Complex Analysis - Maths KU

y
Values of arg
decreasing
Values of arg
increasing
1 + εi
1
1 – εi
y
0
Values of arg
increasing
(a) z = 1 is not a branch point
(b) z = 0 is a branch point
Figure 2.56 G (z) = arg(z)
x
x
2.6 Limits and Continuity 127
being analogous to the positive and negative square roots of a positive real
number.
Other branches of F (z) =z1/2 can be defined in a manner similar to
(23) using anyrayemanating from the origin as a branch cut. For example,
f3(z) = √ reiθ/2 , −3π/4 0 such that |f(z) − L| < ε
whenever |z| > 1/δ.
Using this definition it is not hard to show that:
lim f(z) =L if and onlyif lim
z→∞ z→0 f
� �
1
= L. (24)
z
Similarly, the infinite limit lim f(z) =∞ is defined by:
z→z0
The limit of f as z tends to z 0 is ∞ if forevery ε > 0 there
is a δ>0 such that |f(z)| > 1/ε whenever 0 < |z − z0|