THE UNIVERSITY OF THE FREE STATE
BLOEMFONTEIN CAMPUS
MATM3714 - Complex Analysis
DEPARTMENT OF MATHEMATICS & APPLIED MATHEMATICS
CONTACT NUMBER: 051 401 9789
EXAMINATION: Additional Examination 2022
PAPER 1
ASSESSOR: Dr. C Budde
MODERATOR: Dr. E Kikianty
Time: 3 hours
Marks: 77
Important instructions:
• This is a closed-book test.
• Read each question carefully before you answer it.
• Unsupported answers are not acceptable, even if they are correct. It is to your advantage to write
your solutions as clearly as possible.
• Write with a pen. Solutions written with a pencil will not be accepted.
• Mobile phones and similar devices (incl. smart watches) are prohibited during the test. A mobile
phone on your desk or in your hand means the immediate end of your test.
• Allocated marks are indicated in squared brackets.
• Answer all questions!
Good luck!
Exercise
1
2
3
4
5
6
7
P
Max.
13
10
10
11
9
14
10
77
Points
1
Mark:
QUESTION 1. The goal of this exercise is to evaluate
∞ X
2n 1
.
n 5n
n=0
To do so, follow the following steps:
Z
(1 + z)n
n
1
dz.
(i) Give a justification that
=
2πi γ(0,1) z k+1
k
n
(ii) Use (i) to prove that 2n
n ≤ 4 for all n ∈ N.
(iii) Show that
∞
X
(z + 1)2n
n=0
(iv) Prove that
∞ X
2n 1
n=0
QUESTION 2.
=
(5z)n
n
5n
=
[2]
[3]
5z
for |z| = 1.
3z − 1 − z 2
√
[4]
5. Justify each step.
(a) Show that the series
[4]
∞
X
2n (z − 1)n
n=1
4n − 1
converges uniformly for |z − 1| < 12 .
(b) Use the ratio test to determine whether or not the following series is convergent:
∞
X
(1 − i)n
n4n
n=1
[5]
[5]
.
i
and determine the image of the unit circle
QUESTION 3. (a) Consider the function f (z) = i−z
{z ∈ C : |z| = 1} under f .
[5]
7
(b) Determine all z ∈ C such that the function f (z) = ez sin
16 (z)
+ z 2 is differentiable.
[5]
QUESTION 4. Find the set of all z ∈ C where the function f is differentiable and at which points f is
holomorphic.
(a) f (z) = |z + 1|2
[5]
(b) f (z) = f (x + iy) = x3 − y 3 + i(2x2 y 2 )
[6]
QUESTION 5.
[3]
(a) State Cauchy’s Integral Theorem.
(b) Evaluate the following integral
eiπz
dz,
2
γ 2z − 5z + 2
Z
where γ denotes the anti-clockwise oriented unit circle.
QUESTION 6.
[6]
(a) Find the power series of the function f (z) =
(b) Find the Laurent series of the function f (z) =
answer as a single series.
z+2
about z = −1.
z(z − 1)
[7]
1
about z = 0 for 1 < |z| < ∞. Express your
z(z − 1)
[7]
QUESTION 7. Show that
Z ∞
1
0
(x2 + 1)2 (x2 + 4)
dx =
π
18
by integrating the function f (z) = (z 2 +1)12 (z 2 +4) over the upper semicircle.
2
[10]