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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Spence's function ： ウィキペディア英語版 Spence's function In mathematics, Spence's function, or dilogarithm, denoted as Li2(''z''), is a particular case of the polylogarithm. Two related special functions are referred to as Spence's function, the dilogarithm itself: ::$$ \operatorname_2(z) = -\int_0^z\, \mathrmu \textz \in\mathbb \setminus [1,\infty) and its reflection. For an infinite series also applies (the integral definition constitutes its analytical extension to the complex plane): ::$$ \operatorname_2(z) = \sum_^\infty . Alternatively, the dilogarithm function is sometimes defined as ::$$ \int_^ \frac \mathrmt = \operatorname_2(1-v). In hyperbolic geometry the dilogarithm $\backslash operatorname\_2(z)$ occurs as the hyperbolic volume of an ideal simplex whose ideal vertices have cross ratio $z$. Lobachevsky's function and Clausen's function are closely related functions. William Spence, after whom the function was named by early writers in the field, was a Scottish mathematician working in the early nineteenth century.〔http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Spence.html〕 He was at school with John Galt,〔http://www.biographi.ca/009004-119.01-e.php?BioId=37522〕 who later wrote a biographical essay on Spence. ==Identities== :$\backslash operatorname\_2(z)+\backslash operatorname\_2(-z)=\backslash frac\backslash operatorname\_2(z^2)$〔Zagier〕 :$\backslash operatorname\_2(1-z)+\backslash operatorname\_2\backslash left(1-\backslash frac\backslash right)=-\backslash frac$ :$\backslash operatorname\_2(z)+\backslash operatorname\_2(1-z)=\backslash frac^2\}-\backslash ln\; z\; \backslash cdot\backslash ln(1-z)$〔 :$\backslash operatorname\_2(-z)-\backslash operatorname\_2(1-z)+\backslash frac\backslash operatorname\_2(1-z^2)=-\backslash frac\; ^2\}-\backslash ln\; z\; \backslash cdot\; \backslash ln(z+1)$〔 :$\backslash operatorname\_2(z)\; +\backslash operatorname\_2(\backslash frac)\; =\; -\; \backslash frac\; -\; \backslash frac\backslash ln^2(-z)$〔 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Spence's function」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.