{"id":4431,"date":"2019-05-09T22:54:17","date_gmt":"2019-05-09T13:54:17","guid":{"rendered":"https:\/\/pandanote.info\/?p=4431"},"modified":"2022-08-07T12:10:26","modified_gmt":"2022-08-07T03:10:26","slug":"sinx-xn%e3%81%ae-%e2%88%9e%e2%88%9e%e3%81%ab%e3%81%8a%e3%81%91%e3%82%8b%e5%ba%83%e7%be%a9%e7%a9%8d%e5%88%86%e3%82%92%e8%a8%88%e7%ae%97%e3%81%97%e3%81%a6%e3%81%bf%e3%81%9f%e3%80%82","status":"publish","type":"post","link":"https:\/\/pandanote.info\/?p=4431","title":{"rendered":"(sin^x\/x)^n\u306e(-\u221e,\u221e)\u306b\u304a\u3051\u308b\u5e83\u7fa9\u7a4d\u5206\u3092\u8a08\u7b97\u3057\u3066\u307f\u305f\u3002"},"content":{"rendered":"

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

\u524d\u306e\u8a18\u4e8b<\/a>\u3067\u3001sinc\u95a2\u6570\u306e2,3,4\u4e57\u306e\u7a4d\u5206\u3092\u90e8\u5206\u8a08\u7b97\u3067\u8a08\u7b97\u3057\u3001\u3055\u3089\u306b5\u4e57\u4ee5\u4e0a\u306b\u3064\u3044\u3066\u306f\u90e8\u5206\u7a4d\u5206\u3067\u306f\u8a08\u7b97\u304c\u3067\u304d\u306a\u3055\u305d\u3046\u3060\u3068\u3044\u3046\u3068\u3053\u308d\u307e\u3067\u3092\u66f8\u304d\u307e\u3057\u305f\u3002<\/p>\n

\u90e8\u5206\u7a4d\u5206\u3092\u7528\u3044\u305f\u5e30\u7d0d\u7684\u306a\u30a2\u30d7\u30ed\u30fc\u30c1\u3067\u306f$\\displaystyle\\left(\\frac{\\sin x}{x}\\right)^n (n \\in \\mathbb{N})$\u306e\u8a08\u7b97\u304c\u96e3\u3057\u305d\u3046\u3060\u306a\u2026 \u3068\u601d\u3044\u306a\u304c\u3089\u53c2\u8003\u6587\u732e\u3092\u3082\u3046\u4e00\u5ea6\u3088\u304f\u8aad\u307f\u76f4\u3057\u3066\u307f\u308b\u3068\u3001\u90e8\u5206\u7a4d\u5206\u3092\u4f7f\u3063\u3066\u5206\u6bcd\u306e\u6b21\u6570\u3092\u4e0b\u3052\u3066\u304b\u3089\u8907\u7d20\u7a4d\u5206\u3092\u3057\u3066\u3044\u308b\u3088\u3046\u306a\u6c17\u304c\u3059\u308b\u306e\u3067\u3001\u3068\u308a\u3042\u3048\u305a\u6570\u5f0f\u306b\u66f8\u304d\u8d77\u3053\u3057\u3066\u307f\u308b\u3053\u3068\u306b\u3057\u307e\u3057\u305f\u3002<\/p>\n

\u4ee5\u4e0b\u3001\u3053\u306e\u8a18\u4e8b\u3067\u306f
\n\\begin{align}
\nI_n &= \\int_{-\\infty}^{\\infty} \\frac{\\sin^n x}{x^n}dx\\label{eq:sincpowerint}
\n\\end{align}
\n\u3068\u304a\u3044\u3066\u8a08\u7b97\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n

\u307e\u305f\u3001$\\displaystyle\\begin{pmatrix}
\nn \\cr
\nk
\n\\end{pmatrix}$\u306f\u4e8c\u9805\u4fc2\u6570(${}_nC_k$)\u3092\u8868\u3059\u3053\u3068\u3068\u3057\u307e\u3059\u3002\u4e8c\u9805\u4fc2\u6570\u306e\u4fc2\u6570\u9593\u306b\u306f(\\ref{eq:binomial})\u5f0f\u306e\u95a2\u4fc2\u304c\u3042\u308a\u307e\u3059\u304c\u3001\u3053\u306e\u8a18\u4e8b\u3067\u306f(\\ref{eq:binomial})\u5f0f\u306e\u95a2\u4fc2\u3092\u7e70\u308a\u8fd4\u3057\u4f7f\u3044\u307e\u3059\u3002
\n\\begin{align}
\n\\begin{pmatrix}
\nn \\cr
\nk
\n\\end{pmatrix} &= \\begin{pmatrix}
\nn \\cr
\nn – k
\n\\end{pmatrix} \\label{eq:binomial}
\n\\end{align}<\/p>\n

\u307e\u305a\u3001\u4e0b\u6e96\u5099\u3092\u3057\u307e\u3059\u3002<\/h2>\n

\u6700\u521d\u306b$\\displaystyle\\frac{\\sin^n x}{x^n}$\u3092(\\ref{eq:sincpower})\u5f0f\u306e\u3088\u3046\u306b\u5909\u5f62\u53ca\u3073\u5c55\u958b\u3057\u307e\u3059\u3002
\n\\begin{align}
\n\\frac{\\sin^n x}{x^n} &= \\frac{(e^{ix}-e^{-ix})^n}{(2ix)^n} \\nonumber \\cr
\n&= \\frac{1}{(2ix)^n}\\sum_{k=0}^n (-1)^{n-k} \\begin{pmatrix}
\nn \\cr
\nk
\n\\end{pmatrix} e^{ixk}e^{-ix(n-k)} \\nonumber \\cr
\n&= \\frac{1}{(2i)^n}\\sum_{k=0}^n (-1)^{n-k} \\begin{pmatrix}
\nn \\cr
\nk
\n\\end{pmatrix} \\frac{e^{ix(2k-n)}}{x^n} \\label{eq:sincpower}
\n\\end{align}<\/p>\n

\u90e8\u5206\u7a4d\u5206\u3092\u3057\u307e\u3059\u3002<\/h2>\n

\u304a\u8a66\u3057\u306b1\u56de\u3060\u3051\u90e8\u5206\u7a4d\u5206\u3002<\/h3>\n

(\\ref{eq:sincpower})\u5f0f\u3092(\\ref{eq:sincpowerint})\u5f0f\u306b\u4ee3\u5165\u3057\u3001\u90e8\u5206\u7a4d\u5206\u306b\u3088\u308b\u5909\u5f62\u30921\u56de\u3060\u3051\u884c\u3044\u307e\u3059\u3002<\/p>\n

\u3059\u308b\u3068\u2026
\n\\begin{align}
\nI_n &= \\frac{1}{(2i)^n} \\int_{-\\infty}^{\\infty} \\sum_{k=0}^n (-1)^{n-k} \\begin{pmatrix}
\nn \\cr
\nk
\n\\end{pmatrix} \\left(-\\frac{1}{(n-1)x^{n-1}}\\right)^{\\prime}e^{ix(2k-n)} dx \\nonumber \\cr
\n&= \\frac{1}{(2i)^n}\\left[\\left[ \\sum_{k=0}^n (-1)^{n-k} \\begin{pmatrix}
\nn \\cr
\nk
\n\\end{pmatrix}\\left[-\\frac{e^{ix(2k-n)}}{(n-1)x^{n-1}}\\right] \\right]_{-\\infty}^{\\infty}\\right. \\nonumber \\cr
\n&+ \\left.\\int_{-\\infty}^{\\infty} \\sum_{k=0}^n (-1)^{n-k} \\begin{pmatrix}
\nn \\cr
\nk
\n\\end{pmatrix} \\left[\\frac{i(2k-n)e^{ix(2k-n)}}{(n-1)x^{n-1}}\\right]\\right]dx
\n\\label{eq:sincpowerintegralpart}
\n\\end{align}
\n\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n

