[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/all2jp\/wiki10\/archives\/18598#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/all2jp\/wiki10\/archives\/18598","headline":"dilogarithmus -wikipedia","name":"dilogarithmus -wikipedia","description":"before-content-x4 \u6570\u5b66\u3067\u306f\u3001\u3055\u307e\u3056\u307e\u306a\u7279\u5225\u306a\u6a5f\u80fd\u304c\u3042\u308a\u307e\u3059 dilogarithmus \u5c02\u7528\u3002\u53e4\u5178\u7684\u306a\u30c7\u30a3\u30ed\u30ac\u30ea\u30ba\u30e0\u306f\u3001Polylogarithm\u306e\u7279\u5225\u306a\u30b1\u30fc\u30b9\u3067\u3059\u3002 \u5b9f\u969b\u306e\u8ef8\u4e0a\u306e\u53e4\u5178\u7684\u306a\u30c7\u30a3\u30ed\u30ac\u30ea\u30ba\u30e0\u306e\u5024\u3002 \uff08\u60f3\u50cf\u4e0a\u306e\u90e8\u5206\u306f\u30bc\u30ed\u3068\u540c\u3058\u3067\u3059\u3002\uff09 after-content-x4 \u30af\u30e9\u30b7\u30c3\u30af\u30c7\u30a3\u30ed\u30ac\u30ea\u30ba\u30e0\u306f\u8907\u96d1\u306a\u6570\u5024\u7528\u3067\u3059 \u3068 {displaystyle with} \u3068 after-content-x4 | \u3068 | < \u521d\u3081 {\u5c55\u793a| z","datePublished":"2023-09-11","dateModified":"2023-09-11","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/all2jp\/wiki10\/archives\/author\/lordneo#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/all2jp\/wiki10\/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\/1\/1e\/Mplwp_dilogarithm.svg\/300px-Mplwp_dilogarithm.svg.png","url":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/1\/1e\/Mplwp_dilogarithm.svg\/300px-Mplwp_dilogarithm.svg.png","height":"200","width":"300"},"url":"https:\/\/wiki.edu.vn\/all2jp\/wiki10\/archives\/18598","wordCount":20162,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4\u6570\u5b66\u3067\u306f\u3001\u3055\u307e\u3056\u307e\u306a\u7279\u5225\u306a\u6a5f\u80fd\u304c\u3042\u308a\u307e\u3059 dilogarithmus \u5c02\u7528\u3002\u53e4\u5178\u7684\u306a\u30c7\u30a3\u30ed\u30ac\u30ea\u30ba\u30e0\u306f\u3001Polylogarithm\u306e\u7279\u5225\u306a\u30b1\u30fc\u30b9\u3067\u3059\u3002  \u5b9f\u969b\u306e\u8ef8\u4e0a\u306e\u53e4\u5178\u7684\u306a\u30c7\u30a3\u30ed\u30ac\u30ea\u30ba\u30e0\u306e\u5024\u3002 \uff08\u60f3\u50cf\u4e0a\u306e\u90e8\u5206\u306f\u30bc\u30ed\u3068\u540c\u3058\u3067\u3059\u3002\uff09        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4\u30af\u30e9\u30b7\u30c3\u30af\u30c7\u30a3\u30ed\u30ac\u30ea\u30ba\u30e0\u306f\u8907\u96d1\u306a\u6570\u5024\u7528\u3067\u3059 \u3068 {displaystyle with} \u3068        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4| \u3068 | < \u521d\u3081 {\u5c55\u793a| z | k=1\u221ezkk2mm\u5974\u96b7\u3068\u30bb\u30ec\u30eb &#8211; \u30da\u30a4\u30ea\u30fc\u30fb\u30cf\u30a4\u30fb\u30cf\u30c3\u30d4\uff09\u30de\u30de\uff09m\u00f6toomm\u00f6toopmkome hym hym hork 22-2 2-4-2 \u3002 \u5206\u6790\u7684\u7d99\u7d9a\u306b\u3088\u3063\u3066\u898b\u3064\u3051\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059 c \u2216 [ \u521d\u3081 \u3001 \u221e ] {displaystyle mathbb {c} setminus left [1\u3001infty\u53f3]} \u7d9a\u304f\uff1a        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4\u305d\u308c\u304b 