{"id":1534,"date":"2025-08-18T20:26:12","date_gmt":"2025-08-18T11:26:12","guid":{"rendered":"https:\/\/daba-no-heya.com\/?p=1534"},"modified":"2025-08-18T20:26:13","modified_gmt":"2025-08-18T11:26:13","slug":"post-1534","status":"publish","type":"post","link":"https:\/\/daba-no-heya.com\/?p=1534","title":{"rendered":"\u30ac\u30f3\u30de\u95a2\u6570"},"content":{"rendered":"\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/daba-no-heya.com\/?p=1534\/#%E5%B0%8E%E5%85%A5\" >\u5c0e\u5165<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/daba-no-heya.com\/?p=1534\/#%E5%AE%9A%E7%BE%A9%E3%82%92%E8%A4%87%E7%B4%A0%E6%95%B0%E3%81%AB%E6%8B%A1%E5%BC%B5%E3%81%99%E3%82%8B\" >\u5b9a\u7fa9\u3092\u8907\u7d20\u6570\u306b\u62e1\u5f35\u3059\u308b<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/daba-no-heya.com\/?p=1534\/#%E8%A7%A3%E6%9E%90%E6%8E%A5%E7%B6%9A\" >\u89e3\u6790\u63a5\u7d9a<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/daba-no-heya.com\/?p=1534\/#%E7%84%A1%E9%99%90%E7%A9%8D%E8%A1%A8%E7%A4%BA\" >\u7121\u9650\u7a4d\u8868\u793a<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E5%B0%8E%E5%85%A5\"><\/span>\u5c0e\u5165<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">\u6b63\u306e\u5b9f\u6570$x$\u306b\u5bfe\u3057\u3066\u4ee5\u4e0b\u306e\u30ac\u30f3\u30de\u95a2\u6570(Gamma function)\u3092\u5b9a\u7fa9\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\Gamma(x)=\\int_{0}^{\\infty}t^{x-1}e^{-t}dt<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u306e\u3088\u3046\u306b$\\Gamma(x)$\u3092\u5b9a\u7fa9\u3059\u308b\u3068\u3001\u4efb\u610f\u306e\u6b63\u306e\u6574\u6570$n$\u306b\u5bfe\u3057\u3066\u3001$\\Gamma(n+1)=n!$\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<br>\u8981\u3059\u308b\u306b\u3001$\\Gamma(x)$\u306f\u968e\u4e57\u306e\u4e00\u822c\u5316\u3067\u3059\u3002<br>$n!$\u306f\u6b63\u306e\u6574\u6570$n$\u306b\u3064\u3044\u3066\u306e\u307f\u5b9a\u7fa9\u3055\u308c\u307e\u3059\u304c\u3001$\\Gamma(x)$\u306b\u3088\u3063\u3066\u6b63\u306e\u5b9f\u6570$x$\u306b\u62e1\u5f35\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$\\Gamma(n+1)=n!$\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u8a08\u7b97\u3057\u3066\u307f\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\Gamma(1)=\\int_{0}^{\\infty}e^{-t}dt=\\left[-e^{-t}\\right]_{0}^{\\infty}=1<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\begin{align}<br>\\Gamma(n+1)&amp;=\\int_{0}^{\\infty}t^ne^{-t}dt \\\\<br>&amp;=\\int_{0}^{\\infty}t^n\\left(-e^{-t}\\right)&#8217;dt \\\\<br>&amp;=\\left[-t^ne^{-t}\\right]_{0}^{\\infty}-\\int_{0}^{\\infty}nt^{n-1}\\left(-e^{-t}\\right)dt \\\\<br>&amp;=n\\int_{0}^{\\infty}t^{n-1}e^{-t}dt \\\\<br>&amp;=n\\Gamma(n)<br>\\end{align}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\begin{align}<br>\\Gamma(n+1)&amp;=n\\Gamma(n) \\\\<br>&amp;=n\\left(n-1\\right)\\Gamma(n-1) \\\\<br>&amp;=n\\left(n-1\\right)\\cdots1\\Gamma(1) \\\\<br>&amp;=n!