Math 534 Homework #7
Autumn 2008
1. Suppose fis a rational function such that if zis not a pole of fand if Rez > 0, then Ref > 0,
and if zis not a pole of fand if Rez= 0, then Ref= 0.Prove that
f(z) = a0z+ib0+
n
X
k=1
ak
z−ibk
,
where ak≥0 and bk∈R. (Hint: partial fractions)
2. Suppose fsatisfies the hypotheses in problem 1. Prove (continued fraction expansion):
f(z) = c0z+id0+1
c1z+id1+1
c2z+id2+1
c3z+id3+...
,
where ck≥0 and dk∈Rand the process terminates after finitely many steps. (Hint: use #1). This
problem says that the above class of functions is generated by z,iand the operations of multiplying
by a positive number, addition, and taking reciprocal. It is an open problem to do the same for
a rational function of two complex variables z,won the region {(z, w) : Rez > 0and Rew > 0},
which has zero real part when Rez =Rew = 0.
Construct a one-to-one analytic map fof the following regions Ω onto D. If z0is given, find
the map with f(z0) = 0 and f0(z0)>0. You may leave your answer as an explicit sequence of
maps, but you must show that each map does what you claim it does.
3. Ω = C\(−∞,0] z0= 1
4. Ω = {z: 0 <Imz < 1}
5. Ω = {z: 0 <Imz < 1 and Rez > 0}
6. Ω = D∩ {z: Imz > 1
2}z0=3
4i
7. Ω = {z:|z|<1} ∩ {z:|z−1|<1} ∩{z: Imz > 0}.
8. Ω = C∗\[0,1] z0=∞. Here f0(∞) means limz→∞ z(f(z)−f(∞)).
9. (Rain = complement of the Seattle Umbrella)
Ω = C∗\({z:|z| ≤ 1 and Imz≥0} ∪ {it :−1≤t≤0})
