2005.pdf - UNIVERSITY COLLEGE LONDON t University of London EXAMINATION FOR INTERNAL STUDENTS For The Following Qualifications B.Sc M.Sci Mathematics

# 2005.pdf - UNIVERSITY COLLEGE LONDON t University of London...

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t UNIVERSITY COLLEGE LONDON University of London EXAMINATION FOR INTERNAL STUDENTS For The Following Qualifications:- B.Sc. M.Sci. Mathematics M211: Analysis 3: Complex Analysis COURSE CODE : MATHM211 UNIT VALUE : 0.50 DATE : 03-MAY-05 TIME : 14.30 TIME ALLOWED : 2 Hours 05-C0961-3-200 © 2005 University College London TURN OVER
,I 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. Let f be a differentiable function on the open disc D(0, R) of centre 0 and radius R. Show (i) If f'(z) = 0 for all z E D(0, R), then f is constant on D(0, R). (ii) If If] is constant on D(0, R), then f is constant on D(0, R). (iii) If If(z)I <~ If(0)] for all z • D(O,R), then f is constant on D(O,R). 2. Define exp z COS Z z, ~ c~ (__ l)nz2,~+ I • ~=0 n!' sinz ~=0 (2n + 1)! = ,~=0 (2n!) ' z E C. Show (i) expz¢0, VzEC (ii) exp(a + b) -- expaexpb, (iii) Va, bE C l ~ iz e-iZ) cosz = ~te + 1 . iz e-iZ) sinz -- ~(e - (iv) cos(a + b) = cos a cos b - sin a sin b, Va, b E C. (v) If sin z = 0, then z = k~r, where k is an integer. MATHM211 PLEASE TURN OVER

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