Complex Analysis Final
Stephen Taylor
June 23, 2005
1. For each of the following, determine whether the statement is true or false.
Justify your answers with a brief explanation when your response is true and a
counterexample when your response is false.
(a) There is a branch cut for
f
(
z
) = log(
z
2
+ 1) so that
f
is analytic on
{
z
∈
C
:

z

>
1
}
.
Solution:
This statement is true. Since the branch points of a complex loga
rithm function correspond to its singularities, we find the branch points to be
±
i
. We therefore take a branch cut along the segment [

i, i
], so
f
(
z
) is analytic
everywhere exterior to the unit disc as desired.
(b) If
f
has a pole of order
m
at
a
, then
f /f
has a pole of order 1 at
a
.
Solution:
This statement is true. If
f
has a pole or order
m
, then it has
representation
f
(
z
) =
a
1
(
a

z
)
m
+
a
2
(
a

z
)
m

1
+
O
(
z

m
+2
)
Differentiating we find
f
(
z
) =

a
1
m
(
a

z
)
m
+1
+

a
2
(
m

1)
(
a

z
)
m
+
O
(
z

m
+1
)
Using synthetic division, we find
f
(
z
)
f
(
z
)
=

m
a

z
+
a
2
a
1
+
O
(
z
1
)
from which we note that the quotient has a pole of order 1 at
z
=
a
.
(c) If
f
(
z
) has an essential singularity at
∞
, then 1
/f
(1
/z
) also has an
essential singularity at
∞
.
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Solution:
This statement is true. Since
f
(
z
) has an essential singularity at
∞
,
f
(
z

1
) has an essential singularity at zero.
f
(
z

1
) =
∞
∞
a
n
z
n
We consider the nonterminating singular part of
f
(
z

1
) given by
f
sing
(
z

1
) =
a

1
z
+
a

2
z
2
+
· · ·
+
a

n
z
n
+
· · ·
Using synthetic division, we find
1
f
sing
=
z
a

1

a
1
a
2
+
O

(
z

1
)
where
O

(
·
) is the reverse order function indicating powers of
z
grow
arbitrarily small. This tells us (
f
(
z

1
))

1
has an essential singularity.
(d) If
f
is analytic and bounded on the unit disk,
D
, then
f
must be a rational
function.
Solution:
This statement is false. Consider the entire function
f
(
z
) =
e
z
. We
find

e
z

=

e
x
+
iy

=
e
x
which is bounded since
x
may only range between
(

1
,
1). Since
f
is clearly not a rational function, we conclude the invalidity of
the statement.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Three '09
 Cong
 Math, lim, dz, Pole, Essential singularity, removable singularity

Click to edit the document details