{ "cells": [ { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# 高校数学の話題\n", "\n", "* 黒木玄 (Gen Kuroki)\n", "* Copyright (c) 2018, 2019, 2020, 2021, 2022, 2023 Gen Kuroki\n", "* MIT License\n", "* 更新: 2018-08-15～2019-09-24, 2020-08-27～2020-08-30, 2021-08-30～2021-09-02, 2022-08-31, 2023-05-29, 2023-09-07～2023-09-27\n", "\n", "このノートでは高校の数学の教科書にあるような話題を扱い, その数学的背景について解説する.\n", "\n", "タイポや自明な誤りは自分で訂正して読むこと. 本質的な誤りがあれば著者に教えて欲しい.\n", "\n", "* Copyright 2018, 2019, 2020, 2021, 2022, 2023 Gen Kuroki\n", "* License: MIT https://opensource.org/licenses/MIT\n", "* Repository: https://github.com/genkuroki/HighSchoolMath\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", "* 高校数学の話題 HTML版\n", "\n", "* 高校数学の話題 PDF版\n", "\n", "このノートの想定読者は大学である程度を数学を学んだで人で高校で習った数学について見直したい人達である. \n", "\n", "このファイルはJulia言語カーネルの Jupyter notebook である. 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\n", "\n", "を参照せよ. このファイル中のJulia言語のコードを理解できれば, Julia言語からSymPyを用いた数式処理や数値計算の結果のプロットの仕方を学ぶことができる.\n", "\n", "$\n", "\\newcommand\\eps{\\varepsilon}\n", "\\newcommand\\ds{\\displaystyle}\n", "\\newcommand\\Z{{\\mathbb Z}}\n", "\\newcommand\\R{{\\mathbb R}}\n", "\\newcommand\\C{{\\mathbb C}}\n", "\\newcommand\\T{{\\mathbb T}}\n", "\\newcommand\\Q{{\\mathbb Q}}\n", "\\newcommand\\QED{\\text{□}}\n", "\\newcommand\\root{\\sqrt}\n", "\\newcommand\\bra{\\langle}\n", "\\newcommand\\ket{\\rangle}\n", "\\newcommand\\d{\\partial}\n", "\\newcommand\\sech{\\operatorname{sech}}\n", "\\newcommand\\cosec{\\operatorname{cosec}}\n", "\\newcommand\\sign{\\operatorname{sign}}\n", "\\newcommand\\sinc{\\operatorname{sinc}}\n", "\\newcommand\\arctanh{\\operatorname{arctanh}}\n", "\\newcommand\\sn{\\operatorname{sn}}\n", "\\newcommand\\cn{\\operatorname{cn}}\n", "\\newcommand\\cd{\\operatorname{cd}}\n", "\\newcommand\\dn{\\operatorname{dn}}\n", "\\newcommand\\real{\\operatorname{Re}}\n", "\\newcommand\\imag{\\operatorname{Im}}\n", "\\newcommand\\Ker{\\operatorname{Ker}}\n", "\\newcommand\\Im{\\operatorname{Im}}\n", "\\newcommand\\Li{\\operatorname{Li}}\n", "\\newcommand\\np[1]{:\\!#1\\!:}\n", "\\newcommand\\PROD{\\mathop{\\coprod\\kern-1.35em\\prod}}\n", "\\newcommand{\\stirlingsecond}[2]{\\genfrac{\\lbrace}{\\rbrace}{0pt}{}{#1}{#2}}\n", "\\newcommand{\\nset}[1]{\\{1,2,\\ldots,#1\\}}\n", "$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" }, "toc": true }, "source": [ "

