w ¼ ρ cos φ þ i sin φ
ð
Þ :
Then, we have
z ¼ w
n
¼ ρ
n cos φ þ i sin φ
ð
Þ
n ¼ ρ
n cos nφ þ i sin nφ
ð
Þ ,
ð6:228Þ
where with the last equality we used de Moivre’s theorem (6.36). Comparing the real
and imaginary parts of (6.32) and (6.228), we have
r ¼ ρ
n or ρ ¼ r
1=n
ð6:229Þ
and
θ ¼ nφ þ 2kπ k ¼ 0, Æ1, Æ2, Á Á Á
ð
Þ :
ð6:230Þ
In (6.229), we define both r and ρ as being positive so that positive ρ can be
uniquely determined for any positive integer n (i.e., either even or odd). Recall once
again that if n is even, we have two n-th roots for ρ (i.e., both positive and negative)
with given positive r. Of these two roots, we discard the negative ρ. Rewriting
(6.230), we have
φ ¼
1
n
θ À 2kπ
ð
Þ k ¼ 0, Æ1, Æ2, Á Á Á
ð
Þ :
Further rewriting (6.227), we get
w k θ
ð Þ ¼ z
1=n
¼ r
1=n cos
1
n
θ À 2kπ
ð
Þþi sin
1
n
θ À 2kπ
ð
Þ
h
i
k ¼ 0, 1, 2, Á Á Á, n À 1
ð
Þ ,
ð6:231Þ
where w k (θ) is said to be a branch. Note that we have
w k θ À 2nπ
ð
Þ¼w k θ
ð Þ:
ð6:232Þ
That is, w k (θ) is a periodic function of the period 2nπ. The number of the total
branches is n; the index k of w k (θ) denotes these different branches. In particular,
when k ¼ 0, we have
w 0 θ
ð Þ ¼ z
1=n
¼ r
1=n cos
1
n
θ þ i sin
1
n
θ
:
ð6:233Þ
Comparing (6.231) and (6.233), we obtain
250
6 Theory of Analytic Functions