{ "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, 2024, 2025 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, 2024-08-28, 2025-09-02\n", "* License: MIT https://opensource.org/licenses/MIT\n", "* Repository: https://github.com/genkuroki/HighSchoolMath\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", "このファイルは次の場所で実行できる:\n", "\n", "* Google Colabでこのノートを開く (ランライム→すべてのセルを実行)\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": {}, "outputs": [ { "data": { "text/plain": [ "@autoadd" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Google Colabと自分のパソコンの両方で使えるようにするための工夫\n", "\n", "using Pkg\n", "\n", "\"\"\"すでにPkg.add済みのパッケージのリスト\"\"\"\n", "_packages_added = [sort!(readdir(Sys.STDLIB));\n", " sort!([info.name for (uuid, info) in Pkg.dependencies() if info.is_direct_dep])]\n", "\n", "\"\"\"_packages_added内にないパッケージをPkg.addする\"\"\"\n", "add_pkg_if_not_added_yet(pkg) = if isnothing(Base.find_package(pkg))\n", " println(stderr, \"# $(pkg).jl is not added yet, so let's add it.\")\n", " Pkg.add(pkg)\n", "end\n", "\n", "\"\"\"expr::Exprからusing内の`.`を含まないモジュール名を抽出\"\"\"\n", "function find_using_pkgs(expr::Expr)\n", " pkgs = String[]\n", " function traverse(expr::Expr)\n", " if expr.head == :using\n", " for arg in expr.args\n", " if arg.head == :. && length(arg.args) == 1\n", " push!(pkgs, string(arg.args[1]))\n", " elseif arg.head == :(:) && length(arg.args[1].args) == 1\n", " push!(pkgs, string(arg.args[1].args[1]))\n", " end\n", " end\n", " else\n", " for arg in expr.args arg isa Expr && traverse(arg) end\n", " end\n", " end\n", " traverse(expr)\n", " pkgs\n", "end\n", "\n", "\"\"\"必要そうなPkg.addを追加するマクロ\"\"\"\n", "macro autoadd(expr)\n", " pkgs = find_using_pkgs(expr)\n", " :(add_pkg_if_not_added_yet.($(pkgs)); $expr)\n", "end" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [], "source": [ "using LinearAlgebra: det\n", "using Printf\n", "\n", "@autoadd begin\n", "using Plots\n", "default(fmt=:png)\n", "using SymPy\n", "using LaTeXStrings\n", "using SpecialFunctions\n", "using Elliptic\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": "markdown", "metadata": {}, "source": [ "" ] }, { "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": "markdown", "metadata": {}, "source": [ "" ] }, { "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": "markdown", "metadata": {}, "source": [ "" ] }, { "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": "markdown", "metadata": {}, "source": [ "" ] }, { "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": {}, "source": [ "### 三角関数の加法定理と複素数\n", "\n", "実数 $a,b,c,d$ から複素数 $z, w$ を\n", "\n", "$$\n", "z = a + ci, \\quad w = b + di\n", "$$\n", "\n", "と定める. これらは次のように表される:\n", "\n", "$$\n", "z = |z| e^{i\\alpha}, \\quad w = |w| e^{i\\beta} \\quad (\\alpha, \\beta\\in\\R)\n", "$$\n", "\n", "このとき\n", "\n", "$$\n", "zw = ab - cd + i(ad + bc)\n", "$$\n", "\n", "かつ\n", "\n", "$$\n", "zw = |z||w| e^{i(\\alpha+\\beta)}.\n", "$$\n", "\n", "ここで, 一般に\n", "\n", "$$\n", "e^{i\\theta} = \\cos\\theta + i \\sin\\theta\n", "$$\n", "\n", "となることを使うと,\n", "\n", "$$\n", "\\begin{aligned}\n", "& a = |z|\\cos\\alpha, \\quad & & c = |z|\\sin\\alpha, \\\\\n", "& b = |z|\\cos\\beta, \\quad & & d = |w|\\sin\\beta, \\\\\n", "\\end{aligned}\n", "$$\n", "\n", "かつ\n", "\n", "$$\n", "zw = |z||w|(\\cos(\\alpha+\\beta) + i\\sin(\\alpha+\\beta))\n", "$$\n", "\n", "となるので, ここからも三角函数の加法定理が得られる.\n", "\n", "以上と本質的に同じ話になるが, $zw$ 代わりに\n", "\n", "$$\n", "\\bar{z}w = (a-ci)(b+di)\n", "$$\n", "\n", "を考えると自然にベクトルの内積 $ab+cd$ と外積 $ad-bc$ が自然に出て来る:\n", "\n", "$$\n", "\\bar{z}w = ab+cd + i(ad-bc).\n", "$$" ] }, { "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": 3, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle ω =\\frac{i \\left(\\sqrt{3} + i\\right)}{2}$" ], "text/plain": [ "L\"$\\displaystyle ω =\\frac{i \\left(\\sqrt{3} + i\\right)}{2}$\"" ] }, "execution_count": 3, "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\"\\displaystyle ω =\", sympy.latex(ω₃))" ] }, { "cell_type": "code", "execution_count": 4, "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": 4, "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": 5, "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": 5, "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": 6, "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": 6, "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": 7, "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": [ "$\\displaystyle m = \\frac{q - \\sqrt{- 4 p^{3} + q^{2}}}{2},\\frac{q + \\sqrt{- 4 p^{3} + q^{2}}}{2}$" ], "text/plain": [ "L\"$\\displaystyle m = \\frac{q - \\sqrt{- 4 p^{3} + q^{2}}}{2},\\frac{q + \\sqrt{- 4 p^{3} + q^{2}}}{2}$\"" ] }, "execution_count": 7, "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\"\\displaystyle m = \", sympy.latex(sol[1]), \",\", sympy.latex(sol[2]))" ] }, { "cell_type": "code", "execution_count": 8, "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": [ "$\\displaystyle x_1 = - \\frac{\\sqrt[3]{2} p}{\\sqrt[3]{q + \\sqrt{- 4 p^{3} + q^{2}}}} - \\frac{2^{\\frac{2}{3}} \\sqrt[3]{q + \\sqrt{- 4 p^{3} + q^{2}}}}{2}$" ], "text/plain": [ "L\"$\\displaystyle x_1 = - \\frac{\\sqrt[3]{2} p}{\\sqrt[3]{q + \\sqrt{- 4 p^{3} + q^{2}}}} - \\frac{2^{\\frac{2}{3}} \\sqrt[3]{q + \\sqrt{- 4 p^{3} + q^{2}}}}{2}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle x_2 = - \\frac{\\sqrt[3]{2} p ω^{2}}{\\sqrt[3]{q + \\sqrt{- 4 p^{3} + q^{2}}}} - \\frac{2^{\\frac{2}{3}} ω \\sqrt[3]{q + \\sqrt{- 4 p^{3} + q^{2}}}}{2}$" ], "text/plain": [ "L\"$\\displaystyle x_2 = - \\frac{\\sqrt[3]{2} p ω^{2}}{\\sqrt[3]{q + \\sqrt{- 4 p^{3} + q^{2}}}} - \\frac{2^{\\frac{2}{3}} ω \\sqrt[3]{q + \\sqrt{- 4 p^{3} + q^{2}}}}{2}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle x_3 = - \\frac{\\sqrt[3]{2} p ω}{\\sqrt[3]{q + \\sqrt{- 4 p^{3} + q^{2}}}} - \\frac{2^{\\frac{2}{3}} ω^{2} \\sqrt[3]{q + \\sqrt{- 4 p^{3} + q^{2}}}}{2}$" ], "text/plain": [ "L\"$\\displaystyle x_3 = - \\frac{\\sqrt[3]{2} p ω}{\\sqrt[3]{q + \\sqrt{- 4 p^{3} + q^{2}}}} - \\frac{2^{\\frac{2}{3}} ω^{2} \\sqrt[3]{q + \\sqrt{- 4 p^{3} + q^{2}}}}{2}$\"" ] }, "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\"\\displaystyle x_1 = \", sympy.latex(X[1])) |> display\n", "latexstring(raw\"\\displaystyle x_2 = \", sympy.latex(X[2])) |> display\n", "latexstring(raw\"\\displaystyle x_3 = \", sympy.latex(X[3])) |> display" ] }, { "cell_type": "code", "execution_count": 9, "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": [ "$\\displaystyle x_1 = - \\frac{3 p}{\\sqrt[3]{\\frac{27 q}{2} + \\frac{\\sqrt{- 2916 p^{3} + 729 q^{2}}}{2}}} - \\frac{\\sqrt[3]{\\frac{27 q}{2} + \\frac{\\sqrt{- 2916 p^{3} + 729 q^{2}}}{2}}}{3}$" ], "text/plain": [ "L\"$\\displaystyle x_1 = - \\frac{3 p}{\\sqrt[3]{\\frac{27 q}{2} + \\frac{\\sqrt{- 2916 p^{3} + 729 q^{2}}}{2}}} - \\frac{\\sqrt[3]{\\frac{27 q}{2} + \\frac{\\sqrt{- 2916 p^{3} + 729 q^{2}}}{2}}}{3}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle x_2 = - \\frac{3 p}{\\left(- \\frac{1}{2} - \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{27 q}{2} + \\frac{\\sqrt{- 2916 p^{3} + 729 q^{2}}}{2}}} - \\frac{\\left(- \\frac{1}{2} - \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{27 q}{2} + \\frac{\\sqrt{- 2916 p^{3} + 729 q^{2}}}{2}}}{3}$" ], "text/plain": [ "L\"$\\displaystyle x_2 = - \\frac{3 p}{\\left(- \\frac{1}{2} - \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{27 q}{2} + \\frac{\\sqrt{- 2916 p^{3} + 729 q^{2}}}{2}}} - \\frac{\\left(- \\frac{1}{2} - \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{27 q}{2} + \\frac{\\sqrt{- 2916 p^{3} + 729 q^{2}}}{2}}}{3}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle x_3 = - \\frac{3 p}{\\left(- \\frac{1}{2} + \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{27 q}{2} + \\frac{\\sqrt{- 2916 p^{3} + 729 q^{2}}}{2}}} - \\frac{\\left(- \\frac{1}{2} + \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{27 q}{2} + \\frac{\\sqrt{- 2916 p^{3} + 729 q^{2}}}{2}}}{3}$" ], "text/plain": [ "L\"$\\displaystyle x_3 = - \\frac{3 p}{\\left(- \\frac{1}{2} + \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{27 q}{2} + \\frac{\\sqrt{- 2916 p^{3} + 729 q^{2}}}{2}}} - \\frac{\\left(- \\frac{1}{2} + \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{27 q}{2} + \\frac{\\sqrt{- 2916 p^{3} + 729 q^{2}}}{2}}}{3}$\"" ] }, "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\"\\displaystyle x_1 = \", sympy.latex(sol[1])) |> display\n", "latexstring(raw\"\\displaystyle x_2 = \", sympy.latex(sol[2])) |> display\n", "latexstring(raw\"\\displaystyle 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": 10, "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": 11, "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{smallmatrix}1\\\\0\\\\- 2 x^{2} - 2 y^{2} - 2 z^{2}\\\\8 x y z\\\\x^{4} - 2 x^{2} y^{2} - 2 x^{2} z^{2} + y^{4} - 2 y^{2} z^{2} + z^{4}\\end{smallmatrix}\\right]$" ], "text/plain": [ "5-element Vector{Sym{PyCall.PyObject}}:\n", " 1\n", " 0\n", " -2*x^2 - 2*y^2 - 2*z^2\n", " 8*x*y*z\n", " x^4 - 2*x^2*y^2 - 2*x^2*z^2 + y^4 - 2*y^2*z^2 + z^4" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# 展開結果の 0,1,2,3,4 次の係数の確認.\n", "# 3次の係数は0になっている.\n", "\n", "display(L\"\\text{coefficients:}\\ \")\n", "c4 = sympy.Poly(f4, w).all_coeffs()" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "text/latex": [ "$p=x^{2} + y^{2} + z^{2}$" ], "text/plain": [ "L\"$p=x^{2} + y^{2} + z^{2}$\"" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p = c4[3]/(-2)\n", "latexstring(\"p=\", sympy.latex(p))" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "text/latex": [ "$q=x y z$" ], "text/plain": [ "L\"$q=x y z$\"" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "q = c4[4]/8\n", "latexstring(\"q=\", sympy.latex(q))" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "simplify(c4[5] - (p ^ 2 - 4r)) = 0\n" ] }, { "data": { "text/latex": [ "$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": 14, "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": 15, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "text/latex": [ "$0$" ], "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": 16, "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": 16, "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": 17, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "text/latex": [ "$m^{3} - m^{2} p + m r - q^{2}$" ], "text/plain": [ " 3 2 2\n", "m - m *p + m*r - q " ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$M^{3} - \\frac{M p^{2}}{3} + M r - \\frac{2 p^{3}}{27} + \\frac{p r}{3} - q^{2}$" ], "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": [ "$\\left[\\begin{smallmatrix}1\\\\0\\\\- \\frac{p^{2}}{3} + r\\\\- \\frac{2 p^{3}}{27} + \\frac{p r}{3} - q^{2}\\end{smallmatrix}\\right]$" ], "text/plain": [ "4-element Vector{Sym{PyCall.PyObject}}:\n", " 1\n", " 0\n", " -p^2/3 + r\n", " -2*p^3/27 + p*r/3 - q^2" ] }, "execution_count": 17, "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": 18, "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": 19, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\left[\\begin{smallmatrix}w & x & y & z\\\\x & w & z & y\\\\y & z & w & x\\\\z & y & x & w\\end{smallmatrix}\\right]$" ], "text/plain": [ "4×4 Matrix{Sym{PyCall.PyObject}}:\n", " w x y z\n", " x w z y\n", " y z w x\n", " z y x w" ] }, "execution_count": 19, "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": 20, "metadata": {}, "outputs": [ { "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": [ " 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": 20, "metadata": {}, "output_type": "execute_result" } ], "source": [ "d4 = det(A)" ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "true" ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" } ], "source": [ "d4 == f4" ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\left[\\begin{smallmatrix}1\\\\0\\\\- 2 x^{2} - 2 y^{2} - 2 z^{2}\\\\8 x y z\\\\x^{4} - 2 x^{2} y^{2} - 2 x^{2} z^{2} + y^{4} - 2 y^{2} z^{2} + z^{4}\\end{smallmatrix}\\right]$" ], "text/plain": [ "5-element Vector{Sym{PyCall.PyObject}}:\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": 22, "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": 23, "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_{$k}(x)=\", sympy.latex(B(k,x))) |> display\n", "end" ] }, { "cell_type": "code", "execution_count": 24, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "text/latex": [ "$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": 25, "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": 26, "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": 27, "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\"\u001b[93m\u001b[1m ⋯ 454 bytes ⋯ \u001b[22m\u001b[39m\"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 @.(string(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": 28, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "text/latex": [ "$\\left[\\begin{smallmatrix}1\\\\\\frac{1}{2}\\\\\\frac{1}{6}\\\\0\\\\- \\frac{1}{30}\\\\0\\\\\\frac{1}{42}\\\\0\\\\- \\frac{1}{30}\\\\0\\\\\\frac{5}{66}\\end{smallmatrix}\\right]$" ], "text/plain": [ "11-element Vector{Sym{PyCall.PyObject}}:\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": 28, "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", "$$\n", "\n", "この公式はHurwitzのゼータ函数の解析接続によって論理的に正当化される.\n", "\n", "一方, ガンマ函数 $\\Gamma(s)=\\int_0^\\infty e^{-x}x^{s-1}\\,dx$ の応用としてよく使われる公式\n", "\n", "$$\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", "$$\n", "\n", "この公式の右辺は $\\real s > -N$, $\\real x > 0$ で意味を持ち, そこへの $\\zeta(s,x)$ の解析接続を与える. $m\\in\\Z$, $0\\leqq m < N$ のとき $s\\to -m$ とすることによって, \n", "\n", "$$\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": 29, "metadata": {}, "outputs": [ { "data": { "image/png": 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", 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"image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ], "text/html": [ "" ] }, "execution_count": 30, "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, 400))\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\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ], "text/html": [ "" ] }, "execution_count": 31, "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": 32, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "image/png": 