Midterm Math 534
Autumn 2006
Do as many problems as you can. I don’t expect anyone will be able to do them all. Complete
problems are worth more than several half completed problems. Leave some time at the end of the
hour to reread your solutions to eliminate errors. If you use a theorem with a name, please be sure
to give the name of the theorem you are using and to indicate that you’ve checked the hypotheses.
You may use any result proved in class or proved by you in the homework.
1. Prove that there is no function analytic in the unit disc such that |f(z)| → ∞ as |z| → 1.
2. Suppose K > 1. Find all solutions to the functional equation f(Kz) = zf (z) which are analytic
in C\ {0}.
3. Suppose fis analytic in the strip S={z:−1<Imz < 1}and suppose that
|f(x+iy)| ≤ 1
(1 + x2),
for x+iy ∈Sand |x|>100. Prove that there is a function F+analytic in the half-plane
S+={z: Imz > −1}and a function F−analytic in the half-plane S−={z: Imz < 1}such that
f=F+−F−. Hint: Recall the proof of the Laurent expansion.
4. Suppose fis a one-to-one analytic function mapping Conto C. Prove that f(z) = az +bfor
some constants aand b. Hint: What kind of singularity can f(1/z) have at 0?
5. Suppose fis analytic on the unit disk Dwith f(D)⊂Dand f(0) = 0. Let fn=f◦f◦ · · · ◦ f
denote the n-fold composition of fwith itself. Prove that either fnconverges to 0 uniformly on
compact subsets of Dor fis a rotation: f(z) = cz where cis a constant with |c|= 1. Hint: Fix
r < 1 and estimate how big |f(z)|can be on |z|=r.
6. Suppose Ω is a simply connected region and supp ose fis analytic on Ω. Fix z0∈Ω. Define
g(z) = Zγz
f(ζ)dζ,
where γzis any curve in Ω from z0to z. Prove that gis well-defined (there are such curves and
the definition does not depend on the choice of the curves γz) and analytic in Ω with g0(z) = f(z).