Complex Analysis I: Basic Concepts

Abstract

In this chapter, we will develop the basic principles of the analysis of complex functions of one complex variable. As we will see, using the results of Chapter 8, these developments come almost for free. Yet, the results are of great significance. On the one hand, complex analysis gives a perfect computation of the convergence of a Taylor expansion, which is of use even if we are looking at functions of one real variable (for example, power functions with a real power). On the other hand, the very rigid, almost “algebraic”, behavior of holomorphic functions is a striking mathematical phenomenon important for the understanding of areas of higher mathematics such as algebraic geometry ([8]). In this chapter, the reader will also see a proof of the Fundamental Theorem of Algebra and, in Exercise (4), a version of the famous Jordan Theorem on simple curves in the plane.