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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 Neumann polynomial ： ウィキペディア英語版 Neumann polynomial In mathematics, a Neumann polynomial, introduced by Carl Neumann for the special case $\backslash alpha=0$, is a polynomial in 1/''z'' used to expand functions in term of Bessel functions.〔Abramowitz and Stegun, (p. 363, 9.1.82 ) ff.〕 The first few polynomials are :$O\_0^(t)=\backslash frac\; 1\; t,$ :$O\_1^(t)=2\backslash frac\; ,$ :$O\_2^(t)=\backslash frac\; +\; 4\backslash frac\; ,$ :$O\_3^(t)=2\backslash frac\; +\; 8\backslash frac\; ,$ :$O\_4^(t)=\backslash frac\; +\; 4\backslash frac\; +\; 16\backslash frac\; .$ A general form for the polynomial is :$O\_n^(t)=\; \backslash frac\; \backslash sum\_^\; (-1)^\backslash frac\; \backslash left(\backslash frac\; 2\; t\; \backslash right)^,$ they have the generating function :$\backslash frac\; \backslash frac\; 1\; =\; \backslash sum\_O\_n^(t)\; J\_(z),$ where ''J'' are Bessel functions. To expand a function ''f'' in form :$f(z)=\backslash sum\_\; a\_n\; J\_(z)\backslash ,$ for compute :$a\_n=\backslash frac\; 1\; \backslash oint\_\; \backslash fracf(z)\; O\_n^(z)\backslash mathrm\; d\; z,$ where ==Examples== An example is the extension :$\backslash left(\backslash tfracz\backslash right)^s=\; \backslash Gamma(s)\backslash cdot\backslash sum\_(-1)^k\; J\_(z)(s+2k)$ or the more general Sonine formula〔 II.7.10.1, p.64〕 :$e^=\; \backslash Gamma(s)\backslash cdot\backslash sum\_i^k\; C\_k^(\backslash gamma)(s+k)\backslash frac.$ where $C\_k^$ is Gegenbauer's polynomial. Then, :$\backslash fracJ\_s(z)=\; \backslash sum\_(-1)^(s+2i)J\_(z),$ :$\backslash sum\_\; t^n\; J\_(z)=\; \backslash frac\}\; \backslash sum\_\backslash frac\backslash right)^j\}\backslash frac\backslash right)\}=\; \backslash int\_0^\backslash infty\; e^\}\backslash frac\; \backslash frac\backslash ,dx,$ the confluent hypergeometric function :$M(a,s,z)=\; \backslash Gamma\; (s)\; \backslash sum\_^\backslash infty\; \backslash left(-\backslash frac\backslash right)^k\; L\_k^(t)\; \backslash frac\backslash right)\}\}$ and in particular :$\backslash frac=\; \backslash frace^\backslash sum\_L\_k^\backslash left(\backslash frac4\backslash right)(4\; i\; z)^k\; \backslash frac\backslash right)\}\},$ the index shift formula :$\backslash Gamma(\backslash nu-\backslash mu)\; J\_\backslash nu(z)=\; \backslash Gamma(\backslash mu+1)\; \backslash sum\_\backslash frac\; \backslash left(\backslash frac\; z\; 2\backslash right)^J\_(z),$ the Taylor expansion (addition formula) :$\backslash frac\}=\; \backslash sum\_\backslash frac\backslash frac\; J\_(z)$ are of the same type. 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「Neumann polynomial」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.