Introduction

There is a transform that is closely related to a special case of the Fourier transform, known as the Laplace transform. While the Laplace transform is very similar, historically it has come to have a separate identity, and one can often find separate tables of the two sets of transforms. Furthermore, it is very appropriate to make a separate assessment of both its inversion, and its applications to differential equations. In the latter context, Laplace transforms are particularly useful when dealing with ODEs and PDEs defined on a half-space – in this setting its differential properties are slightly different from the Fourier transform due to the influence of the boundary.

The goal of this chapter is to define the Laplace transform and explain the basic results and links to complex variable theory. It should be appreciated that there is an extensive knowledge base of known transforms and their inverses. Sadly, many of the excellent books of tables of transforms are old and hard to find if not actually out of print. You might like to check if your library has copies of the old works by Erdelyi. One notable exception is the extraordinarily comprehensive series of books by Prudnikov, Brychkov and Marichev, in which volumes 4 and 5 (Prudnikov et al, 1998, 2002) give tables of transforms and their inverses.