UNIVERSITY COLLEGE LONDON
t
University of London
EXAMINATION FOR INTERNAL STUDENTS
For The Following
Qualifications:-
B.Sc.
M.ScL
Mathematics M211 : Analysis 3: Complex Analysis
COURSE CODE
:
MATHM211
UNIT VALUE
:
0.50
DATE
:
22-MAY-03
TIME
:
14.30
TIME ALLOWED
:
2 Hours
03-C0948-3-150
© 2003 University College London
TURN OVER

All
questions may be attempted but only marks obtained on the best
four
solutions will
count.
The use of an electronic calculator is
not
permitted in this examination.
1.
(i)
(ii)
Let
P(z) = a~z '~ + a~_lZ ~-t + ... + ao
be a non-constant polynomial with
complex coefficients a0, ..., a~, n ) 1. Show that P has at least one root in the
complex plane.
(Any results used must be clearly stated).
If a0, ..., a~ are real and
P(w)
= 0, show that P(~) = 0, where ~ is the complex
conjugate of w.
Is the same result necessarily true if a0, ..., a~ are complex
numbers?
.
(i) Let
S={z:z=x+iy,
xy> l}
T={z:z=x+iy,
xy = 1}.
Show that S is an open subset of C and T is a closed subset of C.
C,
OO
(ii) Let { ~}n=l be a nested sequence of non-empty sets in C i.e. Cn D Cn+l for
n = 1, 2,
....
oo
Decide, with justification, whether it is always true that N C~ ~ ~b •

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