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
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,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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