目次

\n", "
" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [], "source": [ "using Logging; disable_logging(Logging.Warn)\n", "using Printf\n", "using Base64\n", "\n", "using Plots\n", "pythonplot(fmt=:png)\n", "\n", "showimg(mime, fn; scale=\"\") = open(fn) do f\n", " base64 = base64encode(f)\n", " option = ifelse(scale == \"\", \"\", \"\"\" width=\"$scale\" \"\"\")\n", " display(\"text/html\", \"\"\"\"\"\")\n", "end\n", "\n", "using SymPy\n", "using LaTeXStrings\n", "using SpecialFunctions\n", "using QuadGK\n", "using Elliptic.Jacobi: cd, sn\n", "using LinearAlgebra: det" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "# Override the Base.show definition of SymPy.jl:\n", "# https://github.com/JuliaPy/SymPy.jl/blob/29c5bfd1d10ac53014fa7fef468bc8deccadc2fc/src/types.jl#L87-L105\n", "\n", "@eval SymPy function Base.show(io::IO, ::MIME\"text/latex\", x::SymbolicObject)\n", " print(io, as_markdown(\"\\\\displaystyle \" * sympy.latex(x, mode=\"plain\", fold_short_frac=false)))\n", "end\n", "@eval SymPy function Base.show(io::IO, ::MIME\"text/latex\", x::AbstractArray{Sym})\n", " function toeqnarray(x::Vector{Sym})\n", " a = join([\"\\\\displaystyle \" * sympy.latex(x[i]) for i in 1:length(x)], \"\\\\\\\\\")\n", " \"\"\"\\\\left[ \\\\begin{array}{r}$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": { "slideshow": { "slide_type": "slide" } }, "source": [ "## 三角函数の加法定理\n", "\n", "**三角函数の加法定理:**\n", "\n", "$$\n", "\\begin{aligned}\n", "&\n", "\\cos(x+y) = \\cos x\\;\\cos y - \\sin x\\;\\sin y, \n", "\\\\ &\n", "\\sin(x+y) = \\cos x\\;\\sin y + \\sin x\\;\\cos y.\n", "\\qquad \\QED\n", "\\end{aligned}\n", "$$\n", "\n", "この公式を**認めて**使えば, 三角函数に関する他の多くの公式が導かれることは知っているだろう. だからよく\n", "\n", ">三角函数の加法定理だけは覚えておいて, 他の公式はそれから導けばよい.\n", "\n", "というような教え方がされている場合がある. しかし, この教え方は数学の理解という観点からはひどく中途半端である. なぜならば, 三角函数の加法定理自体がそう難しくない結果だからである. しかもその本質は中学校レベルの幾何の問題に過ぎない.\n", "\n", "**三角函数の加法定理は簡単に導けるので, 三角函数の加法定理さえ覚える必要はない.**" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### 三角函数の加法定理の導出は易しい\n", "\n", "以下の図を見て欲しい." ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "text/html": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "showimg(\"image/jpeg\", \"images/trigonometric1.jpg\", scale=\"40%\")" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "-" } }, "source": [ "縦の青の点線が黒線と重なっていることを嫌うなら次のように図を描けばよい." ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "text/html": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "showimg(\"image/jpeg\", \"images/trigonometric2.jpg\", scale=\"40%\")" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### 三角函数の加法定理の導出は中学校レベル\n", "\n", "三角函数 $\\cos\\theta$, $\\sin\\theta$ の正式な定義のためには, まず弧度法の意味での角度を定義し, 弧度法の意味ので角度の函数としてそれらを定義しなければいけなくなる. 角度の測り方を固定しない場合には $\\cos(a\\theta)$, $\\sin(a\\theta)$ のように角度の測り方の不定性によって角度に定数倍 ($a$ 倍)の違いが生じる. \n", "\n", "しかし, $\\cos\\theta$, $\\sin\\theta$ の代わりに, $c(\\theta)=\\cos(a\\theta)$, $s(\\theta)=\\sin(a\\theta)$ を使っても, 三角函数の加法定理の形は変わらない:\n", "\n", "$$\n", "c(x+y) = c(x)c(y)-s(x)s(y), \\quad s(x+y)=c(x)s(y)+s(x)c(y).\n", "$$\n", "\n", "このことから, 三角函数の加法定理を理解するためには弧度法による角度の定義を知っている必要がないことがわかる.\n", "\n", "以下の図の問題を見て欲しい. それは実質的に三角函数の加法定理を示せという内容の問題である. そのことから, $\\cos$, $\\sin$ という記号を使わずに, 三角函数の加法定理と同等のことを述べることができることがわかる. そして, その問題の解答は完全に中学校数学の範囲内の議論で可能である. " ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "text/html": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "showimg(\"image/jpeg\", \"images/trigonometric3.jpg\", scale=\"80%\")" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### 三角函数の加法定理は複数の方法で得られる\n", "\n", "自力で何も知らない状態から三角函数の加法定理を証明しようとすれば, 図の描き方に複数の選択肢があることに気付く. 実際にやってみればわかるようにどのように図を描いても, 結果的に三角函数の加法定理が得られる. 要するに, 三角函数の加法定理の証明のためには, 知らなければできそうもないテクニカルな議論をする必要はなく, どのように図を描いても証明できる. ああやっても証明できるし, こうやっても証明できる. そのようなことに気付けば, 三角函数の加法定理は真に易しい結果であることを納得できるはずである." ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "text/html": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "showimg(\"image/jpeg\", \"images/trigonometric4.jpg\", scale=\"80%\")" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "-" } }, "source": [ "$\\sin$ の倍角の公式はこれの右上の図を使うと容易に証明可能である. 右上の図で $\\alpha=\\beta$ のとき $a=1$ となるので, \n", "\n", "$$\\sin(2\\alpha)=\\cos\\alpha\\,(\\sin\\alpha + 1 \\sin\\alpha)=2\\cos\\alpha\\sin\\alpha.$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 三角函数の加法定理と内積の関係\n", "\n", "三角函数の加法定理:\n", "\n", "$$\n", "\\begin{aligned}\n", "&\n", "\\begin{cases}\n", "\\cos(\\alpha+\\beta) = \\cos\\alpha\\,\\cos\\beta - \\sin\\alpha\\,\\sin\\beta, \\\\\n", "\\cos(\\alpha-\\beta) = \\cos\\alpha\\,\\cos\\beta + \\sin\\alpha\\,\\sin\\beta, \\\\\n", "\\end{cases}\n", "\\\\ &\n", "\\begin{cases}\n", "\\sin(\\alpha+\\beta) = \\sin\\alpha\\,\\cos\\beta + \\cos\\alpha\\,\\sin\\beta, \\\\\n", "\\sin(\\alpha-\\beta) = \\sin\\alpha\\,\\cos\\beta - \\cos\\alpha\\,\\sin\\beta. \\\\\n", "\\end{cases}\n", "\\end{aligned}\n", "$$\n", "\n", "これらを成分が極座標された2つの2次元ベクトル\n", "\n", "$$\n", "\\vec{a} = \n", "\\begin{bmatrix}\n", "a\\\\\n", "c\\\\\n", "\\end{bmatrix} =\n", "\\begin{bmatrix}\n", "\\|\\vec{a}\\|\\cos\\alpha \\\\\n", "\\|\\vec{a}\\|\\sin\\alpha \\\\\n", "\\end{bmatrix},\n", "\\quad\n", "\\vec{b} =\n", "\\begin{bmatrix}\n", "b\\\\\n", "d\\\\\n", "\\end{bmatrix} =\n", "\\begin{bmatrix}\n", "\\|\\vec{b}\\|\\cos\\beta \\\\\n", "\\|\\vec{b}\\|\\sin\\beta \\\\\n", "\\end{bmatrix}\n", "$$\n", "\n", "に適用してみよう. これらの対応する成分の積の和を計算すると $\\cos$ の加法定理より,\n", "\n", "$$\n", "ab+cd = \n", "\\|\\vec{a}\\|\\|\\vec{b}\\|(\\cos\\alpha\\,\\cos\\beta+\\sin\\alpha\\,\\sin\\beta) =\n", "\\|\\vec{a}\\|\\|\\vec{b}\\|\\cos(\\beta-\\alpha)\n", "$$\n", "\n", "となる. これで, 2つの2次元ベクトルの対応する成分の和は2つのベクトルのあいだの角度の $\\cos$ の $\\|\\vec{a}\\|\\|\\vec{b}\\|$ 倍になることがわかった. 我々はこれを「内積」と呼ぶのであった.\n", "\n", "次に $ad-bc$ を計算してみよう:\n", "\n", "$$\n", "ad-bc = \n", "\\|\\vec{a}\\|\\|\\vec{b}\\|(\\cos\\alpha\\,\\sin\\beta-\\sin\\alpha\\,\\cos\\beta) =\n", "\\|\\vec{a}\\|\\|\\vec{b}\\|\\sin(\\beta-\\alpha).\n", "$$\n", "\n", "これの絶対値は2つのベクトルを辺とする平行四辺形の面積である. 我々は $ad-bc$ を行列\n", "\n", "$$\n", "\\begin{bmatrix}\n", "a & b \\\\\n", "c & d \\\\\n", "\\end{bmatrix}\n", "$$\n", "\n", "の行列式と呼んでいるのであった. $2\\times2$ の行列式は平行四辺形の面積の $\\pm1$ 倍という幾何学的意味を持っている. 2つの2次元ベクトルの**外積**を $ad-bc$ で定義することもできる.\n", "\n", "このように, 三角函数の加法定理は2次元ベクトルの内積や $2\\times 2$ の行列式(もしくは2次元ベクトルの外積)の幾何学的意味を記述している公式ともみなされる." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## 3次方程式と4次方程式の解法\n", "\n", "高校数学レベルでの3次方程式と4次方程式の解法を解説する." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "-" } }, "source": [ "### ある3次式の因数分解から3次方程式の解法へ\n", "\n", "高校で次の因数分解の公式を習う:\n", "\n", "$$\n", "x^3+y^3+z^3 - 3xyz = (x+y+z)(x^2+y^2+z^2-xy-xz-yz).\n", "$$\n", "\n", "1の原始3乗根を $\\omega$ と書く: $\\omega^2+\\omega+1=0$, \n", "\n", "$$\n", "\\omega = e^{\\pm 2\\pi i/3} = \\frac{-1\\pm\\sqrt{3}\\;i}{2}.\n", "$$\n", "\n", "以下では, $\\omega^2+\\omega+1=0$ とそれから導かれる $\\omega^3=1$, $\\omega\\ne 1$ のみを使う.\n", "\n", "1の原始3乗根 $\\omega$ を使うと上の因数分解の公式は\n", "\n", "$$\n", "x^3+y^3+z^3-3xyz = (x+y+z)(x+\\omega y+\\omega^2 z)(x+\\omega^2 y+\\omega z)\n", "$$\n", "\n", "と書き直される. 最初の因数分解の公式の右辺の $-xy$, $-xz$, $-yz$ の係数 $-1$ は $\\omega^2+\\omega=-1$ によって再現される.\n", "\n", "さらに, $p = yz$, $q=y^3+z^3$ とおくと, 上の因数分解の公式は\n", "\n", "$$\n", "x^3 -3px + q = (x+y+z)(x+\\omega y+\\omega^2 z)(x+\\omega^2 y+\\omega z)\n", "$$\n", "\n", "と書き直される. この公式を使うと3次方程式の解の公式を作れる.\n", "\n", "任意に与えられた $p$, $q$ に対して, $p=yz$, $q=y^3+z^3$ を満たす $y,z$ は以下のようにして求めることができる. $y^3 z^3=p^3$ と $y^3+z^3=q$ より\n", "\n", "$$\n", "\\lambda^2 - q\\lambda + p^3 = (\\lambda - y^3)(\\lambda - z^3).\n", "$$\n", "\n", "ゆえに, 必要ならば $y,z$ の立場を交換すれば\n", "\n", "$$\n", "y^3 = \\frac{q + \\sqrt{q^2-4p^3}}{2}, \\quad z^3 = \\frac{q-\\sqrt{q^2-4p^3}}{2}\n", "$$\n", "\n", "が成立している. 右辺の3乗根を取れば $y,z$ も求まる:\n", "\n", "$$\n", "y = \\sqrt[3]{\\frac{q + \\sqrt{q^2-4p^3}}{2}}, \\quad z = \\sqrt[3]{\\frac{q-\\sqrt{q^2-4p^3}}{2}}.\n", "$$\n", "\n", "ただし, $y$ と $z$ は $yz=p$ を満たすように取る. (すなわち, $y$ が得られたときに $z$ を $z=p/y$ とおけばよい.)\n", "\n", "上の $x^3 -3px + q$ の因数分解の公式より, $x^3-3px+q=0$ の解はこの $y,z$ を使って,\n", "\n", "$$\n", "x = -y-z,\\ -\\omega y-\\omega^2 z,\\ -\\omega^2 y-\\omega z. \n", "$$\n", "\n", "と表わされる. これは本質的に所謂**Cardanoの公式**(カルダノの公式)である." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "-" } }, "source": [ "**問題:** 以上の計算を確認し, さらに解の公式を実際に作ってみよ. $\\QED$\n", "\n", "解答略." ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "text/latex": [ "$\\ds ω =\\frac{i \\left(\\sqrt{3} + i\\right)}{2}$" ], "text/plain": [ "L\"$\\ds ω =\\frac{i \\left(\\sqrt{3} + i\\right)}{2}$\"" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# 1の原始3乗根\n", "\n", "x, y, z = symbols(\"x y z\")\n", "ω₃ = factor((-1+√Sym(-3))/2)\n", "latexstring(raw\"\\ds ω =\", sympy.latex(ω₃))" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "text/latex": [ "$\\left(x + y + z\\right) \\left(x + y ω + z ω^{2}\\right) \\left(x + y ω^{2} + z ω\\right)=x^{3} - 3 x y z + y^{3} + z^{3}$" ], "text/plain": [ "L\"$\\left(x + y + z\\right) \\left(x + y ω + z ω^{2}\\right) \\left(x + y ω^{2} + z ω\\right)=x^{3} - 3 x y z + y^{3} + z^{3}$\"" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# 因数分解の公式の確認\n", "\n", "ω = symbols(\"ω\")\n", "f3_factored = (x+y+z)*(x+ω*y+ω^2*z)*(x+ω^2*y+ω*z)\n", "f3 = simplify(f3_factored(ω=>ω₃))\n", "latexstring(sympy.latex(f3_factored), \"=\", sympy.latex(f3))" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "text/latex": [ "$p=y z$" ], "text/plain": [ "L\"$p=y z$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$q=y^{3} + z^{3}$" ], "text/plain": [ "L\"$q=y^{3} + z^{3}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$- 3 p x + q + x^{3}=x^{3} - 3 x y z + y^{3} + z^{3}$" ], "text/plain": [ "L\"$- 3 p x + q + x^{3}=x^{3} - 3 x y z + y^{3} + z^{3}$\"" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "pp = y*z\n", "qq = y^3+z^3\n", "latexstring(\"p=\", sympy.latex(pp)) |> display\n", "latexstring(\"q=\", sympy.latex(qq)) |> display\n", "\n", "p, q = symbols(\"p q\")\n", "latexstring(sympy.latex(x^3 - 3p*x + q), \"=\", sympy.latex((x^3 - 3pp*x + qq)))" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "text/latex": [ "$\\left(m - y^{3}\\right) \\left(m - z^{3}\\right)=m^{2} - m y^{3} - m z^{3} + y^{3} z^{3}$" ], "text/plain": [ "L\"$\\left(m - y^{3}\\right) \\left(m - z^{3}\\right)=m^{2} - m y^{3} - m z^{3} + y^{3} z^{3}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\text{coefficients:}\\ \\left[\\begin{matrix}1\\\\- y^{3} - z^{3}\\\\y^{3} z^{3}\\end{matrix}\\right]$" ], "text/plain": [ "L\"$\\text{coefficients:}\\ \\left[\\begin{matrix}1\\\\- y^{3} - z^{3}\\\\y^{3} z^{3}\\end{matrix}\\right]$\"" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# (m-y^3)(m-z^3) の係数\n", "\n", "m = symbols(\"m\")\n", "f2_org = (m-y^3)*(m-z^3)\n", "f2 = expand((m-y^3)*(m-z^3))\n", "latexstring(sympy.latex(f2_org), \"=\", sympy.latex(f2)) |> display\n", "c2 = sympy.Poly(f2, m).all_coeffs()\n", "latexstring(raw\"\\text{coefficients:}\\ \", sympy.latex(c2))" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "text/latex": [ "$m^{2} - m q + p^{3}=0$" ], "text/plain": [ "L\"$m^{2} - m q + p^{3}=0$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\ds m = \\frac{q - \\sqrt{- 4 p^{3} + q^{2}}}{2},\\frac{q + \\sqrt{- 4 p^{3} + q^{2}}}{2}$" ], "text/plain": [ "L\"$\\ds m = \\frac{q - \\sqrt{- 4 p^{3} + q^{2}}}{2},\\frac{q + \\sqrt{- 4 p^{3} + q^{2}}}{2}$\"" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# 2次方程式の解\n", "\n", "p, q = symbols(\"p q\")\n", "equ = m^2-q*m+p^3\n", "sol = factor.(solve(equ, m))\n", "latexstring(sympy.latex(equ), \"=0\") |> display\n", "latexstring(raw\"\\ds m = \", sympy.latex(sol[1]), \",\", sympy.latex(sol[2]))" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "text/latex": [ "$x_{1}^3-3px_{1}+q=0$" ], "text/plain": [ "L\"$x_{1}^3-3px_{1}+q=0$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$x_{2}^3-3px_{2}+q=0$" ], "text/plain": [ "L\"$x_{2}^3-3px_{2}+q=0$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$x_{3}^3-3px_{3}+q=0$" ], "text/plain": [ "L\"$x_{3}^3-3px_{3}+q=0$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\ds x_1 = - \\frac{\\sqrt[3]{2} p}{\\sqrt[3]{q + \\sqrt{- 4 p^{3} + q^{2}}}} - \\frac{2^{\\frac{2}{3}} \\sqrt[3]{q + \\sqrt{- 4 p^{3} + q^{2}}}}{2}$" ], "text/plain": [ "L\"$\\ds x_1 = - \\frac{\\sqrt[3]{2} p}{\\sqrt[3]{q + \\sqrt{- 4 p^{3} + q^{2}}}} - \\frac{2^{\\frac{2}{3}} \\sqrt[3]{q + \\sqrt{- 4 p^{3} + q^{2}}}}{2}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\ds x_2 = - \\frac{\\sqrt[3]{2} p ω^{2}}{\\sqrt[3]{q + \\sqrt{- 4 p^{3} + q^{2}}}} - \\frac{2^{\\frac{2}{3}} ω \\sqrt[3]{q + \\sqrt{- 4 p^{3} + q^{2}}}}{2}$" ], "text/plain": [ "L\"$\\ds x_2 = - \\frac{\\sqrt[3]{2} p ω^{2}}{\\sqrt[3]{q + \\sqrt{- 4 p^{3} + q^{2}}}} - \\frac{2^{\\frac{2}{3}} ω \\sqrt[3]{q + \\sqrt{- 4 p^{3} + q^{2}}}}{2}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\ds x_3 = - \\frac{\\sqrt[3]{2} p ω}{\\sqrt[3]{q + \\sqrt{- 4 p^{3} + q^{2}}}} - \\frac{2^{\\frac{2}{3}} ω^{2} \\sqrt[3]{q + \\sqrt{- 4 p^{3} + q^{2}}}}{2}$" ], "text/plain": [ "L\"$\\ds x_3 = - \\frac{\\sqrt[3]{2} p ω}{\\sqrt[3]{q + \\sqrt{- 4 p^{3} + q^{2}}}} - \\frac{2^{\\frac{2}{3}} ω^{2} \\sqrt[3]{q + \\sqrt{- 4 p^{3} + q^{2}}}}{2}$\"" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# 3次方程式の解が得られていることの確認\n", "\n", "y = sol[2]^(Sym(1)/3)\n", "z = p/y\n", "X = -[\n", " y+z\n", " ω*y+ω^2*z\n", " ω^2*y+ω*z\n", "]\n", "equ = @. X^3 - 3p*X + q\n", "res = @.(simplify((f->f(ω=>ω₃))(equ)))\n", "for i in 1:3\n", " latexstring(\"x_{$i}^3-3px_{$i}+q=\", sympy.latex(res[i])) |> display\n", "end\n", "latexstring(raw\"\\ds x_1 = \", sympy.latex(X[1])) |> display\n", "latexstring(raw\"\\ds x_2 = \", sympy.latex(X[2])) |> display\n", "latexstring(raw\"\\ds x_3 = \", sympy.latex(X[3])) |> display" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "text/latex": [ "$x^3-3px+q=0$" ], "text/plain": [ "L\"$x^3-3px+q=0$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$x = x_1,x_2,x_3$" ], "text/plain": [ "L\"$x = x_1,x_2,x_3$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\ds x_1 = - \\frac{3 p}{\\sqrt[3]{\\frac{27 q}{2} + \\frac{\\sqrt{- 2916 p^{3} + 729 q^{2}}}{2}}} - \\frac{\\sqrt[3]{\\frac{27 q}{2} + \\frac{\\sqrt{- 2916 p^{3} + 729 q^{2}}}{2}}}{3}$" ], "text/plain": [ "L\"$\\ds x_1 = - \\frac{3 p}{\\sqrt[3]{\\frac{27 q}{2} + \\frac{\\sqrt{- 2916 p^{3} + 729 q^{2}}}{2}}} - \\frac{\\sqrt[3]{\\frac{27 q}{2} + \\frac{\\sqrt{- 2916 p^{3} + 729 q^{2}}}{2}}}{3}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\ds x_2 = - \\frac{3 p}{\\left(- \\frac{1}{2} - \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{27 q}{2} + \\frac{\\sqrt{- 2916 p^{3} + 729 q^{2}}}{2}}} - \\frac{\\left(- \\frac{1}{2} - \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{27 q}{2} + \\frac{\\sqrt{- 2916 p^{3} + 729 q^{2}}}{2}}}{3}$" ], "text/plain": [ "L\"$\\ds x_2 = - \\frac{3 p}{\\left(- \\frac{1}{2} - \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{27 q}{2} + \\frac{\\sqrt{- 2916 p^{3} + 729 q^{2}}}{2}}} - \\frac{\\left(- \\frac{1}{2} - \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{27 q}{2} + \\frac{\\sqrt{- 2916 p^{3} + 729 q^{2}}}{2}}}{3}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\ds x_3 = - \\frac{3 p}{\\left(- \\frac{1}{2} + \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{27 q}{2} + \\frac{\\sqrt{- 2916 p^{3} + 729 q^{2}}}{2}}} - \\frac{\\left(- \\frac{1}{2} + \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{27 q}{2} + \\frac{\\sqrt{- 2916 p^{3} + 729 q^{2}}}{2}}}{3}$" ], "text/plain": [ "L\"$\\ds x_3 = - \\frac{3 p}{\\left(- \\frac{1}{2} + \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{27 q}{2} + \\frac{\\sqrt{- 2916 p^{3} + 729 q^{2}}}{2}}} - \\frac{\\left(- \\frac{1}{2} + \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{27 q}{2} + \\frac{\\sqrt{- 2916 p^{3} + 729 q^{2}}}{2}}}{3}$\"" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# 3次方程式の解 (SymPyのsolve函数による直接計算)\n", "\n", "x, p, q = symbols(\"x p q\")\n", "equ = x^3-3p*x+q\n", "sol = solve(equ, x)\n", "latexstring(\"x^3-3px+q=0\") |> display\n", "display(L\"x = x_1,x_2,x_3\")\n", "latexstring(raw\"\\ds x_1 = \", sympy.latex(sol[1])) |> display\n", "latexstring(raw\"\\ds x_2 = \", sympy.latex(sol[2])) |> display\n", "latexstring(raw\"\\ds x_3 = \", sympy.latex(sol[3])) |> display" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### 巡回行列式\n", "\n", "1の原始3乗根 $\\omega$ を用いた因数分解の公式\n", "\n", "$$\n", "x^3+y^3+z^3-3xyz = (x+y+z)(x+\\omega y+\\omega^2 z)(x+\\omega^2 y+\\omega z)\n", "$$\n", "\n", "は行列式を用いて次のように書き直される:\n", "\n", "$$\n", "\\begin{vmatrix}\n", "x & y & z \\\\\n", "z & x & y \\\\\n", "y & z & x \\\\\n", "\\end{vmatrix} =\n", "\\prod_{k=0}^2 (x + \\omega^k y + \\omega^{2k} z).\n", "$$\n", "\n", "この公式は $1$ の原始 $n$ 乗根 $\\zeta$ を用いた公式\n", "\n", "$$\n", "\\begin{vmatrix}\n", "x_0 & x_1 & x_2 & \\ddots & x_{n-1} \\\\\n", "x_{n-1} & x_0 & x_1 & \\ddots & \\ddots \\\\\n", "\\ddots & x_{n-1} & x_0 & \\ddots & x_2 \\\\\n", "x_2 & \\ddots & \\ddots & \\ddots & x_1 \\\\\n", "x_1 & x_2 & \\ddots & x_{n-1} & x_0 \\\\\n", "\\end{vmatrix} =\n", "\\prod_{k=0}^{n-1} (x_0 + \\zeta^k x_1 + \\zeta^{2k} x_2 + \\cdots + \\zeta^{(n-1)k} x_{n-1}).\n", "$$\n", "\n", "に一般化される. この公式の証明は例えば\n", "\n", "* 佐武一郎, 線型代数学, 数学選書1, 裳華房\n", "\n", "の第II章の研究課題「1) 巡回行列式」に書いてある." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "### ある4次式の展開公式から4次方程式の解法へ\n", "\n", "$f$ を次のように定める:\n", "\n", "$$\n", "f = (w+x+y+z)(w+x-y-z)(w-x+y-z)(w-x-y+z).\n", "$$\n", "\n", "このとき,\n", "\n", "$$\n", "p = x^2+y^2+z^2, \\quad\n", "q = xyz, \\quad\n", "r = x^2y^2 + x^2z^2 + y^2z^2\n", "\\tag{$*$}\n", "$$\n", "\n", "とおくと,\n", "\n", "$$\n", "f = w^4 - 2pw^2 + 8qw + p^2-4r.\n", "$$\n", "\n", "ゆえに, もしも与えられた $p,q,r$ に対して, 条件($*$)を満たす $x,y,z$ を求めることができたならば, $w$ に関する4次方程式 $f=0$ は次のように解ける:\n", "\n", "$$\n", "w = -x-y-z, \\ -x+y+z, \\ x-y+z, \\ x+y-z.\n", "$$\n", "\n", "与えられた $p,q,r$ に対して条件 ($*$) を満たす $x,y,z$ を求めるためには, 条件\n", "\n", "$$\n", "x^2+y^2+z^2 = p, \\quad\n", "x^2y^2+x^2z^2+y^2z^2 = r, \\quad\n", "x^2y^2z^2 = q^2\n", "\\tag{$**$}\n", "$$\n", "\n", "を満たす $x^2,y^2,z^2$ を求め, それらの平方根を取ればよい. ただし, 平方根の取り方には $\\pm1$ 倍の不定性があることに注意せよ. 条件 $xyz=q$ より, $x,y,z$ のうち2つが決まれば残りは一意的に決まる.\n", "\n", "条件 ($**$) を満たす $x^2,y^2,z^2$ は3次方程式の解と係数の関係より, 次の3次方程式の解である:\n", "\n", "$$\n", "\\lambda^3 - p\\lambda^2 + r\\lambda - q^2 = 0.\n", "$$\n", "\n", "この3次方程式は $\\lambda = \\Lambda+p/3$ とおけば次の形になる:\n", "\n", "$$\n", "\\Lambda^3 - P\\Lambda + Q = 0, \\quad\n", "P = \\frac{p^2}{3} - r, \\quad\n", "Q = -\\frac{2p^3}{27} + \\frac{pr}{3} - q^2.\n", "$$\n", "\n", "この形の3次方程式は前節の結果を用いれば解ける.\n", "\n", "以上の方法は**Eulerの方法**と呼ばれているらしい(https://en.wikipedia.org/wiki/Quartic_function)." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "-" } }, "source": [ "**問題:** 以上の計算を確認し, さらに解の公式を実際に作ってみよ. $\\QED$\n", "\n", "解答略." ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "text/latex": [ "$\\left(w - x - y + z\\right) \\left(w - x + y - z\\right) \\left(w + x - y - z\\right) \\left(w + x + y + z\\right)$" ], "text/plain": [ "L\"$\\left(w - x - y + z\\right) \\left(w - x + y - z\\right) \\left(w + x - y - z\\right) \\left(w + x + y + z\\right)$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$=w^{4} - 2 w^{2} x^{2} - 2 w^{2} y^{2} - 2 w^{2} z^{2} + 8 w x y z + x^{4} - 2 x^{2} y^{2} - 2 x^{2} z^{2} + y^{4} - 2 y^{2} z^{2} + z^{4}$" ], "text/plain": [ "L\"$=w^{4} - 2 w^{2} x^{2} - 2 w^{2} y^{2} - 2 w^{2} z^{2} + 8 w x y z + x^{4} - 2 x^{2} y^{2} - 2 x^{2} z^{2} + y^{4} - 2 y^{2} z^{2} + z^{4}$\"" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# 展開結果の確認\n", "\n", "w, x, y, z = symbols(\"w x y z\")\n", "f4_org = (w+x+y+z)*(w+x-y-z)*(w-x+y-z)*(w-x-y+z)\n", "f4 = expand(f4_org)\n", "latexstring(sympy.latex(f4_org)) |> display\n", "latexstring(\"=\", sympy.latex(f4)) |> display" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "text/latex": [ "$\\text{coefficients:}\\ $" ], "text/plain": [ "L\"$\\text{coefficients:}\\ $\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\left[ \\begin{array}{r}\\displaystyle 1\\\\\\displaystyle 0\\\\\\displaystyle - 2 x^{2} - 2 y^{2} - 2 z^{2}\\\\\\displaystyle 8 x y z\\\\\\displaystyle x^{4} - 2 x^{2} y^{2} - 2 x^{2} z^{2} + y^{4} - 2 y^{2} z^{2} + z^{4}\\end{array} \\right]$\n" ], "text/plain": [ "5-element Vector{Sym}:\n", " 1\n", " 0\n", " -2*x^2 - 2*y^2 - 2*z^2\n", " 8*x*y*z\n", " x^4 - 2*x^2*y^2 - 2*x^2*z^2 + y^4 - 2*y^2*z^2 + z^4" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# 展開結果の 0,1,2,3,4 次の係数の確認.\n", "# 3次の係数は0になっている.\n", "\n", "display(L\"\\text{coefficients:}\\ \")\n", "c4 = sympy.Poly(f4, w).all_coeffs()" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "text/latex": [ "$p=x^{2} + y^{2} + z^{2}$" ], "text/plain": [ "L\"$p=x^{2} + y^{2} + z^{2}$\"" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p = c4[3]/(-2)\n", "latexstring(\"p=\", sympy.latex(p))" ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "text/latex": [ "$q=x y z$" ], "text/plain": [ "L\"$q=x y z$\"" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" } ], "source": [ "q = c4[4]/8\n", "latexstring(\"q=\", sympy.latex(q))" ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "simplify(c4[5] - (p ^ 2 - 4r)) = 0\n" ] }, { "data": { "text/latex": [ "$r=x^{2} y^{2} + x^{2} z^{2} + y^{2} z^{2}$" ], "text/plain": [ "L\"$r=x^{2} y^{2} + x^{2} z^{2} + y^{2} z^{2}$\"" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "r = x^2*y^2 + x^2*z^2 + y^2*z^2\n", "@show simplify(c4[5] - (p^2-4r))\n", "latexstring(\"r=\", sympy.latex(r))" ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle 0$\n" ], "text/plain": [ "0" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$w^4-2pw^2+8qw+(p^2-4r)$" ], "text/plain": [ "L\"$w^4-2pw^2+8qw+(p^2-4r)$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$=w^{4} - 2 w^{2} x^{2} - 2 w^{2} y^{2} - 2 w^{2} z^{2} + 8 w x y z + x^{4} - 2 x^{2} y^{2} - 2 x^{2} z^{2} + y^{4} - 2 y^{2} z^{2} + z^{4}$" ], "text/plain": [ "L\"$=w^{4} - 2 w^{2} x^{2} - 2 w^{2} y^{2} - 2 w^{2} z^{2} + 8 w x y z + x^{4} - 2 x^{2} y^{2} - 2 x^{2} z^{2} + y^{4} - 2 y^{2} z^{2} + z^{4}$\"" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# p,q,rを使った公式の確認\n", "\n", "ff4 = w^4 - 2p*w^2 + 8q*w + (p^2-4r)\n", "simplify(f4 - ff4) |> display\n", "latexstring(\"w^4-2pw^2+8qw+(p^2-4r)\") |> display\n", "latexstring(\"=\", sympy.latex(f4)) |> display" ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "text/latex": [ "$m^{3} - m^{2} \\left(x^{2} + y^{2} + z^{2}\\right) + m \\left(x^{2} y^{2} + x^{2} z^{2} + y^{2} z^{2}\\right) - x^{2} y^{2} z^{2}=\\left(m - x^{2}\\right) \\left(m - y^{2}\\right) \\left(m - z^{2}\\right)$" ], "text/plain": [ "L\"$m^{3} - m^{2} \\left(x^{2} + y^{2} + z^{2}\\right) + m \\left(x^{2} y^{2} + x^{2} z^{2} + y^{2} z^{2}\\right) - x^{2} y^{2} z^{2}=\\left(m - x^{2}\\right) \\left(m - y^{2}\\right) \\left(m - z^{2}\\right)$\"" ] }, "execution_count": 20, "metadata": {}, "output_type": "execute_result" } ], "source": [ "m = symbols(\"m\")\n", "equ = m^3 - p*m^2 + r*m - q^2\n", "sol = simplify(factor(equ))\n", "latexstring(sympy.latex(equ), \"=\", sympy.latex(sol))" ] }, { "cell_type": "code", "execution_count": 21, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle m^{3} - m^{2} p + m r - q^{2}$\n" ], "text/plain": [ " 3 2 2\n", "m - m *p + m*r - q " ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle M^{3} - \\frac{M p^{2}}{3} + M r - \\frac{2 p^{3}}{27} + \\frac{p r}{3} - q^{2}$\n" ], "text/plain": [ " 2 3 \n", " 3 M*p 2*p p*r 2\n", "M - ---- + M*r - ---- + --- - q \n", " 3 27 3 " ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\left[ \\begin{array}{r}\\displaystyle 1\\\\\\displaystyle 0\\\\\\displaystyle - \\frac{p^{2}}{3} + r\\\\\\displaystyle - \\frac{2 p^{3}}{27} + \\frac{p r}{3} - q^{2}\\end{array} \\right]$\n" ], "text/plain": [ "4-element Vector{Sym}:\n", " 1\n", " 0\n", " -p^2/3 + r\n", " -2*p^3/27 + p*r/3 - q^2" ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# 補助的な3次方程式の形の確認\n", "\n", "p, q, r, m, M = symbols(\"p q r m M\")\n", "(f3 = m^3 - p*m^2 + r*m - q^2) |> display\n", "(g3 = simplify(expand(f3(m => M+p/3)))) |> display\n", "c3 = sympy.Poly(g3, M).all_coeffs()" ] }, { "cell_type": "code", "execution_count": 22, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [], "source": [ "# w, p, q, r = symbols(\"w p q r\")\n", "# factor.(solve(w^4 - 2p*w^2 + 8q*w + (p^2-4r), w))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 4次方程式の解法で使える4次式の行列式表示\n", "\n", "前節で用いた $f$ は次のように表される:\n", "\n", "$$\n", "f = \\begin{vmatrix}\n", "w & x & y & z \\\\\n", "x & w & z & y \\\\\n", "y & z & w & x \\\\\n", "z & y & x & w \\\\\n", "\\end{vmatrix}.\n", "$$\n", "\n", "右辺の行列式はKleinの四元群 $\\{1, (1,2)(3,4), (1,3)(2,4), (1,4)(2,3)\\}\\lhd S_4$ (4次の置換群の非自明な正規部分群)の群行列式と呼ばれるものになっている.\n", "\n", "このノートで詳しく説明はできないが, 4次方程式が四則演算とべき根を取る操作だけでいつでも解ける理由は, 4次の置換群がKleinの四元群という非自明な正規部分群を持つからである. そのKleinの四元群由来の行列式を考えれば, 4次方程式が解けるわけである.\n", "\n", "__注意:__ 上の形の行列式は\n", "\n", "* 佐武一郎, 線型代数学, 数学選書1, 裳華房\n", "\n", "の第II章の研究課題の問1にある.\n", "\n", "この佐武一郎著『線型代数学』は数学的にかなり強力な本なので, 数学的教養を深めたい人は熟読する価値がある." ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\left[ \\begin{array}{rrrr}\\displaystyle w&\\displaystyle x&\\displaystyle y&\\displaystyle z\\\\\\displaystyle x&\\displaystyle w&\\displaystyle z&\\displaystyle y\\\\\\displaystyle y&\\displaystyle z&\\displaystyle w&\\displaystyle x\\\\\\displaystyle z&\\displaystyle y&\\displaystyle x&\\displaystyle w\\end{array}\\right]$\n" ], "text/plain": [ "4×4 Matrix{Sym}:\n", " w x y z\n", " x w z y\n", " y z w x\n", " z y x w" ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" } ], "source": [ "A = [\n", " w x y z\n", " x w z y\n", " y z w x\n", " z y x w\n", "]" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle w^{4} - 2 w^{2} x^{2} - 2 w^{2} y^{2} - 2 w^{2} z^{2} + 8 w x y z + x^{4} - 2 x^{2} y^{2} - 2 x^{2} z^{2} + y^{4} - 2 y^{2} z^{2} + z^{4}$\n" ], "text/plain": [ " 4 2 2 2 2 2 2 4 2 2 2 2 4 \n", "w - 2*w *x - 2*w *y - 2*w *z + 8*w*x*y*z + x - 2*x *y - 2*x *z + y - 2\n", "\n", " 2 2 4\n", "*y *z + z " ] }, "execution_count": 24, "metadata": {}, "output_type": "execute_result" } ], "source": [ "d4 = det(A)" ] }, { "cell_type": "code", "execution_count": 25, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "true" ] }, "execution_count": 25, "metadata": {}, "output_type": "execute_result" } ], "source": [ "d4 == f4" ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\left[ \\begin{array}{r}\\displaystyle 1\\\\\\displaystyle 0\\\\\\displaystyle - 2 x^{2} - 2 y^{2} - 2 z^{2}\\\\\\displaystyle 8 x y z\\\\\\displaystyle x^{4} - 2 x^{2} y^{2} - 2 x^{2} z^{2} + y^{4} - 2 y^{2} z^{2} + z^{4}\\end{array} \\right]$\n" ], "text/plain": [ "5-element Vector{Sym}:\n", " 1\n", " 0\n", " -2*x^2 - 2*y^2 - 2*z^2\n", " 8*x*y*z\n", " x^4 - 2*x^2*y^2 - 2*x^2*z^2 + y^4 - 2*y^2*z^2 + z^4" ] }, "execution_count": 26, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sympy.Poly(d4, w).all_coeffs()" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## べき乗和とベルヌイ多項式\n", "\n", "高校数学ではべき乗和\n", "\n", "$$\n", "\\begin{aligned}\n", "&\n", "1+2+\\cdots+n = \\frac{n(n+1)}{2}, \n", "\\\\ &\n", "1^2+2^2+\\cdots+n^2 = \\frac{n(n+1)(2n+1)}{6}, \n", "\\\\ &\n", "1^3+2^3+\\cdots+n^3 = \\frac{n^2(n+1)^2}{4}\n", "\\end{aligned}\n", "$$\n", "\n", "について習う. これらの公式の背景にベルヌイ多項式が控えていることを解説したい.\n", "\n", "ベルヌイ多項式については以下のツイッターにおける以下のスレッドも参照せよ:\n", "\n", "* https://twitter.com/genkuroki/status/1111938896844095488" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Bernoulli多項式\n", "\n", "**Bernoulli多項式**(ベルヌイ多項式) $B_k(x)$ が次のTaylor展開で定義される:\n", "\n", "$$\n", "\\frac{ze^{zx}}{e^z - 1} = \\sum_{k=0}^\\infty \\frac{B_k(x)}{k!}z^k.\n", "$$\n", "\n", "これの左辺をBernoulli多項式の**母函数**と呼ぶ.\n", "\n", "**Bernoulli数:** Bernoulli多項式の定数項 $B_k=B_k(0)$ は**Bernoulli数**と呼ばれている.\n", "\n", "$$\n", "\\begin{aligned}\n", "\\frac{ze^{zx}}{e^z - 1} &= \n", "\\frac{z}{e^z - 1} e^{xz} =\n", "\\sum_{i=0}^\\infty \\frac{B_i z^i}{i!} \\sum_{j=0}^\\infty\\frac{x^j z^j}{j!} \n", "\\\\ &=\n", "\\sum_{i,j=0}^\\infty \\frac{(i+j)!}{i!j!}B_i x^j \\frac{z^{i+j}}{(i+j)!} =\n", "\\sum_{k=0}^\\infty \\sum_{j=0}^k \\binom{k}{j} B_{k-j}x^j \\frac{z^k}{k!}\n", "\\end{aligned}\n", "$$\n", "\n", "より, Bernoulli多項式はBernoulli数によって次のように表わされることがわかる:\n", "\n", "$$\n", "B_k(x) = \n", "\\sum_{j=0}^k \\binom{k}{j} B_{k-j}x^j.\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "**Bernoulli多項式の微分:** Benoulli多項式の定義式の両辺を $x$ で偏微分すると\n", "\n", "$$\n", "\\frac{z^2 e^{zx}}{e^z - 1} = \n", "\\sum_{k=0}^\\infty \\frac{B_k'(x)}{k!}z^k\n", "$$\n", "\n", "となり, これの左辺は\n", "\n", "$$\n", "\\frac{z^2 e^{zx}}{e^z - 1} = \n", "\\sum_{m=0}^\\infty \\frac{B_m(x)}{m!}z^{m+1} =\n", "\\sum_{k=1}^\\infty \\frac{B_{k-1}(x)}{(k-1)!}z^k\n", "$$\n", "\n", "と書けるので, 上と比較して\n", "\n", "$$\n", "B_k'(x) = k B_{k-1}(x)\n", "$$\n", "\n", "が得られる. $B_0(x)=1$ なので $B_k(x)$ は最高次の係数が $1$ である $k$ 次の多項式になる." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "**Bernoulli多項式の積分:** 上の結果より,\n", "\n", "$$\n", "\\frac{d}{dx}\\frac{B_{k+1}(x)}{k+1}=B_k(x)\n", "$$\n", "\n", "であるから, \n", "\n", "$$\n", "\\int_a^b B_k(x)\\,dx = \\frac{B_{k+1}(b)-B_{k+1}(a)}{k+1}.\n", "$$\n", "\n", "これの母函数表示は次の通り:\n", "\n", "$$\n", "\\sum_{k=0}^\\infty \\int_a^b B_k(x)\\,dx\\;\\frac{z^k}{k!} =\n", "\\int_a^b \\frac{ze^{zx}}{e^z - 1}\\,dx =\n", "\\left[\\frac{e^{zx}}{e^z - 1}\\right]_{x=a}^{x=b} = \n", "\\frac{e^{zb}-e^{za}}{e^z - 1}.\n", "$$\n", "\n", "特に $a=0$, $b=1$ のとき, 右辺は $1$ になるので, \n", "\n", "$$\n", "\\int_0^1 B_k(x)\\,dx = \\delta_{k0}\n", "$$\n", "\n", "となることもわかる. $a=x$, $b=x+1$ の場合には\n", "\n", "$$\n", "\\sum_{k=0}^\\infty \\int_x^{x+1} B_k(y)\\,dy\\;\\frac{z^k}{k!} =\n", "\\frac{e^{z(x+1)}-e^{zx}}{e^z - 1} =\n", "e^{zx} =\n", "\\sum_{k=0}^\\infty x^k \\frac{z^k}{k!}\n", "$$\n", "\n", "となるので, \n", "\n", "$$\n", "\\int_x^{x+1} B_k(y)\\,dy = x^k.\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Bernoulli多項式とべき乗和の関係\n", "\n", "べき乗和 $S_k(n)$ が次のように定義される:\n", "\n", "$$\n", "S_k(n) = \\sum_{j=1}^n j^k = 1^k+2^k+\\cdots+n^k.\n", "$$\n", "\n", "べき乗和の母函数は次のように計算される:\n", "\n", "$$\n", "\\begin{aligned}\n", "\\sum_{k=0}^\\infty \\frac{S_k(n)}{k!}z^k &=\n", "\\sum_{k=0}^\\infty \\sum_{j=1}^n \\frac{j^k}{k!}z^k =\n", "\\sum_{j=1}^n \\sum_{k=0}^\\infty \\frac{j^k}{k!}z^k \n", "\\\\ &=\n", "\\sum_{j=1}^n e^{jz} =\n", "\\frac{e^{(n+1)z}-e^z}{e^z-1}.\n", "\\end{aligned}\n", "$$\n", "\n", "Bernoulli多項式の積分に関する結果より,\n", "\n", "$$\n", "\\begin{aligned}\n", "\\frac{e^{(n+1)z}-e^z}{e^z-1} &=\n", "\\sum_{k=0}^\\infty \\int_1^{n+1}B_k(x)\\,dx\\;\\frac{z^k}{k!} \n", "\\\\ &=\n", "\\sum_{k=0}^\\infty \\frac{B_{k+1}(n+1)-B_{k+1}(1)}{k+1} \\frac{z^k}{k!}.\n", "\\end{aligned}\n", "$$\n", "\n", "したがって, \n", "\n", "$$\n", "S_k(n) = \\int_1^{n+1} B_k(x)\\,dx = \\frac{B_{k+1}(n+1) - B_{k+1}(1)}{k+1}.\n", "$$\n", "\n", "特に $S_k(n)$ は $n$ について $k+1$ 次の多項式になり, 最高次の係数は $1/(k+1)$ になることがわかる.\n", "\n", "以上では母函数表示を経由して計算したが, \n", "\n", "$$\n", "\\int_x^{x+1} B_k(y)\\,dy = x^k, \\quad\n", "\\int B_k(x) dx = \\frac{B_{k+1}(x)}{k+1}\n", "$$\n", "\n", "を使えば\n", "\n", "$$\n", "\\begin{aligned}\n", "S_k(n) &= \\sum_{j=1}^n j^k =\n", "\\sum_{j=1}^n \\int_j^{j+1} B_k(x)\\,dx \\\\ &=\n", "\\int_1^{n+1} B_k(x)\\,dx =\n", "\\frac{B_{k+1}(n+1) - B_{k+1}(1)}{k+1}\n", "\\end{aligned}\n", "$$\n", "\n", "と同じ公式がより平易に得られる." ] }, { "cell_type": "code", "execution_count": 27, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "text/latex": [ "$B_{0}(x)=1$" ], "text/plain": [ "L\"$B_{0}(x)=1$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$B_{1}(x)=x - \\frac{1}{2}$" ], "text/plain": [ "L\"$B_{1}(x)=x - \\frac{1}{2}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$B_{2}(x)=x^{2} - x + \\frac{1}{6}$" ], "text/plain": [ "L\"$B_{2}(x)=x^{2} - x + \\frac{1}{6}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$B_{3}(x)=x^{3} - \\frac{3 x^{2}}{2} + \\frac{x}{2}$" ], "text/plain": [ "L\"$B_{3}(x)=x^{3} - \\frac{3 x^{2}}{2} + \\frac{x}{2}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$B_{4}(x)=x^{4} - 2 x^{3} + x^{2} - \\frac{1}{30}$" ], "text/plain": [ "L\"$B_{4}(x)=x^{4} - 2 x^{3} + x^{2} - \\frac{1}{30}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$B_{5}(x)=x^{5} - \\frac{5 x^{4}}{2} + \\frac{5 x^{3}}{3} - \\frac{x}{6}$" ], "text/plain": [ "L\"$B_{5}(x)=x^{5} - \\frac{5 x^{4}}{2} + \\frac{5 x^{3}}{3} - \\frac{x}{6}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$B_{6}(x)=x^{6} - 3 x^{5} + \\frac{5 x^{4}}{2} - \\frac{x^{2}}{2} + \\frac{1}{42}$" ], "text/plain": [ "L\"$B_{6}(x)=x^{6} - 3 x^{5} + \\frac{5 x^{4}}{2} - \\frac{x^{2}}{2} + \\frac{1}{42}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$B_{7}(x)=x^{7} - \\frac{7 x^{6}}{2} + \\frac{7 x^{5}}{2} - \\frac{7 x^{3}}{6} + \\frac{x}{6}$" ], "text/plain": [ "L\"$B_{7}(x)=x^{7} - \\frac{7 x^{6}}{2} + \\frac{7 x^{5}}{2} - \\frac{7 x^{3}}{6} + \\frac{x}{6}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$B_{8}(x)=x^{8} - 4 x^{7} + \\frac{14 x^{6}}{3} - \\frac{7 x^{4}}{3} + \\frac{2 x^{2}}{3} - \\frac{1}{30}$" ], "text/plain": [ "L\"$B_{8}(x)=x^{8} - 4 x^{7} + \\frac{14 x^{6}}{3} - \\frac{7 x^{4}}{3} + \\frac{2 x^{2}}{3} - \\frac{1}{30}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$B_{9}(x)=x^{9} - \\frac{9 x^{8}}{2} + 6 x^{7} - \\frac{21 x^{5}}{5} + 2 x^{3} - \\frac{3 x}{10}$" ], "text/plain": [ "L\"$B_{9}(x)=x^{9} - \\frac{9 x^{8}}{2} + 6 x^{7} - \\frac{21 x^{5}}{5} + 2 x^{3} - \\frac{3 x}{10}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$B_{10}(x)=x^{10} - 5 x^{9} + \\frac{15 x^{8}}{2} - 7 x^{6} + 5 x^{4} - \\frac{3 x^{2}}{2} + \\frac{5}{66}$" ], "text/plain": [ "L\"$B_{10}(x)=x^{10} - 5 x^{9} + \\frac{15 x^{8}}{2} - 7 x^{6} + 5 x^{4} - \\frac{3 x^{2}}{2} + \\frac{5}{66}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$B_{11}(x)=x^{11} - \\frac{11 x^{10}}{2} + \\frac{55 x^{9}}{6} - 11 x^{7} + 11 x^{5} - \\frac{11 x^{3}}{2} + \\frac{5 x}{6}$" ], "text/plain": [ "L\"$B_{11}(x)=x^{11} - \\frac{11 x^{10}}{2} + \\frac{55 x^{9}}{6} - 11 x^{7} + 11 x^{5} - \\frac{11 x^{3}}{2} + \\frac{5 x}{6}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$B_{12}(x)=x^{12} - 6 x^{11} + 11 x^{10} - \\frac{33 x^{8}}{2} + 22 x^{6} - \\frac{33 x^{4}}{2} + 5 x^{2} - \\frac{691}{2730}$" ], "text/plain": [ "L\"$B_{12}(x)=x^{12} - 6 x^{11} + 11 x^{10} - \\frac{33 x^{8}}{2} + 22 x^{6} - \\frac{33 x^{4}}{2} + 5 x^{2} - \\frac{691}{2730}$\"" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# ベルヌイ多項式のリスト k = 0,1,2,…,12\n", "\n", "B(k,x) = sympy.bernoulli(k,x)\n", "SS(k,x) = (B(k+1,x+1) - B(k+1,Sym(1)))/(k+1)\n", "x = symbols(\"x\")\n", "for k in 0:12\n", " latexstring(\"B_{$k}(x)=\", sympy.latex(B(k,x))) |> display\n", "end" ] }, { "cell_type": "code", "execution_count": 28, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "text/latex": [ "$S_{0}(x)=x$" ], "text/plain": [ "L\"$S_{0}(x)=x$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$S_{1}(x)=\\frac{x \\left(x + 1\\right)}{2}$" ], "text/plain": [ "L\"$S_{1}(x)=\\frac{x \\left(x + 1\\right)}{2}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$S_{2}(x)=\\frac{x \\left(x + 1\\right) \\left(2 x + 1\\right)}{6}$" ], "text/plain": [ "L\"$S_{2}(x)=\\frac{x \\left(x + 1\\right) \\left(2 x + 1\\right)}{6}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$S_{3}(x)=\\frac{x^{2} \\left(x + 1\\right)^{2}}{4}$" ], "text/plain": [ "L\"$S_{3}(x)=\\frac{x^{2} \\left(x + 1\\right)^{2}}{4}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$S_{4}(x)=\\frac{x \\left(x + 1\\right) \\left(2 x + 1\\right) \\left(3 x^{2} + 3 x - 1\\right)}{30}$" ], "text/plain": [ "L\"$S_{4}(x)=\\frac{x \\left(x + 1\\right) \\left(2 x + 1\\right) \\left(3 x^{2} + 3 x - 1\\right)}{30}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$S_{5}(x)=\\frac{x^{2} \\left(x + 1\\right)^{2} \\left(2 x^{2} + 2 x - 1\\right)}{12}$" ], "text/plain": [ "L\"$S_{5}(x)=\\frac{x^{2} \\left(x + 1\\right)^{2} \\left(2 x^{2} + 2 x - 1\\right)}{12}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$S_{6}(x)=\\frac{x \\left(x + 1\\right) \\left(2 x + 1\\right) \\left(3 x^{4} + 6 x^{3} - 3 x + 1\\right)}{42}$" ], "text/plain": [ "L\"$S_{6}(x)=\\frac{x \\left(x + 1\\right) \\left(2 x + 1\\right) \\left(3 x^{4} + 6 x^{3} - 3 x + 1\\right)}{42}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$S_{7}(x)=\\frac{x^{2} \\left(x + 1\\right)^{2} \\left(3 x^{4} + 6 x^{3} - x^{2} - 4 x + 2\\right)}{24}$" ], "text/plain": [ "L\"$S_{7}(x)=\\frac{x^{2} \\left(x + 1\\right)^{2} \\left(3 x^{4} + 6 x^{3} - x^{2} - 4 x + 2\\right)}{24}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$S_{8}(x)=\\frac{x \\left(x + 1\\right) \\left(2 x + 1\\right) \\left(5 x^{6} + 15 x^{5} + 5 x^{4} - 15 x^{3} - x^{2} + 9 x - 3\\right)}{90}$" ], "text/plain": [ "L\"$S_{8}(x)=\\frac{x \\left(x + 1\\right) \\left(2 x + 1\\right) \\left(5 x^{6} + 15 x^{5} + 5 x^{4} - 15 x^{3} - x^{2} + 9 x - 3\\right)}{90}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$S_{9}(x)=\\frac{x^{2} \\left(x + 1\\right)^{2} \\left(x^{2} + x - 1\\right) \\left(2 x^{4} + 4 x^{3} - x^{2} - 3 x + 3\\right)}{20}$" ], "text/plain": [ "L\"$S_{9}(x)=\\frac{x^{2} \\left(x + 1\\right)^{2} \\left(x^{2} + x - 1\\right) \\left(2 x^{4} + 4 x^{3} - x^{2} - 3 x + 3\\right)}{20}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$S_{10}(x)=\\frac{x \\left(x + 1\\right) \\left(2 x + 1\\right) \\left(x^{2} + x - 1\\right) \\left(3 x^{6} + 9 x^{5} + 2 x^{4} - 11 x^{3} + 3 x^{2} + 10 x - 5\\right)}{66}$" ], "text/plain": [ "L\"$S_{10}(x)=\\frac{x \\left(x + 1\\right) \\left(2 x + 1\\right) \\left(x^{2} + x - 1\\right) \\left(3 x^{6} + 9 x^{5} + 2 x^{4} - 11 x^{3} + 3 x^{2} + 10 x - 5\\right)}{66}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$S_{11}(x)=\\frac{x^{2} \\left(x + 1\\right)^{2} \\left(2 x^{8} + 8 x^{7} + 4 x^{6} - 16 x^{5} - 5 x^{4} + 26 x^{3} - 3 x^{2} - 20 x + 10\\right)}{24}$" ], "text/plain": [ "L\"$S_{11}(x)=\\frac{x^{2} \\left(x + 1\\right)^{2} \\left(2 x^{8} + 8 x^{7} + 4 x^{6} - 16 x^{5} - 5 x^{4} + 26 x^{3} - 3 x^{2} - 20 x + 10\\right)}{24}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$S_{12}(x)=\\frac{x \\left(x + 1\\right) \\left(2 x + 1\\right) \\left(105 x^{10} + 525 x^{9} + 525 x^{8} - 1050 x^{7} - 1190 x^{6} + 2310 x^{5} + 1420 x^{4} - 3285 x^{3} - 287 x^{2} + 2073 x - 691\\right)}{2730}$" ], "text/plain": [ "L\"$S_{12}(x)=\\frac{x \\left(x + 1\\right) \\left(2 x + 1\\right) \\left(105 x^{10} + 525 x^{9} + 525 x^{8} - 1050 x^{7} - 1190 x^{6} + 2310 x^{5} + 1420 x^{4} - 3285 x^{3} - 287 x^{2} + 2073 x - 691\\right)}{2730}$\"" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# べき乗和の公式のリスト k = 0,1,2,…,12\n", "\n", "x = symbols(\"x\")\n", "for k in 0:12\n", " latexstring(\"S_{$k}(x)=\", sympy.latex(factor(SS(k,x)))) |> display\n", "end" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "**問題:** $k$ が3以上の整数のとき $B_k(0)=0$ となることを示せ. \n", "\n", "**証明:** $B_k(x)$ の母函数で $x=0$ とおくと,\n", "\n", "$$\n", "\\begin{aligned}\n", "\\sum_{k=0}^\\infty \\frac{B_k(0)}{k!}z^k + \\frac{z}{2} = \n", "\\frac{z}{e^z - 1} + \\frac{z}{2} = \n", "\\frac{z}{2}\\frac{e^z + 1}{e^z - 1} = \n", "\\frac{z}{2}\\frac{e^{z/2} + e^{-z/2}}{e^{z/2} - e^{-z/2}}.\n", "\\end{aligned}\n", "$$\n", "\n", "これは $z$ の偶函数なので, 左辺のべき級数の奇数次の係数はすべて消える. ゆえに $k$ が3以上の奇数ならば $B_k(0)=0$ となる. $\\QED$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "**問題:** 以下を示せ:\n", "\n", "(1) $B_k(1-x)=(-1)^k B_k(x)$.\n", "\n", "(2) $k$ が奇数のとき $B_k(1/2)=0$.\n", "\n", "(3) $k$ が $3$ 以上の奇数のとき $B_k(1)=B_k(0)=0$.\n", "\n", "(4) $k\\geqq 2$ ならば $B_k(0)=B_k(1)$.\n", "\n", "**証明:** (1) $B_k(x)$ の母函数で $x$ に $1-x$ を代入すると,\n", "\n", "$$\n", "\\begin{aligned}\n", "\\sum_{k=0}^\\infty \\frac{B_k(1-x)}{k!}z^k &= \n", "\\frac{ze^{z(1-x)}}{e^z - 1} = \n", "\\frac{ze^{-zx}}{1 - e^{-z}} \n", "\\\\ &= \n", "\\frac{(-z)e^{(-z)x}}{e^{-z}-1} = \n", "\\sum_{k=0}^\\infty \\frac{B_k(x)}{k!}(-z)^k.\n", "\\end{aligned}\n", "$$\n", "\n", "なので両辺を比較すると, $B_k(1-x)=(-1)^k B_k(x)$ となることがわかる. \n", "\n", "(2) $x=1/2$ のとき $1-x = x$ であり, $k$ が奇数のとき $B_k(1-x)=-B_k(x)$ なので $B_k(1/2)=0$ となることがわかる. \n", "\n", "(3) 1つ前の問題より, $k$ が3以上の奇数のとき $B_k(0)=0$ なので $B_k(1)=B_k(1-0)=-B_k(0)=0$ となる.\n", "\n", "(4) $k\\geqq 2$ であるとする. $k$ が奇数ならば $B_k(0)=0=B_k(1)$ となり, $k$ が偶数ならば $B_k(0)=B_k(1-1)=B_k(1)$ となる. \n", "$\\QED$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "**問題:** 以下を示せ:\n", "\n", "(1) $S_k(x)$ は $x$ で割り切れる.\n", "\n", "(2) $k\\geqq 1$ のとき $S_k(x)$ は $x+1$ で割り切れる.\n", "\n", "(3) $k$ が2以上の偶数ならば $S_k(x)$ は $x(x+1)(2x+1)$ で割り切れる.\n", "\n", "(4) $k$ が3以上の奇数ならば $S_k(x)$ は $x^2(x+1)^2$ で割り切れる.\n", "\n", "**証明:** (1) $S_k(0)=0$ を示せばよいが, \n", "\n", "$$\n", "S_k(0) = \\frac{B_{k+1}(1)-B_{k+1}(1)}{k+1} = 0.\n", "$$\n", "\n", "(2) $k\\geqq 1$ のとき, 1つ前の問題の(4)より $B_{k+1}(0)=B_{k+1}(1)$ が成立するので, \n", "\n", "$$\n", "S_k(-1) = \\frac{B_{k+1}(0)-B_{k+1}(1)}{m+1} = 0.\n", "$$\n", "\n", "ゆえに $S_k(x)$ は $k+1$ で割り切れる.\n", "\n", "(3) $k$ が2以上の偶数のとき, 1つ前の問題の(2),(3)より $B_{k+1}(1/2)=0$, $B_{k+1}(1)=0$ が成立するので\n", "\n", "$$\n", "S_k(-1/2) = \\frac{B_{k+1}(1/2)-B_{k+1}(1)}{k+1} = 0.\n", "$$\n", "\n", "ゆえに $S_k(x)$ は $2x+1$ で割り切れる. 上の(1),(2)より, $S_k(x)$ は $x$ と $x+1$ でも割り切れるので, $x(x+1)(2x+1)$ で割り切れることがわかる.\n", "\n", "(4) $\\ds S_k(x) = \\int_1^{x+1} B_k(t)\\,dt = \\int_0^x B_k(t+1)\\,dt$ より,\n", "\n", "$$\n", "S_k'(x) = B_k(x+1)\n", "$$\n", "\n", "なので, 1つ前の問題の(3)より, $k$ が3以上の奇数ならば $S_k'(0)=B_k(1)=0$, $S_k'(-1)=B_k(0)=0$ となる. ゆえに $S_k'(x)$ は $x$ と $x+1$ で割り切れる. 上の(1),(2)より, $S_k(x)$ は $x$ と $x+1$ で割り切れる. これらより, $S_k(x)$ が $x^2(x+1)^2$ で割り切れることがわかる. $\\QED$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "**問題:** $S_1(x)$ が $x(x+1)$ で割り切れることを用いて, $\\ds S_1(x)=\\frac{x(x+1)}{2}$ となることを示せ.\n", "\n", "**証明:** $S_1(x)$ は最高次の係数が $1/2$ の2次の多項式になるので, それが $x(x+1)$ で割り切れることを使えば, $\\ds S_1(x)=\\frac{x(x+1)}{2}$ となることがただちに導かれる. $\\QED$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "**問題:** $S_2(x)$ が $x(x+1)(2x+1)$ で割り切れることを用いて, $\\ds S_2(x)=\\frac{x(x+1)(2x+1)}{6}$ となることを示せ.\n", "\n", "**証明:** $S_2(x)$ は最高次の係数が $1/3$ の3次の多項式になるので, それが $x(x+1)(2x+1)$ で割り切れることを使えば, $\\ds S_2(x)=\\frac{x(x+1)(2x+1)}{6}$ となることがただちに導かれる. $\\QED$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "**問題:** $S_3(x)$ が $x^2(x+1)^2$ で割り切れることを用いて, $\\ds S_3(x)=\\frac{x^2(x+1)^2}{4}$ となることを示せ.\n", "\n", "**証明:** $S_3(x)$ は最高次の係数が $1/4$ の4次の多項式になるので, それが $x^2(x+1)^2$ で割り切れることを使えば, $\\ds S_3(x)=\\frac{x^2(x+1)^2}{4}$ となることがただちに導かれる. $\\QED$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "**問題:** $S_4(x)$ が $x(x+1)(2x+1)$ で割り切れることを用いて, $S_4(x)$ を求めよ.\n", "\n", "**解答例:** $S_4(x)$ は最高次の係数が $1/5$ の5次の多項式になり, $x(x+1)(2x+1)$ で割り切れるので,\n", "\n", "$$\n", "S_4(x) = \\frac{1}{10}x(x+1)(2x+1)(x^2+ax+b)\n", "$$\n", "\n", "と書ける. $S_4(1)=1$, $S_4(2)=17$ より,\n", "\n", "$$\n", "\\frac{3}{5}(1+a+b)=1, \\quad 3(4+2a+b)=17.\n", "$$\n", "\n", "左の等式の5倍を→の等式から引くと $3(3+a)=12$ となるので, $a=1$ が得られ, それを左の等式に代入すると $3(2+b)=5$ となり, $b=-1/3$ が得られる. したがって, \n", "\n", "$$\n", "S_4(x) = \n", "\\frac{1}{10}x(x+1)(2x+1)(x^2+x-1/3) =\n", "\\frac{1}{30}x(x+1)(2x+1)(3x^2+3x-1)\n", "$$\n", "\n", "であることがわかる. $\\QED$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "**問題:** $S_5(x)$ が $x^2(x+1)^2$ で割り切れることを用いて, $S_5(x)$ を求めよ.\n", "\n", "**解答例:** $S_5(x)$ は最高次の係数が $1/6$ の6次多項式になり, $x^2(x+1)^2$ で割り切れるので,\n", "\n", "$$\n", "S_5(x) = \\frac{1}{6}x^2(x+1)^2(x^2+ax+b)\n", "$$\n", "\n", "と書ける. $S_5(1)=1$, $S_5(2)=33$ より,\n", "\n", "$$\n", "\\frac{2}{3}(1+a+b)=1, \\quad\n", "6(4+2a+b)=33.\n", "$$\n", "\n", "前者の9倍を後者から引くと $6(3+a)=24$ となるので, $a=1$ が得られ, それを前者に代入すると, $2(2+b)=3$ となるので, $b=-1/2$ が得られる. ゆえに\n", "\n", "$$\n", "S_5(x)=\n", "\\frac{1}{6}x^2(x+1)^2(x^2+x-1/2)=\n", "\\frac{1}{12}x^2(x+1)^2(2x^2+2x-1).\n", "\\qquad\\QED\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### べき乗和の直接的な取り扱い\n", "\n", "ツイッターでぴよぴよさんにBernoulli多項式に頼らない直接的で平易なべき乗和の取り扱いについて教わったのでその内容を以下で説明する.\n", "\n", "**定理:** べき乗和 $S_k(n)$ は $n$ に関する $k+1$ 次の多項式になる.\n", "\n", "**証明:** 多項式 $f(x)$ で $f(0)=0$, $f(x)=f(x-1)+x^k$ を満たすものが唯一存在することを示せば十分である. (そのような $f(x)$ について $S_k(n) = f(n)$ となる.)\n", "\n", "$f(x)$ の次数が $k+1$ 次より大きいならば $f(x)-f(x-1)$ の次数は $k+1$ 次以上になるので, $f(x)-f(x-1)=x^k$ が成立することはありえない. $f(x)$ の次数は $k+1$ 次以下でなければいけない.\n", "\n", "$\\ds f(x)=\\sum_{m=0}^{k+1} a_m x^m$ とおき, $f(0)=0$, $f(x)=f(x-1)+x^k$ を満たす $a_m$ 達が一意に定まることを示そう. $f(x-1)$ は\n", "\n", "$$\n", "f(x-1) = \\sum_{m=0}^{k+1} a_i \\sum_{i=0}^m\\binom{m}{i}(-1)^{m-i}x^i =\n", "\\sum_{i=0}^{k+1}\\left(\\sum_{m=i}^{k+1}(-1)^{m-i}\\binom{m}{i} a_m\\right) x^i\n", "$$\n", "\n", "と書けることから, 条件 $f(0)=0$, $f(x)=f(x-1)+x^k$ は係数 $a_m$ 達に関する以下の連立一次方程式に書き直される:\n", "\n", "$$\n", "\\begin{aligned}\n", "&\n", "-(k+1)a_{k+1} + 1 = 0,\n", "\\\\ &\n", "(i+1)a_{i+1} = \\sum_{m=i+2}^{k+1} (-1)^{m-i} \\binom{m}{i} a_m \n", "\\quad (i=k-1,k-2,\\ldots,0)\n", "\\\\ &\n", "a_0 = 0.\n", "\\end{aligned}\n", "$$\n", "\n", "これより, $\\ds a_{k+1}=\\frac{1}{k+1}$ から順番に $a_k,a_{k-1},\\ldots,a_1, a_0$ が決まり, この連立一次方程式の解 $(a_0,a_1,\\ldots,a_m)$ が唯一つ存在することがわかる. $\\QED$\n", "\n", "**注意:** 上の証明より, $S_k(n)$ の最高次の係数は $\\ds a_{k+1}=\\frac{1}{k+1}$ になることがわかる. さらに, $a_k$ を求めるための式は $i=k-1$ の場合から得られ, \n", "\n", "$$\n", "k a_k = \\frac{(k+1)k}{2}a_{k+1}\n", "$$\n", "\n", "になるので, $a_k=1/2$ となることもわかる. $\\QED$\n", "\n", "**注意:** $f(x)$ の次数が $k+1$ 次より大きいならば $f(x)-f(x-1)$ の次数は $k+1$ 次以上になるので, $f(x)-f(x-1)=x^k$ が成立することはありえない. このことから $S_k(n)$ は $n$ について $k+2$ 次以上の多項式になることはありえないことがわかる. $\\QED$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "**注意:** 上の証明は直接的な計算にこだわりすぎて煩雑になってしまっている. \n", "\n", "上の証明の内容は, 有理数体上の1変数多項式環 $\\Q[x]$ からそれ自身への線形写像 $D:\\Q[x]\\to\\Q[x]$ を\n", "\n", "$$\n", "Df(x) = f(x) - f(x-1) \\qquad(f(x)\\in\\Q[x])\n", "$$\n", "\n", "と定めると, $\\Ker D = \\Q$ かつ $\\Im D = \\Q[x]$ ($D$ は全射)になることに一般化される. 証明しておこう.\n", "\n", "**証明:** $f(x)\\in\\Q[x]$ であるとする. $f(x)\\in\\Q$ ならば $f(x)=f(x-1)$ となるので $Df(x)=0$. $f(x)\\not\\in\\Q$ ならば $f(x)$ は $1$ 次以上になり, $f(x)$ の次数を $n\\geqq 1$ と書き, 最高次の項を $ax^n$ と書くと, $Df(x)$ の最高次の項は $ax^n - a(x-1)^n$ の最高次の項 $anx^{n-1}\\ne 0$ になるので, $Df(x)\\ne 0$. ゆえに, $\\Ker D = \\Q$ である.\n", "\n", "$D:\\Q[x]\\to\\Q[x]$ の全射性を示そう. $n$ 次以下の $Q[x]$ の元全体のなす $\\Q[x]$ の部分空間を $V_n$ と書くことにする. 前段楽の議論によって, $DV_n\\subset V_{n-1}$ となることがわかる. $DV_n = V_{n-1}$ であることを示せば $D:\\Q[x]\\to\\Q[x]$ が全射であることがわかるので, それを示したい. 線形代数における次元定理より, $\\dim DV_n = \\dim V_n - \\Ker D = (n+1) - 1 = n = \\dim V_{n-1}$ となる． ゆえに $DV_n = V_{n-1}$. $\\QED$\n", "\n", "この証明より, $D$ の $xV_k$ (定数項のない $V_{k*1}$ の元全体のなす $V_{k+1}$ の部分空間) への制限が $V_k$ への全単射を定めることもわかる. 特に $k+1$ 次の多項式 $f(x)\\in\\Q[x]$ で $f(0)=0$ と $Df(x) = f(x) - f(x-1) = x^k$ を満たすものが唯一存在することがわかる. この $f(x)$ が上の証明中の $f(x)$ である.\n", "\n", "このように線形代数の抽象的な議論を使いこなすことができれば, 具体的な計算抜きに欲しい $f(x)$ の唯一存在をシンプルな議論で示せる. もちろん, その具体形を求めたい場合には具体的な計算に関わる議論が必要になってしまうが, 欲しいものの存在と一意性が論理的に確定していれば, 具体形を求める議論では存在と一意性を自由に用いる相対的に楽な方法を採用し易くなる. $\\QED$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "**定理:** べき乗和を表す多項式 $S_k(x)$ は以下を満たしている.\n", "\n", "(1) $S_k(x)$ は $S_k(0)=0$, $S_k(x)=S_k(x-1)+x^k$ という条件で一意的に特徴付けられ, その次数は $k+1$ であり, その最高次の係数は $1/(k+1)$ になる.\n", "\n", "(2) $k\\geqq 1$ ならば $S_k(-1)=0$ となり, $S_k(x)$ は $x(x+1)$ で割り切れる.\n", "\n", "(3) $k\\geqq 1$ ならば $S_k(x) = (-1)^{k+1}S_k(-1-x)$.\n", "\n", "(4) $k$ が2以上の偶数ならば $S_k(-1/2)=0$ となり, $S_k(x)$ は $x(x+1)(2x+1)$ で割り切れる.\n", "\n", "(5) $S_k'(x) = (-1)^k S_k'(-1-x)$.\n", "\n", "(6) $k$ が3以上の奇数ならば $S_k'(0)=S_k'(-1)=0$ となり, $S_k(x)$ は $x^2(x+1)^2$ で割り切れる.\n", "\n", "**証明:** (1)はすでに証明されている.\n", "\n", "(2) $k\\geqq 1$ のとき, $S_k(x)=S_k(x-1)+x^k$ で $x=0$ とおくと, $0=S_k(-1)$ が得られる. そのとき, $S_k(x)$ は $x+1$ で割り切れ, $S_k(0)=0$ より $x$ でも割り切れる.\n", "\n", "(3) $S_k(x)=S_k(x-1)+x^k$ の $x$ に $-x$ を代入すると, $S_k(-x)=S_k(-x-1)+(-1)^k x^k$. これは $-S_k(-x-1)=-S_k(-1-(x-1))+(-1)^k x^k$, $(-1)^{k+1}S_k(-x-1)=(-1)^{k+1}S_k(-1-(x-1))+x^k$ と書き直される. $k\\geqq 1$ のとき, $S_k(-1-x)$ で $x=0$ とおくと $S_k(-1-0)=S_k(-1)=0$ となる. ゆえに, $(-1)^{k+1}S_k(-1-x)$ は $S_k(x)$ を一意的に特徴付ける条件を満たしているので, $(-1)^{k+1}S_k(-1-x)=S_k(x)$ となることがわかる.\n", "\n", "(4) $k$ は2以上の偶数であると仮定する. このとき, (3)より $S_k(x)=-S_k(-1-x)$ となる. ゆえに $x=-1/2$ とおくと $S_k(-1/2)=0$ が得られる. そのとき $S_k(x)$ は $2x+1$ で割り切れ, (2)より $x(x+1)$ でも割り切れる.\n", "\n", "(5)は(3)からただちに得られる.\n", "\n", "(6) $k$ は3以上の奇数であると仮定する. このとき, (5)より $S_k'(x)=-S_k'(-1-x)$ となり, 特に $S_k'(0)=-S_k'(-1)$ が得られる. $S_k(x)=S_k(x-1)+x^k$ の両辺を微分すると $S_k'(x)=S_k'(x-1)+kx^{k-1}$ なので, 特に $S_k'(0)=S_k'(-1)$ が得らえる. それらより, $S_k'(0)=S_k'(-1)=0$ が得られる. $S_k(0)=S_k(-1)=0$ と合わせると, $S_k(x)$ は $x^2$ と $(x+1)^2$ で割り切れることがわかる. $\\QED$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### 第2種Stirling数とべき乗和\n", "\n", "以下の内容については https://twitter.com/genkuroki/status/1052837557732433921 も参照せよ. \n", "\n", "集合 $\\{1,2,\\ldots,n\\}$ を空でない $k$ 個の部分集合への分割($k$ 分割)の個数を**第2種Stirling数**と呼び, $\\ds \\stirlingsecond{n}{k}$ と書くことにする. 集合 $\\{1,2,\\ldots,n,n+1\\}$ の $k$ 分割は, $\\{1,2,\\ldots,n\\}$ の $k-1$ 分割と $\\{n+1\\}$ で構成された分割と $\\{1,2,\\ldots,n\\}$ の $k$ 分割中の $k$ 個の部分集合のどれかに $n+1$ を付け加えてできる分割のどちらかになるので, \n", "\n", "$$\n", "\\stirlingsecond{n+1}{k} = \\stirlingsecond{n}{k-1} + k\\stirlingsecond{n}{k}\n", "$$\n", "\n", "を満たしている. この漸化式と $\\ds \\stirlingsecond{0}{k}=\\delta_{k0}$ によって第2種Stirling数は一意的に特徴付けられる." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "__例:__ $n=4$, $k=2$ の場合. 集合 $\\{1,2,3,4\\}$ の2分割全体は\n", "\n", "$$\n", "\\begin{aligned}\n", "&\n", "\\{\\{1\\}, \\{2,3,4\\}\\},\\;\n", "\\{\\{2\\}, \\{1,3,4\\}\\},\\;\n", "\\{\\{3\\}, \\{1,2,4\\}\\},\\;\n", "\\{\\{4\\}, \\{1,2,3\\}\\},\n", "\\\\ &\n", "\\{\\{1,2\\}\\,\\{3,4\\}\\},\\;\n", "\\{\\{1,3\\}\\,\\{2,4\\}\\},\\;\n", "\\{\\{1,4\\}\\,\\{2,3\\}\\}.\n", "\\end{aligned}\n", "$$\n", "\n", "である. ゆえに $\\ds\\stirlingsecond{4}{2}=7$ である.\n", "\n", "集合 $\\{1,2,3\\}$ の1分割は $\\{\\{1,2,3\\}\\}$ しか存在しないので $\\ds\\stirlingsecond{3}{1}=1$. $\\{1,2,3\\}$ の1分割に $\\{4\\}$ を追加すれば $\\{1,2,3,4\\}$ の2分割 $\\{\\{4\\}, \\{1,2,3\\}\\}$ が得られる.\n", "\n", "集合 $\\{1,2,3\\}$ の2分割全体は $\\{\\{1\\}, \\{2,3\\}\\}$, $\\{\\{2\\}, \\{1,3\\}\\}$, $\\{\\{3\\}, \\{1,2\\}\\}$ なので $\\ds\\stirlingsecond{3}{2}=3$. 集合 $\\{1,2,3\\}$ の2分割のどれかの元に $4$ を追加して得られる $\\{1,2,3,4\\}$ の2分割の全体は\n", "\n", "$$\n", "\\begin{aligned}\n", "&\n", "\\{\\{1\\}, \\{2,3,4\\}\\},\\;\n", "\\{\\{2\\}, \\{1,3,4\\}\\},\\;\n", "\\{\\{3\\}, \\{1,2,4\\}\\},\\;\n", "\\\\ &\n", "\\{\\{1,2\\}\\,\\{3,4\\}\\},\\;\n", "\\{\\{1,3\\}\\,\\{2,4\\}\\},\\;\n", "\\{\\{1,4\\}\\,\\{2,3\\}\\}.\n", "\\end{aligned}\n", "$$\n", "\n", "の6通りになる.\n", "\n", "以上の $1+6=7$ 通りで$\\{1,2,3,4\\}$ の2分割の全体が尽くされている. $\\QED$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "**定理:** $\\partial=d/dx$ とおく. 微分作用素 $x\\partial$ の $n$ 乗は以下のように表わされる:\n", "\n", "$$\n", "(x\\partial)^n = \\sum_{k=0}^n \\stirlingsecond{n}{k} x^k\\partial^k.