDOAEwkRUU5Srm/YMdewE4ubijEJX7cOlQmsKoGlUixwRY+LjBMJEIyluPgIJinS8MbCuVCLXJjkXUqGo4KLI0TAmPk4gTBREBHl5cwJxC3H80Z5ViuajpNycXBMlGeseOIEwURRUA9F12gSiFqSQKoLNLCE+1xZJriyfr4Mwxj3RmSiyygn3RzyFmPo83BpYgX85Of4AhYExcVj998044xKO3r79wIP6sO6OEwgTRVY54axLOM66hCNlsOMnDwCI7RWGV4LuBACMUjnuLceM2QufwmKiyCxrnu7jI1YUbSOXCQj30E6fUBKIeGBF1r1xAmGi0BvGXd55fs3389XGWtYAFNWIHAxjIuMEwkThn5OKaeX7MVR1FoGyRrHDsVh/uQZRDUUYX/knsnIuiB0OY6LiayDM7qpUhBn5X2BozUntDPV2QNo5NsVrL/6ER06vAgAcOf5voN+NIkfEmHj4CITZXWYZoVe99hbeClcFJK4eIkdkucCI5r4g6kt8Ky/r3jiBMLs7VVAGv8YKAECNwvE7ELbUs1dzvG6lnEBY98YJhNldyfkc3bQ0OFq0ONpD7usDpdQbABBYlcd3YrFujRMIs7uGohzdtCIyWrQ42uuyl/Y0VpCqFAWl1SJHw5h4OIEwu3MrOa+bDoqOEjGS9qlvMSbWuWweE4t1X5xAmF1VqwihVc0JxDm48yUQ5+Dm6yAluTniBcKYyDiBMLvKUGoQW6/91a50D4TE1V3kiNrOPypaN60uOideIIyJjBMIs6vsvGJ4a7RduGv8Ot/RBwD0jOmhm/ZscTqOse6mc/TeYl3GuZIakEcC+tTlQtYJT18BgLevL845+8C1sQ7FGhmIyKGf586YrXACYXb1O0Xg6ejlAICcGzrvTve10auw6YI3SJBgVCXQw1vsiBizPz6FxezqhFLbb8LTGYiUd45h3A3pEaoACdqvz9ES7gvCuidOIMxuqlSEnErtdLyP0KlP+yT5Ncd+rJQTCOueOIEwuzleotH13E5QiBxMByUqmhMIH4Gw7oqvgTC7OXM2D8czFyLDNRrOftcCmCx2SO3Ww4vw7OUNiKk5C5cCCTDxBbFDYszuOIEwuyk7fxZyTTWG15xAKQ0UO5wOcZJIMLXqAEJrClAnyKCsVcPXjb9OrHvhU1jMbiRF2brp0F69RIzEOip8owEArtSAzOxCcYNhTAScQJhdaIigUJ7VvfaN7i1iNNYhCWnuUFh49qyJkox1TZxAmF3kVBDiarQ72Upnb0jkfiJH1HF+0T110w35Z0SMhDFxcAJhdnEytwT+jeUAgHK/np36Ft4mveKaj6I8L2ebKMlY18QJhNnFxXPNO1hJaOe//gEAHn4BUDr7AAAiK7JRr9aIGxBjdsYJhNlFQ0HzNQK/6K6RQARBwEVf7Wfxa6xAxvnLIkfEmH1xAmF24dXiFE9Y754mSnYuFNKcDHOyTosYCWP2xwmE2VxFAyGqSnsEUi9xgUtwhJklOg+/Hs3XQapz+UI661645xOzuSPFhH9FLkXfunO4KaAcvSSddxDFK8X0i8Obv05FumsvVHj2w2yxA2LMjjiBMJs7XEzIcQlFjksoZgzvOskDALwCAvF533lILwWc6oAaNcFd2vnvMGPMEkZPYdXV1WH69OmIjY1FUlISJk+ejJycHDuGxrqKQ5ebBxscEtD1dq5D/LWfqZF4YEXWvZi8BjJv3jxkZWXh6NGjmDJlCubNm2evuFgXcvifBOLmBMT7iBuLLbRMii2TJWNdndEE4urqiuuvv17X4Wv48OE4y8M1sDYqrtXgpuxPMaX8V0z0uAyppAsegQRI4Kcuw8SKVChPHBE7HMbsxuJrIO+88w6mTp1qssz8+fMhl8uRkpKCGTNmGCyjVCrbFiGziKO268HsMjxyeSsA4FzdAJSWPi1yRG1jSbuGVpbgaNb/AQAOlyeitLRr9HOxJUfdXjs7S9tVobDOA3ksSiArVqzA6dOnsWbNGpPlVq9ejYEDzQ/Tba3gmT5HbNey/cd105Kovg4ZoznmYiZfX6TLFFA0lKJP1SnAzRsKHtrdrM64LXQG9mxXs/1AVq5ciS+++ALfffcd3N3d7RET60LUuVm66aCYPiJGYjuCIKA4MA4A4KWpxdGT50WOiDH7MJlA3njjDWzevBk//fQTfHx87BQS60r8LjcnkKi4rplAAMAlOl43XZCVIWIkjNmP0QSSn5+PhQsXoqysDOPGjUNSUhKGDRtmz9hYJ1dYpUafam3v7BKXADj7dP4h3I3p2a85gQi5mSJGwpj9GD1RGx4eDiK+JZG1X9qJc0jU1AEAlEGxIkdjW8G9Y5EjSOFMakQoM1CnJrhyh0LWxfFYWMxmijPSddPuvfqLGIntCc4yXPDRDhLZq74AaXkVIkfEmO1xAmE2457fnEB6JQwQMRL70IQ3n8Y6c4Kvg7CujxMIs4nyeg36KLW38FZJPaCI6mFmic4vqE9f3XTD2eMmSjLWNXACYTbxZ2E9NiiuR6p7P+QFJ0PoQiPwGhOdMAAqOOGwWxxS1UHQ8DVE1sVxbydmE/tLZHg9cA4AYPO47vE7ReqtwFMTt2JbgQsA4KFSIKnr3njGGB+BMNv47ULzr++rgrvPZjY6yk03/VM+PyOddW3d55vN7KahkfDHJW0CifIEwj27z+2sE8Oav1I/FvApLNa1cQJhVnfkvBI9qs4CRLgquPskDwCIkWuTJohwLrcINWpOIqzr4msgzOryDu7DD2fX4qLUFwXRjwDoPiMYCIKAFys2I+709whWl+JA9mZM6OMrdliM2QQfgTCrczn7FwAgSK1EUs8AkaOxvxi3egSrSwEA2ceOiRwNY7bDCYRZVXl1PeJL/wYAFMsUiOjV9ft/XKlnYpJuWpr9l3iBMGZjnECYVaX9lQ43agAAFIYO0j3Rsjvx75uIOon2Vt5BxX+ioFItckSM2QYnEGZVyvRDumnPvoNEjEQ8grMMRWHaB6v5N5bj18M8Oi/rmjiBMKvRaDQIzU0FAKjghMSh5p9O2VX5J4/QTZf/nSpiJIzZDicQZjUZmTkIq78IADilGAC5j7fIEYknZsgQaKA9fRdT+AdK6/h2Xtb1cAJhVnPuj99106q4ESZKdn1SL19c8NeOzhtbn4ef0wtEjogx6+MEwqxGnn1AN91/ZPdOIADgPqC5/0vhXwdFjIQx2+AEwqwir1yF35z7QOnkhSzPGESHB4odkuhih48EAJQ4eSOrHKhs4NNYrGvhnujMKradl+CZ0IewJHgeXolTYrzYATkAWVAENo15CYsux0EtSDEih3BnbPe7rZl1XXwEwqxiS7b217VK4owbEkNFjsZxXDUyEWpB+zvt49M8Oi/rWjiBsA7LriAcLtYmkGQ/INaHf2U3GREoIOafm9H2FBLOV/JpLNZ1cAJhHbZ7zx9QqMsBALf25E2qJUEQMDdW2yZBqhJ8cYTvxmJdB18DYR2iqqrAuL0rcAgabPO9FnfELBA7JIdzR3AZgvJWY0LFH9hbPBKq0YvgLOGjNNb58c9F1iFHfv4ZrtQAGakR5i1FqAfvGK8UEeCNkXUZkEKDMaUH8e2JUrFDYswqOIGwdqPGRrj8sUP3OvjqG0SMxnEJUmeokq8FADijEXm/fC1yRIxZBycQ1m7ZB/bDv1Y7dEmqdzLGJUWJHJHj6jtpKtSCEwDgmryd+O18tcgRMdZxnEBYuxARyn7+TPe6duTNkPJ5faOcFYG4FDsOAODTWIXUnbtEjoixjuMEwtql4M9fEVx+FgCQ7tYbU8cliRtQJzDgxpm66QnZn+PA+SoRo2Gs4ziBsDajRjVKvlmve3126B3wceFNyRy30GgU9R4DQPuckNQvt0ND3C+EdV78rWdtlvPT1/CrKgQA/OE5ALdMHiJyRJ3HgFvu1F0LmXL+C3xx9KLIETHWfpxAWJtoNBq9YdsvXv0vBLjxZmQp18AwlCdp71Zzp3oc/elHlPCzQlgnxR0JWZt8kQPcFrIc98i+Ri+hFP8eGy92SJ3OgFvm4o/s43jFOwU7vK/GuQON2HgNfxVZ58M/HZnFCqsJ//69EY2CE973vwkxt86Dl4zvvGoriZsHop5cjX0BYwBBwKZswkdZPNAi63w4gTCLqBoacMfeRhTXaV9PjxIwPZqTR3uFekjw7ign3esHf2/EnxcbRYyIsbbjBMLMqj+fhayl96Lq9HEAQJgH8L+rnSAInEA6YlZvCf4dr/0KTi/+EbmrFyHjUq3IUTFmOU4gzKS6M+nIX/00fGouYdP5ZzGq9jg2j3OCnysnD2t4c4QEr9Z/gZWFqzC88ihOvfMs0vK5fwjrHDiBMIOICMr93+LCf5+Gi6oGAJDm1gfzr+2D0SG82ViLi5OAeyYnoMbJDQAwqOoEqlY9jK/+OCtyZIyZx3sC1orqUj7O/fc5VH+xGlJSAwD2eSRDedsy3BbvLnJ0XY+idzwUD76MSmftk6d61BcgYcsC/G/1epwvrRc5OsaM4wTCAGiPOBryTuPcupUoWjEPsjOHde+t958G57nP474EDxEj7Nr8e/VBxGNvoci7BwBARmpMPrMFxSv+hU8+3IJ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\r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", " \r\n", "\r\n" ], "text/html": [ "" ] }, "execution_count": 16, "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", "#### ガンマ函数の1と1/2での値.\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", "$$\n", "\n", "**注意:** 上の問題の解答より, $\\Gamma(1/2)$ は本質的にGauss積分に等しい. その意味でガンマ函数はGauss積分の一般化になっていると言える. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### ガンマ函数の函数等式\n", "\n", "**問題(ガンマ函数の函数等式):** $s>0$ のとき $\\Gamma(s+1)=s\\Gamma(s)$ となることを示せ.\n", "\n", "**解答例:** 部分積分を使う. $s>0$ と仮定する. このとき\n", "\n", "$$\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", "$$\n", "\n", "すなわち, $\\Gamma(s+1)$ は階乗 $n!$ の連続変数 $s$ への拡張になっていることがわかる. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### ガンマ函数の正の半整数での値\n", "\n", "**問題(ガンマ函数の正の半整数での値):** 次を示せ: 非負の整数 $k$ に対して\n", "\n", "$$\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", "$$\n", "\n", "なので, 上の方のGauss積分の応用問題に関する結果から欲しい公式が得られる. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### ガンマ函数の Ramanujan's master theorem 型解析接続\n", "\n", "**問題:** $N=0,1,2,\\ldots$ とする. 次の公式を示せ:\n", "\n", "$$\n", "\\Gamma(s) = \\int_0^\\infty \\left(e^{-x}-\\sum_{k=0}^N\\frac{(-x)^k}{k!}\\right)x^{s-1}\\,dx\n", "\\qquad (-(N+1)<\\real s<-N).\n", "$$\n", "\n", "**解答例:** $\\real s > 0$ のとき\n", "\n", "$$\n", "\\begin{aligned}\n", "&\n", "\\int_0^1 e^{-x}x^{s-1}\\,dx \n", "\\\\ &= \n", "\\int_0^1 \\left(e^{-x}-\\sum_{k=0}^N\\frac{(-x)^k}{k!}\\right)x^{s-1}\\,dx +\n", "\\sum_{k=0}^N\\frac{(-1)^k}{k!}\\int_0^1 x^{s+k-1}\\,dx\n", "\\\\ &=\n", "\\int_0^1 \\left(e^{-x}-\\sum_{k=0}^N\\frac{(-x)^k}{k!}\\right)x^{s-1}\\,dx +\n", "\\sum_{k=0}^N\\frac{(-1)^k}{k!}\\frac{1}{s+k}.\n", "\\end{aligned}\n", "$$\n", "\n", "この計算の最後の結果中の積分は $\\real s > -(N+1)$ で収束し, この計算結果は $\\int_0^1 e^{-x}x^{s-1}\\,dx$ の $\\real s > -(N+1)$ への解析接続を与える.\n", "\n", "$\\int_1^\\infty x^{s+k-1}\\,dx = -1/(s+k)$ ($\\real s < -k$) を使った上と同様の計算によって, $\\real s < -N$ のとき\n", "\n", "$$\n", "\\begin{aligned}\n", "&\n", "\\int_1^\\infty e^{-x}x^{s-1}\\,dx\n", "\\\\ &=\n", "\\int_1^\\infty \\left(e^{-x}-\\sum_{k=0}^N\\frac{(-x)^k}{k!}\\right)x^{s-1}\\,dx -\n", "\\sum_{k=0}^N\\frac{(-1)^k}{k!}\\frac{1}{s+k}\n", "\\end{aligned}\n", "$$\n", "\n", "となることがわかる.\n", "\n", "$-(N+1)<\\real s<-N$ のとき, 以上の2つの結果を足し合わせると, $\\sum$ の項がキャンセルして消えて, \n", "\n", "$$\n", "\\Gamma(s) = \\int_0^\\infty \\left(e^{-x}-\\sum_{k=0}^N\\frac{(-x)^k}{k!}\\right)x^{s-1}\\,dx\n", "$$\n", "\n", "が得られる. $\\QED$\n", "\n", "**注意:** この公式は [Ramanujan's master theorem](https://www.google.com/search?q=Ramanujan+master+theorem) の特別な場合になっている. 論文\n", "\n", "* Tewodros Amdeberhen, Olivier Espinosa, Ivan Gonzalez, Marshall Harrison, Victor H. Moll, and Armin Straub. Ramanujan’s Master Theorem. The Ramanujan Journal 29(1-3). DOI: 10.1007/s11139-011-9333-y ([ResearchGate](https://www.researchgate.net/publication/257643116_Ramanujan's_Master_Theorem))\n", "\n", "の Example 8.2 は上の公式の $N=0$ の場合になっている. