Fourier Series

−fxsinmx dx ∑ a n −n1cosnxsinmx dx ∑n1b n sinnxsinmx dx.−These expressions contain integrals like−cosnxcosmx dx, cosnxsinmx dx and− sinnxsinmx dx.−They are usually evaluated using either addition formulae or by employing a complexrepresentation. Take the first integral as an example, utilising an additional formula, cosnxcosmx dx −− 1 21 cosn mx cosn − mx dx2sinn mx sinn − mxn m n − m 0.Thepresenceofthen − m term in the denominator of the integral of the second termimposes the condition such an approach is only valid when n ≠ m,and thus the case n mshould be considered separately; this is done by starting directly with n m, cos 2 mx dx −−12 1 2 .2 .cos2mx 1 dx−The integral sinnxsinmx dx is evaluated similarly, with the same non-zero value when−m n and possessing a zero value when m ≠ n.Both sets of results can be summarised as−cosnxcosmx dx −sinnxsinmx dx mn ,where mn is called the Kronecker delta and defined by mn 1 m n 0 m ≠ n . (1.2.3)The remaining integrals are of type sinnxcosmx dx, possessing an integrand that is a−product of a sine and a cosine. As cosine and sine are even and odd respectively and therange of integration possesses x 0 as its mid-point, then the value of the integral must bezero, i.e.12