Complex Analysis - Maths KU

110 Chapter 2 Complex Functions and Mappings
(c) If the image of S is a circle S ′ , then what are center and radius of S ′ ?How
do these values compare with the center and radius of S?
29. Consider the complex function f(z) = 1+i
+ 2 defined on the annulus
z
1 ≤|z| ≤2.
(a) Use mappings to determine upper and lower bounds on the modulus of
f(z) = 1+i
+ 2.
� z
�
L ≤ �
�
That is, find real values L and M such that
1+i
z +2
�
�
�
� ≤ M.
(b) Find values of z that attain your bounds in (a). In other words, find z0
and z1 such that z0 and z1 are in the annulus 1 ≤|z| ≤2 and |f(z0)| = L
and |f(z1)| = M.
30. Consider the complex function f(z) = 1
+ i defined on the half-plane x ≥ 2.
z
(a) Use mappings to determine an upper bound M on the modulus of f(z).
(b) Find a value of z that attains your bound in (a). That is, find z0 such that
z0 is in the half-plane x ≥ 2 and |f(z0)| = M.
2.6 Limits and Continuity
2.6
The most important concept in elementarycalculus is that of the limit. Recall that
f(x) =L intuitivelymeans that values f(x) of the function f can be made arbi-
lim
x→x0
trarilyclose to the real number L if values of x are chosen sufficientlyclose to, but not
equal to, the real number x0. In real analysis, the concepts of continuity, the derivative,
and the definite integral were all defined using the concept of a limit. Complex limits play
an equallyimportant role in studyof complex analysis. The concept of a complex limit is
similar to that of a real limit in the sense that lim f(z) =L will mean that the values f(z)
z→z0
of the complex function f can be made arbitrarilyclose the complex number L if values
of z are chosen sufficientlyclose to, but not equal to, the complex number z0. Although
outwardlysimilar, there is an important difference between these two concepts of limit. In
a real limit, there are two directions from which x can approach x0 on the real line, namely,
from the left or from the right. In a complex limit, however, there are infinitelymany
directions from which z can approach z0 in the complex plane. In order for a complex limit
to exist, each wayin which z can approach z0 must yield the same limiting value.
In this section we will define the limit of a complex function, examine some of its
properties, and introduce the concept of continuityfor functions of a complex variable.
2.6.1 Limits
Real Limits The description of a real limit given in the section introduction
is onlyan intuitive definition of this concept. In order to give the
rigorous definition of a real limit, we must preciselystate what is meant by
the phrases “arbitrarilyclose to” and “sufficientlyclose to.” The first thing to