102 Chapter 1 Fourier Series and Integrals
Figure 11 Preparation of a function for numerical integration of Fourier co-
efficients. (a) Graph of sectionally smooth functionf(x)given on−a<x<a.
(b) Graph off 1 (x), which has jumps of the same magnitude and position asf(x).
Coefficients can be found analytically. (c) Graph off(x)−f 1 (x).Thisfunctionhas
no jumps in−a<x<a.(d)Graphoff 2 (x). The periodic extensions off 2 (x)and
off(x)−f 1 (x)have jumps of the same magnitude atx=±a, and so forth. The co-
efficients off 2 can be found analytically. (e) Graph off 3 (x)=f(x)−f 1 (x)−f 2 (x).
The Fourier series off 3 (x)converges uniformly (the coefficients tend to zero
rapidly).