Fourier Series

c n 120e i1−nx dx in − 1 −1n−1 − 1and this is only non-zero when n is even, so we proceed by writing n 2m in the usualway. The final series isfx ~ 1 2 eix− 2i ∑m−e 2imx2m − 1The above method is not the only way of considering the FS of a periodic complex-valuedfunction of a real variable. However, it is the simplest representation and should beemployed whenever possible.The alternative method is to write fx in the formfx f 1 x if 2 x (1.7.6)with both f 1 x and f 2 x being real-valued functions. The condition that fx is periodic isequivalent to both f 1 x and f 2 x being periodic. If this approach is preferred, then the FSfor f 1 x and f 2 x can be found in the usual way and their sum combined via (1.7.6) todetermine a FS for fx. It is important to recognise that this is rather a complicatedapproach and the direct method is much shorter and significantly easier to implement in themajority of cases.The Dirichlet Conditions hold in exactly the same way for a complex-valued function as fora real-valued function. Since a representation of the form (1.7.6) is always applicable, thenthe requirement that f 1 x and f 2 x satisfy the Dirichlet Conditions ensures that fx does.Another structure that occurs quite commonly in complex-valued functions is associatedwith the product of functions. Suppose that fx and gx are periodic complex-valuedfunctions, of period 2, with Fourier representationsfx ~ ∑ n e inx ,n−gx ~ ∑ n e inxn−and that the coefficients n and n are assumed known. If a third function hx is definedby38