Advanced Engineering Mathematics, 7th Edition

18.2 Cauchy–Goursat Theorem

INTRODUCTION

In this section we shall concentrate on contour integrals where the contour C is a simple closed curve with a positive (counterclockwise) orientation. Specifically, we shall see that when f is analytic in a special kind of domain D, the value of the contour integral f(z) dz is the same for any simple closed curve C that lies entirely within D. This theorem, called the Cauchy–Goursat theorem, is one of the fundamental results in complex analysis. Preliminary to discussing the Cauchy–Goursat theorem and some of its ramifications, we first need to distinguish two kinds of domains in the complex plane: simply ...