{"id":4264,"date":"2019-04-20T23:19:46","date_gmt":"2019-04-20T14:19:46","guid":{"rendered":"https:\/\/pandanote.info\/?p=4264"},"modified":"2022-08-07T12:13:22","modified_gmt":"2022-08-07T03:13:22","slug":"%e3%83%87%e3%82%a3%e3%83%aa%e3%82%af%e3%83%ac%e7%a9%8d%e5%88%86%e3%82%92%e3%83%95%e3%83%bc%e3%83%aa%e3%82%a8%e5%a4%89%e6%8f%9b%e3%82%92%e4%bd%bf%e3%81%a3%e3%81%a6%e8%a8%88%e7%ae%97%e3%81%97%e3%82%88","status":"publish","type":"post","link":"https:\/\/pandanote.info\/?p=4264","title":{"rendered":"\u30c7\u30a3\u30ea\u30af\u30ec\u7a4d\u5206\u3092\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u3092\u4f7f\u3063\u3066\u8a08\u7b97\u3057\u3088\u3046\u3068\u3057\u305f\u3089\u3044\u308d\u3044\u308d\u3068\u51fa\u3066\u304d\u305f\u306e\u3067\u3001\u307e\u3068\u3081\u3066\u30e1\u30e2\u3002"},"content":{"rendered":"

\u306f\u3058\u3081\u306b<\/h2>\n

\u30c7\u30a3\u30ea\u30af\u30ec\u7a4d\u5206
\n\\begin{align}
\n\\int_{0}^{\\infty} \\frac{\\sin x}{x} dx &= \\displaystyle\\frac{\\pi}{2} \\label{eq:dirichletint}
\n\\end{align}
\n\u3092\u8a3c\u660e\u3059\u308b\u65b9\u6cd5\u306f\u8907\u7d20\u7a4d\u5206\u3092\u4f7f\u3046\u65b9\u6cd5\u3092\u306f\u3058\u3081\u3068\u3057\u3066\u3044\u308d\u3044\u308d\u3042\u308a\u307e\u3059\u3002<\/p>\n

\u305d\u308c\u4ee5\u5916\u306e\u65b9\u6cd5\u306f\u306a\u3044\u304b\u3068\u3001\u3044\u308d\u3044\u308d\u3068\u8abf\u3079\u3066\u56de\u3063\u3066\u3044\u308b\u3046\u3061\u306b\u3069\u3053\u304b\u3067\u898b\u304b\u3051\u305f\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u3092\u4f7f\u3046\u65b9\u6cd5\u304c\u9762\u767d\u305d\u3046\u3060\u306a\u3068\u601d\u3063\u305f\u307e\u307e\u3001\u305d\u306eURL\u3092\u5931\u5ff5\u3057\u3066\u3057\u307e\u3063\u305f\u306e\u3067\u3001\u81ea\u529b\u3067\u8a08\u7b97\u3057\u3066\u307f\u308b\u3053\u3068\u306b\u3057\u307e\u3057\u305f\u3002<\/p>\n

\u8a08\u7b97\u3092\u3059\u308b\u306b\u3042\u305f\u308a\u3001\u95a2\u9023\u3059\u308b\u60c5\u5831\u304c\u3044\u308d\u3044\u308d\u3068\u51fa\u3066\u304d\u305f\u306e\u3067\u3001\u3064\u3044\u3067\u306b\u30e1\u30e2\u3057\u3066\u304a\u304d\u307e\u3059\u3002<\/p>\n

\u3055\u3063\u305d\u304f\u3001\u8a08\u7b97\u3057\u3066\u307f\u307e\u3059\u3002<\/h2>\n