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Riemannのゼータ函数の積分表示と函数等式と負の整数と正の偶数における特殊値\n", "\n", "この節はこのノートを最初に読むときには飛ばして読んでも構わない. ガンマ函数の理論がRiemannのゼータ函数の理論と密接に関係していることを認識しておけば問題ない. \n", "\n", "Bernoulli数やBernoulli多項式に関してはノート「13 Euler-Maclaurinの和公式」により詳しい解説がある." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Riemannのゼータ函数の積分表示1\n", "\n", "**問題(Riemannのゼータ函数の積分表示1):** 次をが成立することを示せ.\n", "\n", "$$\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", "$$\n", "\n", "と書けるので, $s-2 > -1$ すなわち $s>1$ ならば収束している. $\\QED$\n", "\n", "**解答例:** ガンマ函数のスケール変換に関する問題の結果より, $\\ds\\frac{1}{n^s} = \\frac{1}{\\Gamma(s)}\\int_0^\\infty e^{-nx}x^{s-1}\\,dx$ なので,\n", "\n", "$$\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", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**問題:** $B_0=1$, $\\ds B_1=-\\frac{1}{2}$ であり, $n$ が3以上の奇数のとき $B_n=0$ となることを示せ.\n", "\n", "**解答例:** $z\\to 0$ のとき, $\\ds \\frac{z}{e^z-1}\\to 1$ より $B_0=1$ となる. さらに, \n", "\n", "$$\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", "$$\n", "\n", "であることと, これが偶函数であることから, $\\ds B_1=-\\frac{1}{2}$ でかつ $n$ が3以上の奇数ならば $B_n=0$ となることがわかる. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**問題(Riemannのゼータ函数の積分表示1'):** 非負の整数 $N$ に対して, 次を示せ:\n", "\n", "$$\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", "$$\n", "\n", "であり, $\\ds\\int_0^1 x^{N+1}x^{s-2}\\,dx=\\int_0^1 x^{s+N-1}\\,dx$ が $s>-N$ で絶対収束していることから, 右辺の括弧の内側の2つ目の積分もそこで収束している. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Riemannのゼータ函数の0以下での整数での値\n", "\n", "**問題:** Riemannのゼータ函数の積分表示1'の右辺で $\\zeta(s)$ を $s>-N$ まで拡張しておくとき, \n", "\n", "$$\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", "$$\n", "\n", "が得られる. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### 交代版のRiemannのゼータ函数の積分表示\n", "\n", "**問題:** 次を示せ.\n", "\n", "$$\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", "$$\n", "\n", "**注意:** この公式の積分は $\\real s > 0$ で収束している. $\\QED$\n", "\n", "**解答例:** 1つ目の等号を示そう:\n", "\n", "$$\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", "$$\n", "\n", "**注意:** 以上の計算は統計力学におけるFermi-Dirac統計に関する議論に登場する. ゼータ函数は数論の基本であるだけではなく, 統計力学的にも意味を持っている. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Hurwitzのゼータ函数0以下の整数での特殊値がBernoulli多項式で書けること\n", "\n", "**問題:** **Hurwitzのゼータ函数** $\\zeta(s,x)$ と**Bernoulli多項式** $B_k(x)$ を\n", "\n", "$$\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", "$$\n", "\n", "と定める. $\\zeta(s)=\\zeta(s,1)$ なのでHurwitzのゼータ函数はRiemannのゼータ函数の拡張になっている. 以下を示せ:\n", "\n", "(1) $\\quad\\ds \\zeta(s,x) = \\frac{1}{\\Gamma(s)}\\int_0^\\infty \\frac{e^{(1-x)t}t^{s-1}}{e^t-1}\\,dt$.\n", "\n", "(2) $\\quad\\ds \\zeta(s,x) = \\frac{1}{\\Gamma(s)}\\left[\n", "\\int_1^\\infty \\frac{e^{(1-x)t}t^{s-1}}{e^t-1}\\,dt +\n", "\\int_0^1\\left(\\frac{t e^{(1-x)t}}{e^t-1}-\\sum_{k=0}^N\\frac{B_k(1-x)}{k!}t^k\\right)t^{s-2}\\,dt +\n", "\\sum_{k=0}^N \\frac{B_k(1-x)}{k!}\\frac{1}{s+k-1}\n", "\\right].\n", "$\n", "\n", "(3) Hurwitzのゼータ函数を(2)によって $s<1$ に拡張すると, $0$ 以上の整数 $m$ について\n", "\n", "$$\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", "$$\n", "\n", "となることから, $\\ds\\zeta(-m,x)=\\frac{(-1)^m B_{m+1}(1-x)}{m+1}$ が得られる. さらに, $\\ds\\frac{te^{(1-x)t}}{e^t-1} = \\frac{(-t)e^{a(-t)}}{e^{-t}-1}$ によって, $B_k(1-x)=(-1)^k B_k(x)$ となることがわかるので, $\\zeta(-m,x) = \\ds \\frac{(-1)^m B_{m+1}(1-x)}{m+1}=-\\frac{B_{m+1}(x)}{m+1}$ も得られる. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### べき乗和のBernoulli多項式による表示\n", "\n", "**注意:** 上の結果より, 累乗和\n", "\n", "$$\n", "S_m(n) = 1^m + 2^m + \\cdots + n^m\n", "$$\n", "\n", "を次のようにして, Bernoulli多項式を使って表すことができることがわかる:\n", "\n", "$$\n", "S_m(n) = \\zeta(-m,1) - \\zeta(-m,n+1) = \\frac{B_{m+1}(n+1)-B_{m+1}(1)}{m+1}.