Fourier cosine series

The even-function mirror image of the Fourier sine series -- pure cosines plus a constant.

For an even function, orthogonality forces every sine coefficient $b_n$ to vanish, leaving a constant term plus cosine terms only. This is the Fourier cosine series of $f$ , the mirror image of the Fourier sine series for functions with the opposite symmetry.

Examples

The Fourier series of $f(x)=x^2$ on $(-\pi,\pi)$ is a pure Fourier cosine series, since $f$ is even.