\u3068\u3053\u308d\u3067\u3001(\\ref{eq:sincpowerintegralpart})\u5f0f\u306e\u7b2c1\u9805($J_n$\u3068\u304a\u304d\u307e\u3059\u3002)\u306f$n$\u304c\u5947\u6570\u306e\u5834\u5408\u306b\u306f\u7dcf\u548c\u3092\u3068\u308b\u90e8\u5206\u3092$k
\n=0 \\cdots \\displaystyle\\frac{n-1}{2}$\u307e\u3067\u306e\u524d\u534a\u90e8\u3068\u305d\u308c\u4ee5\u964d\u306e\u5f8c\u534a\u90e8\u306b\u5206\u5272\u3057\u3001\u3055\u3089\u306b\u5f8c\u534a\u90e8\u306b\u3064\u3044\u3066\u306f$k=n-l$\u3068\u304a\u3044\u3066\u5909\u5f62\u3057\u3066\u304b\u3089\u3001$l$\u3092\u518d\u5ea6$k$\u306b\u66f8\u304d\u63db\u3048\u308b\u3068\u3001(\\ref{eq:sincboundarytermatoddorder})\u5f0f\u306e\u3088\u3046\u306b\u5909\u5f62\u3067\u304d\u307e\u3059\u3002
\n\\begin{align}
\nJ_n &= \\sum_{k=0}^{\\frac{n-1}{2}} (-1)^{n-k} \\begin{pmatrix}
\nn \\cr
\nk
\n\\end{pmatrix}\\left[-\\frac{e^{ix(2k-n)}}{(n-1)x^{n-1}}\\right] + \\sum_{k=\\frac{n+1}{2}}^n (-1)^{n-k} \\begin{pmatrix}
\nn \\cr
\nk
\n\\end{pmatrix}\\left[-\\frac{e^{ix(2k-n)}}{(n-1)x^{n-1}}\\right] \\nonumber \\cr
\n&= \\sum_{k=0}^{\\frac{n-1}{2}} (-1)^{n-k} \\begin{pmatrix}
\nn \\cr
\nk
\n\\end{pmatrix}\\left[-\\frac{e^{ix(2k-n)}}{(n-1)x^{n-1}}\\right] + \\sum_{l=0}^{\\frac{n-1}{2}} (-1)^{l} \\begin{pmatrix}
\nn \\cr
\nn-l
\n\\end{pmatrix}\\left[-\\frac{e^{ix(n-2l)}}{(n-1)x^{n-1}}\\right] \\nonumber \\cr
\n&= \\sum_{k=0}^{\\frac{n-1}{2}} (-1)^{n-k} \\begin{pmatrix}
\nn \\cr
\nk
\n\\end{pmatrix}\\left[-\\frac{e^{ix(2k-n)}}{(n-1)x^{n-1}}\\right] + \\sum_{k=0}^{\\frac{n-1}{2}} (-1)^{k} \\begin{pmatrix}
\nn \\cr
\nn-k
\n\\end{pmatrix}\\left[-\\frac{e^{ix(n-2k)}}{(n-1)x^{n-1}}\\right] \\nonumber \\cr
\n&= \\sum_{k=0}^{\\frac{n-1}{2}} (-1)^{n-k} \\begin{pmatrix}
\nn \\cr
\nk
\n\\end{pmatrix}\\left[-\\frac{e^{ix(2k-n)}}{(n-1)x^{n-1}}\\right] + \\sum_{k=0}^{\\frac{n-1}{2}} (-1)^{k} \\begin{pmatrix}
\nn \\cr
\nk
\n\\end{pmatrix}\\left[-\\frac{e^{ix(n-2k)}}{(n-1)x^{n-1}}\\right] \\nonumber \\cr
\n&= \\sum_{k=0}^{\\frac{n-1}{2}} \\begin{pmatrix}
\nn \\cr
\nk
\n\\end{pmatrix}\\left[-\\frac{(-1)^{n-k}e^{ix(2k-n)}+(-1)^ke^{ix(n-2k)}}{(n-1)x^{n-1}}\\right] \\nonumber \\cr
\n&= \\sum_{k=0}^{\\frac{n-1}{2}} \\begin{pmatrix}
\nn \\cr
\nk
\n\\end{pmatrix}\\left[-\\frac{(-1)^k\\left[(-1)^{n-2k}e^{ix(2k-n)}+e^{ix(n-2k)}\\right]}{(n-1)x^{n-1}}\\right] \\nonumber \\cr
\n&= \\sum_{k=0}^{\\frac{n-1}{2}} \\begin{pmatrix}
\nn \\cr
\nk
\n\\end{pmatrix}\\left[\\frac{(-1)^{k+1}\\left[-e^{ix(2k-n)}+e^{ix(n-2k)}\\right]}{(n-1)x^{n-1}}\\right] \\nonumber \\cr
\n&= \\sum_{k=0}^{\\frac{n-1}{2}} \\begin{pmatrix}
\nn \\cr
\nk
\n\\end{pmatrix}\\left[\\frac{2i(-1)^{k+1}\\sin(n-2k)x}{(n-1)x^{n-1}}\\right] \\label{eq:sincboundarytermatoddorder}
\n\\end{align}
\n(\\ref{eq:sincboundarytermatoddorder})\u5f0f\u53f3\u8fba\u306e\u7dcf\u548c\u3092\u3068\u308b\u90e8\u5206\u306e\u5404\u9805\u306f$x \\to \\infty$\u307e\u305f\u306f$x \\to -\\infty$\u306e\u3068\u304d\u306b0\u306b\u8fd1\u3065\u304d\u307e\u3059\u306e\u3067\u3001$J_n \\to 0$\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n

