Throughout this book references have been made to results ‘which are derived from the theory of complex variables’. This theory thus becomes an integral part of the mathematics appropriate to physical applications. The difficulty with it, from the point of view of a book such as the present one, is that, although the applications for which it is needed are very real and applied, the underlying basis of complex variable theory has a distinctly pure mathematics flavour.

To adopt this more rigorous approach correctly would involve developing a large amount of groundwork in analysis, for example, precise definitions of continuity and differentiability, the theory of sets and a detailed study of boundedness. It has been decided not to do so here, but rather to pursue only those parts of the formal theory which are needed to establish the results used elsewhere in this book and some others of general utility. Specifically, the subjects treated are,

1. (i) complex potentials for two-dimensional potential problems,

2. (ii) location of zeros of a function, in particular a polynomial,

3. (iii) summation of series and evaluation of integrals,

4. (iv) the inverse Laplace transform integral.

In this spirit, the proofs that are adopted for some of the standard results of complex variable theory have been chosen with an eye to simplicity rather than sophistication. This means that in some cases the imposed conditions are more stringent than would be strictly necessary if more sophisticated proofs were used; where this happens the less restrictive results are usually stated as well.