(\u521d\u51fa2017\/05\/26, tumblr \u3088\u308a\u79fb\u690d\u8a18\u4e8b)<\/p>\n\n\n\n
\u30b0\u30b0\u3063\u3066\u3082\u7121\u9650\u9060\u3067\u306e\u6f38\u8fd1\u5c55\u958b\u3070\u304b\u308a\u3067\u306a\u304b\u306a\u304b\u82e6\u6226\u3057\u305f\u306e\u3067\u30e1\u30e2\u3057\u3066\u304a\u304f<\/p>\n\n\n\n
\u307e\u305aWikipedia\u306b\u3082\u3042\u308b\u30ac\u30f3\u30de\u95a2\u6570\u306e\u7121\u9650\u4e57\u7a4d\u8868\u793a\uff08\u30aa\u30a4\u30e9\u30fc\u306e\u7121\u9650\u4e57\u7a4d\u8868\u793a\uff09\u304b\u3089\u51fa\u767a\u3057\u3088\u3046\u3002\u305d\u308c\u306f<\/p>\n\n\n\n
\\[
\\frac{1}{\\Gamma(z)}=\\frac{\\prod_{k=0}^n (z+k)}{n^z n!},~~~{\\small n\\rightarrow\\infty}
\\]<\/p>\n\n\n\n
\u3068\u3044\u3046\u3082\u306e\u3002\u539f\u70b9\u8fd1\u304f\u306a\u306e\u3067z=\u03b5\u3068\u66f8\u3053\u3046\u3002\u3053\u308c\u3092\u5bfe\u6570\u306b\u3068\u308b\u3068<\/p>\n\n\n\n
\\[
\\begin{align}
-\\log\\Gamma(\\epsilon)&=
\\sum_{k=0}^n\\log (\\epsilon+k) – \\log n! -\\epsilon \\log n\\newline
&=\\sum_{k=0}^n\\log (\\epsilon+k) – \\log \\prod_{k=1}^n k – \\epsilon\\log n\\newline
&=\\sum_{k=0}^n\\log (\\epsilon+k) – \\sum_{k=1}^n \\log k – \\epsilon\\log n – \\gamma \\epsilon +\\gamma \\epsilon\\newline
&~~{\\small \\left( \\mathrm{where} ~~ \\gamma=\\sum_{k=1}^n \\frac{1}{k}-\\log n \\right)}\\newline
&=\\log \\epsilon+\\sum_{k=1}^n\\left( \\log(\\epsilon+k)-\\log k \\right) – \\sum_{k=1}^n \\frac{\\epsilon}{k} + \\gamma \\epsilon\\newline
&=\\log \\epsilon+\\sum_{k=1}^n\\left( \\log(1+\\frac{\\epsilon}{k})-\\frac{\\epsilon}{k} \\right) + \\gamma \\epsilon
\\end{align}
\\]<\/p>\n\n\n\n
\uff08\u3053\u306e\u6bb5\u968e\u306e\u7121\u9650\u4e57\u7a4d\u8868\u793a\\[ \\frac{1}{\\Gamma(z)}=z e^{\\gamma z}\\prod_{k=1}^n (1+\\frac{z}{k})e^{-\\frac{z}{k}} \\]<\/p>\n\n\n\n
\u304b\u3089\u51fa\u767a\u3057\u305f\u307b\u3046\u304c\u901f\u3044\u3002\uff08\u30ef\u30a4\u30a8\u30eb\u30b7\u30e5\u30c8\u30e9\u30b9\u306e\u4e57\u7a4d\u8868\u793a\uff09\uff09<\/p>\n\n\n\n
\u3053\u3053\u304b\u3089\\[ \\log (1 \\pm x)= \\pm x + \\mathcal{O}(x) \\]<\/p>\n\n\n\n
\u306e\u8fd1\u4f3c\u3092\u4f7f\u3063\u3066\u3044\u304f\u3068\\[ \\begin{align} -\\log\\Gamma(\\epsilon)&=\\log \\epsilon + \\gamma \\epsilon +\\sum_{k=1}^n\\left( \\log(1+\\frac{\\epsilon}{k})-\\frac{\\epsilon}{k} \\right)\\newline &=\\log \\epsilon + \\gamma \\epsilon+\\sum_{k=1}^n\\left(\\frac{\\epsilon}{k}-\\frac{\\epsilon}{k}+\\mathcal{O}(\\epsilon)\\right)\\newline &=\\log \\epsilon – \\log (1-\\gamma \\epsilon) + \\mathcal{O} (\\epsilon)\\newline &=\\log \\left( \\frac{\\epsilon}{1-\\gamma \\epsilon} \\right) + \\mathcal{O}(\\epsilon)\\newline &=-\\log \\left( \\frac{1}{\\epsilon}-\\gamma \\right) + \\mathcal{O}(\\epsilon) \\end{align} \\]<\/p>\n\n\n\n