\u307e\u305f$n$\u304c\u5076\u6570\u306e\u6642\u3082\u7dcf\u548c\u3092\u3068\u308b\u90e8\u5206\u3092$k = 0 \\cdots \\displaystyle\\frac{n}{2}-1$\u307e\u3067\u306e\u524d\u534a\u90e8\u3068\u300c\u4e2d\u592e\u306e\u9805$\\left(k = \\displaystyle\\frac{n}{2}\\right)$\u300d\u53ca\u3073\u305d\u308c\u4ee5\u964d\u306e\u5f8c\u534a\u90e8\u306b\u5206\u5272\u3057\u3001\u3055\u3089\u306b\u5f8c\u534a\u90e8\u306b\u3064\u3044\u3066\u306f$k=n-l$\u3068\u304a\u3044\u3066\u5909\u5f62\u3057\u3066\u304b\u3089\u3001$l$\u3092\u518d\u5ea6$k$\u306b\u66f8\u304d\u63db\u3048\u308b\u3068\u3001\u5947\u6570\u306e\u5834\u5408\u3068\u540c\u69d8\u306b(\\ref{eq:sincboundarytermatevenorder})\u5f0f\u306e\u3088\u3046\u306b\u5909\u5f62\u3067\u304d\u307e\u3059\u3002
\n\\begin{align}
\nJ_n &= \\sum_{k=0}^{\\frac{n}{2}-1} (-1)^{n-k} \\begin{pmatrix}
\nn \\cr
\nk
\n\\end{pmatrix}\\left[-\\frac{e^{ix(2k-n)}}{(n-1)x^{n-1}}\\right] + \\sum_{k=\\frac{n}{2}+1}^n (-1)^{n-k} \\begin{pmatrix}
\nn \\cr
\nk
\n\\end{pmatrix}\\left[-\\frac{e^{ix(2k-n)}}{(n-1)x^{n-1}}\\right] + (-1)^{\\frac{n}{2}} \\begin{pmatrix}
\nn \\cr
\nn\/2
\n\\end{pmatrix}\\left[-\\frac{1}{(n-1)x^{n-1}}\\right] \\nonumber \\cr
\n&= \\sum_{k=0}^{\\frac{n}{2}-1} (-1)^{n-k} \\begin{pmatrix}
\nn \\cr
\nk
\n\\end{pmatrix}\\left[-\\frac{e^{ix(2k-n)}}{(n-1)x^{n-1}}\\right] + \\sum_{k=0}^{\\frac{n}{2}-1} (-1)^{k} \\begin{pmatrix}
\nn \\cr
\nk
\n\\end{pmatrix}\\left[-\\frac{e^{ix(n-2k)}}{(n-1)x^{n-1}}\\right] + (-1)^{\\frac{n}{2}} \\begin{pmatrix}
\nn \\cr
\nn\/2
\n\\end{pmatrix}\\left[-\\frac{1}{(n-1)x^{n-1}}\\right] \\nonumber \\cr
\n&= \\sum_{k=0}^{\\frac{n-1}{2}} \\begin{pmatrix}
\nn \\cr
\nk
\n\\end{pmatrix}\\left[-\\frac{(-1)^{n-k}e^{ix(2k-n)}+(-1)^ke^{ix(n-2k)}}{(n-1)x^{n-1}}\\right] + (-1)^{\\frac{n}{2}} \\begin{pmatrix}
\nn \\cr
\nn\/2
\n\\end{pmatrix}\\left[-\\frac{1}{(n-1)x^{n-1}}\\right] \\nonumber \\cr
\n&= \\sum_{k=0}^{\\frac{n-1}{2}} \\begin{pmatrix}
\nn \\cr
\nk
\n\\end{pmatrix}\\left[-\\frac{(-1)^k\\left[(-1)^{n-2k}e^{ix(2k-n)}+e^{ix(n-2k)}\\right]}{(n-1)x^{n-1}}\\right] + (-1)^{\\frac{n}{2}} \\begin{pmatrix}
\nn \\cr
\nn\/2
\n\\end{pmatrix}\\left[-\\frac{1}{(n-1)x^{n-1}}\\right] \\nonumber \\cr
\n&= \\sum_{k=0}^{\\frac{n-1}{2}} \\begin{pmatrix}
\nn \\cr
\nk
\n\\end{pmatrix}\\left[\\frac{(-1)^{k+1}\\left[e^{ix(2k-n)}+e^{ix(n-2k)}\\right]}{(n-1)x^{n-1}}\\right] + (-1)^{\\frac{n}{2}} \\begin{pmatrix}
\nn \\cr
\nn\/2
\n\\end{pmatrix}\\left[-\\frac{1}{(n-1)x^{n-1}}\\right] \\nonumber \\cr
\n&= \\sum_{k=0}^{\\frac{n-1}{2}} \\begin{pmatrix}
\nn \\cr
\nk
\n\\end{pmatrix}\\left[\\frac{2(-1)^{k+1}\\cos(n-2k)x}{(n-1)x^{n-1}}\\right] + (-1)^{\\frac{n}{2}} \\begin{pmatrix}
\nn \\cr
\nn\/2
\n\\end{pmatrix}\\left[-\\frac{1}{(n-1)x^{n-1}}\\right]\\label{eq:sincboundarytermatevenorder}
\n\\end{align}
\n(\\ref{eq:sincboundarytermatevenorder})\u5f0f\u53f3\u8fba\u7b2c1\u9805\u306e\u7dcf\u548c\u3092\u3068\u308b\u90e8\u5206\u306e\u5404\u9805\u53ca\u3073\u53f3\u8fba\u7b2c2\u9805\u306f$x \\to \\infty$\u307e\u305f\u306f$x \\to -\\infty$\u306e\u3068\u304d\u306b0\u306b\u8fd1\u3065\u304d\u307e\u3059\u306e\u3067\u3001$J_n \\to 0$\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n

\u5c11\u3005\u9577\u304f\u306a\u308a\u307e\u3057\u305f\u304c\u3001\u3053\u3053\u307e\u3067\u306e\u8b70\u8ad6\u3067(\\ref{eq:sincpowerintegralpart})\u5f0f\u306e\u7b2c1\u9805\u306f0\u3068\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3057\u305f\u306e\u3067\u3001
\n\\begin{align}
\nI_n &= \\frac{1}{(2i)^n}\\int_{-\\infty}^{\\infty} \\sum_{k=0}^n (-1)^{n-k} \\begin{pmatrix}
\nn \\cr
\nk
\n\\end{pmatrix} \\frac{i(2k-n)e^{ix(2k-n)}}{(n-1)x^{n-1}}dx
\n\\label{eq:sincpowerintegralpartresult}
\n\\end{align}
\n\u3068\u306a\u3063\u3066\u3001\u90e8\u5206\u7a4d\u5206\u3092\u884c\u3046\u3053\u3068\u306b\u3088\u308a\u3001<\/p>\n

    \n
  1. \u5883\u754c\u7a4d\u5206$J_n$\u306e\u5024\u304c0\u306b\u306a\u308b\u3053\u3068\u3002<\/li>\n
  2. \u5206\u6bcd\u306e$x$\u306e\u6b21\u6570\u30921\u3064\u5c0f\u3055\u304f\u3067\u304d\u308b\u3053\u3068\u3002<\/li>\n<\/ol>\n

    \u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n

    \u7e70\u308a\u8fd4\u3057\u90e8\u5206\u7a4d\u5206\u3057\u307e\u3059\u3002<\/h3>\n

    \u524d\u7bc0\u306e\u624b\u9806\u3092\u5206\u6bcd\u306e$x$\u306e\u6b21\u6570\u304c1\u306b\u306a\u308b\u307e\u3067\u7e70\u308a\u8fd4\u3059\u3068\u3001(\\ref{eq:sincpowerintegralforcomplex})\u5f0f\u3092\u5c0e\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002
    \n\\begin{align}
    \nI_n &= \\frac{1}{(2i)^n}\\int_{-\\infty}^{\\infty} \\sum_{k=0}^n (-1)^{n-k} \\begin{pmatrix}
    \nn \\cr
    \nk
    \n\\end{pmatrix} \\frac{[i(2k-n)]^{n-1}e^{ix(2k-n)}}{(n-1)!x}dx \\label{eq:sincpowerintegralforcomplex}
    \n\\end{align}
    \n\u307e\u305f\u3001(\\ref{eq:sincpowerintegralforcomplex})\u5f0f\u53f3\u8fba\u306e\u88ab\u7a4d\u5206\u95a2\u6570\u306f\u6709\u9650\u548c\u3067\u3059\u306e\u3067\u3001$x$\u306b\u3064\u3044\u3066\u306e\u7a4d\u5206\u8a08\u7b97\u3092\u5148\u306b\u884c\u3046\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u305d\u3053\u3067\u3001
    \n\\begin{align}
    \nI_n &= \\frac{1}{(2i)^n} \\sum_{k=0}^n (-1)^{n-k} \\begin{pmatrix}
    \nn \\cr
    \nk
    \n\\end{pmatrix} \\frac{[i(2k-n)]^{n-1}}{(n-1)!}\\int_{-\\infty}^{\\infty}\\frac{e^{ix(2k-n)}}{x}dx \\label{eq:sincpowerintegralfirst}
    \n\\end{align}
    \n\u3068\u5909\u5f62\u3067\u304d\u307e\u3059\u3002<\/p>\n

