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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 Whipple's transformation of Legendre functions ： ウィキペディア英語版 Whipple formulae In the theory of special functions, Whipple's transformation for Legendre functions, named after Francis John Welsh Whipple, arise from a general expression, concerning associated Legendre functions. These formulae have been presented previously in terms of a viewpoint aimed at spherical harmonics, now that we view the equations in terms of toroidal coordinates, whole new symmetries of Legendre functions arise. For associated Legendre functions of the first and second kind, :$P\_^\backslash biggl(\backslash frace^\; Q\_\backslash nu^\backslash mu(z)\}$ and :$Q\_^\backslash biggl(\backslash frac\backslash Gamma(-\backslash nu-\backslash mu)(z^2-1)^e^\; P\_\backslash nu^\backslash mu(z).$ These expressions are valid for all parameters $\backslash nu,\; \backslash mu,$ and $z$. By shifting the complex degree and order in an appropriate fashion, we obtain Whipple formulae for general complex index interchange of general associated Legendre functions of the first and second kind. These are given by :$$ P_^\mu(z)=\frac}\biggl(P_^\nu\biggl(\fracQ_^\nu\biggl(\frac^\mu(z)=\frac}^\nu\biggl(\frace^\sin\nu\pi" TITLE=" P_^\nu\biggl(\frace^\sin\nu\pi">Q_^\nu\biggl(\frac^n(\cosh\eta)=\frac\sqrt}Q_^m(\coth\eta) and :$$ Q_^n(\cosh\eta)=\frac\sqrt}P_^m(\coth\eta) . These are the Whipple formulae for toroidal harmonics. They show an important property of toroidal harmonics under index (the integers associated with the order and the degree) interchange. ==External links== *() 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「Whipple formulae」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.