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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 Abel–Plana formula ： ウィキペディア英語版 Abel–Plana formula In mathematics, the Abel–Plana formula is a summation formula discovered independently by and . It states that :$\backslash sum\_^\backslash infty\; f(n)=\; \backslash int\_0^\backslash infty\; f(x)\; \backslash ,\; dx+\; \backslash frac\; 1\; 2\; f(0)+i\; \backslash int\_0^\backslash infty\; \backslash frac\; \backslash ,\; dt.$ It holds for functions ''f'' that are holomorphic in the region Re(''z'') ≥ 0, and satisfy a suitable growth condition in this region; for example it is enough to assume that |''f''| is bounded by ''C''/|''z''|1+ε in this region for some constants ''C'', ε > 0, though the formula also holds under much weaker bounds. . An example is provided by the Hurwitz zeta function, :$\backslash zeta(s,\backslash alpha)=\; \backslash sum\_^\backslash infty\; \backslash frac\}\; +\; \backslash frac\; 1\; +\; 2\backslash int\_0^\backslash infty\backslash frac\backslash frac.$ Abel also gave the following variation for alternating sums: :$\backslash sum\_^\backslash infty\; (-1)^nf(n)=\; \backslash frac\; f(0)+i\; \backslash int\_0^\backslash infty\; \backslash frac\; \backslash ,\; dt.$ == See also == *Euler–Maclaurin summation formula 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「Abel–Plana formula」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.