Mathematical Methods for Physics and Engineering : A Comprehensive Guide

COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS

Rez

Imz

r 1 r 2 ei(θ^1 +θ^2 )

r 2 eiθ^2

r 1 eiθ^1

Figure 3.8 The multiplication of two complex numbers. In this caser 1 and

r 2 are both greater than unity.

The algebra of the polar representation is different from that of the real and

imaginary component representation, though, of course, the results are identical.

Some operations prove much easier in the polar representation, others much more

complicated. The best representation for a particular problem must be determined

by the manipulation required.

3.3.1 Multiplication and division in polar form

Multiplication and division in polar form are particularly simple. The product of

z 1 =r 1 eiθ^1 andz 2 =r 2 eiθ^2 is given by

z 1 z 2 =r 1 eiθ^1 r 2 eiθ^2

=r 1 r 2 ei(θ^1 +θ^2 ). (3.25)

The relations|z 1 z 2 |=|z 1 ||z 2 |and arg(z 1 z 2 ) = argz 1 +argz 2 follow immediately.

An example of the multiplication of two complex numbers is shown in figure 3.8.

Division is equally simple in polar form; the quotient ofz 1 andz 2 is given by

z 1

z 2

=

r 1 eiθ^1

r 2 eiθ^2

=

r 1

r 2

ei(θ^1 −θ^2 ). (3.26)

The relations|z 1 /z 2 |=|z 1 |/|z 2 |and arg(z 1 /z 2 )=argz 1 −argz 2 are again