    \u6b21\u306b\u3001\u8907\u7d20\u7a4d\u5206\u3057\u307e\u3059\u3002<\/h2>\n

    \u3053\u3053\u3067\u3001\u3044\u3063\u305f\u3093$2k-n=m$\u3068\u304a\u3044\u3066\u3001\u8907\u7d20\u7a4d\u5206
    \n\\begin{align}
    \nf(z) &= \\int_{-\\infty}^{\\infty} \\frac{e^{izm}}{z} dz \\label{eq:eizmdef}
    \n\\end{align}
    \n\u3092\u8a08\u7b97\u3059\u308b\u3053\u3068\u306b\u3057\u307e\u3059\u3002<\/p>\n

    \u7a4d\u5206\u7d4c\u8def\u306f$m$\u306e\u5024\u306b\u5fdc\u3058\u3066\u5909\u3048\u307e\u3059\u306e\u3067\u3001\u3053\u3053\u304b\u3089\u306f$m$\u306e\u7b26\u53f7\u306b\u5fdc\u3058\u3066\u5834\u5408\u5206\u3051\u3068\u306a\u308a\u307e\u3059\u3002\u306a\u304a\u3001\u3053\u306e\u8a18\u4e8b\u3067\u306f\u4f8b\u306b\u3088\u3063\u3066\u30b8\u30e7\u30eb\u30c0\u30f3\u306e\u88dc\u984c<\/a>\u306b\u3064\u3044\u3066\u306f\u8a3c\u660e\u7121\u3057\u3067\u5229\u7528\u3057\u307e\u3059\u3002<\/p>\n

    $m \\gt 0$\u306e\u5834\u5408<\/h3>\n

    \u7a4d\u5206\u7d4c\u8def\u3068\u3057\u3066\u4ee5\u4e0b\u306e\u7d4c\u8def\u3092\u8003\u3048\u307e\u3059\u3002
    \n    \"\"<\/a><\/p>\n

    \u4e0a\u8a18\u306e\u7a4d\u5206\u7d4c\u8def\u304c\u56f2\u3080\u9818\u57df\u5185\u306b\u306f$f(z)$\u306b\u3064\u3044\u3066\u306e\u7279\u7570\u70b9\u306f\u5b58\u5728\u3057\u306a\u3044\u3053\u3068\u304b\u3089\u3001(\\ref{eq:noresidue})\u5f0f\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002
    \n\\begin{align}
    \n\\left( \\int_{C_1} + \\int_{C_2} + \\int_{C_3} + \\int_{C_4} \\right) f(z)dz &= 0 \\label{eq:noresidue}
    \n\\end{align}
    \n\u3053\u3053\u3067\u3001\u7d4c\u8def$C_1$\u53ca\u3073$C_3$\u306f\u5b9f\u8ef8\u4e0a\u306e\u7d4c\u8def\u3001$C_2$\u306f\u534a\u5f84$R$\u306e\u534a\u5186\u4e0a\u306e\u7d4c\u8def\u3001$C_4$\u306f\u534a\u5f84$r$\u306e\u534a\u5186\u4e0a\u306e\u7d4c\u8def\u3092\u8868\u3059\u3082\u306e\u3068\u3057\u307e\u3059\u3002<\/p>\n

    $C_1$\u53ca\u3073$C_3$\u306f\u5b9f\u8ef8\u4e0a\u306e\u7d4c\u8def\u306a\u306e\u3067\u3001\u7a4d\u5206\u5909\u6570$z$\u306f$x (\\in {\\mathbb R})$\u306b\u7f6e\u304d\u63db\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u3066\u3001
    \n\\begin{align}
    \n\\left( \\int_{C_1} + \\int_{C_3} \\right) f(z)dz &= \\int_{r}^{R} \\frac{e^{imx}}{x} dx + \\int_{-R}^{-r} \\frac{e^{imx}}{x}dx \\label{eq:onrealaxis}
    \n\\end{align}
    \n\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n

    \u307e\u305f\u3001$C_2$\u306f$g(z)=\\displaystyle\\frac{1}{z}$\u3068\u7f6e\u304f\u3068\u3001$f(z) = e^{imz}g(x)$\u3068\u3044\u3046\u5f62\u3067\u66f8\u3051\u3066$m \\gt 0$\u3067\u3042\u308a\u3001\u304b\u3064$|z| \\to \\infty$\u306e\u3068\u304d\u306b$g(z)$\u304c\u8907\u7d20\u5e73\u9762\u4e0a\u3067\u4e00\u69d8\u306b\u53ce\u675f\u3059\u308b\u306e\u3067\u3001\u30b8\u30e7\u30eb\u30c0\u30f3\u306e\u88dc\u984c\u3092\u4f7f\u3046\u3053\u3068\u304c\u3067\u304d\u3066\u3001
    \n\\begin{align}
    \n\\int_{C_2} f(z)dz &\\to 0 \\,\\,(R \\to \\infty) \\label{eq:withjordanlemma}
    \n\\end{align}
    \n\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n

    \u6700\u5f8c\u306b\u3001\u7d4c\u8def$C_4$\u306b\u6cbf\u3063\u305f\u7a4d\u5206\u306b\u3064\u3044\u3066\u8003\u3048\u307e\u3059\u3002
    \n$f(z)$\u3092$z=0$\u306e\u307e\u308f\u308a\u3067\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u3059\u308b\u3068\u3001
    \n\\begin{align}
    \n\\frac{e^{imz}}{z} &= \\frac{1}{z}\\sum_{n=0}^{\\infty}\\frac{(imz)^n}{n!}\\nonumber\\cr
    \n&= \\frac{1}{z}+\\sum_{n=1}^{\\infty}\\frac{(im)^nz^{n-1}}{n!}\\nonumber\\cr
    \n&= \\frac{1}{z}+\\sum_{n=0}^{\\infty}\\frac{(im)^{n+1}z^n}{n!}\\label{eq:cfourtaylor}
    \n\\end{align}
    \n\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n

    \u3053\u3053\u3067$z=re^{i\\theta}$\u3068\u7f6e\u304d\u307e\u3059\u3002$dz=ire^{i\\theta}d\\theta$\u3068\u306a\u308b\u3053\u3068\u3068\u3001\u7a4d\u5206\u5909\u6570$\\theta$\u306b\u3064\u3044\u3066\u306e$\\pi$\u304b\u3089$0$\u307e\u3067\u306e\u7a4d\u5206\u3068\u306a\u308b\u3053\u3068\u306b\u7559\u610f\u3057\u3064\u3064(\\ref{eq:cfourtaylor})\u5f0f\u306e\u53f3\u8fba\u7b2c1\u9805\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002<\/p>\n

