## 18

## Argument Principle and Rouche's Theorem

#### 18.1 Introduction

In this chapter, we state and prove the argument principle and Rouche's theorem, which are consequences of Cauchy's residue theorem.

We shall also prove the fundamental theorem of algebra, which we have been using without proof since the beginning of the study of algebra.

The argument principle helps us find the number of zeros or poles of a function in a given region and is useful in determining the stability criteria of linear systems.

The argument principle applies to a class of complex functions called meromorphic functions defined in a domain *D* whose only singularities in *D* are poles and so we define meromorphic functions.

#### 18.2 Meromorphic Function

**Definition A** function *f* of the ...