Problem Set 2
Due February 5
th
, 2008
1. Consider following sets:
a)
A
=
{
z
:

Arg
z

< π/
4
}
b)
B
=
{
z
:

1
<
Im
z <
1
}
c)
C
=
{
z
:

z
 ≥
1
}
d)
D
=
{
z
: (Re
z
)
2
≥
1
}
Which of these sets are open?
Which are closed?
Which are bounded?
Which are
domains? (No need to prove your statements.)
2. Suppose
u
(
x, y
) is a realvalued function defined in a domain
D
. If
∂u
∂x
=
y
and
∂u
∂y
=
x
at all points of
D
, show that
u
(
x, y
) =
xy
+
c
for some constant
c
.
3. Prove that if

z
0

<
1, then
z
n
0
→
0 as
n
→ ∞
.
4. Find each of the following limits.
(a)
lim
z
→
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 Spring '08
 Hou
 Calculus, Sets, lim, Continuous function, ∂u =x ∂y, nonpositive real axis

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