Page 28 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Richa Nandra, Lovely Professional University Unit 3: Elementary Functions

Unit 3: Elementary Functions Notes

CONTENTS
Objectives

Introduction
3.1 The Exponential Function
3.2 Trigonometric Functions
3.3 Logarithms and Complex Exponents
3.4 Summary

3.5 Keywords
3.6 Self Assessment
3.7 Review Questions

3.8 Further Readings

Objectives

After studying this unit, you will be able to:
Define exponential function

Discuss the trigonometric functions

Describe the logarithms and complex exponents

Introduction

As we know, Complex functions are, of course, quite easy to come by — they are simply ordered
pairs of real-valued functions of two variables. We have, however, already seen enough to
realize that it is those complex functions that are differentiable are the most interesting. It was
important in our invention of the complex numbers that these new numbers in some sense
included the old real numbers — in other words, we extended the reals. We shall find it most
useful and profitable to do a similar thing with many of the familiar real functions. That is, we
seek complex functions such that when restricted to the reals are familiar real functions. As we
have seen, the extension of polynomials and rational functions to complex functions is easy; we
simply change x’s to z’s. Thus, for instance, the function f defined by :

2
z  z 1

f(z) =
z 1

has a derivative at each point of its domain, and for z = x + 0i, becomes a familiar real rational
function :
x 1
2
x  
f(x)  .
x 1

What happens with the trigonometric functions, exponentials, logarithms, etc., is not so obvious.
Let us begin.

LOVELY PROFESSIONAL UNIVERSITY 21