    \u3059\u308b\u3068\u2026
    \n\\begin{align}
    \n\\int_{C_4} \\frac{1}{z} dz &= \\int_{\\pi}^{0} \\frac{1}{re^{i\\theta}}ire^{i\\theta}d\\theta \\nonumber \\cr
    \n&= \\int_{\\pi}^{0}id\\theta \\nonumber \\cr
    \n&= \\left[ i\\theta \\right]_{\\pi}^{0} \\nonumber \\cr
    \n&= -i\\pi \\label{eq:cfourfirstterm}
    \n\\end{align}
    \n\u3068\u8a08\u7b97\u3067\u304d\u3001$r$\u306b\u4f9d\u5b58\u3057\u306a\u3044\u5024\u3068\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002\u307e\u305f\u3001(\\ref{eq:cfourtaylor})\u5f0f\u306e\u53f3\u8fba\u7b2c2\u9805\u306f\u3001\u548c\u3068\u7a4d\u5206\u306e\u9806\u5e8f\u3092\u5165\u308c\u66ff\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u306e\u3067\u3001
    \n\\begin{align}
    \n\\int_{C_4} \\sum_{n=0}^{\\infty}\\frac{(im)^{n+1}z^n}{n!} dz &= \\int_{\\pi}^{0} \\sum_{n=0}^{\\infty}\\frac{(im)^{n+1}r^ne^{in\\theta}}{n!}ire^{i\\theta}d\\theta \\nonumber \\cr
    \n&= \\int_{\\pi}^{0} \\sum_{n=0}^{\\infty}\\frac{(im)^nr^{n+1}e^{i(n+1)\\theta}}{n!}d\\theta \\nonumber \\cr
    \n&= \\sum_{n=0}^{\\infty} \\frac{(im)^nr^{n+1}}{n!}\\int_{\\pi}^{0}e^{i(n+1)\\theta}d\\theta \\label{eq:cfoursecondterm}
    \n\\end{align}
    \n\u3068\u5909\u5f62\u3067\u304d\u3066\u3001\u548c\u306e\u5404\u9805\u306b$r$\u306e\u3079\u304d\u4e57\u304c\u73fe\u308c\u308b\u306e\u3067\u3001$r \\to 0$\u3068\u3059\u308b\u3068\u3001\u5404\u9805\u304c\u3059\u3079\u3066$0$\u3068\u306a\u308a\u307e\u3059\u3002
    \n(\\ref{eq:onrealaxis}),(\\ref{eq:withjordanlemma}),(\\ref{eq:cfourfirstterm})\u53ca\u3073(\\ref{eq:cfoursecondterm})\u5f0f\u3092(\\ref{eq:noresidue})\u5f0f\u306b\u4ee3\u5165\u3057\u3001$R \\to \\infty, r \\to 0$\u3068\u3059\u308b\u3068\u3001
    \n\\begin{align}
    \n\\int_{-\\infty}^{\\infty} \\frac{e^{imx}}{x}dx \\,- \\,i\\pi &= 0 \\label{eq:putvalues}
    \n\\end{align}
    \n\u3068\u306a\u308b\u306e\u3067\u3001(\\ref{eq:putvalues})\u5f0f\u306e\u5de6\u8fba\u7b2c2\u9805\u3092\u53f3\u8fba\u306b\u79fb\u9805\u3059\u308b\u3068\u3001(\\ref{eq:emxatpositive})\u5f0f\u304c\u6c42\u307e\u308a\u307e\u3059\u3002
    \n\\begin{align}
    \n\\int_{\\infty}^{\\infty} \\displaystyle\\frac{e^{imx}}{x}dx &= i\\pi \\label{eq:emxatpositive}
    \n\\end{align}<\/p>\n

    $m \\lt 0$\u306e\u5834\u5408<\/h3>\n

    \u4ee5\u4e0b\u306e\u3088\u3046\u306a\u7a4d\u5206\u8def\u3092\u8003\u3048\u307e\u3059\u3002
    \n    \"\"<\/a>
    \n\u4e0a\u8a18\u306e\u7a4d\u5206\u8def\u3092\u8a2d\u5b9a\u3059\u308b\u3068$m \\lt 0$\u306e\u5834\u5408\u3067\u3082\u7d4c\u8def$C_2$\u306b\u304a\u3044\u3066\u30b8\u30e7\u30eb\u30c0\u30f3\u306e\u88dc\u984c\u304c\u9069\u7528\u3067\u304d\u3066\u3001(\\ref{eq:withjordanlemma})\u5f0f\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002\u307e\u305f\u3001$C_1$\u53ca\u3073$C_3$\u306f\u5b9f\u8ef8\u4e0a\u306e\u7d4c\u8def\u306b\u306a\u308a\u307e\u3059\u304c\u3001\u5411\u304d\u304c\u8ca0\u306e\u5411\u304d\u306b\u306a\u308a\u307e\u3059\u306e\u3067\u3001
    \n\\begin{align}
    \n\\left( \\int_{C_1} + \\int_{C_3} \\right) f(z)dz &= \\int_{R}^{r} \\frac{e^{imx}}{x} dx + \\int_{-r}^{-R} \\frac{e^{imx}}{x}dx \\label{eq:onrealaxisatnegative}
    \n\\end{align}
    \n\u306b\u306a\u308a\u307e\u3059\u3002\u3055\u3089\u306b\u3001\u7d4c\u8def$C_4$\u306f\u539f\u70b9\u306e\u307e\u308f\u308a\u306e\u5186\u3092$2\\pi$\u304b\u3089$\\pi$\u307e\u3067\u56de\u308a\u307e\u3059\u306e\u3067\u3001
    \n\\begin{align}
    \n\\int_{C_4} \\frac{1}{z} dz &= \\int_{\\pi}^{0} \\frac{1}{re^{i\\theta}}ire^{i\\theta}d\\theta \\nonumber \\cr
    \n&= \\int_{2\\pi}^{\\pi}id\\theta \\nonumber \\cr
    \n&= \\left[ i\\theta \\right]_{2\\pi}^{\\pi} \\nonumber \\cr
    \n&= -i\\pi \\label{eq:cfouratnegative}
    \n\\end{align}
    \n(\\ref{eq:onrealaxisatnegative}),(\\ref{eq:withjordanlemma})\u53ca\u3073(\\ref{eq:cfouratnegative})\u5f0f\u3092(\\ref{eq:noresidue})\u5f0f\u306b\u4ee3\u5165\u3057\u3001$R \\to \\infty, r \\to 0$\u3068\u3059\u308b\u3068\u3001
    \n\\begin{align}
    \n-\\int_{-\\infty}^{\\infty} \\frac{e^{imx}}{x}dx \\,- \\,i\\pi &= 0 \\label{eq:putvaluesatnegative}
    \n\\end{align}
    \n\u3068\u306a\u308b\u306e\u3067\u3001(\\ref{eq:putvalues})\u5f0f\u306e\u5de6\u8fba\u7b2c2\u9805\u3092\u53f3\u8fba\u306b\u79fb\u9805\u3059\u308b\u3068\u3001(\\ref{eq:emxatnegative})\u5f0f\u304c\u6c42\u307e\u308a\u307e\u3059\u3002
    \n\\begin{align}
    \n\\int_{\\infty}^{\\infty} \\displaystyle\\frac{e^{imx}}{x}dx &= -i\\pi \\label{eq:emxatnegative}
    \n\\end{align}<\/p>\n

    $m = 0$\u306e\u5834\u5408<\/h3>\n

    $m = 0$\u306e\u5834\u5408\u3082$m \\gt 0$\u306e\u5834\u5408\u3068\u540c\u3058\u7a4d\u5206\u7d4c\u8def\u3092\u8003\u3048\u308b\u3068(\\ref{eq:putvalues})\u5f0f\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u306e\u3067\u3001\u7a4d\u5206\u5024\u306f(\\ref{eq:emxatpositive})\u5f0f\u3068\u540c\u3058\u5024\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n

    \u4ed5\u4e0a\u3052\u3002<\/h2>\n

    \u2026\u306e\u524d\u306b\u3061\u3087\u3063\u3068\u7d30\u5de5\u3002<\/h3>\n

    (\\ref{eq:emxatpositive})\u53ca\u3073(\\ref{eq:emxatnegative})\u5f0f\u3092(\\ref{eq:sincpowerintegralfirst})\u5f0f\u306b\u4ee3\u5165\u3059\u308b\u3068\u7a4d\u5206\u306e\u7d50\u679c\u304c\u6c42\u307e\u308b\u306e\u3067\u3059\u304c\u3001\u305d\u306e\u524d\u306b(\\ref{eq:sincpowerintegralfirst})\u5f0f\u306b\u3061\u3087\u3063\u3068\u7d30\u5de5\u3092\u3057\u307e\u3059\u3002<\/p>\n

