Complex Analysis

6

Singularities and Residues

6.1 INTRODUCTION

In Chapter 2, we have learnt about singularities in brief. In this chapter, we will classify the singularities into different types using Laurent series. Here, we will also introduce an important notion of residue of a function at a singularity which can be used to evaluate certain types of integrals. We know that according to the Cauchy–Goursat theorem, if a function f (z) is analytic at all the points inside and on a simple closed contour C, then the value of the function’s integral is 0 around C. However, if f (z) is not analytic at a finite number of points inside C, then there exists a special number called residue, which each of these points contributes to the value of integral. In this chapter, ...