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CAN702S - COMPLEX ANALYSIS - 2ND OPP - JANUARY 2024 |
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n Am I BI A u nl VE Rs ITY
OF SCIEnCEAnDTECHnOLOGY
FacultyofHealthN, atural
ResourceasndApplied
Sciences
Schoool f NaturalandApplied
Sciences
Departmentof Mathematics,
StatisticsandActuarial Science
13JacksonKaujeuaStreet
PrivateBag13388
Windhoek
NAMIBIA
T: •264 612072913
E: msas@nust.na
W: www.nust.na
QUALIFICATION: BACHELOR OF SCIENCE IN APPLIED MATHEMATICS AND STATISTICS
QUALIFICATIONCODE: 07BSAM; 07BSOC
LEVEL:7
COURSE:COMPLEX ANALYSIS
COURSECODE: CAN702S
DATE: JANUARY 2024
SESSION: 1
DURATION: 3 HOURS
MARKS: 100
SECONDOPPORTUNITY/SUPPLEMENTAREYXAMINATION: MEMORANDUM
EXAMINER:
DR. NEGACHERE
MODERATOR:
PROF.FORTUNEMASSAMBA
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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:~i- 1. (a) Find the real and imiginary part of
(6)
(b) Compute lim =2 - 2i=+~-i if it exists.
(4)
::-tl+i
:-2+t
2. Let z1 = -1 - i, z2 = 1 - i y'3. Then find the polar representation of~·
(10)
3. Find the image of the disk I z + l I< 2 under the transformation
w = (l + 2i) z + 2 - i.
(5)
4.
Show that
lim
2
f'i'I.:
docs not
exist,
(z
=
x
+ iy).
(8)
=-+0 1
5. Let f (z) = J(x +iy) = 3x2 - 2xy + x - 3y2 + 2y + i (-x 2 - 6xy - 2x + y2 + y). Determine
if f is analytic in C or it is not analytic in C.
(12)
6. Show that u(x, y) = y3 - 4xy - 3x2y is harmonic and find its harmonic conjugate v(x, y)
for which J(z) = u(x, y) + i v(x, y) is analytic.
(15)
7. Evaluate Jc(xy- i y2)dz where (z = x + i y) and C is the counter joining Oto 1 +i, 1+i
toiandito-1-i.
(20)
8. Evaluate the follwoing integrals.
(a) fc( 4:: 2 ) dz where C is the circle I z + l I= 2 oriented positively.
(12)
Jc z (b) =~(:~i) where C is the circle I I= ½oriented positively.
(8)
END OF SECOND OPPORTUNITY/SUPPLEMENTARY
QUESTION PAPER
EXAMINATION