Final Exam - Complex Analysis
MATH 440/508 – M.P. Lamoureux
Due Thursday, Dec 17 by noon
Note: Your work will be graded on correctness of the mathematics, and
clarity of presentation. Do not submit your rough work, only the final clean
version. Prove your statements, or justify your calculations, as appropriate.
You are required to do individual work. No consulting with other people,
including other students. This also means no consulting with experts online,
and no Googling for answers. You may, however, make use of textbooks,
including electronic texts online, and our course lecture notes.
C denotes the complex plane, D the open unit disk, and D its closure.
1. Given two complex numbers a, b, show there exists complex numbers c
satisfying |c − a| + |c + a| = 2|b| if and only if |a| ≤ |b|. When there are such
numbers c, determine the largest value possible for |c|. (In terms of a, b.)
2. Use the calculus of residues to compute the following integrals:
Z ∞
Z ∞
1 − cos2 at
log t
dt
dt.
2
t
(1 + t2 )2
0
0
3. a) Suppose P (z) is a polynomial whose roots all lie in some half plane in
C. Show that the roots of the derivative P 0 (z) all lie in the same half plane.
b) Is this true for entire functions? Prove, or give a counterexample.
4. a) Suppose F : D → D is holomorphic. Show that
|F 0 (z)|
1
≤
for all z ∈ D.
2
1 − |F (z)|
1 − |z|2
b) If you have equality for any z, show that F is a linear map.
5. Suppose F : D → C is holomorphic whose derivative is nonzero on D, and
n > 0 is an integer. Prove there exists a holomorphic function G(z) defined
1
in a (small) open disk at the origin, so that
F (z n ) = F (0) + G(z)n
for all z in the small disk.
6.
a) Find the largest open disk centred at the origin, on which
f (z) = z 2 + z is a conformal map.
b) Find the largest open disk centred at the origin, on which
g(z) = exp(z) is a conformal map.
7. Show that the integral
Z
F (z) =
0
z
dω
(1 − ω 4 )1/2
defines a homomorphic, conformal map of the open unit disk onto the interior of a square.
8. Using only methods we learned in the course, find all possible bijective,
conformal holomorphisms f : C → C. Prove that you have found them all.
2