\n", "$$\n", "\n", "ここで $\\zeta(s,x)$ の解析接続を使った. 形式的には\n", "\n", "$$\n", "\\begin{aligned}\n", "\\zeta(-m,1) - \\zeta(-m,n+1) &= \n", "(1^m + \\cdots + n^m + (n+1)^m + (n+1)^m + \\cdots) - ((n+1)^m + (n+1)^m + \\cdots)\n", "\\\\ & = 1^m + \\cdots + n^m = S_m(n)\n", "\\end{aligned}\n", "$$\n", "\n", "という計算が実行されたことになる. この計算は $m < -1$ ならばそのまま正しい. それ以外の場合には解析接続によって正当化される. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Riemannのゼータ函数の積分表示2(テータ函数のMellin変換)と函数等式\n", "\n", "**問題(Riemannのゼータ函数の積分表示2):** $\\theta(t)$ を\n", "\n", "$$\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", "$$\n", "\n", "$\\theta(t)$ に関する上の公式の証明についてはノート「12 Fourier解析」におけるPoissonの和公式の解説を見よ.\n", "\n", "**注意:** 上の問題の公式の右辺の積分は $s$ が任意の複素数であってもしているので, 右辺は左辺の複素平面上への解析接続を与える. さらに, 右辺は $s$ を $1-s$ で置き換える操作で不変であるから,\n", "\n", "$$\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", "$$\n", "\n", "と書き直されることを示せ.\n", "\n", "Legendre's duplication formula と Euler's reflection formula はこのノートの下の方で初等的に証明される. Euler's reflection formulaの証明についてはノート「12 Fourier解析」のガンマ函数とsinの関係の節も参照せよ.\n", "\n", "**解答例:** $\\hat\\zeta(s)=\\pi^{-s/2}\\Gamma(s/2)\\zeta(s)$, $\\hat\\zeta(s)=\\hat\\zeta(1-s)$ より,\n", "\n", "$$\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ガンマ分布の中心極限定理とStirlingの公式\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": [ "#### ガンマ函数の1/2での値をベータ函数経由で計算\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. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### ベータ函数の応用の雑多の例\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": 17, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\ds \\int_0^1 x^5 (1-x^2)^{3/2} \\,dx =\\frac{8}{315}$" ], "text/plain": [ "L\"$\\ds \\int_0^1 x^5 (1-x^2)^{3/2} \\,dx =\\frac{8}{315}$\"" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" } ], "source": [ "x = symbols(\"x\", real=true)\n", "sol = integrate(x^5*(1-x^2)^(Sym(3)/2), (x,0,1))\n", "latexstring(raw\"\\ds \\int_0^1 x^5 (1-x^2)^{3/2} \\,dx =\", latex(sol))" ] }, { "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", "\n", "#### ガンマ函数に関するGaussの公式\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": [ "#### 指数函数の上からと下からの評価を用いた厳密な議論\n", "\n", "**問題:** 上の問題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", "$$\n", "\\left(1+\\frac{t}{a}\\right)^a \\leqq e^t \\leqq \\left(1-\\frac{t}{b}\\right)^{-b}\\qquad(-a0)\n", "\\tag{1}\n", "$$\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数を取り, $a=m$, $b=n$ とおくと, \n", "\n", "$$\n", "\\left(1-\\frac{t}{n}\\right)^n \\leqq e^{-t} \\leqq \\left(1+\\frac{t}{m}\\right)^{-m} \\qquad (-m0$, $m>s$ のとき, \n", "\n", "$$\n", "n^s B(s,n+1) = \\int_0^n t^{s-1}\\left(1-\\frac{t}{n}\\right)^n\\,dt, \\quad \n", "m^s B(s,m-s) = \\int_0^\\infty 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", "$$\n", "\n", "なので, $n^s B(s,n+1)$, $ m^s B(s,m-s)$ は $n,m\\to\\infty$ で同じ値に収束する. これと不等式($*$)を合わせると, $n^s B(s,n+1)$, $ m^s B(s,m-s)$ は $n,m\\to\\infty$ で $\\Gamma(s)$ に収束することがわかる. $\\QED$\n", "\n", "**注意:** 不等式(1),(2)と $a,b$, $m,n$ に関する単調性は極限で指数函数が現われる結果を初等的に正当化するために非常に便利である. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### ガンマ函数に関するWeierstrassの公式\n", "\n", "**問題(Weierstrassの公式):** 上の問題の結果を用いて, 次の公式を示せ.\n", "\n", "$$\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", "$$\n", "\n", "となり, この無限積は任意の複素数 $s$ について収束する. したがって, Weierstrassの公式は $1/\\Gamma(s)$ のすべての複素数 $s$ への自然な拡張を与える. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### sinとガンマ函数の関係\n", "\n", "sinの無限積表示と Euler's reflection formulaの証明についてはノート「12 Fourier解析」のガンマ函数とsinの関係の節も参照せよ. 以下ではsinの奇数倍角の公式を用いた証明を紹介する." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### sinの奇数倍角の公式を用いたsinの無限積表示の導出\n", "\n", "sinの無限積表示\n", "\n", "$$\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 kView Archive\n", "\n", "を参照せよ. $\\QED$" ] }, { "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": [ "#### cosの偶数倍角の公式を用いたcosの無限積表示の導出\n", "\n", "この節の内容は\n", "\n", "* https://twitter.com/genkuroki/status/1094856745485139971\n", "\n", "の詳述になっている. 