Exercise 1. Exercise 2. Exercise 3.

TECHNICAL UNIVERSITY OF DENMARK
Page 1 of 2 pages
Written test, date 27 May 2004 Course no. 01141
Course: Complex Analysis
Aids allowed: All allowed by DTU.
Weighting: Exercise 1 approx. 15%, each of the exercises 2, 3 and 4 approx. 20%, exercise 5
approx. 25%. Your answers will be judged as a whole.
All answers must be argued.
The answers may be given in English or in Danish.
Exercise 1.
Let f.z/ D u.x; y/ C iv.x; y/ be an analytic function in a domain D. Assume there exist
real constants A; B; C, not all equal to zero, such that
Au.x; y/ C Bv.x; y/ D C in D:
Show that f.z/ is constant in D.
Exercise 2.
Consider the function
f.z/ D
:
2z
z 2 1
(1) Determine the constants a and b such that
f.z/ D
a
z 1 C
:
b
z C 1
(2) Determine the Laurent series of f in the largest annulus of the form
0 0 let 0 D C CC denote the simple closed positively oriented curve
consisting of , the linesegment T ; C U on the real axis, and CC , the semi-circle
in the upper half plane with center at 0 and radius .
PLEASE TURN OVER!

–2–
(1) Determine for> 2 the value of
Z
0
(2) Show that Z
Z C1
(3) State the values of the improper integrals
e
p.v.
1 ix
C C
f.z/ dz:
f.z/ dz ! 0 as ! C1:
x 2 C 4 dx and p.v. Z C1
cos x
1 : x 2 C 4 dx
Exercise 4.
Consider the function
h.z/ D f.z/ g.z/
where f.z/ D e z 1 and g.z/ D cos z:
(1) Determine the zeros of f and g and their order. Determine the isolated singularities
of h and state their type.
(2) State the radius of convergence of the Maclaurin series of h, and find the coefficients
a 0 ; a 1 ; a 2 ; a 3 of the series.
Exercise 5.
Consider the Möbius transformation
f.z/ D 1 z C 1 2 i 2
:
Let C 1 denote the circle jz ij D 1 and C 2 the circle jz 1j D 1 . Let D 1 denote the
open disc bounded by C 1 and D 2 the open disc bounded by C 2 . Set A D D 1 [ D 2 .
(1) Sketch the circles C 1 and C 2 . Show that
f.C 1 / D R [ f1g and f.C 2 / D iR [ f1g:
(2) Determine the images f.D 1 / and f.D 2 / of the discs and the image f.A/ of their
union.
(3) Let
L 0 V C n f0g ! C
denote the branch of the logarithm defined as
L 0 .z/ D ln jzj C i arg 0 .z/
where 0