Fourier sine series

The pure-sine Fourier series an odd periodic function's expansion reduces to.

For an odd function, orthogonality forces every cosine coefficient $a_n$ to vanish, leaving only sine terms. What remains is called the Fourier sine series of $f$ -- a Fourier series with no constant term and no cosine terms at all, built entirely from the odd extension of a function originally defined only on half an interval.

Examples

The Fourier series of $f(x)=x$ on $(-\pi,\pi)$ is a pure Fourier sine series, since $f$ is odd.