    $n$\u304c\u5076\u6570\u306e\u5834\u5408\u306b\u306f$2k-n=0$\u3068\u306a\u308b\u5834\u5408\u304c\u3042\u308a\u307e\u3059\u304c\u3001(\\ref{eq:sincpowerintegralfirst})\u5f0f\u53f3\u8fba\u306e\u548c\u3092\u3068\u308b\u969b\u306b\u5bfe\u5fdc\u3059\u308b\u9805\u304c0\u306b\u306a\u308a\u307e\u3059\u306e\u3067\u3001\u305d\u306e\u9805\u306f\u6700\u521d\u304b\u3089\u306a\u304b\u3063\u305f\u3053\u3068\u306b\u3057\u307e\u3059\u3002\u3059\u306a\u308f\u3061\u3001
    \n\\begin{align}
    \nI_n &= \\frac{1}{(2i)^n} \\sum_{k=0,k \\ne \\frac{n}{2}}^n (-1)^{n-k} \\begin{pmatrix}
    \nn \\cr
    \nk
    \n\\end{pmatrix} \\frac{[i(2k-n)]^{n-1}}{(n-1)!}\\int_{-\\infty}^{\\infty}\\frac{e^{ix(2k-n)}}{x}dx \\label{eq:sincpowerintegralremovedatzero}
    \n\\end{align}
    \n\u3068\u3057\u307e\u3059\u3002<\/p>\n

    \u3053\u308c\u306b\u3088\u308a$n$\u304c\u5076\u6570\u3067\u3042\u308b\u304b\u5947\u6570\u3067\u3042\u308b\u304b\u306b\u95a2\u4fc2\u306a\u304f\u3001\u548c\u3092\u3068\u308b\u5bfe\u8c61\u3068\u306a\u308b\u9805\u306e\u6570\u3092\u5076\u6570\u306b\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n

    \u7a4d\u5206\u306e\u7d50\u679c\u3092\u5143\u306e\u5f0f\u306b\u53cd\u6620\u3055\u305b\u307e\u3059\u3002<\/h3>\n

    \u3053\u3053\u307e\u3067\u306e\u8a08\u7b97\u3067\u3001\u3088\u3046\u3084\u304f(\\ref{eq:emxatpositive})\u53ca\u3073(\\ref{eq:emxatnegative})\u5f0f\u3092(\\ref{eq:sincpowerintegralremovedatzero})\u5f0f\u306b\u4ee3\u5165\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n

    $2k-n$\u306e\u7b26\u53f7\u306b\u3088\u3063\u3066\u4ee3\u5165\u3059\u308b\u5f0f\u304c\u7570\u306a\u308b\u3053\u3068\u306b\u6ce8\u610f\u3057\u306a\u304c\u3089\u8a08\u7b97\u3059\u308b\u3068\u2026
    \n\\begin{align}
    \nI_n &= \\frac{1}{(2i)^n} \\sum_{k=\\left\\lfloor\\frac{n}{2}\\right\\rfloor+1}^{n} (-1)^{n-k} \\begin{pmatrix}
    \nn \\cr
    \nk
    \n\\end{pmatrix} \\frac{[i(2k-n)]^{n-1}}{(n-1)!}\\pi i + \\frac{1}{(2i)^n} \\sum_{k=0}^{\\left\\lfloor\\frac{n}{2}\\right\\rfloor-1} (-1)^{n-k} \\begin{pmatrix}
    \nn \\cr
    \nk
    \n\\end{pmatrix} \\frac{[i(2k-n)]^{n-1}}{(n-1)!}(-\\pi i) \\nonumber \\cr
    \n&= \\frac{\\pi}{(n-1)!2^n} \\left[\\sum_{k=\\left\\lfloor\\frac{n}{2}\\right\\rfloor+1}^{n} (-1)^{n-k}(2k-n)^{n-1} \\begin{pmatrix}
    \nn \\cr
    \nk
    \n\\end{pmatrix} – \\sum_{k=0}^{\\left\\lfloor\\frac{n}{2}\\right\\rfloor-1} (-1)^{n-k}(2k-n)^{n-1} \\begin{pmatrix}
    \nn \\cr
    \nk
    \n\\end{pmatrix} \\right]
    \n\\label{eq:sincpowerfirstresult}
    \n\\end{align}
    \n(\\ref{eq:sincpowerfirstresult})\u5f0f\u306e\u53f3\u8fba\u7b2c2\u9805\u3067$l=n-k$\u3068\u304a\u304f\u3068\u3001$n$\u304c\u5076\u6570\u306e\u3068\u304d\u306f\u8ca0\u3067\u306a\u3044\u6574\u6570$r$\u3092\u7528\u3044\u3066$n = 2r$\u3068\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u3066\u3001$r = \\left\\lfloor\\displaystyle\\frac{n}{2}\\right\\rfloor$\u3067\u3059\u306e\u3067\u3001
    \n\\begin{align}
    \nn – \\left( \\left\\lfloor\\frac{n}{2}\\right\\rfloor-1 \\right) &= r + 1 \\nonumber \\cr
    \n&= \\left\\lfloor\\frac{n}{2}\\right\\rfloor+1 \\label{eq:niseven}
    \n\\end{align}
    \n\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n

    \u307e\u305f\u3001$n$\u304c\u5947\u6570\u306e\u3068\u304d\u306f$n = 2r+1$\u3068\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u3066\u3001$r = \\left\\lfloor\\displaystyle\\frac{n-1}{2}\\right\\rfloor = \\left\\lfloor\\displaystyle\\frac{n}{2}\\right\\rfloor$\u3067\u3059\u306e\u3067\u3001
    \n\\begin{align}
    \nn – \\left( \\left\\lfloor\\frac{n}{2}\\right\\rfloor-1 \\right) &= r + 1 \\nonumber \\cr
    \n&= \\left\\lfloor\\frac{n}{2}\\right\\rfloor+1 \\label{eq:nisodd}
    \n\\end{align}
    \n\u3068\u306a\u308a\u307e\u3059\u306e\u3067\u3001(\\ref{eq:niseven})\u5f0f\u53ca\u3073(\\ref{eq:nisodd})\u5f0f\u304b\u3089\u3001$l$\u306f$\\left\\lfloor\\displaystyle\\frac{n}{2}\\right\\rfloor+1$\u304b\u3089$n$\u307e\u3067\u306e\u6574\u6570\u5024\u3092\u3068\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n