\n", "\\tag{$*$}\n", "$$\n", "\n", "**証明:** $n$ に関する数学的帰納法. $n=1$ のとき $\\ds \\stirlingsecond{1}{k}=\\delta_{k1}$ より($*$)は成立している. $n$ について($*$)が成立していると仮定する. このとき\n", "\n", "$$\n", "\\begin{aligned}\n", "(x\\partial)^{n+1} &= \n", "\\sum_{k=0}^n \\stirlingsecond{n}{k} x\\partial x^k\\partial^k = \n", "\\sum_{k=0}^n \\stirlingsecond{n}{k} (x^{k+1}\\partial^{k+1} + k x^k\\partial^k) \n", "\\\\ &=\n", "\\sum_{l=1}^{n+1} \\stirlingsecond{n}{l-1} x^l\\partial^l +\n", "\\sum_{k=0}^n k\\stirlingsecond{n}{k} x^k\\partial^k\n", "\\\\ &=\n", "\\sum_{k=0}^{n+1} \\left(\\stirlingsecond{n}{k-1} + k\\stirlingsecond{n}{k}\\right) x^k\\partial^k =\n", "\\sum_{k=0}^{n+1} \\stirlingsecond{n+1}{k} x^k\\partial^k.\n", "\\end{aligned}\n", "$$\n", "\n", "これで $n$ が $n+1$ の場合にも($*$)が成立することがわかった. $\\QED$\n", "\n", "**注意:** 上の定理の公式は帰納法によらずに「項と $k$ 分割の一対一対応」を構成することによっても証明可能である.\n", "\n", "* https://twitter.com/genkuroki/status/1052837562342027264\n", "\n", "に簡単な説明がある. $\\QED$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "**系:** 次が成立している:\n", "\n", "$$\n", "a^n = \\sum_{k=0}^n \\stirlingsecond{n}{k}a(a-1)\\cdots(a-k+1).\n", "$$\n", "\n", "**証明:** $x^a$ に上の定理の公式の両辺を作用させ, さらに両辺を $x^a$ で割ればこの公式が得られる. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "__別証明:__ 微分作用素に関する上の定理の公式を使わずに, 組合せ論的に上の系を証明しておこう. \n", "\n", "系の公式の両辺は $a$ について多項式なので $a$ が0以上の整数の場合に証明すれば十分である. 以下では $a$ は0以上の整数であると仮定する.\n", "\n", "左辺の $a^n$ は写像 $f:\\nset{n}\\to\\nset{a}$ 全体の個数に等しい.\n", "\n", "右辺の第 $k$ 項の因子の第2種Stirling数 $\\ds\\stirlingsecond{n}{k}$ は集合 $\\nset{n}$ の $k$ 分割全体の個数に等しい.\n", "\n", "集合 $\\nset{n}$ の $k$ 分割 $P$ ($P$ は集合 $\\nset{n}$ の互いに交わらない空でない $k$ 個の部分集合の集合で和集合が全体の $\\nset{n}$ に一致するもの)が与えられたとき, 右辺の第 $k$ 項の因子 $a(a-1)\\cdots(a-k+1)$ は単射 $\\varphi:P\\to\\nset{k}$ 全体の個数に等しい.\n", "\n", "像の元の個数がちょうど $k$ 個の写像 $f:\\nset{n}\\to\\nset{a}$ に対して, 集合 $\\nset{n}$ の $k$ 分割 $P$ が \n", "\n", "$$\n", "P=\\{\\,f^{-1}(i)\\mid i\\in\\nset{a},\\; f^{-1}(i)\\ne\\emptyset\\,\\}\n", "$$\n", "\n", "によって定まり, 単射 $\\varphi:P\\to\\nset{a}$ が\n", "\n", "$$\n", "\\varphi(f^{-1}(i)) = i \\quad (i\\in\\nset{a},\\; f^{-1}(i)\\ne\\emptyset)\n", "$$\n", "\n", "と定まる. 逆にこのような $P$ と $\\varphi$ から写像 $f:\\nset{n}\\to\\nset{a}$ が一意に定まる.\n", "\n", "ゆえに, 右辺の第 $k$ 項 $\\ds\\stirlingsecond{n}{k}a(a-1)\\cdots(a-k+1)$ は像の元の個数が $k$ であるような任意の写像 $f:\\nset{n}\\to\\nset{a}$ 全体の個数に等しい.\n", "\n", "以上から上の系の公式 \n", "\n", "$$\n", "a^n = \\sum_{k=0}^n \\stirlingsecond{n}{k}a(a-1)\\cdots(a-k+1).\n", "$$\n", "\n", "が成立することがわかる. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "__注意:__ 要するに上の系の公式\n", "\n", "$$\n", "a^n = \\sum_{k=0}^n \\stirlingsecond{n}{k}a(a-1)\\cdots(a-k+1).\n", "$$\n", "\n", "は, 写像 $f:\\nset{n}\\to\\nset{a}$ の全体の集合が, $k=0,1,\\ldots,n$ に関する\n", "\n", "* 像の元の個数がちょうど $k$ 個の写像 $f:\\nset{n}\\to\\nset{a}$ 全体の集合\n", "\n", "達に分割されることから得られる. $\\QED$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "__べき乗和の第2種Stirling数による表示:__ $(a+1)a\\cdots(a-r+1)$, $a(a-1)\\cdots(a-r)$ ($r+1$ 個の因子の積達)と $a(a-1)\\cdots(a-r+1)$ ($r$ 個の因子の積)ついて\n", "\n", "$$\n", "(a+1)a\\cdots(a-r+1) - a(a-1)\\cdots(a-r) = (r+1)a(a-1)\\cdots(a-r+1).\n", "$$\n", "\n", "これの両辺を $a=n,n-1,\\ldots,1,0$ について足し上げて, 全体を $r+1$ で割ると,\n", "\n", "$$\n", "\\frac{(n+1)n(n-1)\\cdots(n-r+1)}{r+1} = \\sum_{a=0}^n a(a-1)\\cdots(a-r+1).\n", "$$\n", "\n", "したがって, 上の系の公式を書き直した\n", "\n", "$$\n", "a^k = \\sum_{r=0}^k \\stirlingsecond{k}{r}a(a-1)\\cdots(a-r+1)\n", "$$\n", "\n", "を $a=0,1,2,\\ldots,n$ について足し上げると, \n", "\n", "$$\n", "\\sum_{a=0}^n a^k = \\sum_{r=0}^k \\stirlingsecond{k}{r} \\frac{(n+1)n(n-1)\\cdots(n-r+1)}{r+1}.\n", "$$\n", "\n", "左辺は $k=0$ のとき $1+S_0(n)=n+1$ に一致し, $k\\geqq 1$ のとき $S_k(n)$ に一致する. $\\ds\\stirlingsecond{k}{0}=\\delta_{k0}$ であることに注意せよ. $\\QED$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "**注意:** 第2種Stirling数は次の母函数表示を持つ:\n", "\n", "$$\n", "\\exp(x(e^t - 1)) = \\sum_{n=0}^\\infty\\sum_{k=0}^n\\stirlingsecond{n}{k} x^k \\frac{t^n}{n!}.\n", "$$\n", "\n", "このことは $\\exp(x)$ に $\\exp(tx\\partial)$ を作用させると,\n", "\n", "$$\n", "\\exp(tx\\partial)\\exp(x) = \\exp(x e^t)\n", "$$\n", "\n", "となり, 上の定理の公式より,\n", "\n", "$$\n", "\\begin{aligned}\n", "\\exp(tx\\partial)\\exp(x) &= \\sum_{n=0}^\\infty (x\\partial)^n\\exp(x) \\frac{t^n}{n!} \n", "\\\\&=\n", "\\sum_{n=0}^\\infty \\sum_{k=0}^n \\stirlingsecond{n}{k}x^k\\partial^k\\exp(x) \\frac{t^n}{n!} =\n", "\\exp(x)\\sum_{n=0}^\\infty \\sum_{k=0}^n \\stirlingsecond{n}{k}x^k\\frac{t^n}{n!}\n", "\\end{aligned}\n", "$$\n", "\n", "となることからわかる. $\\QED$" ] }, { "cell_type": "code", "execution_count": 29, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "text/latex": [ "$\\sum_{a=0}^{n} 1=n + 1$" ], "text/plain": [ "L\"$\\sum_{a=0}^{n} 1=n + 1$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\sum_{a=0}^{n} a=\\frac{n \\left(n + 1\\right)}{2}$" ], "text/plain": [ "L\"$\\sum_{a=0}^{n} a=\\frac{n \\left(n + 1\\right)}{2}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\sum_{a=0}^{n} a \\left(a - 1\\right)=\\frac{n \\left(n - 1\\right) \\left(n + 1\\right)}{3}$" ], "text/plain": [ "L\"$\\sum_{a=0}^{n} a \\left(a - 1\\right)=\\frac{n \\left(n - 1\\right) \\left(n + 1\\right)}{3}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\sum_{a=0}^{n} a \\left(a - 2\\right) \\left(a - 1\\right)=\\frac{n \\left(n - 2\\right) \\left(n - 1\\right) \\left(n + 1\\right)}{4}$" ], "text/plain": [ "L\"$\\sum_{a=0}^{n} a \\left(a - 2\\right) \\left(a - 1\\right)=\\frac{n \\left(n - 2\\right) \\left(n - 1\\right) \\left(n + 1\\right)}{4}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\sum_{a=0}^{n} a \\left(a - 3\\right) \\left(a - 2\\right) \\left(a - 1\\right)=\\frac{n \\left(n - 3\\right) \\left(n - 2\\right) \\left(n - 1\\right) \\left(n + 1\\right)}{5}$" ], "text/plain": [ "L\"$\\sum_{a=0}^{n} a \\left(a - 3\\right) \\left(a - 2\\right) \\left(a - 1\\right)=\\frac{n \\left(n - 3\\right) \\left(n - 2\\right) \\left(n - 1\\right) \\left(n + 1\\right)}{5}$\"" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# a(a-1)…(a-r+1) の和\n", "\n", "a = symbols(\"a\", integer=true)\n", "n = symbols(\"n\", integer=true, positive=true)\n", "ff(a,r) = iszero(r) ? typeof(a)(1) : prod(a-i for i in 0:r-1)\n", "\n", "for r in 0:4\n", " s = sympy.Sum(ff(a,r), (a,0,n))\n", " latexstring(sympy.latex(s), \"=\", sympy.latex(s.doit().factor())) |> display\n", "end" ] }, { "cell_type": "code", "execution_count": 30, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "text/latex": [ "$\\sum_{j=0}^nj^{0} =n + 1$" ], "text/plain": [ "L\"$\\sum_{j=0}^nj^{0} =n + 1$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\sum_{j=0}^nj^{1} =\\frac{n \\left(n + 1\\right)}{2}$" ], "text/plain": [ "L\"$\\sum_{j=0}^nj^{1} =\\frac{n \\left(n + 1\\right)}{2}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\sum_{j=0}^nj^{2} =\\frac{n \\left(n + 1\\right) \\left(2 n + 1\\right)}{6}$" ], "text/plain": [ "L\"$\\sum_{j=0}^nj^{2} =\\frac{n \\left(n + 1\\right) \\left(2 n + 1\\right)}{6}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\sum_{j=0}^nj^{3} =\\frac{n^{2} \\left(n + 1\\right)^{2}}{4}$" ], "text/plain": [ "L\"$\\sum_{j=0}^nj^{3} =\\frac{n^{2} \\left(n + 1\\right)^{2}}{4}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\sum_{j=0}^nj^{4} =\\frac{n \\left(n + 1\\right) \\left(2 n + 1\\right) \\left(3 n^{2} + 3 n - 1\\right)}{30}$" ], "text/plain": [ "L\"$\\sum_{j=0}^nj^{4} =\\frac{n \\left(n + 1\\right) \\left(2 n + 1\\right) \\left(3 n^{2} + 3 n - 1\\right)}{30}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\sum_{j=0}^nj^{5} =\\frac{n^{2} \\left(n + 1\\right)^{2} \\left(2 n^{2} + 2 n - 1\\right)}{12}$" ], "text/plain": [ "L\"$\\sum_{j=0}^nj^{5} =\\frac{n^{2} \\left(n + 1\\right)^{2} \\left(2 n^{2} + 2 n - 1\\right)}{12}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\sum_{j=0}^nj^{6} =\\frac{n \\left(n + 1\\right) \\left(2 n + 1\\right) \\left(3 n^{4} + 6 n^{3} - 3 n + 1\\right)}{42}$" ], "text/plain": [ "L\"$\\sum_{j=0}^nj^{6} =\\frac{n \\left(n + 1\\right) \\left(2 n + 1\\right) \\left(3 n^{4} + 6 n^{3} - 3 n + 1\\right)}{42}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\sum_{j=0}^nj^{7} =\\frac{n^{2} \\left(n + 1\\right)^{2} \\left(3 n^{4} + 6 n^{3} - n^{2} - 4 n + 2\\right)}{24}$" ], "text/plain": [ "L\"$\\sum_{j=0}^nj^{7} =\\frac{n^{2} \\left(n + 1\\right)^{2} \\left(3 n^{4} + 6 n^{3} - n^{2} - 4 n + 2\\right)}{24}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\sum_{j=0}^nj^{8} =\\frac{n \\left(n + 1\\right) \\left(2 n + 1\\right) \\left(5 n^{6} + 15 n^{5} + 5 n^{4} - 15 n^{3} - n^{2} + 9 n - 3\\right)}{90}$" ], "text/plain": [ "L\"$\\sum_{j=0}^nj^{8} =\\frac{n \\left(n + 1\\right) \\left(2 n + 1\\right) \\left(5 n^{6} + 15 n^{5} + 5 n^{4} - 15 n^{3} - n^{2} + 9 n - 3\\right)}{90}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\sum_{j=0}^nj^{9} =\\frac{n^{2} \\left(n + 1\\right)^{2} \\left(n^{2} + n - 1\\right) \\left(2 n^{4} + 4 n^{3} - n^{2} - 3 n + 3\\right)}{20}$" ], "text/plain": [ "L\"$\\sum_{j=0}^nj^{9} =\\frac{n^{2} \\left(n + 1\\right)^{2} \\left(n^{2} + n - 1\\right) \\left(2 n^{4} + 4 n^{3} - n^{2} - 3 n + 3\\right)}{20}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\sum_{j=0}^nj^{10} =\\frac{n \\left(n + 1\\right) \\left(2 n + 1\\right) \\left(n^{2} + n - 1\\right) \\left(3 n^{6} + 9 n^{5} + 2 n^{4} - 11 n^{3} + 3 n^{2} + 10 n - 5\\right)}{66}$" ], "text/plain": [ "L\"$\\sum_{j=0}^nj^{10} =\\frac{n \\left(n + 1\\right) \\left(2 n + 1\\right) \\left(n^{2} + n - 1\\right) \\left(3 n^{6} + 9 n^{5} + 2 n^{4} - 11 n^{3} + 3 n^{2} + 10 n - 5\\right)}{66}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\sum_{j=0}^nj^{11} =\\frac{n^{2} \\left(n + 1\\right)^{2} \\left(2 n^{8} + 8 n^{7} + 4 n^{6} - 16 n^{5} - 5 n^{4} + 26 n^{3} - 3 n^{2} - 20 n + 10\\right)}{24}$" ], "text/plain": [ "L\"$\\sum_{j=0}^nj^{11} =\\frac{n^{2} \\left(n + 1\\right)^{2} \\left(2 n^{8} + 8 n^{7} + 4 n^{6} - 16 n^{5} - 5 n^{4} + 26 n^{3} - 3 n^{2} - 20 n + 10\\right)}{24}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\sum_{j=0}^nj^{12} =\\frac{n \\left(n + 1\\right) \\left(2 n + 1\\right) \\left(105 n^{10} + 525 n^{9} + 525 n^{8} - 1050 n^{7} - 1190 n^{6} + 2310 n^{5} + 1420 n^{4} - 3285 n^{3} - 287 n^{2} + 2073 n - 691\\right)}{2730}$" ], "text/plain": [ "L\"$\\sum_{j=0}^nj^{12} =\\frac{n \\left(n + 1\\right) \\left(2 n + 1\\right) \\left(105 n^{10} + 525 n^{9} + 525 n^{8} - 1050 n^{7} - 1190 n^{6} + 2310 n^{5} + 1420 n^{4} - 3285 n^{3} - 287 n^{2} + 2073 n - 691\\right)}{2730}$\"" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Σ_{a=0}^n a^k を a(a-1)…(a-r+1) の和に帰着する方法で計算\n", "\n", "stirlingsecond(n,k) = sympy.functions.combinatorial.numbers.stirling(n, k, kind=2)\n", "n = symbols(\"n\", integer=true, positive=true)\n", "SSS(k,n) = expand(sum(stirlingsecond(k,r)*ff(n+1, r+1)/(r+1) for r in 0:k))\n", "\n", "for k in 0:12\n", " latexstring(raw\"\\sum_{j=0}^n\", \"j^{$k} =\", sympy.latex(factor(SSS(k,n)))) |> display\n", "end" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 第2種Stirling数の二項係数を用いた表示\n", "\n", "__定理:__ 次の公式が成立している:\n", "\n", "$$\n", "k!\\stirlingsecond{n}{k} = \\sum_{a=0}^k (-1)^{k-a}\\binom{k}{a} a^n.\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "__証明:__ $\\partial = \\partial/\\partial x$ とおき, $\\Delta = e^\\partial - 1$ とおく. $\\Delta$ は多項式 $f(x)$ に $\\Delta f(x) = f(x+1) - f(x)$ と作用する.\n", "\n", "$\\Delta^k = (e^\\partial - 1)^k$ に二項定理を適用すると,\n", "\n", "$$\n", "\\Delta^k f(x) =\n", "\\sum_{a=0}^k (-1)^{k-a}\\binom{k}{a} e^{a\\partial} f(x) =\n", "\\sum_{a=0}^k (-1)^{k-a}\\binom{k}{a} f(x+a).\n", "$$\n", "\n", "特に,\n", "\n", "$$\n", "\\Delta^k x^n = \\sum_{a=0}^k (-1)^{k-a}\\binom{k}{a} (x+a)^n.\n", "$$\n", "\n", "前節で示した第2種Stirling数の母函数表示\n", "\n", "$$\n", "\\exp(x(e^t - 1)) =\n", "\\sum_{m=0}^\\infty\\sum_{k=0}^m\\stirlingsecond{m}{k} x^k \\frac{t^m}{m!} =\n", "\\sum_{k=0}^\\infty \\frac{x^k}{k!} k! \\sum_{m=k}^\\infty \\stirlingsecond{m}{k}\\frac{t^m}{m!}\n", "$$\n", "\n", "を使おう. この公式中の $x$ を $z$ で置き換え, $t$ に $\\partial$ を代入すると, \n", "\n", "$$\n", "\\exp(z\\Delta) = \n", "\\sum_{k=0}^\\infty \\frac{z^k}{k!}\\Delta^k =\n", "\\sum_{k=0}^\\infty \\frac{x^k}{k!} k! \\sum_{m=k}^\\infty \\stirlingsecond{m}{k}\\frac{\\partial^m}{m!}.\n", "$$\n", "\n", "これの両辺を $x^n$ に作用させると,\n", "\n", "$$\n", "\\sum_{k=0}^n \\frac{z^k}{k!}\\Delta^k x^n =\n", "\\sum_{k=0}^n \\frac{z^k}{k!} k! \\sum_{m=k}^n \\stirlingsecond{m}{k}\\binom{n}{m}x^{n-m}.\n", "$$\n", "\n", "$z^k/k!$ の係数を比較すると,\n", "\n", "$$\n", "\\Delta^k x^n = k! \\sum_{m=k}^n \\stirlingsecond{m}{k}\\binom{n}{m}x^{n-m}.\n", "$$\n", "\n", "したがって, \n", "\n", "$$\n", "\\Delta^k x^n =\n", "\\sum_{a=0}^k (-1)^{k-a}\\binom{k}{a} (x+a)^n =\n", "k! \\sum_{m=k}^n \\stirlingsecond{m}{k}\\binom{n}{m}x^{n-m}.\n", "$$\n", "\n", "特に $x=0$ とおけば\n", "\n", "$$\n", "\\sum_{a=0}^k (-1)^{k-a}\\binom{k}{a} a^n = k!\\stirlingsecond{n}{k}.\n", "$$\n", "\n", "が得られる. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "__別証明:__ 前節の系の別証明と似た組合せ論的方法で証明しよう.\n", "\n", "全射 $g:\\nset{n}\\to\\nset{k}$ から, 集合 $\\nset{n}$ の $k$ 分割 $P$ が\n", "\n", "$$\n", "P = \\{\\,g^{-1}(i)\\mid i\\in\\nset{k}\\,\\}\n", "$$\n", "\n", "と得られ, 全単射 $\\psi:P\\to\\nset{k}$ が\n", "\n", "$$\n", "\\psi(g^{-1}(i)) = i \\quad (i\\in\\nset{k})\n", "$$\n", "\n", "と得られる. 逆にこのような $P$ と $\\psi$ から $g$ が一意的に定まる. \n", "\n", "集合 $\\nset{n}$ の $k$ 分割 $P$ 全体の個数は $\\ds\\stirlingsecond{n}{k}$ で, 各 $P$ ごとに全単射 $\\psi:P\\to\\nset{k}$ 全体の個数は $k!$ 個なので, 全射 $g:\\nset{n}\\to\\nset{k}$ 全体の個数は $\\ds k!\\stirlingsecond{n}{k}$ に等しい.\n", "\n", "写像 $g:\\nset{n}\\to\\nset{k}$ 全体の集合を $Y$ と書く.\n", "\n", "各 $i\\in\\nset{k}$ に対して, $Y$ の部分集合 $Y_i$ を次のように定める:\n", "\n", "$$\n", "Y_i = \\{\\,g:\\nset{n}\\to\\nset{k}\\mid g(\\nset{n})\\not\\ni i\\,\\}.\n", "$$\n", "\n", "このとき, 有限集合 $A$ の元の個数を $|A|$ と書くことにすると, $1\\le i_10$ ならば $0$ になる).\n", "\n", "したがって, \n", "\n", "* [包除原理](https://ja.wikipedia.org/wiki/%E5%8C%85%E9%99%A4%E5%8E%9F%E7%90%86)\n", "([inclusion–exclusion principle](https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle))\n", "\n", "より, 単射でない写像 $g:\\nset{n}\\to\\nset{k}$ 全体の集合 $Y_1\\cup Y_2\\cup\\cdots\\cup Y_k$ の元の個数は次のように表される:\n", "\n", "$$\n", "\\begin{aligned}\n", "&\n", "|Y_1\\cup Y_2\\cup\\cdots\\cup Y_k|\n", "\\\\ &=\n", "\\sum_{1\\le i_1\\le k} |Y_{i_1}| -\n", "\\sum_{1\\le i_1Google Scholar" ] }, { "cell_type": "code", "execution_count": 31, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle\\begin{matrix}\n", "1&\\frac{1}{2}&\\frac{1}{3}&\\frac{1}{4}&\\frac{1}{5}&\\frac{1}{6}&\\frac{1}{7}&\\frac{1}{8}&\\frac{1}{9}&\\frac{1}{10}&\\frac{1}{11}\\\\\n", "\\frac{1}{2}&\\frac{1}{3}&\\frac{1}{4}&\\frac{1}{5}&\\frac{1}{6}&\\frac{1}{7}&\\frac{1}{8}&\\frac{1}{9}&\\frac{1}{10}&\\frac{1}{11}& \\\\\n", "\\frac{1}{6}&\\frac{1}{6}&\\frac{3}{20}&\\frac{2}{15}&\\frac{5}{42}&\\frac{3}{28}&\\frac{7}{72}&\\frac{4}{45}&\\frac{9}{110}& & \\\\\n", "0&\\frac{1}{30}&\\frac{1}{20}&\\frac{2}{35}&\\frac{5}{84}&\\frac{5}{84}&\\frac{7}{120}&\\frac{28}{495}& & & \\\\\n", "- \\frac{1}{30}&- \\frac{1}{30}&- \\frac{3}{140}&- \\frac{1}{105}&0&\\frac{1}{140}&\\frac{49}{3960}& & & & \\\\\n", "0&- \\frac{1}{42}&- \\frac{1}{28}&- \\frac{4}{105}&- \\frac{1}{28}&- \\frac{29}{924}& & & & & \\\\\n", "\\frac{1}{42}&\\frac{1}{42}&\\frac{1}{140}&- \\frac{1}{105}&- \\frac{5}{231}& & & & & & \\\\\n", "0&\\frac{1}{30}&\\frac{1}{20}&\\frac{8}{165}& & & & & & & \\\\\n", "- \\frac{1}{30}&- \\frac{1}{30}&\\frac{1}{220}& & & & & & & & \\\\\n", "0&- \\frac{5}{66}& & & & & & & & & \\\\\n", "\\frac{5}{66}& & & & & & & & & & \\\\\n", "\\end{matrix}\n", "$" ], "text/plain": [ "\"\\$\\\\displaystyle\\\\begin{matrix}\\n1&\\\\frac{1}{2}&\\\\frac{1}{3}&\\\\frac{1}{4}&\\\\frac{1}{5}&\\\\frac{1}{6}&\\\\frac{1}{7}&\\\\frac{1}{8}&\\\\frac{1}{9}&\\\\frac{1}{10}&\\\\frac{1}{11}\\\\\\\\\\n\\\\frac{1}{2}&\\\\frac{1}{3}&\\\\frac{1}{4}&\\\\frac{1}{5}&\\\\frac{1}{6}&\\\\frac{1}{7}&\\\\frac{1}{8}&\\\\frac{1}{9}&\\\\frac{1}{10}&\\\\frac{1\" ⋯ 454 bytes ⋯ \"42}&\\\\frac{1}{140}&- \\\\frac{1}{105}&- \\\\frac{5}{231}& & & & & & \\\\\\\\\\n0&\\\\frac{1}{30}&\\\\frac{1}{20}&\\\\frac{8}{165}& & & & & & & \\\\\\\\\\n- \\\\frac{1}{30}&- \\\\frac{1}{30}&\\\\frac{1}{220}& & & & & & & & \\\\\\\\\\n0&- \\\\frac{5}{66}& & & & & & & & & \\\\\\\\\\n\\\\frac{5}{66}& & & & & & & & & & \\\\\\\\\\n\\\\end{matrix}\\n\\$\"" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "struct AkiyamaTanigawa{T,S<:Integer}\n", " a::Array{T,2}\n", " N::S\n", "end\n", "\n", "Base.length(AT::AkiyamaTanigawa) = AT.N\n", "\n", "function AkiyamaTanigawa(a0::AbstractArray{T,1}) where T\n", " N = size(a0,1)\n", " a = zeros(eltype(a0), N, N)\n", " a[0+1,:] = a0\n", " for n in 0:N-2\n", " for m in 0:N-n-2\n", " a[(n+1)+1,m+1] = (m+1) * (a[n+1, m+1] - a[n+1, (m+1)+1])\n", " end\n", " end\n", " AkiyamaTanigawa(a, N)\n", "end\n", "\n", "function displayAT(AT::AkiyamaTanigawa)\n", " N = length(AT)\n", " s = raw\"\\begin{matrix}\" * \"\\n\"\n", " for n in 0:N-1\n", " s *= prod(x * raw\"&\" for x in @.(sympy.latex(Sym(AT.a[n+1,1:N-n]))))\n", " s = replace(s, r\"&$\"=>\"\")\n", " s *= \"& \"^n * raw\"\\\\\\\\\" * \"\\n\"\n", " end\n", " s *= raw\"\\end{matrix}\" * \"\\n\"\n", " l = latexstring(raw\"\\displaystyle\", s)\n", " display(l)\n", "end\n", "\n", "L = 10\n", "a0 = 1 .// collect(1:L+1)\n", "AT = AkiyamaTanigawa(a0)\n", "displayAT(AT)" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "-" } }, "source": [ "以上の第 $n+1$ 行目が $a_{n0},a_{n1},\\ldots,$ である. 隣り合った $a_{nm}$ と $a_{n,m+1}$ の差を取り, $m+1$ 倍したものが $a_{n+1,m}$ になっている. 例えば, 2行目の $a_{12}=1/4$ と $a_{13}=1/5$ の差 $1/20$ の3倍が3行目の $3/20$ になっている. そのように計算した結果の左端の $a_{n0}$ が $B_n(1)$ に一致していることを, 以下のセルの計算結果と比較するとわかる." ] }, { "cell_type": "code", "execution_count": 32, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "text/latex": [ "$\\left[ \\begin{array}{r}\\displaystyle 1\\\\\\displaystyle \\frac{1}{2}\\\\\\displaystyle \\frac{1}{6}\\\\\\displaystyle 0\\\\\\displaystyle - \\frac{1}{30}\\\\\\displaystyle 0\\\\\\displaystyle \\frac{1}{42}\\\\\\displaystyle 0\\\\\\displaystyle - \\frac{1}{30}\\\\\\displaystyle 0\\\\\\displaystyle \\frac{5}{66}\\end{array} \\right]$\n" ], "text/plain": [ "11-element Vector{Sym}:\n", " 1\n", " 1/2\n", " 1/6\n", " 0\n", " -1/30\n", " 0\n", " 1/42\n", " 0\n", " -1/30\n", " 0\n", " 5/66" ] }, "execution_count": 32, "metadata": {}, "output_type": "execute_result" } ], "source": [ "[B(n,1) for n in 0:L]" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### べき乗和とHurwitzのゼータ函数の関係\n", "\n", "$s>1$, $x\\ne 0,-1,-2,\\ldots$ に対して, Hurwitz(フルヴィッツ)のゼータ函数 $\\zeta(s,x)$ が\n", "\n", "$$\n", "\\zeta(s,x) = \\sum_{k=0}^\\infty\\frac{1}{(x+k)^s} = \n", "\\frac{1}{x^s} + \\frac{1}{(x+1)^s} + \\frac{1}{(x+2)^s} + \\cdots\n", "$$\n", "\n", "によって定義される. これは形式的には $s=-m$ とおくと,\n", "\n", "$$\n", "\\zeta(-m,x) = x^m + (x+1)^m + (x+2)^m + \\cdots.\n", "$$\n", "\n", "と書け, さらに, $m=0,1,2,\\ldots$ に対して, 形式的には\n", "\n", "$$\n", "\\begin{aligned}\n", "\\zeta(-m,x) - \\zeta(-m,x+n) &=\n", "(x^m + \\cdots + (x+n-1)^m + (x+n)^m + (x+n+1)^m + \\cdots) \n", "\\\\ &\\, - ((x+n)^m + (x+n+1)^m + \\cdots)\n", "\\\\ & = x^m + (x+1)^m + \\cdots + (x+n-1)^m\n", "\\end{aligned}\n", "$$\n", "\n", "なので, 特に\n", "\n", "$$\n", "\\zeta(-m,1) - \\zeta(-m,n+1) = 1^m+2^m+\\cdots+n^m = S_m(n).\n", "$$\n", "\n", "この公式はHurwitzのゼータ函数の解析接続によって論理的に正当化される.\n", "\n", "一方, ガンマ函数 $\\Gamma(s)=\\int_0^\\infty e^{-t}x^{s-1}\\,dx$ の応用としてよく使われる公式\n", "\n", "$$\n", "\\frac{1}{a^s} = \\frac{1}{\\Gamma(s)}\\int_0^\\infty e^{-at} t^{s-1}\\,dt\n", "$$\n", "\n", "の $a=x,x+1,x+2,\\ldots$ の場合をHurwitzのゼータ函数の定義式に代入して, 無限和と積分の順序を交換して, 等比級数の和の公式を使うと, \n", "\n", "$$\n", "\\begin{aligned}\n", "\\zeta(s,x) &= \n", "\\frac{1}{\\Gamma(s)} \\sum_{k=0}^\\infty \\int_0^\\infty e^{-(x+k)t} t^{s-1}\\,dt\n", "\\\\ &=\n", "\\frac{1}{\\Gamma(s)} \\int_0^\\infty \\frac{e^{-xt}}{1-e^{-t}} t^{s-1}\\,dt\n", "\\\\ &=\n", "\\frac{1}{\\Gamma(s)} \\int_0^\\infty \\frac{t e^{(1-x)t}}{e^t-1} t^{s-2}\\,dt.\n", "\\end{aligned}\n", "$$\n", "\n", "このようにして, 自然にBernoulli多項式に $1-x$ を代入したものの母函数\n", "\n", "$$\n", "\\frac{t e^{(1-x)t}}{e^t-1} = \\sum_{k=0}^\\infty B_k(1-x)\\frac{t^k}{k!}\n", "$$\n", "\n", "が出て来る. この結果はべき乗和を無限和に拡張して得られるHurwitzのゼータ函数の中に自然にBernoulli多項式の母函数が現われることを意味している. さらに, $0$ から $\\infty$ までの積分を $0$ から $1$ までの積分と $1$ から $\\infty$ までの積分の和に分解し, $0$ から $1$ までの積分の中の $B_k(1-x)$ の母函数 $\\ds \\frac{t e^{(1-x)t}}{e^t-1}$ をそれから $\\ds\\sum_{k=0}^N B_k(1-x)\\frac{t^k}{k!}$ を引いて足したもので置き換え, 足した分から得らえる項を $0$ から $1$ まで積分することによって, 次が得られる:\n", "\n", "$$\n", "\\begin{aligned}\n", "\\zeta(s,x) = \\frac{1}{\\Gamma(s)}\\biggl[&\n", "\\int_1^\\infty \\frac{t e^{(1-x)t}}{e^t-1} t^{s-2}\\,dt \n", "\\\\ &\\, +\n", "\\int_0^1 \\left(\\frac{t e^{(1-x)t}}{e^t-1} - \\sum_{k=0}^N B_k(1-x)\\frac{t^k}{k!}\\right)t^{s-2}\\,dt \n", "\\\\ &\\, +\n", "\\sum_{k=0}^N \\frac{B_k(1-x)}{k!}\\frac{1}{s+k-1}\n", "\\biggr].\n", "\\end{aligned}\n", "$$\n", "\n", "この公式の右辺は $\\imag s > -N$, $\\imag x > 0$ で意味を持ち, そこへの $\\zeta(s,x)$ の解析接続を与える. $m\\in\\Z$, $0\\leqq m < N$ のとき $s\\to -m$ とすることによって, \n", "\n", "$$\n", "\\zeta(-m,x) = \\frac{(-1)^m B_{m+1}(1-x)}{m+1} = -\\frac{B_{m+1}(x)}{m+1}.\n", "$$\n", "\n", "ここで $B_k(1-x)=(-1)^k B_k(x)$ および $\\Gamma(s+1)=s\\Gamma(s)$ より\n", "\n", "$$\n", "\\frac{1}{\\Gamma(s)} = \\frac{s(s+1)\\cdots(s+m)}{\\Gamma(s+m+1)}\n", "$$\n", "\n", "となること(これは $s\\to-1$ で $0$ になる)を使った($k=m+1$ の分母の $s+m$ と $1/\\Gamma(s)$ の分子の $s+m$ がキャンセルすることに注意せよ). この結果を使っても, べき乗和をBernoulli多項式で表す公式\n", "\n", "$$\n", "S_m(n) = \\zeta(-m,1) - \\zeta(-m,n+1) = \\frac{B_{m+1}(n+1)-B_{m+1}(1)}{m+1}\n", "$$\n", "\n", "が得られる. \n", "\n", "以上の経路でのこの公式の証明はHurwitzのゼータ函数という解析学の対象を用いた分だけ難しくなっているが, Bernoulli多項式の母函数がどのような形で自然に現われるかがよくわかる証明になっている.\n", "\n", "べき乗和を真に理解するためにはHurwitzのゼータ函数のような解析学の対象にまで視界を広げる必要がある." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## 平面上の点と直線の距離\n", "\n", "$a,b,c$ は実数であり, $(a,b)\\ne(0,0)$ であると仮定する.\n", "\n", "高校の数学の教科書には, $xy$ 平面上の直線 $ax+by+c=0$ と $xy$ 平面上の点 $(X,Y)$ の距離 $d$ が\n", "\n", "$$\n", "d = \\frac{|aX+bY+c|}{\\sqrt{a^2+b^2}}\n", "$$\n", "\n", "と表わされることが書いてある. これは, そこに登場する様々な量の幾何学的な意味を理解していれば自明な公式に過ぎないことを以下で説明したい.\n", "\n", "$xyz$ 空間内の傾いた平面 $z=ax+by+c$ を考えよう. この平面の傾きは $(a,b)$ で決まっている.\n", "\n", "**定理:** $xyz$ 空間内における $z=ax+by+c$ のグラフはベクトル $(a,b)$ の方向が登り方向の傾いた平面になり, その方向の傾きの大きさは $\\ds\\sqrt{a^2+b^2}$ になる. すなわち, 単位ベクトル $\\ds\\frac{(a,b)}{\\sqrt{a^2+b^2}}$ の分だけ $(x,y)$ をずらすと高さが $\\sqrt{a^2+b^2}$ だけ増す.\n", "\n", "**証明:** $z=ax+by+c$ は $(x,y)$ を $(\\Delta x, \\Delta y)$ だけずらすと, $a\\Delta x+b\\Delta y$ の分だけ変化する. Cauchy-Schwarzの不等式より,\n", "\n", "$$\n", "\\begin{aligned}\n", "&\n", "-\\sqrt{a^2+b^2}\\sqrt{(\\Delta x)^2+(\\Delta y)^2} \n", "\\\\ & \\qquad\\qquad \\leqq\n", "a\\Delta x+b\\Delta y \n", "\\\\ & \\qquad\\qquad\\qquad \\leqq\n", "\\sqrt{a^2+b^2}\\sqrt{(\\Delta x)^2+(\\Delta y)^2}\n", "\\end{aligned}\n", "$$\n", "\n", "が成立している. ゆえに $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}=1$ という条件のもとでの $a\\Delta x+b\\Delta y$ の最大値は $\\ds(\\Delta x,\\Delta y)=\\frac{(a,b)}{\\sqrt{a^2+b^2}}$ のときの $\\sqrt{a^2+b^2}$ である. $\\QED$\n", "\n", "このように, ベクトル $(a,b)$ は傾いている平面 $z = ax+by+c$ の傾きの方向と大きさを記述している. \n", "\n", "$xy$ 平面上の点 $(X,Y)$ から直線 $ax+by+c=0$ への距離を $d$ と書き, 点 $(X,Y)$ から直線 $ax+by+c=0$ におろした垂線の足を点 $(X_0, Y_0)$ と書くことにする. このとき, 点 $(X,Y)$ は $(X_0, Y_0)$ からベクトル $\\pm (a,b)$ と同じ方向に距離 $d$ の位置にある. したがって, $z = ax+by+c$ の点 $(X,Y)$ における値は\n", "\n", "$$\n", "aX+bY+c = \\pm\\sqrt{a^2+b^2}\\;d\n", "$$\n", "\n", "になる. $(X,Y)$ が $(X_0,Y_0)$ から見てベクトル $(a,b)$ と同じ方向にあれば符号は $+$ になり, その反対側にあれば符号は $-$ になる. これより, \n", "\n", "$$\n", "d = \\frac{|aX+bY+c|}{\\sqrt{a^2+b^2}}.\n", "$$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "**補足:** 平面 $z=ax+by+c$ の傾き方を調べることは2つのベクトル $(a,b)$ と $(\\Delta x, \\Delta y)$ の内積\n", "\n", "$$\n", "a\\Delta x + b\\Delta y = \\sqrt{a^2+b^2}\\sqrt{(\\Delta x)^2+(\\Delta y)^2}\\;\\cos\\theta\n", "$$\n", "\n", "を調べることに他ならない. ここで $\\theta$ はそれら2つのベクトルのなす角度である. この公式から, $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}=1$ という条件のもとで $a\\Delta x+b\\Delta y$ が最大になるのは $\\cos\\theta=1$ のときであり, 最大値は $\\sqrt{a^2+b^2}$ であることがわかる. さらに, $\\cos\\theta=1$ は 2つのベクトル $(a,b)$ と $(\\Delta x, \\Delta y)$ が同じ方向を向いていることを意味するので, そのことから, $\\sqrt{(\\Delta x)^2+(\\Delta y)^2}=1$ という条件のもとで $a\\Delta x+b\\Delta y$ が最大になるのは, $\\ds(\\Delta x,\\Delta y)=\\frac{(a,b)}{\\sqrt{a^2+b^2}}$ のときであることもわかる. \n", "\n", "高校での授業で「点と直線の距離の公式は内積を使えば容易に導ける」と習った人がいるかもしれないが, 内積の使用は実質的に「平面 $z=ax+by+c$ の傾き方」を調べていることに他ならない. $\\QED$" ] }, { "attachments": { "ax+b.jpg": { "image/jpeg": 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} }, "cell_type": "markdown", "metadata": {}, "source": [ "![ax+by+c.jpg](attachment:ax+by+c.jpg)" ] }, { "cell_type": "code", "execution_count": 33, "metadata": {}, "outputs": [ { "data": { "image/png": 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", "text/html": [ "" ] }, "execution_count": 33, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a, b, c = 0.8, 0.5, 1.0\n", "f(x,y) = a*x + b*y + c\n", "x = -10:0.1:5\n", "y = -10:0.1:15\n", "surface(x, y, f.(x',y), colorbar=false, size=(400,300))\n", "plot!(x, @.(-(a*x + c)/b), zero(x); label=\"\", c=:black, lw=0.5)\n", "plot!(; ylim=extrema(y))\n", "title!(\"\\$z = $a x + $b y + $c\\$\")" ] }, { "cell_type": "code", "execution_count": 34, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "image/png": 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", "text/html": [ "" ] }, "execution_count": 34, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a, b, c = 0.8, 0.5, 1.0\n", "f(x,y) = a*x + b*y + c\n", "x = -10:0.1:5\n", "y = -10:0.1:15\n", "surface(x, y, abs.