\u77e9\u5f62\u95a2\u6570\u306e\u30d5\u30fc\u30ea\u30a8\u5909\u63db<\/h3>\n

\u3068\u3044\u3046\u308f\u3051\u3067\u3001\u304b\u306a\u308a\u5510\u7a81\u304b\u3082\u3057\u308c\u307e\u305b\u3093\u304c\u3001(\\ref{eq:squarewave})\u5f0f\u306e\u3088\u3046\u306a\u95a2\u6570\u3092\u8003\u3048\u307e\u3059\u3002
\n\\begin{align}
\nf(x) &= \\left\\{
\n\\begin{array}{l}
\n1 & \\left(|x| \\le \\displaystyle\\frac{1}{2}\\right) \\cr
\n0 & \\left(|x| \\gt \\displaystyle\\frac{1}{2}\\right)
\n\\end{array}
\n\\right. \\label{eq:squarewave}
\n\\end{align}
\n\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u30b0\u30e9\u30d5\u306b\u306a\u308a\u307e\u3059\u3002\u77e9\u5f62\u6ce21\u500b\u5206\u3067\u3059\u3002\ud83d\ude42<\/p>\n

\"\"<\/a><\/p>\n

(\\ref{eq:squarewave})\u5f0f\u3067\u8868\u3055\u308c\u308b\u95a2\u6570$f(x)$\u306f$\\bigg(|x| = \\displaystyle\\frac{1}{2}$\u3067\u4e0d\u9023\u7d9a\u3067\u3059\u304c\u3001$\\bigg)$\u7d76\u5bfe\u53ef\u7a4d\u5206\u3067\u3042\u308a\u3001
\n\\begin{align}
\n\\int_{-\\infty}^{\\infty} |f(x)|dx &= 1 \\lt \\infty \\label{eq:squarewavearea}
\n\\end{align}
\n\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n

\u305d\u3053\u3067\u3001\u30d5\u30fc\u30ea\u30a8\u5909\u63db$F(\\xi)$\u3092\u8a08\u7b97\u3057\u3066\u307f\u308b(\u9593\u9055\u3048\u3084\u3059\u3044\u3068\u3053\u308d\u306a\u306e\u3067\u3001\u3042\u307e\u308a\u7701\u7565\u3057\u306a\u3044\u3067\u8a08\u7b97\u3057\u307e\u3059\u3002)\u3068\u2026
\n\\begin{align}
\nF(\\xi) = \\int_{-\\infty}^{\\infty} f(x) e^{-2\\pi x\\xi i} dx &= \\int_{-\\frac{1}{2}}^{\\frac{1}{2}} e^{-2\\pi x\\xi i} dx \\nonumber \\cr
\n&= \\left[ \\frac{e^{-2\\pi x\\xi i}}{-2\\pi\\xi i} \\right]_{-\\frac{1}{2}}^{\\frac{1}{2}} \\nonumber \\cr
\n&= \\frac{e^{-\\pi \\xi i} – e^{\\pi \\xi i}}{-2\\pi\\xi i} \\nonumber \\cr
\n&= \\frac{\\cos\\pi \\xi-i\\sin\\pi\\xi – \\cos\\pi\\xi -i\\sin\\pi\\xi}{-2\\pi\\xi i} \\nonumber \\cr
\n&= \\frac{2i\\sin\\pi\\xi}{2\\pi\\xi i} \\nonumber \\cr
\n&= \\frac{\\sin\\pi\\xi}{\\pi\\xi} \\label{eq:fouriertransform}
\n\\end{align}
\n\u3068\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002<\/p>\n

\u9006\u5909\u63db\u3057\u3066\u307f\u307e\u3059\u3002<\/h3>\n

\u6b21\u306b\u3001$F(\\xi)$\u306e\u30d5\u30fc\u30ea\u30a8\u9006\u5909\u63db\u3092\u8003\u3048\u307e\u3059\u3002<\/p>\n

$F(\\xi)$\u306e\u30d5\u30fc\u30ea\u30a8\u9006\u5909\u63db$f(x)$\u306f(\\ref{eq:fourierinverse})\u5f0f\u3067\u8868\u3055\u308c\u307e\u3059\u3002
\n\\begin{align}
\nf(x) &= \\int_{-\\infty}^{\\infty} F(\\xi)e^{2\\pi\\xi xi} d\\xi \\nonumber \\cr
\n&= \\int_{-\\infty}^{\\infty} \\frac{\\sin\\pi\\xi}{\\pi\\xi}e^{2\\pi\\xi xi} d\\xi \\label{eq:fourierinverse}
\n\\end{align}
\n\u3053\u3053\u3067\u30d5\u30fc\u30ea\u30a8\u9006\u5909\u63db\u3092\u8a08\u7b97\u3059\u308b\u524d\u306b\u3001(\\ref{eq:fourierinverse})\u5f0f\u306f$\\xi$\u306b\u3064\u3044\u3066\u306e\u7a4d\u5206\u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3057\u3064\u3064\u3001$x=0$\u3092\u4ee3\u5165\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n