    \u305d\u3053\u3067\u3001(\\ref{eq:sincpowerfirstresult})\u5f0f\u3092\u5909\u5f62\u3057\u3001$l$\u3092$k$\u306b\u66f8\u304d\u76f4\u3059\u3068\u2026
    \n\\begin{align}
    \nI_n &= \\frac{\\pi}{(n-1)!2^n} \\left[\\sum_{k=\\left\\lfloor\\frac{n}{2}\\right\\rfloor+1}^{n} (-1)^{n-k}(2k-n)^{n-1} \\begin{pmatrix}
    \nn \\cr
    \nk
    \n\\end{pmatrix} – \\sum_{l=\\left\\lfloor\\frac{n}{2}\\right\\rfloor+1}^{n} (-1)^{l}(n-2l)^{n-1} \\begin{pmatrix}
    \nn \\cr
    \nn – l
    \n\\end{pmatrix} \\right] \\nonumber \\cr
    \n&= \\frac{\\pi}{(n-1)!2^n} \\left[\\sum_{k=\\left\\lfloor\\frac{n}{2}\\right\\rfloor+1}^{n} (-1)^{n-k}(2k-n)^{n-1} \\begin{pmatrix}
    \nn \\cr
    \nk
    \n\\end{pmatrix} – \\sum_{k=\\left\\lfloor\\frac{n}{2}\\right\\rfloor+1}^{n} (-1)^{k}(n-2k)^{n-1} \\begin{pmatrix}
    \nn \\cr
    \nk
    \n\\end{pmatrix} \\right] \\nonumber \\cr
    \n&= \\frac{\\pi}{(n-1)!2^n} \\left[\\sum_{k=\\left\\lfloor\\frac{n}{2}\\right\\rfloor+1}^{n}\\begin{pmatrix}
    \nn \\cr
    \nk
    \n\\end{pmatrix} \\left[ (-1)^{n-k}(2k-n)^{n-1} – (-1)^{k}(n-2k)^{n-1} \\right] \\right] \\nonumber \\cr
    \n&= \\frac{\\pi}{(n-1)!2^n} \\left[\\sum_{k=\\left\\lfloor\\frac{n}{2}\\right\\rfloor+1}^{n}\\begin{pmatrix}
    \nn \\cr
    \nk
    \n\\end{pmatrix} \\left[ (-1)^{n-k}(2k-n)^{n-1} – (-1)^{k+n-1}(2k-n)^{n-1} \\right] \\right] \\nonumber \\cr
    \n&= \\frac{\\pi}{(n-1)!2^n} \\left[\\sum_{k=\\left\\lfloor\\frac{n}{2}\\right\\rfloor+1}^{n}\\begin{pmatrix}
    \nn \\cr
    \nk
    \n\\end{pmatrix} (2k-n)^{n-1} \\left[ (-1)^{n-k} + (-1)^{k+n} \\right] \\right] \\nonumber \\cr
    \n&= \\frac{\\pi}{(n-1)!2^n} \\left[\\sum_{k=\\left\\lfloor\\frac{n}{2}\\right\\rfloor+1}^{n}\\begin{pmatrix}
    \nn \\cr
    \nk
    \n\\end{pmatrix} (2k-n)^{n-1} (-1)^{n-k} \\left[ 1 + (-1)^{2k} \\right] \\right] \\nonumber \\cr
    \n&= \\frac{\\pi}{(n-1)!2^n} \\left[\\sum_{k=\\left\\lfloor\\frac{n}{2}\\right\\rfloor+1}^{n}\\begin{pmatrix}
    \nn \\cr
    \nk
    \n\\end{pmatrix} (2k-n)^{n-1} (-1)^{n-k} \\cdot 2 \\right] \\nonumber \\cr
    \n&= \\frac{\\pi}{2^{n-1}}\\sum_{k=\\left\\lfloor\\frac{n}{2}\\right\\rfloor+1}^n (-1)^{n-k} \\begin{pmatrix}
    \nn \\cr
    \nk
    \n\\end{pmatrix} \\frac{(2k-n)^{n-1}}{(n-1)!} \\nonumber \\cr
    \n&= \\frac{n\\pi}{2^{n-1}}\\sum_{k=\\left\\lfloor\\frac{n}{2}\\right\\rfloor+1}^n (-1)^{n-k} \\frac{(2k-n)^{n-1}}{(n-k)!k!} \\label{eq:sincpowerfinalresult}
    \n\\end{align}
    \n\u3068\u5909\u5f62\u3067\u304d\u307e\u3059\u3002<\/p>\n

    \u9577\u304b\u3063\u305f\u3067\u3059\u306d\u3002\ud83d\ude00<\/p>\n

    \u304a\u597d\u307f\u3067\u3001\u3061\u3087\u3063\u3068\u5909\u5f62\u3057\u307e\u3059\u3002<\/h3>\n

    (\\ref{eq:sincpowerfinalresult})\u5f0f\u306e\u548c\u3092\u3068\u308b\u969b\u306e\u5909\u6570$k$\u306e\u5024\u306e\u7bc4\u56f2\u304c$\\left\\lfloor\\displaystyle\\frac{n}{2}\\right\\rfloor+1$\u304b\u3089$n$\u307e\u3067\u306e\u6574\u6570\u306a\u306e\u306f\u5c11\u3005\u6c17\u6301\u3061\u304c\u60aa\u3044\u3068\u3044\u3046\u65b9\u5411\u3051\u306b\u3061\u3087\u3063\u3068\u5909\u5f62\u3057\u307e\u3059\u3002<\/p>\n

    \u4f8b\u306b\u3088\u3063\u3066$l=n-k$\u3068\u304a\u304f\u3068\u3001$l$\u306e\u3068\u308a\u5f97\u308b\u5024\u306e\u7bc4\u56f2\u304c$0$\u304b\u3089$\\left\\lfloor\\displaystyle\\frac{n}{2}\\right\\rfloor-1$\u307e\u3067\u306e\u6574\u6570\u306b\u306a\u308a\u307e\u3059\u306e\u3067\u3001
    \n\\begin{align}
    \nI_n &= \\frac{n\\pi}{2^{n-1}}\\sum_{l=0}^{\\left\\lfloor\\frac{n}{2}\\right\\rfloor-1} (-1)^{l} \\frac{(n-2l)^{n-1}}{l!(n-l)!} \\nonumber \\cr
    \n&= \\frac{n\\pi}{2^{n-1}}\\sum_{l=0}^{\\left\\lfloor\\frac{n}{2}\\right\\rfloor-1} (-1)^{l} \\frac{(n-2l)^{n-1}}{l!(n-l)!} \\nonumber \\cr
    \n&= \\frac{n\\pi}{2^{n-1}}\\sum_{k=0}^{\\left\\lfloor\\frac{n}{2}\\right\\rfloor-1} (-1)^{k} \\frac{(n-2k)^{n-1}}{k!(n-k)!}\\label{eq:sincpowerfinalresultfromzero}
    \n\\end{align}
    \n\u3068\u5909\u5f62\u3067\u304d\u307e\u3059\u3002<\/p>\n

    \u691c\u7b97\u3057\u307e\u3059\u3002<\/h2>\n

    (\\ref{eq:sincpowerfinalresult})\u5f0f\u53ca\u3073(\\ref{eq:sincpowerfinalresultfromzero})\u5f0f\u304c\u540c\u3058\u5024\u306b\u306a\u308b\u3053\u3068\u306e\u78ba\u8a8d\u3082\u517c\u306d\u3066$n=2,3,4,5$\u3092\u4ee3\u5165\u3057\u3066\u8a08\u7b97\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n

    $n=2$\u306e\u5834\u5408<\/h3>\n

    $\\displaystyle\\frac{n\\pi}{2^{n-1}} = \\pi$\u306b\u306a\u308b\u3053\u3068\u3068\u3001\u548c\u306e\u90e8\u5206\u306f(\\ref{eq:sincpowerfinalresult})\u5f0f\u53ca\u3073(\\ref{eq:sincpowerfinalresultfromzero})\u5f0f\u306e\u4e21\u65b9\u3068\u3082\u306b$1$\u306b\u306a\u308a\u307e\u3059\u306e\u3067\u3001$I_2 = \\pi$\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n

    $n=3$\u306e\u5834\u5408<\/h3>\n

    \u307e\u305a\u3001$\\displaystyle\\frac{n\\pi}{2^{n-1}}$\u306f(\\ref{eq:preludeatthree})\u5f0f\u306e\u3088\u3046\u306b\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002
    \n\\begin{align}
    \n\\frac{n\\pi}{2^{n-1}} &= \\frac{3}{4}\\pi \\label{eq:preludeatthree}
    \n\\end{align}
    \n\u6b21\u306b\u3001(\\ref{eq:sincpowerfinalresult})\u5f0f\u306b\u6cbf\u3063\u3066\u8a08\u7b97\u3057\u307e\u3059\u3002\u548c\u306e\u90e8\u5206\u306f$k=2,3$\u306b\u3064\u3044\u3066\u8a08\u7b97\u3059\u308b\u3068\u3001\u305d\u306e\u7d50\u679c\u306f1\u306b\u306a\u308a\u307e\u3059\u306e\u3067\u3001$I_3 = \\displaystyle\\frac{3}{4}\\pi$\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n

