Elementary Analytic Functions

Abstract

Conformal map**s are of immense importance in various branches of mathematics and in many applications. To solve many problems, one needs to be able to construct a bijective conformal map** from one domain onto another in the complex plane. In this chapter we study how to construct such bijective conformal map**s. We will consider various elementary analytic functions, find domains of univalence and images of these domains. In addition, for many elementary analytic functions in $\mathbb C$ we find their inverses, which in some cases turn out to be multivalued. We introduce the first (intuitive) concept of a Riemann surface for multivalued functions and show how to construct Riemann surfaces for the inverses of elementary analytic functions. As a result of these studies, we will establish facts that are incorrect in real analysis, for example, we can calculate the logarithm of negative numbers and solve the equation

$$\displaystyle \sin z = 2 . $$