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Sachin Kaushal, Lovely Professional University Unit 6: Cauchy’s Integral Formula

Unit 6: Cauchy’s Integral Formula Notes

CONTENTS
Objectives

Introduction
6.1 Cauchy’s Integral Formula
6.2 Functions defined by Integrals
6.3 Liouville’s Theorem
6.4 Maximum Moduli

6.5 Summary
6.6 Keywords
6.7 Self Assessment

6.8 Review Questions
6.9 Further Readings

Objectives

After studying this unit, you will be able to:

Define Cauchy's integral formula

Discuss functions defined by integrals

Describe liouville's theorem

Explain maximum moduli

Introduction

In last unit, you have studied about concept of Cauchy's theorem. A very important concept of
analytic functions which is useful in many application of the complex variable theory. This unit
provides you information related to Cauchy's integral formula, functions defined by integrals
and maximum moduli.

6.1 Cauchy’s Integral Formula

Suppose f is analytic in a region containing a simple closed contour C with the usual positive
orientation and its inside, and suppose z is inside C. Then it turns out that
0
1 f(z)

f(z )  2 i z z 0 dz.
0


C
This is the famous Cauchy Integral Formula. Let’s see why it’s true.

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