Fourier Series

is required.It is instructive to identify the two differences between the form of the Fourier coefficientsin (1.2.4) and (1.7.5) for real-valued and complex-valued functions respectively. Thecomparison shows that the 1/ factor is replaced by 1/2, moving from real to complex,and that a minus sign is present in the exponential term of the complex exponential term.The relationship c n c −n , from (1.7.2) and valid for all n, can be extended in two specialcases. If fx is an even function, then the c n are all real and if fx is an odd function, thenthe c n are all purely imaginary. These results are stated without proof but can be readilyverified via (1.7.5).1.7.2 Complex-valued functionsSuppose that f is complex-valued, so that f : R → C, then how do we determine its FourierSeries? The approach is exactly the same as in the previous section and can be summarisedasfx ~ ∑n−c n e inx , where c n 12 fx e −inx dx .−In general the coefficients c n will all be complex but there is one special case. If fx ispurely imaginary, then c −n −c n and this can be shown from (1.7.5).ExampleDetermine the Fourier Series of the complex-valued periodic function of period 2, definedbyfx 0 − x 0 e ix0 x The first step should be to sketch the function, although this is omitted here. Note that twodiagrams would be required: one each for the real and imaginary parts. The coefficients aregiven byc n 12 fx e −inx dx 1−2 e i1−nx dx0It cannot be assumed in this case that the relationship (1.7.4) is applicable, since the rangesof integration do not coincide. However, it is clear that the cases n 1andn ≠ 1 must betreated separately. When n 1, the formula produces c 1 1/2, whereas for n ≠ 1,37