Page 106 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Richa Nandra, Lovely Professional University Unit 10: Residues and Singularities

Unit 10: Residues and Singularities Notes

CONTENTS
Objectives

Introduction
10.1 Residues
10.2 Poles and other Singularities
10.3 Summary
10.4 Keywords

10.5 Self Assessment
10.6 Review Questions
10.7 Further Readings

Objectives

After studying this unit, you will be able to:
Discuss residues

Describe the concept of singularities

Introduction

In last unit, you have studied about the Taylor series. Taylor series representation of a complex
valued function is discussed. In earlier unit, we have introduced the concept of absolute and
uniform convergence of power series and defined its radius of convergence. This unit will
explain zeros and singularities of complex valued functions and use the Laurent series to
classify these singularities.

10.1 Residues

A point z is a singular point of a function f if f is not analytic at z , but is analytic at some point
0
0
of each neighborhood of z . A singular point z of f is said to be isolated if there is a neighborhood
0
0
of z which contains no singular points of f save z . In other words, f is analytic on some region
0
0
0 < |z – z | < .
0
Example:
The function f given by
1
f(z) 
2
z(z  4)
has isolated singular points at z = 0, z = 2i, and z = –2i.

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