\u7279\u6b8a\u95a2\u6570\u306e\u7d1a\u6570\u5c55\u958b\u306b\u3064\u3044\u3066\u306e\u8a18\u4e8b\u7b49\u3092\u8aad\u307f\u6f01\u3063\u3066\u3044\u308b\u3046\u3061\u306b\u3001\u30c7\u30a3\u30ac\u30f3\u30de\u95a2\u6570\u3068\u30ac\u30f3\u30de\u95a2\u6570\u306e\u5546\u306e\u6975\u9650\u5024\u306e\u8a08\u7b97\u7d50\u679c\u3060\u3051\u304c\u3055\u3089\u3063\u3068\u66f8\u3044\u3066\u3042\u308b\u306e\u3092\u898b\u304b\u3051\u307e\u3057\u305f\u3002<\/p>\n
\u3053\u308c\u304c\u3044\u307e\u3044\u3061\u81ea\u660e\u306a\u7d50\u679c\u3058\u3083\u306a\u3044\u3088\u3046\u306a\u6c17\u304c\u3057\u305f\u306e\u3067\u3001\u8a08\u7b97\u3057\u3066\u307f\u308b\u3053\u3068\u306b\u3057\u307e\u3057\u305f\u3002<\/p>\n
\u3068\u3044\u3046\u308f\u3051\u3067\u3001\u65e9\u901f\u554f\u984c\u3067\u3059\u3002<\/p>\n
\n$n$\u3092\u6b63\u306e\u6574\u6570\u3068\u3059\u308b\u3068\u304d\u3001(\\ref{eq:psiGamma})\u5f0f\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u305b\u3002<\/p>\n
\\begin{align}
\n\\lim_{z \\to -n}\\frac{\\psi(z)}{\\Gamma(z
\n)} &= (-1)^{n-1}n! \\label{eq:psiGamma}
\n\\end{align}<\/p>\n\u306a\u304a\u3001$\\psi(z)$\u306f\u30c7\u30a3\u30ac\u30f3\u30de\u95a2\u6570<\/p>\n
\\begin{align}
\n\\psi(z) &= \\frac{d}{dz}\\log\\Gamma(z) \\label{eq:digamma}
\n\\end{align}<\/p>\n\u3067\u3042\u308b\u3002\n<\/p><\/blockquote>\n
\u5b9f\u306f\u3001(\\ref{eq:psiGamma})\u5f0f\u306f$n = 0$\u306e\u3068\u304d\u3067\u3082\u6210\u308a\u7acb\u3064\u306e\u3067\u3059\u304c\u3001\u3053\u308c\u306b\u3064\u3044\u3066\u306f\u5f8c\u8ff0\u3057\u307e\u3059\u3002<\/p>\n
\u30b5\u30af\u30b5\u30af\u3068\u89e3\u3044\u3066\u3044\u304d\u307e\u3059\u3002<\/h2>\n
\u307e\u305a\u3001\u5206\u6bcd\u3068\u5206\u5b50\u306b\u767b\u5834\u3059\u308b$\\Gamma(z)$\u53ca\u3073$\\psi(z)$\u3092$z+n$\u3092\u542b\u3080\u5f62\u3067\u8868\u3059\u3053\u3068\u3092\u8003\u3048\u307e\u3059\u3002\u306a\u304a\u3001$z+n \\ne 0$\u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3057\u306a\u304c\u3089\u8a08\u7b97\u3092\u9032\u3081\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n
$\\Gamma(z)$\u306e\u5834\u5408<\/h3>\n
$\\Gamma(z)$\u306f
\n\\begin{align}
\n\\Gamma(z+1) &= z\\Gamma(z) \\label{eq:gammainduction}
\n\\end{align}
\n\u3068\u3044\u3046\u95a2\u4fc2\u304c\u3042\u308a\u307e\u3059\u306e\u3067\u3001\u3053\u308c\u3092\u3055\u3089\u306b$n$\u56de\u7528\u3044\u308b\u3068\u2026
\n\\begin{align}
\n\\Gamma(z+n+1) &= (z+n)(z+n-1)\\cdots z\\Gamma(z) \\nonumber \\cr
\n&= \\Gamma(z)\\prod_{k=0}^n (z+k) \\label{eq:gammainductionntimes}
\n\\end{align}
