Complex Variables MAT334
T
ERM TEST
2
Solution
1. (5 points) State the Liouville’s theorem.
Ans 1:
Let
f
be an entire function.
If there exists a real number
M >
0
such that

f
(
z
)

< M
for all
z
∈
C
then
f
must be constant on
C
.
2. (10 points)
Res
e
π
2
iz
(
z
2
+ 1)(
z

1)
,
1
=
e
π
2
i
(1 + 1)
=
i
2
Here we used that the given function has a pole of order 1 at 1, since the function
e
π
2
iz
/
(
z
2
+ 1)
is analytic at 1 and its value at 1 is not 0.
Res
cos
z
e
z

1
,
0
=
cos 0
e
0
= 1
Here we used the fact that if for some functions
F
(
z
)
and
G
(
z
)
,
F
(
z
0
) = 0
,
G
(
z
0
) = 0
,
G
(
z
0
) = 0
and both
F
and
G
are analytic at
z
0
then Res
(
F
G
, z
0
) =
F
(
z
0
)
G
(
z
0
)
. We applied
this fact to
F
(
z
) = cos
z
and
G
(
z
) =
e
z

1
at
z
0
= 0
.
3. (10 points) Let
f
(
z
) =
z
sin
z
(
z

π
)
3
. Find the first three terms of the Laurent series of the
function at
z
=
π
. Determine the residue of
f
at
π
.
Ans:
Let
h
(
z
) =
z
sin
z
. To determine the Laurent series for
f
we first find the (first few
terms of the) Taylor series for
h
at
z
=
π
. To do that we compute:
h
(
z
) =
z
sin
z
h
(
π
) = 0
h
(
z
) = sin
z
+
z
cos
z
h
(
π
) =

π
h
(
z
) = 2 cos
z

z
sin
z
h
(
π
) =

2
h
(
z
) =

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 Spring '09
 Cos, dz, Laurent

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