1550251515-Classical_Complex_Analysis__Gonzalez_

Complex Numbers 47

where, as before, k denotes an arbitrary integer. This shows that the
modulus of any nth root of w "is uniquely determined as the principal nth
root of r = lwl, while the argument depends on the choice of the integer k.
·Thus it may appear that there are infinitely many roots corresponding to
the infinitely many choices of k. However, the choices k = p and k = p+nq
(p, q integers) lead to the same root since
B + 2(p + nq)1l' B + 2p1l'
~~=-~--'-'--= + 2 q1l'
n n
i.e., the corresponding arguments differ by a multiple of 211'. Hence it
suffices to take n consecutive values of k to get all the roots of w. More
generally, we may choose any set of n incongruent integers modulus n. For

convenience we usually let k = 0, 1, 2, ... , n - 1.

Therefore, we have established the following

Theorem 1.10 Every complex number w = rei^9 =/; 0 has exactly n

different nth roots, which are given by the formula

* ~ = { y'r"ei[(IJ/n)+k(^2 ,,./n)] : k = O, 1, ... , n - 1} (1.12-7)

where the notation* \fW is used to indicate the set of all different roots of w.

Example The cubic roots of -1 + i = ...;2e^3 i,,./^4 are

Briefly,

*..V-1 + i = { {/2ei[(,,./^4 )+k(^2 ,,./^3 )]: k = 0, 1,2,}

Example The nth roots of 1 are given by

*Ji= { e^2 k7ri/n : k = 0, 1, ... , n - 1}

If we let (3 = e^2 ,,.i/n, the nth ioots of unity are

1, (3, (32, ... ; (Jn-1

For instance, the cubic roots of 1 are

1, (3 = e2,,.i/3 = %(-1 + iVs), (32 = e4,,.i/3 = %(-1 -iVs)

Definition 1.6 If -1!' < B :::; 11', the principal nth root of w = rei^9 will

be defined to be

(1.12-8)

Since