Homework 11 04–09–08
Complex Analysis, M381D 59180
0. Optional. (Prelim Spring 2005) Suppose that F(z) and G(z) are entire functions
having no common zeros. Prove that there exist entire functions ϕ(z) and ψ(z) such
that
F(z)ϕ(z) + G(z)ψ(z) = 1
for all z∈C.
1. Assume that fis entire analytic and has nzeros (counting multiplicities) in a disk
|z|< r. Show that for every R > r there exists a complex number zof modulus R,
such that
|f(z)| ≥ |f(0)|R
r−1n
.
2. (∼Ahlfors 5.2.3.2) Prove that
sinπ(z+α)= sin(πα)eπz cot(π α)
∞
Y
n=−∞ 1 + z
n+αe−z/(n+α)
whenever αis not an integer. Hint: Denote the factor in front of the canonical product
by g(z) and determine g0(z)/g(z).
3. (Ahlfors 5.2.4.1) The characterization of the genus given in the first paragraph of
Sec. 3.2 is not literally the same as the definition in Sec. 2.3. Supply the reasoning
necessary to see that the conditions are equivalent.
4. (Ahlfors 5.2.3.3) What is the genus of cos √z?
5. (Ahlfors 5.2.3.4) If f(z) is of genus h, how large and how small can the genus of f(z2)
be?
6. (Conway VII.5.13) Find a non-constant entire function f, such that f(m+in) = 0 for
all possible integers m, n. Find the most elementary solution possible.
7. (Prelim Jan 2001) Suppose that f(z) is an entire function such that |f(z)| ≤ BeA|z|,
z∈C, for some positive constants Aand B. Let ω1, ω2, . . . be the zeros of flisted
with appropriate multiplicity. Prove that
∞
X
n=1
(1 + |ωn|)−α<∞
for all α > 1.
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