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CAN702S - COMPLEX ANALYSIS - 1ST OPP - NOVEMBER 2023 |
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nAm I BI A UnlVERSITY
OF SCIEnCE Ano TECHnOLOGY
FacultoyfHealthN, atural
ResourceasndApplied
Sciences
Schoolof Natural and Applied
Sciences
Department of Mathematics,
StatisticsandActuarial Science
13JacksonKaujeuaStreet
PrivateBag13388
Windhoek
NAMIBIA
T: •264612072913
E: msas@nust.na
W: www.nust.na
QUALIFICATION : BACHELOR OF SCIENCE IN APPLIED MATHEMATICS AND STATISTICS
QUALIFICATION CODE: 07BSAM; 07BSOC
LEVEL: 7
COURSE:COMPLEX ANALYSIS
COURSECODE: CAN702S
DATE: NOVEMBER 2023
SESSION: 1
DURATION: 3 HOURS
MARKS: 100
FIRSTOPPORTUNITY EXAMINATION: QUESTION PAPER
EXAMINER:
DR. NEGA CHERE
MODERATOR:
PROF. FORTUNE MASSAMBA
INSTRUCTIONS:
1. Answer all questions on the separate answer sheet.
2. Please write neatly and legibly with black or blue ink pen.
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.
PERMISSIBLEMATERIALS:
1. Non-Programmable Calculator
ATTACHMENTS:
NONE
This paper consists of 2 pages including this front page.
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r::;:. 1. (a) Find the real and imaginary part of
(6)
(b) Find the image of the disk I z + 1 I< 2 under the transformation
w = (1 + 2i) z + 2 - i.
(5)
(8)
3.
Let
f(z)
=
{
0.0.:.:.:.iiff~:=
:f
0
0
, where z
=
x
+ ·iy. Then
show that
(a) f(z) is not analytic at (0, 0).
(13)
(b) the Cauchy-Riemann Equations are satisfied at (0, 0).
(13)
4. Show that u(x, y) = y3 - 4xy - 3x 2y is harmonic and find its harmonic conjugate v(x, y)
for which f(z) = ·u(x, y) + ·iv(x, y) is analytic.
(15)
J 5. Evaluate c(Y - x - ix 2)dz where C is the counter joining Oto 1 + i, 1 + i to i and i to
-1 + 2i.
(17)
:a~ 6. Without evaluating the integral show that I fc
5
3
dz
I::; 1r,
where
C is the
semicirle
with center the origin and radius 3, oriented positively.
(8)
7. (a) Evaluate Jrc .::14 d~z where C is the circle I z - 21i I= 1.
(4)
(b) Evaluate fc ,J:9 dz where C = C1 + C2 and C1 = {I z + 3i I= 3},
C2 = {I z - 3i I= 3}.
(11)
END OF FIRST OPPORTUNITY EXAMINATION QUESTION PAPER