Fourier Series

1.5 General range Fourier SeriesWe have only considered functions previously that are defined on −, and with period2 or on 0, andextendedtohaveperiod2. Suppose with have a function defined on−L,L with period 2L, how is the corresponding Fourier series obtained?Introduce a new variable z defined by z x/L and define the function gz fx for−L x L, sothatgz is defined on − z . Provided that gz satisfies theDirichlet conditions,gz ~ 1 2 a 0 ∑n1a n cosnz b n sinnzwherea n 1 −gzcosnz dz , bn 1 −gzsinnz dz .Now substitute z x/L fx gz ~ 1 2 a 0 ∑n1a n cos nxL b n sin nxLwitha n 1 L gzcosnz dz 1− −LL 1 L −Lfxcos nxLdxfxcos nxLLdxSimilarlyLb n 1 L −Lfxsin nxL dx .As fx is 2L-periodic, it is enough to consider any interval of length 2L and so we can use0,2L if we wish.ExampleFind the Fourier expansion, valid in 0,2, of the function fx 4 − x 2 . What is the valueof the series when x 0,1?29