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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 Hurwitz zeta function ： ウィキペディア英語版 Hurwitz zeta function In mathematics, the Hurwitz zeta function, named after Adolf Hurwitz, is one of the many zeta functions. It is formally defined for complex arguments ''s'' with Re(''s'') > 1 and ''q'' with Re(''q'') > 0 by :$\backslash zeta(s,q)\; =\; \backslash sum\_^\backslash infty\; \backslash frac\{(q+n)^\{s\}\}.$ This series is absolutely convergent for the given values of ''s'' and ''q'' and can be extended to a meromorphic function defined for all ''s''≠1. The Riemann zeta function is ζ(''s'',1). ==Analytic continuation== If $Re(s)\; \backslash leq\; 1$ the Hurwitz zeta function can be defined by the equation :$\backslash zeta\; (s,q)=\backslash Gamma(1-s)\backslash frac\; \backslash int\_C\; \backslash frac\}dz$ where the contour $C$ is a loop around the negative real axis. This provides an analytic continuation of $\backslash zeta\; (s,q)$. The Hurwitz zeta function can be extended by analytic continuation to a meromorphic function defined for all complex numbers $s$ with $s\; \backslash neq\; 1$. At $s\; =\; 1$ it has a simple pole with residue $1$. The constant term is given by :$\backslash lim\_\; \backslash left(\backslash zeta\; (s,q)\; -\; \backslash frac\backslash right)\; =$ \frac = -\psi(q) where $\backslash Gamma$ is the Gamma function and $\backslash psi$ is the digamma function. 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「Hurwitz zeta 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.