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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 quarter period ： ウィキペディア英語版 quarter period In mathematics, the quarter periods ''K''(''m'') and i''K'' ′(''m'') are special functions that appear in the theory of elliptic functions. The quarter periods ''K'' and i''K'' ′ are given by :$K(m)=\backslash int\_0^\}\; \backslash frac\}K\text{'}(m)\; =\; \}\; u\backslash ,$ and $\}K\text{'}\backslash ,$ . The quarter periods are essentially the elliptic integral of the first kind, by making the substitution $k^2=m\backslash ,$. In this case, one writes $K(k)\backslash ,$ instead of $K(m)\backslash ,$, understanding the difference between the two depends notationally on whether $k\backslash ,$ or $m\backslash ,$ is used. This notational difference has spawned a terminology to go with it: * $m\backslash ,$ is called the parameter * $m\_1=\; 1-m\; \backslash ,$ is called the complementary parameter * $k\backslash ,$ is called the elliptic modulus * $k\text{'}\; \backslash ,$ is called the complementary elliptic modulus, where $^2=m\_1\backslash ,\backslash !$ * $\backslash alpha\backslash ,\backslash !$ the modular angle, where $k=\backslash sin\; \backslash alpha\backslash ,\backslash !$ * $\backslash frac-\backslash alpha\backslash ,\backslash !$ the complementary modular angle. Note that :$m\_1=\backslash sin^2\backslash left(\backslash frac-\backslash alpha\backslash right)=\backslash cos^2\; \backslash alpha.\backslash ,\backslash !$ The elliptic modulus can be expressed in terms of the quarter periods as :$k=\backslash textrm\; (K+\; K\backslash ,$ where ns and dn Jacobian elliptic functions. The nome $q\backslash ,$ is given by :$q=e^\}.\backslash ,$ The complementary nome is given by :$q\_1=e^\}.\backslash ,$ The real quarter period can be expressed as a Lambert series involving the nome: :$K=\backslash frac\; +\; 2\backslash pi\backslash sum\_^\backslash infty\; \backslash frac\{1+q^\{2n\}\}.\backslash ,$ Additional expansions and relations can be found on the page for elliptic integrals. ==References== * Milton Abramowitz and Irene A. Stegun, ''Handbook of Mathematical Functions'', (1964) Dover Publications, New York. ISBN 0-486-61272-4. See chapters 16 and 17. 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「quarter period」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.