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CAN702S - COMPLEX ANALYSIS - 1ST OPP - NOVEMBER 2024 |
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nAm I BIA UnlVERSITY
OF SCIEnCE AnDTECHnOLOGY
FacultyofHealth,Natural
ResourceasndApplied
Sciences
Schoool f NaturalandApplied
Sciences
Departmentof Mathematics,
Statisticsand ActuarialScience
13JacksonKaujeuaStreet
Private Bag13388
Windhoek
NAMIBIA
T: +264612072913
E: msas@nust.na
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QUALIFICATION: BACHELOR of SCIENCE IN APPLIED MATHEMATICS AND STATISTICS
QUALIFICATION CODE: 07BSAM; 07BSOC
LEVEL:7
COURSE:COMPLEX ANALYSIS
COURSECODE: CAN702S
DATE: NOVEMBER 2024
SESSION: 1
DURATION: 3 HOURS
MARKS: 100
FIRSTOPPORTUNITYEXAMINATION: QUESTIONPAPER
EXAMINER:
DR. NEGACHERE
MODERATOR:
PROF.FORTUNE MASSAMBA
INSTRUCTIONS:
1. Answer all questions on the separate answer sheet.
2. Please write neatly and legibly.
3. Do not use the left side margin of the exam paper. This must be allowed for the
examiner.
4. No books, notes and other additional aids are allowed.
5. Mark all answers clearly with their respective question numbers.
PERMISSIBLE MATERIALS:
1. Non-Programmable Calculator
ATTACHMENTS:
NONE
This paper consists of 3 pages including this front page.
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QUESTION 1 [21)
1.1. Express~
-
1
~i
in
the
form
of x
+ iy,
where
x, y
E
IRL
[6]
1-1
I
= 1.2. Find the principal argument of the complex number given by z (l~~) 2 •
[6]
G 1.3. Compute cos + i).
[9]
QUESTION 2 [9]
Let lz+ 11= lz- ii be a curve in the complex plane. Then
2.1. describe the type of curve represented by lz+ 11= lz- ii.
[6]
2.2. determine whether the curve is open or closed, connected or unconnected, bounded or
unbounded. Justify your answer.
[3]
QUESTION 3 [9]
Find cos 48 and sin 48 in terms of sin 8 and cos 8.
[9]
QUESTION 4 [25)
x3-y3+i(x3+y3)
* if Z 0
Let f(z) = {
x2+y 2
0 if z = 0
. Then show that
4.1. Cauchy-Riemann Equations are satisfied at (0, 0).
[15]
4.2. f' (0, 0) does not exist.
[10]
QUESTION 5 [14)
Let f: IR2 IR be twice differentiable function.
5.1. What does it mean to say f is harmonic?
[2]
5.2. Determine whether u(x, y) = 2x - x 3 + 3xy 2 is harmonic or not. If it is harmonic find
= its harmonic conjugate function v(x, y) such that f(z) u(x, y) + iv(x, y) is analytic. [12]
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QUESTION 6 [22]
Evaluate the following contour integrals.
C = 6.1. [
2
;
1
)
dz where
C is the
semicircle
parametrized
by
z(8)
zei9, TT$ 8 $ Zn. [10]
Jc 6.2.
[
(cozshz)
z +z
dz where
C is the
counter
formed
by two
parts: C1 defined
by the
circle
lz + 11= 2 and C2 defined by the circle lzl = ~2 , both oriented counterclockwise.
[12]
END OF FIRST OPPORTUNITY EXAMINATION QUESTION PAPER
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