\u4e45\u3005\u306b\u8a08\u7b97\u7528\u7d19\u306b\u66f8\u3044\u305f\u3053\u3068\u3092\u305d\u306e\u307e\u307e\u516c\u958b\u3059\u308b\u3060\u3051\u306e\u8a18\u4e8b\u3092\u304a\u9001\u308a\u3059\u308b\u6b21\u7b2c\u3067\u3054\u3056\u3044\u307e\u3059\u3002<\/p>\n
\u30cd\u30a4\u30d4\u30a2\u6570$e$\u306e\u5b9a\u7fa9\u5f0f\u306e\u4e00\u3064\u3067\u3042\u308b<\/p>\n
\\begin{align}
\n\\lim_{x\\to\\infty}\\left(1+\\frac{1}{x}\\right)^x&= e \\label{eq:napiersconstantdefinition}
\n\\end{align}<\/p>\n
\u3092\u4f7f\u3046\u3068\u3001$\\lambda$\u304c\u6b63\u306e\u5b9f\u6570\u306e\u5834\u5408\u306b\u306f\u3001<\/p>\n
\\begin{align}
\n\\lim_{x\\to\\infty}\\left(1+\\frac{\\lambda}{x}\\right)^x&= e^{\\lambda} \\label{eq:elambda}
\n\\end{align}<\/p>\n
\u3067\u3042\u308b\u3053\u3068\u3092\u5bb9\u6613\u306b\u5c0e\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u3059(\u672c\u8a18\u4e8b\u672b\u5c3e\u306e\u300c\u304a\u307e\u3051\u300d\u306e\u7bc0\u53c2\u7167)\u3002<\/p>\n
\u6b21\u306b\u3001$\\lambda$\u304c\u8ca0\u306e\u5b9f\u6570\u3067\u3042\u308b\u5834\u5408\u3092\u8003\u3048\u307e\u3059\u3002<\/p>\n
$\\mu = -\\lambda$\u3068\u304a\u3044\u3066(\\ref{eq:elambda})\u306e\u5de6\u8fba\u306b\u4ee3\u5165\u3059\u308b\u3068\u2026<\/p>\n
\\begin{align}
\n\\lim_{x\\to\\infty}\\left(1+\\frac{\\lambda}{x}\\right)^x&=\\lim_{x\\to\\infty}\\left(1-\\frac{\\mu}{x}\\right)^x \\label{eq:elambdamu}
\n\\end{align}<\/p>\n
\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n
\u3057\u304b\u3057\u3001$\\mu$\u306f\u6b63\u306e\u5b9f\u6570\u3067\u3042\u308b\u305f\u3081\u3001(\\ref{eq:napiersconstantdefinition})\u5f0f\u306e\u5de6\u8fba\u53ca\u3073(\\ref{eq:elambda})\u5f0f\u306e\u5de6\u8fba\u306e\u4e00\u90e8\u307e\u305f\u306f\u5168\u90e8\u306b\u5408\u81f4\u3059\u308b\u5f0f\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002<\/p>\n
\u305d\u3053\u3067\u3001\u300c$\\lambda$\u304c\u8ca0\u306e\u5b9f\u6570\u3067\u3042\u308b\u5834\u5408\u306e$\\displaystyle\\lim_{x\\to\\infty}\\left(1+\\displaystyle\\frac{\\lambda}{x}\\right)^x$\u306e\u5024\u3092\u6c42\u3081\u308b\u3053\u3068\u300d\u3092\u8ab2\u984c\u3068\u8a2d\u5b9a\u3057\u307e\u3059\u3002<\/p>\n
\u524d\u7bc0\u3067\u5b9a\u3081\u305f\u8ab2\u984c\u3092\u89e3\u304f\u306b\u3042\u305f\u308a\u3001(\\ref{eq:elambdamu})\u5f0f\u306e\u53f3\u8fba\u306e\u5f0f\u304b\u3089(\\ref{eq:napiersconstantdefinition})\u5f0f\u306e\u53f3\u8fba\u306e\u5f0f\u306e\u5f62\u3092\u5f0f\u5909\u5f62\u306a\u3069\u306b\u3088\u3063\u3066\u4f5c\u308a\u3060\u3059\u3053\u3068\u3092\u6700\u521d\u306e\u76ee\u6a19\u3068\u3057\u307e\u3059\u3002<\/p>\n
\u6700\u521d\u306b\u3001(\\ref{eq:elambdamu})\u5f0f\u306e\u53f3\u8fba\u306e\u62ec\u5f27\u5185\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5909\u5f62\u3057\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n
\\begin{align}
\n1-\\frac{\\mu}{x} &= \\frac{x-\\mu}{x} \\nonumber\\cr
\n&= \\frac{1}{\\dfrac{x}{x-\\mu}} \\nonumber\\cr
\n&= \\frac{1}{1+\\dfrac{\\mu}{x-\\mu}} \\label{eq:fraccalc}
\n\\end{align}<\/p>\n
\u6b21\u306b(\\ref{eq:fraccalc})\u5f0f\u3092(\\ref{eq:elambdamu})\u5f0f\u306e\u53f3\u8fba\u306b\u4ee3\u5165\u3057\u3001$x-\\mu=u$\u3068\u304a\u304d\u307e\u3059\u3002<\/p>\n
