Preliminary Exam in Complex Analysis
August 2012
Instructions
All assertions require written justification. In particular, state and verify the hypotheses of
any theorems you use. In complex analysis the terms ’analytic’ and ’holomorphic’ are used
interchangeably.
In a multipart problem, if you can’t do an earlier part of the problem you may nevertheless
assume it when attempting a later part.
1. Suppose the function fis analytic in a simply connected region Ω. Show that there exists
an analytic function Fin Ω so that F0(z) = f(z) for all z∈Ω.
2. Let Pand Qbe polynomials with deg Q≥deg P+ 2 (here deg stands for degree). Show
that there is a number r > 0 so that if γis a closed curve outside the disc of radius r
centered at the origin, then Zγ
P(z)
Q(z)dz = 0. (If you don’t see how to proceed, you might
first want to do this assuming that γis a circle.)
3. Prove that if fand gare entire functions and (f◦g)(z) is a polynomial, then both fand
gare polynomials.
4. Prove Vitali’s Theorem:
Suppose the sequence of analytic functions {fk}is locally uniformly bounded in a region Ω,
there is a sequence {an}in Ω so that lim
n→∞
an=a∈Ω and for each nthe limits lim
k→∞
fk(an)
exist .
(a) Show that the sequence {fk}converges pointwise everywhere in Ω.
(b) Show that the sequence {fk}converges uniformly on compact subsets in Ω.
5. Let ∆ denote the unit disc centered at the origin and ∆ its closure. A function is analytic
on a closed set if it is analytic in a domain containing the set.
(a) Prove that an analytic function f: ∆ →∆ has exactly one fixed point; that is,
exactly one point z0∈∆ so that f(z0) = z0.
(b) Suppose z0∈∆ is the fixed point of ffrom part (a). Given z∈∆ define the sequence
z1=f(z), z2=f(z1), . . . , f (zn) = zn+1 . . . Prove that for any z∈∆,lim
n→∞
zn=z0.
(If you are stumped, try the case where z0= 0.)
(c) Suppose f: ∆ →∆ is analytic and fis not the identity function f(z) = z. Is it
possible for fto have no fixed points or more than one fixed point?