1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

34 CHAPTER 1 • COMPLEX NUMBERS

Definition 1.12: Primitive nth root
For any natural number n, the value Wn given by
i a,, 27T.. 27T
wn=e " =cos- +ism-
n n
is called the primitive nth root of unity.

By De Moivre's formula (Equation (1-40)), the nth roots of unity can be

expressed as

1 ,Wn,,wn,··^2 · ,Wn n-1. (1-44)

Geometrically, the nth roots of unity a.re equally spaced points that lie on

the unit circle C 1 (0) = {z : lzl = l } and form the vertices of a regular polygon

with n sides.

• EXAMPLE 1.19 The solutions to the equation z^8 = 1 are given by the eight

va. 1 ues Zk = e ; •·• • = cos -^28 ,.-k + ism · · 2,,k 8 , r, or k = o ,^1 ,^2 ,... ,. 7 I n C artesian · ' rorm,

these solu tions a.re ± 1, ±i , ±~,and ±fi-,/12. The primitive 8th root of

unity is ws = ei"f = eit =cos~ + i sin~ = ../{ + i:lf..

From Expressions (1- 44 ) it is clear that ws = z1 of Equation (1- 43 ). Figure

1.18 illustrates this result.

The procedure for solving z" = 1 is easy to generalize in solving zn = c for

any nonzero complex number c. If c = pei~ = p (cos</> + i sin </>) and z = rei^6 ,

then zn = c iff r"einll = pe•<P. But this last equation is satisfied iff

r n =p, and

n8 = </> + 2k7T, where k is ari integer.

y

• i=W~

Figure 1.18 The eight eighth roots of unity.