Complex Analysis I: Basic Concepts

Igor Kriz and Aleš Pultr
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Igor Kriz: University of Michigan, Department of Mathematics
Aleš Pultr: Charles University, Department of Applied Mathematics (KAM) Faculty of Mathematics and Physics

Chapter 10 in Introduction to Mathematical Analysis, 2013, pp 237-266 from Springer

Abstract: 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.

Date: 2013
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DOI: 10.1007/978-3-0348-0636-7_10

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