(f.(x',y)), colorbar=false, size=(400,300))\n", "plot!(title=\"\\$z = |$a x + $b y + $c|\\$\")" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Jensenの不等式と相加相乗調和平均\n", "\n", "相加相乗平均の不等式はそれより圧倒的に一般的なJensenの不等式の特別な場合になっていることを解説する. さらに, 相加相乗調和平均の一般化になっている $p$ 乗平均についても解説する. 最後に単位円に内接する多角形の周長と面積の最大値が正多角形の場合に得られることをJensenの不等式を使って証明する." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Jensenの不等式\n", "\n", "$I$ は実数の区間であるとする. 例えば $I=\\R$, $I=(0,\\infty)$ のような場合を考える. \n", "\n", "$a_1,\\ldots,a_n\\in I$ であるとし, $p_1,\\ldots,p_n\\geqq 0$, $p_1+\\cdots+p_n=1$ と仮定する.\n", "\n", "区間 $I$ 上の実数値函数 $f(x)$ に対して, 実数 $E[f(x)]$ を対応させる函数(汎函数) $E[\\ ]$ を\n", "\n", "$$\n", "E[f(x)] = p_1 f(a_1) + \\cdots + p_n f(a_n)\n", "$$\n", "\n", "と定めると, $I$ 上の実数値函数 $f(x), g(x)$ と実数 $\\alpha$, $\\beta$ に対して以下が成立している.\n", "\n", "(1) 線形性: $E[\\alpha f(x)+\\beta g(x)]=\\alpha E[f(x)] + \\beta E[g(x)]$.\n", "\n", "(2) 単調性: $I$ 全体上で $f(x)\\leqq g(x)$ ならば $E[f(x)]\\leqq E[g(x)]$.\n", "\n", "(3) 規格化条件: $I$ 上の定数値函数 $\\alpha$ に対して, $E[\\alpha]=\\alpha$.\n", "\n", "区間 $I$ 上の実数値函数 $f(x)$ が上に凸な函数であるとは, \n", "\n", "$$\n", "a,b\\in I,\\ 0" ] }, "execution_count": 35, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# log x は上に凸な函数\n", "\n", "x = 0:0.01:2.0\n", "a, b = 0.3, 1.5\n", "f(x) = log(x)\n", "t = 0:0.01:1.0\n", "g(a,b,t) = (1-t)*f(a) + t*f(b)\n", "h(a,b,t) = (1-t)*a + t*b\n", "plot(size=(400,250), legend=:topleft, xlims=(0,2.0), ylims=(-2.0, 0.8))\n", "plot!(x, f.(x), label=L\"y = \\log\\,x\")\n", "plot!(h.(a,b,t), g.(a,b,t), label=\"\")\n", "plot!([a,a], [-10.0, f(a)], label=L\"x = a\", ls=:dash)\n", "plot!([b,b], [-10.0, f(b)], label=L\"x = b\", ls=:dashdot)" ] }, { "cell_type": "code", "execution_count": 36, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "image/png": 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", "text/html": [ "" ] }, "execution_count": 36, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# exp(x) は下に凸な函数\n", "\n", "x = -1:0.01:2\n", "a, b = -0.3, 1.5\n", "f(x) = exp(x)\n", "t = 0:0.01:1.0\n", "g(a,b,t) = (1-t)*f(a) + t*f(b)\n", "h(a,b,t) = (1-t)*a + t*b\n", "plot(size=(400,250), legend=:topleft, xlims=(-1,2), ylims=(0,8))\n", "plot!(x, f.(x), label=L\"y = e^x\")\n", "plot!(h.(a,b,t), g.(a,b,t), label=\"\")\n", "plot!([a,a], [-0.0, f(a)], label=L\"x = a\", ls=:dash)\n", "plot!([b,b], [-0.0, f(b)], label=L\"x = b\", ls=:dashdot)" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "**Jensenの不等式:** $f(x)$ が区間 $I$ 上の上に凸な函数ならば $E[f(x)]\\leqq f(E[x])$. (下に凸ならば不等式の向きが逆になる.)\n", "\n", "**証明:** $f(x)$ が $C^1$ 級の場合に限定して証明する. そのように仮定しない場合には接線の存在を微分に頼らずに直接示す必要が出て来る.\n", "\n", "$f(x)$ は上に凸であると仮定し, $\\mu=E[x]=p_1a_1+\\cdots+p_na_n$ とおく. $E[f(x)]\\leqq f(\\mu)$ を示したい. $x=\\mu$ における $y=f(x)$ の接線を $y=a(x-\\mu)+f(\\mu)$ と書く. $f(x)$ が上に凸であることより, $I$ 全体上で\n", "\n", "$$\n", "f(x) \\leqq a(x-\\mu)+f(\\mu)\n", "$$\n", "\n", "が成立する. ゆえに $E[\\ ]$ の性質より,\n", "\n", "$$\n", "E[f(x)]\\leqq E[a(x-\\mu)+f(\\mu)]=a(E[x]-\\mu)+f(\\mu) = f(\\mu).\n", "$$\n", "\n", "最初の等号で $E[\\ ]$ の単調性を使い, 2つ目の等号で $E[\\ ]$ の線形性と規格化条件を使い, 3つ目の等号で $E[x]=\\mu$ を使った. $\\QED$\n", "\n", "**注意:** 以上の証明法ならば $n$ に関する数学的帰納法を使わずに, しかも $E[\\ ]$ の定義に直接触れずに, その基本性質だけを使って証明をできた. $E[\\ ]$ と同じ性質を持つものの例として, 確率密度函数 $p(x)$ に対する\n", "\n", "$$\n", "E[f(x)] = \\int_I f(x)p(x)\\,dx\n", "$$\n", "\n", "がある. これは確率密度函数 $p(x)$ を持つ確率分布における $f(x)$ の期待値である. $E[\\ ]$ は**期待値汎函数**(expected value functional)と呼ばれる. $\\QED$" ] }, { "cell_type": "code", "execution_count": 37, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "image/png": 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"text/html": [ "" ] }, "execution_count": 37, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# 上に凸な函数 f(x) = log x の接線\n", "\n", "x = 0:0.01:2.0\n", "μ = 0.7\n", "f(x) = log(x)\n", "g(μ,x) = (1/μ)*(x-μ) + f(μ)\n", "plot(size=(500,350), legend=:topleft, xlims=(0,1.7), ylims=(-2.2, 0.8))\n", "plot!(x, f.(x), label=L\"y = \\log\\,x\")\n", "plot!(x, g.(μ,x), label=L\"y = a(x-\\mu)+f(\\mu)\", ls=:dashdot)\n", "plot!([μ, μ], [-3.0, f(μ)], label=L\"x=\\mu\", ls=:dash)\n", "plot!(size=(400,250))" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### 相加相乗調和平均の不等式\n", "\n", "一般の相加相乗平均の不等式は $p_1=p_2=\\cdots=p_n=1/n$ と $f(x)=\\log x$ の場合にJensenの不等式からただちに得られる. そのとき, $a_1,\\ldots,a_n>0$ に対して,\n", "\n", "$$\n", "\\begin{aligned}\n", "&\n", "E[\\log x] = \\frac{\\log a_1+\\cdots+\\log a_n}{n} =\n", "\\log(a_1\\cdots a_n)^{1/n}, \n", "\\\\ &\n", "E[x] = \\frac{a_1+\\cdots+a_n}{n}\n", "\\end{aligned}\n", "$$\n", "\n", "であり, Jensenの不等式より, \n", "\n", "$$\n", "%\\begin{aligned}\n", "%&\n", "\\log(a_1\\cdots a_n)^{1/n} = E[\\log x] \n", "%\\\\ &\n", "\\leqq \\log E[x] = \\log \\frac{a_1+\\cdots+a_n}{n}\n", "%\\end{aligned}\n", "$$\n", "\n", "ゆえに, $\\log$ が単調増加函数であることより,\n", "\n", "$$\n", "(a_1\\cdots a_n)^{1/n} \\leqq \\frac{a_1+\\cdots+a_n}{n}.\n", "$$\n", "\n", "この不等式の $a_i$ 達をそれらの逆数で置き換えて, 全体の分子分母を交換することによって,\n", "\n", "$$\n", "\\frac{n}{\\dfrac{1}{a_1}+\\cdots+\\dfrac{1}{a_n}} \\leqq\n", "(a_1,\\ldots,a_n)^{1/n}\n", "$$\n", "\n", "も得られる." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### p乗平均\n", "\n", "$x_1,\\ldots,x_n>0$ の $p$ 乗平均 $M_p(x_1,\\ldots,x_n)$ を, $p\\ne 0$ に対して\n", "\n", "$$\n", "M_p(x_1,\\ldots,x_n) = \\left(\\frac{1}{n}\\sum_{i=1}^n x_i^p\\right)^{1/p}\n", "$$\n", "\n", "と定め, $p=0$ に対して\n", "\n", "$$\n", "M_0(x_1,\\ldots,x_n) = (x_1,\\ldots,x_n)^{1/n}\n", "$$\n", "\n", "と定める. $M_0$ は相乗平均である. そして, \n", "\n", "$$\n", "\\begin{aligned}\n", "&\n", "M_1(x_1,\\ldots,x_n) = \\frac{x_1+\\cdots+x_n}{n}, \n", "\\\\ &\n", "M_{-1}(x_1,\\ldots,x_n) = \\frac{n}{\\dfrac{1}{x_1}+\\cdots+\\dfrac{1}{x_n}}.\n", "\\end{aligned}\n", "$$\n", "\n", "なので, $M_1$ は加法平均で, $M_{-1}$ は調和平均である." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### p→0でのp乗平均の挙動\n", "\n", "$p\\to 0$ における $p$ 乗平均の挙動を調べよう.\n", "\n", "$$\n", "\\begin{aligned}\n", "\\frac{1}{n}\\sum_{i=1}^n x_i^p &= \n", "\\frac{1}{n}\\sum_{i=1}^n e^{p\\log x_i} \n", "\\\\ &=\n", "\\frac{1}{n}\\sum_{i=1}^n (1 + p\\log x_i + O(p^2)) \n", "\\\\ &=\n", "1 + p\\log(x_1\\cdots x_n)^{1/n} + O(p^2)\n", "\\end{aligned}\n", "$$\n", "\n", "なので, $\\log(1+X)=X+O(X^2)$ を使うと, \n", "\n", "$$\n", "\\begin{aligned}\n", "\\log M_p(x_1,\\ldots,x_n) &=\n", "\\frac{1}{p}\\log \\frac{1}{n}\\sum_{i=1}^n x_i^p \n", "\\\\ &=\n", "\\frac{1}{p}\\log\\left(1 + p\\log(x_1\\cdots x_n)^{1/n} + O(p^2)\\right) \n", "\\\\ &=\n", "\\log(x_1\\cdots x_n)^{1/n} + O(p) \n", "\\\\ &=\n", "\\log M_0(x_1,\\ldots,x_n) + O(p)\n", "\\end{aligned}\n", "$$\n", "\n", "これより, $M_p(x_1,\\ldots,x_n)$ は $p=0$ でも解析的であることがわかる. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "**問題:** $p\\to\\infty$ のとき $M_p(x_1,\\ldots,x_n)\\to \\max\\{x_1,\\ldots,x_n\\}$ となることを示せ.\n", "\n", "**解答例:** $x_1=\\cdots=x_k>x_{k+1}\\geqq\\cdots\\geqq x_n$ と仮定してよい. \n", "\n", "$$\n", "\\sum_{i=1}^n x_i^p = k x_1^p\\left(1 + \\sum_{i>k} \\left(\\frac{x_i}{x_1}\\right)^p\\right)\n", "$$\n", "\n", "なので,\n", "\n", "$$\n", "\\begin{aligned}\n", "\\log M_p(x_1,\\ldots,x_n) &=\n", "\\frac{1}{p}\\log \\frac{1}{n}\\sum_{i=1}^n x_i^p \n", "\\\\ &= -\n", "\\frac{1}{p}\\log n + \\frac{1}{p}\\log k + \\log x_i \n", "\\\\ &\\,+\n", "\\frac{1}{p}\\log\\left(1 + \\sum_{i>k} \\left(\\frac{x_i}{x_1}\\right)^p\\right).\n", "\\end{aligned}\n", "$$\n", "\n", "であり, $i>k$ のとき $00$ なので $f(x)$ は下に凸な函数である. Jensenの不等式を $E[f(x)] = \\frac{1}{n}\\sum_{i=1}^n f(x_i^p)$ と下に凸な函数 $f(x)=x \\log x$ に適用すると, \n", "\n", "$$\n", "\\begin{aligned}\n", "\\frac{1}{n}\\sum_{i=1}^n x_i^p \\log x_i^p &=\n", "E[x\\log x] \\geqq \n", "E[x]\\log E[x] \n", "\\\\ &=\n", "\\frac{1}{n}\\sum_{i=1}^n x_i^p \\log\\frac{1}{n}\\sum_{i=1}^n x_i^p.\n", "\\end{aligned}\n", "$$\n", "\n", "これで $\\ds \\frac{d}{dp}\\log M_p \\geqq 0$ であることがわかった. $M_p$ は $p$ について単調増加函数になる:\n", "\n", "$$\n", "p\\leqq q \\implies M_p(x_1,\\ldots,x_n) \\leqq M_q(x_1,\\ldots,x_n).\n", "$$\n", "\n", "この不等式は相加相乗平均の不等式の大幅な一般化になっている. 例えば $M_{-1}\\leqq M_0\\leqq M_1$ より相加相乗調和平均の不等式\n", "\n", "$$\n", "\\frac{n}{\\dfrac{1}{x_1}+\\cdots+\\dfrac{1}{x_n}} \\leqq\n", "(x_1,\\ldots,x_n)^{1/n} \\leqq\n", "\\frac{x_1+\\cdots+x_n}{n}\n", "$$\n", "\n", "が得られる." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### p乗平均のpに関する依存性(2)\n", "\n", "前節の結果の別証明を与えよう.\n", "\n", "$x>0$ の函数 $f(x)=x^p$ は $p<0$, $p\\geqq 1$ のとき下に凸で, $00 \\implies\n", "\\left(\\frac{1}{n}\\sum_{i=1}^n x_i^{pr}\\right)^{1/(pr)} \\leqq\n", "\\left(\\frac{1}{n}\\sum_{i=1}^n x_i^r\\right)^{1/r} \\leqq\n", "\\left(\\frac{1}{n}\\sum_{i=1}^n x_i^{qr}\\right)^{1/(qr)}\n", "\\quad \\text{and}\\quad pr\\leqq r\\leqq qr,\n", "\\\\ &\n", "r<0 \\implies\n", "\\left(\\frac{1}{n}\\sum_{i=1}^n x_i^{pr}\\right)^{1/(pr)} \\geqq\n", "\\left(\\frac{1}{n}\\sum_{i=1}^n x_i^r\\right)^{1/r} \\geqq\n", "\\left(\\frac{1}{n}\\sum_{i=1}^n x_i^{qr}\\right)^{1/(qr)}\n", "\\quad \\text{and}\\quad pr\\geqq r\\geqq qr.\n", "\\end{aligned}\n", "$$\n", "\n", "これと $M_p$ が $p=0$ でも連続であることを合わせれば, $M_p$ が $p$ について単調増加することが示される." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "**プロット:** $p\\ne 0$ のとき, $\\ds M_p(x,y) = \\left(\\frac{x^p+y^p}{2}\\right)^{1/p}$ なので $M_p(x,y)=1$ と $y=(2-x^p)^{1/p}$ と同値であり, $M_0(x,y)=\\sqrt{xy}$ なので $M_0(x,y)=1$ と $y=1/x$ は同値である. $M_p(x,y)=1$ のグラフをプロットしよう." ] }, { "cell_type": "code", "execution_count": 38, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "image/png": 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", "text/html": [ "" ] }, "execution_count": 38, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# M_p(x,y) = 1 のプロット\n", "\n", "f(p,x) = iszero(p) ? 1/x : (2 - x^p)^(1/p)\n", "P = plot(size=(500,500))\n", "ps = [-10, -2, -1, 0, 1, 2, 10]\n", "for p in ps\n", " a = iszero(p) ? 1/3 : 2^(1/p)\n", " if p > 0\n", " Δx = a/1000\n", " x = Δx:Δx:a\n", " else\n", " Δx = 3/1000\n", " x = a+eps():Δx:3\n", " end\n", " plot!(x, f.(p,x), label=\"p = $p\", ls=:auto)\n", "end\n", "plot(P, xlim=(0,3), ylim=(0,3), size=(300,300))" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### 単位円に内接する多角形の周長と面積の最大値\n", "\n", "正弦函数 $\\sin\\alpha$ が $0\\leqq\\alpha\\leqq\\pi$ において上に凸な函数であることにJensenの不等式を適用することによって, 単位円に内接する $n$ 角形の周長と面積が最大になるのは正 $n$ 角形の場合であることを示そう. 一般に上に凸な函数 $f(x)$ についてJensenの不等式より,\n", "\n", "$$\n", "\\frac{1}{n}\\sum_{i=1}^n f(x_i) \\leqq f\\left(\\frac{1}{n}\\sum_{i=1}^n x_i\\right).\n", "$$\n", "\n", "が成立している. さらに, $f(x)$ が強い意味で上に凸ならば($f(x)$ のグラフに局所的に直線になっている部分が存在しなければ), 等号が成立するための必要十分条件は $x_1=\\cdots=x_n$ となることである.\n", "\n", "$\\theta_1<\\cdots<\\theta_n<\\theta_{n+1}=\\theta_1+2\\pi$ と仮定し, $A_i = (\\cos\\theta_i, \\sin\\theta_i)$ ($i=1,\\ldots,n$) とおき, 単位円に内接する $n$ 角形 $A_1\\cdots A_n$ を考える. \n", "\n", "$\\alpha_i = \\theta_{i+1}-\\theta_i$ とおく. もしも $\\alpha_i > \\pi$ となる $i$ が存在するならば, 直線 $A_i A_{i+1}$ をそれに平行な原点を通る直線で線対称変換して得られる直線と単位円の交点を $A'_i$, $A'_{i+1}$ とし, $A_i,A_{i+1}$ のぞれぞれを $A'_i,A'_{i+1}$ で置き換えて得られる単位円に内接する $n$ 角形を考えることによって, 単位円に内接する $n$ 角形の周長と面積を真に大きくすることができる. ゆえに, 単位円に内接する $n$ 角形で周長の面積の最大化に興味があるならば, すべての $i=1,\\ldots,n$ について $\\alpha_i \\leqq\\pi$ であると仮定してよい. 以下ではそのように仮定する. \n", "\n", "**補足:** $\\alpha_i = \\theta_{i+1} - \\theta_i > \\pi$ のとき, $\\theta'_i = \\theta_i + (\\alpha_i-\\pi) = \\theta_{i+1}-\\pi$, $\\theta'_{i+1}=\\theta_{i+1}-(\\alpha_i-\\pi) = \\theta_i+\\pi$, $A'_i = (\\cos\\theta'_i, \\sin\\theta'_i)$, $A'_{i+1} = (\\cos\\theta'_{i+1}, \\sin\\theta'_{i+1})$ とおくと, 線分 $\\overline{A_i A_{i+1}}$ と線分 $\\overline{A'_i A'_{i+1}}$ は平行で同じ長さになり, $n$ 角形 $A_1\\cdots A'_i A'_{i+1}\\cdots A_n$ の周長と面積は $n$ 角形 $A_1\\cdots A_n$ のそれらよりも真に大きくなる. 図を描いてみよ! $\\QED$\n", "\n", "以上の設定のもとで, $n$ 角形 $A_1\\cdots A_n$ の周長 $L$ と面積 $S$ について以下が成立している.\n", "\n", "線分 $\\overline{A_i A_{i+1}}$ の長さは $2\\sin(\\alpha_i/2)$ に等しいので, \n", "\n", "$$\n", "\\begin{aligned}\n", "L = 2\\sum_{i=1}^n \\sin\\frac{\\alpha_i}{2} = 2n\\,\\frac{1}{n}\\sum_{i=1}^n \\sin\\frac{\\alpha_i}{2} \\leqq\n", "2n \\sin\\left(\\frac{1}{n}\\sum_{i=1}^n\\frac{\\alpha_i}{2}\\right) = 2n\\sin\\frac{\\pi}{n}.\n", "\\end{aligned}\n", "$$\n", "\n", "この計算中の不等号は $\\sin\\alpha$ が $0\\leqq\\alpha\\leqq\\pi$ で上に凸であることとJensenの不等式から従う(注意: 上で述べた仮定を使わなくても, $0< \\alpha_i<2\\pi$ なので 0<$\\alpha_i/2<\\pi$ となる). この不等式の最右辺は単位円に内接する正 $n$ 角形の周長に等しい. $\\sin\\alpha$ が $0\\leqq\\alpha\\leqq\\pi/2$ で強い意味で凸であることを使えば, 逆に周長 $L$ が最大になるのは単位円に内接する $n$ 角形 $A_1\\ldots A_n$ が正 $n$ 角形になるときであることもわかる.\n", "\n", "三角形 $\\triangle A_iOA_{i+1}$ の面積は $(1/2)\\sin\\alpha_i$ に等しいので, \n", "\n", "$$\n", "\\begin{aligned}\n", "S = \\frac{1}{2}\\sum_{i=1}^n \\sin\\alpha_i = \\frac{n}{2}\\,\\frac{1}{n}\\sum_{i=1}^n \\sin\\alpha_i \\leqq\n", "\\frac{n}{2} \\sin\\left(\\frac{1}{n}\\sum_{i=1}^n\\alpha_i\\right) = \\frac{n}{2}\\sin\\frac{2\\pi}{n}.\n", "\\end{aligned}\n", "$$\n", "\n", "この計算中の不等号は $\\sin\\alpha$ が $0\\leqq\\alpha\\leqq\\pi$ で上に凸であることとJensenの不等式から従う. この不等式の最右辺は単位円に内接する正 $n$ 角形の面積に等しい. $\\sin\\alpha$ が $0\\leqq\\alpha\\leqq\\pi$ で強い意味で凸であることを使えば, 逆に面積 $S$ が最大になるのは単位円に内接する $n$ 角形 $A_1\\ldots A_n$ が正 $n$ 角形になるときであることもわかる." ] }, { "cell_type": "code", "execution_count": 39, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "image/png": 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", "text/html": [ "" ] }, "execution_count": 39, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# α_i > π ならば周長と面積をより大きくできること\n", "\n", "t = range(0, 2π, length=400)\n", "theta = 2π * [0, 3/20, 14/20, 16/20, 18/20, 1]\n", "phi = copy(theta)\n", "phi[2] = theta[3] - π\n", "phi[3] = theta[2] + π\n", "plot(size=(300,300), aspect_ratio=1, legend=false)\n", "plot!(cos.(t), sin.(t), lw=0.5, color=:black)\n", "plot!(cos.(theta), sin.(theta), color=:blue)\n", "plot!(cos.(phi), sin.(phi), ls=:dash, color=:red)" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## 三角函数の微積分\n", "\n", "高校数学の範囲内で三角函数の微分積分学を再構成してみせる. その結果は直接楕円函数論に一般化可能である." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "-" } }, "source": [ "### 高校の数学の教科書の方針\n", "\n", "高校の数学の教科書では以下のような筋道で $\\sin x$ の導函数を求めている. \n", "\n", "$0< x <\\pi/2$ のとき, 「面積の大小関係」によって\n", "\n", "$$\n", "\\frac{1}{2}\\sin x < \\frac{1}{2}x < \\frac{1}{2}\\frac{\\sin x}{\\cos x}\n", "$$\n", "\n", "が得られ, 全体に $2$ をかけて, 逆数を取って, $\\sin x$ をかけると, \n", "\n", "$$\n", "1 > \\frac{\\sin x}{x} > \\cos x\n", "$$\n", "\n", "となることから, 挟み撃ちによって\n", "\n", "$$\n", "\\lim_{x\\to 0} \\frac{\\sin x}{x}=1\n", "$$\n", "\n", "を示す. そのとき $\\ds\\frac{\\sin(-x)}{-x}=\\frac{\\sin x}{x}$ であることに注意せよ. これより\n", "\n", "$$\n", "\\frac{\\cos x - 1}{x^2} = \n", "\\frac{\\cos^2 x - 1}{x^2(\\cos x+1)} = \n", "\\frac{-\\sin^2 x}{x^2(\\cos x+1)} \\to\n", "-\\frac{1}{2} \\quad (x\\to 0)\n", "$$\n", "\n", "も得られる:\n", "\n", "$$\n", "\\lim_{x\\to 0} \\frac{\\cos x - 1}{x^2} = -\\frac{1}{2}.\n", "$$\n", "\n", "そして, 三角函数の加法定理\n", "\n", "$$\n", "\\begin{aligned}\n", "&\n", "\\cos(x+y) = \\cos x\\;\\cos y - \\sin x\\;\\sin y, \n", "\\\\ &\n", "\\sin(x+y) = \\cos x\\;\\sin y + \\sin x\\;\\cos y.\n", "\\end{aligned}\n", "$$\n", "\n", "を使って, $h\\to 0$ のとき\n", "\n", "$$\n", "\\begin{aligned}\n", "&\n", "\\frac{\\cos(x+h)-\\cos x}{h} =\n", "\\frac{\\cos x\\;\\cos h - \\sin x\\;\\sin h - \\cos x}{h} \n", "\\\\ &\\quad =\n", "h\\frac{\\cos x\\,(\\cos h - 1)}{h^2} -\n", "\\frac{\\sin x\\;\\sin h}{h} \n", "\\to -\\sin x,\n", "\\\\ &\n", "\\frac{\\sin(x+h) - \\sin x}{h} =\n", "\\frac{\\cos x\\;\\sin h + \\sin x\\;\\cos h - \\sin x}{h}\n", "\\\\ &\\quad =\n", "h\\frac{\\sin x\\,(\\cos h - 1)}{h^2} +\n", "\\frac{\\cos x\\;\\sin h}{h}\n", "\\to \\cos x\n", "\\end{aligned}\n", "$$\n", "\n", "となることを示す. これで\n", "\n", "$$\n", "(\\cos x)' = -\\sin x, \\quad (\\sin x)' = \\cos x\n", "$$\n", "\n", "であることが示された.\n", "\n", "しかし, 以上の方針は次の節の方針と比較すると, 非常に遠回りになっており, 弧度法の意味での角度の定義(単位円弧の長さで角度を定義すること)が不明瞭になっているという問題がある." ] }, { "cell_type": "code", "execution_count": 40, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "text/html": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "showimg(\"image/jpeg\", \"images/Jikkyo20140125limitsinc.jpg\", scale=\"60%\")" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### 曲線の長さが速さの積分になることの応用\n", "\n", "高校数学IIIの教科書には $(x(t),y(t))$, $a\\leqq t\\leqq b$ の軌跡の長さ(曲線の長さ) $L$ が\n", "\n", "$$\n", "L = \\int_a^b \\sqrt{x'(t)^2 + y'(t)^2}\\,dt\n", "$$\n", "\n", "と表せることが説明されている. $t$ を時間変数とみなすとき, 点 $(x(t),y(t))$ の運動の時刻 $t$ における速度ベクトルは $(x'(t), y'(t))$ になり, 速さは $\\sqrt{x'(t)^2 + y'(t)^2}$ と書ける. 上の公式は曲線の長さを速さの積分で表せることを意味している." ] }, { "cell_type": "code", "execution_count": 41, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "text/html": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "showimg(\"image/jpeg\", \"images/Jikkyo20140125ArcLength1.jpg\", scale=\"60%\")" ] }, { "cell_type": "code", "execution_count": 42, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "text/html": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "showimg(\"image/jpeg\", \"images/Jikkyo20140125ArcLength2.jpg\", scale=\"60%\")" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "-" } }, "source": [ "これを使えば(曲線の長さを上の公式で定義すれば), 三角函数の微分の導出を非常に簡潔な議論で行うことができる. そのことを以下で説明しよう.\n", "\n", "$(x(t),y(t)) = (\\sqrt{1-t^2}, t)$, $-1 display\n", "latexstring(raw\"\\ds\\left(\\frac{dx}{dt}\\right)^2+\\left(\\frac{dy}{dt}\\right)^2=\", sympy.latex(sol))" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "**問題:** 直線 $y=tx$ と単位円 $x^2+y^2=1$ の右半分の交点は\n", "\n", "$$\n", "(x(t),y(t)) = \\left(\\frac{1}{\\sqrt{1+t^2}}, \\frac{t}{\\sqrt{1+t^2}}\\right).\n", "$$\n", "\n", "原点を通る直線の傾き $a$ をそれに対応する弧度法の意味での角度 $\\theta$ に対応させる函数 $\\theta=G(a)$ が\n", "\n", "$$\n", "\\theta = G(a) = \\int_0^a \\frac{dt}{1+t^2}\n", "$$\n", "\n", "と書けることを確認せよ. これと逆に角度 $\\theta$ を直線の傾き $a$ に対応させる函数が $\\tan$ の定義なので, $a=\\tan\\theta$ の定義は $\\theta=G(a)$ の逆函数である. $\\QED$\n", "\n", "解答略." ] }, { "cell_type": "code", "execution_count": 44, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "text/latex": [ "$\\text{euqation:}\\; - t x + y=0,\\ x^{2} + y^{2} - 1=0$" ], "text/plain": [ "L\"$\\text{euqation:}\\; - t x + y=0,\\ x^{2} + y^{2} - 1=0$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\text{solution:}\\; x = \\sqrt{\\frac{1}{t^{2} + 1}}, y = t \\sqrt{\\frac{1}{t^{2} + 1}}$" ], "text/plain": [ "L\"$\\text{solution:}\\; x = \\sqrt{\\frac{1}{t^{2} + 1}}, y = t \\sqrt{\\frac{1}{t^{2} + 1}}$\"" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "t, x, y = symbols(\"t x y\")\n", "equ = [y-t*x, x^2+y^2-1]\n", "s = solve(equ, [x,y])\n", "latexstring(raw\"\\text{euqation:}\\; \", sympy.latex(equ[1]), raw\"=0,\\ \", sympy.latex(equ[2]), \"=0\") |> display\n", "latexstring(raw\"\\text{solution:}\\; x = \", sympy.latex(s[2][1]), \", y = \", sympy.latex(s[2][2])) |> display" ] }, { "cell_type": "code", "execution_count": 45, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "text/latex": [ "$\\ds X=\\sqrt{\\frac{1}{t^{2} + 1}},\\quad Y=t \\sqrt{\\frac{1}{t^{2} + 1}}$" ], "text/plain": [ "L\"$\\ds X=\\sqrt{\\frac{1}{t^{2} + 1}},\\quad Y=t \\sqrt{\\frac{1}{t^{2} + 1}}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\ds\\left(\\frac{dX}{dt}\\right)^2+\\left(\\frac{dY}{dt}\\right)^2=\\frac{1}{\\left(t^{2} + 1\\right)^{2}}$" ], "text/plain": [ "L\"$\\ds\\left(\\frac{dX}{dt}\\right)^2+\\left(\\frac{dY}{dt}\\right)^2=\\frac{1}{\\left(t^{2} + 1\\right)^{2}}$\"" ] }, "execution_count": 45, "metadata": {}, "output_type": "execute_result" } ], "source": [ "X, Y= s[2][1], s[2][2]\n", "sol = simplify(diff(X,t)^2 + diff(Y,t)^2)\n", "latexstring(raw\"\\ds X=\", sympy.latex(X), raw\",\\quad Y=\", sympy.latex(Y)) |> display\n", "latexstring(raw\"\\ds\\left(\\frac{dX}{dt}\\right)^2+\\left(\\frac{dY}{dt}\\right)^2=\", sympy.latex(sol))" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "**問題:** 以下のセルの画像を解読して, 双曲線函数の微積分の理論について整理せよ. $\\QED$\n", "\n", "解答略." ] }, { "cell_type": "code", "execution_count": 46, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "text/html": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "showimg(\"image/jpeg\", \"images/sin-sinh.jpg\", scale=\"80%\")" ] }, { "cell_type": "code", "execution_count": 47, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "text/html": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "showimg(\"image/jpeg\", \"images/hyperbolicsine.jpg\", scale=\"80%\")" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### 楕円積分, 楕円函数, 楕円曲線暗号\n", "\n", "$y = \\sin\\theta$ の逆函数 $\\theta = F(y)$ は\n", "\n", "$$\n", "\\theta = F(y) = \\int_0^y \\frac{dt}{\\sqrt{1-t^2}}\n", "$$\n", "\n", "と書けるのであった. これの次の拡張は非常に有名である:\n", "\n", "$$\n", "u = F(y,k) = \\int_0^y \\frac{dt}{\\sqrt{(1-t^2)(1-k^2 t^2)}}.\n", "$$\n", "\n", "これは**第一種楕円積分**と呼ばれており, その逆函数は $y = \\sn(u,k)$ と書かれ, **Jacobiのsn函数**と呼ばれる**楕円函数**の有名な例になっている. $\\sin\\theta=\\sn(\\theta,0)$ なので, sn函数はsinの一般化になっている.\n", "\n", "竹内端三著『楕圓函数論』は著作権が切れており, 現在では無料で入手可能である:\n", "\n", "* 原著の画像ファイル\n", "\n", "* LaTeX化 (PDF)\n", "\n", "* 現代語訳\n", "\n", "以下, $k$ を略して, $y = \\sn u=\\sn(u,k)$ と書く. cn, dn, cd 函数を\n", "\n", "$$\n", "(\\cn u)^2 = 1 - y^2, \\quad\n", "(\\dn u)^2 = 1 - k^2 y^2, \\quad\n", "\\cd u = \\frac{\\cn u}{\\dn u}\n", "$$\n", "\n", "を満たすように定義すれば, $(x,y)=(\\cd u, \\sn u)$ は\n", "\n", "$$\n", "x^2 + y^2 = 1 + k^2 x^2 y^2\n", "$$\n", "\n", "を満たしている. この等式は**楕円曲線のEdwards形式**と呼ばれており, この方程式で定義される平面曲線は**Edwards曲線**と呼ばれている. Edwards曲線は $k=0$ の場合に単位円になるので, Edwards形式のもとで楕円曲線は単位円の一般化になっていることがわかる. \n", "\n", "$(x,y)=(\\cos\\alpha,\\sin\\alpha)$, $(X,Y)=(\\cos\\beta,\\sin\\beta)$ のとき, 三角函数の加法公式より,\n", "\n", "$$\n", "(\\cos(\\alpha+\\beta),\\sin(\\alpha+\\beta)) = (xX - yY, xY + yX).\n", "$$\n", "\n", "この公式は次のように拡張される: $(x,y)=(\\cd u, \\sn u)$, $(X,Y)=(\\cd v, \\sn v)$ のとき,\n", "\n", "$$\n", "(\\cd(u+v), \\sn(u+v)) = \n", "\\left(\\frac{xX-yY}{1-k^2xXyY}, \\frac{xY+yX}{1+k^2xXyY}\\right)\n", "$$\n", "\n", "を満たしている. これは $k=0$ の場合の三角函数の加法公式の拡張になっている. この公式は本質的にJacobiの楕円函数の加法公式である. この公式の代数幾何的な証明については次の論文を見よ:\n", "\n", "* Thomas Hales, The Group Law for Edwards Curves, arXiv:1610.05278\n", "\n", "楕円曲線のEdwards形式の理論は単位円と三角函数の理論の楕円曲線と楕円函数の理論への拡張になっている. \n", "\n", "楕円曲線のEdwards形式における加法公式は楕円曲線暗号に応用されることによって我々の社会の中で役に立っている. 楕円曲線暗号の規格 Ed25519 について検索してみよ. このように, 高校のときに習う三角函数論は楕円函数論に自然に拡張されており, 三角函数の加法公式の楕円函数の加法公式への拡張は楕円曲線暗号の形で我々の社会の中で役に立っている.