\u3059\u308b\u3068\u3001(\\ref{eq:squarewave})\u5f0f\u3088\u308a
\n\\begin{align}
\nf(0) &= \\int_{-\\infty}^{\\infty} \\frac{\\sin\\pi\\xi}{\\pi\\xi} d\\xi \\nonumber \\cr
\n&= 1 \\label{eq:fourierinverseatzero}
\n\\end{align}
\n\u3068\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n

\u305d\u3053\u3067\u3001$t = \\pi\\xi$\u3068\u304a\u304f\u3068\u3001
\n\\begin{align}
\n\\int_{-\\infty}^{\\infty} \\frac{\\sin\\pi\\xi}{\\pi\\xi} d\\xi &= \\int_{-\\infty}^{\\infty} \\frac{1}{\\pi}\\cdot\\frac{\\sin t}{t} dt
\n= 1 \\label{eq:ftparamtrans}
\n\\end{align}
\n(\\ref{eq:ftparamtrans})\u5f0f\u306e\u53f3\u50742\u8fba\u3092$\\pi$\u3067\u5272\u3063\u3066\u7a4d\u5206\u306e\u5909\u6570$t$\u3092$x$\u306b\u7f6e\u304d\u63db\u3048\u308b\u3068\u3001
\n\\begin{align}
\n\\int_{-\\infty}^{\\infty} \\frac{\\sin x}{x} dx
\n&= \\pi \\label{eq:doubleddirichletint}
\n\\end{align}
\n\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n

\u306a\u304a\u3001$f(x) = \\displaystyle\\frac{\\sin x}{x}$\u306e\u30b0\u30e9\u30d5\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u611f\u3058\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n

\"\"<\/a><\/p>\n

\u307e\u305f\u3001$\\displaystyle\\frac{\\sin x}{x}$\u306f\u5076\u95a2\u6570\u3067\u3042\u308b\u305f\u3081\u3001
\n\\begin{align}
\n\\int_{0}^{\\infty} \\frac{\\sin x}{x} dx
\n&= \\frac{\\pi}{2}
\n\\end{align}
\n\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n

\u4ee5\u4e0a\u3067\u3001(\\ref{eq:dirichletint})\u5f0f\u3092\u793a\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002$\\qquad\\blacksquare$<\/p>\n

\u3053\u3053\u304b\u3089\u304c\u672c\u5f53\u306e\u30d5\u30fc\u30ea\u30a8\u9006\u5909\u63db\u3067\u3059\u3002<\/h4>\n

\u3053\u3053\u3067\u3001$F(\\xi)$\u306e\u30d5\u30fc\u30ea\u30a8\u9006\u5909\u63db\u304c$f(x)$\u306b\u306a\u308b\u3053\u3068\u3082\u3064\u3044\u3067\u306b\u78ba\u8a8d\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n

(\\ref{eq:doubleddirichletint})\u5f0f\u3092\u5229\u7528\u3057\u3064\u3064\u3001(\\ref{eq:fourierinverse})\u5f0f\u306e\u53f3\u8fba\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002
\n\\begin{align}
\n\\int_{-\\infty}^{\\infty} \\frac{\\sin\\pi\\xi}{\\pi\\xi}e^{2\\pi\\xi xi} d\\xi &= \\int_{-\\infty}^{\\infty} \\frac{\\sin\\pi\\xi}{\\pi\\xi}(\\cos 2\\pi\\xi x+i\\sin 2\\pi\\xi x) d\\xi \\label{eq:fourierinversetriangle}
\n\\end{align}<\/p>\n

