Complex Analysis Exam 2
This is a take-home exam. You may use the book or your notes as you wish, but you must complete each
problem on your own. Show all your work (and be neat). Due: last day of finals – no exceptions!
1. Perform the following integrations along the indicated contours. You can use any method you like.
a)
∫
−
C
z
dz
z
e
2
, C the unit circle
1|| =z
b)
∫
C
iz
dz
z
e
3
, C the square with corners 1, i, -1, and –i.
c)
∫
−
C
dz
zz
z
)2(
)2cos(
, C the circle
2|3| =−z
d)
∫+
+
C
dz
zz
z
)1(
12
22
, C the circle
2|| =z
e)
dzez
C
z
∫2
4
C the circle
42|| =− iz
2. Find the Taylor series for each given function centered at the point
0
0
=z
. Specify the radius of
convergence for each series.
a)
f z z z( ) cos( )=
3 2
b)
z
z
zf 2
3
)( −
=
c)
)1ln()( zzf +=
3. Find a Laurent series for the given function centered at the given point
z0
that converges in the specified
domain.
a)
fz z e
z
()=
31
,
z
0
0=
, convergent in domain including
1=z
b)
2
43
1
)( zz
zf +
−
=
,
0
0=
z
, convergent in domain including
2=z
4. Consider the function
( )
( )
f z e
z z
z
( ) =− −3 1
2
If you were to find the Laurent series centered at
iz =
converging in the largest annulus
Rizr <−< ||
including the point
2=z
, then what are
r
and
R
?
5. Each of the following functions has one or more isolated singularity. Identify each singularity and
classify it as removable, pole, or essential. If it is a pole, find its order. Also, find the residue at each
singularity.
a)
=z
zzf 1
cos)(
3
b)
( )
( )
2
2
11
)( ++
=zz
z
zg
c)
z
e
zh z1
)( −
=
6. Use the (complex) Residue Theorem to evaluate
∫
∞
∞−
+dx
x
x
4
2
4
. Make sure to justify each step. Hint: the
answer is
2
π
Extra credit: An analytic function
)(zf
is said to have a zero of order m at
0
z
if
∑
∞
=
−
=
mn
n
n
zzazf )()(
0
,
i.e. the first non-zero coefficient in the Taylor series for f is
m
a
. Suppose
)(zf
is analytic near
0
z
with a
zero of order
k
at
0
z
Show that
)(
)('
zf
z
f
has a pole of order 1 at
0
z
. Hint: factor what you can from f(z),
then work out f’(z)/f(z) and use a theorem on what it means to have a pole of order m (or 1 in our case).
