Page 405 - Schaum's Outline of Theory and Problems of Advanced Calculus

396 FUNCTIONS OF A COMPLEX VARIABLE [CHAP. 16

BRANCHES AND BRANCH POINTS
Another type of singularity is a branch point. These points play a vital role in the construction of
single-valued functions from ones that are multiple-valued, and they have an important place in the
computation of integrals.
In the study of functions of a real variable, domains were chosen so that functions were single-
valued. This guaranteed inverses and removed any ambiguities from differentiation and integration.
The applications of complex variables are best served by the approach illustrated below. It is in the
realm of real variables and yet illustrates a pattern appropriate to complex variables.
2
ffiffiffi
p
Let y ¼ x; x > 0, then y ¼ x. In real variables two functions f 1 and f 2 are described by
ffiffiffi p ffiffiffi
p
y ¼þ x on x > 0, and y ¼ x on x > 0, respectively. Each of them is single-valued.
An approach that can be extended to complex variable results by defining the positive x-axis (not
including zero) as a cut in the plane. This creates two branches f 1 and f 2 of a new function on a domain
called the Riemann axis. The only passage joining the spaces in which the branches f 1 and f 2 , respec-
tively, are defined is through 0. This connecting point, zero, is given the special name branch point.
Observe that two points x in the space of f 1 and x in that of f 2 can appear to be near each other in the

ordinary view but are not from the Riemannian perspective. (See Fig. 16-1.)

Fig. 16-1

The above real variables construction suggests one for complex variables illustrated by w ¼ z 1=2 .
In polar coordinates e 2 i ¼ 1; therefore, the general representation of w ¼ z 1=2 in that system is
w ¼ 1=2 ið þ2 kÞ=2 , k ¼ 0; 1.
e
Thus, this function is double-valued.
If k ¼ 0, then w 1 ¼ 1=2 e i =2 ,0 < 2 ; > 0
If k ¼ 1, then w 2 ¼ 1=2 e ið þ2 Þ=2 ¼ 1=2 e i =2 i 1=2 e i =2 ; 2 < 4 ; > 0.
¼
Thus, the two branches of w are w 1 and w 2 , where w 1 ¼ w 2 . (The double valued characteristic of w
is illustrated by noticing that as z traverses a circle, C: jzj¼ through the values to 2 . The functional
values run from 1=2 i =2 to 1=2 i
e
e .In other words, as z navigates the entire circle, the range variable
only moves halfway around the corresponding range circle. In order for that variable to complete the
circuit, z would have to make a second revolution. Thus, we would have coincident positions of z giving
rise to distinct values of w. For example, z 1 ¼ e ð =2Þ=i and z 2 ¼ e ð =2þ2 Þi are coincident points on the unit
p ffiffiffi p ffiffiffi
2 1=2 2
1=2
circle. The distinct functional values are z 1 ¼ ð1 þ iÞ and z 2 ¼ ð1 þ iÞ.
2 2
The following abstract construction replaces the multiple-valued function with a new single-valued
one.
Make a cut in the complex plane that includes all of the positive x-axis except the origin. Think of
two planes, P 1 and P 2 , the first one of infinitesimal distance above the complex plane and the other
infinitesimally below it. The point 0 which connects these spaces is called a branch point. The planes