\u3053\u3053\u3067\u3001$g(\\xi) = \\displaystyle\\frac{\\sin\\pi\\xi\\sin 2\\pi\\xi x}{\\pi\\xi}$\u3068\u7f6e\u304f\u3068\u3001$g(-\\xi) = g(\\xi)$\u3067\u3059\u306e\u3067\u3001$g(\\xi)$\u306f\u5947\u95a2\u6570\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002\u3057\u305f\u304c\u3063\u3066(\\ref{eq:fourierinversetriangle})\u5f0f\u306e\u53f3\u8fba\u306e\u865a\u6570\u90e8\u306f0\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308b\u306e\u3067\u3001
\n\\begin{align}
\n\\int_{-\\infty}^{\\infty} \\frac{\\sin\\pi\\xi}{\\pi\\xi}(\\cos 2\\pi\\xi x+i\\sin2\\pi\\xi x) d\\xi &= \\int_{-\\infty}^{\\infty} \\frac{\\sin\\pi\\xi}{\\pi\\xi}\\cos 2\\pi\\xi x d\\xi \\nonumber \\cr
\n&= \\int_{-\\infty}^{\\infty} \\frac{1}{2\\pi\\xi}[\\sin\\pi\\xi(1+2x)+\\sin\\pi\\xi(1-2x)] d\\xi
\n\\label{eq:fourierinversedirichlet}
\n\\end{align}
\n\u3068\u306a\u308a\u307e\u3059\u3002(\\ref{eq:fourierinverseatzero})\u5f0f\u3068\u4f3c\u305f\u5f0f\u304c\u73fe\u308c\u307e\u3057\u305f\u3002\u305f\u3060\u3001$\\sin$\u306e\u4e2d\u306e\u3046\u3061\u7a4d\u5206\u5909\u6570$\\xi$\u3092\u9664\u304f\u90e8\u5206\u306e\u5024\u304c$x$\u306e\u5024\u306b\u3088\u3063\u3066\u306f\u8ca0\u306e\u5024\u306b\u306a\u308b\u30b1\u30fc\u30b9\u304c\u3042\u308a\u305d\u3046\u3067\u3059\u3002<\/p>\n

(\\ref{eq:fourierinverseatzero})\u5f0f\u3092\u5c11\u3057\u3060\u3051\u4e00\u822c\u5316\u3057\u3066\u3001$h(\\alpha) = \\displaystyle\\int_{-\\infty}^{\\infty} \\displaystyle\\frac{\\sin \\alpha \\xi}{\\alpha \\xi}d\\xi \\,(\\alpha \\ne 0)$\u3068\u7f6e\u304f\u3068\u3001$\\alpha$\u304c\u8ca0\u306e\u5024\u306e\u5834\u5408\u306b\u306f\u7b26\u53f7\u304c\u53cd\u8ee2\u3057\u307e\u3059\u306e\u3067\u3001$\\alpha$\u304c\u6b63\u306e\u5834\u5408\u3068\u5408\u308f\u305b\u3066
\n\\begin{align}
\n\\int_{-\\infty}^{\\infty} \\frac{\\sin \\alpha \\xi}{\\alpha \\xi}d\\xi &= \\frac{\\pi}{\\alpha{\\rm sgn}(\\alpha)} \\label{eq:sincintegral}
\n\\end{align}
\n\u3068\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u305f\u3060\u3057\u3001${\\rm sgn}(x)$\u306f\u7b26\u53f7\u95a2\u6570
\n\\begin{align}
\n{\\rm sgn}(x) &= \\left\\{
\n\\begin{array}{l}
\n1 & \\left(x \\gt 0\\right) \\cr
\n0 & \\left(x = 0\\right) \\cr
\n-1 & \\left(x \\lt 0\\right) \\cr
\n\\end{array}
\n\\right. \\label{eq:signfunction}
\n\\end{align}
\n\u3092\u8868\u3057\u307e\u3059\u3002<\/p>\n

