CHAPTER 3 : COMPLEX VARIABLES

3.1 INTRODUCTION

The inadequacy of the real number system (rational and irrational numbers) in solving algebraic
equations was known to mathematicians in the past. It therefore became necessary to extend the real
number system, so as to obtain meaningful solutions to simple equations such as
x
2+ 1 =0 (3.1-1)
For quite sometime, it appears that equations, which could not be solved in the domain of real
numbers, were solved by accepting V-T as a possible number. This notation, however, has had its
own shortcomings. Euler was the first to introduce the symbol i for i^A with the basic property
i
2 = -1 (3.1-2)
(Electrical engineers use j to denote i^T.)
He also established the relationships between complex numbers and trigonometric functions.
However, in those times, no actual meaning could be assigned to the expression V^T. It was,
therefore, called an "imaginary" (as opposed to real) number. This usage still prevails in the
literature.
It was not until around 1800 that sound footing was given to the complex number system by Gauss,
Wessel, and Argand. Gauss proved that every algebraic equation with real coefficients has complex
roots of the form c + i d. Real roots are special cases, when d is zero. Argand proposed a
graphical representation of complex numbers. The concept of a function was subsequently extended to
complex functions of the type
w = f(z) (3.1-3)
where z (= x +
 i y) is the independent variable.
The concept of complex variables is a powerful and a widely used tool in mathematical analysis.
The theory of differential equations has been extended within the domain of complex variables.
Complex integral calculus has found a wide variety of applications in evaluating integrals, inverting
power series, forming infinite products, and asymptotic expansions. Applied mathematicians,
physicists, and engineers make extensive use of complex variables. It is indispensable for students in
mathematical, physical, and engineering sciences to have some knowledge of the theory of complex
analysis.