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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 Bessel–Clifford function ： ウィキペディア英語版 Bessel–Clifford function In mathematical analysis, the Bessel–Clifford function, named after Friedrich Bessel and William Kingdon Clifford, is an entire function of two complex variables that can be used to provide an alternative development of the theory of Bessel functions. If :$\backslash pi(x)\; =\; \backslash frac\; =\; \backslash frac$ is the entire function defined by means of the reciprocal Gamma function, then the Bessel–Clifford function is defined by the series :$\_n(z)\; =\; \backslash sum\_^\backslash infty\; \backslash pi(k+n)\; \backslash frac$ The ratio of successive terms is ''z''/''k''(''n'' + ''k''), which for all values of ''z'' and ''n'' tends to zero with increasing ''k''. By the ratio test, this series converges absolutely for all ''z'' and ''n'', and uniformly for all regions with bounded |''z''|, and hence the Bessel–Clifford function is an entire function of the two complex variables ''n'' and ''z''. == Differential equation of the Bessel–Clifford function == It follows from the above series on differentiating with respect to ''x'' that $\_n(x)$ satisfies the linear second-order homogeneous differential equation :$xy\text{'}\text{'}\; +\; (n+1)y\text{'}\; =\; y.\; \backslash qquad$ This equation is of generalized hypergeometric type, and in fact the Bessel–Clifford function is up to a scaling factor a Pochhammer–Barnes hypergeometric function; we have :$\_n(z)\; =\; \backslash pi(n)\backslash \; \_0F\_1(;n+1;\; z).$ Unless n is a negative integer, in which case the right hand side is undefined, the two definitions are essentially equivalent; the hypergeometric function being normalized so that its value at ''z'' = 0 is one. 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「Bessel–Clifford function」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.