Page 121 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry Richa Nandra, Lovely Professional University

Notes Unit 12: Fundamental Theorem of Algebra

CONTENTS
Objectives

Introduction
12.1 Fundamental Theorem of Algebra
12.2 Calculus of Residues
12.3 Jordan’s Inequality
12.4 Multivalued Function and its Branches

12.5 Summary
12.6 Keywords
12.7 Self Assessment

12.8 Review Questions
12.9 Further Readings

Objectives

After studying this unit, you will be able to:
Discuss the concept of fundamental theorem on algebra

Describe the calculus of residues

Discuss the multivalued functions and its branches

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. Also you studied about the concept
related to argument principle and Rouche's theorem. This unit will explain fundamental theorem
on algebra.

12.1 Fundamental Theorem of Algebra

Every polynomial of degree n has exactly n zeros.
Proof. Let us consider the polynomial
a + a z + a z + + a z , a  0
n
2
2
0
1
n
n
We take f(z) = a z , g(z) = a + a z + a z + + a z n-1
2
n
1
n-1
2
0
n
Let C be a circle |z| = r, where r > 1.
Now, | f(z)| = |a z | = |a | r n
n
n n
|g(z)|  |a | + |a | r + |a | r + + |a | r n-1
2
0 1 2 n-1
 (|a | + |a | + + |a |) r n-1
1
0
n-1
114 LOVELY PROFESSIONAL UNIVERSITY