Page 16 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Richa Nandra, Lovely Professional University Unit 2: Complex Functions

Unit 2: Complex Functions Notes

CONTENTS
Objectives

Introduction
2.1 Functions of a Real Variable
2.2 Functions of a Complex Variable
2.3 Derivatives
2.4 Summary

2.5 Keywords
2.6 Self Assessment
2.7 Review Questions

2.8 Further Readings

Objectives

After studying this unit, you will be able to:
Explain the function of a complex variable

Describe the functions of a complex variable

Define derivatives

Introduction

There are equations such as x + 3 = 0, x – 10x + 40 = 0 which do not have a root in the real number
2
2
system R . There does not exist any real number whose square is a negative real number. If the
roots of such equations are to be determined then we need to introduce another number system
called complex number system. Complex numbers can be defined as ordered pairs (x, y) of real
numbers and represented as points in the complex plane, with rectangular coordinates x and y.
In this unit, we shall review the function of the complex variable.

2.1 Functions of a Real Variable

A function  : I  C from a set I of reals into the complex numbers C is actually a familiar concept
from elementary calculus. It is simply a function from a subset of the reals into the plane, what
we sometimes call a vector-valued function. Assuming the function  is nice, it provides a vector,
or parametric, description of a curve. Thus, the set of all {(t) : (t) = e = cos t + i sin t = (cos t, sin t),
it
0  t  2} is the circle of radius one, centered at the origin.
We also already know about the derivatives of such functions. If (t) = x(t) + iy(t), then the
derivative of  is simply ’(t) = x’(t) + iy’(t), interpreted as a vector in the plane, it is tangent to the
curve described by  at the point (t).

LOVELY PROFESSIONAL UNIVERSITY 9