\u3088\u3063\u3066\u5bfe\u6570\u3092\u623b\u3059\u3068\\[ \\Gamma (\\epsilon) = \\frac{1}{\\epsilon}-\\gamma +\\mathcal{O}(\\epsilon) \\]<\/p>\n\n\n\n
\u306e\u8fd1\u4f3c\u5f0f\u304c\u5f97\u3089\u308c\u308b\u3002\u03b3\u306f\u30aa\u30a4\u30e9\u30fc\u30fb\u30de\u30b9\u30b1\u30ed\u30fc\u30cb\u5b9a\u6570\u3068\u3044\u3046\u3082\u306e\u3067\u03b3=0.5772156649..\u3068\u3044\u3046\u6570\u3067\u3042\u308b\u3002\u3053\u306e\u7d50\u679c\u306f\\[ \\Gamma (z)=\\frac{\\Gamma (z+1)}{z} \\]<\/p>\n\n\n\n
\u306e\u95a2\u4fc2\u5f0f\u304b\u3089\u5f97\u3089\u308c\u308bz=0\u306b\u7559\u6570\u0393(1)=\uff11\u306e\u4e00\u4f4d\u306e\u6975\u3092\u6301\u3064\u3068\u3044\u3046\u6027\u8cea\u306b\u77db\u76fe\u3057\u306a\u3044\u3002<\/p>\n\n\n\n
<\/p>\n","protected":false},"excerpt":{"rendered":"
(\u521d\u51fa2017\/05\/26, tumblr \u3088\u308a\u79fb\u690d\u8a18\u4e8b) \u30b0\u30b0\u3063\u3066\u3082\u7121\u9650\u9060\u3067\u306e\u6f38\u8fd1\u5c55\u958b\u3070\u304b\u308a\u3067\u306a\u304b\u306a\u304b\u82e6\u6226\u3057\u305f\u306e\u3067\u30e1\u30e2\u3057\u3066\u304a\u304f \u307e\u305aWikipedia\u306b\u3082\u3042\u308b\u30ac\u30f3\u30de\u95a2\u6570\u306e\u7121\u9650\u4e57\u7a4d\u8868\u793a\uff08\u30aa\u30a4\u30e9\u30fc\u306e\u7121\u9650\u4e57\u7a4d\u8868\u793a\uff09\u304b\u3089\u51fa\u767a\u3057…<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_kad_post_transparent":"","_kad_post_title":"","_kad_post_layout":"","_kad_post_sidebar_id":"","_kad_post_content_style":"","_kad_post_vertical_padding":"","_kad_post_feature":"","_kad_post_feature_position":"","_kad_post_header":false,"_kad_post_footer":false,"footnotes":""},"categories":[8],"tags":[10,9],"class_list":["post-65","post","type-post","status-publish","format-standard","hentry","category-8","tag-10","tag-9"],"_links":{"self":[{"href":"https:\/\/blog.shuphy.com\/index.php?rest_route=\/wp\/v2\/posts\/65","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.shuphy.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.shuphy.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.shuphy.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.shuphy.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=65"}],"version-history":[{"count":5,"href":"https:\/\/blog.shuphy.com\/index.php?rest_route=\/wp\/v2\/posts\/65\/revisions"}],"predecessor-version":[{"id":135,"href":"https:\/\/blog.shuphy.com\/index.php?rest_route=\/wp\/v2\/posts\/65\/revisions\/135"}],"wp:attachment":[{"href":"https:\/\/blog.shuphy.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=65"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.shuphy.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=65"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.shuphy.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=65"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}