MATH 5320-001
Final Exam
8 December 2008
Answer the problems on separate paper. You do not need to rewrite the problem statements on your answer sheets.
Work carefully. Do your own work. Show all relevant supporting steps!
w3
3 − i and w = − 2 + 2i . Find 9 and express the value in rectangular form.
z
z=
1.
Let
2.
Let (X,d) be a metric space and let A ⊂ X . Prove: A is closed if and only if
3.
A = A.
Classify the following sets as to whether they are: a) open, b) closed, c) connected,
d) polygonally path-connected, e) compact, f) complete, g) bounded, h) region.
You do not need to provide a rationale for your classification. Use the table on the last page to record your
classifications.
4.
1
1
|sin |, 0 < x < π }
x
x
1.
A = {( x , y ) : 0 < y <
2.
B = B(11
, ) \ B( 1 2 , 1 2)
3.
C = S \ B(0, π ) where S = {z : |Re z| < 1}
4.
D = B(2,2) \ B(2 − 2i , 8 )
Find the radius of convergence of the power series
∞
1.
2n n !n !
(3 z + 1) n
∑
n = 0 (2 n )!
∞
2.
∑ a ( z + i)
n =0
5.
Let
n
n
where the power series represents
f ( z ) = tan z
G = { z = reiθ : 0 < r < 1, 0 < θ < π } and let S = { z : | Re z | < 1} . Find a one-to-one
2
conformal map from G to S.
6.
The line L = { z : Im z = 0} and the circle
C1 = { z : | z − 1| = 1} divide D = { z :| z |< 1} into 4
subregions D1, D2, D3 and D4. See figure to the right. Let
w=
z
z−2
. Find the images Ej of each subregion Dj under w,
i.e., find E j = w( D j ), j = 1, 2, 3, 4.
7.
Let G be a region in ^ and let f ∈ A (G ), f = u + iv . Prove that if u 2 − v 2 = 1 on G, then f is
constant on G.
8.
Evaluate each of the following integrals:
1.
∫γ (Re z + (Im z )
2.
cos z + 1
∫γ z 3 dz where γ is a positively (counter-clockwise) oriented parametrization of the unit
circle
3.
∫γ z
2
) dz where γ is a parametrization of the straight line segment from 1 to i.
C = { z : | z |= 1}
z +1
dz where γ is a parametrization of the
− 2z
2
figure eight curve oriented as given in the figure to
the right
4.
dz
∫γ z where γ is a parametrization of the arc
Γ of the unit circle C = { z : | z |= 1} from i to -i passing through
-1 (see figure to the right) and where z is the branch of square
root z chosen so that the branch cut is taken as the positive real
axis, [0, ∞ ) , such that
5.
∫γ e
−1 = i
1+ z
dz where γ is a positively (counter-clockwise)
− e− z
z
oriented parametrization of the circle
C 2 = { z :| z − 2 |= 1}
9.
Prove that if f is an entire function and if there exist positive constants A and B such that
| f ( z ) |≤ A | z | + B for all z ∈ ^ , then f is a linear function.
10.
Let
11.
12.
D = { z :| z |< 1} and let f ∈ A ( D ) such that f ( D ) ⊂ D . Prove for z ∈ D that
1
. Hint. Suppose z ∈ D . Then, there exists a ρ > 0 such that B ( z , ρ ) ⊂ D .
| f ′( z ) |≤
1− | z |
D = { z :| z |< 1} and let f ∈ A ( D ) such that f ( D ) ⊂ D . Suppose that for each integer n > 1 that
i
i
f satisfies f ( ) = − 3 . Find the value of f ( 12 + 12 i ) .
n
n
Let
State and prove Louiville’s Theorem.
Classification Table for Problem 3
open
A
B
C
D
closed
connected
polygonally
path
connected
compact
complete
bounded
region