UNIVERSITY OF OSLO
Faculty of Mathematics and Natural Sciences
Examination in
MAT2300 — Analysis II.
Day of examination: Monday, December 14, 2009.
Examination hours:
14.30 – 17.30.
This problem set consists of 2 pages.
Appendices:
None.
Permitted aids:
None.
Please make sure that your copy of the problem set is
complete before you attempt to answer anything.
Problem 1
a) Find all the solutions of the equation
π
z 3 = ei 4 .
b) Find the Laurent series for
z2
1
− z3
in the domains 0 < |z| < 1 and |z| > 1.
c) Determine what kind of singularity
f (z) =
1
1 − e−z
has at z = 0 and compute Res(f ; 0).
Problem 2
Use the residue theorem to compute the integral
Z∞
−∞
(Continued on page 2.)
dx
.
1 + x4
Examination in MAT2300, Monday, December 14, 2009.
Page 2
Problem 3
a) What is the maximal value of |3z 2 − 1| in the closed disk |z| ≤ 1? For
which values of z does the maximum occur?
b) Show that |z 6 − 6z 4 | > 4 on the circle |z| = 1. Determine how many
zeros the function f (z) = z 6 − 6z 4 + 3z 2 − 1 has in the domain |z| < 1.
(Counted with multiplicities.)
Problem 4
Let n and k be integers with n ≥ k ≥ 0. Recall the binomial numbers
n
n!
=
k!(n − k)!
k
P
and the binomial theorem which says that (1 + z)n = nk=0 nk z k .
a) Prove that
Z
n
1
(1 + z)n
dz
=
k
2πi
z k+1
C
where C is the positively oriented unit circle |z| = 1.
b) Use the statement in a) to prove that
2n
≤ 4n
n
for all positive integers n.
END