Fourier Series

a n 0, b n 2 0fxsinnx dx .1.4.3 Decomposition into even and odd functions.It has already been shown that a general function fx on an arbitrary interval −a,a can bewritten as the sum of an even and an odd function by the decompositionfx 1 fx f−x 2 1fx − f−x2 f E x f O x Even function Odd function .In the context of a Fourier series, the sums of cosine and sine terms must correspond toeven and odd functions respectively as the individual terms are themselves either even orodd.Suppose that a function fx is defined on −,, is there any benefit to be gained bydecomposing it into an even and an odd component before the Fourier series is sought? Inmost cases the answer is negative, since only the Fourier series is required and the even andodd components are associated with the cosine and sine terms. However, it mayoccasionally be easier to determine the series by considering the even and odd componentsseparately, with the advantage of requiring only the half-range formulae.ExampleConsider the function defined byfx −x − x 0 x − 0 x and decompose this into the sum of even and odd functions.It is not immediately obvious from the schematic in figure 1.4.1 how the even and oddcharacteristics of the function will manifest themselves but the presence of the |x|component suggests that a decomposition should not be difficult to obtain. Follow thestandard procedures, ensuring that the implementation is made separately for x 0andx 0. With the suffices E and O denoting the even and odd components respectively, theeven component is23