上における $\\sin$ の無限積表示の導出と同じ方法で\n", "\n", "$$\n", "\\cos\\frac{\\pi s}{2} = \\prod_{k=1}^\\infty\\left(1-\\frac{s^2}{(2k-1)^2}\\right) =\n", "\\left(1-\\frac{s^2}{1^2}\\right)\n", "\\left(1-\\frac{s^2}{3^2}\\right)\n", "\\left(1-\\frac{s^2}{5^2}\\right)\n", "\\cdots\n", "$$\n", "\n", "を示そう. $e^{inx}=(e^{ix})^n$ の右辺に $e^{ix}=\\cos x+i\\sin x$ を代入して二項定理を適用し, 両辺の実部を取ることによって,\n", "\n", "$$\n", "\\cos(nx) = \\sum_{0\\leqq k\\leqq n/2}(-1)^k \\binom{n}{2k}(\\cos x)^{n-2k}(\\sin x)^{2k}.\n", "$$\n", "\n", "$(\\cos x)^2 = 1 - (\\sin x)^2$ より, $n$ が偶数ならば $\\cos(nx)$ は $\\sin x$ の $n$ 次多項式で表わされることがわかる. 以下では $m$ は非負の整数であるとし, $n=2m$ とおき, $\\cos(nx)=\\cos(2mx)$ について考える. $\\cos(2mx)$ は $\\sin x$ の $2m$ 次の多項式で表わされ, 周期 $\\ds\\frac{2\\pi}{2m}$ を持ち, \n", "\n", "$$\n", "\\pm x_k = \\pm\\frac{(2k-1)\\pi}{4m}, \\quad $k=1,2,\\ldots,m$\n", "$$\n", "\n", "で $0$ になり,\n", "\n", "$$\n", "-\\pi/2 < -x_m < \\cdots < -x_1 < 0 < x_1 < \\cdots < x_m < \\pi/2\n", "$$\n", "\n", "なので, $2m$ 個の $\\sin(\\pm x_k)$ は互いに異なる. $0$ でないある定数 $C$ が存在して, \n", "\n", "$$\n", "\\cos(2mx) = \n", "C\\prod_{k=1}^m\\left[\n", "\\left(\\sin x - \\sin\\frac{(2k-1)\\pi}{4m}\\right)\n", "\\left(\\sin x + \\sin\\frac{(2k-1)\\pi}{4m}\\right)\n", "\\right].\n", "$$\n", "\n", "この等式の両辺に $x=0$ を代入すると,\n", "\n", "$$\n", "1 = C\\prod_{k=1}{m}\\left[\n", "\\left(-\\sin\\frac{(2k-1)\\pi}{4m}\\right)\n", "\\left(\\sin\\frac{(2k-1)\\pi}{4m}\\right)\n", "\\right].\n", "$$\n", "\n", "この等式の両辺で上の等式の両辺をそれぞれ割ると,\n", "\n", "$$\n", "\\cos(2mx) = \n", "\\prod_{k=1}^m\\left[\n", "\\left(1-\\frac{\\sin x}{\\sin\\frac{(2k-1)\\pi}{4m}}\\right)\n", "\\left(1+\\frac{\\sin x}{\\sin\\frac{(2k-1)\\pi}{4m}}\\right)\n", "\\right].\n", "$$\n", "\n", "これに $x=\\frac{\\pi s}{4m}$ を代入すると,\n", "\n", "$$\n", "\\cos\\frac{\\pi s}{2} =\n", "\\prod_{k=1}^m\\left[\n", "\\left(1-\\frac{\\sin\\frac{\\pi s}{4m}}{\\sin\\frac{(2k-1)\\pi}{4m}}\\right)\n", "\\left(1+\\frac{\\sin\\frac{\\pi s}{4m}}{\\sin\\frac{(2k-1)\\pi}{4m}}\\right)\n", "\\right].\n", "$$\n", "\n", "これの両辺の $m\\to\\infty$ の極限を取ると余弦函数の無限積表示\n", "\n", "$$\n", "\\cos\\frac{\\pi s}{2} =\n", "\\prod_{k=1}^\\infty\\left[\n", "\\left(1-\\frac{s}{2k-1}\\right)\n", "\\left(1+\\frac{s}{2k-1}\\right)\n", "\\right] =\n", "\\prod_{k=1}^\\infty\n", "\\left(1-\\frac{s^2}{(2k-1)^2}\\right)\n", "$$\n", "\n", "を得る. この等式の両辺の $s^2$ の係数の $-1$ 倍を比較すると\n", "\n", "$$\n", "\\frac{\\pi^2}{8} = \\sum_{k=1}^\\infty \\frac{1}{(2k-1)^2} =\n", "\\frac{1}{1^2}+\\frac{1}{3^2}+\\frac{1}{5^2}+\\cdots\n", "$$\n", "\n", "ゆえに\n", "\n", "$$\n", "\\zeta(2) = \\sum_{n=1}^\\infty\\frac{1}{n^2} = \n", "\\sum_{k=0}^\\infty\\frac{1}{(2k)^2} + \\sum_{k=1}^\\infty\\frac{1}{(2k-1)^2} =\n", "\\frac{1}{4}\\zeta(2) + \\frac{\\pi^2}{8}\n", "$$\n", "\n", "となるので\n", "\n", "$$\n", "\\zeta(2) = \\frac{\\pi^2}{6}\n", "$$\n", "\n", "も得られる. この公式は\n", "\n", "$$\n", "\\frac{\\sin(\\pi s)}{\\pi s} =\n", "\\prod_{n=1}^\\infty\\left(1-\\frac{s^2}{n^2}\\right)\n", "$$\n", "\n", "の両辺の $s^2$ の係数の $-1$ 倍を比較しても得られる.\n", "\n", "**問題:** 上の議論で現われた定数 $C$ を求めよ.\n", "\n", "**解答例:** \n", "$$\n", "\\cos(2mx) = \\sum_{k=0}^m (-1)^k \\binom{2m}{2k}(\\cos x)^{2m-2k}(\\sin x)^{2k}\n", "$$\n", "\n", "の右辺に $(\\cos x)^2 = 1 - (\\sin x)^2$ を代入すると, 右辺の $\\sin x$ の多項式としての最高次の係数 $C$ は\n", "\n", "$$\n", "\\begin{aligned}\n", "C &=\n", "\\sum_{k=0}^m (-1)^k\\binom{2m}{2k}(-1)^{m-k} = (-1)^m\\sum_{k=0}^m\\binom{2m}{2k}\n", "\\\\ &=\n", "(-1)^m\\left(\\sum_{k=1}^m\\binom{2m-1}{2k-1} + \\sum_{k=0}^{m-1}\\binom{2m-1}{2k}\\right) \\\\ &=\n", "(-1)^m\\sum_{j=0}^{2m-1}\\binom{2m-1}{j} =\n", "(-1)^m(1+1)^{2m-1} = \\frac{(-4)^m}{2}\n", "\\end{aligned}\n", "$$\n", "\n", "となることがわかる. $\\QED$\n", "\n", "**注意:** これで次が示された: 非負の偶数 $n$ に対して,\n", "\n", "$$\n", "\\cos(nx) = \\frac{(-4)^{n/2}}{2}\\prod_{k=1}^{n/2}\\left(\\sin^2 x - \\sin^2\\frac{(2k-1)\\pi}{2n}\\right).