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CAN702S- COMPLEX ANALYSIS - JAN 2020 |
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NAMIBIA UNIVERSITY
OF SCIENCE AND TECHNOLOGY
FACULTY OF HEALTH AND APPLIED SCIENCES
DEPARTMENT OF MATHEMATICS AND STATISTICS
QUALIFICATION: Bachelor of Science in Applied Mathematics and Statistics
QUALIFICATION CODE: O7BAMS
LEVEL: 7
COURSE CODE: CAN7025
COURSE NAME: COMPLEX ANALYSIS
SESSION: JANUARY 2020
DURATION: 3 HOURS
PAPER: THEORY
MARKS: 100
EXAMINER
SECOND OPPORTUNITY EXAMINATION QUESTION PAPER
PROF. G. HEIMBECK
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 4 PAGES (Including this front page)
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Question 1 [11 marks]
Let U be a one-dimensional subspace of the R-vector space C anda € C.
a) Prove that U is uniquely determined by the coset a+ U.
[4]
b) Show that the representatives of a + U are exactly the elements of a+ U.
[5]
c) Prove that any two cosets of a+ U are disjoint or equal.
[2]
Question 2 [13 marks]
a) What are separated subsets of C? Which subsets of C are connected? State the
definitions.
[5]
b) Let c,d € C. You are reminded that the line segment with endpoints c and d is the
set
(c,d) = {(1-—A)e+ Ad] 0<A< 1}.
Explain why (c,d) is connected. Proofs are not required.
[5]
c) Let D C C. Assume that there exists some a € D such that (a,z) C D for all
z € D. Show that D is connected.
[3]
Question 3 [12 marks]
a) What is an argument of a non-zero complex number? State the definition.
[2]
b) Let z € C% and let ~y := Arggz be the principal arggument of z.
i) Show that Rez = |z| cosy.
[2]
ii) When is
yp =_ arccos TRaez
true? State your reasons.
[8]
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Question 4 [13 marks]
Let X CC, f:X — Ca function andaeé X.
a) Prove that f is continuous at a if and only if for each ¢ > 0, there exists some 6
such that f(Ns(a)N X) C N-(f(a)).
[3]
b) Let (zn) € N be a sequence in X which converges to a. If f is continuous at a, prove
that (f(zn))n is convergent and lim f(zn) = f(a).
[4]
c) If f is not cintinuous at a, prove that there exists a sequence (wn)n in X which
coverges to a but (f(w,))n does not converge to f(a).
[6]
Question 5 [18 marks]
a) When does the path integral f f(¢) d¢ exist? Explain!
[5]
¥
b) Let y be a continuously differentiable path. If f f(¢) d¢ exists, show that f f(¢) d¢
=y
exists and
[row=| r-oa.
o
c) Let a and £ be two paths in C. When does a + f exist? State the definition and
show that a + @ is a path.
[8]
Question 6 [17 marks]
a) State and prove the addition theorem of the exponential function.
[5]
b) Let exp: C — C be defined by exp(z) := e?. Show that exp is a homomorphism
from the additive group C+ onto the multiplicative group C% of the field C. (6]
c) What is a period of the function exp? Show that the periods of exp form the
subgroup (277) of Ct.
[6]
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Question 7 [16 marks]
a) In comples analysis, what is an analytic function? State the definition.
[4]
b) Show that every analytic function is a holomorphic function.
[5]
c) Prove that every holomorphic function is an analytic function.
[7]
End of the question paper