13

Integral transforms

In the previous chapter we encountered the Fourier series representation of a

periodic function in a fixed interval as a superposition of sinusoidal functions. It is

often desirable, however, to obtain such a representation even for functions defined

over an infinite interval and with no particular periodicity. Such a representation

is called aFourier transformand is one of a class of representations calledintegral

transforms.

We begin by considering Fourier transforms as a generalisation of Fourier

series. We then go on to discuss the properties of the Fourier transform and its

applications. In the second part of the chapter we present an analogous discussion

of the closely relatedLaplace transform.

13.1 Fourier transforms

The Fourier transform provides a representation of functions defined over an

infinite interval and having no particular periodicity, in terms of a superposition

of sinusoidal functions. It may thus be considered as a generalisation of the

Fourier series representation of periodic functions. Since Fourier transforms are

often used to represent time-varying functions, we shall present much of our

discussion in terms off(t), rather thanf(x), although in some spatial examples

f(x) will be the more natural notation and we shall use it as appropriate. Our

only requirement onf(t) will be that

∫∞

−∞|f(t)|dtis finite.

In order to develop the transition from Fourier series to Fourier transforms, we

first recall that a function of periodTmay be represented as a complex Fourier

series, cf. (12.9),

f(t)=

∑∞

r=−∞

cre^2 πirt/T=

∑∞

r=−∞

creiωrt, (13.1)

whereωr=2πr/T. As the periodTtends to infinity, the ‘frequency quantum’