Mathematical Methods for Physics and Engineering : A Comprehensive Guide

COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS

Rez

Imz

e−^2 iπ/^3

e^2 iπ/^3

2 π/ 3

2 π/ 3

1

Figure 3.10 The solutions ofz^3 =1.

Not surprisingly, given that|z^3 |=|z|^3 from (3.10), all the roots of unity have

unit modulus, i.e. they all lie on a circle in the Argand diagram of unit radius.

The three roots are shown in figure 3.10.

The cube roots of unity are often written 1,ωandω^2. The propertiesω^3 =1

and 1 +ω+ω^2 = 0 are easily proved.

3.4.3 Solving polynomial equations

A third application of de Moivre’s theorem is to the solution of polynomial

equations. Complex equations in the form of a polynomial relationship must first

be solved forzin a similar fashion to the method for finding the roots of real

polynomial equations. Then the complex roots ofzmay be found.

Solve the equationz^6 −z^5 +4z^4 − 6 z^3 +2z^2 − 8 z+8=0.

We first factorise to give

(z^3 −2)(z^2 +4)(z−1) = 0.

Hencez^3 =2orz^2 =−4orz= 1. The solutions to the quadratic equation arez=± 2 i;
to find the complex cube roots, we first write the equation in the form

z^3 =2=2e^2 ikπ,

wherekis any integer. If we now take the cube root, we get

z=2^1 /^3 e^2 ikπ/^3.