2 - Preliminaries in complex analysis

Summary

Complex analysis is a vast and very beautiful subject, and the key to its beauty is the harmonious coexistence of analysis, algebra, geometry and topology in its most fundamental entity, the complex plane. We will assume that the reader is already familiar with the basic facts about analytic functions in one complex variable, such as Cauchy's theorem, the Cauchy–Riemann equations, power series expansions, residues and so on. Holomorphic functions in one complex variable enjoy a double life, as they can be viewed both as analytic objects (power series, integral representations) and as geometric objects (conformal mappings). The topics presented in this book exploit freely this dual character of holomorphic functions. Our purpose in this short chapter is to present some well-known or not so well-known analytic and geometric facts that will be necessary later. The reader is warned that what follows is only a brief collection of facts to be used, not a systematic exposition of the theory. For general background reading in complex analysis, see for instance [A2], [An] or [Rud].

Analytic facts

Let us start with some differential calculus of complex-valued functions defined on some domain in the complex plane (by a domain we mean as usual a non-empty, connected, open set).