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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