\n", "\\qquad \\QED\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Wallisの公式\n", "\n", "#### 2種類のWallisの公式の同値性\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": [ "#### sinの無限積表示を用いたWallisの公式の証明\n", "\n", "**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": 18, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\ds \\frac{1}{1-1/(2n)^2} =\\frac{4 n^{2}}{\\left(2 n - 1\\right) \\left(2 n + 1\\right)}$" ], "text/plain": [ "L\"$\\ds \\frac{1}{1-1/(2n)^2} =\\frac{4 n^{2}}{\\left(2 n - 1\\right) \\left(2 n + 1\\right)}$\"" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "n = symbols(\"n\")\n", "sol = 1/factor(1-1/(2n)^2)\n", "latexstring(raw\"\\ds \\frac{1}{1-1/(2n)^2} =\", latex(sol))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### ベータ函数の極限でガンマ函数が得らえることを用いWallisの公式の証明\n", "\n", "**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": [ "#### Wallisの公式からGauss積分の値を得る方法\n", "\n", "**問題(Wallisの公式からGauss積分の値を方法):** 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", "#### Legendre's duplication formula を用いた 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, The Gamma Function\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ガンマ分布の中心極限定理とStirlingの公式\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", "$$\n", "\\begin{aligned}\n", "&\n", "\\log\\Gamma(x) =\n", "\\lim_{n\\to\\infty}\\left(\n", "\\log n! + x\\log n - \\log x - \\log(x+1) - \\cdots - \\log(x+n)\n", "\\right),\n", "\\\\ &\n", "(\\log\\Gamma(x))' =\n", "\\lim_{n\\to\\infty}\\left(\n", "\\log n - \\frac{1}{x} - \\frac{1}{x+1} - \\cdots - \\frac{1}{x+n}\n", "\\right),\n", "\\\\ &\n", "(\\log\\Gamma(x))'' =\n", "\\lim_{n\\to\\infty}\\left(\n", "\\frac{1}{x^2} + \\frac{1}{(x+1)^2} + \\cdots + \\frac{1}{(x+n)^2}\n", "\\right) = \\zeta(2,x).\n", "\\end{aligned}\n", "$$\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", "\n", "#### digamma, trigamma, polygamma 函数\n", "\n", "上のLerchの定理の証明中に出て来た**対数ガンマ函数** $\\log\\Gamma(x)$ の導函数達を**polygamma函数**と呼ぶ. 例えば, 以下の $\\psi(x)$, $\\psi'(x)$ はそれぞれ**digamma函数**, **trigamma函数**と呼ばれる:\n", "\n", "$$\n", "\\begin{aligned}\n", "&\n", "\\log\\Gamma(x) =\n", "\\lim_{n\\to\\infty}\\left(\n", "\\log n! + x\\log n - \\log x - \\log(x+1) - \\cdots - \\log(x+n)\n", "\\right),\n", "\\\\ &\n", "\\psi(x) = (\\log\\Gamma(x))' =\n", "\\lim_{n\\to\\infty}\\left(\n", "\\log n - \\frac{1}{x} - \\frac{1}{x+1} - \\cdots - \\frac{1}{x+n}\n", "\\right),\n", "\\\\ &\n", "\\psi'(x) = (\\log\\Gamma(x))'' =\n", "\\lim_{n\\to\\infty}\\left(\n", "\\frac{1}{x^2} + \\frac{1}{(x+1)^2} + \\cdots + \\frac{1}{(x+n)^2}\n", "\\right) = \\zeta(2,x).\n", "\\end{aligned}\n", "$$\n", "\n", "これらの式を見ればわかるように, $m\\geqq 3$ に対するtrigamma函数以降の $m$-th polygamma函数 $\\psi^{(m-2)}(x)=(\\log\\Gamma(x))^{(m-1)}$ はHurwitzのゼータ函数の特別な場合 $\\zeta(m-1, x)$ に一致する.\n", "\n", "調和級数 $1+1/2+\\cdots+1/n$ と $\\log n$ の差が $n\\to\\infty$ で収束し, その収束先はEuler定数と呼ばれ, $\\gamma=0.5772\\cdots$ と書かれる. その事実はdigamma函数を使うと\n", "\n", "$$\n", "-\\psi(1) = \\gamma = 0.5772\\cdots\n", "$$\n", "\n", "と表わされる. digamma函数の $-1$ 倍 $-\\psi(x)$ は一般化された調和級数 $1/x+1/(x+1)+\\cdots+1/(x+n)$ と $\\log n$ の差の極限に等しいので, digamma函数の $-1$ 倍はEuler定数の一般化であるとも考えられる.\n", "\n", "digamma函数 $\\psi(\\alpha)$ は\n", "\n", "$$\n", "\\psi(\\alpha) = \\frac{\\Gamma'(\\alpha)}{\\Gamma(\\alpha)} = \n", "\\frac{1}{\\Gamma(\\alpha)}\\int_0^\\infty e^{-x} x^{\\alpha-1}\\log x\\,dx\n", "$$\n", "\n", "とも書ける. これはパラメーター ($\\alpha$, $\\theta=1$) のガンマ分布に従う確率変数 $X$ の対数 $\\log X$ の期待値に等しい. 