Homework 6 Math 8052, Spring 2009
1. Let GโCbe open and aโG.
(1) Show that
Ia={fโHol(G) : f(a)=0}
is a maximal ideal in Hol(G). Hint: Let 1:GโCdenote the constant
function with value 1. If fโHol(G), then f=f(a)1+ (fโf(a)1). Note
that (fโf(a)1)โIa.
(2) Give an example of an ideal JโHol(G), J6=Ia, all whose elements
vanish at a.
2. Let f:GโCbe meromorphic, not identically 0, let Pbe the set of poles of
fand Zthe set of zeros. Show that there are functions rP:Pโ(0,โ) and
rZ:Zโ(0,โ) such that the sets
[
aโP
B(a, rP(a)),[
aโZ
B(a, rZ(a))
are disjoint and contained in G.
3. Let Gbe a proper open subset of C.
(1) Show that for every zโGthere is wโC\Gsuch that |zโw|= dist(z, C\G).
(2) Assume now that Ghas the property that
there is R > 0 such that {z:|z|> R} โ G.
Let {zn}be a sequence in Gwithout points of accumulation in Gand such
that |zn| โค Rfor all R. For each npick wnโC\Gsuch that |znโwn|=
dist(zn,C\G). Show that limnโโ |znโwn|= 0.
4. Give an example of a proper open subset of Cand a sequence {zn}in Gwithout
points of accumulation in Gsuch that with whatever choice of wnโC\G,|znโwn|
does not tend to 0.
5. Let D={zโC:|z|<1}. Show that there is fโHol(D) with the following
property. For each z0โโD and each r > 0 there is no holomorphic function on
B(z0, r) which coincides with fon DโฉB(z0, r).
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