12

Fourier series

We have already discussed, in chapter 4, how complicated functions may be

expressed as power series. However, this is not the only way in which a function

may be represented as a series, and the subject of this chapter is the expression

of functions as a sum of sine and cosine terms. Such a representation is called a

Fourier series. Unlike Taylor series, a Fourier series can describe functions that are

not everywhere continuous and/or differentiable. There are also other advantages

in using trigonometric terms. They are easy to differentiate and integrate, their

moduli are easily taken and each term contains only one characteristic frequency.

This last point is important because, as we shall see later, Fourier series are often

used to represent the response of a system to a periodic input, and this response

often depends directly on the frequency content of the input. Fourier series are

used in a wide variety of such physical situations, including the vibrations of a

finite string, the scattering of light by a diffraction grating and the transmission

of an input signal by an electronic circuit.

12.1 The Dirichlet conditions

We have already mentioned that Fourier series may be used to represent some

functions for which a Taylor series expansion is not possible. The particular

conditions that a functionf(x) must fulfil in order that it may be expanded as a

Fourier series are known as theDirichlet conditions, and may be summarised by

the following four points:

(i) the function must be periodic;

(ii) it must be single-valued and continuous, except possibly at a finite number

of finite discontinuities;

(iii) it must have only a finite number of maxima and minima within one

period;

(iv) the integral over one period of|f(x)|must converge.