(\\ref{eq:sincintegral})\u5f0f\u53f3\u8fba\u306e\u5206\u6bcd\u306f$|\\alpha|$\u3068\u66f8\u3044\u3066\u3082\u3044\u3044\u306e\u3067\u3059\u304c\u3001\u5f8c\u306e\u8a08\u7b97\u306e\u90fd\u5408\u304c\u3042\u308b\u306e\u3067\u3001\u3053\u306e\u3088\u3046\u306a\u5f62\u3067\u8868\u3057\u3066\u304a\u304d\u307e\u3059\u3002<\/p>\n

\u3053\u3053\u3067\u672c\u984c\u306b\u623b\u308a\u307e\u3057\u3066\u3001(\\ref{eq:sincintegral})\u5f0f\u3092\u5229\u7528\u3057\u3064\u3064(\\ref{eq:fourierinversedirichlet})\u5f0f\u306e\u53f3\u8fba\u7b2c1\u9805\u306e\u7a4d\u5206\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002$x = -\\displaystyle\\frac{1}{2}$\u306e\u3068\u304d\u306e\u3053\u3068\u306f\u3044\u3063\u305f\u3093\u5fd8\u308c\u3066\u3001$x \\ne -\\displaystyle\\frac{1}{2}$\u3068\u3057\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5909\u5f62\u3067\u304d\u307e\u3059\u3002
\n\\begin{align}
\n\\int_{-\\infty}^{\\infty} \\frac{1}{2\\pi\\xi}\\sin\\pi\\xi(1+2x) d\\xi &= \\int_{-\\infty}^{\\infty} \\frac{1+2x}{2}\\cdot\\frac{\\sin\\pi\\xi(1+2x)}{\\pi\\xi(1+2x)} d\\xi \\nonumber \\cr
\n&= \\frac{1+2x}{2}\\int_{-\\infty}^{\\infty} \\frac{\\sin\\pi\\xi(1+2x)}{\\pi\\xi(1+2x)} d\\xi \\nonumber \\cr
\n&= \\frac{1+2x}{2}\\cdot\\frac{\\pi}{\\pi(1+2x){\\rm sgn}(1+2x)} \\nonumber \\cr
\n&= \\frac{1}{2{\\rm sgn}(1+2x)} \\nonumber \\cr
\n&= \\frac{{\\rm sgn}(1+2x)}{2} \\label{eq:fourierinverseatfirstsection}
\n\\end{align}
\n\u6700\u5f8c\u306e2\u884c\u306b\u3064\u3044\u3066\u306f${\\rm sgn}(x)$\u304c\u7b26\u53f7\u95a2\u6570\u3067\u3042\u308b\u3053\u3068\u304b\u3089\u3001$\\displaystyle\\frac{1}{{\\rm sgn}(x)} = {\\rm sgn}(x)$\u3067\u3042\u308b\u3053\u3068\u3092\u5229\u7528\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n

\u306a\u304a\u3001\u5909\u5f62\u306e\u9014\u4e2d\u3067$\\pi(1+2x)$\u3092(\\ref{eq:sincintegral})\u5f0f\u306b\u304a\u3051\u308b$\\alpha$\u3060\u3068\u601d\u3063\u3066\u8a08\u7b97\u3057\u3066\u3044\u307e\u3059\u3002\u307e\u305f$x = -\\displaystyle\\frac{1}{2}$\u306e\u3068\u304d\u306b\u306f(\\ref{eq:fourierinversedirichlet})\u5f0f\u306e\u53f3\u8fba\u7b2c1\u9805\u306e\u5206\u5b50\u304c0\u306b\u306a\u308a\u307e\u3059\u306e\u3067\u3001\u7a4d\u5206\u3057\u305f\u7d50\u679c\u30820\u306b\u306a\u308a\u307e\u3059\u3002\u3053\u308c\u3089\u3092\u307e\u3068\u3081\u308b\u3068\u3001
\n\\begin{align}
\n\\int_{-\\infty}^{\\infty} \\frac{1}{2\\pi\\xi}\\sin\\pi\\xi(1+2x) d\\xi &= \\left\\{
\n\\begin{array}{l}
\n\\displaystyle\\frac{{\\rm sgn}(1+2x)}{2} & \\left(x \\ne -\\displaystyle\\frac{1}{2}\\right) \\cr
\n0 & \\left(x = -\\displaystyle\\frac{1}{2}\\right)
\n\\end{array}
\n\\right. \\label{eq:fourierinverseatfirstsectionfinal}
\n\\end{align}
\n\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n

