[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/all2jp\/wiki11\/archives\/19743#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/all2jp\/wiki11\/archives\/19743","headline":"Eulersche\u756a\u53f7-Wikipedia","name":"Eulersche\u756a\u53f7-Wikipedia","description":"before-content-x4 eulersche\u756a\u53f7 \u3001\u30b7\u30f3\u30dc\u30eb\u3067 \u305d\u3046\u3067\u3059 {displaystyle e} after-content-x4 \u8a18\u8f09\u3055\u308c\u3066\u3044\u308b\u306e\u306f\u3001\u5206\u6790\u5168\u4f53\u3068\u6570\u5b66\u306e\u3059\u3079\u3066\u306e\u30b5\u30d6\u30a8\u30ea\u30a2\u3001\u7279\u306b\u5dee\u52d5\u304a\u3088\u3073\u7a4d\u5206\u8a08\u7b97\u3067\u3082\u3001\u77f3\u4f9d\u5b58\u75c7\uff08\u7d44\u307f\u5408\u308f\u305b\u3001\u6b63\u898f\u5206\u5e03\uff09\u306b\u304a\u3044\u3066\u4e2d\u5fc3\u7684\u306a\u5f79\u5272\u3092\u679c\u305f\u3057\u3066\u3044\u308b\u5b9a\u6570\u3067\u3059\u3002\u3042\u306a\u305f\u306e\u6570\u5024\u306f\u3067\u3059 after-content-x4 \u305d\u3046\u3067\u3059 = 2.718 28 18284 59045 23536 02874 71352 66249 77572","datePublished":"2019-06-09","dateModified":"2019-06-09","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/all2jp\/wiki11\/archives\/author\/lordneo#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/all2jp\/wiki11\/archives\/author\/lordneo","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/44a4cee54c4c053e967fe3e7d054edd4?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/44a4cee54c4c053e967fe3e7d054edd4?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/e\/ea\/Disambig-dark.svg\/25px-Disambig-dark.svg.png","url":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/e\/ea\/Disambig-dark.svg\/25px-Disambig-dark.svg.png","height":"19","width":"25"},"url":"https:\/\/wiki.edu.vn\/all2jp\/wiki11\/archives\/19743","wordCount":31402,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4 eulersche\u756a\u53f7 \u3001\u30b7\u30f3\u30dc\u30eb\u3067 \u305d\u3046\u3067\u3059 {displaystyle e}        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4\u8a18\u8f09\u3055\u308c\u3066\u3044\u308b\u306e\u306f\u3001\u5206\u6790\u5168\u4f53\u3068\u6570\u5b66\u306e\u3059\u3079\u3066\u306e\u30b5\u30d6\u30a8\u30ea\u30a2\u3001\u7279\u306b\u5dee\u52d5\u304a\u3088\u3073\u7a4d\u5206\u8a08\u7b97\u3067\u3082\u3001\u77f3\u4f9d\u5b58\u75c7\uff08\u7d44\u307f\u5408\u308f\u305b\u3001\u6b63\u898f\u5206\u5e03\uff09\u306b\u304a\u3044\u3066\u4e2d\u5fc3\u7684\u306a\u5f79\u5272\u3092\u679c\u305f\u3057\u3066\u3044\u308b\u5b9a\u6570\u3067\u3059\u3002\u3042\u306a\u305f\u306e\u6570\u5024\u306f\u3067\u3059        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4\u305d\u3046\u3067\u3059 = 2.718 28 18284 59045 23536 02874 71352 66249 77572 47093 69995 &#8230; {displaystyle e = 2 {\u3001} 71828,18284,59045,23536,02874,71352,66249,77572,47093,69995\u3001\u30c9\u30c3\u30c8} [\u521d\u3081] \u305d\u3046\u3067\u3059 {displaystyle e} \u306f\u8d85\u8d8a\u7684\u3067\u3042\u308a\u3001\u3057\u305f\u304c\u3063\u3066\u4e0d\u5408\u7406\u306a\u5b9f\u6570\u3067\u3059\u3002\u3053\u308c\u306f\u3001\u81ea\u7136\u5bfe\u6570\u3068\uff08\u81ea\u7136\u306a\uff09\u6307\u6570\u95a2\u6570\u306e\u57fa\u790e\u3067\u3059\u3002\u9069\u7528\u3055\u308c\u305f\u6570\u5b66\u3001\u6307\u6570\u95a2\u6570\u3001\u3057\u305f\u304c\u3063\u3066 \u305d\u3046\u3067\u3059 {displaystyle e} \u653e\u5c04\u6027\u5d29\u58ca\u3084\u81ea\u7136\u6210\u9577\u306a\u3069\u306e\u30d7\u30ed\u30bb\u30b9\u306e\u8aac\u660e\u306b\u304a\u3051\u308b\u91cd\u8981\u306a\u5f79\u5272\u3002 \u306e\u591a\u6570\u306e\u540c\u7b49\u306e\u5b9a\u7fa9\u304c\u3042\u308a\u307e\u3059        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4\u305d\u3046\u3067\u3059 {displaystyle e} \u3001\u6700\u3082\u3088\u304f\u77e5\u3089\u308c\u3066\u3044\u308b\u306e\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3059\u3002 \u305d\u3046\u3067\u3059 = \u521d\u3081 + 11+ 11\u22c52+ 11\u22c52\u22c53+ 11\u22c52\u22c53\u22c54+ \u22ef = \u2211 k=0\u221e1k!{displaystyle e = 1+{frac {1} {1}}+{frac {1} {1cdot 2}}+{frac {1} {1cdot 2cdot 3}}+{frac {1} {1} {1cdot 2cdot 3cdot 4}}}}}}}+dotb}+ {frac {1} {k\uff01}}} \u3053\u306e\u756a\u53f7\u306f\u3001\u30b9\u30a4\u30b9\u306e\u6570\u5b66\u8005\u30ec\u30aa\u30f3\u30cf\u30eb\u30c8\u30aa\u30a4\u30e9\u30fc\u306b\u3061\u306a\u3093\u3067\u540d\u4ed8\u3051\u3089\u308c\u307e\u3057\u305f\u3001 [2] \u306e\u591a\u6570\u306e\u7279\u6027 \u305d\u3046\u3067\u3059 {displaystyle e} \u8aac\u660e\u3055\u308c\u305f\u3002\u6642\u3005\u5f7c\u5973\u306f\u30b9\u30b3\u30c3\u30c8\u30e9\u30f3\u30c9\u306e\u6570\u5b66\u8005\u30b8\u30e7\u30f3\u30fb\u30cd\u30a4\u30d4\u30a2\u306e\u5f8c\u306b \u30cd\u30a4\u30d4\u30a2\u306e\u5b9a\u6570 \uff08\u307e\u305f Nepersche\u306f\u6c38\u4e45\u306b \uff09 \u5c02\u7528\u3002\u6570\u5b66\u306e\u6700\u3082\u91cd\u8981\u306a\u5b9a\u6570\u306e1\u3064\u3067\u3059\u3002 \u30aa\u30a4\u30e9\u30fc\u756a\u53f7\u306e\u56fd\u969b\u7684\u306a\u65e5\u304c\u3042\u308a\u307e\u3059 \u305d\u3046\u3067\u3059 {displaystyle e} \u3002\u30c9\u30a4\u30c4\u306e\u3088\u3046\u306b\u3001\u305d\u306e\u6708\u524d\uff081\u670827\u65e5\uff09\u306e\u524d\u65e5\uff081\u670827\u65e5\uff09\u306b\u3042\u308b\u56fd\u3067\u306f\u30011\u670827\u65e5\u3067\u3059\u3002 [3] 2\u67087\u65e5\u306b1\u65e5\u524d\uff082\/7\uff09\u304c\u66f8\u304b\u308c\u305f\u56fd\u3067\u3002 \u4eba\u6570\u3001\u500b\u6570\u3001\u7dcf\u6570 \u305d\u3046\u3067\u3059 {displaystyle e} Leonhard Euler\u306b\u3088\u3063\u3066\u6b21\u306e\u30b7\u30ea\u30fc\u30ba\u3067\u5b9a\u7fa9\u3055\u308c\u307e\u3057\u305f\u3002 [4] e=1+11+11\u22c52+11\u22c52\u22c53+11\u22c52\u22c53\u22c54+\u22ef=10!+11!+12!+13!+14!+\u22ef=\u2211k=0\u221e1k!