Engineering Mathematics, Volume III

4

Complex Integration

4.1 Introduction

Complex integration is useful in calculating certain real integrals that cannot be evaluated by the usual methods as, for instance, in finding the inverse Laplace, Fourier or Z-transforms in applications. It also enables us to establish some basic properties of analytic functions such as the existence of higher order derivatives, in the case of analytic functions.

In this chapter, we define and explain the concept of complex integral, state and prove Cauchy’s integral formula. We also prove Cauchy’s generalised integral formula, which shows that if a function is analytic in a domain D, it possesses the derivatives of all orders in D.

4.2 Basic Concepts

The concept of line integral is a natural generalisation ...