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vyaUVgsACwASoGmrEohhZ1rUH3hE7Bn55ZI6HyYyyPwMCCwAjpii7JxbFKMryeS9NPT79+wPVgvgpLA+TmMHwXBBYABghjGW5N8WJpxiqhR81x+GNjZuzWBeqcNI0tisXVV15ZBfIF3hb67tKzYPAAsCYYKVCmnFVnHSa5NkIouZyh7zEIOK9m3RGAxZN/W3ljfSRwtpdpe9KtQICCwDjwEhaqZRYSeoFjneg/aIPLX0HIYRUDHr7Gl0pwfFTfl/V7D/PnWVpmjOWILAAMHSqx7VUUoz0TjI3+GXn975ku3i0Py9TodcSaQbjS5P+sGMMy8o00wpBYAFgyBQVBeLEmLaSHP6YKcJN+1kC+46XWhRoxhWVJ5/4McJYd+PrgZ4E1vbt2zkcztatWzVeDQAAIYQwlt9PFydE0y2P+ZGzHRavIzhWT79eJ0OvXlZFuBJfhT1np2MYdP/doUOHfvjhBxcXFwgsADQOKxXSzARx0hnS0koQNY87NByRz3afilrw5Mv024HkluDn96wktW1coaXAW/vl6px6gVVbW/vll19u2bLl+++/11JBAJgnRiqmUs9LUmItPALs56+29B/y3Ga3GvCsq/Snw8nlgWZzHfgU9QJr1apVn332mVwu11I1AJghVWM9lXRampnIHTzG6W+7LFz/tGt0oQq/fU11MII91dNkp4a+mBqBdeLECaVSOWfOnOPHjz+3wYMHD2JjYxMTE387NJv9ww8/9O3bVwNlgj+SSCQYY31XYRYoitLewemHJYrUX5XFOZwR43mrd5MCezlCcrH4uY1/LGF9cY99Mlw53O7PmvxGpVLJZDKx2Ji6YAzDEETXKdzdwHr8+PH27duTk5Nf0Mbb23vChAlr167teGbgwIFsNnwRqXkEQfD5fH1XYS4EAoGGj4ixvCBTnHBK9aROEDmLt/gjwvJFuxdjhDbdos+U49QZLF8Bp8vDs9nNXC5X82VrE8MwMpmsy2bdTZPc3NyKioqgoCCEkEKhkMvlDg4O1dXV1ta/fxNhaWkpFAqHDx/es4oBMHmYVkkzE6mk04jFEkTN4wa/TLC6+B2U0+jNZPqhFN+YYfS3kOi97gZWVFSUQqFo//n48eNffvnl3bt3tVYVAKaGkUskqReoa+cs3HzsZq+07DesO+96JEOzrqp8bYj4V9mWxndjec2D6zUAtItubqCSzkhuxVsNHOW08nML9+6O6uY14RlX6KUBxI4Qllpj7ASLINR7h9HoSWDNmzdv2rRpGi8FABOjrCkRJ8bICzJ5oROFH3/HsnPq/nsvVuFl11Rfh7EW+ak9dg5rCf+Aw+FwOF2P/AFgtuSFd8QJp1T1lfzI2XbzV5NW6k06353LfJXHnJvADnPpSUcJ1hICALqGaZXsTrI4KQZhLIiayw0Z2+WY+jPaaLTyOp3TiG/OYHma7j58PQaBBYAGMHKp5OYlKvkM28XDdtrbVv2Ho27MKnpGjQTPTaB9+MT16WzrXvxqwlpCAMDz0S1PqGtnJWlxVv2HO/3l7xYefj07zs1HeH4C/bcB5KbgzsuZ1QMb+AEAnqWsLReLYuT30q1Hjhd+tIdl79LjQ+0rYHbcpg9GsKdoYs0NDLoDAH7XVpQlToxR1pTwI2bazVpJWvd81YGcRqtS6YwGfH06W1O3EYRBdwAAQgwtzUqhRDFY2caPmuu4fDvBtuj6XX+uXIznJtD9bIm0mWwe/C52A3xIAHQNt8kkaXHi5NNsB6HN5CVWA0f1YEz9GReq8PJrqs3BrPcHabhDBIPuAJgpRtzUkhwtSbtsFRDsuGwrx7Nf749JY7Qtk/6pGJ+ZwB7do5lWLwYb+AFgdpT1lZQoRpqdyhsxzmXtN2xHV40c9qEULxbRHBLdns12tuq6PXgaBBYAz2oryRUnRiuringvTxes+dZG6K6pI8dV42XXVH8bwNocTPZ28oJZgsAC4P8xjCwnVZwYzcglgqi5jm9tISw44hfvlddtSgZtv00fK8Y/j2NHuEJW9RAEFgAIK9okGVco0WnSxkEwYSE3KKz3Y+pPKxPj10W0oyW6M5vtBJeBvQCBBcwaLW6SpMRSNy5Z+gU5vLGB49Nf46c4XsKsTaM3DWV9EARXgb0FgQXMlKqhRiyKkWVd4w6LdPlgN9u5j8ZP0apEq1PpzMc4bjI72BHCSgMgsIDZUZTdF4uiFWX5vJemum45SPJstHGWlDq8NJl+1YO4PesP95HXAdjADwDjh7Es94ZYFMOIm/lRcxze2EhYaGVbNwWDdtymjxThfeGsaV56CA5YSwiAEcNKhfRWvFgUQ1oLBOPmcoeEa3ZM/Wk5jXhpEt1XQGTN0ds0K1hLCIBRYiStVEqsJPUCx6e//aIPLX0Hae9cNEb/ymG+yqX/NYr1Vj+TjQz9gsACpkn1uJZKipHeSeIOfdl59b/YQk+tnu5+M377Gi2wQJmz2F58PY8fwVpCAIyGorJQnBDdVpLDHzNFuOl7lsBeq6dTMWh3LrM7l/50OGvlAIOYuAAb+AFg8DCW3UunRDF0cwM/crbD4nUER+tjSLmNeHkKbcdBt2axvfXdseoAg+4AGC6sUkozE8SiGJLD5Y+baz00HJFav+loG412ZtH7CpgvRrKW9TOIjlUHGHQHwBAxUjGVel6S8quFh7/9/Pcs/Yfo5rwpdXjFdXqgHZE128LNBEeKDBcEFjBKqsZ6Kum0NDORO3i009++sHDV0eZPTW1owy36chX+dgw5y9tAOzIw6A6AoVBUPaASY+QP7vJGvyrcuI9l46CzUx8rZj7OoOf6kHnz2Da92htZu2ADPwD0DWN5fqZYFK16XCuInGW/cA1hydXZyQua8aobdLMCnZvAHulsUANW5gUCCxg6TKtkd5LEidGIJAVj51gPj9LBmHoHKU384xb9QyGzdRhr1UDSRJfoGQ0ILGC4GBkluXGRunbOws3Hbva7lv2CdVzAqTLmw5ucse4oew4MrhsECCxgiOjmBirpjORWvNXAkU4rPrPo46vjAvKa8Ac36SdydCBMOclXK2ukQQ9AYAHDoqwuEYti5AWZvNCJwo+/Y9k56biAxja04zb9SxmzYxhr5QBSSsl0XAB4AQgsYCjkhXeoxGhlfSU/Ypbd/FWkFU/HBSgZtDef2ZlFz+9L3p9n4Wip4/ODrkFgAX3DWH4/vTXuOFYq+C/PcBw5XkvbVL3Yr5XMx+mMNx8lTmEPsjfuoXXYwA8AzWPkUsnNS1TyWbZLH5spb1oFhmhvm6oXuPMYf5xB10nRV2GsVz1N4fcc1hICoEl0yxPq2llJWpxVYIjTX3ZYePjrpYwyMd52mxE9xDtCyLf7kWxT+R2HtYQAaIaytlwsipHnpVmPmiD8aA/L3kUvZTyWo8+z6J+KmPeDWPvCWTz4PTAS8BcFdKStKEucGKN8WMp/eYbd1h9Ja75eyhAr0Vd5zP/coxf5kffnWbjobra87sBaQgB6iqGlWSmUKAYr2/hR8xz/soNg6edfnUyFvstn/pVDT/YgM2ay+wpMYbjquWADPwDUhttkkrTL4uQzbAehzatvWA0YqZcxdYRQG42+L2R2ZTOhzoQJfAnYJRh0B0ANdGsTlXJOcvOSpf8Qx7e2crz66asSBYMOFjJfZDNDHFDsBFaIk4lHVTsYdAegW5R1lZQoRpZ7w3rEOJe137AdXfVVSRuNDj5gdmUzQfbo1CusUbDFgkmAwAKa0VaSK06MUVYV8l6a7rrlBy3dTrk7ZCr0fSHzZQ4T7GimUQWD7gD8CYaR5aaKE6IZuUQQNdfxrc16maferlWJvrvPfJNHjxGS58zmArAz2MAPIYRqamoOHDiQnp5O03RkZOSaNWusrU0wwkE3YUWbJD2OSjrDsnUQTFjIDQrT15g6QuiRDH1zj95fwEz2IOPNYFjdbKkRWKmpqQ0NDatWreJwOJs3by4oKDhy5Ij2KgMGi6GaqZRfqdQLlr6DHN5Yz/EZoMdiSlrx7lzmZCmz0M/EJysApFZgLViwYMGCBe0/W1pazpgxQzslAcOlelQtTjoty7pmPWysy5r/sJ3c9VhM2iO8O5dJrmVWDiDzTXQKKHhGD8ew7t+/7+Pjo9FKgEFTlN0XJ0YryvN5L0113fwDybfVVyU0RmfLmf/kMXVStCaIPBRpAQtrzEdP/qrLysq2bt16/PjxZ54vKCg4f/68SCRqf0gQxMGDB/38/HpbI+hEIpHo6EwYK/Mz5CnnsKTVMnw6f85qwoIjRQhRlI4KeEqrkjhcQu4rYrlx8epAepoHwyIQliOtliKRSAj9jc31jEqlksvlFKW7ne97j2GY7nzOagdWTU3NhAkTduzYMXHixGde8vPzmzBhwtq1a9sfEgQxePBgkjTZOWz6xedrdy0eViqkt+LFohiSJ7Abv4A75CU9jqnnN+M995kTJcwUT/LUeFKX963BGGv7o9Y4NrvJysrKuMpmGEYm63pzV/UCq66u7pVXXlm5cuV7773X+VULCwuhUDh8+HC1jgkMDSNppa6fl6T+yvHub7/oQ0vfQfqqhMbofCWz5z5zrwm/E0jemwt3gugW2MAPIYQaGhrGjx+/ZMmSjz/+WHsFAT1SPa6lkmKkd5K4Q192Xv0l28VDX5XUydAPhcz+AsaDh1YNJOf1JTnQU+82WEuIEEI//fTTvXv3tm3btm3btvZnKIri8XS98TbQBkVloTghuq0khz96inDT9yyBvV7KwAglPsT78pn4h8z8vuS5CaxgR9PsKWgVrCVECKG1a9d2jE8BE4Gx7F46lRhNtzzmR852WLyO4FjppZB6GTr0gDlQyPDYaEV/8kCEhSHfCx7oC3whbKawSinNTBCLTpMcK/64udZDw3V5O+UOKgZdrsYHHzDJtcxsH/KnsaxQF+hS9RasJQSmg5GKf7udsoe//fzVlv5D9FJGQTM+VMQcKWL6Coi3+5FHIi340KXSENjAD5gCVWM9lXRampnIHTza6W+7LFy9dF9DUxv6uZQ5UsRUUmiJP5EwhT3ADrpUGgaD7sC4KaqKKFG0vPAuL2yycMP/smwddV0Agy5WMUeLcOJDZrInuW0Ya5KHqX7zrn8w6A6ME8by/EyxKFr1uFYQOcv+tTWEpU5X3DEYpdTh4yVMTBkz2IF4I4A8GGFhq7ftZ4DRg8AyTZhWSW+LKFEMYrEEUXO5wRE6vvXDrQb8cynzSyl2tESv+5N357A9edCh0hEYdAdGg5FLJKkXqWtnLdx87Ga/a9kvWJdnz3qCfyllfinDLAIt9CXiXmUNhCEqnYMN/IARoJsbqKQzkoyrVoNGOa383MK9r85Offsxji5josswg9ECXyL6FZjwCbQCAssUKGtKxKIYeX4mL3SicP13LDtnHZyUwehGPT5TwZwuxxYkmudDnBxnvrsSA92AwDJu8oLb4sRoVX0lP3K23bzVpJXWhy3kNEp8iM9WMLEVjCuXmO1Dxk4gBztATgFdgMAySphWSTMTxaJohLFg7Bzu8Chtj6k3yNGFSuZ8FY6vYYIdiZne5KahsB8x0DUILCPDyKWStMvipDMWLn1spy+3CgzR6jZV2Y34YhX+tYK534zH9yGnexH7wi0cLbV3QgBeBALLaNAtT6jkM5L0K1aBIbzX19v2G6ylE4mVKOEhc7kKX6zCHBaa6kn8fTgrwpWwNKYNLIFpgsAyAsrackoUI7uXbj3iFeG6PSwHF0rTOxRjhLKf4LhqHFfN3H6Mw1yIyZ7k2sFkoC1c9Bkf2MAP6EdbUZY4MVpZU8qPmOk6eyXJ1fCmtw+l+GoNvlqD42sYOw4xyYP4aAhrrBthDf8ujBmsJQS6xdDSrBRKFIOVCn7UHMflOwi2xrYyaFGgpFom8SGOr8H1MjzOnRzfh/h8ONsHRtBNBawlBDqC22SStMvi5DNsB6HNq29YDRipkTF1Somu12PRQ0ZUiwua8WgX4pU+5JGx5DBHgoSYAsYDAstQ0K1NVMo5yc1LlgFDHd/ayvHq18sDtirR9Tp8rY5JrlM5q24AAAl/SURBVMV5TXiEExHlTu4OJUNdCNgf3bTBWkKgRcr6Skp0WpaTaj1inMvab9iOrj0+VK0UXa9nrtfh6/W4qAWPciYi3MhdI8lQF8IKvuMzG7CBH9CKtuJcsShaWfWAFz7DdcsPJM9G3SMwGN1vxqn1OLUO33iEm9rwS0LyZVdizxhyhBNhAT0pswSD7kCjGEaWkypOjGbkEsHYOY5vbSEs1NgjqrENJT0kcyj6Zj1Ob8BCLjFGSES4ERuDyQF2xnaTYqAFMOgONAMr2iTpcVTSGdLGQTBhITcorDtj6goGZT3BGQ044xHOaMC1UjzMgRXuht4bxDrmQjjp5zY3AOgBBJaOMFQzlfIrlXrB0i/I4Y31HJ8BL2isYtD9Zpz5GGc24MzH+F4T7mdLjHImxroT64eSA+0IqYTi82GBDHg+GHQHPadqqBGLYmRZKdbDIl3W/Ift5N65jYJBeY347hN85wm+8xjnNWEvHjHciRjhTCzxJ4c5Elz4iwLdBhv4gZ5QlN0XJ0a3ld3nh09z3XyA5Nt2vNSsQNlPcNYTnN2Is57gghbsJyCGORIhTsRCX3KYIwH3vAKgMwgsLcBYlntTLIpmxM38qDkOb2yg2ZYPWnFOKZPTiHObUE4jfiLHgx2IoQ5EmAvx1wFkkD30oQDoGvyWaBJWKqQZV8VJp5WWgophc284jsltRvcu4MIWpbs1McSBGOyA3gwghjqQfQUwxRwAtUFg9RaNUZkY5zfjgpTd03Pv3BcEfmv/Xql90EAlGtyGxrkTHwSRA+1gOTEAGgC/RuppUaAHLbiw/b9mVNCCi1uxG5fob4fkKM191i5/X/9zdgTceg8AbYDA+lNyGhW34qIWXNSKilpwe05JVKifLdHPluhvS8ztiwJtyf52v616mXSSNXeEpxUbrvQA0BYILIQQalag0lZcIsYlraikFRe34pJW1CDHfQVEgC0RYINGOBOL/Mh+tsgD7gYKDB5s4Gci2mhUSeFyCpWJcZkYl4pRaSsuFWMlg/xsCF8B4WeDRjoTC/1IPwHy4sO4ODBKsJbQyEhUqILClRSqoHCFGFdQqILCZWL0pA178AgfPuorIPoKiFneyFdA+toQzrC6BZgQWEtoiNpoVCPFNRJUQeFqCaqW4EoKVVK4SoLlNPLmE1585MUnvPnEq57Ih0/6CJC7NXSaADBihh5YDXJUJ8XVElQnw1US9FCCa6S4WoIeSnGLArlbEx485MUnPHiovx0xyQN58kgPnn7WA3NIC5Iw2f+zASMCawm1RcmgRzJcJ0O1UvRIjmsk6JEMV0tQvQzXSFGdFNtykJBLePCQqzXhyUPBjsRUL8KDR7hbE0Kufmt/1omZ+zksmM4A9A828OshiQo1yHCdDDXIcYMc1UpRgxw/lqNaKa6XoQY5bmpDLlzClYtcuUjIJfrwUKAdMc4dCblkHx5y5RrTvfD4HJ6+SwAAIRh0776rbnOmxqkey1GdDD2WY4yQsxXhykXOVsiZS7hykTefGOmE3KxJIRc5WxEuBtZLAsAEwKB7d3lISv86YIyTFRJykQuX4Bn6EBkAwJhoOIkHtGZN8yLCXIi+ArNLqy3JO1UMre8qAEBVVx6JK6T6rkIrTLbrqHuZddkqRqnvKgBAktq2tmbT/KcIgQUAMBqaDKyamprq6moNHhD8mVOnTtXX1+u7CtOHMd67d6++qzALlZWVsbGxXTZTL7Cio6OjoqLCw8P379/f+dXi4uLS0lK1Dgh65vDhw/fu3dN3FaZPoVBs375d31WYhbt37544caLLZmoMjKelpb3zzjvHjh0TCASLFi1ycnKaM2dOLyoEAAD1qNHD2rt37zvvvDNlypSXX355w4YN//3vf7VXFgAAdKZGYGVnZ48aNar959DQ0OzsbO2UBAAAz6fGJeGjR4/s7Ozaf7a3t29sbFQqlRYWv9+OqqSkpLCwsOMZgiCEQuHTDUyb4zyPQVsHYRrr4Fx1dXVLliyxsoJtcbQLY6xUKn19ffVdiHom+88s/p+C4sZCfReiBqlU6ujo2GUzNQLLxsZGKv1tNhpFUTwe75kwio+Pv337tq3t73ff8/DwILpxK3bT8a6OztPa2ioQCMzrs9WT1tZWGxsbfVdh+hiGsbTs+mbmagSWj49PcXFx+8/FxcXe3s/eWLZPnz59+vTp/gEBAEAtaoxhLV68+ODBgxKJRKVS7d27d/HixdorCwAAOlMjsBYtWhQSEuLj4+Pp6cnlcj/44APtlQUAAJ0RGKs3SNzU1KRSqZydnbVUEOgSxrimpsbW1lYgEOi7FgA049GjR2w228HB4cXN1F6aY29v32Va/fLLL6NHj7a2tp45c6a6xwcvVl5ePmTIkIiICC8vr507d+q7HJPV0tIya9as9m+NSkpK9F2OKYuJifH09Bw0aFBAQEBoaGhZWdkLGmtl8bO7u/v27dvff/99bRzczK1fvz4qKqq0tDQrK+vf//53Tk6OvisyTSwWa/bs2b/++qu+CzF9Tk5Oly9fbmhoqK+vDwgIWLNmzQsaa2XPqvDwcIRQbm6uNg5uziQSydmzZ/Py8hBC3t7eM2bMOH78+JAhQ/Rdlwni8/lvvvmmUmmam7QYlMjIyPYf2Gz2lClT/vGPf7ygMWwvY0xqampomvbz82t/GBAQUF5erteKANAYjPHx48cnTZr0gjY97GFdunTp6tWrzzxpY2PzySef9OyAoDvEYjGHw2GxfrszB4/Ha2lp0W9JAGjKF198UVpa+uI9G3oYWPb29j4+Ps88yePBbWO0y8XFRS6Xy+Xy9kU5TU1NQqFQ30UBoAHffPPNwYMHk5OTX/zddw8DKywsLCwsrGfvBT3m5ubm7OyckZERERGBEEpPT588ebK+iwKgtw4cOPDVV1+JRKIul8qwtHERV15efvXq1ZSUlLKyMnt7e7lc7u7urvGzmCGSJFtbW/ft2xcSEhIbG3v48OEDBw5YW5vgDX4Nwblz53JycmJiYgICAioqKgIDA0kSxnw17+TJk8uXL9+4caNMJrt//35xcXH//v3/rLHaE0e7Izk5+endsl555ZWVK1dq/CzmSalU7ty5My4uztnZedu2bSNHjtR3RSbr7bffpiiq4+GRI0dgewxtOHLkyPnz5zse8ni8H3/88c8aayWwAABAG6CLCwAwGhBYAACjAYEFADAaEFgAAKMBgQUAMBoQWAAAowGBBQAwGhBYAACjAYEFADAaEFgAAKMBgQUAMBr/BwebilrgDaTUAAAAAElFTkSuQmCC", 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"**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": 33, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "image/png": 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", "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ], "text/html": [ "" ] }, "execution_count": 33, "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", "$$\n", "\n", "これと $M_p$ が $p=0$ でも連続であることを合わせれば, $M_p$ が $p$ について単調増加することが示される." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "**プロット:** $p\\ne 0$ のとき, $\\ds M_p(x,y) = \\left(\\frac{x^p+y^p}{2}\\right)^{1/p}$ なので $M_p(x,y)=1$ と $y=(2-x^p)^{1/p}$ と同値であり, $M_0(x,y)=\\sqrt{xy}$ なので $M_0(x,y)=1$ と $y=1/x$ は同値である. $M_p(x,y)=1$ のグラフをプロットしよう." ] }, { "cell_type": "code", "execution_count": 34, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "image/png": 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単位円に内接する多角形の周長と面積の最大値\n", "\n", "正弦函数 $\\sin\\alpha$ が $0\\leqq\\alpha\\leqq\\pi$ において上に凸な函数であることにJensenの不等式を適用することによって, 単位円に内接する $n$ 角形の周長と面積が最大になるのは正 $n$ 角形の場合であることを示そう. 