Applications of Cauchy’s Theorem

Abstract

Most of the powerful and beautiful theorems proved in this chapter have no analog in real variables. While Cauchy’s theorem is indeed elegant, its importance lies in applications. In this chapter, we prove several theorems that were alluded to in previous chapters. We prove the Cauchy integral formula which gives the value of an analytic function in a disk in terms of the values on the boundary. Also, we show that an analytic function has derivatives of all orders and may be represented by a power series. The fundamental theorem of algebra is proved in several different ways. In fact, there is such a nice relationship between the different theorems in this chapter that it seems any theorem worth proving is worth proving twice.