Page 143 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry Richa Nandra, Lovely Professional University

Notes Unit 13: Schwarz's Reflection Principle

CONTENTS
Objectives

Introduction
13.1 Analytic Function
13.2 Complete Analytic Function
13.3 Schwarz’s Reflection Principle
13.4 Summary

13.5 Keywords
13.6 Self Assessment
13.7 Review Questions

13.8 Further Readings

Objectives

After studying this unit, you will be able to:
Discuss the concept of complex analytic functions

Describe the Schwarz's reflection principle

Introduction

In earlier unit, you studied about the concept related to argument principle and Rouche's theorem.
This unit will explain Schwarz's reflection principle.

13.1 Analytic Function

Definition. An analytic function f(z) with its domain of definition D is called a function element
and is denoted by (f, D). If zD, then (f, D) is called a function element of z. Using this notation,
we may say that (f , D ) and (f , D ) are direct analytic continuations of each other iff D  D  
1
1
2
2
2
1
and f (z) = f (z) for all zD  D . 2
2
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1
Remark. We use the word ‘direct’ because later on we shall deal with continuation along a curve.
i.e. just to distinguish between the two.
Analytic continuation along a chain of Domain. Suppose we have a chain of function elements
(f , D ), (f , D ), , (f , D ), ,(f , D ) such that D and D have the part D in common, D and D 3
K
k
2
n
1
n
2
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2
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have the part D is common and so on. If f (z)= f (z) in D , f (z) = f (z) in D and so on, then we
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2
3
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12
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say that (f , D ) is direct analytic continuation of (f , D ). In this way, (f , D ) is analytic
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K-1
k
n
n
K-1
continuation of (f , D ) along a chain of domains D , D , , D . Without loss of generality, we
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1
may take these domains as open circular discs. Since (f , D ) and (f , D ) are direct analytic
K-1
K1
K
K
continuations of each other, thus, we have defined an equivalence relation and the equivalence
classes are called global analytic functions.
136 LOVELY PROFESSIONAL UNIVERSITY