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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 Riemann's differential equation ： ウィキペディア英語版 Riemann's differential equation In mathematics, Riemann's differential equation, named after Bernhard Riemann, is a generalization of the hypergeometric differential equation, allowing the regular singular points to occur anywhere on the Riemann sphere, rather than merely at 0, 1, and $\backslash infty$. The equation is also known as the Papperitz equation. The hypergeometric differential equation is a second-order linear differential equation which has three regular singular points, 0, 1 and $\backslash infty$. That equation admits two linearly independent solutions; near a singularity $z\_s$, the solutions take the form $x^s\; f(x)$, where $x\; =\; z-z\_s$ is a local variable, and $f$ is locally holomorphic with $f(0)\backslash neq0$. The real number $s$ is called the exponent of the solution at $z\_s$. Let ''α'', ''β'' and ''γ'' be the exponents of one solution solution at 0, 1 and & $\backslash infty$ respectively; and let ''α''', ''β''' and ''γ''' be that of the other. Then :$\backslash alpha\; +\; \backslash alpha\text{'}\; +\; \backslash beta\; +\; \backslash beta\text{'}\; +\; \backslash gamma\; +\; \backslash gamma\text{'}\; =\; 1.$ By applying suitable changes of variable, it is possible to transform the hypergeometric equation: Applying Möbius transformations will adjust the positions of the RSPs, while other transformations (see below) can change the exponents at the RSPs , subject to the exponents adding up to 1. ==Definition== The differential equation is given by : \frac + \frac \right ) \frac :: ■ウィキペディアで「Riemann's differential equation」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.