1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

68 CHAPTER 2 • COMPLEX FUNCTIONS

periodic with period ~, so f is in general an n -to-one function; that is, n points
in the z plane are mapped onto each nonzero point in the w plane.

If we now restrict the domain of w = f (z) = zn to the region

{

E = re• ~ : r > 0 and n -7r < 0 :::; 7r} n ,

then the image of E under the mapping w = zn can be described by the set

F = {pei.P: p > 0 and - w < ¢:::; w},

which consists of all points in the w plane except the point w = 0. The inverse

mapping of/, which we denote g, is then

l. .I.. t

z =g(w) =w• =p"e' • ,

where w E F. That is,

1 l .Andw)

z =g(w)=wn=lw lne• n ,

where w =/= O. As with the principal square root function, we make an analogous

definition for nth roots.

Definition 2.2: Principal nth root
The function

I J. · Ar«-( w }

g (w) = wn = lwl n e' n , for w =/= 0,

is called the principal nth root functio n.

We leave as an exercise to show that f and g are inverses of each other that

map the set E one-to-one and onto the set F and the set F one- to-one and onto

the set E, respectively. F igure 2.16 illustrates this relationship.

y

w=t

x

z= wk

t

Figure 2 .16 The mappings w = z" and z = wn:.

v

u