{displaystyle {begin {aligned} e\uff06= 1+{frac {1} {1}}+{frac {1} {1cdot 2}}+{frac {1} {1cdot 2cdot 3}}}+{frac {1} {1cdot 2cdot 3cdot 3cdot 3} {1cdot 2cdot 3cdot 4 ac {1} {0\uff01}}+{frac {1} {1\uff01}}+{frac {1} {2\uff01}}+{frac {1} {3\uff01}}+{frac {1} {4\uff01}}+dotsb \\\uff06= sum _ {k = 0} {k = 0} {k = 0} {k = 0} { \uff01}} \\ end {aligned}}} \u305f\u3081\u306b k \u2208 n 0 {displaystyle kin mathbb {n} _ {0}} \u5165\u3063\u3066\u3044\u307e\u3059 k \uff01 {displaystyle k\uff01} \u306e\u6559\u54e1 k {displaystyle k} \u3001\u305d\u306e\u5834\u5408 0&#8243;>\u88fd\u54c1 k \uff01 = \u521d\u3081 de 2 de &#8230; de k {displaystyle k\uff01= 1cdot 2cdot ldot cdot k} \u306e\u81ea\u7136\u6570 \u521d\u3081 {displaystyle1} \u305d\u308c\u307e\u3067 k {displaystyle k} \u3001 \u305d\u306e\u9593 0 \uff01 \uff1a= \u521d\u3081 {displaystyle 0\uff01\uff1a= 1} \u5b9a\u7fa9\u3055\u308c\u3066\u3044\u307e\u3059\u3002 \u30aa\u30a4\u30e9\u30fc\u304c\u3059\u3067\u306b\u5b9f\u8a3c\u3057\u3066\u3044\u308b\u3088\u3046\u306b\u3001\u3042\u306a\u305f\u306f\u30aa\u30a4\u30e9\u30fc\u30b7\u30e5\u756a\u53f7\u3092\u53d6\u5f97\u3057\u307e\u3059 \u305d\u3046\u3067\u3059 {displaystyle e} \u6a5f\u80fd\u7684\u306a\u5236\u9650\u3068\u3057\u3066\u3082\u3002 [5] \u4eba\u6570\u3001\u500b\u6570\u3001\u7dcf\u6570 \u305d\u3046\u3067\u3059 {displaystyle e} \u30a8\u30d4\u30bd\u30fc\u30c9\u306e\u9650\u754c\u3067\u3082\u3042\u308a\u307e\u3059 \uff08 a n \uff09\uff09 n \u2208 N{displaystyle\uff08a_ {n}\uff09_ {nin mathbb {n}}}} \u3068 a n \uff1a= \uff08 1+1n\uff09\uff09 n {displaystyle a_ {n}\uff1a= left\uff081+ {frac {1} {n}}\u53f3\uff09^{n}}} \u66f8\u304b\u308c\u308b\uff1a \u305d\u3046\u3067\u3059 = \u30ea\u30e0 n\u2192\u221e(1+1n)n{displaystyle e = lim _ {nto infty}\u5de6\uff081+ {frac {1} {n}}\u53f3\uff09^{n}} \u3053\u308c\u306f\u305d\u308c\u306b\u57fa\u3065\u3044\u3066\u3044\u307e\u3059 \u305d\u3046\u3067\u3059 = exp \u2061 \uff08 \u521d\u3081 \uff09\uff09 = \u305d\u3046\u3067\u3059 1{displaystyle e = exp\uff081\uff09= e^{1}} \u8a72\u5f53\u3059\u308b\u3001 \u305d\u3046\u3067\u3059 {displaystyle e} \u3057\u305f\u304c\u3063\u3066\u3001\u6307\u6570\u95a2\u6570\u306e\u95a2\u6570\u5024\uff08\u307e\u305f\u306f\u300d \u305d\u3046\u3067\u3059 {displaystyle e} -Function\u300d\uff09\u30dd\u30a4\u30f3\u30c8\u3067 \u521d\u3081 {displaystyle1} \u306f\u3002\u4e0a\u8a18\u306e\u30b7\u30ea\u30fc\u30ba\u30d7\u30ec\u30bc\u30f3\u30c6\u30fc\u30b7\u30e7\u30f3 \u305d\u3046\u3067\u3059 {displaystyle e} \u3053\u308c\u306b\u95a2\u9023\u3057\u3066\u3001\u958b\u767a\u6a5f\u95a2\u306e\u5468\u308a\u306e\u6307\u6570\u95a2\u6570\u306e\u30c6\u30a4\u30e9\u30fc\u30b7\u30ea\u30fc\u30ba\u304c\u898b\u3064\u304b\u308b\u3068\u3044\u3046\u4e8b\u5b9f\u306b\u8d77\u56e0\u3057\u307e\u3059 0 {displaystyle 0} \u30dd\u30a4\u30f3\u30c8\u3067 \u521d\u3081 {displaystyle1} \u8a55\u4fa1\u3055\u308c\u305f\u3002 \u30aa\u30a4\u30e9\u30fc\u756a\u53f7\u306e\u5b9a\u7fa9\u3078\u306e\u4ee3\u66ff\u30a2\u30af\u30bb\u30b9\u306f\u3001\u305f\u3068\u3048\u3070\u9014\u4e2d\u3067\u30a4\u30f3\u30bf\u30fc\u30d0\u30eb\u30dc\u30c3\u30af\u30b9\u306b\u95a2\u3059\u308b\u3082\u306e\u3067\u3059\u3002 \u7121\u9650\u884c\u306e\u7406\u8ad6\u3068\u9069\u7528 Konrad Knopp\u306b\u3088\u3063\u3066\u793a\u3055\u308c\u3066\u3044\u307e\u3059\u3002\u305d\u306e\u5f8c\u3001\u3059\u3079\u3066\u306e\u4eba\u306b\u5f53\u3066\u306f\u307e\u308a\u307e\u3059 n \u2208 n {displaystyle nin mathbb {n}} \uff1a [6] (1+1n)n< \u305d\u3046\u3067\u3059 < (1+1n)n+1{displaystyle\u5de6\uff081+ {frac {1} {n}}\u53f3\uff09^{n} \u222b 11+1n1xd \u30d0\u30c4 \u2264 1nde \u521d\u3081 {displaystyle {frac {1} {n}} cdot {frac {n} {n+1}} leq int _ {1}^{frac {1}} {n}}} {frac {1} {x}}}} {d} {d} {d} {d} {d} {d} {n} } \u21d4 1nde nn+1\u2264 ln \u2061 \uff08 1+1n\uff09\uff09 \u2264 1nde \u521d\u3081 {displaystyle leftrightarrow {frac {1} {n}} cdot {frac {n} {n+1}} leq ln left\uff081+ {frac {1} {n}} right\uff09leq {frac {1} {n}} cdot 1}}  \uff08\u7a4d\u5206\u306e\u89e3\u6c7a\u7b56\uff09 \u21d4 nn+1\u2264 n de ln \u2061 \uff08 1+1n\uff09\uff09 \u2264 \u521d\u3081 {displaystyle leftrightarrow {frac {n} {n+1}} leq ncdot ln left\uff081+ {frac {1} {n}}\u53f3\uff09leq 1}}  \uff08n\u3068\u306e\u4e57\u7b97\uff09 \u21d4 nn+1\u2264 ln \u2061 (1+1n)n\u2264 \u521d\u3081 {displaystyle leftrightarrow {frac {n} {n+1}} leq ln left\uff081+ {frac {1} {n}}\u53f3\uff09^{n} leq 1}  \uff08\u5bfe\u6570\u6cd5\u306e\u9069\u7528\uff09 \u21d4 \u30ea\u30e0 n\u2192\u221enn+1\u2264 \u30ea\u30e0 n\u2192\u221e\uff08 ln\u2061(1+1n)n\uff09\uff09 \u2264 \u521d\u3081 {displaystyle leftrightarrow lim _ {nto infty} {frac {n} {n+1}} leq lim _ {nto inf infty}\u5de6\uff081+ {1} {n}}\u53f3\uff09^{n}\u53f31}  \uff08\u5024\u306e\u5f62\u6210\u3092\u5236\u9650\uff09 \u21d4 ln \u2061 \uff08 limn\u2192\u221e(1+1n)n\uff09\uff09 = \u521d\u3081 {displaystyle leftrightarrow ln\u5de6\uff08lim _ {nto infty}\u5de6\uff081+ {frac {1} {n}}\u53f3\uff09^{n}\u53f3\uff09= 1}  \uff08\u5bfe\u6570\u95a2\u6570\u306e\u30b9\u30bf\u30f3\u30c9\u30cd\u30b9\uff09 \u21d4 \u305d\u3046\u3067\u3059 = \u30ea\u30e0 n\u2192\u221e(1+1n)n{displaystyle leftrightarrow e = lim _ {nto infty}\u5de6\uff081+ {frac {1} {n}}\u53f3\uff09^{n}}  \uff08\u53cd\u8ee2\u95a2\u6570\u306e\u6307\u6570\u95a2\u6570\uff09 \u30aa\u30a4\u30e9\u30fc\u756a\u53f7\u306e\u6b74\u53f2 \u305d\u3046\u3067\u3059 {displaystyle e} 16\u4e16\u7d00\u306b\u306f3\u3064\u306e\u554f\u984c\u9818\u57df\u304c\u3042\u308b16\u4e16\u7d00\u306b\u59cb\u307e\u308a\u3001\u305d\u306e\u4e2d\u306b\u6570\u5b66\u8005\u306b\u8fd1\u3065\u3044\u305f\u6570\u5b57\u304c\u73fe\u308c\u307e\u3057\u305f\u3002 \u305d\u3046\u3067\u3059 {displaystyle e} \u3068\u547c\u3070\u308c\u3066\u3044\u307e\u3057\u305f\uff1a John Napier\u3068JostB\u00fcrgi\u306b\u3088\u308b\u5bfe\u6570\u30dc\u30fc\u30c9\u306e\u5bfe\u6570\u306e\u57fa\u790e\u3068\u3057\u3066\u3002\u4e21\u65b9\u3068\u3082\u3001\u30de\u30a4\u30b1\u30eb\u30fb\u30b9\u30c6\u30a3\u30d5\u30a7\u30eb\u306e\u9332\u97f3\u306816\u4e16\u7d00\u306e\u4ed6\u306e\u6570\u5b66\u8005\u306e\u7d50\u679c\u306b\u3088\u308b\u30a2\u30a4\u30c7\u30a2\u3092\u4f7f\u7528\u3057\u3066\u3001\u4e92\u3044\u306b\u72ec\u7acb\u3057\u3066\u30c6\u30fc\u30d6\u30eb\u3092\u958b\u767a\u3057\u3066\u3044\u307e\u3057\u305f\u3002 