Mathematical Methods for Physics and Engineering - Matematica.NET

FOURIER SERIESconverge to the correct values of f(x) =±4 atx = ±2; it converges, instead, tozero, the average of the values at the two ends of the range.12.6 Integration and differentiationIt is sometimes possible to find the Fourier series of a function by integration ordifferentiation of another Fourier series. If the Fourier series of f(x) isintegratedterm by term then the resulting Fourier series converges to the integral of f(x).Clearly, when integrating in such a way there is a constant of integration that mustbe found. If f(x) is a continuous function of x for all x and f(x) is also periodicthen the Fourier series that results from differentiating term by term converges tof ′ (x), provided that f ′ (x) itself satisfies the Dirichlet conditions. These propertiesof Fourier series may be useful in calculating complicated Fourier series, sincesimple Fourier series may easily be evaluated (or found from standard tables)and often the more complicated series can then be built up by integration and/ordifferentiation.◮Find the Fourier series of f(x) =x 3 for 0