Answer ALL questions from Section A.
All questions from Section B may be attempted, but only marks obtained on the best
two
solutions from Section B will count.
The use of an electronic calculator is
not
permitted in this examination.
The following notation is used throughout:
for any
a
∈
C
and
r >
0,
D
(
a, r
) =
{
z
:

z

a

< r
}
,
D
(
a, r
) =
{
z
:

z

a
 ≤
r
}
,
D
0
(
a, r
) =
{
z
: 0
<

z

a

< r
}
.
Section A
1. Let
f
be a function holomorphic on a punctured disk
D
0
(
z
0
, r
),
z
0
∈
C
, r >
0.
Describe the three types of isolated singularities of a function
f
by explaining how
they are related to the principal part of its Laurent expansion.
2.
(a) Suppose that
g
(
z
) has a removable singularity at the point
z
0
.
What is the
residue of
g
at
z
0
? Justify your answer.
(b) What type of singularity does the function
f
(
z
) =
sinh
z
z
4
have at
z
0
= 0? Justify
your answer.
Find the residue of
f
at
z
0
.
3.
(a) Prove that the disk
D
(
a, r
),
a
∈
C
, r >
0, is an open set.
(b) Let
A
and
B
be open subsets of the complex plane. Prove that
A
∪
B
is also
open.
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 Fall '18