Page 115 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry Richa Nandra, Lovely Professional University

Notes Unit 11: Rouche’s Theorem

CONTENTS
Objectives

Introduction
11.1 Argument Principle
11.2 Rouche’s Theorem
11.3 Summary
11.4 Keyword

11.5 Self Assessment
11.6 Review Questions
11.7 Further Readings

Objectives

After studying this unit, you will be able to:
Discuss the concept of argument principle

Describe the Rouche's theorem

Introduction

In last unit, you have studied about the Taylor series, singularities of complex valued functions
and use the Laurent series to classify these singularities. This unit will explain the concept
related to argument principle and Rouche's theorem.

11.1 Argument Principle

Let C be a simple closed curve, and suppose f is analytic on C. Suppose moreover that the only
singularities of f inside C are poles. If f(z)  0 for all z  C, then  = (C) is a closed curve which
does not pass through the origin. If
(t), a  t  b

is a complex description of C, then
(t) = f((t)),   t  
is a complex description of . Now, let’s compute

f'(z)  f'( (t))

 f(z) dz   f( (t))  '(t)dt.
C  

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