1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

82 CHAPTER 2 • COMPLEX FUNCTIONS

v

w=f,/..z)

x

z=w'

Figure 2. 19

l

The branch J .. off (z) = z~.

The corresponding branch, denoted f oo is

(2-30)

where z = rei^9 # 0 and a<(}:::; a+ 211".

The branch cut for f 0 is the ray r ~ 0, (} = a, which includes the origin.

The point z = 0, common to all branch cuts for the multivalued square root

function, is called a bra nch point. The mapping w = f 0 (z) and its branch cut

are illustrated in Figure 2.19.

2.4.1 The Riemann Surface for w = z4

A Riemann surface is a construct useful for visualizing a multivalued function.

It was introduced by G. F. B. Riemann (1826-1866) in 1851. The idea is

ingenious-a geometric construction that permits surfaces to be the domain or

range of a multivalued function. Riemann surfaces depend on the function being

investigated. We now give a nontechnical formulation of the Riemann surface

for the multivalued square root function.

Consider w = f (z) = z~, which has two values for any z # 0. Each function

Ji and f2 in Figure 2.18 is single-valued on the domain formed by cutting the z

plane along the negative x-axis. Let Di and D2 be the domains of Ji and h,

respectively. The range set for fi is the set H i consisting of the right half-plane,

and the positive v-ax.is; the range set for h is the set H2 consisting of the left

half-plane and the negative v-axis. The sets H1 and H2 are "glued together"

along the positive v-ax.is and the negative v-ax.is to form the w plane with the

origin deleted.

We stack Di directly above D 2. The edge of Di in the upper half-plane

is joined to the edge of D 2 in the lower half-plane, and the edge of D 1 in the

lower half-plane is joined to the edge of D 2 in the upper half-plane. When these

domains are glued together in this manner, they form R, which is a Riemann

surface domain for the mapping w = f ( z) = z ~. The portions of Di, D 2 , and

R that lie in {z: lzl < 1} are shown in Figure 2.20.