Complex Analysis - Maths KU

126 Chapter 2 Complex Functions and Mappings
y
Figure 2.55 The domain D of the
x
We call the function f1 defined by(23) the principal branch of F (z) =z 1/2
because the value of θ represents the principal argument of z for all z in
Dom (f1). In the following example we show that f1 is, in fact, a branch of
F .
EXAMPLE 8 A Branch of F (z) =z 1/2
Show that the function f1 defined by(23) is a branch of the multiple-valued
function F (z) =z 1/2 .
Solution The domain of the function f1 is the set Dom(f1) defined by
|z| > 0, −π