1550251515-Classical_Complex_Analysis__Gonzalez_

286 Chapter^5

and n distinct values of w, say w 0 , w 1 , ••• , Wn-l are obtained by taking

k = 0, 1, ... , n - 1 successively. As we have seen, for n > 2 then images

of z are the vertices of a regular polygon of n sides inscribed in the circle

lwl = P·

If we consider in the z-plane, the n angular regions

Gk= {z: 0 < lzl < oo,2k7r :5 argz < 2(k+ l)7r}

where k = 0, 1, ... , n - 1, to each Gk we assign, under w = * y'Z, an
angular region (sector)

Hk={w: O<lwl<oo,k

2

: '.5argw<(k+1)

2

:}

where the value wk is located. Each Gk may be thought of as a z-plane cut
along the positive real axis from 0 to oo, and the function w = * y'Z can be
regarded as decomposed into n single-valued functions or branches, each
mapping in a one-to-one fashion a cut plane Gk onto the corresponding
sector Hr,. For instance, in the case n = 2 we consider two cut z-planes G 0
and Gi and the punctured w-plane decomposed into the sectors Ho and

Hi {Fig. 5.26). If a point is given as z = 4ein:/^3 , it belongs to Go and its

image w 0 = 2ei7r/^5 lies in H 0 , but if z is given as z = 4ei(11'/3+^2 11'), it belongs
to Gi and its image W1 = 2ei(11'/6+11') = -wo lies in H 1 •
For the case n = 3 we need three cut z-planes, G 0 , G 1 and G 2 , each of
which will be mapped onto the corresponding sector H 0 , JI 1 , and H 2 of
the punctured w-plane (Fig. 5.27).

y

Go z

v

'IT/3 --...... ...... .... Ho

....

(^0) x '-

'

'

WO

y u

G1 z - --------

H1

x

Fig. 5.26