一般に上に凸な函数 $f(x)$ についてJensenの不等式より,\n", "\n", "$$\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", "$$\n", "\n", "この計算中の不等号は $\\sin\\alpha$ が $0\\leqq\\alpha\\leqq\\pi$ で上に凸であることとJensenの不等式から従う. この不等式の最右辺は単位円に内接する正 $n$ 角形の面積に等しい. $\\sin\\alpha$ が $0\\leqq\\alpha\\leqq\\pi$ で強い意味で凸であることを使えば, 逆に面積 $S$ が最大になるのは単位円に内接する $n$ 角形 $A_1\\ldots A_n$ が正 $n$ 角形になるときであることもわかる." ] }, { "cell_type": "code", "execution_count": 35, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "image/png": 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", "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ], "text/html": [ "" ] }, "execution_count": 35, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# α_i > π ならば周長と面積をより大きくできること\n", "\n", "t = range(0, 2π, length=400)\n", "theta = 2π * [0, 3/20, 14/20, 16/20, 18/20, 1]\n", "phi = copy(theta)\n", "phi[2] = theta[3] - π\n", "phi[3] = theta[2] + π\n", "plot(size=(300,300), aspect_ratio=1, legend=false)\n", "plot!(cos.(t), sin.(t), lw=0.5, color=:black)\n", "plot!(cos.(theta), sin.(theta), color=:blue)\n", "plot!(cos.(phi), sin.(phi), ls=:dash, color=:red)" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## 三角函数の微積分\n", "\n", "高校数学の範囲内で三角函数の微分積分学を再構成してみせる. その結果は直接楕円函数論に一般化可能である." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "-" } }, "source": [ "### 高校の数学の教科書の方針\n", "\n", "高校の数学の教科書では以下のような筋道で $\\sin x$ の導函数を求めている. \n", "\n", "$0< x <\\pi/2$ のとき, 「面積の大小関係」によって\n", "\n", "$$\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", "$$\n", "\n", "であることが示された.\n", "\n", "しかし, 以上の方針は次の節の方針と比較すると, 非常に遠回りになっており, 弧度法の意味での角度の定義(単位円弧の長さで角度を定義すること)が不明瞭になっているという問題がある." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### 曲線の長さが速さの積分になることの応用\n", "\n", "高校数学IIIの教科書には $(x(t),y(t))$, $a\\leqq t\\leqq b$ の軌跡の長さ(曲線の長さ) $L$ が\n", "\n", "$$\n", "L = \\int_a^b \\sqrt{x'(t)^2 + y'(t)^2}\\,dt\n", "$$\n", "\n", "と表せることが説明されている. $t$ を時間変数とみなすとき, 点 $(x(t),y(t))$ の運動の時刻 $t$ における速度ベクトルは $(x'(t), y'(t))$ になり, 速さは $\\sqrt{x'(t)^2 + y'(t)^2}$ と書ける. 上の公式は曲線の長さを速さの積分で表せることを意味している." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "-" } }, "source": [ "これを使えば(曲線の長さを上の公式で定義すれば), 三角函数の微分の導出を非常に簡潔な議論で行うことができる. そのことを以下で説明しよう.\n", "\n", "$(x(t),y(t)) = (\\sqrt{1-t^2}, t)$, $-1 display\n", "latexstring(raw\"\\displaystyle\\left(\\frac{dx}{dt}\\right)^2+\\left(\\frac{dy}{dt}\\right)^2=\", sympy.latex(sol))" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "**問題:** 直線 $y=tx$ と単位円 $x^2+y^2=1$ の右半分の交点は\n", "\n", "$$\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": 37, "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": 38, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle X=\\sqrt{\\frac{1}{t^{2} + 1}},\\quad Y=t \\sqrt{\\frac{1}{t^{2} + 1}}$" ], "text/plain": [ "L\"$\\displaystyle X=\\sqrt{\\frac{1}{t^{2} + 1}},\\quad Y=t \\sqrt{\\frac{1}{t^{2} + 1}}$\"" ] }, "metadata": {}, "output_type": "display_data" }, { "data": { "text/latex": [ "$\\displaystyle\\left(\\frac{dX}{dt}\\right)^2+\\left(\\frac{dY}{dt}\\right)^2=\\frac{1}{\\left(t^{2} + 1\\right)^{2}}$" ], "text/plain": [ "L\"$\\displaystyle\\left(\\frac{dX}{dt}\\right)^2+\\left(\\frac{dY}{dt}\\right)^2=\\frac{1}{\\left(t^{2} + 1\\right)^{2}}$\"" ] }, "execution_count": 38, "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\"\\displaystyle X=\", sympy.latex(X), raw\",\\quad Y=\", sympy.latex(Y)) |> display\n", "latexstring(raw\"\\displaystyle\\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": "markdown", "metadata": {}, "source": [ "" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "" ] }, { "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": 39, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "text/latex": [ "$0$" ], "text/plain": [ "0" ] }, "execution_count": 39, "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": 40, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "image/png": 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", "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ], "text/html": [ "" ] }, "execution_count": 40, "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!(Jacobi.cd.(u,k²), Jacobi.sn.(u,k²))\n", "\n", "plot(P1, P2, size=(500, 260))" ] }, { "cell_type": "code", "execution_count": 41, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "image/png": 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", "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ], "text/html": [ "" ] }, "execution_count": 41, "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, Jacobi.cd.(u,k²), label=\"cd u\", ls=:dash)\n", "plot!(u, Jacobi.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", "$$\n", "\n", "の高校数学の範囲内での証明である. 高校数学IIIの教科書にも以下のような問題が載っている. (前者の問題はGauss積分と関係しており, 後者の問題はゼータ函数と関係している.)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "問題文では曲線 $y=e^{-x^2}$ を $y$ 軸のまわりに回転しているが, 以下では $xyz$ 空間内の $xz$ 平面上の曲線 $z=e^{-x^2}$ を $z$ 軸のまわりに回転して得られる曲面を扱う. さらに $a$ の代わりに $r$ と書く. \n", "\n", "$xyz$ 空間内の $xz$ 平面 $y=0$ 上の曲線 $z=e^{-x^2}$ を $z$ 軸のまわりに回転して得られる曲面と高さ $00$ であるとする.\n", "\n", "平面 $y=0$ 上の曲線 $z=e^{-x^2}$ と $x$ 軸と直線 $x=r$ で囲まれた領域を $z$ 軸のまわりに1回転してできる回転体を $A(r)$ と書く. 