Mathematical Methods for Physics and Engineering : A Comprehensive Guide

COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS

1

1

2

2

3

3

4

4

5

z

f(z)

Figure 3.1 The functionf(z)=z^2 − 4 z+5.

the first term is called arealterm. The full solution is the sum of a real term

and an imaginary term and is called acomplex number. A plot of the function

f(z)=z^2 − 4 z+ 5 is shown in figure 3.1. It will be seen that the plot does not

intersect thez-axis, corresponding to the fact that the equationf(z)=0hasno

purely real solutions.

The choice of the symbolzfor the quadratic variable was not arbitrary; the

conventional representation of a complex number isz,wherezis the sum of a

real partxanditimes an imaginary party,i.e.

z=x+iy,

whereiis used to denote the square root of−1. The real partxand the imaginary

partyare usually denoted by Rezand Imzrespectively. We note at this point

that some physical scientists, engineers in particular, usejinstead ofi. However,

for consistency, we will useithroughout this book.

In our particular example,

√

−4=2

√

−1=2i, and hence the two solutions of

(3.1) are

z 1 , 2 =2±

2 i

2

=2±i.

Thus, herex= 2 andy=±1.

For compactness a complex number is sometimes written in the form

z=(x, y),

where the components ofzmay be thought of as coordinates in anxy-plot. Such

a plot is called anArgand diagramand is a common representation of complex

numbers; an example is shown in figure 3.2.