Principles of Fourier Analysis, 2nd Edition

15

Derivatives and Integrals of Fourier Series

There are two good reasons for considering the differentiation and integration of Fourier series. One is that derivatives and integrals of Fourier series often arise in applications. In the next chapter, for example, we will use Fourier series to solve differential equations describing flow of heat in a rod and vibrations in a string. The other reason is that some results from this chapter have already been quoted (see page 164). So we had better develop those results.

15.1  Differentiation of Fourier Series

Let f be a piecewise smooth, periodic function. Since, as piecewise continuous functions,

$f\left(t\right)=F.S.{\left[f\right]|}_t={\displaystyle \sum _{k=-\infty }^{\infty }{c}_k{e}^{i2\pi {\omega }_{k}t}},$

we certainly have

$f^{\prime }\left(t\right)=\frac{d}{dt}{\displaystyle \sum _{k=-\infty }^{\infty }{c}_k{e}^{i2\pi {\omega }_{k}t}}$

at all points at which ...