\n", "\n", "19世紀のJacobiによる楕円函数に関する研究が約200年後のコンピューター社会でプライバシーを守るための暗号技術に使われることをJacobiは予想できていなかったはずである. " ] }, { "cell_type": "code", "execution_count": 48, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle 0$\n" ], "text/plain": [ "0" ] }, "execution_count": 48, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Edwards曲線の公式の確認\n", "\n", "k, y = symbols(\"k y\")\n", "x² = (1-y^2)/(1-k^2*y^2)\n", "y² = y^2\n", "simplify(x² + y² - 1 - k^2*x²*y²)" ] }, { "cell_type": "code", "execution_count": 49, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "image/png": 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gIAldCCGECAJuJfSXXnqJiIgIDMNg+/btt91uz5491KpVi8jISKpVq8aOHTvc2a0Qwg9JfSCEtdxK6G3btuXXX3+lePHiqW7XtWtXunTpQkJCAv369aNz587u7FYI4YekPhDCWm4l9Lp161K0aNFUtzl69Cjx8fG0b98egNjYWBITE9m/f787uxbedOECbNuGbe9eqyMRAUTqg+Bz8rLihySTBQcdbD+pUEpZHZJIhcPbO0hKSqJw4cI4HHpXhmEQHh7OgQMHiIiIuO3zevbsSZ48eQBo06YNsbGxnDp1ytvhekWgxB2ybBlZp08nZPlyzLvv5q7ERJylS5PcsiWXX3wRde3z8HeBUt43ulXcYWFhFkTiPRmpD6Qu8L3jVwwGb8/CzD9DcCoDCIX1TsrmdjG4whUaFAqMGdgCpbxvdLu471QfeD2hgz5p/ykt3/LGjh1LpUqVbno8UCs4v47b6YTevWHMGKhcGQYOxP7QQ5zbupVcS5fi+PRTQmfNgnnzoGJFq6NNE78u71QEatzpkd76QOoC31p/1OSR5S6cJoyoZqNNhA3z4ikSku9i5BaDtr/a6VfBxvBqNmw3fJb+yN/L+3YyErfXE3qxYsU4ePAgTqcTh8OBUoqkpCTCw8O9vWuRFi4XdOgAs2bBuHHw4otw7SRNLlZM/23/fmjTBurXh59+0klfiAyQ+sC/rTti0mixi6h8BnMa27k7u64LTjrh4bttPFTU4MNtJn3Xm5xNhvExtpu+oAnreH3YWsGCBYmOjmb69OkAzJkzh4iIiFSb24UPvf46fPstzJwJ3bunJPN/iYiAn3+GyEho0QIOHPB5mCI4SH3gv/aeUbRc6qJSPoOfmv1/Mv8nwzDoU8HOpDp2Juw0eW+LaUGk4nbcSug9evSgaNGiHDx4kMaNG1OyZMmUvzVr1oxNmzYBMHHiRCZOnEhkZCQjRoxgypQp7kUtPGPqVPjgAxg9GmJjU982Tx5YsACyZoVWreDSJd/EKAKG1AeB6+xVRfMlTsKywrwmdkIdqV91dy5t451KNt7YaDInUZK6vzCUn3VbjI+Pp3LlysTFxd103+zkyZMBeT/EL+PetQsqVYKnnoJJk255ZX7LuLdtg2rVoFMn3UTvh/yyvNMgUOP2FqkLfOe5X5zMSVTEt3ZQMk/a6gKlFO2Wu1h+SLG1jYOiOf2v6d1fy/tOMhq3zBSXGSUn60RevDh8/PGtm9lvp3x5+PBDGD8evv/eezEKIXxiTqLJ5wmKT2rZb5nMb8cwDCbWthNqh2d/cWH617VhpiQJPTP66CPYsgW+/BJy5Ej/87t1g6ZN9T338+c9Hp4QwjeOXVJ0XeUitoRBx/vTf4Udls1gWn07yw8pxv0uTe9Wk4Se2SQmwsCB8NJLUKVKxl7DMODTT+HYMXjnHc/GJ4Twmb7rXZjApzH2DPdWb1zERtfSNgZsMjl0Qa7SrSQJPTNRSl9V588P777r3muVKAGDBukm+7g4j4QnhPCdlYdNpu1RvFfNToFQ9+5/D69mI5sdeq8LjAlngpUk9Mxk3jz48UcYOxZy5nT/9V59FcqV018S5P6ZEAEj2VS8+KuLmgUNOpdyvzNb3qwGH9SwM/MPxU9/SdO7VSShZxZXr0K/fvred6tWnnnNkBB9hb5hgx7HLoQICBN3muw8DeNr2z0221v7kga1Cxn0XufCZcoXfCtIQs8sPv0U/vgDRo3y7OvWr6+/IPTvD5cve/a1hRAed+qKYlCcSadSBlH5PDfUzDAMPqxpY9tJmJogCd0KktAzg1OnYPBg6NxZN5F72siRcPAgfPKJ519bCOFRQ38zueyCd6vYPf7aVQvYeOo+g7c3uTifLEnd1yShZwZDh8KVKzqpe0OpUnoo29Chuue7EMIv7Tur+OR3k/4Vbdxzi6ldPWFYVTunrsJImRbW5yShB7s//tCrqL3+Otx9t/f2M3Cg/tdbXxqEEG7rv8FFoVDoXcF7VX/xXAavlrMxaqvJwfNyle5LktCDXf/+UKAA9Onj3f0UKAADBsCECbB7t3f3JYRIt1//NpmdqBhe1U72O8zV7q43omzkDIG3NskwNl+ShB7M1qzRy6IOHQrZs3t/fy+9BEWK6N70Qgi/oZSizzqTyvkNnirp/TnXc2cx+E9lG1/sUfx2XK7SfUUSerBSSl+VR0XpNc19IVs2GDFCr8q2YoVv9imEuKNv/1BsOKYYVd3msWFqd/JCaRul79KTzfjZGmBBSxJ6sJo1C9at08uj2nz4MT/+uF6NrU8fMKVTjBBWu+JS9N/gomW4Qf3CvqsLHDaD96vbWXFY8f0BSei+IAk9GF2+rO+dt2gBDRv6dt+GoVdji4+H6dN9u28hxE3G/W6SdAHeq+b5YWp30qyYQcPCBn03uHDKZDNeJwk9GI0ZAwcOwPvvW7P/mBho2xbefBMuXrQmBiEEJy8r3v3NpEtpG2Xy+n69csMwGFXdzu7TMGmXtNh5myT0YHPsGAwZoseFly5tXRwjRsDRo/pqXQhhiSG/mbgUDKpsXVUfnd/gmfsNBsaZnL0qV+neJAk92PznP7rZ+/q4cKvcd5/u9T5iBBw+bG0sQmRC+84qxu4web2ijYJurqbmriFV7JxLhvdkshmvkoQeTHbt0uPABwzQ48KtNmAAZM0qa6YLYYE3NrgoGAqvlre+mi+a06BPeRsfbjNJkslmvMb6T1p4Tt++EB4OvXpZHYmWN69eM33KFNi61epohMg01h4xmZWoGFrF+5PIpNXrFW3kzgIDNspkM94iCT1Y/O9/sGiRbuLOls3qaP5ft25w//3w2muyZroQPqCU4rX1JlH59JKm/iJXFoPBlW18uVcRd0zqAm+QhB4MXC497rtmTXjsMauj+beQEN3bftkyWLzY6miECHpf71OsOaIYVd2O3eY/CR2gcykb5fJCrzUuTPmC73GS0IPBxImweTOMHq07xPmbli2hUSN4+WVZM10ILzpzVfHaOhePlTBoVMT/qneHzWBsjJ21RxUjNksHOU/zv09cpM/Ro7rz2fPPQ/XqVkdza4ahx8b/+SeMGmV1NEIErUFxJmeT4YMavp9EJq3q3aPTzoBN0kHO0yShB7r+/fXUrsOHWx1J6sqUgd699UIx+/dbHY0QQWfbScWY303eibZRLKcfttT9w5pW+gtH+DdOiyMJLpLQA9nq1TB1qk7m+fNbHc2dvf22jvPll62ORIigopSix2oXJXP7xzC1O6lZyEa+rPr/0/dI07un+P8nL27N6YQePaBqVejc2epo0iZnTn2ff8EC3SNfCOERX+9TrPpbMaaWnSx2/746v+7PJx0AdFghHeQ8RRJ6oPrkEz22e9w4sPvv/bKbxMbCgw/qsfLnz1sdjRAB79QVRZ91LtqWMHiwaOBU6TlCDN6K1vHGLJCx6Z4QOJ+++H9798Jbb+mpVatWtTqa9DEM+PRTOHJEd+YTQril9zoXl5zwUc0A+mJ/zbtVdMzrjio2HZOmd3dJQg80pql7tN99t+5gFojuuw+GDdM931evtjoaIQLWkiSTzxMUH9awUyRHYDS132jv47rpveo8F0qa3t0iCT3QfPYZ/PILTJ4MOXJYHU3G9eoFNWpAp05w6ZLV0QgRcM5dVXT51UXjIgadSgVmMge4L7eRMqNd7E/S9O4OSeiB5MABPV97ly7QsKHV0bjHbtdzvO/fr+d7F0KkS/+NJscvw2e17Rj+OKFUOnzZQF+lz92v2HZSrtIzShJ6oHC5oEMHveDJyJFWR+MZZcro5V7ff1+3Oggh0mTxAZPxO0zer26jRO7ATubXbYvVSb3CHKf0es8gSeiBYsQI+PVXmD4d8uSxOhrP6dsX6taF9u3h1CmroxHC7x25qHhupYvmxQxeLBM8VXi5MIMeZfX7qTBHJpzJiOA5GoLZ+vUwcCC8+aZOfsHEbocvv9RD2Lp0kRXZhEiFUopOK/V95v/WC/ym9huNjdG93n8/Bd8lSq/39JKE7u/OnoWnnoIqVeCdd6yOxjuKFYNJk2D2bD3znRDilsbtMFmcpPi8np2CocGVzK873kE3vcf+5OL0FfmCnx6S0P2ZUno98WPH4Ouv9VKkwaptWz3jXa9esGOH1dEI4XfWHzXpvc7kpQdsPFwseKvufNkMPq+nr9TzfiFN7+nh9lGxZ88eatWqRWRkJNWqVWPHbSrjiIgISpcuTVRUFFFRUcycOdPdXQe/MWPgm2/0ULV777U6Gu/7+GMoUQJat4YzZ6yORqST1AXec+ySou1PLqrkN3i/evAm8+s6Rtqoe7dugai9QJJ6Wrl9ZHTt2pUuXbqQkJBAv3796JzKvOKzZ89m8+bNbN68mccff9zdXQe3lSuhTx+9QtkTT1gdjW/kyAFz58Lff0PHjnoSHREwpC7wDqepeOJ/Lq6aMKtx4MzV7q5fWuqm99VHFFN2SV2QFm4l9KNHjxIfH0/79u0BiI2NJTExkf2yPKZ7Dh2Cdu2gdm147z2ro/Gt++/XPfnnz9c9+0VAkLrAe97aZLLisGJGw8CdDS6jTnfUSf35VS52npL76XficOfJSUlJFC5cGIdDv4xhGISHh3PgwAEiIiJu2v7pp5/GNE2qV6/O8OHDKVCgwG1fu2fPnuS5NjyrTZs2xMbGcipAhzWlK+4LF8jdqhU2m40zEyagzp71XmB3YFl5x8QQ2rcv2d56i/PFipHcvHm6nh5Mx0lYWJgFkaSf1AVpk964v9rv4L0toQwuf5mK2ZI5edJLgd2BleW9sK6dliuzU3a2k8RW58idjq5EwXac3Kk+cCuhAzcNm7jdXLwrV64kPDyc5ORk3nrrLTp27MjixYtv+7pjx46lUqVKNz0eKBXcjdIUt8sFzz2nF19ZtYq8pUp5P7A7sKy8R4yA/fvJ1bUrrFgB1aql6+lBfZz4KakL0iatcf/vL5Pe8S5eKG3wVvWclg9Rs6q8W4TBx8kuXl5rUmJBLlzPO7CloyyC/Tj5J7ea3IsVK8bBgwdxOnWnBaUUSUlJhIeH37Tt9cdCQkJ45ZVXWLVqlTu7Dk69e+t1wmfOhKgoq6Oxls0G06bpcmjZEhITrY5IpELqAs/aeUrR5icXDQobjIsJvvHm6fVSOTuP36vLwD5ZOsndjlsJvWDBgkRHRzN9+nQA5syZQ0RExE1NbBcuXOD06dMpv3/zzTdER0e7s+vgM3q0XuN83Dho1szqaPxDaKi+l54rly6TEyesjkjchtQFnnP4oqL5EidFc+hOcCG2zJ3Mr5vRyEGeLPr/teZLUr8Vt3u5T5w4kYkTJxIZGcmIESOYMmVKyt+aNWvGpk2bOHLkCA0aNKBChQqUL1+eX375hS+++MLdXQeP//5XX53366fHnYv/V6AALF4Mx4/DQw/JcDY/JnWB+45fVjT+3kmyCd83dZAniyTzfzr1jL5LvPaoInaZJPUbuX0PvVSpUqxdu/aWf/vnfbHffvvN3V0Fp2++0eubd+smvbpvJzISli7VK8w1bw5LlgT20rFBSuoC95y5qmj6g5PjV2BlCwfFc0kyv5FhGCR3dhAyxcl3+xUv/uri09p2q8PyG8E/Q4E/mzdPr6D2zDO6qT2T3ydLVXQ0/PgjbNkCjz4Kly9bHZEQHnMhWdHsRxeJ52DZww5K3SV1we04bAaXntPXohN2mry9SdZQv04SulUWLoTHH4fYWJg8WXcCE6mrXl13Gly9Gtq0gUuXrI5ICLedvaqT+daTih8fslMhnyTzO8nmMDhzbYz6kN9M+q2XpA6S0K3x9dd6etOWLfUkKg6373xkHvXqwYIFev30hx/Wi9cIEaBOXFY0Xuxiy0nFkoftVCsoVXJa5c5icPLaPfX3t5p0/kXuqcvR42uffqrX/n7mGZgxI7gXXPGWxo1h2TLYvBkaNdId5oQIMH9fVNRf5CTxnOLn5g5qFZLqOL3yZjU4/6xO6v9NULRakrmTuhxBvjRiBHTvDi+9pJvZ5co842rV0hPOHDigr9r/+svqiIRIs8SzijoLnZy81gEuOr80s2dUjhCDy510XbrwgKLCnGSLI7KOJHRfcDr1sqBvvAEDB+ox53LP3H1RUbBqFZw7pxP8tm1WRyTEHa09YlL92jjqVS0dlMkrydxdWe269zvAtpNgTErGaWa+ud8lq3jb2bPkfPpp3dQ+YQIMGiS92T0pMhLWroV8+SAmBn74weqIhLiteQcdNPjeRak8BmsfcXBvbqkLPMVhM1AvhFAyt/49ZIqTE1cyV/lKQvemP/+E2rVxrF+vE03XrlZHFJyKFNHLzdavDy1akHXSJLjNPOJCWEEpxfDNLjqvD6VtCYOfmtvJny1zJRtf2fN4CC+U1mUbuSgn8cczT10gCd1bli2DypXh3DnO/vgjPPig1REFt5w59VrqL79Mjv799SI3MqxN+IFzVxXtlrt4c6NJ3zJX+LK+nayZZE1zq3xWx8HkOnrCmcpznQz7LXMMa5OE7mmmCcOGQdOmUKUKbNqEWbq01VFlDnY7fPgh5z/9FL79Vt9X/+MPq6MSmdiOU4qq85wsOaj4rrGd/mWvZvqFVnylc2kbKxtfAGDAJpPwr5NvuwJgsJCE7kknTujx5QMGwFtvwfff63u7wqeutmun76ufPatbSRYssDokkQnN3GdSbZ6TEBtsetRB6xJS3fraA3nMlFnlki6AbbKTQxeCN6nLEeYpP/8MFSvqXtcLFsDgwfqKUVijYkXYtAnq1oVHHoEePaQJXvjE+WRF51+cPPE/F49GGKx7xEGkTOVqmWwO3Vnuyfv0Z1DkaycfbQvOJnhJ6O5KToY339QTnNx/P2zdqmeAE9bLm1fPlz92LEyZAlWrytA24VXrj5pEfedk5h+KKXXtfFnfTo4QSeb+4OuGDpY+rC+yXl1nEjI5mSuu4Lpal4Tujt9/10OlRo6EoUPhp5+gaFGroxL/ZBj66nzTJv3/qlXhgw/AFZzf0IU1nKbi3XgXMQtc5M9msLmNg06lbHK/3M88WNTGxWtN8E4F2f7r5Ke/TIuj8hxJ6BnhdOqOb5Uq6UlNVq/Wk8ZIE7v/KlcONmyAF1+Evn2hdm3YtcvqqEQQ2HJCUWO+i0HxJm9G2VjV0k7JPJLI/VXotSb4kdV0+ntwsYt7ZySTHAQT0UhCT69t26BGDXj7bXj1VfjtN70KmPB/oaF6lr5Vq3QHxqgoeO89/QVNiHS64lK8vclFlblOrrgUa1vZGVzFTohNknkg6FvRzqGn9dV64jnIMsXJ3MTAvlqXhJ5WFy5A//76qvzSJd2LesQIyJbN6shEesXE6HXVe/bULSvVqsG6dVZHJQLI6r9NKn3n5L0tJgOibcS1dshKaQHonuz6an1YVf3ZtfnJRbb/JnP2amBercsReCdK6Y5VZcrAxx/rK/P4eJ0EROAKDYVRo/QXM4CaNaFLF33lLsRtHLmoeHaFk9oLXeQIMYhr7WBQZTtZZKKYgPZGlJ0THfTV+hUX5JnmpPfawOtnIwk9NX/8oXust24N5cvrTnDvvANZs1odmfCU6tVh40bdE/7bb6F0aZg6VU8QJMQ1TlPxyXYXpWY5WXhAMbG2nbWt7JQPk0QeLMKy6av1bxvpvlCjt5sYk5JZdThw6gJJ6Ldy6RK8+y488IAehjZ3LixaBPfea3Vkwhvsdt0TftcueOgh6NQJ6tTR/SNEprfqsEnluU5eWWvyxL02Eto56FLGhl3ulQelx+61YT7v4JHi+vOtu8hFzqmB0QwvCf2fTBO+/FKv4PXuu/DKK7BzJzz6qKyQlhncfbf+/H/+GU6f1rPMPfssHDxodWTCAgfPK5752UndRS6y2Q02PGpnQh07+WRRlaBnGAbzmjj46yndDH/BqZvhO//i9OvpYyWhX7dihR6j/Mwzuhf7jh0wfDjkyGF1ZMLX6tfXnebGj4fFi/UXvLff1kMURdA7d1Xx1kYX93/r5MeDikl17Kx9xE6VAlJdZjaFc/y7Gf6/CQrbZCcTdvjn/XU5QnftglatoEEDCAmBX3+FWbOgZEmrIxNWcjigWzfYu1cPTxw1Ss8E+NlnMswtSDlNxYQdLkp+6+SDbSa9y9vY+7iD50vbsEkLXaZ2vRn+pQd0ynxxtb6/vtLP7q9n3oR+9Ki+b1qunB5bPmOG7vEcE2N1ZMKf5M6tZwHcvRuaNNFr2lesqK/c/bjpTaSdUopFf5pUmOPkxdUmDxU1SGjnYGhVO7mzSCIXmmEYfFzLzuVODqoX1MdFvUUujEnJ/HnOP+qCzJfQL13STeklS8JXX+mJRXbtgscfl/vk4vbCw+GLL/QUsgUKQPPmOsFv3mx1ZMINvx1XNF7souVSF3eH6mFo0+o7KJZT6gJxa1ntesGdw9cmpQGImOEkcmYyF53WJvbMk9BNE6ZPh1Kl9NCzTp1g3z7o00eGoYm0q1xZd5qbPx+SkvREQ889B3/9ZXVkIh0OntfjySvPdXLoomJhEzvLm9uplF8SuUibu69NSrPxUX1/fc9ZyDHVyTM/W9dxLnMk9BUr9EQwHTrojm87dsBHH8la5SJjDEP3u9i2TY9fX7RI319/5x3pOOfnzifr6Vojv3WyOEkxLsbGtlgHLYrLQioiY6oUsKFeCGF6A53Yv9yrO859akHHueBO6Hv36klhGjTQY41XrYI5c3TlK4S7QkKge3d9nL38sl51LzIS/vtfmZjGz7hMxeRdJiVnOnl/q8nL5XSHtxfL2nHIeHLhAU+X1Im9d3mdVrtf6zi35ojv6oLgTOhnzugVtcqWhbg4fa987Vq9wpYQnpYnj+6XsXu3HvLWuTNUqQIrV1odmQCWHjSJnuvkhVUuGhU22P2Yg+HVpMOb8I4PauiOc1Wu3b6JWaA7zh2+6P1m+OBK6C4XTJyor8DHj9djh3ftgqeeAltwvVXhh4oXh2++0cvphoRAvXrQti0kJlodWaa054yi2Y9Omv7gIk+IwfpH7HzV0EHxXJLIhXdltRtsbO3g4FP/33Gu8FdOGn/vxPTi/fXgyXK//aYnhOnWDR5+GBISdELPnt3qyERmU6uWbhH68ku9ilvZsnplvuRkqyPLFC47FYPiXJSb7WTnKcWcxnZWtrTLamjC54pcm5hmVUt9f335IYV9spPZf3inGT7wj/BLl3RP9SpV4PJlfXU0bRoUKWJ1ZCIzs9mgfXvdQtSjB7z1lu4Rv3691ZEFtZWHTcrPcTJss8lrFWz8/piDNiWkw5uwVu279f31fhV0yn1suW6Gv5Ds2av1wE7oW7boYUTjx8OwYXpZ01q1rI5KiP+XM6eeZW7TJr1ka0wMDB4ss8152FWX4o0NLuov0uPJt7TRE8Nkd0giF/7jvep2znR0cP2ozPm5k3n7PXe1HrgJfdw4PRQtJERXlq+/rv8vhD+KitKtR2++Cf/5j+48d/iw1VEFhQPnFbUWuBi11WRYVRsrWtgpk1cSufBPubMYmC+EMKG2Tr+tl7lo9L1nvuAHXkI3TT23ds+eehrODRv0MqdC+LuQEH11/ssvuqNczZp6NT+RYZtPKGrMd3LismLdIw76R9llWVMRELqWsXOsg+40979DCmNSstsd5gIrobtc+r7kJ5/oCT0++URmeROBp3Zt3VkuVy7dBB8XZ3VEAWnVYZM6C50Uya6n4qxcQBK5CCz5sxmYzzvIfa1x2T7ZvV7wAZXQQwcPhpkz9U+PHlaHI0TGFSumJzqKjIRmzWTq2HRKPG/QaqmLKvkNVrSwUyi7JHMRmAzD4MyzIdS7Rx/D2admvPk9cBL6qlWEjh2rZ+Nq29bqaIRw3113wcKFehbDrl2tjiZgKKXoFZeNsKwwr4mdHCGSzEXgW9FCN79fccFX+x132PrW3Eroe/bsoVatWkRGRlKtWjV27Njh1napevddnBUr6vvnQgSLAgXg44/h+++xB/jKbb6qD1YcVqw97mBcjJ08MtubCCKXntOJ/KW40Aw9362E3rVrV7p06UJCQgL9+vWjc+fObm13W+fPw/LlXOnQQWZ8E8GnTRsICyPLkiVWR+IWX9UHC/5UFM1u0rSoJHMRXLI5DPJd6xZ21ZX+e+kZzo5Hjx4lPj6e9u3bAxAbG0tiYiL79+/P0HapOnIETBNXiRIZDVcI/2W3Q4kSGAE8jM2X9cFfFxQlcpgyWYwISt3L6rR85FL6n5uxhnogKSmJwoUL43DolzAMg/DwcA4cOEBERES6t7tRz549yZMnDwCPP/wwHQ2Dy7t34zx5MqMhW+bUqVNWh5AhErePmCZ3HTjAperVuXjD8R0WFmZRUOnjzfrgn3VBmzZtuOu+p1h3zuDEiZMEWk4PuGPzGonbd7K6QggxspLl8ilOXv333+5UH2Q4oQM3fUO+3aLuad3un8aOHUulSpX+/4HZs8k7fz4h/fsTcGcxgVMx30ji9oFFi+DYMewtW5I7kOK+gbfqgxvrgmJJJpP2ufj9ajbq3hN4t+AC6tj8B4nb+0yl+Pagk4aFnBTKn/64M3w2FCtWjIMHD+K8NoWlUoqkpCTCw8MztN0dvfUWIevXw+TJGQ1ZCP9z+jS89BI0aICzalWro8kwX9YHTYoaVLzLRffVLi46vb8kpRC+MnqbydaT8ErpKxl6foYTesGCBYmOjmb69OkAzJkzh4iIiJuazdK63R01acLlZ5+F7t1h/vyMhi2E/zh3To9BP31af1ENwJan63xZH9gMg/FVL5N4Dp78nytDnYeE8DdzE036bdCLClXLl8H53ZUbdu3apWrUqKHuv/9+VblyZbV9+/aUvz388MNq48aNd9zuRnFxcQpQcXFxN/3txJEjSj32mFIhIUrNmeNO6D514sQJq0PIEInbi06cUKpOHaVy51bq2nkSEHGnwtP1Qap1wYkTavEBl8oy+apqsDBZnbpseudNeVigfsYSt3dN2OFUtklXVbufkpXTZWY4brfuoZcqVYq1a9fe8m+LFy9O03bp4nDAV19Bhw4QGwtvvAHvvqt7CQsRKH77TR+/Z87Ajz/qpX+DgK/rg4eL2VjWDB5d5iJmgZN5TRzcnydwWzlE5nPVpRiwyWTUVpOeZW18VNPm1loEgdejJCQEvvlGzxj33nvQpAkcOGB1VELcmWnqVQJr1YKwML3cb82aVkcV0OreY2NtKwfJJkR952TiTleaOt0KYbXdp/UqgR9tM/mguo1ParmXzCEQEzroe419+8JPP0FCgl5tbeJEkBNZ+Kt9+6BhQ71KYKdO8OuvULy41VEFhVJ3GcS3cdC+pI1uv5q0XOIi6bzUBcI/uUzFmO0uKs11ci5ZsfYRO70r2D0yr0JgJvTrGjSA7dvhySehWzeoW1evjS6Ev7hwQS+ZWqGCbklavlxfpWfLZnVkQSVniMHEOnYWNLETd1xRZpaTUVtdJJuS2IX/2HTMpPp8Fy+vNXk20kZ8awdVCnguDQd2QgfIkwc++0xXlKdPQ9Wq8MwzcPCg1ZGJzMw04YsvoFQpGDpUrw64dau+Shde07K4jV3tHHQuZeP1DSaV5zr55XAGewwL4SHHLim6/+qi2jwXLqWvysfFeH5hocBP6Nc1bKg7G332GSxZopelfOcdPQ+8EL60YoX+Ytmxo17vfNcu3ecjZ06rI8sU8mQx+LiWnY2POsjuMKi/yEXrpU72nJGrdeFbl52KkVtclJzp5Ot9Jh/UsLHxUQfVC3on9QZPQgfdC/6FF2DPHr0q28iRcP/98N//gstldXQi2O3ZA61b61tBISGwejXMnAmyBoElKuU3WNPKztcN7MSfUJSd5eTVtS5OXpbELrxLKcWMfSalZzl5c6PJM/fb2Pu4g1fL23G42fEtNcGV0K/LnVs3c+7era/cO3eGSpV0s7wQnnbypP4CWbas7rn+zTewdq3uzS4sZTMMnixpY9djDt6tYmPKbpOS3zr5aJtMSCO8Y80Rk5oLXDz5PxcV8xn83tbBmBg7+bN5f0hlcCb064oX1+PW163TzZ2NG0PLlroJVAh3Xb0KH30EJUvClCl6ToRdu+CJJwJ61rdgFOow6B9lZ087B+1K2Oiz3qTsbCdzEk0Z5iY84o+zinY/OYlZoL8s/q+5nflNHJS6y3d1QXAn9OuqV9fDhGbNgt9/h3LloFcvOH7c6shEIFIK5s3TwyX79IF27XRze//+EBpqdXQiFYWyG0yoY2drGwel8hi0/clF7YUu1h2RjnMiY05eVvRe66L0LCdrjiqm1bOzqbWDBoV9n14zR0IHfcXUti3s3AkjRugeyCVLwqhRcCVjE+GLTCg+Xt8jb90a7rsPtmyBCROgUCGrIxPp8ECYwfcPOVjWzM6FZEXNBS7a/eRk31m5Whdpc8WlGL3NRclvnUzabTKwko2Edg6eibRhs6iFLvMk9OuyZoXXXoO9e6F9e31VVaYMzJ4tE9OI2zt4UPdar1JFt+z88IOetrVcOasjE25oXMRGXGsHn9ezs+aoHr/eWzrOiVQopZj9h0nZWU5eW2/SroSNPe0cDIi2k91h7a22zJfQrytQAMaOhW3bdGemxx6DOnUgLs7qyIQ/uXgRBg3SwyB/+AE+/RQ2b4aHHrI6MuEhdptBx0h9dTWwko1Ju03um+nkw60urkjHOfEPa4+YxCxw8dhyF2XuMtgW62BCHTt3Z/ePPjOZN6FfV6YMLFoEy5bpxTKqVtVD344etToyYSWldJ+LMmVg+HC9ZvnevdC1qx4eKYJOdofBgGg7e9s5eOI+G/02mJSZJR3nBCSdVzy+3EmtBS4uuRQ/NbOz6CEHZfP6RyK/ThL6dY0b64lpxoyBOXP0Fdno0ZCcbHVkwte2btX3ydu1g6go3ZFyxAg9HFIEvULZDT6tbWdbrIOyd+mOc01/cLH7tCT1zOaKSzF8s+7wtvKw4vN6duJaO2hUxD9Tp39GZRWHQ0/RmZAATz2l77VXqABLl1odmfCFEyege3eIjoa//9b3yOfP150nRaZTJq/BooccLGxiZ+9ZRfk5Tt7c6OJCsiT2zGBJkkn52U7e3mTSrYyN3e0cdLSww1taSEK/lfz5Yfx43aO5YEFo2lT3kD90yOrIhDcoBZ9/rudd/+oreP99fZXetKnVkQk/0KK4jd/bOhgQZWP0NmmGD3b7zylaL3Xy0I8uiuQw2NLGwQc17OTO4r+J/DpJ6KmpWFHPy/3117Bqlb6f+umneuENERwSEqBRI3juOd3RLSEBeveGLFmsjkz4kVCHwcDKdn5v6yAqn26Gf+gHF4kyzC1oOE0973qZWU42HFN809DO/5rbeSDM/xP5dZLQ78Qw9PKsO3fqe6rdu0Pt2nrZVhG4rl6FIUP0LZX9+/WCPtOny3hykap7cxssaKqb4XefUZSb4+ST7S5MuVoPaFtOKGrMd/HGRpPuZfVUwU/cZ/PIGuW+JAk9rcLCYNIk+OUXOHVKzw0/YoQs+hKItm3TswcOGgSvvKK/nDVpYnVUIoC0KG5je1sHnSJtvLzWpO5C6TQXiK64FG9vclFlrpPLLsXaVnY+qGEnVwA0r9+KJPT0qltXj0Pu3RsGDNBX6wkJVkcl0sLl0l/CKlfWoxc2bNC/Z89udWQiAOUMMRgTY2dlCztHLykqfudk9Da5Wg8U208qqs5zMmKzyYBoG/GtHVTz0rKmvhLY0Vsla1adCFat0rOGRUXpSWrkRPZf+/bpL19vvqlXRouL060sQripzj02tsQ66F7GRu91Jg/94OLQBakL/JWpFB9vd1FlnhOlYFNrB4Mq28liD8yr8n+ShO6OWrX01XqnTnqxl9at9VKawr/MmKGHoh07phfpee89/aVMCA8JdRh8WNPO0oftbD+lqDDHybz90nnW3xy6oHjoBxevrNVD0TY+6qBivsBP5NdJQndXjhz66nz+fFi5Ul+tr1ljdVQC9LStL7ygOzW2aKGHIcoa5cKLHixqY2usgzp3G7Re5qLnall33V8s/8uk4ndOtp9SLHnYzkc17WSzeO51T5OE7imtWumr9fBwfZ/9/felCd5KCQlQrZoeVz55sv5XZnoTPpA/m8F3D9oZH2Pjs10m9Ra5OHhe6gKrmErP9tbkBxfR+fS48iZFgzP1Bee7skp4uB633rcv9OunrwwvXLA6qkwnZNkyncydTti4ETp31sMPhfARwzB4saydVS3tHLygqDTXyc+HpAne185chUeXunhzo8mAKBs/PGSnQGjw1gWS0D3N4dCLecyapRd9iYnBduCA1VFlDkrB8OHkfPJJ3Uqyfj088IDVUYlMrHpB3Xu6fJhB48UuJuwJkRnmfCThtKLx/3Kw6m/FoqZ2BlexY7cFbzIHSeje07YtrF0LZ8+Su1Ejua/ubVeuwNNPw5tvcvm112DePMiTx+qohKBAqMGSh+28Ws7GgK3Z6LXGxGlKUvemlYdNai5wYrcpNrV20Dw8c6S6zPEurVK+PGzahKt0aWjYUK/iJjzv5Ek9MczcuTBrFpf69webHNrCfzhsBqNq2Pkg+jITdpo8utTFeVnkxSu+SDBpvNhFVD6DJfUvcl/u4L4q/yep9bwtLIxzs2frIW2PPaaXZBWes38/xMToJU6XL9ctI0L4qWfvTeb7pnZW/q2os9Ap49U9SCnF4HgXHX9x0aGkwQ8P2cmTyZZkkITuC1mz6l7W/fr9/wxzch/Nfbt26WFoV6/qWxoyJE0EgKbFbKxu5eD4Zai7yMmf56QucJdSij7rTAbGmQypYmNy3eCYKCa9JKH7is2mZ5cbNQqGDYM+fSSpu2PbNqhXT8+xv3o1REZaHZEQaVY+zGBVSwdKQe2FThJkHvgMc5mKLqtcjN5uMi7GxoBoe8AtquIpktB9rU8fPRHN6NHQo4csxZoRcXFQvz4UKaKHCd59t9URCZFuEbl0Us8Voq/Ut52UpJ5eyaai/QoX/01QTKtnp3tZu9UhWUoSuhV69NCTnUyYAM8/L0k9Pdav1+uX33+/vmeeP7/VEQmRYYVzGPzSwkHh7FB/kZPtktTTzGUq2v/sYk6i4ttGdp6JlHQmJWCVzp3hiy/g88/hpZek+T0ttm6Fhx6CcuVg6VLIm9fqiIRwW4FQg+XNHYTnhAcXO9lzRuqCOzGV4vlVOpnPbGgntoSkMpCEbq327WHiRBg3Dt55x+po/NvevXpo2r33wvffyzSuIqjkzWqw5GEHebJA48VOkmSq2NtSSvHyGpNpCYpp9e20lmSeQkrCai+8ACNHwpAh8OGHVkfjn/76Cxo3hrvugh9/lAljRFAqGGrwUzMHBjqpH7koSf1W3tpkMnaHycQ6dp4uKSnsn6Q0/EHfvtC/v+4wN22a1dH4l9On9ZW5aepm9gIFrI5ICK8pmlM3v5+9Ci2XurjolKT+TxN3uhi22WRUdRsvlJb0dSO3SmTPnj3UqlWLyMhIqlWrxo4dO265XUREBKVLlyYqKoqoqChmzpzpzm6D07Bh+r76Cy/oZVgFJCfryXgOH4Zly/TiN8JvSX3gGfflNljU1MHvpxQdfnZhSv8aAH5MMumx2qRnWRu9y0syvxWHO0/u2rUrXbp04dlnn2X27Nl07tyZtWvX3nLb2bNnU65cOXd2F9wMA8aPhz/+0LPKrV8PJUtaHZV1lIJevfSwtKVLoVQpqyMSdyD1gedULmDwdQM7rZe56L/BZGT1zD0ca+sJRbvlLh4uZvBRTVumHWd+Jxn+mnP06FHi4+Np3749ALGxsSQmJrJ//35PxZb5ZMmi53vPnx9atIBTp6yOyDqffKI7DE6cCA0aWB2NuAOpDzzvkQgbH9Sw8f5Wk0m7Mu/Q1iMXFc2XOCmZG75pGPwrprkjw1foSUlJFC5cGIdDv4RhGISHh3PgwAEiIiJu2v7pp5/GNE2qV6/O8OHDKXCHe6E9e/Ykz7XOT23atCE2NpZTAZrg0hu37auvyN2kCc7HHuP8jBmWLTRiVXk7VqwgV+/eXO7Vi0uPPqoXX0mHYDpOwsLCLIgk/bxZH2TmuuCZwrC1RFZ6/BpCccdFqoRZk9itKm+nCbGrQkl22fii+kWunlOkpzYItuPkTvXBbRN6nTp12Llz5y3/9ttvvwHc1Oxxu3V+V65cSXh4OMnJybz11lt07NiRxYsXpxrY2LFjqVSp0k2PB0oFd6N0xR0WBjNnkqVpU8LGj7d0SJvPy/vgQejaFRo3JnT0aELtGWtqzBTHiQ9ZWR9k6roAmNhAsWuRi84bchDf2kGBUGuuUK0o777rXaw7YfJzczvl7snYvBOZ5TiBVBL6qlWrUn1i1qxZOXjwIE6nE4fDgVKKpKQkwm/Rcen6YyEhIbzyyitEyrzbd/bgg/Cf/8DAgVC9OjRtanVE3nf1qu4EFxqqF7PJYDIXnif1gXWy2A1mNbITPdfJ0z+7+OGhzNHsPPsPk1FbTUbXsFHnHukElxYZLqWCBQsSHR3N9OnTAZgzZw4RERE3Na9duHCB06dPp/z+zTffEB0dndHdZi4DBuiZ0Z56Cg4csDoa73vtNT1P++zZMqVrgJH6wLuK5jSY0dDO8kOKQfHBfz894bTiuZUuHr/X4OVykszTyq1e7hMnTuTZZ59l2LBh5M6dm2n/GEPdrFkzBg8eTFhYGLGxsbhcLpRS3HvvvXzxxRduB54p2Gzw5ZdQqZKeVe7nn4P3qnXhQhgzRv9Uq2Z1NCIDpD7wrkZFbAyurHh7k0mjwgb1CwdnorvqUjz1s4vC2WFy3cy7clpGuJXQS5UqddthKf+8J3b9HpvIgHz59JzvDRrA++/rCWiCzZEjegx+ixZ64RoRkKQ+8L7+FW0sPah4ZoWLLbEGebMGX7IbGGey5YRi3SMOcoYE3/vzpuD8ihds6tWD11/XnePi462OxrOU0ivOGQZMmaL/FULckt1m8GUDO2eToftq1207HgaqFYdM3ttiMqSKjcoFpC5IL0nogeI//9GrjD39NFy8aHU0nvPZZ7BokU7mBQtaHY0Qfi88p8GnMXZm7FN8tTd4EvqpK4oOK1zUu8fgtQqSmjJCSi1QZMmie37v3x88ze5790Lv3tCtm25uF0KkyZMlbTxd0qDnGheHLgRHUu+zzsXZq/BF/czRi98bJKEHkjJlYPhwGDsW1qyxOhr3KKXnrS9UCEaNsjoaIQLOJzXtZLMHR9P70oMmUxMUH9SwUyynJPOMkoQeaHr10r3An38erlyxOpqMmzxZz9P+2WeQI4fV0QgRcMKyGYyPsTP/T8W3fwRuQj93VfHCKheNCht0LiXJ3B2S0AON3a6T4d69eoW2QHTokF4y9rnn9DrnQogMaVPCRmwJg15rXBy/HJhJ/Y2NJscvw6Q6MkTNXZLQA1G5cvDGG7r5fft2q6NJv549IVs2+OADqyMRIuCNrWXHqfQ96ECz4ajJ+B0mw6raKJFbkrm7JKEHqjff1Murvviivh8dKL7/HubO1RPI5M3Y3MxCiP93d3aDEVXtfLFHsfrvwJlFzmUqeqw2icoHPctKKvIEKcVAlTUrfPwx/PqrXnI1ECQnQ58+0LAhtG1rdTRCBI3OpQyq5DfosdqF0wyML/iTd5tsOq4YFyO92j1FEnoge/BBaN4c+vWDy5etjubOJkyAhAT48EOZQEYID7LbDMbXtrH1JEzY6f9X6ccvK97YaPJcpEHNQpKGPEVKMtCNGqUXbvnkE6sjSd2pUzBokJ7itWJFq6MRIuhULWDj+dIGb20yOXrJv6/S39zoQikYUS1I16awiCT0QFe6tL6PPnQoHD1qdTS39+67ennUd9+1OhIhgtawqnbsBvTf4L8d5NYfNZm8SzGkio2CFq3tHqwkoQeDQYP0ymwDB1odya3t2aMnw3njDbj7bqujESJo5c9mMLSKjakJirVH/K/p3VSKntc6wnUrI+nH06REg0G+fHrhls8+889hbH37wj33wKuvWh2JEEHvhdI2ovNB73Wm380g980+xabjio9rSkc4b5CEHix69ID77tO9yP3pJP75Z5g/H957D0JDrY5GiKBntxmMqm5n3VHFLD+aQe6SU/HmRhetIwzq3COpxxukVINFlix6vfSlS+GHH6yORnO59FV5zZrw+ONWRyNEptGwiI0W4Qb9N7q44vKPpP7JdpNDF2BEVekI5y2S0INJq1bQoIG+Sk9OtjoamDYNtmyRYWpCWGBkNTsHzsO4362/l37skmLYZpMXy9qIvEvqAm+RhB5MDEMnz9279f10K507BwMGwJNPQo0a1sYiRCZUJq/BC6VtvPubyQmL53kfHK+/VLxTSVKON0npBpuoKL3oycCBeuy3Vd57D06fhhEjrItBiEzuP5VtuBQM+c26q/TdpxUTdpoMiLaRP5tcnXuTJPRgNGSInjluyBBr9n/ggF54pU8fCA+3JgYhBAVDDfpXtDFuh8neM9Zcpfff4KJwDnjpAUk33iYlHIzuuUeP+R4zRo8B97X+/eGuu/S/QghLvVLeRqFQeGOj7yebWXnYZN6fiuFV7WRzyNW5t0lCD1a9e+tJXPr18+1+162Db77RM9flzOnbfQshbpLdYTC0ip3Zib5djc1UitfWm1TJb/DEfZLMfUESerAKDdX3r+fNgxUrfLNPpfQwtago6NjRN/sUQtxR+/sNKuWHPut9N9nMzH2KjccUo2rYsMkoF5+QhB7MnnwSqlfXV+suHzS3zZypr9A//BDsMtZUCH9hMww+qG5n/VHFTB9MNnPJqei/0cUjxQ3qySQyPiMlHcwMA0aPht9+gy++8O6+Ll2C11+HRx7RY+GFEH6lfmEbrYob9N/g4rLTu0n942uTyIyU1dR8ShJ6sKtZE554At58E86f995+Ro+Gw4f1bHVCCL80spqdvy7AJ16cbObIRT2JTI8HZBIZX5OEnhmMGOHdMeGHD8Pw4dCzJ9x/v3f2IYRwW6m7DLqVsTH0N5NjXlozfWCcicMG70RLevE1KfHMoHhxeO01GDUK/vjD86//+uuQLRu8/bbnX1sI4VEDK9swDPhPvOev0n8/qZi02+SdaBthMomMz0lCzyz694eCBT2/hOmaNfDllzBsGOTN69nXFkJ4XP5sBgOibEzYabLrtOeu0pVS9Fnv4t5c0L2spBYrSKlnFjly6Cv0BQvgxx8985oul25mr1wZOnXyzGsKIbyu1wM2iueEXmtcHhvGNne/YslBxQc17GSxy9W5FSShZyaPPaZ7oL/8Mly96v7rTZmie9CPHSvD1IQIINkcBmNq2fnpL8WMfe4n9PPJipfXumgZbtCquKQVq0jJZyaGAZ98Avv2ud9B7u+/dTN+x46ympoQAahZuI22JQxeXefi9BX3kvqgOJMTl+GTWvLF3kqS0DObcuX0PO9DhsDWrRl7DaWgWzcICdGLsAghAtJHNe1cdMKr6zI+8dSaIyajt5u8U8lGRC5pareSJPTM6K23oFQpePbZjDW9f/01zJ8PEyZAvnweD08I4RtFchh8XNPO5wmK2X+kv9f7uauKDj+7qFbA4LUKkk6sJp9AZpQ1K3z+OWzfnv7FWxIS4MUX4amnoHVrr4QnhPCdZyMNYksYdPnVxcHzaW96V0rfNz9yCaY3sOOwydW51SShZ1aVK+vZ3T7+WF9xp8WZMxAbC0WK6KtzIUTAMwyDibXt5HRAy6VOzl1NW1Ifvc1kaoJibIyd+3JLMvcHktAzs+7ddae2jh1h4cLUtz1/Hpo1g4MHYc4cyJXLNzEKIbwuXzaD7x9y8MdZeGy5i2Qz9aT+XaLJa+tN+le08WykpBF/4dYn8dJLLxEREYFhGGzfvv222+3Zs4datWoRGRlJtWrV2LFjhzu7FZ5iGDB5sl5QpW1bmDHj1tsdOaKT+bZtsGQJlC3r2ziF35O6IPCVDzOY+6Cd/x1StFpy+57vU3ebPL7cxeP3GQytKsncn7j1abRt25Zff/2V4sWLp7pd165d6dKlCwkJCfTr14/OnTu7s1vhSQ6HbnJ/7DG93GqTJjBpEhw+jGPNGj1mvXx52L1bT0hTrZrVEQs/JHVBcGhYxMbih+ysPaqoMMfJ1N0mZ64qnCZsOGrSZpmTTitdPBtp8GV9u6xz7mfcSuh169alaNGiqW5z9OhR4uPjad++PQCxsbEkJiayf/9+d3YtPClLFj196/Tp4HTqIWkxMeRu2VI3rz/+OGzeDLVqWR2p8FNSFwSPxkVsbGnjoHJ+g04rXdw1zcndc3NSfb6LuOOKbxramVTXIZ3g/JDD2ztISkqicOHCOBx6V4ZhEB4ezoEDB4iIiLjt83r27EmePHkAaNOmDbGxsZw6dcrb4XpFwMT98MPw8MMYx49j27ePcxcvkr1ePbBd+9538qS18aVRwJT3DW4Vd1hYmAWReIfUBYFzbOYCplSBgWUM1p+wc/zsRSoWCqVaPhcOW8BUBQFT3je6Xdx3qg9um9Dr1KnDzp07b/m33377jWLFiqU5OOOGZpm0zB08duxYKlWqdNPjgVrBBVTcYWEQGYnr5MnAivsfJG7PkbrAswIp7rAwiCoGJ086CQvLY3U4GRJI5f1PGYn7tgl91apVbgVzXbFixTh48CBOpxOHw4FSiqSkJMLDwz3y+kII75K6QIjA4PUuigULFiQ6Oprp06cDMGfOHCIiIlJtYhNCBB+pC4TwLrcSeo8ePShatCgHDx6kcePGlCxZMuVvzZo1Y9OmTQBMnDiRiRMnEhkZyYgRI5gyZYp7UQsh/IrUBUJYz1CeWgzXQ+Lj46lcuTJxcXE33Tc7GaD3dCVu35K4g4PUBf5D4vatjMYtswIIIYQQQUASuhBCCBEEAiqhz5kzx+oQMkTi9i2JO/gFallJ3L6V2eIOqIT+3XffWR1ChkjcviVxB79ALSuJ27cyW9wBldCFEEIIcWten/o1vS5dugRwy5mpzpw5Q3x8vK9DcpvE7VvBFnfp0qXJnj27BRFZS+oC/yFx+1ZqcadaHyg/M336dAXIj/zIz7WfuLg4q09LS0hdID/yc/NPavWB341DP378OEuWLCEiIoLQ0FCrwxHCcpn1Cl3qAiFullp94HcJXQghhBDpJ53ihBBCiCAgCV0IIYQIApLQhRBCiCDg1wn9pZdeIiIiAsMw2L59+22327NnD7Vq1SIyMpJq1aqxY8cOH0bpXkwRERGULl2aqKgooqKimDlzpt/FKOXrGYF8PPuDQC2/QDlWA7U+CJTyvZFXjmcfjUDJkF9++UUlJSWp4sWLq23btt12uwYNGqipU6cqpZSaNWuWqlGjho8ivL20xnSn9+ZNaY1RytczAvl49geBWn6BcqwGan0QKOV7I28cz36d0K9L7Q0fOXJE5cmTRyUnJyullDJNUxUqVEglJib6MMKMx2TVQZbWGKV8PS/Qjmd/E0jlFyjHaqDWB4FSvqnx5PHs103uaZGUlEThwoVxOPSkd4ZhEB4ezoEDBwImpqeffpry5cvz/PPPc+zYMb+KUcrXt/yxvAOJv5VfoByrgVofBEr5ZlR6359lCb1OnTrkz5//lj9JSUnpei3DMP71u/Ly0Pq0xJ7WmFauXMmWLVuIj48nX758dOzY0aux/1NaY/R1+aZFIJRvRvljeXtboNYHwVIXQODWB4FSvhmVrvL2ZNOBt9ypSSJ37tx+0wTkTkyHDh1SOXPm9EGEaY9RytfzAu149jeBVH6BcqwGan0QKOWbGk8ezwHf5F6wYEGio6OZPn06oNeRjYiIICIiwu9junDhAqdPn075/ZtvviE6OtqvYpTy9S1/LO9A4m/lFyjHaqDWB4FSvhmV7vL21rcOT+jevbsqUqSIstvtqlChQuq+++5L+dvDDz+sNm7cqJRSateuXapGjRrq/vvvV5UrV1bbt2+3KuQUqcV0PfZ9+/apqKgoVb58eVWuXDnVqlUrn37TTUuMd9rOKoFQvjcK5OPZHwRq+QXKsRqo9UGglO+NvHE8y1zuQgghRBAI+CZ3IYQQQkhCF0IIIYKCJHQhhBAiCEhCF0IIIYKAJHQhhBAiCEhCF0IIIYKAJHQhhBAiCPwfN/XobjVlXowAAAAASUVORK5CYII=", "text/html": [ "" ] }, "execution_count": 49, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Edwards曲線の二通りのプロット\n", "\n", "k² = -20\n", "\n", "y = -1:0.01:1\n", "x = @. √((1-y^2)/(1-k²*y^2))\n", "P1 = plot(title=\"\\$x^2+y^2=1+k^2x^2y^2\\$ for \\$k^2=$k²\\$\", titlefontsize=10)\n", "plot!(aspectratio=1, legend=false)\n", "plot!(x, y, color=:red)\n", "plot!(-x, y, color=:red)\n", "\n", "u = 0:0.01:3\n", "P2 = plot(title=\"(cd u, sn u) for \\$k^2=$k²\\$\", titlefontsize=10)\n", "plot!(aspectratio=1, legend=false)\n", "plot!(cd.(u,k²), sn.(u,k²))\n", "\n", "plot(P1, P2, size=(500, 260))" ] }, { "cell_type": "code", "execution_count": 50, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "image/png": 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", "text/html": [ "" ] }, "execution_count": 50, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# (cd u, sn u)のプロット\n", "\n", "k² = -20\n", "u = -5:0.01:5\n", "plot(title=\"Jacobi's elliptic functions for \\$k^2=$k²\\$\", titlefontsize=10)\n", "plot!(u, cd.(u,k²), label=\"cd u\", ls=:dash)\n", "plot!(u, sn.(u,k²), label=\"sn u\")\n", "plot!(size=(500, 200), legend=:bottomleft)" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Gauss積分の大学入試問題\n", "\n", "Gauss積分の公式 $\\ds\\int_{-\\infty}^\\infty e^{-x^2}\\,dx=\\sqrt{\\pi}$ は筆者の個人的な意見では新入生が習う定積分の公式の中で最も重要なものである. Gauss積分の公式の問題が大学入試問題として出題されたことがあるので紹介しておく." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "2015年の東京工業大学前期日程の入試問題として次の問題が出題された.\n", "\n", ">\\[3\\] $a>0$ とする. 曲線 $y=e^{-x^2}$ と $x$ 軸, $y$ 軸, および直線 $x=a$ で囲まれた図形を, $y$ 軸のまわりに1回転してできる回転体を $A$ とする.\n", ">\n", ">(1) $A$ の体積 $V$ を求めよ.\n", ">\n", ">(2) 点 $(t,0)$ ($-a\\leqq t\\leqq a$) を通り, $x$ 軸と垂直な平面による $A$ の切り口の面積を $S(t)$ とするとき, 不等式 $\\ds S(t)\\leqq\\int_{-a}^a e^{-(s^2+t^2)}\\,ds$ を示せ.\n", ">\n", ">(3) 不等式 $\\ds \\sqrt{\\pi(1-e^{-a^2})}\\leqq\\int_{-a}^a e^{-x^2}\\,dx$ を示せ.\n", "\n", "この問題の内容は, 本質的に**Gauss積分の公式**\n", "\n", "$$\n", "\\int_{-\\infty}^\\infty e^{-x^2}\\,dx =\n", "\\lim_{a\\to\\infty}\\int_{-a}^a e^{-x^2}\\,dx = \\sqrt{\\pi}\n", "$$\n", "\n", "の高校数学の範囲内での証明である. 高校数学IIIの教科書にも以下のような問題が載っている. (前者の問題はGauss積分と関係しており, 後者の問題はゼータ函数と関係している.)" ] }, { "cell_type": "code", "execution_count": 51, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "text/html": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "showimg(\"image/jpeg\", \"images/Jikkyo20140125GaussZeta.jpg\", scale=\"80%\")" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "問題文では曲線 $y=e^{-x^2}$ を $y$ 軸のまわりに回転しているが, 以下では $xyz$ 空間内の $xz$ 平面上の曲線 $z=e^{-x^2}$ を $z$ 軸のまわりに回転して得られる曲面を扱う. さらに $a$ の代わりに $r$ と書く. \n", "\n", "$xyz$ 空間内の $xz$ 平面 $y=0$ 上の曲線 $z=e^{-x^2}$ を $z$ 軸のまわりに回転して得られる曲面と高さ $00$ であるとする.\n", "\n", "平面 $y=0$ 上の曲線 $z=e^{-x^2}$ と $x$ 軸と直線 $x=r$ で囲まれた領域を $z$ 軸のまわりに1回転してできる回転体を $A(r)$ と書く. 上で述べたことより, $A(r)$ の体積 $V(r)$ は次のように計算される:\n", "\n", "$$\n", "\\begin{aligned}\n", "V(r) &= \n", "\\pi r^2 e^{-r^2} + \\int_{e^{-r^2}}^1 \\pi(-\\log t)\\,dt \n", "\\\\ &=\n", "\\pi r^2 e^{-r^2} - \\pi[t\\log t - t]_{e^{-r^2}}^1 \n", "\\\\ &=\n", "\\pi r^2 e^{-r^2} -\\pi(-1 + r^2 e^{-r^2} + e^{-r^2}) \n", "\\\\ &=\n", "\\pi(1-e^{-r^2}).\n", "\\end{aligned}\n", "$$\n", "\n", "曲面 $z=e^{-(x^2+y^2)}$ と $xy$ 平面 $z=0$ と4つの平面 $x=\\pm r$, $y=\\pm r$ で囲まれた領域を $B(r)$ と書く. そして, $-r\\leqq t\\leqq r$ のとき, $A(r)$ の平面 $y=t$ による断面の面積は\n", "\n", "$$\n", "\\int_{-r}^r e^{-(x^2+t^2)}\\,dx\n", "$$\n", "\n", "になるので, これを $-r\\leqq t\\leqq r$ で積分すれば $B(r)$ の体積 $W(r)$ が求まる. $t$ を $y$ と書くと,\n", "\n", "$$\n", "\\begin{aligned}\n", "W(r) &= \n", "\\int_{-r}^r \\left(\\int_{-r}^r e^{-(x^2+y^2)}\\,dx\\right)\\,dy \n", "\\\\ &=\n", "\\int_{-r}^r e^{-x^2}\\,dx \\int_{-r}^r e^{-y^2}\\,dy \n", "\\\\ &=\n", "\\left(\\int_{-r}^r e^{-x^2}\\,dx\\right)^2.\n", "\\end{aligned}\n", "$$\n", "\n", "$B(r)$ は $A(r)$ を含み, $A(\\sqrt{2}\\;r)$ に含まれる. それらは次の包含関係から導かれる:\n", "\n", "$$\n", "\\begin{aligned}\n", "&\n", "\\{(x,y)\\in\\R^2\\mid x^2+y^2\\leqq r^2\\}\n", "\\\\ & \\qquad \\subset\n", "\\{(x,y)\\in\\R^2\\mid |x|,|y|\\leqq r\\}\n", "\\\\ & \\qquad\\qquad \\subset\n", "\\{(x,y)\\in\\R^2\\mid x^2+y^2\\leqq 2r^2\\}.\n", "\\end{aligned}\n", "$$\n", "\n", "ゆえに, $V(r)\\leqq W(r)\\leqq V(\\sqrt{2}\\,r)$ となる. すなわち,\n", "\n", "$$\n", "\\sqrt{\\pi(1-e^{-r^2})} \\leqq \n", "\\int_{-r}^r e^{-x^2}\\,dx \\leqq \n", "\\sqrt{\\pi(1-e^{-2r^2})}.\n", "$$\n", "\n", "これより,\n", "\n", "$$\n", "\\lim_{r\\to\\infty}\\int_{-r}^r e^{-x^2}\\,dx = \\sqrt{\\pi}\n", "$$\n", "\n", "となることがわかる.\n", "\n", "以上の計算は次のようにまとめられる:\n", "\n", "$$\n", "\\begin{aligned}\n", "\\left(\\int_{-\\infty}^\\infty e^{-x^2}\\,dx\\right)^2 &=\n", "\\int_{-\\infty}^\\infty \\int_{-\\infty}^\\infty e^{-(x^2+y^2)}\\,dx\\,dy \n", "\\\\ &=\n", "\\int_0^1 \\pi(-\\log z)\\,dz = -\\pi[z\\log z - z]_0^1 = \\pi.\n", "\\end{aligned}\n", "$$\n", "\n", "Gauss積分は正規分布の確率密度函数の理解に必須であり, その他にも多くの場面に現われ, 非常に重要な定積分である." ] }, { "cell_type": "code", "execution_count": 52, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "text/plain": [ "0.7071067811865475" ] }, "execution_count": 52, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f(x,y) = exp(-(x^2+y^2))\n", "g(x,y,r) = x^2+y^2 ≤ r^2 ? exp(-(x^2+y^2)) : zero(x)\n", "h(x,y,r) = (abs(x) ≤ r && abs(y) ≤ r) ? exp(-(x^2+y^2)) : zero(x)\n", "\n", "x = range(-2, 2, length=201)\n", "y = range(-2, 2, length=201)\n", "r = 1/√2" ] }, { "cell_type": "code", "execution_count": 53, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "image/png": 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", "text/html": [ "" ] }, "execution_count": 53, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# z = e^{-(x^2+y^2)} のグラフ\n", "surface(x, y, f.(x', y), size=(400,250), colorbar=false)" ] }, { "cell_type": "code", "execution_count": 54, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "image/png": 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", "text/html": [ "" ] }, "execution_count": 56, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# A(√2 r) のグラフ (r=1/√2)\n", "surface(x, y, g.