B\u00fcrgi\u306f1620\u5e74\u306b\u300c\u7b97\u8853\u3068\u5e7e\u4f55\u5b66\u7684\u30d7\u30ed\u30b0\u30e9\u30e0\u306e\u30bf\u30d6\u30fc\u30eb\u300d\u3092\u516c\u958b\u3057\u307e\u3057\u305f\u3002\u5f7c\u306e\u5bfe\u6570\u30b7\u30b9\u30c6\u30e0\u306e\u57fa\u790e\u3068\u3057\u3066\u3001B\u00fcrgi\u306f\u660e\u3089\u304b\u306b\u672c\u80fd\u7684\u306b\u8fd1\u304f\u306e\u6570\u5b57\u3092\u4f7f\u7528\u3057\u3066\u3044\u307e\u3059 \u305d\u3046\u3067\u3059 {displaystyle e} \u5618\u3002 Napier\u306f1614\u5e74\u306b\u5f7c\u306e\u300cMirifici logarithmorum canonis\u8a18\u8ff0\u300d\u3092\u516c\u958b\u3057\u30011\u3064\u3092\u4f7f\u7528\u3057\u307e\u3059 \u521d\u3081 \/ \u305d\u3046\u3067\u3059 {displaystyle1\/e} \u6bd4\u4f8b\u57fa\u6e96\u3002 [8] \u5bfe\u6570\u30c6\u30fc\u30d6\u30eb\u306e\u52a9\u3051\u3092\u501f\u308a\u3066\u3001Napier\u3068B\u00fcrgi\u306f\u3001\u81a8\u5927\u306a\u8acb\u6c42\u66f8\u3092\u3088\u308a\u7c21\u5358\u3067\u6642\u9593\u3092\u77ed\u7e2e\u3059\u308b\u305f\u3081\u306b\u3001\u5897\u52a0\u3092\u52a0\u7b97\u306b\u8ffd\u8de1\u3057\u305f\u3044\u3068\u8003\u3048\u3066\u3044\u307e\u3057\u305f\u3002 \u8907\u5229\u306e\u30b7\u30fc\u30b1\u30f3\u30b9\u306e\u5236\u9650\u3068\u3057\u3066\u3002 1669\u5e74\u3001\u30e4\u30b3\u30d6\u30fb\u30d9\u30eb\u30cc\u30fc\u30ea\u306f\u30bf\u30b9\u30af\u3092\u63d0\u4f9b\u3057\u307e\u3057\u305f\uff1a\u300c\u5e74\u9593\u5229\u606f\u306e\u6bd4\u4f8b\u90e8\u5206\u304c\u500b\u3005\u306e\u77ac\u9593\u306b\u9996\u90fd\u306b\u6253\u3061\u306e\u3081\u3055\u308c\u305f\u3068\u3044\u3046\u5229\u606f\u306e\u91d1\u984d\u304c\u5275\u51fa\u3055\u308c\u307e\u3057\u305f\u3002 [9] Bernoulli\u306f\u3001\u500b\u3005\u306e\u77ac\u9593\u304c\u77ed\u304f\u3066\u77ed\u304f\u306a\u3063\u3066\u3044\u308b\u5951\u7d04\u304c\u958b\u59cb\u984d\u306e\u500d\u6570\u3092\u9054\u6210\u3067\u304d\u308b\u304b\u3069\u3046\u304b\u3092\u5c0b\u306d\u3001\u89e3\u6c7a\u7b56\u3068\u3057\u3066\u3001\u79c1\u305f\u3061\u306f\u4eca\u65e5\u306eEulersche\u756a\u53f7\u3068\u3057\u3066\u306e\u6570\u306b\u9054\u3057\u307e\u3059 \u305d\u3046\u3067\u3059 {displaystyle e} \u77e5\u308b\u3002 [\u5341] \u7121\u9650\u306e\u30b7\u30ea\u30fc\u30ba\u3068\u3057\u3066\uff08\u30da\u30eb\u30b2\u306e\u30a2\u30dd\u30ed\u30cb\u30aa\u306e\u53cc\u66f2\u7dda\u306e\u9818\u57df\uff09\u3002\u305d\u308c\u306f\uff08\u4eca\u65e5\u306e\u8a00\u8a9e\u3067\uff09\u30cf\u30a4\u30d1\u30fc\u30d9\u30eb\u306e\u4e0b\u306e\u9818\u57df\u306e\u554f\u984c\u3067\u3057\u305f \u30d0\u30c4 \u3068 = \u521d\u3081 {displaystyle xy = 1} \u304b\u3089 \u30d0\u30c4 = \u521d\u3081 {displaystyle x = 1} \u53f3\u306b\u4f38\u3073\u307e\u3059\u3002\u3053\u308c\u306f\u3001\u30e6\u30cb\u30c3\u30c8\u30b9\u30af\u30a8\u30a2\u306e\u9762\u7a4d\u3068\u540c\u3058\u5927\u304d\u3055\u3067\u3059\u3002\u30d5\u30e9\u30f3\u30c9\u30eb\u306e\u6570\u5b66\u8005\u30b0\u30ec\u30b4\u30ef\u30fc\u30eb\u30fb\u30c7\u30fb\u30b5\u30f3\u30fb\u30d3\u30f3\u30bb\u30f3\u30c8\uff08\u30e9\u30c6\u30f3\u5316\u30b0\u30ec\u30b4\u30ea\u30a6\u30b9\u30fb\u30b5\u30f3\u30af\u30c8\u30fb\u30f4\u30a3\u30f3\u30bb\u30f3\u30c6\u30a3\u30fc\u30ce\uff09\u306f\u3001\u79c1\u305f\u3061\u304c\u4eca\u65e5\u547c\u3073\u3001\u81ea\u7136\u5bfe\u6570\u3092\u4f7f\u7528\u3057\u3066\u30bd\u30ea\u30e5\u30fc\u30b7\u30e7\u30f3\u306e\u305f\u3081\u306e\u6a5f\u80fd\u3092\u958b\u767a\u3057\u307e\u3057\u305f ln {displaystyle ln} \u8aac\u660e\u3002\u5f7c\u306f\u3001\u4eca\u65e5\u79c1\u305f\u3061\u304c\u5bfe\u6570\u306e\u6a5f\u80fd\u65b9\u7a0b\u5f0f\u3068\u547c\u3076\u65b9\u7a0b\u5f0f\u3092\u542b\u3080\u8208\u5473\u6df1\u3044\u7279\u6027\u3092\u767a\u898b\u3057\u307e\u3057\u305f\u3002 [11] \u5f7c\u304c\u3053\u306e\u5bfe\u6570\u306e\u6839\u62e0\u304c\u5f8c\u306e\u6570\u5b57\u3067\u3042\u308b\u3053\u3068\u3092\u5f7c\u304c\u77e5\u3063\u3066\u3044\u305f\u304b\u3069\u3046\u304b\u306f\u78ba\u304b\u3067\u306f\u3042\u308a\u307e\u305b\u3093 \u305d\u3046\u3067\u3059 {displaystyle e} \u3068\u547c\u3070\u308c\u3066\u3044\u307e\u3057\u305f\u3002\u3053\u308c\u306f\u3001\u5f7c\u306e\u4f5c\u54c1\u304c\u516c\u958b\u3055\u308c\u305f\u5f8c\u306b\u306e\u307f\u6c17\u3065\u304d\u307e\u3057\u305f\u3002 [12\u756a\u76ee] \u6700\u65b0\u306e\u72b6\u614b\u3067\u306f\u3001\u5f7c\u306e\u5b66\u751f\u3067\u3042\u308a\u5171\u8457\u8005\u3067\u3042\u308bAlphonse Antonio de Sarasa\u304c\u5bfe\u6570\u95a2\u6570\u3068\u306e\u3064\u306a\u304c\u308a\u3092\u63d0\u793a\u3057\u307e\u3057\u305f\u3002DeSarasa\u3092\u901a\u3058\u3066Saint-Vincent\u306e\u30a2\u30a4\u30c7\u30a2\u306espread\u5ef6\u3092\u6271\u3046\u30a8\u30c3\u30bb\u30a4\u3067\u306f\u3001\u300c\u30ed\u30eb\u30ac\u30ea\u30c3\u30c8\u3068\u30cf\u30a4\u30d1\u30fc\u30d9\u30eb\u306e\u95a2\u4fc2\u306f\u3001\u30b5\u30f3\u30f4\u30a3\u30f3\u30b7\u30f3\u30c8\u306b\u3088\u3063\u3066\u3042\u3089\u3086\u308b\u7279\u6027\u306b\u898b\u3089\u308c\u305f\u300d\u3068\u8a00\u308f\u308c\u3066\u3044\u307e\u3059\u3002 [13] \u30cb\u30e5\u30fc\u30c8\u30f3\u3068\u30aa\u30a4\u30e9\u30fc\u3067\u50cd\u304f\u3053\u3068\u3067\u3001 \u305d\u3046\u3067\u3059 {displaystyle e} \u57fa\u790e\u306f\u3067\u3059\u3002 [14] \u30e9\u30a4\u30d7\u30cb\u30c3\u30c4\u306f\u3001\u660e\u3089\u304b\u306b\u3053\u306e\u756a\u53f7\u306b\u624b\u7d19\u3092\u4f7f\u7528\u3057\u305f\u6700\u521d\u306e\u4eba\u7269\u3067\u3057\u305f\u3002 1690\/1\u5e74\u304b\u3089\u306eChristiaan Huygens\u3068\u306e\u901a\u4fe1\u3067\u3001\u5f7c\u306f\u6587\u5b57B\u3092\u52b9\u529b\u306e\u57fa\u790e\u3068\u3057\u3066\u4f7f\u7528\u3057\u307e\u3057\u305f\u3002 [15] \u6587\u5b57\u3092\u4f7f\u7528\u3059\u308b\u305f\u3081\u306b\u4f7f\u7528\u3055\u308c\u308b\u6700\u3082\u521d\u671f\u306e\u30c9\u30ad\u30e5\u30e1\u30f3\u30c8\u3068\u3057\u3066 \u305d\u3046\u3067\u3059 {displaystyle e} \u30ec\u30aa\u30f3\u30cf\u30eb\u30c8\u30fb\u30aa\u30a4\u30e9\u30fc\u306b\u3088\u308b\u3053\u306e\u756a\u53f7\u306b\u306f\u30011731\u5e7411\u670825\u65e5\u304b\u3089\u30aa\u30a4\u30e9\u30fc\u304b\u3089\u30af\u30ea\u30b9\u30c1\u30e3\u30f3\u30fb\u30b4\u30fc\u30eb\u30c9\u30d0\u30c3\u30cf\u3078\u306e\u624b\u7d19\u304c\u9069\u7528\u3055\u308c\u307e\u3059\u3002 [16] \u3055\u3089\u306b\u65e9\u304f\u30011727\u5e74\u307e\u305f\u306f1728\u5e74\u3001\u30aa\u30a4\u30e9\u30fc\u306f\u624b\u7d19\u3092\u59cb\u3081\u307e\u3057\u305f \u305d\u3046\u3067\u3059 {displaystyle e} 1862\u5e74\u306b\u306e\u307f\u516c\u958b\u3055\u308c\u305f\u5927\u7832\u306e\u7206\u767a\u529b\u306b\u3064\u3044\u3066\u3001\u300c\u5b9f\u9a13\u7206\u767a\u6027\u7206\u767a\u6027\u30c8\u30eb\u30e1\u30f3\u30c8\u30fc\u30e9\u30e0\u30cc\u30fc\u30d1\u30fc\u7814\u7a76\u6240\u300d\u306e\u8a18\u4e8b\u300c\u7791\u60f3\u300d\u3067\u4f7f\u7528\u3059\u308b\u305f\u3081\u3002 [17] [18] \u3053\u306e\u624b\u7d19\u3092\u4f7f\u7528\u3059\u308b\u305f\u3081\u306e\u6b21\u306e\u5b89\u5168\u306a\u60c5\u5831\u6e90\u306f\u3001\u30aa\u30a4\u30e9\u30fc\u306e\u4f5c\u54c1\u3067\u3059 \u5206\u6790\u7684\u306b\u66b4\u9732\u3055\u308c\u305f\u6a5f\u68b0\u7684\u307e\u305f\u306f\u611f\u60c5\u7684\u306a\u79d1\u5b662 1736\u5e74\u304b\u3089\u3002 [6] 1748\u5e74\u306b\u516c\u958b\u3055\u308c\u305f\u3082\u306e \u5206\u6790\u306e\u7d39\u4ecb \u30aa\u30a4\u30e9\u30fc\u306f\u518d\u3073\u3053\u306e\u540d\u524d\u3092\u62fe\u3044\u307e\u3059\u3002 [19] \u3053\u306e\u624b\u7d19\u306e\u9078\u629e\u306e\u8a3c\u62e0\u306f\u3042\u308a\u307e\u305b\u3093 \u305d\u3046\u3067\u3059 {displaystyle e} \u5f7c\u306e\u540d\u524d\u306b\u57fa\u3065\u3044\u3066\u3044\u307e\u3059\u3002\u307e\u305f\u3001\u5f7c\u304c\u6307\u6570\u95a2\u6570\u306b\u57fa\u3065\u3044\u3066\u3044\u308b\u306e\u304b\u3001\u305d\u308c\u3068\u3082\u4f7f\u7528\u3055\u308c\u3066\u3044\u308b\u6587\u5b57\u3078\u306e\u5883\u754c\u306e\u5b9f\u969b\u7684\u306a\u8003\u616e\u4e8b\u9805\u306b\u57fa\u3065\u3044\u3066\u3044\u308b\u306e\u304b\u306f\u4e0d\u660e\u3067\u3059\u3002 A\u3001B\u3001c \u307e\u305f d \u4f5c\u308b\u3002\u305f\u3068\u3048\u3070\u3001\u4ed6\u306e\u540d\u524d\u3082\u4f7f\u7528\u3055\u308c\u3066\u3044\u307e\u3057\u305f\u304c c \u30c0\u30ec\u30f3\u30d9\u30fc\u30eb\u30ba\u3067 \u30a2\u30ab\u30c7\u30df\u30fc\u306e\u6b74\u53f2\u3001 \u3082\u3063\u3066\u3044\u308b \u305d\u3046\u3067\u3059 {displaystyle e} \u5f37\u5236\u3002 \u30d5\u30a9\u30fc\u30df\u30e5\u30e9\u30bb\u30c3\u30c8\u306f\u3067\u3059 \u305d\u3046\u3067\u3059 {displaystyle e} DIN 1338\u304a\u3088\u3073ISO 80000-2\u306b\u3088\u308b\u3068\u3001\u6570\u3092\u5909\u6570\u3068\u533a\u5225\u3059\u308b\u305f\u3081\u306e\u659c\u4f53\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002 [20] \u305f\u3060\u3057\u3001\u30b3\u30fc\u30b9\u3082\u5e83\u7bc4\u56f2\u306b\u53ca\u3073\u307e\u3059\u3002 Eulersche\u756a\u53f7 \u305d\u3046\u3067\u3059 {displaystyle e} \u8d85\u8d8a\u7684\u3067\u3059\uff08 \u8a3c\u62e0 \u30c1\u30e3\u30fc\u30eb\u30ba\u30fb\u30d8\u30eb\u30df\u30b9\u30011873\u5e74\u306b\u3088\u308b\u3068\uff09\u3001\u3057\u305f\u304c\u3063\u3066\u975e\u5408\u7406\u7684\u306a\u6570\uff08\u30c1\u30a7\u30fc\u30f3\u30d6\u30ec\u30a4\u30af\u306b\u3088\u308b\u8a3c\u660e \u305d\u3046\u3067\u3059 2 {displaystyle e^{2}} \u3057\u305f\u304c\u3063\u3066 \u305d\u3046\u3067\u3059 {displaystyle e} \u3059\u3067\u306b1737\u5e74\u306b\u30aa\u30a4\u30e9\u30fc\u306b\u3088\u3063\u3066\u3001 [21] \u8a3c\u660e \u8a3c\u660e &#8211;  \u307e\u305f\u306f\u8a18\u4e8b\uff09\u3002\u3060\u304b\u3089\u305d\u308c\u306f\uff08\u305d\u3057\u3066\u56de\u8def\u306e\u6570\u3068\u540c\u69d8\u306b\u305d\u3046\u304b\u3082\u3057\u308c\u307e\u305b\u3093 pi {displaystylepi} Ferdinand von Lindemann 1882\uff09\u306b\u3088\u308b\u3068\u30012\u3064\u306e\u81ea\u7136\u6570\uff08\u4ee3\u6570\u65b9\u7a0b\u5f0f\u306e\u89e3\u6c7a\u7b56\u3068\u3057\u3066\u3082\uff09\u306e\u4f11\u61a9\u3068\u3057\u3066\u3067\u306f\u306a\u304f\u3001\u305d\u306e\u7d50\u679c\u3001\u7121\u9650\u306e\u975e\u5468\u671f\u30d5\u30e9\u30b0\u30e1\u30f3\u30c8\u958b\u767a\u304c\u3042\u308a\u307e\u3059\u3002\u306e\u4e0d\u5408\u7406\u6027 \u305d\u3046\u3067\u3059 {displaystyle e} \u7279\u306b\u3001\u4e0d\u5408\u7406\u306a\u6570\u306e\u5834\u5408\u306f\u3067\u304d\u308b\u3060\u3051\u5c0f\u3055\u30442\u3064\u3067\u3059 \u305d\u3046\u3067\u3059 {displaystyle e} liouvillesch\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002\u304b\u3069\u3046\u304b\u306f\u4e0d\u660e\u3067\u3059 \u305d\u3046\u3067\u3059 {displaystyle e} \u4f55\u3089\u304b\u306e\u6839\u62e0\u304c\u6b63\u5e38\u3067\u3059\u3002 [22] \u30aa\u30a4\u30e9\u30fc\u306e\u30a2\u30a4\u30c7\u30f3\u30c6\u30a3\u30c6\u30a3\u3067 \u305d\u3046\u3067\u3059 i\u22c5\u03c0=  &#8211;  \u521d\u3081 {displaystyle e^{mathrm {i} cdot pi} = -1} \u57fa\u672c\u7684\u306a\u6570\u5b66\u5b9a\u6570\u306b\u95a2\u9023\u3057\u3066\u3044\u307e\u3059\uff1a\u6574\u65701\u3001eulersche\u756a\u53f7 \u305d\u3046\u3067\u3059 {displaystyle e} \u3001\u60f3\u50cf\u4e0a\u306e\u30e6\u30cb\u30c3\u30c8 \u79c1 {displaystyle mathrm {i}} \u8907\u96d1\u306a\u6570\u3068\u56de\u8def\u306e\u6570 pi {displaystylepi} \u3002 Eulersche\u756a\u53f7\u306f\u3001\u6559\u54e1\u306e\u6f38\u8fd1\u8a55\u4fa1\u3067\u3082\u767a\u751f\u3057\u307e\u3059\uff08\u30b9\u30bf\u30fc\u30ea\u30f3\u30b0\u30d5\u30a9\u30fc\u30df\u30e5\u30e9\u3092\u53c2\u7167\uff09\u3002 [23] 2\u03c0n(ne)n\u2264 n \uff01 \u2264 2\u03c0n(ne)nde \u305d\u3046\u3067\u3059 112n{displaystyle {sqrt {2pi n}}\u5de6\uff08{frac {n} {e}}\u53f3\uff09^{n} leq n\uff01leq {sqrt {2pi n}} 2\u3064\u306e\uff08\u7d76\u5bfe\u306b\u53ce\u675f\uff09\u5217\u3068\u30d3\u30ce\u30df\u30c3\u30af\u6559\u80b2\u7387\u306e\u30b3\u30fc\u30b7\u30fc\u88fd\u54c1\u306e\u5f0f \u2211 k=0\u221e1k!de \u2211 k=0\u221e(\u22121)kk!= \u2211 k=0\u221e\u2211 j=0k1j!(\u22121)k\u2212j(k\u2212j)!= \u2211 k=0\u221e1k!