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Sachin Kaushal, Lovely Professional University Unit 5: Cauchy’s Theorem

Unit 5: Cauchy’s Theorem Notes

CONTENTS
Objectives

Introduction
5.1 Homotopy
5.2 Cauchy’s Theorem
5.3 Summary
5.4 Keywords

5.5 Self Assessment
5.6 Review Questions
5.7 Further Readings

Objectives

After studying this unit, you will be able to:
Define homotopy

Discuss the Cauchy's theorem

Describe examples of Cauchy's theorem

Introduction

In earlier unit, you have studied about complex functions and complex number. Cauchy-Riemann
equations which under certain conditions provide the necessary and sufficient condition for the
differentiability of a function of a complex variable at a point. A very important concept of
analytic functions which is useful in many application of the complex variable theory. Let's
discuss the concept of Cauchy's theorem.

5.1 Homotopy

Suppose D is a connected subset of the plane such that every point of D is an interior point—we
call such a set a region—and let C and C be oriented closed curves in D. We say C is homotopic
1
2
1
to C in D if there is a continuous function H : S  D, where S is the square S = {(t, s) : 0  s, t  1},
2
such that H(t,0) describes C and H(t,1) describes C , and for each fixed s, the function H(t, s)
2
1
describes a closed curve C in D.
s
The function H is called a homotopy between C and C . Note that if C is homotopic to C in D,
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1
2
1
then C is homotopic to C in D. Just observe that the function K(t, s) = H(t,1 – s) is a homotopy.
2
1
It is convenient to consider a point to be a closed curve. The point c is a described by a constant
function (t) = c. We thus speak of a closed curve C being homotopic to a constant—we sometimes
say C is contractible to a point.
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