<br>\\end{align}<br>$$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E5%AE%9A%E7%BE%A9%E3%82%92%E8%A4%87%E7%B4%A0%E6%95%B0%E3%81%AB%E6%8B%A1%E5%BC%B5%E3%81%99%E3%82%8B\"><\/span>\u5b9a\u7fa9\u3092\u8907\u7d20\u6570\u306b\u62e1\u5f35\u3059\u308b<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">\u30ac\u30f3\u30de\u95a2\u6570\u306e\u5b9a\u7fa9\u3092$\\Re(z)&gt;0$\u306e\u8907\u7d20\u6570$z$\u306b\u62e1\u5f35\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\Gamma(z)=\\int_{0}^{\\infty}t^{z-1}e^{-t}dt\\qquad\\left(\\Re(z)&gt;0\\right)<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u307e\u305a\u306f\u3001$\\Gamma(z)$\u304c$\\Re(z)&gt;0$\u3067\u53ce\u675f\u3059\u308b\u3053\u3068\u3092\u793a\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\left|t^{z-1}\\right|=\\left|e^{(z-1)\\log t}\\right|=e^{(\\Re(z)-1)\\log t}=t^{\\Re(z)-1}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$\\Re(z)=R$\u3068\u304a\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\begin{align}<br>\\int_0^\\infty\\left|t^{z-1}e^{-t}\\right|dt&amp;=\\int_0^\\infty t^{R-1}e^{-t}dt \\\\<br>&amp;=\\int_0^1 t^{R-1}e^{-t}dt+\\int_1^\\infty t^{R-1}e^{-t}dt<br>\\end{align}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$\\int_0^1 t^{R-1}e^{-t}dt$\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$0\\le t\\le 1 \\Rightarrow t^{R-1}e^{-t}\\le t^{R-1}$\u306a\u306e\u3067\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\int_0^1 t^{R-1}e^{-t}dt\\le\\int_0^1 t^{R-1}dt=\\left[\\frac{1}{R}t^R\\right]_0^1=\\frac{1}{R}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$\\int_0^1 t^{R-1}e^{-t}dt$\u306f\u6b63\u306e\u5024\u3092\u3068\u308b\u306e\u3067\u3001$\\frac{1}{R}&gt;0$\u3001\u3064\u307e\u308a$R&gt;0$\u3067\u3042\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002<br>\u524d\u63d0\u6761\u4ef6\u306b\u3042\u308b$\\Re(z)=R&gt;0$\u306f\u3053\u3053\u304c\u7406\u7531\u3060\u3068\u601d\u3044\u307e\u3059\u3002<br>$R&gt;0$\u306e\u5834\u5408\u3001\u7a4d\u5206\u306f\u6709\u9650\u306e\u5024\u306b\u53ce\u675f\u3059\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u6b21\u306b\u3001$\\int_1^\\infty t^{R-1}e^{-t}dt$\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$1\\le t$\u3067$R\\le n$\u3068\u306a\u308b\u81ea\u7136\u6570$n$\u3092\u3068\u308b\u3068\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\frac{t^{R-1}}{e^t}\\le\\frac{t^{n-1}}{e^t}=\\frac{t^{n+1}}{e^t}\\frac{1}{t^2}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u306a\u306e\u3067\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\lim_{t\\to\\infty}\\frac{t^{R-1}}{e^t}\\le\\lim_{t\\to\\infty}\\frac{t^{n+1}}{e^t}\\frac{1}{t^2}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u30ed\u30d4\u30bf\u30eb\u306e\u5b9a\u7406\u3088\u308a\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\lim_{t\\to\\infty}\\frac{t^{n+1}}{e^t}=\\lim_{t\\to\\infty}\\frac{(n+1)!