統計学におけるガンマ分布の取り扱いではdigamma函数などが必要になる. Lerchの定理の証明を通して, 対数ガンマ函数の導函数の様子がよくわかるという感覚を身に付けておくと, その感覚はガンマ分布の統計学でも役に立つことになる.\n", "\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": [ "#### Lerchの定理を用いたBinetの公式の証明\n", "\n", "上でLerchの定理を示した.\n", "\n", "**Lerchの定理:** $\\ds \\zeta_s(0,x) = \\log\\frac{\\Gamma(x)}{\\sqrt{2\\pi}}$. $\\QED$\n", "\n", "これを用いて次のBinetの公式を示そう. \n", "\n", "**Binetの公式:** $x>0$ のとき, \n", "\n", "$$\n", "\\log\\Gamma(x+1) = x\\log x - x +\\frac{1}{2}\\log x + \\log\\sqrt{2\\pi} + \\varphi(x).\n", "$$\n", "\n", "ここで\n", "\n", "$$\n", "\\varphi(x) = \\int_0^\\infty \\left(\\frac{1}{e^t-1}-\\frac{1}{t}+\\frac{1}{2}\\right) e^{-xt}t^{-1}\\,dt.\n", "\\qquad \\QED\n", "$$\n", "\n", "**証明:** $x>0$ と仮定する.\n", "\n", "$\\real s > 1$ のとき, $a>0$ について $\\ds \\frac{1}{a^s} =\\frac{1}{\\Gamma(s)}\\int_0^\\infty e^{-at}t^{s-1}\\,dt$ より,\n", "\n", "$$\n", "\\begin{aligned}\n", "\\zeta(s,x) &= \\sum_{k=0}^\\infty \\frac{1}{(x+k)^s} =\n", "\\frac{1}{\\Gamma(s)}\\sum_{k=0}^\\infty \\int_0^\\infty e^{-(x+k)t} t^{s-1}\\,dt =\n", "\\frac{1}{\\Gamma(s)}\\int_0^\\infty \\frac{e^{-xt}}{1-e^{-t}} t^{s-1}\\,dt.\n", "\\end{aligned}\n", "$$\n", "\n", "ゆえに\n", "\n", "$$\n", "\\zeta(s,x+1) = \\frac{1}{\\Gamma(s)}\\int_0^\\infty \\frac{e^{-xt}}{e^t-1}t^{s-1}\\,dt.\n", "$$\n", "\n", "$F(s,x)$ を\n", "\n", "$$\n", "F(s,x) = \\frac{1}{\\Gamma(s)} \\int_0^\\infty \\left(\\frac{1}{e^t-1}-\\frac{1}{t}+\\frac{1}{2}\\right) e^{-xt} t^{s-1}\\,dt\n", "$$\n", "\n", "と定めると, \n", "\n", "$$\n", "\\frac{1}{e^t-1}-\\frac{1}{t}+\\frac{1}{2} = O(t) \\quad (t\\to 0)\n", "$$\n", "\n", "なので, $F(s,x)$ の定義式は $\\real s > -1$ で収束している. $\\real s > 1$ のとき\n", "\n", "$$\n", "\\begin{aligned}\n", "&\n", "\\frac{1}{\\Gamma(s)}\\int_0^\\infty \\frac{1}{t} e^{-xt} t^{s-1}\\,dt = \n", "\\frac{\\Gamma(s-1)}{\\Gamma(s)}\\frac{1}{x^{s-1}} =\n", "\\frac{x^{1-s}}{s-1}, \n", "\\\\ &\n", "\\frac{1}{\\Gamma(s)}\\int_0^\\infty \\left(-\\frac{1}{2}\\right) e^{-xt} t^{s-1}\\,dt = -\\frac{x^{-s}}{2}\n", "\\end{aligned}\n", "$$\n", "\n", "であるから, \n", "\n", "$$\n", "\\zeta(s,x+1) = \\frac{x^{1-s}}{s-1} - \\frac{x^{-s}}{2} + F(s,x).\n", "$$\n", "\n", "この等式は $\\real s > -1$ でも解析接続によって成立している. そこで以下では $\\real s > -1$ と仮定する.\n", "\n", "$\\ds\\lim_{s\\to 0}\\frac{1}{\\Gamma(s)} = 0$ より, $F(0,x)=0$ となるので, \n", "\n", "$$\n", "\\begin{aligned}\n", "&\n", "\\left.\\frac{\\d}{\\d s}\\right|_{s=0} \\frac{x^{1-s}}{s-1} =\n", "\\left.\\left(\\frac{-x^{1-s}\\log x}{s-1} - \\frac{x^{1-s}}{(s-1)^2}\\right)\\right|_{s=0} =\n", "x\\log x - x,\n", "\\\\ &\n", "\\left.\\frac{\\d}{\\d s}\\right|_{s=0} \\left(-\\frac{x^{-s}}{2}\\right) = \n", "\\left.\\left(\\frac{x^{-s}\\log x}{2}\\right)\\right|_{s=0} =\n", "\\frac{1}{2}\\log x,\n", "\\\\ &\n", "\\left.\\frac{\\d}{\\d s}\\right|_{s=0}F(s,x) = \n", "\\lim_{s\\to 0}\\frac{F(s,x)}{s} =\n", "\\lim_{s\\to 0}\\frac{1}{\\Gamma(s+1)}\n", "\\int_0^\\infty \\left(\\frac{1}{e^t-1}-\\frac{1}{t}+\\frac{1}{2}\\right) e^{-xt} t^{s-1}\\,dt \n", "\\\\ & \\qquad =\n", "\\int_0^\\infty \\left(\\frac{1}{e^t-1}-\\frac{1}{t}+\\frac{1}{2}\\right) e^{-xt}t^{-1}\\,dt =\n", "\\varphi(x).\n", "\\end{aligned}\n", "$$\n", "\n", "ゆえに,\n", "\n", "$$\n", "\\zeta_s(0, x+1) = x\\log x - x +\\frac{1}{2}\\log x + \\varphi(x).\n", "$$\n", "\n", "したがって, Lerchの定理より,\n", "\n", "$$\n", "\\log\\Gamma(x+1) = \\zeta_s(0,x+1) + \\log\\sqrt{2\\pi} =\n", "x\\log x - x +\\frac{1}{2}\\log x + \\log\\sqrt{2\\pi} + \\varphi(x).\n", "$$\n", "\n", "これで, Lerchの定理からBinetの公式が導かれることがわかった. $\\QED$\n", "\n", "Binetの公式は本質的にHurwitzのゼータ函数 $\\zeta(s,x)$ の $s$ に関する偏微分係数 $\\zeta_s(0,x)$ に関する公式であるとみなせる. Binetの公式の右辺の積分表示はHurwitzのゼータ函数の積分表示式から得られる." ] }, { "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, Kullback-Leibler情報量とSanovの定理\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": 19, "metadata": {}, "outputs": [], "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, g.