Fourier Series

1.6 Integration and Differentiation of Fourier SeriesNo attempt at proof will be given in this section. Plausible explanations may be given butthe derivations are mainly heuristic. As a general rule integration is more useful in FStheory than differentiation.1.6.1 IntegrationResult 1If fx satisfies the Dirichlet conditions in the interval −, and if a n ,b n are the Fouriercoefficients of fx thenx fu du a 0x − xx020 ∑n11n b n cosnx 0 − cosnx a n sinnx − sinnx 0 when − ≤ x 0 x ≤ .Some comments:• The usual interpretation is that x 0 is a fixed value and that x is a variable, so all of theterms containing x 0 can be lumped together onto a single constant provided that anyrequirements on convergence are satisfied.• This is the series that would be obtained using term-by-term integration and thepresence of the O1/n factor in the series, introduced by integration, will generallyaccelerate convergence.• It is not a FS in x unless either a 0 0 or steps are taken to make it into a FS. Theexample in §1.2 provides the FS for the function fx x, viewed as a periodicfunction on −,, and this may be substituted as required.ExampleFind the Fourier series of the function fx defined on −, byand deduce thatfx − 2 − x 0 20 x 1 13 2 15 2 ..... 28 .Only sine terms will appear in the FS, since fx is an odd function,32