\n\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n(\\ref{eq:gammainductionntimes})\u5f0f\u306e\u5de6\u8fba\u306e\u95a2\u6570\u306e\u5f15\u6570\u304c$z+n$\u304b\u3089\u5148\u306b\u3061\u3087\u3044\u3068\u884c\u304d\u904e\u304e\u3066\u3044\u308b\u306e\u306f\u6c17\u306b\u3057\u306a\u3044\u65b9\u5411\u3067\u304a\u9858\u3044\u3059\u308b\u3053\u3068\u306b\u3057\u3066\u3001(\\ref{eq:gammainductionntimes})\u5f0f\u306e\u4e21\u8fba\u3092$\\displaystyle\\prod_{k=0}^n (z+k)$\u3067\u5272\u308b\u3068\u2026
\n\\begin{align}
\n\\frac{\\Gamma(z+n+1)}{\\displaystyle\\prod_{k=0}^n (z+k)} &= \\Gamma(z) \\label{eq:gammabygammazn1}
\n\\end{align}
\n\u3068\u306a\u308a\u307e\u3059\u3002(\\ref{eq:gammabygammazn1})\u5f0f\u306e\u4e21\u8fba\u306b$z+n$\u3092\u304b\u3051\u308b\u3068\u3001\u5de6\u8fba\u306e\u5206\u6bcd\u304b\u3089$z+n$\u304c1\u500b\u3060\u3051\u6d88\u3048\u307e\u3059\u306e\u3067\u3001
\n\\begin{align}
\n\\frac{\\Gamma(z+n+1)}{\\displaystyle\\prod_{k=0}^{n-1} (z+k)} &= (z+n)\\Gamma(z) \\label{eq:gammabygammazn1zn}
\n\\end{align}
\n\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n$\\psi(z)$\u306e\u5834\u5408<\/h3>\n
\u5b9f\u306f\u30c7\u30a3\u30ac\u30f3\u30de\u95a2\u6570$\\psi(z)$\u306b\u3082
\n\\begin{align}
\n\\psi(z+1) &= \\psi(z) + \\frac{1}{z} \\label{eq:psiinduction}
\n\\end{align}
\n\u3068\u3044\u3046\u95a2\u4fc2\u304c\u3042\u308a\u307e\u3059\u3002\u305d\u3053\u3067\u3001\u3053\u308c\u3092\u3055\u3089\u306b$n$\u56de\u7528\u3044\u308b\u3068\u3001
\n\\begin{align}
\n\\psi(z+n+1) &= \\psi(z) + \\sum_{k=0}^{n}\\frac{1}{z+k} \\label{eq:psiinductionntimes}
\n\\end{align}
\n\u3068\u306a\u308a\u307e\u3059\u3002\u305d\u3053\u3067\u3001(\\ref{eq:psiinductionntimes})\u5f0f\u306e\u4e21\u8fba\u304b\u3089$\\displaystyle\\sum_{k=0}^{n}\\frac{1}{z+k}$\u3092\u5f15\u304f\u3068\u3001\u4ee5\u4e0b\u306e(\\ref{eq:psisubsum})\u5f0f\u3092\u5c0e\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002
\n\\begin{align}
\n\\psi(z+n+1) – \\sum_{k=0}^{n}\\frac{1}{z+k} &= \\psi(z) \\label{eq:psisubsum}
\n\\end{align}
\n(\\ref{eq:psisubsum})\u5f0f\u306e\u4e21\u8fba\u306b$z+n$\u3092\u304b\u3051\u308b\u3068\u3001
\n\\begin{align}
\n(z+n)\\left[ \\psi(z+n+1) – \\sum_{k=0}^{n}\\frac{1}{z+k} \\right] &= (z+n)\\psi(z) \\label{eq:psisubsumzn}
\n\\end{align}
\n\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\u7d44\u307f\u5408\u308f\u305b\u3066\u307f\u307e\u3059\u3002<\/h3>\n
(\\ref{eq:psisubsumzn})\u5f0f\u306e\u4e21\u8fba\u3092(\\ref{eq:gammabygammazn1zn})\u5f0f\u306e\u4e21\u8fba\u3067\u305d\u308c\u305e\u308c\u5272\u308b\u3068\u3001(\\ref{eq:product})\u5f0f\u306e\u3088\u3046\u306b\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002
\n\\begin{align}
\n\\frac{\\psi(z)}{\\Gamma(z)} &= \\frac{\\psi(z)(z+n)}{\\Gamma(z)(z+n)} \\nonumber \\cr
\n&= \\frac{(z+n)\\left[ \\psi(z+n+1) – \\displaystyle\\sum_{k=0}^{n}\\frac{1}{z+k} \\right]}{\\displaystyle\\frac{\\Gamma(z+n+1)}{\\displaystyle\\prod_{k=0}^{n-1} (z+k)}} \\nonumber \\cr
