Fourier Series

defining our new function Fx to beFx − cosx − x 0 cosx 0 x so that this is an odd function by construction and shown in figure 1.4.3 below.Figure 1.4.3. Extension of half-range sine series.Now calculate the sine coefficients for an odd function,b n 2 0sinnxcosx dx.Although the integrand is the product of an odd and an even function, the resulting integralwill not generally be zero because x 0 is not the midpoint of the domain of integration.Using addition formulaeb n 1 0sinn 1x sinn − 1x dxand this suggests that we need to consider the cases where n 1andn 1 separately,since sinn − 1x 0ifn 1. This givesand for n 1b 1 1 0sin2x dx 026