Math 535 Homework #2
Winter 2008
1. Prove directly from the definition (don’t just quote a theorem or proposition... it’s practice with
making estimates):
∞
Y
n=3
n2+π
n2−1
converges.
2. Find an explicit analytic function f6≡ 0 defined on the unit disk with zeros {zn}such that every
point of the unit circle is a cluster point of {zn}and such that
X(1 − |zn|)<∞.
3. Find an explicit entire function gwith g(nlog n) = nπ, n = 1,2,3,.... Prove that your function
works.
4. Find an entire function of least possible genus, with simple zeros at the Gaussian integers:
{m+in :m, n integers },
and no other zeros. (prove your result).
5. If Ω is a region, then H(Ω) is the algebra of analytic functions on Ω. One of the first questions
that you might ask about an algebra are: what are its ideals? Suppose g1, g2∈H(Ω) with no
common zeros. Prove that there are functions f1, f2∈H(Ω) with
(∗)f1(z)g1(z) + f2(z)g2(z) = 1
for all z∈Ω. If g1and g2had a common zero then we could not find f1and f2satisfying (∗). This
says that the ideal generated by g1and g2consists of all analytic functions on Ω if and only if g1
and g2have no common zeros.
Challenge Problem: Prove that if g1, g2,...,gn∈H(Ω) have no common zero then 1 = Pfjgj
for some f1,...,fn∈H(Ω). (this result can be used to determine the maximal ideals).