Fourier Series

1.7 Complex Fourier SeriesIt is often useful to take a complex representation of a real-valued function and it may alsooccur that the function is a complex-valued function of a real variable. The onlydiscriminant is whether the function fx is real-valued or complex valued; in both cases itis taken to be defined and bounded on −,. Although we shall treat them separately,these are very similar to deal with and depend upon being able to relate trigonometricfunctions to a complex representation.From De Moivre’s Theorem, for any integer me imx cosmx isinmx ,and it is straightforward to show from this expression thatcosmx 1 2 eimx e −imx , sinmx − i 2 eimx − e −imx . (1.7.1)This provides the means of linking the real-valued and complex-valued Fourier series.1.7.1 Real-valued functionsThis simply corresponds to a re-writing of the representation that has been utilised thus far.In the existing representation the function fx is taken to be of the formfx ~ 1 2 a 0 ∑n1a n cosnx b n sinnxwith the real constants a n , b n given bya n 1 −fxcosnx dx , bn 1 −fxsinnx dx , n 0,1,2,.....Now substitute into the main series for a n and b n from (1.7.1),fx ~ 12a 0 1 2 ∑ n1 12a 0 1 2 ∑ n1a n e inx e −inx − ib n e inx − e −inxa n − ib n e inx a n ib n e −inx 1 2∑ n1a n ib n e −inx 1 2 a 0 1 2 ∑ n1a n − ib n e inx .Introduce the complex coefficients c n : n −,..., defined by35