\u3059\u308b\u3068\u3001$x \\to \\infty$\u306e\u3068\u304d\u3001$u$\u306b\u3064\u3044\u3066\u3082$u \\to \\infty$\u3068\u306a\u308b\u3053\u3068\u304b\u3089\u2026<\/p>\n
\\begin{align}
\n\\lim_{x\\to\\infty}\\frac{1}{\\left(1+\\dfrac{\\mu}{x-\\mu}\\right)^x}&= \\lim_{u\\to\\infty}\\frac{1}{\\left(1+\\dfrac{\\mu}{u}\\right)^{u+\\mu}}\\nonumber\\cr
\n&= \\lim_{u\\to\\infty}\\frac{1}{\\left(1+\\dfrac{\\mu}{u}\\right)^u}\\frac{1}{\\left(1+\\dfrac{\\mu}{u}\\right)^{\\mu}}\\label{eq:firstform}
\n\\end{align}<\/p>\n
\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n
(\\ref{eq:firstform})\u5f0f\u306e\u53f3\u8fba\u306e\u7a4d\u306e\u7b2c1\u56e0\u5b50\u306b\u306f(\\ref{eq:elambda})\u5f0f\u3092\u9069\u7528\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u307e\u305f\u3001\u7b2c2\u56e0\u5b50\u306b\u3064\u3044\u3066\u306f$u \\to \\infty$\u306e\u6642\u306b$1+\\dfrac{\\mu}{u}$\u304c1\u306b\u53ce\u675f\u3057\u3001\u304b\u3064$\\mu$\u306f\u5b9a\u6570\u306a\u306e\u3067\u3001\u7b2c2\u56e0\u5b50\u5168\u4f53\u3068\u3057\u3066\u30821\u306b\u53ce\u675f\u3057\u307e\u3059\u3002<\/p>\n
\u3057\u305f\u304c\u3063\u3066\u2026<\/p>\n
\\begin{align}
\n\\lim_{u\\to\\infty}\\frac{1}{\\left(1+\\dfrac{\\mu}{u}\\right)^u}\\frac{1}{\\left(1+\\dfrac{\\mu}{u}\\right)^{\\mu}}&= \\frac{1}{e^{\\mu}}\\cdot 1 \\nonumber\\cr
\n&= e^{-\\mu} \\label{eq:emu}
\n\\end{align}<\/p>\n
\u3068\u306a\u308a\u3001<\/p>\n
\\begin{align}
\n\\lim_{x\\to\\infty}\\left(1-\\frac{\\mu}{x}\\right)^x &=e^{-\\mu}\\quad(\\mu \\gt 0) \\label{eq:emufinal}
\n\\end{align}<\/p>\n
\u3068\u306a\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002$\\blacksquare$<\/p>\n
(\\ref{eq:emufinal})\u5f0f\u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u304c\u793a\u305b\u305f\u306e\u3067\u3001(\\ref{eq:elambda})\u5f0f\u306f$\\lambda \\lt 0$\u306e\u5834\u5408\u3067\u3082\u6210\u308a\u7acb\u3064\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3057\u305f\u3002<\/p>\n
\u307e\u305f\u3001$\\lambda = 0$\u306e\u5834\u5408\u306b\u3064\u3044\u3066\u3082$e^0 = 1$\u3068\u8a08\u7b97\u3067\u304d\u308b\u3053\u3068\u304b\u3089(\\ref{eq:elambda})\u5f0f\u306f\u6210\u308a\u7acb\u3064\u306e\u3067\u3001(\\ref{eq:elambda})\u5f0f\u306f\u3059\u3079\u3066\u306e\u5b9f\u6570$\\lambda$\u306b\u304a\u3044\u3066\u6210\u7acb\u3059\u308b\u3053\u3068\u3082\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n
\u2026\u3068\u3001\u3053\u3053\u307e\u3067\u8003\u5bdf\u3057\u3066\u304d\u305f\u3068\u3053\u308d\u3067\u3001\u300c(\\ref{eq:elambda})\u5f0f\u304c\u3059\u3079\u3066\u306e\u5b9f\u6570$\\lambda$\u306b\u304a\u3044\u3066\u6210\u308a\u7acb\u3064\u306e\u306a\u3089\u3001\u305d\u308c\u3092\u305d\u306e\u307e\u307e\u6697\u8a18\u3057\u3066\u304a\u3051\u3070\u3044\u3044\u3058\u3083\u3093\u3002\u300d\u3068\u7d50\u8ad6\u3065\u3051\u305f\u304f\u306a\u308b\u3068\u3053\u308d\u3067\u3059\u304c\u3001\u6570\u5b66\u7cfb\u306e\u4e38\u6697\u8a18\u3057\u305f\u77e5\u8b58\u306f\u3044\u305a\u308c\u5fd8\u308c\u3066\u3057\u307e\u3046\u3070\u304b\u308a\u304b\u3001\u4e38\u6697\u8a18\u81ea\u4f53\u304c\u9577\u7d9a\u304d\u3057\u306a\u3044\u3082\u306e\u3067\u3059\u3002<\/p>\n
