Introduction

In this chapter we shall explore the notion of a ‘transform’ of a function, where an integral mapping is used to construct a ‘transformed’ function out of an original function. The continuous Fourier transform is one of a family of such mappings, which also includes the Laplace transform and the discrete Fourier transform. The Laplace transform will be discussed in Chapter 18. Numerical methods for the discrete Fourier transform and for the inversion of Laplace transforms will be given in Chapter 21.

What is the point of such transforms? Perhaps the most important lies in the solution of linear differential equations. Here the operation of a transform can convert differential equations into algebraic equations. In the case of an ordinary differential equation (ODE), one such transform can produce a single algebraic condition that can be solved for the transform by elementary means, leaving one with the problem of inversion – the means by which the transformed solution is turned into the function that is desired. In the case of a partial differential equation (PDE), for example in two variables, one transform can be used to reduce the PDE into an ODE, which may be solved by standard methods, or, perhaps, by the application of a further transform to an algebraic condition. Again one proceeds to a solution of the transformed problem. One or more inversions is required to obtain the solution.