Page 37 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry Richa Nandra, Lovely Professional University

Notes Unit 4: Integration

CONTENTS
Objectives

Introduction
4.1 Integral
4.2 Evaluating Integrals
4.3 Antiderivatives
4.4 Summary

4.5 Keywords
4.6 Self Assessment
4.7 Review Questions

4.8 Further Readings

Objectives

After studying this unit, you will be able to:
Explain the evaluation of integrals

Discuss the anti derivatives

Introduction



If  : D  C is simply a function on a real interval D = [, ], then the integral  (t)dt of course,

simply an ordered pair of everyday 3 grade calculus integrals:
rd
  
  (t)dt   x(t)dt i y(t)dt,


  
where g(t) = x(t) + iy(t). Thus, for example,
Nothing really new here. The excitement begins when we consider the idea of an integral of an
honest-to-goodness complex function f : D  C, where D is a subset of the complex plane. Let’s
define the integral of such things; it is pretty much a straightforward extension to two dimensions
of what we did in one dimension back in Mrs. Turner’s class.

4.1 Integral

Suppose f is a complex-valued function on a subset of the complex plane and suppose a and b are
complex numbers in the domain of f. In one dimension, there is just one way to get from one
number to the other; here we must also specify a path from a to b. Let C be a path from a to b, and
we must also require that C be a subset of the domain of f.

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