Mathematical Methods for Physics and Engineering : A Comprehensive Guide

COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS

Expresszin the formx+iy,when

z=

3 − 2 i

−1+4i

.

Multiplying numerator and denominator by the complex conjugate of the denominator
we obtain

z=

(3− 2 i)(− 1 − 4 i)

(−1+4i)(− 1 − 4 i)

=

− 11 − 10 i

17

=−

11

17

−

10

17

i.

In analogy to (3.10) and (3.11), which describe the multiplication of two

complex numbers, the following relations apply to division:

∣

∣

∣

∣

z 1

z 2

∣

∣

∣

∣=

|z 1 |

|z 2 |

, (3.17)

arg

(

z 1

z 2

)

=argz 1 −argz 2. (3.18)

The proof of these relations is left until subsection 3.3.1.

3.3 Polar representation of complex numbers

Although considering a complex number as the sum of a real and an imaginary

part is often useful, sometimes thepolar representationproves easier to manipulate.

This makes use of the complex exponential function, which is defined by

ez=expz≡1+z+

z^2

2!

+

z^3

3!

+···. (3.19)

Strictly speaking it is the function expzthat is defined by (3.19). The numbere

is the value of exp(1), i.e. it is just a number. However, it may be shown thatez

and expzare equivalent whenzis real and rational and mathematicians then

definetheir equivalence for irrational and complexz. For the purposes of this

book we will not concern ourselves further with this mathematical nicety but,

rather, assume that (3.19) is valid for allz. We also note that, using (3.19), by

multiplying together the appropriate series we may show that (see chapter 24)

ez^1 ez^2 =ez^1 +z^2 , (3.20)

which is analogous to the familiar result for exponentials of real numbers.