(\\ref{eq:fourierinversedirichlet})\u5f0f\u306e\u53f3\u8fba\u7b2c2\u9805\u306b\u3064\u3044\u3066\u3082\u540c\u69d8\u306e\u8b70\u8ad6\u304c\u3067\u304d\u3066\u3001
\n\\begin{align}
\n\\int_{-\\infty}^{\\infty} \\frac{1}{2\\pi\\xi}\\sin\\pi\\xi(1-2x) d\\xi &= \\left\\{
\n\\begin{array}{l}
\n\\displaystyle\\frac{{\\rm sgn}(1-2x)}{2} & \\left(x \\ne \\displaystyle\\frac{1}{2}\\right) \\cr
\n0 & \\left(x = \\displaystyle\\frac{1}{2}\\right)
\n\\end{array}
\n\\right. \\label{eq:fourierinverseatsecondsectionfinal}
\n\\end{align}
\n\u3068\u306a\u308a\u307e\u3059\u306e\u3067\u3001(\\ref{eq:fourierinverseatfirstsectionfinal})\u5f0f\u53ca\u3073(\\ref{eq:fourierinverseatsecondsectionfinal})\u5f0f\u3092\u8db3\u3057\u5408\u308f\u305b\u308b\u3068\u3001
\n\\begin{align}
\nf(x) &= \\left\\{
\n\\begin{array}{l}
\n1 & \\left(|x| \\lt \\displaystyle\\frac{1}{2}\\right) \\cr
\n\\!\\displaystyle\\frac{1}{2} & \\left(|x| = \\displaystyle\\frac{1}{2}\\right) \\cr
\n0 & \\left(|x| \\gt \\displaystyle\\frac{1}{2}\\right) \\cr
\n\\end{array}
\n\\right. \\label{eq:fourierinversefinal}
\n\\end{align}
\n\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n

\u5143\u306e\u95a2\u6570\u304c\u518d\u3073\u73fe\u308c\u307e\u3057\u305f\u306d\u3002$\\qquad\\blacksquare$<\/p>\n

$|x| = \\displaystyle\\frac{1}{2}$\u306e\u3068\u3053\u308d\u306b\u3064\u3044\u3066\u306f\u30d5\u30fc\u30ea\u30a8\u9006\u5909\u63db\u306e\u7d50\u679c\u304c\u53f3\u6975\u9650\u3068\u5de6\u6975\u9650\u306e\u5e73\u5747\u5024\u306b\u306a\u3063\u3066\u3044\u307e\u3059\u304c\u3001\u6c17\u306b\u3057\u306a\u3044\u65b9\u5411\u3067\u304a\u9858\u3044\u3044\u305f\u3057\u307e\u3059\u3002\ud83d\ude01<\/p>\n

\u4ee5\u4e0b\u3001\u4f59\u8ac7\u304c\u7d9a\u304d\u307e\u3059\u3002<\/h2>\n

\u30c7\u30a3\u30ea\u30af\u30ec\u95a2\u6570<\/h3>\n

\u4f55\u304b\u3068\u30c7\u30a3\u30ea\u30af\u30ec\u7a4d\u5206\u3068\u9593\u9055\u3048\u3089\u308c\u3084\u3059\u3044\u30c7\u30a3\u30ea\u30af\u30ec\u95a2\u6570\u3067\u3059\u304c\u3001\u30c7\u30a3\u30ea\u30af\u30ec\u95a2\u6570\u306f\u300c\u3059\u3079\u3066\u306e\u5b9f\u6570\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u6709\u7406\u6570\u306e\u6642\u306b1\u3001\u7121\u7406\u6570\u306e\u6642\u306b0\u3092\u3068\u308b\u95a2\u6570\u300d\u3067\u3001(\\ref{eq:dirichletfunction})\u5f0f\u3067\u8868\u3055\u308c\u307e\u3059\u3002
\n\\begin{align}
\nf(x) &= \\lim_{n \\to \\infty}\\lim_{k \\to \\infty}\\cos^{2k}(n!\\pi x) \\label{eq:dirichletfunction}
\n\\end{align}<\/p>\n

