DefinitionThe half-range sine and cosine expansions, also called the half range expansion as a composition of an even and odd function, respectively, are terms in a Fourier series where a function defined on a finite interval [0,L], is extended as an even or odd part of the function on the interval [−L,L]. This extension is convenient for the analysis and solution of boundary value problems because it allows the function to be represented by either a sine (odd extension) or cosine (even extension) series. Partial differential equations, heat conductance, wave equations Half-range expansions Example: Suppose we have a function f(x) defined on [0,L] which can be expanded as f(x)=∑n=1∞Bnsin(Lnπx) for an odd extension.