    \u4e00\u65b9\u3001(\\ref{eq:sincpowerfinalresultfromzero})\u5f0f\u306b\u6cbf\u3063\u3066\u8a08\u7b97\u3059\u308b\u5834\u5408\u306b\u306f\u3001\u548c\u306e\u90e8\u5206\u306f$k=0,1$\u306b\u3064\u3044\u3066\u8a08\u7b97\u3057\u307e\u3059\u3002\u3059\u308b\u3068\u305d\u306e\u7d50\u679c\u30821\u306b\u306a\u308a\u307e\u3059\u306e\u3067\u3001$I_3 = \\displaystyle\\frac{3}{4}\\pi$\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n

    $n=4$\u306e\u5834\u5408<\/h3>\n

    \u307e\u305a\u3001$\\displaystyle\\frac{n\\pi}{2^{n-1}}$\u306f(\\ref{eq:preludeatfour})\u5f0f\u306e\u3088\u3046\u306b\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002
    \n\\begin{align}
    \n\\frac{n\\pi}{2^{n-1}} &= \\frac{\\pi}{2} \\label{eq:preludeatfour}
    \n\\end{align}
    \n\u6b21\u306b\u3001(\\ref{eq:sincpowerfinalresult})\u5f0f\u306b\u6cbf\u3063\u3066\u8a08\u7b97\u3057\u307e\u3059\u3002\u548c\u306e\u90e8\u5206\u306f$k=3,4$\u306b\u3064\u3044\u3066\u8a08\u7b97\u3059\u308b\u3068\u3001\u305d\u306e\u7d50\u679c\u306f$\\displaystyle\\frac{4}{3}$\u306b\u306a\u308a\u307e\u3059\u306e\u3067\u3001$I_4 = \\displaystyle\\frac{2}{3}\\pi$\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n

    \u4e00\u65b9\u3001(\\ref{eq:sincpowerfinalresultfromzero})\u5f0f\u306b\u6cbf\u3063\u3066\u8a08\u7b97\u3059\u308b\u5834\u5408\u306b\u306f\u3001\u548c\u306e\u90e8\u5206\u306f$k=0,1$\u306b\u3064\u3044\u3066\u8a08\u7b97\u3057\u307e\u3059\u3002\u3059\u308b\u3068\u3001\u305d\u306e\u7d50\u679c\u3082$\\displaystyle\\frac{4}{3}$\u306b\u306a\u308a\u307e\u3059\u306e\u3067\u3001$I_4 = \\displaystyle\\frac{2}{3}\\pi$\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n

    $n=5$\u306e\u5834\u5408<\/h3>\n

    \u307e\u305a\u3001$\\displaystyle\\frac{n\\pi}{2^{n-1}}$\u306f(\\ref{eq:preludeatfive})\u5f0f\u306e\u3088\u3046\u306b\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002
    \n\\begin{align}
    \n\\frac{n\\pi}{2^{n-1}} &= \\frac{5}{16}\\pi \\label{eq:preludeatfive}
    \n\\end{align}
    \n\u6b21\u306b\u3001(\\ref{eq:sincpowerfinalresult})\u5f0f\u306b\u6cbf\u3063\u3066\u8a08\u7b97\u3057\u307e\u3059\u3002\u548c\u306e\u90e8\u5206\u306f$k=3,4,5$\u306b\u3064\u3044\u3066\u8a08\u7b97\u3057\u307e\u3059\u3002<\/p>\n

    \u3060\u3093\u3060\u3093\u8a08\u7b97\u304c\u3057\u3093\u3069\u304f\u306a\u3063\u3066\u304d\u307e\u3059\u3002<\/p>\n

    \u305d\u306e\u7d50\u679c\u306f\u2026
    \n\\begin{align}
    \n\\frac{1}{12} – \\frac{27}{8} + \\frac{125}{24} &= \\frac{23}{12} \\label{eq:sumatfive}
    \n\\end{align}
    \n\u306b\u306a\u308a\u307e\u3059\u306e\u3067\u3001(\\ref{eq:preludeatfive})\u5f0f\u53ca\u3073(\\ref{eq:sumatfive})\u5f0f\u3092\u307e\u3068\u3081\u3066\u3001
    \n\\begin{align}
    \nI_5 &= \\frac{23}{12}\\cdot\\frac{5}{16}\\pi \\nonumber \\cr
    \n&= \\frac{115}{192}\\pi \\label{eq:resultatfive}
    \n\\end{align}
    \n\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n

    \u4e00\u65b9\u3001(\\ref{eq:sincpowerfinalresultfromzero})\u5f0f\u306b\u6cbf\u3063\u3066\u8a08\u7b97\u3059\u308b\u5834\u5408\u306b\u306f\u3001\u548c\u306e\u90e8\u5206\u306f$k=0,1,2$\u306b\u3064\u3044\u3066\u8a08\u7b97\u3057\u307e\u3059\u3002\u305d\u306e\u7d50\u679c\u306f\u2026
    \n\\begin{align}
    \n\\frac{1}{12} – \\frac{27}{8} + \\frac{125}{24} &= \\frac{23}{12} \\label{eq:sumatfivefromzero}
    \n\\end{align}
    \n\u306b\u306a\u308a\u307e\u3059\u306e\u3067\u3001(\\ref{eq:resultatfive})\u5f0f\u3068\u540c\u69d8\u306e\u7d50\u679c\u3092\u5f97\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n

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

    \u3053\u3053\u307e\u3067\u306e\u8a08\u7b97\u3067\u3001\u4efb\u610f\u306e$n \\in \\mathbb{N}$\u306b\u3064\u3044\u3066\u306e$I_n$\u306e\u5024\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002<\/p>\n

    (\\ref{eq:sincpowerint})\u5f0f\u3067$n$\u3092\u5927\u304d\u304f\u3057\u305f\u3068\u304d\u306e\u5024\u3092\u8a08\u7b97\u3059\u308b\u884c\u70ba\u81ea\u4f53\u306b\u306f\u8ab0\u5f97\u611f\u306f\u3069\u3046\u3057\u3066\u3082\u4ed8\u304d\u307e\u3068\u3063\u3066\u3057\u307e\u3044\u307e\u3059\u304c\u3001\u8a08\u7b97\u306e\u9014\u4e2d\u3067\u8907\u7d20\u7a4d\u5206\u306e\u8a08\u7b97\u6642\u306b\u3088\u304f\u767b\u5834\u3059\u308b\u30b8\u30e7\u30eb\u30c0\u30f3\u306e\u88dc\u984c\u306e\u4f7f\u3044\u65b9\u3084\u3001\u8907\u7d20\u5e73\u9762\u4e0a\u306e\u6975\u3092\u5b8c\u5168\u306b\u56f2\u307e\u306a\u3044\u7d4c\u8def\u306b\u304a\u3051\u308b\u8907\u7d20\u7a4d\u5206\u306e\u8a08\u7b97\u6642\u306e\u6ce8\u610f\u70b9(\u4f8b: 2\u4f4d\u4ee5\u4e0a\u306e\u6975\u3092\u4e2d\u5fc3\u3068\u3059\u308b\u5c0f\u3055\u3044\u534a\u5186\u4e0a\u3067\u306e\u7a4d\u5206\u306f\u305d\u306e\u6975\u306b\u304a\u3051\u308b\u7559\u6570\u3092\u4f7f\u3063\u3066\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3068\u306f\u9650\u3089\u306a\u3044\u3053\u3068\u3001\u306a\u3069\u3002)\u3092\u601d\u3044\u51fa\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3068\u3044\u3046\u70b9\u3067\u306f\u3001\u5358\u306a\u308b\u982d\u306e\u4f53\u64cd\u4ee5\u4e0a\u306e\u4fa1\u5024\u306f\u3042\u308b\u306e\u3067\u306f\u306a\u3044\u304b\u3068\u601d\u3044\u307e\u3059\u3002<\/p>\n