\u2211 j=0k(kj)\u521d\u3081 j\uff08  &#8211;  \u521d\u3081 \uff09\uff09 k\u2212j= \u2211 k=0\u221e\uff08 \u521d\u3081 + \uff08  &#8211;  \u521d\u3081 \uff09\uff09 \uff09\uff09 k= \u2211 k=0\u221e0 k= \u521d\u3081 = \u305d\u3046\u3067\u3059 de \u305d\u3046\u3067\u3059 \u22121{displaystyle sum _ _ {k = 0 ^^ {inflty} {frac {1} {k\uff01}} cdot sum _ _ _ _ _ _ q = 0 {^{infty} {frac {\uff08-1\uff09{k}} {frac {\uff08-1\uff09{k} {k} {k}} {k} {k} {k} {k} {k} {k} {k} {k} {k} {k} {k} } \u305d\u3057\u3066\u3053\u308c\u304b\u3089\u3059\u3050\u306b\u7d9a\u304d\u307e\u3059\uff1a \u305d\u3046\u3067\u3059 \u22121= 1e= \u2211 k=0\u221e(\u22121)kk!mm\u30b9\u30ec\u30fc\u30d3\u30fc\u3002 \u30aa\u30a4\u30e9\u30fc\u6570\u306e\u5e7e\u4f55\u5b66\u7684\u89e3\u91c8\u306f\u3001\u7a4d\u5206\u8a08\u7b97\u3092\u63d0\u4f9b\u3057\u307e\u3059\u3002\u305d\u306e\u5f8c\u3067\u3059 \u305d\u3046\u3067\u3059 {displaystyle e} \u660e\u78ba\u306b\u6c7a\u5b9a\u3055\u308c\u305f\u6570  1&#8243;>\u3001\u5b9f\u969b\u306e\u76f8\u4e92\u95a2\u6570\u306e\u6a5f\u80fd\u30b0\u30e9\u30d5\u306e\u4e0b\u306e\u9818\u57df\u306e\u5185\u5bb9 \u3068 = 1x{displaystyle y = {tfrac {1} {x}}} \u9593\u9694\u3067 [ \u521d\u3081 \u3001 b ] {displaystyle [1\u3001b]} \u307e\u3063\u305f\u304f\u540c\u3058 \u521d\u3081 {displaystyle1} \u306f\uff1a [24] \u222b 1e1xd \u30d0\u30c4 = \u521d\u3081 {displaystyle int _ {1}^{e} {frac {1} {x}}\u3001mathrm {d} x = 1} Eulersche\u756a\u53f7\u3082\u5b9f\u884c\u3067\u304d\u307e\u3059 \u305d\u3046\u3067\u3059 = \u30ea\u30e0 n\u2192\u221enn!nmm\u30b9\u30ec\u30fc\u30d6\u30f3 \u307e\u305f\u306f\u3001\u6559\u54e1\u3068\u526f\u7701\u304b\u3089\u306e\u5546\u306e\u9650\u754c\u3067\u8aac\u660e\u3057\u3066\u304f\u3060\u3055\u3044\u3002 \u305d\u3046\u3067\u3059 = \u30ea\u30e0 n\u2192\u221en!!n\u3002 {displaystyle e = lim _ {nto infty} {frac {n\uff01} {\uff01n}}\u3002}\u3002 \u7d20\u6570\u306e\u5206\u5e03\u3078\u306e\u63a5\u7d9a\u306f\u5f0f\u3092\u4ecb\u3057\u3066\u884c\u308f\u308c\u307e\u3059 \u305d\u3046\u3067\u3059 = \u30ea\u30e0 n\u2192\u221e\uff08 nn\uff09\uff09 \u03c0(n)M TUME SLELE\u3001MMMEMB\u00e9MartyMALHYM HALM HOUPE HUPE HUPE HUPE\uff09MJOYI HUPE HUPE HYM\uff09 \u305d\u3046\u3067\u3059 = \u30ea\u30e0 n\u2192\u221en#nM Tume Slee Tele YmEmb\u00e9TufritssMalm M h reppie \u660e\u3089\u304b\u306b\u3001\u3057\u304b\u3057 pi \uff08 n \uff09\uff09 {displaystyle pi\uff08n\uff09} \u7d20\u6570\u95a2\u6570\u3068\u30b7\u30f3\u30dc\u30eb n \uff03 {displaystyle n\uff03} \u6570\u5b57\u306e\u539f\u59cb n {displaystyle n} \u610f\u5473\u3002 \u307e\u305f\u3001\u5b9f\u969b\u7684\u306b\u91cd\u8981\u306a\u30a8\u30ad\u30be\u30c1\u30c3\u30af\u306a\u523a\u6fc0\u306e\u65b9\u304c \u30ab\u30bf\u30ed\u30cb\u30a2\u306e\u8868\u73fe  \u305d\u3046\u3067\u3059 = 211de 432de 6\u22c585\u22c574de 10\u22c512\u22c514\u22c5169\u22c511\u22c513\u22c5158\u22ef {displaystyle e = {sqrt [{1}] {frac {2} {1}}} cdot {sqrt [{2}] {frac {4} {3}}} cdot {sqrt [{4}] {cdot 8} {6cdot 8} {} {} {} {} {} {6cdot 7t {8}] {frac {10cdot 12cdot 14cdot 16} {9cdot 11cdot 13cdot 15}}} cdots} \u756a\u53f7\u306b\u95a2\u9023\u3057\u3066 \u305d\u3046\u3067\u3059 {displaystyle e} \u6700\u65b0\u306e\u30ec\u30aa\u30f3\u30cf\u30eb\u30c8\u30aa\u30a4\u30e9\u30fc\u30ba\u306e\u51fa\u7248\u4ee5\u6765\u5b58\u5728\u3057\u3066\u3044\u307e\u3059 \u5206\u6790\u306e\u7d39\u4ecb 1748\u5e74\u3001\u591a\u304f\u306e\u30c1\u30a7\u30fc\u30f3\u30d6\u30ec\u30a4\u30af\u958b\u767a\u304c \u305d\u3046\u3067\u3059 {displaystyle e} \u305d\u3057\u3066\u304b\u3089 \u305d\u3046\u3067\u3059 {displaystyle e} \u6d3e\u751f\u53ef\u80fd\u306a\u30b5\u30a4\u30ba\u3002 \u3057\u305f\u304c\u3063\u3066\u3001\u30aa\u30a4\u30e9\u30fc\u306b\u306f\u6b21\u306e\u53e4\u5178\u7684\u306a\u30a2\u30a4\u30c7\u30f3\u30c6\u30a3\u30c6\u30a3\u304c\u3042\u308a\u307e\u3059 \u305d\u3046\u3067\u3059 {displaystyle e} \u898b\u3064\u304b\u3063\u305f\uff1a \uff08 \u521d\u3081 \uff09\uff09 e=[2;1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,\u2026]=2+11+12+11+11+14+11+11+16+\u22ef{displayStyle\uff081\uff09{begin {aligned} e\uff06= [2; 1,2,1,1,1,1,6,1,1,8,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1, = 2+ {cfrac {1} {1+ {cfrac {1} {{cfrac {1} {1 {1} {1 {1} {1 {1} cfrac {1} {4+ {cfrac {1} {1+ {cfrac {1} {1+ {cfrac {1} {6+dotsb}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} \uff08\u7d50\u679c A003417 OEIS\u3067\uff09 \u30a2\u30a4\u30c7\u30f3\u30c6\u30a3\u30c6\u30a3\uff081\uff09\u306b\u306f\u3001\u7121\u9650\u306b\u7d9a\u304f\u898f\u5247\u7684\u306a\u30d1\u30bf\u30fc\u30f3\u304c\u3042\u308b\u3088\u3046\u3067\u3059\u3002\u3053\u308c\u306f\u3001\u30aa\u30a4\u30e9\u30fc\u306b\u3088\u3063\u3066\u4ee5\u4e0b\u304b\u3089\u6d3e\u751f\u3057\u305f\u5b9a\u671f\u7684\u306a\u30c1\u30a7\u30fc\u30f3\u30d6\u30ec\u30a4\u30af\u3092\u53cd\u6620\u3057\u3066\u3044\u307e\u3059\u3002 [25] \uff08 2 \uff09\uff09 e+1e\u22121=[2;6,10,14,\u2026]=2+16+110+114+1\u22f1\u22482,1639534137386{displaystyle\uff082\uff09{begin {aligned} {frac {e+1} {e-1}}\uff06= [2; 6,10,14\u3001dotsc] \\\uff06= {2+ {cfrac {1} {6+ {cfrac {1} {{1} {{1} {1} {1} {1} {1} {1} {1} }}}}}}}} \\\uff06comprx 2 {\u3001} 1639534137386end {aligned}}}} \uff08\u7d50\u679c A016825 OEIS\u3067\uff09 \u6b21\u306b\u3001\u3053\u306e\u30c1\u30a7\u30fc\u30f3\u30d6\u30ec\u30a4\u30af\u306f\u6b21\u306e\u7279\u5225\u306a\u30b1\u30fc\u30b9\u3067\u3059 k = 2 {displaystyle k = 2} \uff1a \uff08 3 \uff09\uff09 coth\u20611k=e2k+1e2k\u22121=[k;3k,5k,7k,\u2026]=k+13k+15k+17k+1\u22f1{displaystyle\uff083\uff09{begin {aligned} {coth {frac {1} {k}}}\uff06= {frac {frac {2} {k}}+1} {e^{frac {2} {k}}}}}}}}}}}}}}}}}} +{cfrac {1} {3k+{cfrac {1} {5k+{cfrac {1} {7k+{cfrac {1} {;\u3001ddots}}}}}}}}}} \\ {aligned}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}  \uff08 k = \u521d\u3081 \u3001 2 \u3001 3 \u3001 &#8230; \uff09\uff09 {displaystyle\uff08k = 1,2,3\u3001dots\uff09} \u3057\u304b\u3057\u3001\u5225\u306e\u53e4\u5178\u7684\u306a\u30c1\u30a7\u30fc\u30f3\u30d6\u30ec\u30a4\u30af\u958b\u767a\u3001\u3057\u304b\u3057\u3001\u305d\u308c\u306f \u898f\u5236\u3067\u306f\u3042\u308a\u307e\u305b\u3093 \u30aa\u30a4\u30e9\u30fc\u304b\u3089\u306e\u3082\u306e\u3067\u3082\u3042\u308a\u307e\u3059\uff1a [26] \uff08 4 \uff09\uff09 1e\u22121=0+11+22+33+4\u22f1\u22480,58197670686932{displayStyle\uff084\uff09{begin {aligned} {frac {1} {e-1}}\uff06= {0+ {cfrac {1} {1+ {cfrac {2} {2+ {cfrac {3} {3} {3} {3+ {cfrac {4} {{; ddot}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} {\u3001} 58197670686932END {aligned}}} \uff08\u7d50\u679c A073333 OEIS\u3067\uff09 \u306e\u5225\u306e\u30c1\u30a7\u30fc\u30f3\u7834\u58ca Eulerschen\u756a\u53f7 \u30d0\u30c3\u30af\u3001\u3053\u308c\u306f\uff081\uff09\u4ee5\u5916\u306e\u30d1\u30bf\u30fc\u30f3\u304b\u3089\u306e\u3082\u306e\u3067\u3059\u3002 [27] \uff08 5 \uff09\uff09 e=2+11+12+23+34+45+56+67+78+\u22ef{displaystyle\uff085\uff09{begin {aligned} e\uff06= 2+{cfrac {1} {1+ {cfrac {1} {2+ {cfrac {2} {3+ {cfrac {3} {4+ {4+ {cfrac {4} {{{{{6 {{{6 {{{{5} {5+ {{5} {5+ {5+ {5+ {{5+ {5+ {5+ {5+ {5+ {5+ {5+ {5+ {{5+ {{5+ {{6 {{6 {{6 {{6 {{6 {{6 {{6 {{6 {{{6 {{6 {6} {cfrac {7} {8+dotsb}}}}}}}}}}}}}}}}}}} ent {aligned}}}}}}}}}}}}}}}}}}}}} \u306b\u95a2\u9023\u3057\u3066 Eulerschen\u756a\u53f7 \u307e\u305f\u3001\u591a\u6570\u306e\u4e00\u822c\u7684\u306a\u30c1\u30a7\u30fc\u30f3\u30eb\u30fc\u30c1\u7406\u8ad6\u65b9\u7a0b\u5f0f\u304c\u3042\u308a\u307e\u3059\u3002 Oskar Perron\u306f\u6b21\u306e\u4e00\u822c\u7684\u306b\u9069\u7528\u53ef\u80fd\u306a\u30d7\u30ec\u30bc\u30f3\u30c6\u30fc\u30b7\u30e7\u30f3\u306b\u540d\u524d\u3092\u4ed8\u3051\u307e\u3059 \u305d\u3046\u3067\u3059 {displaystyle e} -\u95a2\u6570\uff1a [27] \uff08 6 \uff09\uff09 ez=1+z1\u22121z2+z\u22122z3+z\u22123z4+z\u22124z5+z\u22125z6+z\u22126z7+z\u22127z8+z\u2212\u22ef{displaystyle\uff086\uff09{begin {aligned} {e^{from}}\uff06= 1+ {cfrac {z} {1- {cfrac {1z} {2+z- {2z} {3+z- {cfrac {cfrac {{cfrac} {5z} {6+z-{x-cfrac {{6 7z} {8+z-d-dotsb}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}  \uff08 \u3068 \u2208 c \uff09\uff09 {displaystyle\uff08zin mathbb {c}\uff09} \u3053\u306e\u5225\u306e\u4f8b\u306f\u3001\u30e8\u30cf\u30f3\u30fb\u30cf\u30a4\u30f3\u30ea\u30c3\u30d2\u30fb\u30e9\u30f3\u30d0\u30fc\u30c8\u304b\u3089\u6765\u3066\u3044\u308b\u3001\u30bf\u30f3\u30b8\u30f3\u30b0\u30cf\u30a4\u30d1\u30fc\u30dc\u30ea\u30af\u30b9\u306e\u767a\u5c55\u3067\u3059 \u30e9\u30f3\u30d0\u30fc\u30c8\u306e\u30c1\u30a7\u30fc\u30f3\u304c\u58ca\u308c\u307e\u3059 \u671f\u5f85\u3055\u308c\u3066\u3044\u307e\u3059\uff1a [28] [29] \uff08 7 \uff09\uff09 tanh\u2061z=ez\u2212e\u2212zez+e\u2212z=e2z\u22121e2z+1=0+z1+z23+z25+z27+z29+z211+z213+z215+\u22ef{displayStyle\uff087\uff09{begin {aligned} {tanh from}\uff06= {frac {e {from} -e {z}} {e {e}+e {-z}}} \\\uff06= {frac {e}}}\uff06= 0+ {cfrac {{x} {{x} {x}} {{{x} {{{x} {x}} {3+ {cfrac {z {2}} {5+ {cfrac {z {2}} {7+ {cfrac {cfrac {z {z {2}} {13+ {cfrac {z {z {2}}} {15+ dotsb}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} aligned}}}}  \uff08 z\u2208C\u2216{i\u03c02+k\u03c0:k=0,1,2,3,\u2026}\uff09\uff09 {displaystyle left\uff08zin mathbb {c} setminus left {{frac {mathrm {i} pi} {2}}+kpi\u30b3\u30ed\u30f3k = 0,1,2,3\u3001dotsc\u53f3}\u53f3\uff09} Srinivasa Ramanujan\u304c\u30b3\u30f3\u30d4\u30e5\u30fc\u30bf\u30fc\u30d7\u30ed\u30b0\u30e9\u30e0\u306e\u52a9\u3051\u306b\u306a\u3063\u305f\u306e\u306f2019\u5e74\u3060\u3051\u3067\u3057\u305f \u30e9\u30de\u30cc\u30b8\u30e3\u30f3\u30de\u30b7\u30f3 \u6700\u7d42\u7684\u306b\u306f\u3001\u88c1\u5224\u3068\u30c6\u30ed\u30ea\u30b9\u30c8\u306e\u65b9\u6cd5\u306b\u57fa\u3065\u3044\u3066\u3001Gal Raayoni\u304c\u7387\u3044\u308b\u30c1\u30fc\u30e0\u304c\u30c6\u30af\u30cb\u30aa\u30f3\u3067\u7387\u3044\u308b\u30c1\u30fc\u30e0\u306f\u3001Eulersche\u756a\u53f7\u306e\u5225\u306e\u3001\u4ee5\u524d\u306f\u672a\u77e5\u306e\u30c1\u30a7\u30fc\u30f3\u30d6\u30ec\u30a4\u30af\u958b\u767a\u306e\u958b\u767a\u65b9\u6cd5\u306b\u57fa\u3065\u3044\u3066\u540d\u524d\u304c\u4ed8\u3051\u3089\u308c\u307e\u3057\u305f\u3002\u4ee5\u524d\u306b\u65e2\u77e5\u306e\u3059\u3079\u3066\u306e\u30c1\u30a7\u30fc\u30f3\u30d6\u30ec\u30a4\u30ad\u30f3\u30b0\u958b\u767a\u3068\u6bd4\u8f03\u3057\u3066\u3001\u30aa\u30a4\u30e9\u30fc\u6570\u3088\u308a\u3082\u5c0f\u3055\u3044\u6574\u6570\u306e\u3059\u3079\u3066\u304c\u521d\u3081\u3066\u3067\u3059 3 \u3001\u30aa\u30a4\u30e9\u30fc\u30b7\u30e5\u6570\u3088\u308a\u3082\u5927\u304d\u3044\u6574\u6570\u3002 [30] \u30aa\u30a4\u30e9\u30fc\u756a\u53f7\u3088\u308a\u3082\u5927\u304d\u3044\u6574\u6570\u304b\u3089\u305d\u306e\u3088\u3046\u306a\u4e0b\u964d\u30c1\u30a7\u30fc\u30f3\u30d6\u30ec\u30a4\u30af\u3092\uff08\u5358\u4e00\uff09\u898b\u3064\u3051\u308b\u3060\u3051 e)}&#8221;>\u305d\u306e\u3088\u3046\u306a\u964d\u9806\u306e\u30c1\u30a7\u30fc\u30f3\u30d6\u30ec\u30fc\u30af\u304c\u7121\u9650\u306b\u3042\u308b\u3053\u3068\u3092\u793a\u5506\u3057\u3066\u3044\u307e\u3059 n {displaystyle n} \u3068 e}&#8221;>\u307e\u305f\u3001\u30aa\u30a4\u30e9\u30fc\u30b7\u30e5\u756a\u53f7\u306b\u3064\u306a\u304c\u308a\u307e\u3059\u3002 \uff08 8 \uff09\uff09 e=3+\u221214+\u221225+\u221236+\u221247+\u221258+\u22ef{displayStyle\uff088\uff09{begin {aligned} e\uff06= 3+{cfrac {-1} {4+ {cfrac {-2} {5+ {cfrac {-3} {6+ {cfrac {-4} {7+ {cfrac {emd {8+dotsb}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} {aligned}}} \u91d1\u5229\u8a08\u7b97 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u6b21\u306e\u4f8b\u3067\u306f\u3001\u30aa\u30a4\u30e9\u30fc\u6570\u306e\u8a08\u7b97\u304c\u3088\u308a\u8a18\u8ff0\u7684\u306b\u306a\u308b\u3060\u3051\u3067\u306a\u304f\u3001\u30aa\u30a4\u30e9\u30fc\u6570\u306e\u767a\u898b\u306e\u6b74\u53f2\u306b\u3064\u3044\u3066\u3082\u8aac\u660e\u3057\u3066\u3044\u307e\u3059\u3002\u5f7c\u3089\u306e\u6700\u521d\u306e\u5834\u6240\u306f\u3001\u8907\u5229\u3092\u8abf\u3079\u308b\u969b\u306b\u30e4\u30b3\u30d6\u30fb\u30a2\u30a4\u30fb\u30d9\u30eb\u30cc\u30fc\u30ea\u306b\u3088\u3063\u3066\u767a\u898b\u3055\u308c\u307e\u3057\u305f\u3002 \u6700\u521d\u306e\u5f0f\u306e\u9650\u754c\u306f\u6b21\u306e\u3088\u3046\u306b\u89e3\u91c8\u3067\u304d\u307e\u3059\u3002\u8ab0\u304b\u304c1\u67081\u65e5\u306b\u30d9\u30f3\u30c1\u30671\u30e6\u30fc\u30ed\u3092\u652f\u6255\u3046\u3067\u3057\u3087\u3046\u3002\u9280\u884c\u306f\u5f7c\u306b\u91d1\u5229\u306b\u95a2\u3059\u308b\u73fe\u5728\u306e\u5229\u606f\u3092\u4fdd\u8a3c\u3057\u307e\u3059 \u3068 = 100 \uff05 {\u5c55\u793az = 100\u3001\uff05} 1\u5e74\u5f53\u305f\u308a\u3002\u6765\u5e74\u306e1\u67081\u65e5\u306e\u5f7c\u304c\u540c\u3058\u6761\u4ef6\u3067\u95a2\u5fc3\u3092\u751f\u307f\u51fa\u3057\u305f\u3068\u304d\u3001\u5f7c\u306e\u4fe1\u7528\u306f\u3069\u308c\u304f\u3089\u3044\u5927\u304d\u3044\u3067\u3059\u304b\uff1f \u8907\u5229\u5f0f\u306e\u5f8c k 0 {displaystyle k_ {0}} \u5f8c n {displaystyle n} \u91d1\u5229\u306e\u5229\u606f \u3068 {displaystyle with} \u9996\u90fd k n= k 0\uff08 \u521d\u3081 + \u3068 \uff09\uff09 n\u3002 {displaystyle k_ {n} = k_ {0}\uff081+z\uff09^{n}\u3002} \u3053\u306e\u4f8b\u3067\u306f k 0 = \u521d\u3081 {displaystyle k_ {0} = 1} \u3068 \u3068 = 100 \uff05 = \u521d\u3081 {\u5c55\u793az = 100\u3001\uff05= 1} \u5229\u606f\u8ffd\u52a0\u6599\u91d1\u304c\u6bce\u5e74\u884c\u308f\u308c\u308b\u5834\u5408\u3001\u307e\u305f\u306f \u3068 = \u521d\u3081 \/ n {displaystyle z = 1\/n} \u5229\u606f\u8ffd\u52a0\u6599\u91d1\u306e\u5834\u5408 n {displaystyle n}  &#8211; \u30d7\u30e9\u30b9\u306f\u4eca\u5e74\u3001\u3064\u307e\u308a\u4eca\u5e74\u306b\u8208\u5473\u3092\u6301\u3063\u3066\u884c\u308f\u308c\u307e\u3059\u3002 \u6bce\u5e74\u306e\u8ffd\u52a0\u6599\u91d1\u304c\u3042\u308a\u307e\u3059 k 1= \u521d\u3081 de \uff08 \u521d\u3081 + \u521d\u3081 \uff09\uff09 1= 2 \u3001 00\u3002 {displaystyle k_ {1} = 1cdot\uff081+1\uff09^{1} = 2 {\u3001} 00\u3002} \u534a\u5e74\u9593\u306e\u8ffd\u52a0\u6599\u91d1\u3067\u3042\u306a\u305f\u304c\u6301\u3063\u3066\u3044\u307e\u3059 \u3068 = \u521d\u3081 2 {displaystyle z = {frac {1} {2}}} \u3001 k 2= \u521d\u3081 de (1+12)2= 2 \u3001 25 {displaystyle k_ {2} = 1cdot\u5de6\uff081+ {frac {1} {2}}\u53f3\uff09^{2} = 2 {\u3001} 25} \u3082\u3046\u5c11\u3057\u3002\u6bce\u65e5\u306e\u91d1\u5229 \uff08 \u3068 = \u521d\u3081 \/ 365 \uff09\uff09 {displastyle\uff08z = 1\/365\uff09} \u3042\u306a\u305f\u306f\u5f97\u307e\u3059 k 365= \u521d\u3081 de (1+1365)365= 2.714 567\u3002 {displaystyle k_ {365} = 1cdot\u5de6\uff081+ {frac {1} {365}}\u53f3\uff09^{365} = 2 {\u3001} 714567\u3002} \u95a2\u5fc3\u304c\u6bce\u56de\u7d99\u7d9a\u7684\u306b\u884c\u308f\u308c\u3066\u3044\u308b\u5834\u5408 n {displaystyle n} \u7121\u9650\u306b\u5927\u304d\u304f\u3001\u4e0a\u8a18\u306e\u6700\u521d\u306e\u5f0f\u3092\u53d6\u5f97\u3057\u307e\u3059 \u305d\u3046\u3067\u3059 {displaystyle e} \u3002 \u78ba\u7387\u8a08\u7b97 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u305d\u3046\u3067\u3059 {displaystyle e} \u307e\u305f\u3001\u78ba\u7387\u7406\u8ad6\u3067\u3082\u3088\u304f\u898b\u3089\u308c\u307e\u3059\u3002\u305f\u3068\u3048\u3070\u3001\u30d1\u30f3\u5c4b\u306f\u5404\u30ed\u30fc\u30eb\u306e\u751f\u5730\u306b\u30ec\u30fc\u30ba\u30f3\u3092\u4e0e\u3048\u3001\u305d\u308c\u3092\u3088\u304f\u3053\u306d\u308b\u3068\u60f3\u5b9a\u3055\u308c\u3066\u3044\u307e\u3059\u3002\u305d\u306e\u5f8c\u3001\u7d71\u8a08\u7684\u306b\u306f\u3001\u305d\u308c\u305e\u308c\u306b\u305d\u308c\u305e\u308c\u304c\u542b\u307e\u308c\u307e\u3059 \u305d\u3046\u3067\u3059 {displaystyle e} -te\u30d1\u30f3\u306f\u30ec\u30fc\u30ba\u30f3\u306a\u3057\u3002\u78ba\u7387 p {displaystyle p} \u305d\u308c\u3067 n {displaystyle n} \u30d1\u30f3\u306f\u3069\u308c\u3082\u3042\u308a\u307e\u305b\u3093 n {displaystyle n} \u30ec\u30fc\u30ba\u30f3\u306f\u9078\u3070\u308c\u305f\u30d5\u30a7\u30b9\u30c6\u30a3\u30d0\u30eb\u306b\u3042\u308a\u3001\u305d\u306e\u7d50\u679c n \u2192 \u221e {displaystyle nto infty} \uff0837\uff05-rule\uff09\uff1a p = \u30ea\u30e0 n\u2192\u221e(n\u22121n)n= \u30ea\u30e0 n\u2192\u221e(1\u22121n)n= 1e\u3002 {displaystyle p = lim _ {nto infty}\u5de6\uff08{frac {n-1} {n-1} {n}}\u53f3\uff09^{n} = lim _ {nto infty}\u5de6\uff081- {frac {1} {n}}\u53f3\uff09^{n} = {frac {1} {e}} \u4f4f\u6240\u306b\u95a2\u9023\u3059\u308b\u95a2\u9023\u3059\u308b\u5c01\u7b52\u306f\u3001\u4e92\u3044\u306b\u72ec\u7acb\u3057\u3066\u66f8\u304b\u308c\u3066\u3044\u307e\u3059\u3002\u305d\u308c\u304b\u3089\u898b\u305a\u306b\u3001\u7d14\u7c8b\u306b\u5076\u7136\u306b\u3001\u6587\u5b57\u306f\u5c01\u7b52\u306b\u5165\u308c\u3089\u308c\u307e\u3059\u3002\u9069\u5207\u306a\u5c01\u7b52\u306b\u624b\u7d19\u304c\u306a\u3044\u53ef\u80fd\u6027\u306f\u3069\u308c\u304f\u3089\u3044\u3067\u3059\u304b\uff1fEuler\u306f\u3053\u306e\u30bf\u30b9\u30af\u3092\u89e3\u6c7a\u3057\u30011751\u5e74\u306b\u30a8\u30c3\u30bb\u30a4\u300cCalcul da laprobabilit\u00e9dansle jeu de rencontre\u300d\u3067\u516c\u958b\u3057\u307e\u3057\u305f\u3002\u305d\u308c\u306f\u975e\u5e38\u306b\u3046\u307e\u304f\u3084\u308a\u307e\u3059 \u521d\u3081 \/ \u305d\u3046\u3067\u3059 = 0.367 879 \u22ef \u2248 36,787 9 \uff05 {displaystyle 1\/e = 0 {\u3001} 367879dots\u7d0436 {\u3001} 7879\u3001\uff05} \u5feb\u9069\u306a\u3001\u6587\u5b57\u306e\u6570\u304c\u5927\u304d\u304f\u306a\u3063\u3066\u3044\u308b\u3068\u304d\u306e\u78ba\u7387\u306e\u9650\u754c\u3002 \u30cf\u30f3\u30bf\u30fc\u306f1\u30b7\u30e7\u30c3\u30c8\u306e\u307f\u304c\u5229\u7528\u3067\u304d\u307e\u3059\u3002\u305d\u308c\u306f\u7fa4\u8846\u304b\u3089\u9ce9\u306b\u8a00\u308f\u308c\u3066\u3044\u307e\u3059\u3001\u305d\u306e\u6570 n {displaystyle n} \u5f7c\u306f\u8ab0\u304c\u30e9\u30f3\u30c0\u30e0\u306a\u9806\u5e8f\u3067\u5f7c\u3092\u901a\u308a\u904e\u304e\u3066\u98db\u3076\u3053\u3068\u3092\u77e5\u3063\u3066\u3044\u307e\u3059\u3001\u5049\u5927\u306a\u3082\u306e\u3092\u6483\u3061\u307e\u3059\u3002\u3069\u306e\u6226\u7565\u3067\u6700\u5927\u306e\u30cf\u30c8\u3092\u4f5c\u308b\u53ef\u80fd\u6027\u306f\u3042\u308a\u307e\u3059\u304b\uff1f\u3053\u306e\u9ce9\u306e\u554f\u984c\u306f\u3001\u30a2\u30e1\u30ea\u30ab\u306e\u6570\u5b66\u8005\u30cf\u30fc\u30d0\u30fc\u30c8\u30fb\u30ed\u30d3\u30f3\u30ba\uff08*1915\uff09\u306b\u3088\u3063\u3066\u7b56\u5b9a\u3055\u308c\u307e\u3057\u305f\u3002\u540c\u3058\u6c7a\u5b9a &#8211; \u5236\u4f5c\u306e\u554f\u984c\u306f\u3001N\u7533\u8acb\u8005\uff08\u79d8\u66f8\u554f\u984c\uff09\u304a\u3088\u3073\u540c\u69d8\u306e\u8863\u670d\u3067\u6700\u9ad8\u306e\u5f93\u696d\u54e1\u3092\u96c7\u7528\u3059\u308b\u3068\u304d\u306b\u3082\u5b58\u5728\u3057\u307e\u3059\u3002\u89e3\u6c7a\u7b56\uff1a\u6700\u9069\u306a\u6226\u7565\u306f\u3001\u6700\u521d\u306b\u3067\u3059 k {displaystyle k} \u8074\u899a\u969c\u304c\u3044 \uff08 k < n \uff09\uff09 {displaystyle\uff08k \u98db\u3073\u56de\u3063\u3066\u304b\u3089\u3001\u305d\u308c\u307e\u3067\u306b\u5927\u304d\u3044\u3082\u306e\u304c\u98db\u3093\u3067\u3044\u306a\u3044\u5834\u5408\u306f\u3001\u3053\u308c\u307e\u3067\u3088\u308a\u3082\u5927\u304d\u3044\u3001\u307e\u305f\u306f\u6700\u5f8c\u306e\u9ce9\u3092\u6483\u3061\u307e\u3059\u3002\u6700\u5927\u306e\u9ce9\u3092\u6355\u307e\u3048\u308b\u53ef\u80fd\u6027\u306f\u3001\u3053\u306e\u6700\u9069\u306a\u6226\u7565\u306b\u307b\u307c\u5b58\u5728\u3057\u307e\u3059 \u521d\u3081 \/ \u305d\u3046\u3067\u3059 {displaystyle1\/e} N\u306b\u95a2\u4fc2\u306a\u304f\u3001\u5c0f\u3055\u3059\u304e\u3066\u306f\u306a\u308a\u307e\u305b\u3093\u3002\u3082\u3057\u79c1\u9054 k \/ n {displaystyle k\/n} \u306e\u63a8\u5b9a\u5024\u3068\u3057\u3066 \u521d\u3081 \/ \u305d\u3046\u3067\u3059 {displaystyle1\/e} \u9078\u629e\u3057\u3066\u304b\u3089\u3001\u6b21\u3092\u7d9a\u3051\u307e\u3059\u3002 k \u2248 \u521d\u3081 \/ \u305d\u3046\u3067\u3059 \u2217 n {displaystyle kapprox 1\/e*n} \u3002\u3057\u305f\u304c\u3063\u3066\u300127\u5339\u306e\u9ce9\u306710\u3092\u98db\u3070\u3059\u3060\u3051\u3067\u3059\u3002\u3067\u304d\u308b\u3053\u3068\u306f\u9a5a\u304f\u3079\u304d\u3053\u3068\u3067\u3059 2 \/ 3 {displaystyle 2\/3} \u3059\u3079\u3066\u306e\u30b1\u30fc\u30b9\u3067\u306f\u3001\u76ee\u7684\u306e\u6700\u9069\u30bd\u30ea\u30e5\u30fc\u30b7\u30e7\u30f3\u3092\u53d7\u3051\u53d6\u308a\u307e\u305b\u3093\u3002 [\u6700\u521d\u306b30] \u30dd\u30a2\u30bd\u30f3\u3067\u306f\u3001\u6307\u6570\u304a\u3088\u3073\u6b63\u898f\u5206\u5e03 \u305d\u3046\u3067\u3059 {displaystyle e} \u4ed6\u306e\u30b5\u30a4\u30ba\u306b\u52a0\u3048\u3066\u5206\u5e03\u3092\u8aac\u660e\u3059\u308b\u305f\u3081\u306b\u4f7f\u7528\u3055\u308c\u307e\u3059\u3002 Eulersche\u756a\u53f7\u306f\u3001\u6570\u5b66\u306e\u3055\u307e\u3056\u307e\u306a\u91cd\u8981\u306a\u5834\u6240\u306b\u8868\u793a\u3055\u308c\u307e\u3059\u3002 Eulersche\u6570\u306f\u3001\u5dee\u5206\u8a08\u7b97\u3067\u3082\u767a\u751f\u3057\u307e\u3059\u3002\u30dd\u30a4\u30f3\u30c8\u3067 \u305d\u3046\u3067\u3059 {displaystyle e} \u6700\u5927\u95a2\u6570\u3067\u3059 f \uff08 \u30d0\u30c4 \uff09\uff09 = \u30d0\u30c4 1x= \u305d\u3046\u3067\u3059 ln\u2061(x)x{displaystyle f\uff08x\uff09= x^{tfrac {1} {x}} = e^{tfrac {ln\uff08x\uff09} {x}}}} \u3002\u3055\u3089\u306b\u3001\u5834\u6240\u304c\u3042\u308a\u307e\u3059 \u305d\u3046\u3067\u3059  &#8211;  \u521d\u3081 {displaystyle e^{ &#8211;  1}} \u95a2\u6570\u306e\u6700\u5c0f f \uff08 \u30d0\u30c4 \uff09\uff09 = \u30d0\u30c4 \u30d0\u30c4 = \u305d\u3046\u3067\u3059 \u30d0\u30c4 de ln \u2061 \uff08 \u30d0\u30c4 \uff09\uff09 {displaystyle f\uff08x\uff09= x^{x} = e^{xcdot ln\uff08x\uff09}}} \u3002\u3053\u308c\u306f\u3001\u6d3e\u751f\u95a2\u6570\u3092\u4f7f\u7528\u3057\u3066\u8868\u793a\u3067\u304d\u307e\u3059\u3002 1850\u5e74\u306eCrelles Journal\u306e\u6700\u3082\u5e78\u904b\u306a\u30dc\u30ea\u30e5\u30fc\u30e0\u3067\u3001\u30b9\u30a4\u30b9\u306e\u6570\u5b66\u8005Jakob Steiner\u306f Eulerschen\u756a\u53f7  \u305d\u3046\u3067\u3059 {displaystyle e} \u3001 \u4f55 \u305d\u3046\u3067\u3059 {displaystyle e} \u6975\u5ea6\u306e\u4fa1\u5024\u30bf\u30b9\u30af\u306e\u89e3\u6c7a\u7b56\u3068\u3057\u3066\u3002\u30b7\u30e5\u30bf\u30a4\u30ca\u30fc\u306f\u305d\u306e\u6570\u3092\u793a\u3057\u307e\u3057\u305f \u305d\u3046\u3067\u3059 {displaystyle e} \u305d\u308c\u306f\u3001\u30eb\u30fc\u30c8\u3092\u5f15\u304f\u3068\u304d\u306b\u6700\u5927\u306e\u30eb\u30fc\u30c8\u3092\u63d0\u4f9b\u3059\u308b\u660e\u3089\u304b\u306b\u7279\u5b9a\u306e\u6b63\u306e\u5b9f\u6570\u3068\u3057\u3066\u7279\u5fb4\u4ed8\u3051\u3089\u308c\u307e\u3059\u3002\u30b7\u30e5\u30bf\u30a4\u30ca\u30fc\u306f\u6587\u5b57\u901a\u308a\u6b21\u306e\u3088\u3046\u306b\u66f8\u3044\u3066\u3044\u307e\u3059\uff1a\u300c\u5404\u6570\u5024\u304c\u305d\u308c\u81ea\u4f53\u3092\u6839\u672c\u7684\u306b\u6839\u672c\u7684\u306b\u4ecb\u3057\u3066\u3001\u305d\u306e\u6570\u306f\u6700\u5927\u306e\u30eb\u30fc\u30c8\u3092\u4ed8\u4e0e\u3057\u307e\u3059\u3002\u300d [32] \u30b7\u30e5\u30bf\u30a4\u30ca\u30fc\u306f\u3001\u6a5f\u80fd\u306e\u305f\u3081\u306b\u7591\u554f\u3092\u6271\u3063\u3066\u3044\u307e\u3059 f \uff1a \uff08 0 \u3001 \u221e \uff09\uff09 \u2192 \uff08 0 \u3001 \u221e \uff09\uff09 \u3001 \u30d0\u30c4 \u21a6 f \uff08 \u30d0\u30c4 \uff09\uff09 = xx= \u30d0\u30c4 1x{displaystyle fcolon\uff080\u3001infty\uff09to\uff080\u3001infty\uff09\u3001; xmapsto f\uff08x\uff09= {sqrt [{x}] {x}} = x^{frac {1} {x}}}}} \u30b0\u30ed\u30fc\u30d0\u30eb\u306a\u6700\u5927\u5024\u304c\u5b58\u5728\u3057\u3001\u3069\u306e\u3088\u3046\u306b\u6c7a\u5b9a\u3055\u308c\u308b\u304b\u3002\u5f7c\u306e\u58f0\u660e\u306f\u3001\u305d\u308c\u304c\u5b58\u5728\u3057\u3001\u305d\u308c\u306f\u3067\u53d7\u3051\u5165\u308c\u3089\u308c\u3066\u3044\u308b\u3068\u3044\u3046\u3053\u3068\u3067\u3059 \u30d0\u30c4 max= \u305d\u3046\u3067\u3059 {displaystyle x_ {mathrm {max}} = e} \u3002 \u5f7c\u306e\u672c\u3067 \u6570\u5b66\u306e\u52dd\u5229 HeinrichD\u00f6rrie\u306f\u3001\u3053\u306e\u6975\u7aef\u306a\u4fa1\u5024\u306e\u30bf\u30b9\u30af\u306b\u5bfe\u3059\u308b\u57fa\u672c\u7684\u306a\u30bd\u30ea\u30e5\u30fc\u30b7\u30e7\u30f3\u3092\u63d0\u4f9b\u3057\u307e\u3059\u3002\u5f7c\u306e\u30a2\u30d7\u30ed\u30fc\u30c1\u306f\u3001\u5b9f\u969b\u306e\u6307\u6570\u95a2\u6570\u306b\u95a2\u3059\u308b\u6b21\u306e\u771f\u306e\u58f0\u660e\u306b\u57fa\u3065\u3044\u3066\u3044\u307e\u3059\u3002 "},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/all2jp\/wiki11\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/all2jp\/wiki11\/archives\/19743#breadcrumbitem","name":"Eulersche\u756a\u53f7-Wikipedia"}}]}]