上で述べたことより, $A(r)$ の体積 $V(r)$ は次のように計算される:\n", "\n", "$$\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", "$$\n", "\n", "Gauss積分は正規分布の確率密度函数の理解に必須であり, その他にも多くの場面に現われ, 非常に重要な定積分である." ] }, { "cell_type": "code", "execution_count": 42, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "text/plain": [ "0.7071067811865475" ] }, "execution_count": 42, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f(x,y) = exp(-(x^2+y^2))\n", "g(x,y,r) = x^2+y^2 ≤ r^2 ? exp(-(x^2+y^2)) : zero(x)\n", "h(x,y,r) = (abs(x) ≤ r && abs(y) ≤ r) ? exp(-(x^2+y^2)) : zero(x)\n", "\n", "x = range(-2, 2, length=201)\n", "y = range(-2, 2, length=201)\n", "r = 1/√2" ] }, { "cell_type": "code", "execution_count": 43, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "image/png": 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", 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"image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ], "text/html": [ "" ] }, "execution_count": 46, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# A(√2 r) のグラフ (r=1/√2)\n", "surface(x, y, g.(x', y, √2*r), size=(400, 300), colorbar=false)" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## ガンマ函数の応用\n", "\n", "高校数学では $\\cos x$, $\\sin x$, $e^x$, $\\log x$ などの初等函数について習うが, 大学新入生が新たに習う特殊函数として, ゼータ函数 $\\ds\\zeta(s)$, ガンマ函数 $\\Gamma(s)$, ベータ函数 $B(p,q)$ は特に重要である. この節では高校数学にも密かにガンマ函数にあたるものが現われていたことについて解説する. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### 多項式×指数函数の積分\n", "\n", "$n$ は0以上の整数であるとする. 高校数学IIIの教科書にはよく次の形の不定積分を求める問題が書いてある:\n", "\n", "$$\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", "$$\n", "\n", "を扱う. $(-e^{-x})'=e^{-x}$ を用いた部分積分によって得られる公式\n", "\n", "$$\n", "\\int x^k e^{-x}\\,dx = -x^k e^{-x} + k\\int x^{k-1}e^{-x}\\,dx\n", "$$\n", "\n", "の $k=n,n-1,n-2,\\ldots$ の場合を次々に使うと, もしくは $(-e^{-x})'=e^{-x}$ を用いた部分積分を次々に使うと,\n", "\n", "$$\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(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(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(n-1)\\cdots 2 x e^{-x} - n!e^{-x}\n", "\\\\ &= -(x^n + nx^{n-1} + n(n-1)x^{n-2}+ \\cdots \n", "%\\\\ &\\qquad\n", "+ 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", "$$\n", "\n", "の形で使われることも多い. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Stirlingの公式\n", "\n", "**準備の準備:** $|X|<1$ のとき\n", "\n", "$$\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": {}, "source": [ "__ガンマ函数の被積分関数のグラフ__" ] }, { "cell_type": "code", "execution_count": 47, "metadata": {}, "outputs": [ { "data": { "image/png": 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[]\n", "for n in (10, 20, 50, 100)\n", " P = plot(x -> f(x, n), 0, 2.5n; title=\"\\$e^{-x} x^{$n}\\$\", legend=false)\n", " push!(PP, P)\n", "end\n", "plot(PP...; layout=(2, 2))" ] }, { "cell_type": "code", "execution_count": 48, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "f(n, n) = 0.0\n" ] }, { "data": { "image/png": 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", "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ], "text/html": [ "" ] }, "execution_count": 48, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# 上の方法だと n = 200 で破綻する\n", "f(x, n) = exp(-x) * x^n\n", "n = 200\n", "@show f(n, n)\n", "plot(x -> f(x, n), 0, 2n; title=\"\\$e^{-x} x^{$n}\\$\", size=(300, 200), legend=false)" ] }, { "cell_type": "code", "execution_count": 49, "metadata": {}, "outputs": [ { "data": { "image/png": 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log(f(x)) # これは x = n で最大になる\n", "g(x, n) = exp(logf(x, n) - logf(n, n)) # = f(x, n) / f(n, n) = f(x, n) / (e^{-n} n^n))\n", "PP = []\n", "for n in (100, 200, 500, 1000)\n", " P = plot(x -> g(x, n), 0, 2n; title=\"\\$e^{-x} x^{$n} / \\\\max\\$\", legend=false)\n", " push!(PP, P)\n", "end\n", "plot(PP...; layout=(2, 2))" ] }, { "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": 50, "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": 51, "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": 52, "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": 53, "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": "markdown", "metadata": {}, "source": [ "" ] }, { "cell_type": "code", "execution_count": 54, "metadata": { "slideshow": { "slide_type": "subslide" } }, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle\\lim_{n\\to\\infty}\\left(\\frac{\\binom{3n}{n}}{\\binom{2n}{n}}\\right)^{1/n}=\\frac{27}{16}$" ], "text/plain": [ "L\"$\\displaystyle\\lim_{n\\to\\infty}\\left(\\frac{\\binom{3n}{n}}{\\binom{2n}{n}}\\right)^{1/n}=\\frac{27}{16}$\"" ] }, "execution_count": 54, "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\"\\displaystyle\\lim_{n\\to\\infty}\\left(\\frac{\\binom{3n}{n}}{\\binom{2n}{n}}\\right)^{1/n}=\", sympy.latex(sol))" ] }, { "cell_type": "code", "execution_count": 55, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle\\lim_{n\\to\\infty}\\frac{1}{n}\\left(\\frac{(2n)!}{n!}\\right)^{1/n}=\\frac{4}{e}$" ], "text/plain": [ "L\"$\\displaystyle\\lim_{n\\to\\infty}\\frac{1}{n}\\left(\\frac{(2n)!