Qualitative Properties of Analytic Functions

Abstract

Inspired by the properties of analytic functions proved in the previous sections, in the last section we are ready to explore new, no less amazing properties of such functions. In Sect. 9.1 we show that analyticity is sufficient for a nonconstant function being an open map. This property indicates that the modulus of a non-constant analytic function cannot have a strict local maximum. A direct application of the maximum modulus principle is Schwarz’s Lemma, established by the German mathematician K. A. Schwarz (1943–1921) in 1869, which is important in the theory of bounded analytic functions, where it is fundamental to most estimates. Sect. 9.2 shows how methods of complex analysis can be used to efficiently find inverse functions and expand them into Lagrange series (for single-valued inverse functions) and Puiseux series (for multi-valued inverse functions). Sections 9.3 and 9.4 are a preparation for the proof of Riemann’s theorem, namely here we are interested in the conformal classification of domains of the complex plane and the finding of a sufficient condition for the precompactness of a family of analytic functions (Montel’s theorem). In the last section there is a proof of the Riemann map** theorem, which is undoubtedly one of the most beautiful theorems in mathematics.