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.The Fourier sine transform of  f (t), sometimes denoted by either f ^ s {displaystyle {hat {f}}^{s}}   or F s ( f ) {displaystyle {mathcal {F}}_{s}(f)}  , isThe original function  f  can be recovered from its transform under the usual hypotheses, that  f  and both of its transforms should be absolutely integrable. For more details on the different hypotheses, see Fourier inversion theorem.The form of the Fourier transform used more often today is