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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 or , is : 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 : : 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 The Fourier cosine transform of , sometimes denoted by either or , is : It is necessarily an even function of frequency, i.e. for all : : 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, : Similarly, if is an odd function, then the cosine transform is zero and the sine transform can be simplified to : 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
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