Fourier Series

Figure 1.4.2. Decomposition into even and odd componentsThe resulting components of the FS are given bya n 2 0fE xcosnx dx, b n 2 0fO x sinnx dx ,which are straightforward to evaluate.1.4.4 Half-range SeriesSuppose that the function is defined on 0,. Any function defined on just this domainwill not be 2-periodic and not amenable to existing techniques, as we do not yet have thetechniques available for a domain of arbitrary length.However we can easily construct a new function Fx that is 2-periodic, defined upon−, and coincides with fx on 0,. There are an infinite number of possibilities andin the first instance we employ either a Half-range Sine Series or a Half-range CosineSeries, as these are the simplest such extensions and can have evenness or oddness bychoice. Each of these representations only represents the original function in 0,, hence"half-range". This is easiest seen by an example.Example 1Suppose that fx cosx in 0,. We extend this to −, as a half-range sine series by25