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.
3.2 BASIC PROPERTIES OF COMPLEX NUMBERS
We can write a complex number z as
where x and y are real numbers.
The numbers x and y are the real and imaginary parts of z respectively and are denoted by Re (z)
and Im(z).
We can also regard z as an ordered pair of real numbers. As in vector algebra, we write z as
(x, y).
Just as a real number can be represented by a point on a line, a complex number can be represented by
a point in a
plane. This representation is the Argand diagram and is shown in Figure 3.2-1
Two complex numbers are equal if and only if (iff) their real and imaginary parts are equal. If z^
(=Xj +iyi) and Z2 (=X2 + iy2) are equal, it implies
M = x2, yi = y2 (3.2-2a,b)
Thus a complex equation is equivalent to two real equations. The addition and multiplication can be
handled in the same way as for real numbers and, whenever i
2 appears, it is replaced by -1. The
commutative, associative, and distributive laws hold. We list some of the results
zj + z2 = (xt + x2, yi + y2) (3.2-3a)
z
2 - z2 = (x2 - x2, y} - y2) (3.2-3b)
•LX • z2 = (xT x2 - yj y2, xj y2 + x2 y{) = z2 • zx (3.2-3c,d)
zl = xl + iyi = (xi+iyi)(x2-iy2) = [x1x2 + y1y2 + i(x2y1-x1y2)] (3 2 3e f }
Z
2 x2 + iy2 (x2 + iy2)(x2-iy2) X2 + y2
The complex conjugate z of the complex number z (= x +
i y) is defined as
z = x-iy (3.2-4)
In the Argand diagram, it is the reflection of z about the x-axis and it is shown in Figure
3.2-1.
So far we used only the rectangular Cartesian system. We can also use the polar coordinate
system (r, 9). From Figure
3.2-1, we find
x = r cos 0, y = r sin 6 (3.2-5a,b)
Inverting Equations (3.2-5a, b), we obtain
r = Vx
2 + y2 (3.2-6a)
tan9 = y/x (3.2-6b)
The number r is called the modulus or the absolute value of z and is denoted by
I z I. It can
be regarded as the length of the vector represented by z. The absolute value of z is also given by
|z| = VzT (3.2-7)
0 is the argument or amplitude of z. It is written as arg z = 0. Hence we can write
z = x + iy = r (cos 0 + i sin 0) (3.2-8a,b)
Since any multiple of 2K radians may be added to 0 without changing the value of z, we specify
-K < 0 < n as the principal value of arg z, and denote it by Arg z. The polar representation is
useful for computational purposes, as shown next.
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