Math 621 Homework Assignment 5 Spring 2006
Due: Monday, April 10
1. (a) Ahlfors, page 130 Problem 2: Show that a function which is analytic in the
whole plane and has a non-essential singularity at ∞reduces to a polynomial.
(You may use Problem 7 in Homework assignment 4).
(b) Lang, page 171 Problem 10: Show that any function, which is meromorphic
on the extended complex plane, is a rational function.
2. (a) Show that the functions cos(z) and sin(z) have essential singularities at ∞.
(b) Let f(z) = cos 1 + z
1−z,|z|<1. Find the set Zfof zeroes of f. Does Zf
have any accumulation points? Explain. (See Lang, page 21 for the definition
of an accumulation point).
3. Lang, page 171 Problem 11: Define the order Ordpfof a meromorphic function f
at a point pto be Ordpf:= mif pis a zero of fof order m
−mif pis a pole of fof order m
Above, mcould be zero, meaning that fis analytic at pand f(p)6= 0.
Let fbe a meromorphic function on the extended complex plane CP1(so a rational
function by problem 1a).
(a) Prove that X
p∈CP1
Ordpf= 0.
In other words, the number of points in the fiber f−1(0), counted with multi-
plicity, is equal to the number of points in f−1(∞), counted with multiplicity.
(b) Prove that all fibers f−1(λ), λ ∈CP1, of fconsist of the same number of
points, provided they are counted with multiplicity,
4. Ahlfors, page 130 Problem 5: Let z0be an isolated singularity of an analytic
function f. Prove that if Re(f) is bounded from above or below, then z0is a
removable singularity. Ahlfors’ Hint: Apply a linear l.f.t. Note: Personally, I find
it easier to avoid using a l.f.t (which does not seem to help rule-out the case of a
pole). Instead, a short proof can be obtained using both the Casorati-Weirstrass
and the Open Mapping Theorems.
5. Let τ∈Cbe a complex number and assume that Im(τ)6= 0. A function fis said
to be doubly periodic with periods 1and τif
f(z+ 1) = f(z) and f(z+τ) = f(z),for all z∈C.
Show that every entire function, which is doubly periodic with periods 1 and τ, is
necessarily constant. (We will see that there exist non-constant, doubly periodic,
meromorphic functions f:C→CP1).
6. Jan 96 Basic Exam, Problem 5: Find the maximum value of the function g(z) =
|z3−z|on the disk |z| ≤ 2. Justify your answer!
1