(x', y, √2*r), size=(400,250), colorbar=false)" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## ガンマ函数の応用\n", "\n", "高校数学では $\\cos x$, $\\sin x$, $e^x$, $\\log x$ などの初等函数について習うが, 大学新入生が新たに習う特殊函数として, ゼータ函数 $\\ds\\zeta(s)$, ガンマ函数 $\\Gamma(s)$, ベータ函数 $B(p,q)$ は特に重要である. この節では高校数学にも密かにガンマ函数にあたるものが現われていたことについて解説する. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### 多項式×指数函数の積分\n", "\n", "$n$ は0以上の整数であるとする. 高校数学IIIの教科書にはよく次の形の不定積分を求める問題が書いてある:\n", "\n", "$$\n", "\\int x^n e^x\\,dx.\n", "$$\n", "\n", "以下ではこれと本質的に同じ($x$ を $-x$ で置き換えて得られる)\n", "\n", "$$\n", "\\int x^n e^{-x}\\,dx\n", "$$\n", "\n", "を扱う. 部分積分を次々に使うと,\n", "\n", "$$\n", "\\begin{aligned}\n", "&\n", "\\int x^n e^{-x}\\,dx \n", "\\\\ &= -x^n e^{-x} + n\\int x^{n-1}e^{-x}\\,dx\n", "\\\\ &= -x^n e^{-x} - nx^{n-1}e^{-x} \n", "\\\\ &+ n(n-1)\\int x^{n-2}e^{-x}\\,dx\n", "\\\\ &= -x^n e^{-x} - nx^{n-1}e^{-x} - n(n-1)x^{n-2}e^{-x} \n", "\\\\ &\\qquad + n(n-1)(n-2)\\int x^{n-3}e^{-x}\\,dx\n", "\\\\ &=\n", "\\cdots\\cdots\\cdots\\cdots\n", "\\\\ &= -x^n e^{-x} - nx^{n-1}e^{-x} - n(n-1)x^{n-2}e^{-x} - \\cdots \n", "\\\\ &\\qquad + n(n-1)\\cdots 2 x e^{-x} - n!e^{-x}\n", "\\\\ &= -(x^n + nx^{n-1}e^{-x} + n(n-1)x^{n-2}+ \\cdots \n", "\\\\ &\\qquad+ n(n-1)\\cdots 2 x + n!)e^{-x}.\n", "\\end{aligned}\n", "$$\n", "\n", "積分定数は省略した. これは $x\\to\\infty$ で $0$ に収束し, $x=0$ のとき $-n!$ になる. ゆえに\n", "\n", "$$\n", "\\lim_{a\\to\\infty}\\int_0^a x^n e^{-n}\\,dx = 0 - (-n!) = n!.\n", "$$\n", "\n", "大学1年のときの解析学の授業でガンマ函数\n", "\n", "$$\n", "\\Gamma(s) = \\int_0^\\infty e^{-x} x^{s-1}\\,dx \\quad (s > 0)\n", "$$\n", "\n", "について習う. 上の高校数学の範囲内の結果は $\\Gamma(n+1)=n!$ が成立することを意味している.\n", "\n", "以上のように高校数学IIIの教科書にある $\\ds\\int x^n e^x\\,dx$ 型の不定積分を求める問題は本質的にガンマ函数に関する問題だとみなされる.\n", "\n", "以下のセルに実際の教科書の様子を引用しておく. 例題はガンマ函数に研究課題は三角函数の**Laplace変換**(ラプラス変換)を実質的に扱っているとみなされる. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "**注意:** $a>0$, $s>0$ に関する次の公式はよく使われる:\n", "\n", "$$\n", "\\int_0^\\infty e^{-at} t^{s-1}\\,dt = \\frac{\\Gamma(s)}{a^s}.\n", "$$\n", "\n", "この公式は $t=x/a$ という置換積分によって容易に示される. この公式は\n", "\n", "$$\n", "\\frac{1}{a^s} = \\frac{1}{\\Gamma(s)} \\int_0^\\infty e^{-at} t^{s-1}\\,dt\n", "$$\n", "\n", "の形で使われることも多い. $\\QED$" ] }, { "cell_type": "code", "execution_count": 57, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "text/html": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "showimg(\"image/jpeg\", \"images/Jikkyo20140125GammaLaplace.jpg\", scale=\"80%\")" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Stirlingの公式\n", "\n", "**準備の準備:** $|X|<1$ のとき\n", "\n", "$$\n", "\\log(1 + X) = X - \\frac{X^2}{2} + \\frac{X^3}{3} - \\frac{X^4}{4} + \\cdots.\n", "$$\n", "\n", "正値函数の漸近挙動は, 対数を取ってから, Taylor展開などを使って調べることが多い. $\\QED$\n", "\n", "**準備:** $n\\to\\infty$ のとき, Taylor展開 $\\log(1+X)=X-X^2/2+O(X^3)$ を使うと,\n", "\n", "$$\n", "\\begin{aligned}\n", "&\n", "\\log\\left(e^{-\\sqrt{n}\\,y}\\;\\left(1+\\frac{y}{\\sqrt{n}}\\right)^n\\right) =\n", "-\\sqrt{n}\\;y + n\\log\\left(1+\\frac{y}{\\sqrt{n}}\\right) \n", "\\\\ &\\qquad=\n", "-\\sqrt{n}\\;y + n\\left(\\frac{y}{\\sqrt{n}}-\\frac{y^2}{2n} + O(n^{-3/2})\\right) \n", "\\\\ &\\qquad= -\n", "\\frac{y^2}{2} + O(n^{-1/2}) \\to -\\frac{y^2}{2}.\n", "\\end{aligned}\n", "$$\n", "\n", "なので, \n", "\n", "$$\n", "\\int_{-\\sqrt{n}}^\\infty e^{-\\sqrt{n}\\,y}\\;\\left(1+\\frac{y}{\\sqrt{n}}\\right)^n\\,dy \\to\n", "\\int_{-\\infty}^\\infty e^{-y^2/2}\\,dy = \\sqrt{2\\pi}.\n", "$$\n", "\n", "最後の等号で $a>0$ のとき\n", "\n", "$$\n", "\\int_{-\\infty}^\\infty e^{-y^2/a}\\,dy = \\sqrt{a\\pi}\n", "$$\n", "\n", "となることを使った. この公式はGauss積分の公式\n", "\n", "$$\n", "\\int_{-\\infty}^\\infty e^{-x^2}\\,dx = \\sqrt{\\pi}\n", "$$\n", "\n", "で $x=y/\\sqrt{a}$ とおけば得られる. $\\QED$\n", "\n", "正の整数 $n$ の階乗 $n!$ をガンマ函数で表すと,\n", "\n", "$$\n", "n! = \\Gamma(n+1) = \\int_0^\\infty e^{-x}x^n\\,dx\n", "$$\n", "\n", "なので, $\\ds x = n + \\sqrt{n}\\;y = n(1+y/\\sqrt{n})$ と積分変数を変換すると,\n", "\n", "$$\n", "n! = \n", "n^n e^{-n}\\sqrt{n}\n", "\\int_{-\\sqrt{n}}^\\infty e^{-\\sqrt{n}\\,y}\\;\\left(1+\\frac{y}{\\sqrt{n}}\\right)^n\\,dy.\n", "$$\n", "\n", "したがって, 上で準備した結果を使うと, $n\\to\\infty$ のとき\n", "\n", "$$\n", "n! \\sim n^n e^{-n}\\sqrt{n}\\sqrt{2\\pi} = n^n e^{-n} \\sqrt{2\\pi n}.\n", "$$\n", "\n", "これを**Stirlingの公式**(スターリングの公式)と呼ぶ. ここで $a_n\\sim b_n$ は $a_n/b_n\\to 1$ となることを意味する." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "**問題:** Stirlingの公式の誤差が $n=1,2,\\ldots,10$ でどの程度であるかを確認せよ. $\\QED$\n", "\n", "次のセルを参照せよ." ] }, { "cell_type": "code", "execution_count": 58, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ " n n! Stirling Error Rel.Err.\n", "------------------------------------------------------------\n", " 1 1 0.9221 -0.0779 -0.0779\n", " 2 2 1.9190 -0.0810 -0.0405\n", " 3 6 5.8362 -0.1638 -0.0273\n", " 4 24 23.5062 -0.4938 -0.0206\n", " 5 120 118.0192 -1.9808 -0.0165\n", " 6 720 710.0782 -9.9218 -0.0138\n", " 7 5040 4980.3958 -59.6042 -0.0118\n", " 8 40320 39902.3955 -417.6045 -0.0104\n", " 9 362880 359536.8728 -3343.1272 -0.0092\n", "10 3628800 3598695.6187 -30104.3813 -0.0083\n" ] } ], "source": [ "stirling_approx(n) = n^n * exp(-n) * √(2π*n)\n", "error(x, x₀) = x - x₀ # 誤差\n", "relative_error(x, x₀) = x/x₀ - 1 # 相対誤差\n", "\n", "@printf(\"%2s %10s %13s %13s %13s\\n\", \"n\", \"n!\", \"Stirling\", \"Error\", \"Rel.Err.\")\n", "println(\"-\"^60)\n", "for n in 1:10\n", " ft = factorial(n)\n", " s = stirling_approx(n)\n", " err = error(s, ft)\n", " relerr = relative_error(s, ft)\n", " @printf(\"%2d %10d %13.4f %13.4f %13.4f\\n\", n, ft, s, err, relerr)\n", "end" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "$n$ が大きくなるほどStirlingの公式の相対誤差は小さくなる. $n=5$ の段階ですでに相対誤差は $2\\%$ を切っており, $n=9$ で相対誤差は $1\\%$ を切っている.\n", "\n", "Stirlingの公式よりも精密に\n", "\n", "$$\n", "n! = n^n e^{-n} \\sqrt{2\\pi n}\\left(1 + \\frac{1}{12n} + O(n^{-2})\\right)\n", "$$\n", "\n", "が成立することが知られている. $1/(12n)$ で補正されたStirlingの公式は $n=1$ の段階ですでに相対誤差が $0.1\\%$ 程度になっている:\n", "\n", "$$\n", "\\frac{\\sqrt{2\\pi}}{e} = 0.92213\\cdots, \\quad\n", "\\frac{\\sqrt{2\\pi}}{e}\\frac{13}{12} = 0.99898\\cdots.\n", "$$\n", "\n", "$\\sqrt{2\\pi} = 2.50662\\cdots$ の $13/12$ 倍の $2.71551\\cdots$ が $e=2.71828\\cdots$ に近いのは偶然ではなく, 補正されたStirlingの公式の $n=1$ の場合だということである." ] }, { "cell_type": "code", "execution_count": 59, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ " n n! Improved Stirling Error Rel.Err.\n", "-----------------------------------------------------------------\n", " 1 1 0.99898 -0.00102 -0.00102\n", " 2 2 1.99896 -0.00104 -0.00052\n", " 3 6 5.99833 -0.00167 -0.00028\n", " 4 24 23.99589 -0.00411 -0.00017\n", " 5 120 119.98615 -0.01385 -0.00012\n", " 6 720 719.94038 -0.05962 -0.00008\n", " 7 5040 5039.68626 -0.31374 -0.00006\n", " 8 40320 40318.04541 -1.95459 -0.00005\n", " 9 362880 362865.91796 -14.08204 -0.00004\n", "10 3628800 3628684.74890 -115.25110 -0.00003\n" ] } ], "source": [ "stirling_approx1(n) = n^n * exp(-n) * √(2π*n) * (1 + 1/(12n))\n", "\n", "@printf(\"%2s %10s %20s %13s %13s\\n\", \"n\", \"n!\", \"Improved Stirling\", \"Error\", \"Rel.Err.\")\n", "println(\"-\"^65)\n", "for n in 1:10\n", " ft = factorial(n)\n", " s₁ = stirling_approx1(n)\n", " err = error(s₁, ft)\n", " relerr = relative_error(s₁, ft)\n", " @printf(\"%2d %10d %20.5f %13.5f %13.5f\\n\", n, ft, s₁, err, relerr)\n", "end" ] }, { "cell_type": "code", "execution_count": 60, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "stirling_approx(1) = 0.9221370088957891\n", "stirling_approx1(1) = 0.9989817596371048\n" ] } ], "source": [ "@show stirling_approx(1)\n", "@show stirling_approx1(1);" ] }, { "cell_type": "code", "execution_count": 61, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "√(2π) = 2.5066282746310002\n", "(√(2π) * 13) / 12 = 2.7155139641835837\n", "float(ℯ) = 2.718281828459045\n" ] } ], "source": [ "@show √(2π)\n", "@show √(2π) * 13/12\n", "@show float(ℯ);" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Stirlingの公式を使うと簡単に解ける大学入試問題\n", "\n", "1988年の東京工業大学の入試問題に次の問題があった.\n", "\n", ">\\[5\\] $\\ds\\lim_{n\\to\\infty}\\left(\\frac{_{3n}C_n}{_{2n}C_n}\\right)^{1/n}$ を求めよ.\n", "\n", "その他にも1968年の東京工業大学の入試問題に次の問題があった.\n", "\n", ">\\[5\\] 次の極限値を求めよ. \n", ">$$\\ds\\lim_{n\\to\\infty}\\frac{1}{n}\\sqrt[n]{_{2n}P_n}$$\n", "\n", "これらの問題はStirlingの公式を使うとほぼただちに答えを得ることができる. \n", "\n", "\n", "**前者の問題の解答例:** Stirlingの公式を使うと, $n\\to\\infty$ のとき\n", "\n", "$$\n", "\\begin{aligned}\n", "\\left(\\frac{_{3n}C_n}{_{2n}C_n}\\right)^{1/n} &=\n", "\\left(\\frac{(3n)!/(2n)!}{(2n)!/n!}\\right)^{1/n} =\n", "\\left(\\frac{(3n)!n!}{((2n)!)^2}\\right)^{1/n} \n", "\\\\ &\\sim\n", "\\left(\\frac\n", "{(3n)^{3n}e^{-3n}\\sqrt{2\\pi n}\\cdot n^n e^{-n}\\sqrt{2\\pi n}}\n", "{(2n)^{4n}e^{-4n}2\\pi n}\n", "\\right)^{1/n} =\n", "\\frac{3^3}{2^4}.\n", "\\end{aligned}\n", "$$\n", "\n", "ゆえに $\\ds\\lim_{n\\to\\infty}\\left(\\frac{_{3n}C_n}{_{2n}C_n}\\right)^{1/n} = \\frac{3^3}{2^4} = \\frac{27}{16}$. $\\QED$\n", "\n", "**後者の問題の解答例:** Stirlingの公式を使うと, $n\\to\\infty$ のとき\n", "\n", "$$\n", "\\begin{aligned}\n", "\\frac{1}{n}\\sqrt[n]{_{2n}P_n} &=\n", "\\frac{1}{n}\\left(\\frac{(2n)!}{n!}\\right)^{1/n} \\sim\n", "\\frac{1}{n}\\left(\\frac{(2n)^{2n} e^{-2n} \\sqrt{2\\pi n}}{n^n e^{-n} \\sqrt{2\\pi n}}\\right)^{1/n} =\n", "2^2 e^{-1}.\n", "\\end{aligned}\n", "$$\n", "\n", "ゆえに $\\ds \\ds\\lim_{n\\to\\infty}\\frac{1}{n}\\sqrt[n]{_{2n}P_n} = 2^2 e^{-1} = \\frac{4}{e}$. $\\QED$\n", "\n", "高校の教科書にもこの後者の問題が掲載されている." ] }, { "cell_type": "code", "execution_count": 62, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "text/html": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "showimg(\"image/jpeg\", \"images/Jikkyo20140125Stirling.jpg\", scale=\"80%\")" ] }, { "cell_type": "code", "execution_count": 63, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "text/latex": [ "$\\ds\\lim_{n\\to\\infty}\\left(\\frac{\\binom{3n}{n}}{\\binom{2n}{n}}\\right)^{1/n}=\\frac{27}{16}$" ], "text/plain": [ "L\"$\\ds\\lim_{n\\to\\infty}\\left(\\frac{\\binom{3n}{n}}{\\binom{2n}{n}}\\right)^{1/n}=\\frac{27}{16}$\"" ] }, "execution_count": 63, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# 前者の問題の SymPy による解\n", "\n", "n = symbols(\"n\", positive=true)\n", "binom(n,k) = gamma(n+1)/(gamma(k+1)*gamma(n-k+1))\n", "sol = limit((binom(3n,n)/binom(2n,n))^(1/n), n=>oo)\n", "latexstring(raw\"\\ds\\lim_{n\\to\\infty}\\left(\\frac{\\binom{3n}{n}}{\\binom{2n}{n}}\\right)^{1/n}=\", sympy.latex(sol))" ] }, { "cell_type": "code", "execution_count": 64, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "text/latex": [ "$\\ds\\lim_{n\\to\\infty}\\frac{1}{n}\\left(\\frac{(2n)!}{n!}\\right)^{1/n}=\\frac{4}{e}$" ], "text/plain": [ "L\"$\\ds\\lim_{n\\to\\infty}\\frac{1}{n}\\left(\\frac{(2n)!}{n!}\\right)^{1/n}=\\frac{4}{e}$\"" ] }, "execution_count": 64, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# 後者の問題の SymPy による解\n", "\n", "n = symbols(\"n\", positive=true)\n", "sol = limit((1/n)*(gamma(2n+1)/gamma(n+1))^(1/n), n=>oo)\n", "latexstring(raw\"\\ds\\lim_{n\\to\\infty}\\frac{1}{n}\\left(\\frac{(2n)!}{n!}\\right)^{1/n}=\", sympy.latex(sol))" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "**問題:** 上で扱った問題をStirlingの公式を使わずに解け. $\\QED$\n", "\n", "**ヒント:** 大学入試問題レベルなのでヒントは必要ないと思われるが, 念のためにヒントを与えておく. 対数を取ってから極限を取ると, 区分求積法によって定積分の計算に極限の計算が帰着する. $\\QED$\n", "\n", "**注意:** 対数を取ってから積分で近似するというアイデアでStirlingの公式を証明することもできる. $\\QED$\n", "\n", "**注意:** 以上の話題に関する詳しい解説については\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4節を参照せよ. $\\QED$\n", "\n", "**注意:** Stirlingの公式は場合の数の漸近挙動の分析で基本的な役目を果たす. 二項係数 $\\ds\\binom{n}{k}$ の $n,k$ が大きなときの漸近挙動**はエントロピー**やその $-1$ 倍の**情報量**と関係がある. この点に関しては\n", "\n", "* 黒木玄, Kullback-Leibler情報量とSanovの定理\n", "\n", "の解説が詳しい. 統計学の授業で「尤度」(ゆうど)の概念について習うが, 尤度がどうして「尤もらしさ」(もっともらしさ)だと解釈できるかはSanovの定理について学ばないと理解不可能である. この意味でSanovの定理は大数の法則や中心極限定理に匹敵するほど統計学の基礎付けにおいて基本的な結果である. 以上の点は赤池弘次氏の論説\n", "\n", "* 赤池弘次, エントロピーとモデルの尤度(〈講座〉物理学周辺の確率統計), 日本物理学会誌, 1980年35巻7号, pp. 608-614\n", "\n", "* 赤池弘次, 統計的推論のパラダイムの変遷について, 統計数理研究所彙報, 27巻1号, pp. 5-12, 1980-03\n", "\n", "を参照せよ. 将来統計学を理解することが必要になったら, これらの文献を最初に眺めるとよい. $\\QED$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## ベータ函数の応用\n", "\n", "この節では高校数学とベータ函数の関係について解説する." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### 1/6公式\n", "\n", "大学受験のために次の公式を「1/6公式」などと呼んで「暗記せよ」と教えている場合もあるようだ:\n", "\n", "$$\n", "\\int_a^b (x-a)(b-x)\\,dx = \\frac{(b-a)^3}{6}.\n", "$$\n", "\n", "もちろんそのような数学の教え方はよくない. 実はこの公式は大学1年のときに習うベータ函数に関する公式の特殊な場合だとみなされる. ベータ函数は\n", "\n", "$$\n", "B(p,q) = \\int_0^1 x^{p-1}(1-x)^{q-1}\\,dx \\quad(p,q>0)\n", "$$\n", "\n", "と定義される. \n", "\n", "ベータ函数はガンマ函数によって次のように表わされるのであった(後で証明する):\n", "\n", "$$\n", "B(p,q) = \\frac{\\Gamma(p)\\Gamma(q)}{\\Gamma(p+q)}.\n", "$$\n", "\n", "例えば $B(2,2)=\\Gamma(2)^2/\\Gamma(4)=(1!)^2/3!=1/6$ となることがわかる. 以下が成立している: $x=y+a$, $y=(b-a)z$ とおくと, \n", "\n", "$$\n", "\\begin{aligned}\n", "&\n", "\\int_a^b (x-a)^{p-1}(b-x)^{q-1}\\,dx =\n", "\\int_0^{b-a} y^{p-1}(b-a-y)^{q-1}\\,dy \n", "\\\\ &\\qquad=\n", "\\int_0^1 (b-a)^{p-1}z^{p-1}(b-a)^{q-1}(1-z)^{q-1}(b-a)\\,dz\n", "\\\\ &\\qquad=\n", "(b-a)^{p+q-1}B(p,q).\n", "\\end{aligned}\n", "$$\n", "\n", "これは「1/6公式」の大幅な一般化になっている." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "**問題:** $B(p,q)=B(q,p)$ を示せ.\n", "\n", "**略解1:** $B(p,q)=\\Gamma(p)\\Gamma(q)/\\Gamma(p+q)$ より $B(p,q)=B(q,p)$ であることがわかる. $\\QED$\n", "\n", "**略解2:** $\\ds B(p,q)=\\int_0^1 x^{p-1}(1-x)^{q-1}\\,dx$ において $x=1-y$ とおくと, $B(p,q)=B(q,p)$ であることがわかる. $\\QED$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "**問題:** 高校の教科書にある次の問題をベータ函数を用いて解け." ] }, { "cell_type": "code", "execution_count": 65, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "text/html": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "showimg(\"image/jpeg\", \"images/Jikkyo20140125Beta.jpg\", scale=\"80%\")" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "-" } }, "source": [ "**解答例:** 求めるべき面積を $S$ と書くと, \n", "\n", "$$\n", "\\begin{aligned}\n", "S &= 2\\int_0^1 \\sqrt{x(1-x)^2}\\;dx \n", "\\\\ &=\n", "2\\int_0^1 x^{3/2-1}(1-x)^{2-1}\\,dx =\n", "2B(3/2,2).\n", "\\end{aligned}\n", "$$\n", "\n", "そして, \n", "\n", "$$\n", "\\begin{aligned}\n", "&\n", "\\Gamma(3/2)=\\frac{1}{2}\\Gamma(1/2)=\\frac{\\sqrt{\\pi}}{2}, \n", "\\\\ &\n", "\\Gamma(2)=1!=1, \n", "\\\\ &\n", "\\Gamma(3/2+2)=\\Gamma(7/2)=\n", "\\frac{5}{2}\n", "\\frac{3}{2}\n", "\\frac{1}{2}\n", "\\sqrt{\\pi},\n", "\\\\ &\n", "S=2B(3/2,2) = \\frac{2\\Gamma(3/2)\\Gamma(2)}{\\Gamma(3/2+2)} \n", "\\\\ & \\;\\, =\n", "2\\times\\frac{\\sqrt{\\pi}}{2}\\times 1 \\times \\frac{2^3}{1\\cdot3\\cdot5\\sqrt{\\pi}} =\n", "\\frac{8}{15}.\n", "\\qquad\\QED\n", "\\end{aligned}\n", "$$" ] }, { "cell_type": "code", "execution_count": 66, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "text/latex": [ "$\\ds 2\\int_0^1\\sqrt{x(1-x^2)}\\,dx=\\frac{8}{15}$" ], "text/plain": [ "L\"$\\ds 2\\int_0^1\\sqrt{x(1-x^2)}\\,dx=\\frac{8}{15}$\"" ] }, "execution_count": 66, "metadata": {}, "output_type": "execute_result" } ], "source": [ "x = symbols(\"x\", real=true)\n", "sol = 2integrate(√(x*(1-x)^2), (x,0,1))\n", "latexstring(raw\"\\ds 2\\int_0^1\\sqrt{x(1-x^2)}\\,dx=\", sympy.latex(sol))" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### sinのべきの定積分\n", "\n", "$n$ は0以上の整数であるとする. 高校数学では\n", "\n", "$$\n", "S_n = \\int_0^{\\pi/2} \\sin^n\\theta\\,d\\theta\n", "$$\n", "\n", "の形の定積分を扱うことがある. これは本質的にベータ函数の特別な場合 $B(1/2, q)$ に一致する. 実際, $x=\\cos^2\\theta$ とおくと,\n", "\n", "$$\n", "\\begin{aligned}\n", "B(1/2, q) &= \\int_0^1 x^{-1/2}(1-x)^{q-1}\\,dx \n", "\\\\ &=\n", "\\int_0^{\\pi/2} (\\cos\\theta)^{-1} \\;(\\sin\\theta)^{2q-2}\\;2\\cos\\theta\\;\\sin\\theta\\;d\\theta\n", "\\\\ &=\n", "2\\int_0^{\\pi/2} (\\sin\\theta)^{2q-1}\\,d\\theta.\n", "\\end{aligned}\n", "$$\n", "\n", "特に\n", "\n", "$$\n", "\\frac{1}{2}B\\left(\\frac{1}{2}, \\frac{n+1}{2}\\right) = \\int_0^{\\pi/2}\\sin^n\\theta\\,d\\theta. \n", "$$\n", "\n", "より一般に\n", "\n", "$$\n", "\\frac{1}{2}B\\left(\\frac{m+1}{2}, \\frac{n+1}{2}\\right) = \n", "\\int_0^{\\pi/2}\\cos^m\\theta\\;\\sin^n\\theta\\,d\\theta. \n", "$$\n", "\n", "ベータ函数の応用範囲は結構広い." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### ガンマ函数とベータ函数の関係\n", "\n", "$p,q>0$ であると仮定する. \n", "\n", "ベータ函数は $x=1/(1+t)$, $dx=-dt/(1+t)^2$ という積分変数の変換によって\n", "\n", "$$\n", "\\begin{aligned}\n", "B(p,q)& = \\int_0^1 x^{p-1}(1-x)^{q-1}\\,dx \n", "\\\\&=\n", "\\int_0^\\infty \\frac{1}{(1+t)^{p-1}}\\frac{t^{q-1}}{(1+t)^{q-1}}\\frac{dt}{(1+t)^2} \n", "\\\\&=\n", "\\int_0^\\infty \\frac{t^{q-1}}{(1+t)^{p+q}}\\,dt.\n", "\\end{aligned}\n", "$$\n", "\n", "この計算における最後の行によるベータ函数の表示は統計学における第2種ベータ分布で使用され, $\\ds B(p,q)=\\int_0^1 x^{p-1}(1-x)^{q-1}\\,dx$ という表示は第1種ベータ分布で使用される. どちらの表示も重要である.\n", "\n", "$a,b>0$ のとき, 積分変数の変換 $x=y/a$ によって,\n", "\n", "$$\n", "\\begin{aligned}\n", "\\int_0^\\infty e^{-ax}x^{b-1}\\,dx &= \n", "\\int_0^\\infty e^{-y}\\frac{x^{b-1}}{a^{b-1}}\\frac{dy}{a} \n", "\\\\ &= \n", "\\frac{1}{a^b}\\int_0^\\infty e^{-y}x^{b-1}\\,dy = \n", "\\frac{\\Gamma(b)}{a^b}.\n", "\\end{aligned}\n", "$$\n", "\n", "ガンマ函数はこの形式で自然に現われることが非常に多い.\n", "\n", "以上の準備のもとで, \n", "\n", "$$\n", "\\Gamma(p)=\\int_0^\\infty e^{-x}x^{p-1}\\,dx, \\quad\n", "\\Gamma(q)=\\int_0^\\infty e^{-y}y^{p-1}\\,dy\n", "$$\n", "\n", "の積は以下のように計算される. $y=tx$ という積分変数の変換(これは平面上の $(x,y)$ を傾き $t$ の原点を通る直線と $x$ の値で表示する変数変換であり, それなりの自然さを持っている)と積分順序の交換によって,\n", "\n", "$$\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", "\\\\ &=\n", "\\int_0^\\infty \\left(\\int_0^\\infty e^{-x-y}x^{p-1}(tx)^{q-1}x\\,dt\\right)\\,dx \n", "\\\\ &=\n", "\\int_0^\\infty \\left(\\int_0^\\infty e^{-x-tx}x^{p-1}(tx)^{q-1}x\\,dt\\right)\\,dx \n", "\\\\ &=\n", "\\int_0^\\infty \\left(\\int_0^\\infty t^{q-1} e^{-(1+t)x}x^{p+q-1}\\,dt\\right)\\,dx \n", "\\\\ &=\n", "\\int_0^\\infty \\left(\\int_0^\\infty t^{q-1} e^{-(1+t)x}x^{p+q-1}\\,dx\\right)\\,dt \n", "\\\\ &=\n", "\\int_0^\\infty t^{q-1}\\frac{\\Gamma(p+q)}{(1+t)^{p+q}}\\,dt \n", "\\\\ &=\n", "\\Gamma(p+q) \\int_0^\\infty \\frac{t^{q-1}}{(1+t)^{p+q}}\\,dt \n", "\\\\ &=\n", "\\Gamma(p+q)B(p,q).\n", "\\end{aligned}\n", "$$\n", "\n", "これで\n", "\n", "$$\n", "\\Gamma(p)\\Gamma(q) = \\Gamma(p+q)B(p,q)\n", "$$\n", "\n", "が示された. これと $x=\\cos^2 \\theta$ とおくことによって得られる\n", "\n", "$$\n", "\\begin{aligned}\n", "B(p,q) &= \\int_0^1 x^{p-1}(1-x)^{q-1}\\,dx \n", "\\\\ &=\n", "2\\int_0^{\\pi/2} (\\cos\\theta)^{2p-1} (\\sin\\theta)^{2q-1}\\,d\\theta\n", "\\end{aligned}\n", "$$\n", "\n", "より, $p,q=1/2$ のとき $B(1/2,1/2)=\\pi$ であるから, $\\ds\\Gamma(1)=\\int_0^\\infty e^{-x}\\,dx=[-e^{-x}]_0^\\infty =1$ を使うと, \n", "\n", "$$\n", "\\Gamma(1/2)^2 = \\Gamma(1)B(1/2,1/2) = \\pi, \\quad\n", "\\therefore\\quad\n", "\\Gamma(1/2)=\\sqrt{\\pi}.\n", "$$\n", "\n", "さらに, $x=\\sqrt{y}$ とおくと, \n", "\n", "$$\n", "\\begin{aligned}\n", "\\int_{-\\infty}^\\infty e^{-x^2}\\,dx &= \n", "2\\int_0^\\infty e^{-x^2}\\,dx \n", "\\\\ &=\n", "\\int_0^\\infty e^{-y} y^{-1/2}\\,dx =\n", "\\Gamma(1/2)=\\sqrt{\\pi}.\n", "\\end{aligned}\n", "$$\n", "\n", "要するにガンマ函数とベータ函数の関係はGauss積分の公式 $\\ds\\int_{-\\infty}^\\infty e^{-x^2}\\,dx=\\sqrt{\\pi}$ を含んでいる. \n", "\n", "正規分布の確率密度函数を理解するためには, Gauss積分の公式を理解しておかないといけない. Stirlingの公式の導出でもGauss積分の公式を利用した. Gauss積分は多くの数学的場面に普遍的に現われる重要な積分である." ] }, { "cell_type": "code", "execution_count": 67, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "text/html": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "showimg(\"image/jpeg\", \"images/Gamma-Beta-01.jpg\", scale=\"60%\")" ] }, { "cell_type": "code", "execution_count": 68, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "text/html": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "showimg(\"image/jpeg\", \"images/Gamma-Beta-02.jpg\", scale=\"60%\")" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### ベータ函数の極限によるガンマ函数の表示とWallisの公式\n", "\n", "$\\ds B(p,q)=\\int_0^1 x^{p-1}(1-x)^{q-1}\\,dx$ において, $p=s$, $q=n+1$, $x = t/n$ とおいて, $n\\to\\infty$ とすると,\n", "\n", "$$\n", "\\begin{aligned}\n", "n^s B(s,n+1) &= \n", "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\\,dx \n", "\\\\ &\\to\n", "\\int_0^\\infty t^{s-1} e^{-t}\\,dt = \\Gamma(s).\n", "\\end{aligned}\n", "$$\n", "\n", "特に $s=1/2$ のとき\n", "\n", "$$\n", "\\sqrt{n}\\;B(1/2,n+1)\\to \\Gamma(1/2) = \\sqrt{\\pi}.\n", "$$\n", "\n", "ベータ函数の三角函数を用いた表示を使うと, \n", "\n", "$$\n", "2 n^s \\int_0^{\\pi/2} (\\cos\\theta)^{2s-1}(\\sin\\theta)^{2n+1}\\,d\\theta \\to \\Gamma(s), \\quad\n", "2 \\sqrt{n} \\int_0^{\\pi/2} (\\sin\\theta)^{2n+1}\\,d\\theta \\to \\Gamma(1/2).\n", "$$\n", "\n", "$\\sin$ のべきの $0$ から $\\pi/2$ での定積分はGauss積分 $\\ds \\Gamma(1/2)=\\int_{-\\infty}^\\infty e^{-x^2}\\,dx$ の計算にこのような形で関係している.\n", "\n", "$n$ が正の整数のとき, \n", "\n", "$$\n", "\\begin{aligned}\n", "&\n", "\\Gamma(n+1)=n!, \\quad \\Gamma(1/2)=\\sqrt{\\pi},\n", "\\\\ &\n", "\\Gamma(n+1/2)=\\frac{2n-1}{2}\\cdots\\frac{3}{2}\\frac{1}{2}\\sqrt{\\pi} \n", "\\\\&\\qquad=\n", "\\frac{1\\cdot3\\cdots(2n-1)}{2^n}\\frac{2\\cdot4\\cdots(2n)}{2^n n!}\\sqrt{\\pi} =\n", "\\frac{(2n)!}{2^{2n} n!}\\sqrt{\\pi},\n", "\\\\ &\n", "\\Gamma(n+1+1/2) = \n", "(n+1/2)\\Gamma(n+1/2) = \n", "\\frac{2n+1}{2}\\frac{(2n)!}{2^{2n} n!}\\sqrt{\\pi},\n", "\\\\ &\n", "\\frac{1}{\\sqrt{n}B(1/2,n+1)} = \n", "\\frac{\\Gamma(n+1+1/2)}{\\sqrt{n}\\;\\Gamma(1/2)\\Gamma(n+1)} \n", "\\\\ &\\qquad=\n", "\\frac{2n+1}{2}\\frac{(2n)!}{2^{2n} n!}\\sqrt{\\pi}\\cdot\n", "\\frac{1}{\\sqrt{n}\\sqrt{\\pi}\\;n!}\n", "\\\\ &\\qquad=\n", "\\frac{2n+1}{2n}\\sqrt{n}\\;\\frac{1}{2^{2n}}\\binom{2n}{n} \\to \n", "\\frac{1}{\\Gamma(1/2)}=\\frac{1}{\\sqrt{\\pi}}\n", "\\quad (n\\to\\infty).\n", "\\end{aligned}\n", "$$\n", "\n", "ここで, $\\ds\\frac{2n+1}{2\\sqrt{n}} = \\frac{2n+1}{2n}\\sqrt{n}$, $\\ds\\frac{(2n)!}{n!n!}=\\binom{2n}{n}$ を使った. $\\ds\\frac{2n+1}{2n}\\to 1$ より, \n", "\n", "$$\n", "\\sqrt{n}\\;\\frac{1}{2^{2n}}\\binom{2n}{n} \\to \\frac{1}{\\sqrt{\\pi}}.\n", "$$\n", "\n", "すなわち\n", "\n", "$$\n", "\\frac{1}{2^{2n}}\\binom{2n}{n} \\sim \\frac{1}{\\sqrt{\\pi n}}\n", "$$\n", "\n", "が示された. これを**Wallisの公式**(ウォリスの公式)と呼ぶ. これとは見掛け上異なる同値な結果\n", "\n", "$$\n", "\\prod_{n=1}^\\infty \\frac{(2n)(2n)}{(2n-1)(2n+1)} = \\frac{\\pi}{2}\n", "$$\n", "\n", "もWallisの公式と呼ぶこともある. \n", "\n", "**問題:** すぐ上の後者のWallisの公式を証明せよ. $\\QED$\n", "\n", "**ヒント:** すでに証明した前者のWallisの公式を使えば後者を示せる. $B(1/2,n+1)/B(1/2,n+1/2)\\to 1$ を書き直しても後者のWallisの公式が得られる. もしくは高校の教科書の掲載されている $\\sin$ のべきの $0$ から $\\pi/2$ までの定積分の計算結果と上で述べたことを合わせて使ってみよ. 偶数べきと奇数べきの比を考えよ. $\\QED$" ] }, { "cell_type": "code", "execution_count": 69, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "text/html": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "showimg(\"image/jpeg\", \"images/Jikkyo20140125Wallis.jpg\", scale=\"50%\")" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "**参考:** 以上で扱った大学レベルの微分積分学については\n", "\n", "* 黒木玄, 微分積分学のノート\n", "\n", "を参照せよ. 例えば, Wallisの公式については「10 Gauss積分, ガンマ函数, ベータ函数」「12 Fourier解析」に非常に詳しい解説がある. $\\QED$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Gaussの超幾何函数への一般化\n", "\n", "$\\ds \\int_a^b (x-a)^A (b-x)^B \\,dx$ 型の積分は本質的にベータ函数とみなせるのであった. これを\n", "\n", "$$\n", "I = \\int_a^b (x-a)^A (b-x)^B (c-x)^C \\,dx\n", "$$\n", "\n", "に一般化するとどうなるか. このような一般化は高校生でも自然に思い付きそうである. \n", "\n", "$$\n", "x = (1-t)a + t b = a + (b-a)t = b - (b-a)(1-t), \\quad z = \\frac{b-a}{c-a}\n", "$$\n", "\n", "とおくと, \n", "\n", "$$\n", "I = (b-a)^{A+B+1}(c-a)^C \\int_0^1 t^A (1-t)^B (1-zt)^C \\,dt. \n", "$$\n", "\n", "Gaussの超幾何函数 ${}_2F_1(a,b,c;z)$ が\n", "\n", "$$\n", "{}_2F_1(a,b,c;z) = \\frac{1}{B(a,c-a)} \\int_0^1 t^{a-1}(1-t)^{c-a-1}(1-zt)^{-b}\\,dt\n", "$$\n", "\n", "と定義される. 上の積分 $I$ は本質的にGaussの超幾何函数である.\n", "\n", "このように高校生が取り扱いに挑戦しそうなちょっとした積分であっても, 本質的にGaussの超幾何函数になってしまうことがある. 高校生に微積分を教える予定がある人はGaussの超幾何函数についても知っておいた方がよいだろう." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Kummerの超幾何函数\n", "\n", "$\\ds \\int_a^b (x-a)^A (b-x)^B \\,dx$ 型の積分は\n", "\n", "$$\n", "J = \\int_a^b (x-a)^A (b-x)^B e^{rx} dx\n", "$$\n", "\n", "という型の積分にも一般化される. $r=0$ の場合が本質的にベータ函数の場合である. この積分は\n", "\n", "$$\n", "x = (1-t)a + t b = a + (b-a)t = b - (b-a)(1-t), \\quad z = (b-a)r\n", "$$\n", "\n", "とおくと, 次のように書き直される:\n", "\n", "$$\n", "J = (b-a)^{A+B+1} e^{ra} \\int_0^1 t^A (1-t)^B e^{zt}\\,dt.\n", "$$\n", "\n", "Kummerの超幾何函数 ${}_1F_1(a,c;z)$ が\n", "\n", "$$\n", "{}_1F_1(a,c;z) = \\frac{1}{B(a,c-a)}\\int_0^1 t^{a-1}(1-t)^{c-a-1}e^{zt}\\,dt\n", "$$\n", "\n", "と定義される. 上の積分 $J$ は本質的にKummerの超幾何函数である.\n", "\n", "Kummerの超幾何函数はGaussの超幾何函数で\n", "\n", "$$\n", "z = \\frac{z}{b}\n", "$$\n", "\n", "とおいて $b\\to\\infty$ とすれば得られる: $b\\to\\infty$ のとき\n", "\n", "$$\n", "\\begin{aligned}\n", "{}_2F_1\\left(a,b,c;\\frac{z}{b}\\right) &=\n", "\\frac{1}{B(a,c-a)}\\int_0^1 t^{a-1}(1-t)^{c-a-1}\\left(1-\\frac{zt}{b}\\right)^{-b}\\,dt \\\\\n", "& \\to\n", "\\frac{1}{B(a,c-a)}\\int_0^1 t^{a-1}(1-t)^{c-a-1}e^{zt}\\,dt =\n", "{}_1F_1(a,c;z).\n", "\\end{aligned}\n", "$$\n", "\n", "この手続きは $z = b/t$ における特異点を $z=\\infty$ における特異点に合流させる手続きになっており, Kummerの超幾何函数は合流型超幾何函数と呼ばれる超幾何函数の一族のうちの1つになっている.\n", "\n", "Gauss積分($\\Gamma(1/2)$ に等しい), ガンマ函数 $\\Gamma(s)$, ベータ函数 $B(a,b)$, Gaussの超幾何函数 ${}_2F_1(a,b,c;z)$, Kummerの超幾何函数 ${}_1F_1(a,c;z)$ などは特殊函数の広い一族の一部分になっており, 高校数学でも自然に出て来てしまうものだと言える.\n", "\n", "この点に関してはツイッターにおける以下のスレッドも参照せよ:\n", "\n", "* https://twitter.com/genkuroki/status/1093510712125583360" ] }, { "cell_type": "code", "execution_count": 70, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "text/html": [ "