}{n!}\\right)^{1/n}=\\frac{4}{e}$\"" ] }, "execution_count": 55, "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\"\\displaystyle\\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", "$$\n", "\n", "これは「1/6公式」の大幅な一般化になっている." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "**問題:** $B(p,q)=B(q,p)$ を示せ.\n", "\n", "**略解1:** $B(p,q)=\\Gamma(p)\\Gamma(q)/\\Gamma(p+q)$ より $B(p,q)=B(q,p)$ であることがわかる. $\\QED$\n", "\n", "**略解2:** $\\ds B(p,q)=\\int_0^1 x^{p-1}(1-x)^{q-1}\\,dx$ において $x=1-y$ とおくと, $B(p,q)=B(q,p)$ であることがわかる. $\\QED$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "**問題:** 高校の教科書にある次の問題をベータ函数を用いて解け." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "-" } }, "source": [ "**解答例:** 求めるべき面積を $S$ と書くと, \n", "\n", "$$\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", "$$" ] }, { "cell_type": "code", "execution_count": 56, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle 2\\int_0^1\\sqrt{x(1-x^2)}\\,dx=\\frac{8}{15}$" ], "text/plain": [ "L\"$\\displaystyle 2\\int_0^1\\sqrt{x(1-x^2)}\\,dx=\\frac{8}{15}$\"" ] }, "execution_count": 56, "metadata": {}, "output_type": "execute_result" } ], "source": [ "x = symbols(\"x\", real=true)\n", "sol = 2integrate(√(x*(1-x)^2), (x,0,1))\n", "latexstring(raw\"\\displaystyle 2\\int_0^1\\sqrt{x(1-x^2)}\\,dx=\", sympy.latex(sol))" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### sinのべきの定積分\n", "\n", "$n$ は0以上の整数であるとする. 高校数学では\n", "\n", "$$\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", "$$\n", "\n", "要するにガンマ函数とベータ函数の関係はGauss積分の公式 $\\ds\\int_{-\\infty}^\\infty e^{-x^2}\\,dx=\\sqrt{\\pi}$ を含んでいる. \n", "\n", "正規分布の確率密度函数を理解するためには, Gauss積分の公式を理解しておかないといけない. Stirlingの公式の導出でもGauss積分の公式を利用した. Gauss積分は多くの数学的場面に普遍的に現われる重要な積分である." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### ベータ函数の極限によるガンマ函数の表示とWallisの公式\n", "\n", "$\\ds B(p,q)=\\int_0^1 x^{p-1}(1-x)^{q-1}\\,dx$ において, $p=s$, $q=n+1$, $x = t/n$ とおいて, $n\\to\\infty$ とすると,\n", "\n", "$$\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", "$$\n", "\n", "もWallisの公式と呼ぶこともある. \n", "\n", "**問題:** すぐ上の後者のWallisの公式を証明せよ. $\\QED$\n", "\n", "**ヒント:** すでに証明した前者のWallisの公式を使えば後者を示せる. $B(1/2,n+1)/B(1/2,n+1/2)\\to 1$ を書き直しても後者のWallisの公式が得られる. もしくは高校の教科書の掲載されている $\\sin$ のべきの $0$ から $\\pi/2$ までの定積分の計算結果と上で述べたことを合わせて使ってみよ. 偶数べきと奇数べきの比を考えよ. $\\QED$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "**参考:** 以上で扱った大学レベルの微分積分学については\n", "\n", "* 黒木玄, 微分積分学のノート\n", "\n", "を参照せよ. 例えば, Wallisの公式については「10 Gauss積分, ガンマ函数, ベータ函数」「12 Fourier解析」に非常に詳しい解説がある. $\\QED$" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Gaussの超幾何函数への一般化\n", "\n", "$\\ds \\int_a^b (x-a)^A (b-x)^B \\,dx$ 型の積分は本質的にベータ函数とみなせるのであった. これを\n", "\n", "$$\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", "$$\n", "\n", "この手続きは $z = b/t$ における特異点を $z=\\infty$ における特異点に合流させる手続きになっており, Kummerの超幾何函数は合流型超幾何函数と呼ばれる超幾何函数の一族のうちの1つになっている.\n", "\n", "Gauss積分($\\Gamma(1/2)$ に等しい), ガンマ函数 $\\Gamma(s)$, ベータ函数 $B(a,b)$, Gaussの超幾何函数 ${}_2F_1(a,b,c;z)$, Kummerの超幾何函数 ${}_1F_1(a,c;z)$ などは特殊函数の広い一族の一部分になっており, 高校数学でも自然に出て来てしまうものだと言える.\n", "\n", "この点に関してはツイッターにおける以下のスレッドも参照せよ:\n", "\n", "* https://twitter.com/genkuroki/status/1093510712125583360" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 超幾何函数関連ノート集" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### ベータ函数の現れ方(1)\n", "\n", "" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### ベータ函数の現れ方(2)\n", "\n", "" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### ガンマ函数の基礎\n", "\n", "" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Gaussの超幾何函数の現れ方(1)\n", "\n", "" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Gaussの超幾何函数の現れ方(2)\n", "\n", "" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Kummerの超幾何函数の現れ方\n", "\n", "" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### ガンマ函数と正弦函数の関係\n", "\n", "" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "#### Hurwitzのゼータ函数とガンマ函数の関係\n", "\n", "" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Taylor展開\n", "\n", "例えば, 実教出版の高校数学の教科書『数学III』2014年1月25日発行の終わりの方にはCauchyの平均値の定理の応用としてTaylorの公式を示す議論が載っている. その証明法は高木貞治『解析概論』におけるTaylorの公式の証明法と同じである. 以下ではより「初等的」な証明を解説する." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Taylorの公式の証明\n", "\n", "以下では微分積分学の基本定理(微分して積分するとものとの函数に戻るという意味の公式)\n", "\n", "$$\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", "$$\n", "\n", "これを**Taylorの公式**と呼び, $R_n$ を**剰余項**と呼ぶ.\n", "\n", "Taylorの公式において $\\ds\\frac{(x-a)^k}{k!}$ の項が出て来る理由も以上の議論では明瞭である. 定数函数 $1$ を $a$ から $x$ まで積分することを $k$ 回繰り返すと, $\\ds\\frac{(x-a)^k}{k!}$ が出て来る. $n$ 階の導函数を $n$ 回積分するだけなので簡単である." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "subslide" } }, "source": [ "### Taylorの公式の剰余項の評価 (1)\n", "\n", "$R_n$ の絶対値の大きさの評価不等式を作ろう. ある定数 $M_n$ が存在して, $a$ と $x$ のあいだのすべての実数 $x_n$ について $|f^{(n)}(x_n)|\\leqq M_n$ が成立しているとする. このとき,\n", "\n", "$$\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}^{n-1} 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}^{n-1} 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