\n&= \\frac{-1+(z+n)\\left[ \\psi(z+n+1) – \\displaystyle\\sum_{k=0}^{n-1}\\frac{1}{z+k} \\right]}{\\displaystyle\\frac{\\Gamma(z+n+1)}{\\displaystyle\\prod_{k=0}^{n-1} (z+k)}} \\label{eq:product}
\n\\end{align}
\n(\\ref{eq:product})\u5f0f\u3067$z$\u304c$-n$\u306b\u8fd1\u3065\u304f\u6642\u306e\u6319\u52d5\u3092\u8abf\u3079\u3066\u307f\u307e\u3059\u3002<\/p>\n\u5206\u5b50\u306e\u7b2c2\u9805\u306f$z+n$\u3092\u304b\u3051\u3066\u3044\u308b\u3053\u3068\u3068\u3001\u62ec\u5f27\u306e\u4e2d\u306f$\\psi(1) = -\\gamma$($\\gamma$\u306fEuler\u306e\u5b9a\u6570)\u3067\u3042\u308b\u3053\u3068\u3068$\\displaystyle\\sum_{k=0}^{n-1}\\frac{1}{z+k}$\u304c\u6709\u9650\u306e\u5024\u3092\u3068\u308b\u3053\u3068\u304b\u3089\u3001$z$\u304c$-n$\u306b\u8fd1\u3065\u304f\u3068\u304d\u306b\u306f\u5168\u4f53\u3068\u3057\u3066$0$\u306b\u8fd1\u3065\u304f\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u5206\u5b50\u5168\u4f53\u3067\u306f$-1$\u306b\u8fd1\u3065\u304d\u307e\u3059\u3002<\/p>\n
\u307e\u305f\u3001\u5206\u6bcd\u306b\u3064\u3044\u3066\u306f$\\Gamma(z+n+1)$\u306f$\\Gamma(1) = 1$\u306b\u8fd1\u3065\u304d\u3001\u3055\u3089\u306b$\\displaystyle\\prod_{k=0}^{n-1} (z+k)$\u306b\u3064\u3044\u3066\u306f\u3001
\n\\begin{align}
\n\\prod_{k=0}^{n-1} (-n+k) &= (-1)^n\\prod_{k=0}^{n-1}(n-k) \\nonumber \\cr
\n&= (-1)^n\\cdot n(n-1) \\cdots 1 \\nonumber \\cr
\n&= (-1)^n n! \\label{eq:nfactorial}
\n\\end{align}
\n\u3068\u306a\u308a\u307e\u3059\u306e\u3067\u3001\u5206\u6bcd\u5168\u4f53\u3068\u3057\u3066\u306f$\\displaystyle\\frac{1}{(-1)^n n!} = \\displaystyle\\frac{(-1)^n}{n!}$\u306b\u8fd1\u3065\u304d\u307e\u3059\u3002<\/p>\n\u3088\u3063\u3066\u3001
\n\\begin{align}
\n\\lim_{z \\to -n}\\frac{\\psi(z)}{\\Gamma(z
\n)} &= (-1)^{n+1} n! \\nonumber \\cr
\n&= (-1)^{n-1} n! \\nonumber
\n\\end{align}
\n\u3068\u306a\u308b\u3002\u3053\u308c\u306f(\\ref{eq:psiGamma})\u5f0f\u3068\u4e00\u81f4\u3057\u307e\u3059\u3002$\\qquad\\blacksquare$<\/p>\n\u307e\u3068\u3081<\/h2>\n
\u306a\u3093\u304b\u3001\u3069\u3053\u304b\u306e\u5927\u5b66\u9662(\u5927\u5b66\u3067\u306f\u3042\u308a\u307e\u305b\u3093<\/strong>\u3001\u5ff5\u306e\u70ba)\u306e\u5165\u8a66\u306b\u51fa\u305d\u3046\u306a\u554f\u984c\u3067\u3059\u306d\u3002<\/p>\n
\u300c\u5927\u5b66\u3078\u306e\u6570\u5b66\u300d\u3063\u3066\u3044\u3046\u96d1\u8a8c\u304c\u3042\u308a\u307e\u3059\u304c\u3001\u300c\u5927\u5b66\u9662\u3078\u306e\u6570\u5b66\u300d\u306a\u3093\u3066\u3044\u3046\u53c2\u8003\u66f8\u3082\u3042\u308a\u307e\u3059\u3002<\/p>\n
\u4e88\u5099\u6821\u306e\u6570\u5b66\u306e\u5148\u751f\u304c\u7de8\u96c6\u306b\u52a0\u308f\u3063\u3066\u3044\u3066\u3001\u5927\u5b66\u5408\u683c\u5f8c\u306b\u4e00\u518a\u3044\u305f\u3060\u304d\u307e\u3057\u305f\u3002<\/p>\n
