Sine and cosine transforms

Summary

The sine and cosine transforms are the real forms of the Fourier transform for functions on the half-line $[0, \infty)$. The cosine transform integrates $f$ against $\cos(2\pi x \xi)$, and the sine transform integrates it against $\sin(2\pi x \xi)$. They are the Fourier transforms of the even and of the odd extension of $f$, up to constant factors. They are used to solve boundary value problems on a half-line, where the boundary condition picks the even or the odd extension. See the definition.