# MATH 4356-001 Exam I September 25, 2003 ```MATH 4356-001
Exam I
September 25, 2003
Answer the problems on separate paper. You do not need to rewrite the problem
Notation:
1.
DOOTF: Write each number in a + bi form:
a.
2.
i
1
−
1+ i i −1
b.
( 2 + i ) ( −3 + 4 i )
b.
z+2
= 2i − 3
z
2
DOOTF: Solve for z :
a.
3.
Q1 , Q2 , Q3 , Q4 will denote the four standard quadrants.
DOOTF will stand for Do One Of The Following
z
= 3−4i
1− z
a.
1
1
and − belong to if :
z
z
z ∈ Q3
b.
z ∈ Q4
1+ z
=0.
1− z
4.
Show that if | z | = 1, z ≠ 1 , then Re
5.
DOOTF: Describe the set of points z ∈ &pound; which satisfy:
a.
6.
| z − 1 −4 i | = 6
| z − 3i | &gt; 2
b.
Find the argument of the following number and write it in polar form reiθ :
(1 + i ) (
3 −i
)
7.
Sketch the curve: z (t ) = e( i −1)t , 0 ≤ t ≤ π
8.
DOOTF: Find (all) value(s) of :
a.
( 2 −2
3i
)
11
b.
( i − 1)
1
3
9.
Let
a.
b.
c.
10.
A = { z :| z − 3 + i | = 2}
B = { z : | z − i − 1| &gt; 1}
C = { z : 0 &lt; | z − 3 i | ≤ 1}
D = z : ( Re z )2 &gt; 0
{
}
Which of the above sets are open?
Which of the above sets are domains?
Which of the above sets are bounded?
Suppose u ( x, y ) is a real-valued function defined on a domain D. If
∂u
∂u
= x and
= −2 y for all points in D,
∂x
∂y
then show that u ( x , y ) =
11.
DOOTF: Describe the range of :
a.
12.
g( z) =
1
for | z | &gt; 2
z
b.
F ( H ( z ))
G ( F( z))
DOOTF: Decide whether the sequence converges; if so, find its limit:
zn
 i
i 1+ 
= e  n
n
b.
 1− i 
zn = 

 2 
b.
z 2 − (3 + i) z
lim
z → i −1
iz
DOOTF: Find the limit, if it exists:
a.
15.
b.
iπ
a.
14.
f ( z ) = z 2 for z ∈ Q2
DOOTF: Let F ( z ) = z = 3i , G( z ) = e 3 z , H ( z ) = 2 z . Consider the set
S = {z :| z | &lt; 1 and Re z &gt; 0} . Sketch the image of S under:
a.
13.
x2
− y 2 + c for some constant c.
2
lim
z → 2−3 i
(z
2
+ 6z
)
2
DOOTF: Find the limit, if it exists:
a.
ez + e−z
lim
z→ π i
2i
b.
lim
z → −2 i
z2 + 8
2z2 + 8
```