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ACA801S - ADVANCED COMPLEX ANALYSIS - 1ST OPP - JUNE 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: O08BSMH
LEVEL: 8
COURSE CODE: ACA801S
COURSE NAME: ADVANCED COMPLEX ANALYSIS
SESSION:
DURATION:
JUNE 2022
3 HOURS
PAPER: THEORY
MARKS: 100
EXAMINER
FIRST OPPORTUNITY EXAMINATION QUESTION PAPER
Dr S.N. NEOSSI 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]
Let O Cc C be an open set and let f: O + C be a holomorphic function.
1.1 What is an isolate singularity of f?
[3]
1.2 When is c € O a removable singularity and how does one remove such a singularity?
[4]
1.3 What is an essential singularity of f?
[3]
1.4 What is a pole of f and what is the order of a pole?
[5]
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(x) = zsi— nz at 2 =0;
[6]
2.1.2 f(z) = e 2 ——j— 1 at 2 =0;
2.2 Show that the functions given by f(x) = —sin— z at z = 0 and g(x) = e= 1-1]i
Zz
_—
possess a removable singularity at the indicated point.
[6]
at z = 1
[9]
2?
2.3 For the given functions f(z) = (z? — 1) zZ- i and g(x) = zi Z-
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]
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
fiz) = S > ax(z —c)*.
k=0
1
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3.1 Prove that f is n-times differentiable for all n € N and that
f(z) = So k(k-1)-+-(k- nt Lax(z - 0)
k=n
for all n € N and all « € B,(c). With respect to differentiability what kind of function is f? [15]
3.2 Show that
a =aQ,, for alln € No.
What does this mean for the power series?
.
[7]
3.2 What is the Taylor series of f at c?
[3]
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 = 5 for 1 < |z|.
[7]
4.3 Let f: C\\ {0} > C be defined by
f(z) =e.
4.3.1 Find the Laurent series of f about zp = 0.
[5]
4.3.2 What kind of singularity is 29 = 0? How does f behave in the vicinity of z) = 0?
[5]
4.3.3 State the residue Theorem.
[4]
4.3.4 Find
(5]