Complex Analysis - Maths KU

112 Chapter 2 Complex Functions and Mappings
y
δ
z
z 0
(a) Deleted δ -neighborhood of z0 v
ε
L
f(z)
(b) ε-neighborhood
of L
Figure 2.51 The geometric meaning
of a complex limit
y
4
3
2
1
–2 –1
1 2
–1
Figure 2.52 The limit of f does not
exist as x approaches 0.
y
z 0
Figure 2.53 Different ways to approach
z0 in a limit
x
u
x
x
Definition 2.8 Limit of a Complex Function
Suppose that a complex function f is defined in a deleted neighborhood
of z0 and suppose that L is a complex number. The limit of f as z
tends to z 0 exists and is equal to L, written as lim f(z) =L, if
z→z0
for every ε>0 there exists a δ>0 such that |f(z) − L|