}{e^t}=0<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3057\u305f\u304c\u3063\u3066\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\lim_{t\\to\\infty}\\frac{t^{R-1}}{e^t}\\le\\lim_{t\\to\\infty}\\frac{t^{n+1}}{e^t}\\frac{1}{t^2}=0<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3088\u308a\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\lim_{t\\to\\infty}\\frac{t^{R-1}}{e^t}=0<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3068\u306a\u308a\u3001$\\int_1^\\infty t^{R-1}e^{-t}dt$\u304c\u6709\u754c\u306a\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u4ee5\u4e0a\u3088\u308a\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\left|\\int_0^\\infty t^{z-1}e^{-t}dt\\right|\\le\\int_0^\\infty\\left|t^{z-1}e^{-t}\\right|dt&lt;\\infty<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u306a\u306e\u3067\u3001\u30ac\u30f3\u30de\u95a2\u6570$\\Gamma(z)$\u306f$\\Re(z)&gt;0$\u306e\u8907\u7d20\u6570$z$\u306b\u3064\u3044\u3066\u3082\u5b9a\u7fa9\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E8%A7%A3%E6%9E%90%E6%8E%A5%E7%B6%9A\"><\/span>\u89e3\u6790\u63a5\u7d9a<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">\u8907\u7d20\u5e73\u9762\u4e0a\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u6b63\u5247\u306a\u95a2\u6570\u306e\u5b9a\u7fa9\u57df\u3092\u62e1\u5f35\u3059\u308b\u64cd\u4f5c\u306e\u3053\u3068\u3092<strong>\u89e3\u6790\u63a5\u7d9a<\/strong>(analytic continuation)\u3068\u547c\u3073\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u5148\u306b\u793a\u3057\u305f\u3088\u3046\u306b\u3001\u30ac\u30f3\u30de\u95a2\u6570$\\Gamma(z)$\u306f$\\Re(z)>0$\u3092\u6e80\u305f\u3059\u8907\u7d20\u6570$z$\u306b\u3064\u3044\u3066\u5b9a\u7fa9\u3055\u308c\u307e\u3059\u3002<br>\u3053\u306e\u5b9a\u7fa9\u57df\u3092$\\Re(z)&lt;0$\u306e\u7bc4\u56f2\u306b\u62e1\u5f35\u3059\u308b\u3053\u3068\u3092\u8003\u3048\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$\\Gamma(z+1)=z\\Gamma(z)$\u3088\u308a\u3001$\\Gamma(z)=\\frac{\\Gamma(z+1)}{z}$\u3067\u3059\u3002<br>$\\Gamma(z+1)$\u304c\u5b9a\u7fa9\u3067\u304d\u308b\u3001\u3064\u307e\u308a$\\Re(z+1)>0$\u3067\u3042\u308c\u3070\u5de6\u8fba\u306e$\\Gamma(z)$\u306e\u5024\u304c\u308f\u304b\u308b\u306e\u3067\u3001\u3053\u306e\u64cd\u4f5c\u306b\u3088\u3063\u3066$\\Gamma(z)$\u306e\u5b9a\u7fa9\u57df\u3092$\\Re(z)>-1$\u307e\u3067\u62e1\u5f35\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u540c\u69d8\u306e\u64cd\u4f5c\u3092\u7e70\u308a\u8fd4\u3059\u3053\u3068\u3067\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\Gamma(z)=\\frac{\\Gamma(z+m)}{z(z+1)(z+2)\\cdots(z+m-1)}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3068\u306a\u308b\u306e\u3067\u3001$\\Gamma(z)$\u306e\u5b9a\u7fa9\u57df\u3092$\\Re(z)&gt;-m$\u307e\u3067\u62e1\u5f35\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E7%84%A1%E9%99%90%E7%A9%8D%E8%A1%A8%E7%A4%BA\"><\/span>\u7121\u9650\u7a4d\u8868\u793a<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">\u30ac\u30f3\u30de\u95a2\u6570\u306e\u7121\u9650\u7a4d\u8868\u793a\u3092\u5c0e\u51fa\u3059\u308b\u3053\u3068\u3067\u3001\u30ac\u30f3\u30de\u95a2\u6570\u304c\u3069\u3053\u3067\u6975\u3092\u3082\u3064\u304b\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$x$\u3092\u6b63\u306e\u5b9f\u6570\u3001$n$\u3092\u6574\u6570\u3068\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\begin{align}<br>\\Gamma(x)&amp;=\\lim_{n\\to\\infty}\\int_0^n t^{x-1}e^{-t}dt \\\\<br>&amp;=\\lim_{n\\to\\infty}\\int_0^nt^{x-1}\\left(1-\\frac{t}{n}\\right)^ndt<br>\\end{align}<br>$$<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\">$e^{-t}=\\left(1-\\frac{t}{n}\\right)^n$\u306e\u5909\u63db\u306b\u3064\u3044\u3066\u306f\u4ee5\u4e0b\u306e\u3068\u304a\u308a\u3067\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\left(1-\\frac{t}{n}\\right)^n=e^{n\\log\\left(1-\\frac{t}{n}\\right)}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$\\log(1+x)$\u3092\u30de\u30af\u30ed\u30fc\u30ea\u30f3\u5c55\u958b\u3059\u308b\u3068<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\log(1+x)=x+\\mathcal{O}(x^2)\\qquad(x\\to0)<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u306a\u306e\u3067\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\begin{align}<br>n\\log\\left(1-\\frac{t}{n}\\right)&amp;=n\\left(-\\frac{t}{n}+\\mathcal{O}{\\left(\\frac{t}{n}\\right)^2}\\right) \\\\<br>&amp;=-t+\\mathcal{O}\\left({\\frac{t^2}{n}}\\right)\\qquad(n\\to\\infty)<br>\\end{align}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3088\u308a\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\lim_{n\\to\\infty}\\left(1-\\frac{t}{n}\\right)^n=\\lim_{n\\to\\infty}e^{n\\log\\left(1-\\frac{t}{n}\\right)}=e^{-t}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>a_n=\\int_0^n t^{x-1}\\left(1-\\frac{t}{n}\\right)^ndt<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3068\u304a\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$u=\\frac{t}{n}$\u3068\u304a\u304f\u3068\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\begin{align}<br>a_n&amp;=\\int_0^1(nu)^{x-1}(1-u)^n ndu \\\\<br>&amp;=n^x\\int_0^1 u^{x-1}(1-u)^n du<br>\\end{align}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\begin{align}<br>\\int_0^1 u^{x-1}(1-u)^n du&amp;=\\int_0^1\\left(\\frac{1}{x}u^x\\right)'(1-u)^n du \\\\<br>&amp;=\\frac{1}{x}\\int_0^1 (u^x)'(1-u)^n du<br>\\end{align}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\begin{align}<br>\\int_0^1 (u^x)'(1-u)^n du&amp;=[u^x(1-u)^n]_0^1-\\int_0^1u^x\\cdot n(1-u)^{n-1}\\cdot (-1)du \\\\<br>&amp;=n\\int_0^1 u^x(1-u)^{n-1}du<br>\\end{align}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3088\u308a\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\int_0^1u^x(1-u)^ndu=\\frac{n}{x}\\int_0^1u^x(1-u)^{n-1}du \\tag{1}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u540c\u69d8\u306e\u8a08\u7b97\u3067\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\begin{align}<br>\\int_0^1u^x(1-u)^{n-1}du&amp;=\\int_0^1\\left(\\frac{1}{x+1}u^{x+1}\\right)'(1-u)^{n-1}du \\\\<br>&amp;=\\frac{1}{x+1}\\int_0^1(u^{x+1})'(1-u)^{n-1}du \\\\<br>&amp;=\\frac{n-1}{x+1}\\int_0^1u^{x+1}(1-u)^{n-2}du \\tag{2}<br>\\end{align}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u306a\u306e\u3067\u3001\u5f0f2\u3092\u5f0f1\u306b\u4ee3\u5165\u3059\u308c\u3070\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\int_0^1u^{x-1}(1-u)^ndu=\\frac{n}{x}\\cdot\\frac{n-1}{x+1}\\int_0^1 u^{x+1}(1-u)^{n-2}du<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u306e\u64cd\u4f5c\u3092$n$\u56de\u7e70\u308a\u8fd4\u3059\u3068<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\int_0^1u^{x-1}(1-u)^n=\\frac{n(n-1)\\cdots 1}{x(x+1)\\cdots (x+n-1)}\\int_0^1 u^{x+n-1}(1-u)^0du<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3068\u306a\u308b\u306e\u3067\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\int_0^1 u^{x+n-1}du=\\left[\\frac{1}{x+n}u^{x+n}\\right]_0^1=\\frac{1}{x+n}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3088\u308a\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\int_0^1u^{x-1}(1-u)^ndu=\\frac{n!