Mathematical Methods for Physics and Engineering - Matematica.NET

24Complex variablesThroughout this book references have been made to results derived from the theoryof complex variables. This theory thus becomes an integral part of the mathematicsappropriate to physical applications. Indeed, so numerous and widespreadare these applications that the whole of the next chapter is devoted to a systematicpresentation of some of the more important ones. This current chapter developsthe 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 underlyingbasis 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 definitionsof continuity and differentiability, developing the theory of sets and making adetailed study of boundedness. Instead, we will be selective and pursue only thoseparts of the formal theory that are needed to establish the results used in the nextchapter 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 simplicityrather than sophistication. This means that in some cases the imposed conditionsare more stringent than would be strictly necessary if more sophisticated proofswere used; where this happens the less restrictive results are usually stated aswell. The reader who is interested in a fuller treatment should consult one of themany 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 ‘canbe made as small as we like’ are used, rather than a careful treatment in termsof ‘given ɛ>0, there exists a δ>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 Applications 7th edn (Edinburgh: Oliver and Boyd, 1951); E.C. Titchmarsh, The Theory of Functions (Oxford: Oxford University Press, 1952).824