\u3080\u3057\u308d\u3001\u6697\u8a18\u3059\u308b\u306e\u306f\u30cd\u30a4\u30d4\u30a2\u6570\u306e\u5b9a\u7fa9\u3067\u3042\u308b(\\ref{eq:napiersconstantdefinition})\u5f0f\u3060\u3051\u306b\u3057\u3066\u304a\u3044\u3066\u3001\u4ed6\u306e\u5f0f\u306f\u3059\u3079\u3066(\\ref{eq:napiersconstantdefinition})\u5f0f\u304b\u3089\u5c0e\u3051\u308b\u3088\u3046\u306b\u3057\u3066\u304a\u3044\u305f\u65b9\u304c\u3001<\/p>\n
\u300c(\\ref{eq:elambda})\u5f0f\u304c\u3059\u3079\u3066\u306e\u5b9f\u6570$\\lambda$\u306b\u304a\u3044\u3066\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u8a3c\u660e\u305b\u3088\u3002\u300d<\/p>\n
\u306e\u3088\u3046\u306a\u8a3c\u660e\u554f\u984c\u306b\u3082\u5bfe\u5fdc\u3067\u304d\u308b\u306e\u3067\u3001\u3088\u308a\u610f\u5473\u304c\u3042\u308b\u5b66\u3073\u306b\u306a\u308b\u3068\u8003\u3048\u3066\u304a\u308a\u307e\u3059\u3002<\/p>\n
\u3053\u306e\u8a18\u4e8b\u306f\u4ee5\u4e0a\u3067\u3059\u3002<\/p>\n
(\\ref{eq:elambda})\u5f0f\u3067$u = \\dfrac{x}{\\lambda}$\u3068\u304a\u304f\u3068\u3001$x \\to \\infty$\u306e\u3068\u304d\u3001$u$\u306b\u3064\u3044\u3066\u3082$u \\to \\infty$\u3068\u306a\u308b\u306e\u3067\u2026<\/p>\n
\\begin{align}
\n\\lim_{x\\to\\infty}\\left(1+\\frac{\\lambda}{x}\\right)^x&=\\lim_{u\\to\\infty}\\left(1+\\frac{1}{u}\\right)^{u\\lambda}\\nonumber\\cr
\n&=\\left[\\lim_{u\\to\\infty}\\left(1+\\frac{1}{u}\\right)^{u}\\right]^{\\lambda}\\nonumber\\cr
\n&=e^{\\lambda} \\label{eq:elambdafinal}
\n\\end{align}<\/p>\n
\u306e\u3088\u3046\u306b\u5909\u5f62\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"
\u306f\u3058\u3081\u306b \u4e45\u3005\u306b\u8a08\u7b97\u7528\u7d19\u306b\u66f8\u3044\u305f\u3053\u3068\u3092\u305d\u306e\u307e\u307e\u516c\u958b\u3059\u308b\u3060\u3051\u306e\u8a18\u4e8b\u3092\u304a\u9001\u308a\u3059\u308b\u6b21\u7b2c\u3067\u3054\u3056\u3044\u307e\u3059\u3002 \u8ab2\u984c \u8ab2\u984c\u306e\u8a2d\u5b9a \u30cd\u30a4\u30d4\u30a2\u6570$e$\u306e\u5b9a\u7fa9\u5f0f\u306e\u4e00\u3064\u3067\u3042\u308b \\begin{align} \\lim_{x\\to\\infty}\\left(1+\\frac{1}{x}\\right)^x&= e \\label{eq:napiersconstantdefinition} \\end{align} \u3092\u4f7f\u3046\u3068\u3001$\u2026 Read More »<\/a><\/span><\/p>\n","protected":false},"author":1,"featured_media":12615,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[13],"tags":[],"class_list":["post-12622","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\/12622","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=12622"}],"version-history":[{"count":9,"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/posts\/12622\/revisions"}],"predecessor-version":[{"id":12631,"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/posts\/12622\/revisions\/12631"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/pandanote.info\/index.php?rest_route=\/wp\/v2\/media\/12615"}],"wp:attachment":[{"href":"https:\/\/pandanote.info\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=12622"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/pandanote.info\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=12622"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/pandanote.info\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=12622"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}