2003.pdf - UNIVERSITY COLLEGE LONDON t University of London EXAMINATION FOR INTERNAL STUDENTS For The Following Qualifications B.Sc M.ScL Mathematics

2003.pdf - UNIVERSITY COLLEGE LONDON t University of London...

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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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• Fall '18
• Taylor Series, University College London, UNIVERSITY OF LONDON

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