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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 Two-sided Laplace transform ： ウィキペディア英語版 Two-sided Laplace transform In mathematics, the two-sided Laplace transform or bilateral Laplace transform is an integral transform equivalent to probability's moment generating function. Two-sided Laplace transforms are closely related to the Fourier transform, the Mellin transform, and the ordinary or one-sided Laplace transform. If ''ƒ''(''t'') is a real or complex valued function of the real variable ''t'' defined for all real numbers, then the two-sided Laplace transform is defined by the integral :$\backslash mathcal\backslash \; =\; F(s)\; =\; \backslash int\_^\backslash infty\; e^\; f(t)\backslash ,\; dt.$ The integral is most commonly understood as an improper integral, which converges if and only if each of the integrals :$\backslash int\_0^\backslash infty\; e^\; f(t)\; \backslash ,\; dt,\backslash quad\; \backslash int\_^0\; e^\; f(t)\backslash ,\; dt$ exists. There seems to be no generally accepted notation for the two-sided transform; the $\backslash mathcal$ used here recalls "bilateral". The two-sided transform used by some authors is :$\backslash mathcal\backslash \; =\; s\backslash mathcal\backslash \; =\; sF(s)\; =\; s\; \backslash int\_^\backslash infty\; e^\; f(t)\backslash ,\; dt.$ In pure mathematics the argument ''t'' can be any variable, and Laplace transforms are used to study how differential operators transform the function. In science and engineering applications, the argument ''t'' often represents time (in seconds), and the function ''ƒ''(''t'') often represents a signal or waveform that varies with time. In these cases, the signals are transformed by filters, that work like a mathematical operator, but with a restriction. They have to be causal, which means that the output in a given time ''t'' cannot depend on an output which is a higher value of ''t''. In population ecology, the argument ''t'' often represents spatial displacement in a dispersal kernel. When working with functions of time, ''ƒ''(''t'') is called the time domain representation of the signal, while ''F''(''s'') is called the s-domain (or ''Laplace domain'') representation. The inverse transformation then represents a ''synthesis'' of the signal as the sum of its frequency components taken over all frequencies, whereas the forward transformation represents the ''analysis'' of the signal into its frequency components. ==Relationship to other integral transforms== If ''u''(''t'') is the Heaviside step function, equal to zero when ''t'' is less than zero, to one-half when ''t'' equals zero, and to one when ''t'' is greater than zero, then the Laplace transform $\backslash mathcal$ may be defined in terms of the two-sided Laplace transform by :$\backslash mathcal\backslash \; =\; \backslash mathcal\backslash .$ On the other hand, we also have :$\backslash mathcal\backslash (s)\; =\; \backslash mathcal\backslash (s)\; +\; \backslash mathcal\backslash (-s),$ so either version of the Laplace transform can be defined in terms of the other. The Mellin transform may be defined in terms of the two-sided Laplace transform by :$\backslash mathcal\backslash (s)\; =\; \backslash mathcal\backslash (s),$ and conversely we can get the two-sided transform from the Mellin transform by :$\backslash mathcal\backslash (s)\; =\; \backslash mathcal\backslash (s).$ The Fourier transform may also be defined in terms of the two-sided Laplace transform; here instead of having the same image with differing originals, we have the same original but different images. We may define the Fourier transform as :$\backslash mathcal\backslash \; =\; F(s\; =\; i\backslash omega)\; =\; F(\backslash omega).$ Note that definitions of the Fourier transform differ, and in particular :$\backslash mathcal\backslash \; =\; F(s\; =\; i\backslash omega)\; =\; \backslash frac\backslash (s)$ is often used instead. In terms of the Fourier transform, we may also obtain the two-sided Laplace transform, as :$\backslash mathcal\backslash (s)\; =\; \backslash mathcal\backslash (-is).$ The Fourier transform is normally defined so that it exists for real values; the above definition defines the image in a strip which may not include the real axis. The moment-generating function of a continuous probability density function ''ƒ''(''x'') can be expressed as $\backslash mathcal\backslash (-s)$. 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「Two-sided Laplace 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.