Page 97 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry Richa Nandra, Lovely Professional University

Notes Unit 9: Taylor and Laurent Series

CONTENTS
Objectives

Introduction
9.1 Taylor Series
9.2 Laurent Series
9.3 Summary
9.4 Keywords

9.5 Self Assessment
9.6 Review Questions
9.7 Further Readings

Objectives

After studying this unit, you will be able to:
Discuss Taylor series

Describe the concept of Laurent series

Introduction

In last unit, you have studied about concept of power series and also discussed basic facts
regarding the convergence of sequences and series of complex numbers. We shall show that if f
(z) is analytic in some domain D then it can be represented as a power series at any point z  D
0
in powers of (z - z ) which is the Taylor series of f (z) . If f (z) fails to be analytic at a point z , we
0
0
cannot find Taylor series expansion of f (z) at that point. However, it is often possible to expand
f (z) in an infinite series having both positive and negative powers of (z - z ) . This series is called
0
the Laurent series.
9.1 Taylor Series

Suppose f is analytic on the open disk |z – z | < r. Let z be any point in this disk and choose C to
0
be the positively oriented circle of radius , where |z – z | <  < r. Then for s  C we have
0
 
1 1 1  1   (z z ) j

0

j 0 (s z )
s z  (s z ) (z z )  (s z )  1  z z 0      0 j 1






0 
0
0

  s z 

0 
z z

since 0 < 1. The convergence is uniform, so we may integrate
s z 0

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