\u8a73\u7d30\u306a\u89e3\u8aac\u306b\u3064\u3044\u3066\u306f\u53c2\u8003\u6587\u732e\u306b\u793a\u3057\u305fURL\u307e\u305f\u306fWikipedia<\/a>\u3092\u3054\u53c2\u7167\u3044\u305f\u3060\u3051\u308c\u3070\u3068\u601d\u3044\u307e\u3059\u3002<\/p>\n

\u5b9a\u7fa9\u304c\u308f\u304b\u308a\u3084\u3059\u304b\u3063\u305f\u308a\u3001\u6574\u6570\u306e\u968e\u4e57\u3092\u6709\u7406\u6570\u3092\u9078\u3073\u51fa\u3059\u305f\u3081\u306b\u4f7f\u3063\u305f\u308a\u3057\u3066\u3044\u308b\u3068\u3053\u308d\u306f\u975e\u5e38\u306b\u9762\u767d\u3044\u3068\u601d\u3046\u306e\u3067\u3059\u304c\u3001\u3069\u306e\u3088\u3046\u306b\u5fdc\u7528\u3057\u305f\u3089\u3088\u3044\u304b\u304c\u3088\u304f\u308f\u304b\u3089\u306a\u3044\u95a2\u6570\u3067\u306f\u3042\u308a\u307e\u3059\u3002\ud83d\udc3c<\/p>\n

\u30c7\u30a3\u30ea\u30af\u30ec\u7a4d\u5206\u304c\u7d76\u5bfe\u53ef\u7a4d\u5206\u3067\u306a\u3044\u3053\u3068\u306e\u8a3c\u660e<\/h3>\n

\u6b21\u306b\u3001\u30c7\u30a3\u30ea\u30af\u30ec\u7a4d\u5206\u306f\u7d76\u5bfe\u53ef\u7a4d\u5206\u3067\u306a\u3044\u3053\u3068\u3001\u3059\u306a\u308f\u3061\u3001
\n\\begin{align}
\n\\int_{-\\infty}^{\\infty} \\left|\\frac{\\sin x}{x}\\right| dx &\\to \\infty \\label{eq:dirichletinfty}
\n\\end{align}
\n\u3067\u3042\u308b\u3053\u3068\u3092\u8a3c\u660e\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n

$\\displaystyle\\frac{\\sin x}{x}$\u306f\u5076\u95a2\u6570\u306a\u306e\u3067\u3001
\n\\begin{align}
\n\\int_{-\\infty}^{\\infty} \\left|\\frac{\\sin x}{x}\\right| dx &= 2\\int_{0}^{\\infty} \\left|\\frac{\\sin x}{x}\\right|dx\\nonumber \\cr
\n&= 2\\sum_{n=0}^{\\infty}\\int_{n\\pi}^{(n+1)\\pi} \\left|\\frac{\\sin x}{x}\\right|dx\\nonumber \\cr
\n&= 2\\sum_{n=0}^{\\infty}\\left(\\int_{2n\\pi}^{(2n+1)\\pi} \\frac{\\sin x}{x}-\\int_{(2n+1)\\pi}^{(2n+2)\\pi} \\frac{\\sin x}{x}\\right)dx\\nonumber \\cr
\n&\\gt 2\\sum_{n=0}^{\\infty}\\left(\\int_{2n\\pi}^{(2n+1)\\pi} \\frac{\\sin x}{(2n+1)\\pi}-\\int_{(2n+1)\\pi}^{(2n+2)\\pi} \\frac{\\sin x}{(2n+2)\\pi}\\right)dx\\nonumber \\cr
\n&= 2\\sum_{n=0}^{\\infty}\\left(\\frac{2}{(2n+1)\\pi}+\\frac{2}{(2n+2)\\pi}\\right) \\nonumber \\cr
\n&= 2\\sum_{n=0}^{\\infty}\\frac{2}{(n+1)\\pi}\\nonumber \\cr
\n&= 4\\sum_{n=1}^{\\infty}\\frac{1}{n\\pi}\\label{eq:sumathalf}
\n\\end{align}
\n\u3068\u5909\u5f62\u3067\u304d\u307e\u3059\u304c\u3001(\\ref{eq:sumathalf})\u5f0f\u306e\u53f3\u8fba\u306f\u767a\u6563\u3057\u3066\u3057\u307e\u3044\u307e\u3059\u3002\u3088\u3063\u3066\u3001(\\ref{eq:dirichletinfty})\u5f0f\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3001\u3059\u306a\u308f\u3061\u3001\u7d76\u5bfe\u53ef\u7a4d\u5206\u3067\u306f\u306a\u3044\u3053\u3068\u304c\u793a\u3055\u308c\u307e\u3057\u305f\u3002$\\qquad\\blacksquare$<\/p>\n

