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CAN702S - COMPLEX ANALYSIS - 2ND OPP - JANUARY 2025 |
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p
I
nAml BIA UnlVERSITY
OF SCIEnCE AnDTECHnOLOGY
FacultyofHealth,Natural
ResourceasndApplied
Sciences
Schoolof NaturalandApplied
Sciences
Departmentof Mathematics,
StatisticsandActuarialScience
13JacksonKaujeuaStreet
PrivateBag13388
Windhoek
NAMIBIA
T: +26461207 2913
E: msas@nust.na
W: www.nust.na
QUALIFICATION: BACHELOR of SCIENCEIN APPLIED MATHEMATICS AND STATISTICS
QUALIFICATION CODE: 07BSAM; 07BSOC
LEVEL:7
COURSE:COMPLEX ANALYSIS
COURSECODE: CAN702S
DATE: JANUARY 2025
SESSION: 2
DURATION: 3 HOURS
MARKS: 100
SECOND OPPORTUNITY/ SUPPLEMENTARY: EXAMINATION QUESTION PAPER
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 [15]
I 1.1. Find ( ...)3- i)100. 1
[4]
1.2. Determine the principal argument of the complex number l-i.•
[6]
-1-1
= ( 1.3. Compute the principal logarithm of the complex number z -1 + i-v'3).
[S]
QUESTION 2 [13]
2.1. Express f(z) = 1-z in the form of u(x, y) + iv(x, y).
[S]
2.2. Find the image of the disk lz- 1 + ii < 1 under the transformation
w = (1 - i)z + 1 + 2i.
[8]
QUESTION 3 [10]
x3-y3+i(x3+y3)
if Z -=I=0
Let f(z) = {
x2+y 2
0 if z = 0
. Then show that f' (0, 0) does not exist.
[10]
QUESTION 4 [20]
= 4.1. Use the Cauchy-Riemann conditions to show that f(z) eY cos x + i eY sin x is nowhere
analytic.
[8]
= 4.2. Determine where f(z) x 3 + 3x y 2 + i(y 3 + 3 x 2 y) is differentiable. Is f analytic? Justify
your answer.
[12]
QUESTION 5 [12]
= Let u(x, y) 3x 2 + 10xy - 3y 2 • Determine whether u(x, y) is harmonic or not. If it is
harmonic find all analytic function f(z) = u(x, y) + iv(x, y).
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QUESTION 6 [30]
Evaluate the following counter integrals.
r(:z) Jr z 6.1.
r = dz where is the counter as shown in the figure below and C1: lzl and
z -z
= r C2 : lz- 11 ¼are interior to oriented counterclockwise.
[10]
y
6.2. J/e 2 + z)dz where C is the boundary of the triangle with vertices at the points 0, 3i
and 4 oriented positively. See the figure below.
[20]
3i
X
END OF SECOND OPPORTUNITY/ SUPPLEMENTARY EXAMINATION QUESTION PAPER
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