\u5fae\u5206\u7a4d\u5206\u7de8\u306f\u3053\u3061\u3089<\/a>\u3067\u3001\u7dda\u5f62\u4ee3\u6570\u7de8\u306f\u3053\u3061\u3089<\/a>\u3067\u3059\u3002<\/p>\n
\u95a2\u6570\u306e\u7d1a\u6570\u5c55\u958b\u3092\u3059\u308b\u3068\u3001\u601d\u308f\u306c\u3068\u3053\u308d\u3067$\\Gamma$\u95a2\u6570\u306e\u3088\u3046\u306a\u7279\u6b8a\u95a2\u6570\u304c\u767b\u5834\u3059\u308b\u3053\u3068\u304c\u3042\u308a\u307e\u3059\u3002<\/p>\n
\u7279\u6b8a\u95a2\u6570\u304c\u767b\u5834\u3057\u3066\u9014\u65b9\u306b\u66ae\u308c\u305d\u3046\u306b\u306a\u3063\u305f\u3068\u304d\u306b\u3053\u306e\u8a18\u4e8b\u3092\u601d\u3044\u51fa\u3057\u3066\u3044\u305f\u3060\u3051\u308b\u3068\u5e78\u3044\u3067\u3059\u3002\ud83d\ude0e<\/p>\n
\u3053\u306e\u8a18\u4e8b\u306e\u672c\u7de8\u306f\u4ee5\u4e0a\u3067\u3059\u3002<\/p>\n
\u4ed8\u9332<\/h2>\n
\u305d\u306e1: $\\Gamma$\u95a2\u6570\u306e\u7559\u6570\u3068\u6975<\/h3>\n
(\\ref{eq:product})\u5f0f\u306e\u5206\u6bcd\u306e$z = -n$\u3067\u306e\u6975\u9650\u5024
\n\\begin{align}
\n\\lim_{z \\to -n}(z+n)\\Gamma(z) &= \\frac{(-1)^n}{n!} \\label{eq:gammaresidue}
\n\\end{align}
\n\u306f$z = -n$\u306b\u304a\u3051\u308b$\\Gamma(z)$\u306e\u7559\u6570\u306b\u306a\u308a\u307e\u3059\u3002\u307e\u305f\u3001$\\Gamma(z)$\u306f\u6b63\u3067\u306a\u3044\u6574\u6570$z$\u306b\u304a\u3044\u3066\u4f4d\u65701\u306e\u6975\u3092\u6301\u3064\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n\u305d\u306e2: $z = 0$\u3067\u306e\u6975\u9650\u5024 [2019\/03\/02\u8ffd\u8a18]<\/h3>\n
\u3053\u306e\u8a18\u4e8b<\/a>\u3092\u66f8\u304f\u305f\u3081\u306b\u3053\u306e\u8a18\u4e8b\u3082\u898b\u76f4\u3057\u3066\u3044\u305f\u3068\u3053\u308d\u3001$z = 0$\u3067\u306e\u6975\u9650\u5024\u3082\u78ba\u8a8d\u3057\u3066\u304a\u304b\u306d\u3070\u306a\u3089\u306a\u3044\u3053\u3068\u306b\u6c17\u304c\u4ed8\u3044\u305f\u306e\u3067\u3001\u8ffd\u52a0\u3057\u307e\u3059\u3002<\/p>\n
(\\ref{eq:psiGamma})\u5f0f\u5de6\u8fba\u306e\u5206\u6bcd\u306b\u3064\u3044\u3066\u306f(\\ref{eq:gammainduction})\u5f0f\u306e\u95a2\u4fc2\u304c\u3001\u5206\u5b50\u306b\u3064\u3044\u3066\u306f(\\ref{eq:psiinduction})\u5f0f\u306e\u95a2\u4fc2\u304c\u76f4\u63a5\u4f7f\u3048\u307e\u3059\u306e\u3067\u3001<\/p>\n
\\begin{align}
\n\\frac{\\psi(z)}{\\Gamma(z)} &= \\frac{\\psi(z+1)-\\displaystyle\\frac{1}{z}}{\\displaystyle\\frac{\\Gamma(z+1)}{z}} \\nonumber \\cr
\n&{} \\frac{z\\psi(z+1)-1}{\\Gamma(z+1)} \\label{eq:psiGammaNearZero}
\n\\end{align}<\/p>\n$\\gamma$\u3092Euler\u306e\u5b9a\u6570\u3068\u3057\u3066\u3001(\\ref{eq:psiGammaNearZero})\u5f0f\u306e\u4e21\u8fba\u306e\u6975\u9650\u3092\u3068\u308b\u3068\u2026<\/p>\n
\\begin{align}
\n\\lim_{z \\to 0}\\frac{\\psi(z)}{\\Gamma(z)} &= \\lim_{z \\to 0} \\frac{z\\psi(z+1)-1}{\\Gamma(z+1)} \\nonumber \\cr
