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ACA801S - ADVANCED COMPLEX ANALYSIS - 2ND OPP - JULY 2022 |
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NAMIBIA UNIVERSITY
OF SCIENCE AND TECHNOLOGY
FACULTY OF HEALTH, APPLIED SCIENCES AND NATURAL RESOURCES
DEPARTMENT OF MATHEMATICS AND STATISTICS
QUALIFICATION:
Bachelor of Science Honours in Applied Mathematics
QUALIFICATION CODE: 08BSMH
LEVEL: 8
COURSE CODE: ACA801S
COURSE NAME: ADVANCED COMPLEX ANALYSIS
SESSION:
DURATION:
JULY 2022
3 HOURS
PAPER: THEORY
MARKS: 100
SUPPLEMENTARY/SECOND OPPORTUNITY EXAMINATION QUESTION PAPER
EXAMINER
Dr S.N. NEOSS] NGUETCHUE
MODERATOR:
Prof F. MASSAMBA
INSTRUCTIONS
1. Answer ALL the questions in the booklet provided.
2. Show clearly all the steps used in the calculations
3. All written work must be done in blue or black ink and sketches must be done in
pencil.
PERMISSIBLE MATERIALS
1. Non-programmable calculator without a cover.
THIS QUESTION PAPER CONSISTS OF 3 PAGES (Including this front page)
Attachments
None
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Problem 1 [15 marks]
1.1 Define the Cauchy Principal Value and hence
[2]
Evaluate the following:
Lad PY: [ aoe“sina
6
1.1.2 Pv. [a— (x?—+-4)8
a
Problem 2 [30 marks]
2.1 Determine the order of the pole of each of the following functions at the indicated point:
2.1.1 f(z) = zintg2 ?= 0;
66]
2.1.2 f(x) = e 2 z =—j— 1 at 2% = 0;
[6]
2.2 Show that the functions given by f(x) = sin z at z = 0 and g(x) = e F+T_] at z = 1
possess a removable singularity at the indicated point.
[9]
2.3
For
the
given
functions
f(z)
=
(z* 9 —
1)
1
zZ-
i
and
g(x)
=ezo -e%
possess:
(i) Removable singularity;
(ii) Pole(s), or
(iii) Essential singularity.
If it is a pole, then determine the order of the pole.
determine whether they
[9]
Problem 3 [25 marks]
co
Let S> a;,(z—c)* be a convergent power series and ¢ > 0 such that B.(c) C D(c, R), where D(c, R)
is thek=0disk of convergence of the power series.
Let f: B.(c) + C be defined by
= S> ax(z —c)*.
k=0
3.1 Prove that f is n-times differentiable for all n € N and that
f(z) = YM —1)-+-(k—n+1)a—yc(e)z
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for alln € N and all x € B,(c). With respect to differentiability what kind of function is f? [15]
3.2 Show that
f(c)
n!
=a,,
What does this mean for the power series?
for alln € No.
3.2 What is the Taylor series of f at c?
Problem 4 [30 marks]
4.1 State the Laurent series Theorem for a function of complex variable.
[4]
4.2 Find the Laurent
series of f(z) =
i
1
=2
for 1 < |z|.
[7]
4.3 Let f: C \\ {0} — C be defined by
fet.
4.3.1 Find the Laurent series of f about zp = 0.
4.3.2 What kind of singularity is z3 = 0? How does f behave in the vicinity of z9 = 0?
4.3.3 State the residue Theorem.
4.3.4 Find
[ etd¢
C,(0)