Complex Integration and Cauchy’s Theorem

Abstract

One of the most important theorems in calculus is properly named the fundamental theorem of integral calculus. On the one hand it relates integration to differentiation, and on the other hand it gives a method for evaluating integrals. In this chapter, we mainly look for a complex analog to develop a machinery of integration along arcs and contours in the complex plane. The problem, of course, is that between any two points there are an infinite variety of paths along which to integrate. The antiderivative of a complex-valued function f(z) of a complex variable z is completely analogous to that for a real function; it is indeed a complex function F whose derivative is f. Cauchy’s theorem, the fundamental theorem of complex integration says that for analytic functions, one path over special domains is as good as another.