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P. Vojta
Math 185: Review Problems
Fall 2007
This sheet is adapted from a similar review sheet by M. Christ, which in turn contains some
problems from
Notes on Complex Function Theory
by Donald Sarason.
1. Suppose that
∑
n
a
n
z
n
and
∑
n
b
n
z
n
have radii of convergence
R
1
,R
2
respectively. Suppose that
there exists a ﬁnite constant
M
such that

a
n
 ≤
M

b
n

for all but ﬁnitely many
n
. Prove that
R
1
≥
R
2
.
2. Evaluate
∑
∞
n
=1
n
2
z
n
.
3. Evaluate (a)
R
[

i,i
]

z

dz
and (b)
R
C
+

z

dz
where
C
+
is the right half of the unit circle, oriented
counterclockwise. You should get two diﬀerent answers. Explain why this doesn’t contradict the
theorem on page 135.
4. Let
a >
0 . Calculate
R
∞
0
t
2
t
4
+
a
4
dt
and
R
∞
0
t
2
cos
t
t
4
+
a
4
dt
.
5. Show that
Z
∞
0
e

t
2
cos(
t
2
)
dt
=
√
π
4
q
1 +
√
2 .
6. Let
C
be a circle bounding an open disk
D
, and suppose that
f
is analytic on
D
. Suppose that
z
0
/
∈
D
. Evaluate
R
C
f
(
z
)
z

z
0
dz
.
7. Show that if
f
is analytic on the disk of radius
R
centered at
z
0
then for any 0
< r < R
,
f
(
z
0
) = (
πr
2
)

1
RR

z

z
0

<r
f
(
z
)
dxdy
where
z
=
x
+
iy
.
8. Evaluate
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 Fall '07
 Lim
 Math

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