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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 Mellin transform ： ウィキペディア英語版 Mellin transform In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is often used in number theory, mathematical statistics, and the theory of asymptotic expansions; it is closely related to the Laplace transform and the Fourier transform, and the theory of the gamma function and allied special functions. The Mellin transform of a function ''f'' is :$\backslash left\backslash (s)\; =\; \backslash varphi(s)=\backslash int\_0^\; x^\; f(x)dx.$ The inverse transform is :$\backslash left\backslash \backslash varphi\backslash right\backslash \}(x)\; =\; f(x)=\backslash frac\; \backslash int\_^\; x^\; \backslash varphi(s)\backslash ,\; ds.$ The notation implies this is a line integral taken over a vertical line in the complex plane. Conditions under which this inversion is valid are given in the Mellin inversion theorem. The transform is named after the Finnish mathematician Hjalmar Mellin. ==Relationship to other transforms== The two-sided Laplace transform may be defined in terms of the Mellin transform by :$\backslash left\backslash (s)\; =\; \backslash left\backslash (s)$ and conversely we can get the Mellin transform from the two-sided Laplace transform by :$\backslash left\backslash (s)\; =\; \backslash left\backslash )\backslash right\backslash \}(s).$ The Mellin transform may be thought of as integrating using a kernel ''x''''s'' with respect to the multiplicative Haar measure, $\backslash frac$, which is invariant under dilation $x\; \backslash mapsto\; ax$, so that $\backslash frac\; =\; \backslash frac;$ the two-sided Laplace transform integrates with respect to the additive Haar measure $dx$, which is translation invariant, so that $d(x+a)\; =\; dx$. We also may define the Fourier transform in terms of the Mellin transform and vice versa; if we define the two-sided Laplace transform as above, then :$\backslash left\backslash (-s)\; =\; \backslash left\backslash (-is)$ = \left\(-is). We may also reverse the process and obtain :$\backslash left\backslash (s)\; =\; \backslash left\backslash )\backslash right\backslash \}(s)\; =\; \backslash left\backslash )\backslash right\backslash \}(-is).$ The Mellin transform also connects the Newton series or binomial transform together with the Poisson generating function, by means of the Poisson–Mellin–Newton cycle. 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「Mellin transform」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.