Wikipedia\u306b\u306f$\\displaystyle\\frac{\\sin x}{x}$\u306e\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u306f\u77e9\u5f62\u95a2\u6570\u306b\u306a\u308b\u3053\u3068\u304c\u3057\u308c\u3063\u3068\u66f8\u3044\u3066\u3042\u308a\u307e\u3059\u304c\u3001\u3053\u306e\u8a18\u4e8b\u3067\u306f\u305d\u308c\u304c\u53ef\u80fd\u3067\u3042\u308b\u304b\u3069\u3046\u304b\u306e\u8b70\u8ad6\u306b\u306f\u7acb\u3061\u5165\u3089\u305a\u3001$\\displaystyle\\frac{\\sin x}{x}$\u306e\u30d5\u30fc\u30ea\u30a8\u9006\u5909\u63db\u306e\u307f\u3092\u8003\u3048\u308b\u3053\u3068\u3068\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n

\u307e\u3068\u3081<\/h2>\n

\u30c7\u30a3\u30ea\u30af\u30ec\u7a4d\u5206\u306f\u8907\u7d20\u7a4d\u5206\u3092\u5229\u7528\u3057\u3066\u8a08\u7b97\u3055\u308c\u308b\u3053\u3068\u304c\u591a\u3044\u3067\u3059\u3002<\/p>\n

\u8907\u7d20\u7a4d\u5206\u3092\u4f7f\u3046\u3068\u521d\u7b49\u7684\u306b\u306f\u8a08\u7b97\u3067\u304d\u306a\u3044\u7a4d\u5206\u3082\u8a08\u7b97\u3067\u304d\u308b\u5834\u5408\u304c\u3042\u308b\u306e\u3067\u3001\u30c4\u30fc\u30eb\u3068\u3057\u3066\u306e\u6c4e\u7528\u6027\u304c\u9ad8\u3044\u3068\u601d\u3044\u307e\u3059\u304c\u3001\u524d\u63d0\u3068\u306a\u308b\u77e5\u8b58\u3082\u305d\u306e\u5206\u591a\u304f\u306a\u308a\u307e\u3059\u3002\u3088\u3063\u3066\u3001\u624b\u3063\u53d6\u308a\u65e9\u304f\u7d50\u679c\u3060\u3051\u3092\u77e5\u308a\u305f\u3044\u3068\u304d\u306b\u306a\u3069\u306b\u3053\u306e\u8a18\u4e8b\u3067\u66f8\u3044\u305f\u3088\u3046\u306a\u6280\u5de7\u7684\u306a\u8a08\u7b97\u65b9\u6cd5\u304c\u3042\u308b\u3053\u3068\u3092\u982d\u306b\u5165\u308c\u3066\u304a\u304f\u3068\u4f55\u304b\u306e\u5f79\u306b\u306f\u7acb\u3064\u304b\u3082\u3057\u308c\u307e\u305b\u3093\u3002<\/p>\n

\u3053\u306e\u8a18\u4e8b\u306f\u4ee5\u4e0a\u3067\u3059\u3002<\/p>\n

References \/ \u53c2\u8003\u6587\u732e<\/h2>\n