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CHAPTER 7

Complex Variables and the Laplace

Inversion Integral

7.1 BASIC PROPERTIES
A complex number z can be defined as an ordered pair of real numbers, say x and y, where x is
therealpartof z and y is the real value of the imaginary part:

z = x + iy (7.1)
√
where i = −1
I am going to assume that the reader is familiar with the elementary properties of
addition, subtraction, multiplication, etc. In general, complex numbers obey the same rules as
real numbers. For example

(x 1 + iy 1)(x 2 + iy 2) = x 1 x 2 − y 1 y 2 + i (x 1 y 2 + x 2 y 1) (7.2)

The conjugate of z is

¯ z = x − iy (7.3)

It is often convenient to represent complex numbers on Cartesian coordinates with x and
y as the axes. In such a case, we can represent the complex number (or variable) z as

z = x + iy = r(cos θ + i sin θ) (7.4)

as shown in Fig. 7.1. We also define the exponential function of a complex number as cos θ +
x
i sin θ = e iθ which is suggested by replacing x in series e = ∞ x n by iθ.
n=0 n!
Accordingly,

e iθ = cos θ + i sin θ (7.5)
and

e −iθ = cos θ − i sin θ (7.6)