MATH 4356-001
Exam I
September 25, 2003
Answer the problems on separate paper. You do not need to rewrite the problem
statements on your answer sheets. Do your own work. Show all relevant steps which
lead to your solutions. Retain this question sheet for your records.
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
DOOTF: Determine which quadrant(s)
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 ∈ £ which satisfy:
a.
6.
| z − 1 −4 i | = 6
| z − 3i | > 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| > 1}
C = { z : 0 < | z − 3 i | ≤ 1}
D = z : ( Re z )2 > 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 | > 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 | < 1 and Re z > 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