Mathematical Methods for Physics and Engineering - Matematica.NET

12.7 COMPLEX FOURIER SERIESwhere the Fourier coefficients are given byc r = 1 ∫ x0+L(f(x)exp − 2πirx )dx. (12.10)LLx 0This relation can be derived, in a similar manner to that of section 12.2, by multiplying(12.9) by exp(−2πipx/L) before integrating and using the orthogonalityrelation∫ x0+L(exp − 2πipx ) ( ) {2πirx L for r = p,exp dx =LL0 for r ≠ p.x 0The complex Fourier coefficients in (12.9) have the following relations to the realFourier coefficients:c r = 1 2 (a r − ib r ),c −r = 1 2 (a r + ib r ).(12.11)Note that if f(x) isrealthenc −r = c ∗ r , where the asterisk represents complexconjugation.◮Find a complex Fourier series for f(x) =x in the range −2