Half-range expansion

The sine or cosine series obtained by extending a function, defined only on half an interval, oddly or evenly.

A half-range expansion is the Fourier sine or cosine series obtained by choosing to extend a function $f$ defined only on $(0,L)$ -- oddly, to get a half-range sine expansion, or evenly, to get a half-range cosine expansion -- onto $(-L,L)$ , when $f$ itself was never given a value there. The same function on $(0,L)$ has two entirely different, equally valid half-range expansions, one of each kind.

Examples

$f(x)=1$ on $(0,\pi)$ has a half-range cosine expansion that is just the constant $1$ , but a half-range sine expansion that is an infinite series.