(n,x), label=\"Gaussian\")\n", " plot!(x, f.(n,x), label=\"approx.\", ls=:dash)\n", " push!(PP, P)\n", "end" ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [ { "data": { "image/png": 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}, { "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, 11 Kullback-Leibler情報量\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": 22, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "10-element Vector{Tuple{Int64, Int64, Float64, Float64}}:\n", " (1, 1, 0.9221370088957891, -0.07786299110421091)\n", " (2, 2, 1.9190043514889832, -0.04049782425550841)\n", " (3, 6, 5.836209591345864, -0.027298401442355957)\n", " (4, 24, 23.50617513289329, -0.02057603612944625)\n", " (5, 120, 118.01916795759008, -0.01650693368674927)\n", " (6, 720, 710.0781846421849, -0.013780299108076544)\n", " (7, 5040, 4980.395831612461, -0.01182622388641652)\n", " (8, 40320, 39902.3954526567, -0.010357255638474672)\n", " (9, 362880, 359536.87284194824, -0.00921276223008094)\n", " (10, 3628800, 3.5986956187410364e6, -0.008295960443938433)" ] }, "execution_count": 22, "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": 23, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "10-element Vector{Tuple{Int64, Int64, Float64, Float64}}:\n", " (1, 1, 0.9989817596371048, -0.0010182403628952175)\n", " (2, 2, 1.9989628661343577, -0.0005185669328211517)\n", " (3, 6, 5.998326524438804, -0.000278912593532632)\n", " (4, 24, 23.995887114828566, -0.0001713702154764185)\n", " (5, 120, 119.98615409021657, -0.00011538258152854475)\n", " (6, 720, 719.9403816511041, -8.280326235543534e-5)\n", " (7, 5040, 5039.686258179276, -6.225036125484529e-5)\n", " (8, 40320, 40318.04540528855, -4.847705137533964e-5)\n", " (9, 362880, 362865.9179608552, -3.880632480379731e-5)\n", " (10, 3628800, 3.6286847488972116e6, -3.1760114304502096e-5)" ] }, "execution_count": 23, "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, 単純ランダムウォークの逆正弦法則 (手描きのノートのPDF)\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": 24, "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": 24, "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. PDF\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の方法の解説\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": [ "#### Laplaceの方法によるStirlingの公式の導出\n", "\n", "**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, 一般化されたLaplaceの方法\n", "\n", "を参照せよ. 一般化されたLaplaceの方法は\n", "\n", "* 渡辺澄夫, ベイズ統計の理論と方法, 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. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**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." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**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, $0 < e^{-nf_0(x)}0$ 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. $\\QED$" ] } ], "metadata": { "@webio": { "lastCommId": null, "lastKernelId": null }, "_draft": { "nbviewer_url": "https://gist.github.com/8e4e14edc28f77b6669165d1af460c30" }, "gist": { "data": { "description": "10 Gauss, Gamma, Beta.ipynb", "public": true }, "id": "8e4e14edc28f77b6669165d1af460c30" }, "jupytext": { "cell_metadata_json": true, "formats": "ipynb,md" }, "kernelspec": { "display_name": "Julia 1.9.3", "language": "julia", "name": "julia-1.9" }, "language_info": { "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", "version": "1.9.3" }, "toc": { "base_numbering": 1, "nav_menu": {}, "number_sections": true, "sideBar": true, "skip_h1_title": true, "title_cell": "目次", "title_sidebar": "目次", "toc_cell": true, "toc_position": {}, "toc_section_display": true, "toc_window_display": false } }, "nbformat": 4, "nbformat_minor": 4 }