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Solution: Problem Set 2
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.)
Answer.
a
and
b
are open,
c
and
d
are closed. None of them are bounded.
a
and
b
are domains.
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
.
Proof.
Let
v
(
x,y
) =
u
(
x,y
)

xy
, then
∂v
∂x
=
∂u
∂x

y
=
y

y
= 0
,
and
∂v
∂y
=
∂u
∂y

x
=
x

x
= 0
.
So
v
(
x,y
) must be a constant, hence
u
(
x,y
) =
xy
+
v
(
x,y
) =
xy
+
c
for some constant
c
.
3. Prove that if

z
0

<
1, then
z
n
0
→
0 as
n
→ ∞
.
Proof.
For any
ε >
0, let
N
= log

z
0

ε,
then for any
n > N
, because

z
0

<
1, so the function

z
0

x
is monotonic decreasing, so

z
x
0

0

=

z
x
0

=

z
0

x
≤ 
z
0

N
=
ε.
Hence
lim
n
→∞
z
n
0
= 0
.
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View Full Document4. Find each of the following limits.
(a)
lim
z
→
2+3
i
(
z

5
i
)
2
(b) lim
z
→
3
i
z
2
+ 9
z

3
i
(c)
lim
z
→
1+2
i

z
2

1

(d) lim
z
→
i
z
2
+
i
z
4

1
Answer.

8
i,
6
i,
4
√
2 and ‘not exists’.
5. Show that the function Arg
z
is discontinuous at each point on the nonpositive real
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 Spring '08
 Hou
 Sets

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