2\u2061 \uff08 \u3068 \uff09\uff09 =  &#8211;  \u222b 0zln\u2061(1\u2212t)td t {displaystyle operatorname {li} _ {2}\uff08z\uff09=  &#8211;  int _ {0}^{z} {frac {ln\uff081-t\uff09} {t}}\u3001mathrm {d} t} \u3002 \uff08\u3053\u3053\u306b\u30d1\u30b9\u306b\u6cbf\u3063\u3066\u3044\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059 c \u2216 [ \u521d\u3081 \u3001 \u221e ] {displaystyle mathbb {c} setminus left [1\u3001infty\u53f3]} \u7d71\u5408\u3059\u308b\u305f\u3081\u306b\u3002\uff09 Bloch-wartner Dilogarithm\u306e\u5411\u3051\u3067\u3059 \u3068 \u2208 c {displaystyle zin mathbb {c}} \u306b\u3088\u3063\u3066\u5b9a\u7fa9\u3055\u308c\u307e\u3059 d 2\u2061 \uff08 \u3068 \uff09\uff09 = \u306e\u4e2d\u306b \u2061 \uff08 \u305d\u308c\u304b 2\u2061 \uff08 \u3068 \uff09\uff09 \uff09\uff09 + arg \u2061 \uff08 \u521d\u3081  &#8211;  \u3068 \uff09\uff09 ln \u2061 \uff08 | \u3068 | \uff09\uff09 {displaystyle operatorname {d} _ {2}\uff08z\uff09= operatorname {im}\uff08operatorname {li} _ {2}\uff08z\uff09\uff09+arg\uff081-z\uff09ln\uff08| z |\uff09} \u3002 \u5f7c\u306f\u3001\u660e\u78ba\u3067\u5b89\u5b9a\u3057\u3066\u3044\u307e\u3059 [ \u521d\u3081 \u3001 \u221e ] {displaystyle left [1\u3001infty\u53f3]} \u3002 \u5f7c\u306f\u5206\u6790\u7684\u3067\u3059 c \u2216 { 0 \u3001 \u521d\u3081 } {displaystyle mathbb {c} setminus left {0,1right}} \u30010\u30681\u3067\u5f7c\u306f\u30bf\u30a4\u30d7\u306e\u7279\u7570\u70b9\u3092\u6301\u3063\u3066\u3044\u307e\u3059 r ln \u2061 \uff08 r \uff09\uff09 {displaystyle rln\uff08r\uff09} \u3002 Rogers Dilogarithm\u306f\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u3066\u3044\u307e\u3059 l \uff08 \u30d0\u30c4 \uff09\uff09 = 6\u03c02\uff08 Li2\u2061(x)+12ln\u2061(x)ln\u2061(1\u2212x)\uff09\uff09 {displaystyle l\uff08x\uff09= {frac {6} {pi ^{2}}}\u5de6\uff08operatorname {li} _ {2}\uff08x\uff09+{frac {1} {2}} ln\uff08x\uff09ln\uff081-x\uff09}}} \u305f\u3081\u306b 0 < \u30d0\u30c4 < \u521d\u3081 {displaystyle 0 \uff08 \u30d0\u30c4 \uff09\uff09 ln \u2061 \uff08 \u521d\u3081  &#8211;  \u30d0\u30c4 \uff09\uff09  &#8211;  \u03c026{displaystyle r\uff08x\uff09= operatorname {li} _ {2}\uff08x\uff09+{frac {1} {2}} ln\uff08x\uff09ln\uff081-x\uff09 &#8211;  {frac {pi ^{2}} {6}}}}} \u3002 \u3053\u308c\u306f\u3001\u6700\u521d\u306b\u8a00\u53ca\u3055\u308c\u305f\u3082\u306e\u306b\u4f9d\u5b58\u3057\u307e\u3059 r \uff08 \u30d0\u30c4 \uff09\uff09 = \u03c026\uff08 l \uff08 \u30d0\u30c4 \uff09\uff09  &#8211;  \u521d\u3081 \uff09\uff09 {displaystyle r\uff08x\uff09= {frac {pi ^{2}} {6}}\uff08l\uff08x\uff09-1\uff09} \u4e00\u7dd2\u3002 \u3067\u304d\u307e\u3059 r {displaystyle r} \uff08\u4e0d\u5b89\u5b9a\u306a\uff09\u5168\u4f53\u306b r {displaystyle mathbb {r}} \u7d9a\u304f r \uff08 \u521d\u3081 \uff09\uff09 = 0 \u3001 r \uff08 0 \uff09\uff09 =  &#8211;  \u03c026 {displaystyle r\uff081\uff09= 0\u3001r\uff080\uff09=  &#8211;  {frac {pi ^{2}} {6}}}} \u3068 r \uff08 \u30d0\u30c4 \uff09\uff09 = { \u2212R(1\/x)\u00a0f\u00fcr\u00a0x>1\u2212R(x\/(x\u22121))\u00a0f\u00fcr\u00a0x 