24 Complex variables
Throughout this book references have been made to results derived from the the-
ory of complex variables. This theory thus becomes an integral part of the mathe-
matics appropriate to physical applications. Indeed, so numerous and widespread
are these applications that the whole of the next chapter is devoted to a systematic
presentation of some of the more important ones. This current chapter develops
the general theory on which these applications are based. The difficulty with it,
from the point of view of a book such as the present one, is that the underlying
basis has a distinctly pure mathematics flavour.
Thus, to adopt a comprehensive rigorous approach would involve a largeamount of groundwork in analysis, for example formulating precise definitions
of continuity and differentiability, developing the theory of sets and making a
detailed study of boundedness. Instead, we will be selective and pursue only those
parts of the formal theory that are needed to establish the results used in the next
chapter and elsewhere in this book.
In this spirit, the proofs that have been adopted for some of the standardresults 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. The reader who is interested in a fuller treatment should consult one of the
many excellent textbooks on this fascinating subject.§
One further concession to ‘hand-waving’ has been made in the interests ofkeeping the treatment to a moderate length. In several places phrases such as ‘can
be made as small as we like’ are used, rather than a careful treatment in terms
of ‘given>0, there exists aδ>0 such that’. In the authors’ experience, some
§For example, K. Knopp,Theory of Functions, Part I(New York: Dover, 1945); E. G. Phillips,
Functions of a Complex Variable with Applications7th edn (Edinburgh: Oliver and Boyd, 1951); E.
C. Titchmarsh,The Theory of Functions(Oxford: Oxford University Press, 1952).