ベータ函数の現れ方(1)

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ベータ函数の現れ方(1)

\")\n", "showimg(\"image/jpeg\", \"images/Beta-Gamma-Gauss-Kummer-01.jpg\", scale=\"50%\")" ] }, { "cell_type": "code", "execution_count": 71, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "text/html": [ "

ベータ函数の現れ方(2)

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ベータ函数の現れ方(2)

\")\n", "showimg(\"image/jpeg\", \"images/Beta-Gamma-Gauss-Kummer-02.jpg\", scale=\"50%\")" ] }, { "cell_type": "code", "execution_count": 72, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "text/html": [ "

ガンマ函数の基礎

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ガンマ函数の基礎

\")\n", "showimg(\"image/jpeg\", \"images/Beta-Gamma-Gauss-Kummer-03.jpg\", scale=\"80%\")" ] }, { "cell_type": "code", "execution_count": 73, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "text/html": [ "

Gaussの超幾何函数の現れ方(1)

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Gaussの超幾何函数の現れ方(1)

\")\n", "showimg(\"image/jpeg\", \"images/Beta-Gamma-Gauss-Kummer-04.jpg\", scale=\"60%\")" ] }, { "cell_type": "code", "execution_count": 74, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "text/html": [ "

Gaussの超幾何函数の現れ方(2)

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Gaussの超幾何函数の現れ方(2)

\")\n", "showimg(\"image/jpeg\", \"images/Beta-Gamma-Gauss-Kummer-05.jpg\", scale=\"70%\")" ] }, { "cell_type": "code", "execution_count": 75, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "text/html": [ "

Kummerの超幾何函数の現れ方

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Kummerの超幾何函数の現れ方

\")\n", "showimg(\"image/jpeg\", \"images/Beta-Gamma-Gauss-Kummer-06.jpg\", scale=\"70%\")" ] }, { "cell_type": "code", "execution_count": 76, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "text/html": [ "

ガンマ函数と正弦函数の関係

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ガンマ函数と正弦函数の関係

\")\n", "showimg(\"image/jpeg\", \"images/Beta-Gamma-Gauss-Kummer-07.jpg\", scale=\"70%\")" ] }, { "cell_type": "code", "execution_count": 77, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "text/html": [ "

Hurwitzのゼータ函数とガンマ函数の関係

" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/html": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "display(\"text/html\", \"

Hurwitzのゼータ函数とガンマ函数の関係

\")\n", "showimg(\"image/jpeg\", \"images/Hurwitz-Gamma.jpg\", scale=\"70%\")" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Taylor展開\n", "\n", "例えば, 実教出版の高校数学の教科書『数学III』2014年1月25日発行の終わりの方にはCauchyの平均値の定理の応用としてTaylorの公式を示す議論が載っている. その証明法は高木貞治『解析概論』におけるTaylorの公式の証明法と同じである. 以下ではより「初等的」な証明を解説する." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Taylorの公式の証明\n", "\n", "以下では微分積分学の基本定理(微分して積分するとものとの函数に戻るという意味の公式)\n", "\n", "$$\n", "f(x) = f(a) + \\int_a^x f'(x_1)\\,dx_1\n", "$$\n", "\n", "のみを用いたTaylorの公式のシンプルな証明法を紹介する. 以下の方針であれば高木貞治『解析概論』におけるTaylorの公式の証明法と違って誰でも容易に理解できるものと思われる.\n", "\n", "以下では繰り返し函数を積分する. そのとき括弧の使用量を減らすために積分を\n", "\n", "$$\n", "\\int_a^x g(x_1)\\,dx_1 = \\int_a^x dx_1\\, g(x_1)\n", "$$\n", "\n", "の右辺のように書く場合もある. 例えば,\n", "\n", "$$\n", "\\begin{aligned}\n", "&\n", "\\int_a^x \\left(\\int_a^{x_1}\\left(\\int_a^{x_2}g(x_3)\\,dx_3\\right)\\,dx_2\\right)\\,dx_1 \n", "\\\\ & \\qquad=\n", "\\int_a^x dx_1 \\int_a^{x_1}dx_2 \\int_a^{x_2}dx_3\\,g(x_3).\n", "\\end{aligned}\n", "$$\n", "\n", "右辺の書き方であれば括弧の使用量を大幅に減らすことができる.\n", "\n", "以下, $n=4$ であると仮定し, $f(x)$ は $C^n$ 級($n$ 回微分可能で $f^{(n)}$ は連続)であると仮定する. 一般の $n$ についても以下の議論は同様に適用できる. $f^{(4)}(x_4)$ を $x_4=a$ から $x_4=x_3$ まで積分すると\n", "\n", "$$\n", "f'''(x_3) = f'''(a) + \\int_a^{x_3}dx_4\\,f^{(4)}(x_4).\n", "$$\n", "\n", "両辺を $x_3=a$ から $x_3=x_2$ まで積分すると\n", "\n", "$$\n", "\\begin{aligned}\n", "f''(x_2) &= f''(a) + \\int_a^{x_2}dx_3\\,f'''(x_3)\n", "\\\\ &=\n", "f''(a) + f'''(a)(x_2-a) + \\int_a^{x_2}dx_3\\int_a^{x_3}dx_4\\,f^{(4)}(x_4).\n", "\\end{aligned}\n", "$$\n", "\n", "両辺を $x_2=a$ から $x_2=x_1$ まで積分すると\n", "\n", "$$\n", "\\begin{aligned}\n", "f'(x_1) &= f'(a) + \\int_a^{x_1}dx_2\\,f''(x_2)\n", "\\\\ &=\n", "f'(a) + f''(a)(x_1-a) + f'''(a)\\frac{(x_1-a)^2}{2} + Q,\n", "\\\\ \n", "Q &=\\int_a^{x_1}dx_2\\int_a^{x_2}dx_3\\int_a^{x_3}dx_4\\,f^{(4)}(x_4).\n", "\\end{aligned}\n", "$$\n", "\n", "両辺を $x_1=a$ から $x_1=x$ まで積分すると\n", "\n", "$$\n", "\\begin{aligned}\n", "f(x) &= f(a) + \\int_a^{x}dx_1\\,f'(x_1)\n", "\\\\ &=\n", "f(a) + f'(a)(x-a) + f''(a)\\frac{(x-a)^2}{2} + f'''(a)\\frac{(x-a)^3}{3!} + R_4,\n", "\\\\ \n", "R_4 &=\n", "\\int_a^x dx_1\\int_a^{x_1}dx_2\\int_a^{x_2}dx_3\\int_a^{x_3}dx_4\\,f^{(4)}(x_4).\n", "\\end{aligned}\n", "$$\n", "\n", "一般の $n$ では以下が成立する:\n", "\n", "$$\n", "\\begin{aligned}\n", "&\n", "f(x) = \\sum_{k=0}^{n-1}f^{(k)}(a)\\frac{(x-a)^k}{k!} + R_n, \n", "\\\\ &\n", "R_n = \\int_a^x dx_1\\int_a^{x_1}dx_2\\cdots\\int_a^{x_{n-1}}dx_n\\,f^{(n)}(x_n).\n", "\\end{aligned}\n", "$$\n", "\n", "これを**Taylorの公式**と呼び, $R_n$ を**剰余項**と呼ぶ.\n", "\n", "Taylorの公式において $\\ds\\frac{(x-a)^k}{k!}$ の項が出て来る理由も以上の議論では明瞭である. 定数函数 $1$ を $a$ から $x$ まで積分することを $k$ 回繰り返すと, $\\ds\\frac{(x-a)^k}{k!}$ が出て来る. $n$ 階の導函数を $n$ 回積分するだけなので簡単である." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Taylorの公式の剰余項の評価 (1)\n", "\n", "$R_n$ の絶対値の大きさの評価不等式を作ろう. ある定数 $M_n$ が存在して, $a$ と $x$ のあいだの実数 $x_n$ について $|f^{(n)}(x_n)|\\leqq M_n$ が成立しているとする. このとき,\n", "\n", "$$\n", "|R_n| \\leqq \\left|\\int_a^x dx_1\\int_a^{x_1}dx_2\\cdots\\int_a^{x_{n-1}}dx_n\\,M_n\\right| =\n", "\\frac{M_n|x-a|^n}{n!}.\n", "$$\n", "\n", "したがって, もしも $n\\to\\infty$ のとき $\\ds\\frac{M_n|x-a|^n}{n!}\\to 0$ が成立しているならば, **Taylor展開**\n", "\n", "$$\n", "f(x) = \\sum_{k=0}^\\infty f^{(k)}(a)\\frac{(x-a)^k}{k!}\n", "$$\n", "\n", "が成立する. \n", "\n", "**例:** $f(x)=e^x$, $a=0$ の場合を考えよう. このとき, $f'(x)=f(x)$ と $f(0)=1$ より, $f^{(k)}(0)=1$ となる. $r>0$ であるとし, $|x|\\leqq r$ であると仮定する. $0$ と $x$ のあいだの実数 $x_n$ について, $0< f^{(n)}(x_n) = f(x_n) \\leqq f(r)=e^r$ となる. したがって,\n", "\n", "$$\n", "e^x = \n", "f(x) =\n", "\\sum_{k=0}^{n-1} f^{(k)}(0)\\frac{x^k}{k!} + R_n =\n", "\\sum_{k=0}^{n-1} \\frac{x^k}{k!} + R_n\n", "$$\n", "\n", "でかつ\n", "\n", "$$\n", "|R_n| \\leqq \n", "\\frac{e^r |x|^n}{n!} \\leqq \n", "\\frac{e^r r^n}{n!} \\to 0 \\quad (n\\to\\infty).\n", "$$\n", "\n", "$r$ は幾らでも大きくできるので, $|x|$ がどんなに大きくても, \n", "\n", "$$\n", "e^x = \\sum_{k=0}^\\infty \\frac{x^k}{k!}\n", "$$\n", "\n", "が成立している. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Taylorの公式の剰余項の評価 (2)\n", "\n", "$R_n$ 自体の大きさの評価式も同様にして作れる. \n", "\n", "簡単のため $a < x$ であると仮定する.\n", "\n", "$a\\leqq t \\leqq x$ において $A\\leqq f^{(n)}(t)\\leqq B$ が成立しているとする. このとき, \n", "\n", "$$\n", "\\begin{aligned}\n", "&\n", "\\int_a^x dx_1\\int_a^{x_1}dx_2\\cdots\\int_a^{x_{n-1}}dx_n\\,A\n", "\\\\ &\\qquad \\leqq\n", "R_n = \\int_a^x dx_1\\int_a^{x_1}dx_2\\cdots\\int_a^{x_{n-1}}dx_n\\,f^{(n)}(x_n)\n", "\\\\ &\\qquad\\qquad \\leqq\n", "\\int_a^x dx_1\\int_a^{x_1}dx_2\\cdots\\int_a^{x_{n-1}}dx_n\\,B.\n", "\\end{aligned}\n", "$$\n", "\n", "すなわち\n", "\n", "$$\n", "A\\frac{(x-a)^n}{n!}\n", "\\leqq\n", "R_n\n", "\\leqq\n", "B\\frac{(x-a)^n}{n!}.\n", "$$\n", "\n", "ゆえに,\n", "\n", "$$\n", "\\sum_{k=0}^\\infty f^{(k)}(a)\\frac{(x-a)^k}{k!} + A\\frac{(x-a)^n}{n!}\n", "\\leqq\n", "f(x)\n", "\\leqq\n", "\\sum_{k=0}^\\infty f^{(k)}(a)\\frac{(x-a)^k}{k!} + B\\frac{(x-a)^n}{n!}.\n", "$$\n", "\n", "$A$, $B$ の値を具体的に求められる場合にはこの不等式を用いて $f(x)$ が含まれる範囲が分かる." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "__例:__ $n=2$ のとき, $a