1 \\ -r\uff08x\/\uff08x-1\uff09\uff09\uff06{mbox {f\u00fcr}} x \u3002 \u591a\u5206 \u3068 {displaystyle e} 1\u3064\u4ee5\u4e0a Q {displaystyle mathbb {q}} \u5b9a\u7fa9\u3055\u308c\u305f\u6955\u5186\u66f2\u7dda\u3002 weierstra\u00dfefunction\u3092\u4f7f\u7528\u3059\u308b\u3068\u3001\u30b0\u30ea\u30c3\u30c9\u3067\u4f7f\u7528\u3067\u304d\u307e\u3059 l = { \u521d\u3081 \u3001 t } {displaystyle lambda =\u5de6{1\u3001\u53f3}}} \u3067\u30d1\u30e9\u30e1\u30fc\u30bf\u30fc\u5316\u3057\u307e\u3059 c \/ l \u2192 \u3068 \uff08 c \uff09\uff09 {displaystyle mathbb {c} \/lambda rightarrow e\uff08mathbb {c}\uff09} \u306e {displaystyleu} \u306b\u5bfe\u3057\u3066 l \u21a6 \uff08 p \uff08 \u306e \uff09\uff09 \u3001 p \u2032\uff08 \u306e \uff09\uff09 \uff09\uff09 {displaystyle lambda mapsto\uff08p\uff08u\uff09\u3001p^{prime}\uff08u\uff09\uff09} \u3002 \u6955\u5186\u5f62\u306e\u30c7\u30a3\u30ed\u30ac\u30ea\u30ba\u30e0 d \u3068 \uff1a \u3068 \uff08 c \uff09\uff09 \u2192 c {displaystyle d^{e} colon e\uff08mathbb {c}\uff09rightArrow mathbb {c}} \u6b21\u306b\u3001\u6b21\u306b\u5b9a\u7fa9\u3055\u308c\u307e\u3059 d E\uff08 p \uff08 \u306e \uff09\uff09 \u3001 p \u2032\uff08 \u306e \uff09\uff09 \uff09\uff09 = \u2211 n=\u2212\u221e\u221ed 2\uff08 \u305d\u3046\u3067\u3059 2\u03c0i(n\u03c4+u)\uff09\uff09 {displaystyle d^{e}\uff08p\uff08u\uff09\u3001p^{prime}\uff08u\uff09\uff09= sum _ {n = -infty}^{infty} d_ {2}\uff08e^{2pi i\uff08ntau +u\uff09}\uff09}} \u3001 \u3057\u305f\u304c\u3063\u3066 d 2 {displaystyle d_ {2}} Bloch-Bartner Dilogarithm\u3092\u793a\u3057\u307e\u3057\u305f\u3002 \u6955\u5186\u5f62\u306e\u30c7\u30a3\u30ed\u30ac\u30ea\u30ba\u30e0\u306f\u3001\u306e\u5408\u7406\u7684\u306a\u500d\u6570\u3092\u9664\u3044\u3066\u6b63\u3057\u3044 pi {displaystylepi} \u5024\u3067 l \uff08 \u3068 \u3001 2 \uff09\uff09 {displaystyle l\uff08e\u30012\uff09} L\u6a5f\u80fd\u3002 [\u521d\u3081] Table of Contents\u30af\u30e9\u30b7\u30c3\u30af\u30c7\u30a3\u30ed\u30ac\u30ea\u30ba\u30e0 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] Bloch-Wigner-Dilogarithmus [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] Rogers-dilogarithmus [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u30d0\u30b9\u30e9\u30fc\u306e\u554f\u984c [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u30af\u30e9\u30b7\u30c3\u30af\u30c7\u30a3\u30ed\u30ac\u30ea\u30ba\u30e0 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] Bloch-Wigner-Dilogarithmus [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] Rogers-dilogarithmus [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u30af\u30e9\u30b7\u30c3\u30af\u30c7\u30a3\u30ed\u30ac\u30ea\u30ba\u30e0 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u6b21\u306e\u6570\u5b57\u306b\u3064\u3044\u3066\u306f \u3068 {displaystyle with} \u3068 \u305d\u308c\u304b 2 \u2061 \uff08 \u3068 \uff09\uff09 {displaystyle operatorname {li} _ {2}\uff08z\uff09} \u9589\u3058\u305f\u5f62\u5f0f\u3067\u8868\u73fe\u3057\u307e\u3059\uff1a \u305d\u308c\u304b 2\u2061 \uff08  &#8211;  \u521d\u3081 \uff09\uff09 =  &#8211;  \u03c0212\u3001 \u305d\u308c\u304b 2\u2061 \uff08 0 \uff09\uff09 = 0 \u3001 {displaystyle operatorname {li} _ {2}\uff08 &#8211;  1\uff09=  &#8211;  {frac {{pi}^{2}} {12}}\u3001qquad operatorname {li} _ {2}\uff080\uff09= 0\u3001} \u305d\u308c\u304b 2\u2061 (12)= \u03c0212 &#8211;  12ln 2\u2061 \uff08 2 \uff09\uff09 \u3001 \u305d\u308c\u304b 2\u2061 \uff08 \u521d\u3081 \uff09\uff09 = \u03c026{displaystyle operatorname {li} _ {2} {biggl\uff08} {frac {1} {2}} {biggr\uff09} = {frac {{pi} ^{2}} {12}}  &#8211;  {frac {1} {2}} {2}} {2}} {2}} {2}}}} {2}\uff081\uff09= {frac {{pi}^{2}} {6}}}}} \u3001 \u305d\u308c\u304b 2\u2061 \uff08  &#8211;  \u30d5\u30a1\u30a4 \u22121\uff09\uff09 =  &#8211;  \u03c0215+ 12ln 2\u2061 \uff08 \u30d5\u30a1\u30a4 \uff09\uff09 \u3001 \u305d\u308c\u304b 2\u2061 \uff08  &#8211;  \u30d5\u30a1\u30a4 \uff09\uff09 =  &#8211;  \u03c0210 &#8211;  ln 2\u2061 \uff08 \u30d5\u30a1\u30a4 \uff09\uff09 \u3001 {displaystyle operaatorname {li} _ {2}\uff08-phi ^{ &#8211;  1}\uff09=  &#8211;  {frac {pi} ^{2}}}}}}+{frac {1}}}}}}}}}} {2}}\uff08Phi\uff09\u3001Qquad operator {} {{{2} {{{2}} {2}} ln ^{2}\uff08phi\uff09\u3001} \u305d\u308c\u304b 2\u2061 \uff08 \u30d5\u30a1\u30a4 \u22122\uff09\uff09 = \u03c0215 &#8211;  ln 2\u2061 \uff08 \u30d5\u30a1\u30a4 \uff09\uff09 \u3001 \u305d\u308c\u304b 2\u2061 \uff08 \u30d5\u30a1\u30a4 \u22121\uff09\uff09 = \u03c0210 &#8211;  ln 2\u2061 \uff08 \u30d5\u30a1\u30a4 \uff09\uff09 {displaystyle operatualName {li} _ {2}\uff08phi ^{-2}\uff09= {frac {pi} ^{2}}} {15}}  &#8211;  ln ^{2}\uff08phi\uff09\u3001qquad operatorname {li}} _ {f}\uff08phi ^{1}}}}}}}}}}}} {2}\uff08phi\uff09} \u3002 \u7565\u8a9e\u03c6\u3067\u306f\u3001\u30b4\u30fc\u30eb\u30c7\u30f3\u30ab\u30c3\u30c8\u306e\u6570\u304c\u8868\u73fe\u3055\u308c\u307e\u3059\u3002 \u30d5\u30a1\u30a4 = \uff08 5 + \u521d\u3081 \uff09\uff09 \/ 2 {displaystyle non- =\uff08{sqrt {5}}+1\uff09\/2}  6\u756a\u76ee\u306e\u30e6\u30cb\u30c3\u30c8\u30eb\u30fc\u30c8\u4ed8\u304d \u304a\u304a = \u521d\u3081 2 + 32 \u79c1 {displaystyle omega = {frac {1} {2}}+{frac {sqrt {3}} {}}} i} i} \u305d\u3057\u3066\u3001\u30ae\u30fc\u30ba\u306e\u5b9a\u6570 \u306e 0 = 1.014 9 &#8230; {displaystyle v_ {0} = 1 {\u3001} 0149ldots} \u3042\u306a\u305f\u3082\u6301\u3063\u3066\u3044\u307e\u3059 \u305d\u308c\u304b 2\u2061 \uff08 \u304a\u304a \uff09\uff09 = \u03c0236+ \u306e 0\u79c1 \u3001 \u305d\u308c\u304b 2\u2061 \uff08 \u304a\u304a 2\uff09\uff09 =  &#8211;  \u03c0218+ 23\u306e 0\u79c1 \u3001 {Displaystyle operatorName {Li} _ {2} (omega) = {Frac {pi ^{2}}}}}}+V_ {0} i, qquad operatorName {Li} _ {2} (Omega ^{2}) =-{Fract {2} {3}} V_ {0} i,} \u305d\u308c\u304b 2\u2061 \uff08 \u521d\u3081 + \u304a\u304a \uff09\uff09 = \u03c029+ \uff08 23V0+13ln\u2061(3)\u03c0\uff09\uff09 \u79c1 \u3001 \u305d\u308c\u304b 2\u2061 \uff08 11+\u03c9\uff09\uff09 = 5\u03c0272 &#8211;  18ln \u2061 \uff08 3 \uff09\uff09 + \uff08 \u221223V0+112ln\u2061(3)\u03c0\uff09\uff09 \u79c1 {displaystyle operatorname {Li} _{2}(1+omega )={frac {pi ^{2}}{9}}+left({frac {2}{3}}V_{0}+{frac {1}{3}}ln(3)pi right)i,qquad operatorname {Li} _{2}left({frac {1}{1+omega }}right)={frac {5pi ^{2}}{72}}-{frac {1}{8}}ln(3)+left(-{frac {2}{3}}V_{0}+{frac {1}{12}}ln(3)pi right)i} \u3002 Bloch-Wigner-Dilogarithmus [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u3053\u308c\u307e\u3067\u306e\u3068\u3053\u308d\u3001Bloch-Bartner Dilogarithm\u306e\u5024\u306f\u6570\u5024\u7684\u306b\u306e\u307f\u8a08\u7b97\u3067\u304d\u3001Bloch-Bartner Dilogarithm\u306e\u5024\u306e\u9593\u306b\u3044\u304f\u3064\u304b\u306e\u4ee3\u6570\u7684\u95a2\u4fc2\u3057\u304b\u308f\u304b\u308a\u307e\u305b\u3093\u3002\u30b8\u30e7\u30f3\u30fb\u30df\u30eb\u30ca\u30fc\u306b\u3088\u308b\u63a8\u5b9a\u306f\u8a00\u3044\u307e\u3059 n \u2265 3 {displaystyle ngeq 3} \uff1a \u652f\u6255\u3044 d 2\uff08 \u305d\u3046\u3067\u3059 2\u03c0ijN\uff09\uff09 {displaystyle d_ {2}\uff08e^{2pi i {frac {j} {n}}}\uff09} \u305f\u3081\u306b 0 < j < N2{displaystyle 0 52\uff09\uff09 = 25\u3001 l \uff08 5\u221212\uff09\uff09 = 35{l\uff080\uff09= 0\u306e\u8868\u793a\u30b9\u30bf\u30a4\u30eb\u3001queft\uff08{1 {{sql {{sql {{3} {1} {2} {{1} c {3} {5}}}}} \u3002 6\u756a\u76ee\u306e\u30e6\u30cb\u30c3\u30c8\u30eb\u30fc\u30c8\u4ed8\u304d \u304a\u304a = \u521d\u3081 2 + 32 \u79c1 {displaystyle omega = {frac {1} {2}}+{frac {sqrt {3}} {}}} i} i} \u305d\u3057\u3066\u3001\u30ae\u30fc\u30ba\u306e\u5b9a\u6570 \u306e 0 = \u521d\u3081 \u3001 0149 &#8230; {displaystyle v_ {0} = 1,0149 &#8230;} 1\u3064\u3042\u308a\u307e\u3059 r \uff08 \u304a\u304a \uff09\uff09 =  &#8211;  \u03c0212+ \u306e 0\u79c1 \u3001 r \uff08 \u304a\u304a 2\uff09\uff09 =  &#8211;  \u03c026+ \uff08 23V0+16ln\u2061(3)\u03c0\uff09\uff09 \u79c1 \u3001 {displaystyle r\uff08omega\uff09=  &#8211;  {frac {pi ^{2}} {12}+v_ {0} i\u3001qquad r\uff08omega ^{2}\uff09=  &#8211;  {frac {pi ^{2}} {6}}+let\uff08{{2}}}}}}}}}}+ } {6}} ln\uff083\uff09pi right\uff09i\u3001} r \uff08 \u521d\u3081 + \u304a\u304a \uff09\uff09 = \uff08 23V0+16ln\u2061(3)\u03c0\uff09\uff09 \u79c1 \u3001 r \uff08 11+\u03c9\uff09\uff09 =  &#8211;  \u03c0212 &#8211;  18ln \u2061 \uff08 3 \uff09\uff09 + 18ln 2\u2061 \uff08 3 \uff09\uff09 + \uff08 \u221223V0+112ln\u2061(3)\u03c0\uff09\uff09 \u79c1 {displaystyle r\uff081+omega\uff09= left\uff08{frac {2} {3} {3}} v_ {0}+{frac {1} {6}} ln\uff083\uff09pi right\uff09i\u3001qquad reft\uff08{frac {1} {1} {1+of {pi {pi {pi {pi} {pi {pi} {pi} }  &#8211;  {frac {1} {8}} ln\uff083\uff09+{frac {1} {8}} ln ^{2}\uff083\uff09+left\uff08 &#8211;  {frac {2} {3}} v_ {0}+{frac {1} {12}} {3\uff09pi reag \u30d0\u30b9\u30e9\u30fc\u306e\u554f\u984c [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] 1\u3064\u306e\u30c7\u30a3\u30ed\u30ac\u30ea\u30ba\u30e0\u306e\u4fa1\u5024\u306e\u8a3c\u62e0\u306f\u3001SO -Caled Basel\u306e\u554f\u984c\u3067\u6271\u308f\u308c\u307e\u3059\u3002\u3053\u306e\u8a3c\u62e0\u306f\u3001\u6b21\u306e\u65b9\u6cd5\u3067\u5b8c\u4e86\u3067\u304d\u307e\u3059\u3002 \u6b21\u306e\u95a2\u6570\u306b\u306f\u3001\u6b21\u306e\u5fae\u5206\u304c\u3042\u308a\u307e\u3059\u3002 ddx[2 \u305d\u308c\u304b 2\u2061 (xx2+1+1) &#8211;  12\u305d\u308c\u304b 2\u2061 (x2+1\u22121x2+1+1)]= arsinh\u2061(x)xx2+1{displaystyle {frac {mathrm {d}} {mathrm {d} x}} {biggl [} 2operatorname {li} _ {2} {biggl\uff08} {frac {x} {{sqrt {{sqrt {x^{2} +1}}}}}}}}}}} } {2}} operatorname {li} _ {2} {biggl\uff08} {frac {{sqrt {x^{2} +1}}  &#8211;  1} {{sqrt {x^{2} +1}}}}} {biggr\uff09{biggr\uff09{biggr\uff09{biggr\uff09{biggr\uff09 }\uff08x\uff09} {x {sqrt {x^{2} +1}}}}}} \u3057\u305f\u304c\u3063\u3066\u3001\u6b21\u306e\u7a4d\u5206\u304c\u9069\u7528\u3055\u308c\u307e\u3059\u3002 \u222b 0\u221earsinh\u2061(x)xx2+1d \u30d0\u30c4 = [2 \u305d\u308c\u304b 2\u2061 (xx2+1+1) &#8211;  12\u305d\u308c\u304b 2\u2061 (x2+1\u22121x2+1+1)]x=0x=\u221e= 32Li2\uff08 \u521d\u3081 \uff09\uff09 {displaystyle int _ {0}^{infty} {frac {operatorname {arsinh}\uff08x\uff09} {x {sqrt {x^{2} +1}}}}\u3001Mathrm {d} x = {biggl [} {{} {{} {} {{} {} {bigggl [} {bigggl [} {bigggl [} {biggl] } {{sqrt {x^{2} +1}}+1}} {biggr\uff09}  &#8211;  {frac {1} {2}} operatorname {li} _ {2} {biggl\uff08} {frac {{sqrt {x^{2} {2} {{2} {{2} {{2} {{2} {{2} {{2} {{2} {{2} {{2} {{2} {{2} {{2} {{2} {{2} {{2}\uff09 } +1}}+1}} {biggr\uff09} {biggr]}} _ {x = 0}^{x = infty} = {frac {3} {2}}\u3001{text {li}} _ {2}\uff081\uff09}}}}}}}}}}}}}}}} Fubini\u306e\u6587\u306f\u3053\u306e\u63a5\u7d9a\u3092\u63d0\u4f9b\u3057\u307e\u3059\u3002 \u222b 0\u221earsinh\u2061(x)xx2+1d \u30d0\u30c4 = \u222b 0\u221e\u222b 011\u2212x2y2+x2+1d \u3068 d \u30d0\u30c4 = \u222b 01\u222b 0\u221e1\u2212x2y2+x2+1d \u30d0\u30c4 d \u3068 = \u222b 01\u03c021\u2212y2d \u3068 = \u03c024{displaystyle int _ {0}^{infty} {frac {operatorname {arsinh}\uff08x\uff09} {x {sqrt {x^{2} +1}}}} mathrm {d} x = int _ {0} {{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{\u300d\uff09\uff09 x^{2} y^{2}+x^{2} +1}}\u3001mathrm {d} y\u3001mathrm {d} x = int _ {0}^{1} int _ {0}^{infty} {frac {1}} {-x^{2} {2} {2} {2} {2} {2} {2} {2} {d} x\u3001mathrm {d} y = int _ {0}^{1} {frac {pi} {2 {sqrt {1-y^{2}}}}}}\u3001mathrm {d} y = {frac {pi^{2}} {4}}}}}} \u4e0a\u8a18\u306e\u6700\u5f8c\u306e2\u3064\u306e\u5f0f\u3092\u7b49\u3057\u304f\u3059\u308b\u3053\u3068\u306b\u3088\u308a\u3001\u7d50\u679c\u304c\u53d6\u5f97\u3055\u308c\u307e\u3059\u3002 32Li2\uff08 \u521d\u3081 \uff09\uff09 = \u03c024{displaystyle {frac {3} {2}}\u3001{text {li}} _ {2}\uff081\uff09= {frac {pi ^{2}}}}}}}}}}}}}}} \u4e0a\u8a18\u306e\u5024\u306f\u89e3\u6c7a\u3055\u308c\u3066\u3044\u307e\u3059\uff1a Li2\uff08 \u521d\u3081 \uff09\uff09 = \u03c026{displaystyle {text {li}} _ {2}\uff081\uff09= {frac {pi ^{2}}}}}}}}} \u307e\u3055\u306b\u3053\u306e\u5024\u306f\u3001\u6b63\u65b9\u5f62\u306e\u6570\u5b57\u306e\u6383\u9664\u5024\u306e\u7121\u9650\u306e\u5408\u8a08\u3067\u3082\u3042\u308a\u307e\u3059\u3002 \u2211 n=1\u221e1n2= \u03c026{displaystyle sum _ {n = 1}^{infty} {frac {1} {n^{2}}} = {frac {pi^{2}}}}}}}}}}}}} \u3053\u306e\u4e8b\u5b9f\u306f\u3001Dilogarithm\u304b\u3089Maclaurin\u30b7\u30ea\u30fc\u30ba\u304b\u3089\u76f4\u63a5\u51fa\u73fe\u3057\u3066\u3044\u307e\u3059\u3002 \u30af\u30e9\u30b7\u30c3\u30af\u30c7\u30a3\u30ed\u30ac\u30ea\u30ba\u30e0 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u305f\u3068\u3048\u3070\u3001\u53e4\u5178\u7684\u306adilogarithm\u306f\u591a\u6570\u306e\u6a5f\u80fd\u65b9\u7a0b\u5f0f\u306b\u5341\u5206\u3067\u3059 \u305d\u308c\u304b 2\u2061 \uff08 \u3068 \uff09\uff09 + \u305d\u308c\u304b 2\u2061 \uff08  &#8211;  \u3068 \uff09\uff09 = 12\u305d\u308c\u304b 2\u2061 \uff08 \u3068 2\uff09\uff09 \u3001 {displaystyle operaatorname {li} _ {2}\uff08z\uff09+operatorname {li} _ {2}\uff08-z\uff09= {frac {1} {2}} operacame {li} _ {2}\uff08z^{2}\uff09\u3001}\u3001}} \u305d\u308c\u304b 2\u2061 \uff08 \u521d\u3081  &#8211;  \u3068 \uff09\uff09 + \u305d\u308c\u304b 2\u2061 \uff08 1\u22121z\uff09\uff09 =  &#8211;  ln2\u2061(z)2\u3001 {displaystyle operatorname {li} _ {2}\uff081-z\uff09+operatorname {li} _ {2}\u5de6\uff081- {frac {1} {z}}\u53f3\uff09=  &#8211;  {frac {ln ^{2}\uff08z\uff09} {2}} \u305d\u308c\u304b 2\u2061 \uff08 \u3068 \uff09\uff09 + \u305d\u308c\u304b 2\u2061 \uff08 \u521d\u3081  &#8211;  \u3068 \uff09\uff09 = \u03c026 &#8211;  ln \u2061 \uff08 \u3068 \uff09\uff09 de ln \u2061 \uff08 \u521d\u3081  &#8211;  \u3068 \uff09\uff09 \u3001 {displaystyle operaatorname {li} _ {2}\uff08z\uff09+operatorname {li} _ {2}\uff081-z\uff09= {frac {pi}^{2}}}}}}} -Ln\uff08z\uff09cdot ln\uff081-z\uff09\u3001}} \u305d\u308c\u304b 2\u2061 \uff08  &#8211;  \u3068 \uff09\uff09  &#8211;  \u305d\u308c\u304b 2\u2061 \uff08 \u521d\u3081  &#8211;  \u3068 \uff09\uff09 + 12\u305d\u308c\u304b 2\u2061 \uff08 \u521d\u3081  &#8211;  \u3068 2\uff09\uff09 =  &#8211;  \u03c0212 &#8211;  ln \u2061 \uff08 \u3068 \uff09\uff09 de ln \u2061 \uff08 \u3068 + \u521d\u3081 \uff09\uff09 \u3001 {displaystyle operaatorname {li} _ {2}\uff08-z\uff09-peratorname {li} _ {2}\uff081-z\uff09+{frac {1}}}} operatorne {li} _ {2}\uff081-z^{2}\uff09=-{pi {z {{2 \u3001} \u305d\u308c\u304b 2\u2061 \uff08 \u3068 \uff09\uff09 + \u305d\u308c\u304b 2\u2061 \uff08 1z\uff09\uff09 =  &#8211;  \u03c026 &#8211;  12ln 2\u2061 \uff08  &#8211;  \u3068 \uff09\uff09 \u3001 {displaystyle operatorname {li} _ {2}\uff08z\uff09+operatorname {li} _ {2} left\uff08{frac {1} {z}}\u53f3\uff09=  &#8211;  {frac {pi ^{2}} {6}}}  &#8211;  {1} {2}}}}}}}}}} \u305d\u308c\u304b 2\u2061 \uff08 \u3068 \uff09\uff09  &#8211;  14\u305d\u308c\u304b 2\u2061 \uff08 \u3068 2\uff09\uff09 = \u222b 01arcsin\u2061(xz)1\u2212x2d \u30d0\u30c4 {displaystyle operatorname {li} _ {2}\uff08z\uff09 &#8211;  {frac {1} {4}} operatorname {li} _ {2}\uff08z^{2}\uff09= int _ {0}^{1} {frac {sidin} {arcsin}\uff08x-x}\uff08x-sin} {arcsin}\uff09 }}}\u3001mathrm {d} x} \u3002\u3053\u308c\u3082\u6b21\u306e\u3068\u304a\u308a\u3067\u3059\u3002 \u305d\u308c\u304b 2\u2061 \uff08 \u521d\u3081 \uff09\uff09 = \u03c026{displaystyle operatorname {li} _ {2}\uff081\uff09= {frac {{pi}^{2}} {6}}}}} \u3002 Bloch-Wigner-Dilogarithmus [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] Bloch-Bartner Dilogarithm\u306fID\u3092\u6e80\u305f\u3057\u3066\u3044\u307e\u3059 