MATH 530 Qualifying Exam
August 2008 (S. Bell, A. Eremenko)
Each problem is worth 25 points
1.
Suppose
u
and
v
are real valued harmonic functions on a domain Ω.
a)
(10 pts.)
If
u
and
v
agree on a set with a limit point in Ω, does it follow that
u
=
v
on all of Ω? Explain.
b)
(15 pts.)
If
u
and
v
satisfy the CauchyRiemann equations on a set with a
limit point in Ω, does it follow that
u
+
iv
is analytic on Ω? Explain.
2.
Suppose that
f
is continuous on
{
z
:

z
≤
1
}
and analytic in
{
z
:

z

<
1
}
.
a)
(15 pts.)
Prove that if
f
vanishes on
{
z
:
z
=
e
iθ
,
0
≤
θ
≤
π
2
}
,then
f
≡
0on
the unit disc. Hint: Where do
f
(
i
n
z
) vanish for
n
=1
,
2
,
3?
b)
(10 pts.)
Does the same result hold for harmonic functions? Explain.
3.
Suppose
f
and
g
are analytic in a neighborhood of a point
a
, and suppose
g
has
a simple zero at
a
. Find a formula for the residue of
f
(
z
)
g
(
z
)
2
at
a
in terms of
f
(
a
)
and derivatives of
f
and
g
at
a
.
4.
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 Spring '09
 Math, Complex number, Cr, Logarithm, unit disc

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