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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 Sine and cosine transforms ： ウィキペディア英語版 Sine and cosine transforms In mathematics, the Fourier sine and cosine transforms are forms of the Fourier integral transform that do not use complex numbers. They are the forms originally used by Joseph Fourier and are still preferred in some applications, such as signal processing or statistics. ==Definition== The Fourier sine transform of , sometimes denoted by either $^s$ or $\_s\; (f)$, is :$\backslash int\_^\backslash infty\; f(t)\backslash sin(2\backslash pi\; \backslash nu\; t)\; \backslash ,dt.$ If means time, then is frequency in cycles per unit time, but in the abstract, they can be any pair of variables which are dual to each other. This transform is necessarily an odd function of frequency, i.e. for all : :$^s(\backslash nu)\; =\; -\; ^s(-\backslash nu).$ The numerical factors in the Fourier transforms are defined uniquely only by their product. Here, in order that the Fourier inversion formula not have any numerical factor, the factor of 2 appears because the sine function has norm of $\backslash tfrac.$ The Fourier cosine transform of , sometimes denoted by either $^c$ or $\_c\; (f)$, is :$\backslash int\_^\backslash infty\; f(t)\backslash cos(2\backslash pi\; \backslash nu\; t)\; \backslash ,dt.$ It is necessarily an even function of frequency, i.e. for all : :$^c(\backslash nu)\; =\; ^c(-\backslash nu).$ Some authors〔Mary L. Boas, ''Mathematical Methods in the Physical Sciences'', 2nd Ed, John Wiley & Sons Inc, 1983. ISBN 0-471-04409-1〕 only define the cosine transform for even functions of , in which case its sine transform is zero. Since cosine is also even, a simpler formula can be used, :$2\; \backslash int\_0^\backslash infty\; f(t)\backslash cos(2\backslash pi\; \backslash nu\; t)\; \backslash ,dt.$ Similarly, if is an odd function, then the cosine transform is zero and the sine transform can be simplified to :$2\backslash int\_0^\backslash infty\; f(t)\backslash sin(2\backslash pi\; \backslash nu\; t)\; \backslash ,dt.$ 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「Sine and cosine transforms」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.