d 2\u2061 \uff08 \u3068 \uff09\uff09 = d 2\u2061 \uff08 1\u22121z\uff09\uff09 = d 2\u2061 \uff08 11\u2212z\uff09\uff09 =  &#8211;  d 2\u2061 \uff08 1z\uff09\uff09 =  &#8211;  d 2\u2061 \uff08 \u521d\u3081  &#8211;  \u3068 \uff09\uff09 =  &#8211;  d 2\u2061 \uff08 \u2212z1\u2212z\uff09\uff09 {displaystyle operatorname {d} _ {2}\uff08z\uff09= operatorname {d} _ {2}\u5de6\uff081- {frac {1} {z}}\u53f3\uff09= operatorname {d} _ {2}\u5de6\uff08{1} {1-z} {1-z} {d-z}} {1-z} {d-z} {d-z}} {1-z}} {frac {1} {z}} right\uff09=  &#8211;  operatorname {d} _ {2}\uff081-z\uff09= -operatorname {d} _ {2}\u5de6\uff08{frac {z} {1-z}}\u53f3\uff09}} \u304a\u3088\u30735\u671f\u306e\u95a2\u4fc2 d 2\u2061 \uff08 \u30d0\u30c4 \uff09\uff09 + d 2\u2061 \uff08 \u3068 \uff09\uff09 + d 2\u2061 \uff08 1\u2212x1\u2212xy\uff09\uff09 + d 2\u2061 \uff08 \u521d\u3081  &#8211;  \u30d0\u30c4 \u3068 \uff09\uff09 + d 2\u2061 \uff08 1\u2212y1\u2212xy\uff09\uff09 = 0 {displaystyle operatorname {d} _ {2}\uff08x\uff09+operatorname {d} _ {2}\uff08y\uff09+operatorname {d} _ {2} left\uff08{frac {1-x} {1-xy}}\u53f3\uff09 {2}\u5de6\uff08{frac {1-y} {1-xy}}\u53f3\uff09= 0} \u3002 Rogers-dilogarithmus [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] Rogers Dilogarithm\u304c\u95a2\u4fc2\u3092\u6e80\u305f\u3057\u3066\u3044\u307e\u3059 l \uff08 \u30d0\u30c4 \uff09\uff09 + l \uff08 \u521d\u3081  &#8211;  \u30d0\u30c4 \uff09\uff09 = \u521d\u3081 {displaystyle l\uff08x\uff09+l\uff081- x\uff09= 1} \u30a2\u30d9\u30eb\u306e\u6a5f\u80fd\u65b9\u7a0b\u5f0f l \uff08 \u30d0\u30c4 \uff09\uff09 + l \uff08 \u3068 \uff09\uff09 = l \uff08 \u30d0\u30c4 \u3068 \uff09\uff09 + l \uff08 x(1\u2212y)1\u2212xy\uff09\uff09 + l \uff08 y(1\u2212x)1\u2212xy\uff09\uff09 {l\uff08x\uff09 + l\uff08y\uff09= l\uff08^ ll {{1-xyc}}}}}}}}}}}\u306e\u8868\u793a\u30b9\u30bf\u30a4\u30eb\uff09}} \u3002 \u305f\u3081\u306b r {displaystyle r} 1\u3064\u3042\u308a\u307e\u3059 r \uff08 \u30d0\u30c4 \uff09\uff09 + r \uff08 \u521d\u3081  &#8211;  \u30d0\u30c4 \uff09\uff09 =  &#8211;  \u03c026{displaystyle r\uff08x\uff09+r\uff081-x\uff09=  &#8211;  {frac {pi ^{2}} {6}}}} \u304a\u3088\u30735\u671f\u306e\u95a2\u4fc2 r \uff08 \u30d0\u30c4 \uff09\uff09  &#8211;  r \uff08 \u3068 \uff09\uff09 + r \uff08 yx\uff09\uff09  &#8211;  r \uff08 1\u2212x\u221211\u2212y\u22121\uff09\uff09 + r \uff08 1\u2212x1\u2212y\uff09\uff09 = 0 {displaystyle r\uff08x\uff09-r\uff08y\uff09+rleft\uff08{frac {y} {x}} right\uff09-rleft\uff08{frac {1-x^{-1}} {1-y^{-1}}}}+rleft\uff08{frac {1-x} {1-y-y}}}}} \u3001 \u7279\u306b\u305d\u3046\u3067\u3059 r {displaystyle r} Bloch\u30b0\u30eb\u30fc\u30d7\u306e\u660e\u78ba\u306b\u5b9a\u7fa9\u3055\u308c\u305f\u95a2\u6570\u3002 \u6b21\u306e\u65b9\u7a0b\u5f0f\u304c\u9069\u7528\u3055\u308c\u307e\u3059 \u306e \u2260 0 {displaystyle vnot = 0} \u3068 "},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/all2jp\/wiki10\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/all2jp\/wiki10\/archives\/18598#breadcrumbitem","name":"dilogarithmus -wikipedia"}}]}]