2006.pdf - UNIVERSITY COLLEGE LONDON University of London EXAMINATION FOR INTERNAL STUDENTS For The Following Qualifications B.Sc M.Sci Mathematics M211

# 2006.pdf - UNIVERSITY COLLEGE LONDON University of London...

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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 08-MAY-06 TIME : 14.30 TIME ALLOWED : 2 Hours 06-C1043-3-190 © 2006 Universi~ 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. . a) Find all complex numbers z such that e z = -1. b) Show that the Cauchy-Riemann equations are satisfied for f(z) = ! on C \ {0}. z c) Is there a holomorphic function F • C \ {0} ~ C such that F'(z) = !7 Justify Z " your answer. d) Is there a holomorphic function F" C \ {0} --, C such that F'(z) = cos~? Justify ~-7~- • your answer. . a) Define the terms path and contour. b) Define the path integral f~ f(z)dz. c) Compute f~ ½ dz for 7(t) = e~t,O < t < 7r. d) Prove: If F is holomorphic on the open set G, F'(z) = f(z) and 7" [a, b] --~ G is a path then f~ f(z)dz = F(7(b)) - F(7(a)).

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