}{x(x+1)\\cdots (x+n-1)(x+n)}=\\frac{n!}{\\prod_{k=0}^n(x+k)}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3057\u305f\u304c\u3063\u3066\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>a_n=\\frac{n^xn!}{\\prod_{k=0}^n(x+k)}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u306a\u306e\u3067\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\Gamma(x)=\\lim_{n\\to\\infty}\\frac{n^xn!}{\\prod_{k=0}^{n}(x+k)}\\qquad(x&gt;0)<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u3053\u3067\u9818\u57df$D$\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>D=\\mathbb{C}\\setminus\\{z\\in\\mathbb{Z}\\mid z\\le0\\}\\quad(z\\ne0,-1,-2,\\cdots)<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u306e\u3068\u304d\u3001$\\Gamma(z)$\u306f$D$\u5185\u3067\u6b63\u5247\u3067\u3059\u3002<br>\u307e\u305f\u3001\u5b9f\u8ef8\u306e\u6b63\u306e\u90e8\u5206($x&gt;0$)\u306f\u9818\u57df$D$\u306b\u542b\u307e\u308c\u3066\u3044\u3066\u3001\u3053\u306e\u90e8\u5206\u3067$\\Gamma(z)$\u3068$\\Gamma(x)$\u306f\u4e00\u81f4\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3057\u305f\u304c\u3063\u3066\u3001\u4e00\u81f4\u306e\u5b9a\u7406\u3088\u308a\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">$$<br>\\Gamma(z)=\\lim_{z\\to\\infty}\\frac{n^zn!}{\\prod_{k=0}^n(z+k)}<br>$$<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u306f$z\\ne 0,-1,-2,\\cdots$\u3092\u6e80\u305f\u3059\u8907\u7d20\u6570$z$\u306b\u3064\u3044\u3066\u6210\u7acb\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u5c0e\u5165 \u6b63\u306e\u5b9f\u6570$x$\u306b\u5bfe\u3057\u3066\u4ee5\u4e0b\u306e\u30ac\u30f3\u30de\u95a2\u6570(Gamma function)\u3092\u5b9a\u7fa9\u3057\u307e\u3059\u3002 $$\\Gamma(x)=\\int_{0}^{\\infty}t^{x-1}e^{-t}dt$$ \u3053\u306e\u3088\u3046\u306b$\\Gamma(x)$\u3092 [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[30],"tags":[],"class_list":["post-1534","post","type-post","status-publish","format-standard","hentry","category-30"],"_links":{"self":[{"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/posts\/1534","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1534"}],"version-history":[{"count":19,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/posts\/1534\/revisions"}],"predecessor-version":[{"id":1588,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=\/wp\/v2\/posts\/1534\/revisions\/1588"}],"wp:attachment":[{"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1534"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1534"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/daba-no-heya.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1534"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}