\n&= \\frac{0 \\cdot -\\gamma – 1}{1} \\nonumber \\cr
\n&= -1 \\label{eq:psiZero}
\n\\end{align}
\n\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\u3053\u308c\u306f\u3001(\\ref{eq:psiGamma})\u5f0f\u306e\u53f3\u8fba\u3067$n = 0$\u3068\u3057\u305f\u3068\u304d\u3068\u4e00\u81f4\u3059\u308b\u3002<\/p>\n
\u3088\u3063\u3066\u3001(\\ref{eq:psiGamma})\u5f0f\u306f$n = 0$\u306e\u3068\u304d\u3067\u3082\u6210\u308a\u7acb\u3064\u3002$\\qquad\\blacksquare$<\/p>\n","protected":false},"excerpt":{"rendered":"
\u306f\u3058\u3081\u306b \u7279\u6b8a\u95a2\u6570\u306e\u7d1a\u6570\u5c55\u958b\u306b\u3064\u3044\u3066\u306e\u8a18\u4e8b\u7b49\u3092\u8aad\u307f\u6f01\u3063\u3066\u3044\u308b\u3046\u3061\u306b\u3001\u30c7\u30a3\u30ac\u30f3\u30de\u95a2\u6570\u3068\u30ac\u30f3\u30de\u95a2\u6570\u306e\u5546\u306e\u6975\u9650\u5024\u306e\u8a08\u7b97\u7d50\u679c\u3060\u3051\u304c\u3055\u3089\u3063\u3068\u66f8\u3044\u3066\u3042\u308b\u306e\u3092\u898b\u304b\u3051\u307e\u3057\u305f\u3002 \u3053\u308c\u304c\u3044\u307e\u3044\u3061\u81ea\u660e\u306a\u7d50\u679c\u3058\u3083\u306a\u3044\u3088\u3046\u306a\u6c17\u304c\u3057\u305f\u306e\u3067\u3001\u8a08\u7b97\u3057\u3066\u307f\u308b\u3053\u3068\u306b\u3057\u307e\u3057\u305f\u3002 \u554f\u984c\u3067\u3059\u3002 \u3068\u3044\u3046\u308f\u3051\u3067\u3001\u65e9\u901f\u554f\u984c\u3067\u3059\u3002 $n$\u3092\u6b63\u306e\u6574\u6570\u3068\u3059\u308b\u3068\u304d\u3001(\\ref{eq:psiGamma})\u5f0f\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u305b\u3002 \\begin{align\u2026 Read More »<\/a><\/span><\/p>\n","protected":false},"author":1,"featured_media":3923,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[13],"tags":[],"class_list":["post-3905","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-13"],"_links":{"self":[{"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/posts\/3905","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/pandanote.info\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=3905"}],"version-history":[{"count":28,"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/posts\/3905\/revisions"}],"predecessor-version":[{"id":11915,"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/posts\/3905\/revisions\/11915"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/media\/3923"}],"wp:attachment":[{"href":"https:\/\/pandanote.info\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3905"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/pandanote.info\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3905"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/pandanote.info\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3905"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}