Preliminary Exam in Complex Analysis
January 2013
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.
1. Suppose the function fis analytic in the unit disc ∆, f(0) = 0 and |f(z)|<1 for all
z∈∆.Show that the series
∞
X
n=1
f(zn) converges to an analytic function gin ∆.
2. Let fnbe a sequence of entire functions converging uniformly on compact subsets of C
to a function f. Furthermore, suppose that for each nthe zeros of fnlie on the real axis.
Show that fis identically 0 or fhas zeros only on the real axis.
3. Let gbe a meromorphic function on the Riemann sphere ˆ
C.
(a) Prove that gis a rational function.
(b) Suppose ghas a removable singularity at ∞. Show that lim
z→∞ g0(z) = 0.
4. Use residues to evaluate the integral Z2π
0
cos θ
3−2 cos θdθ.
5. Suppose fis an entire function and Mis a positive real number. Prove that there is at
most one connected component in the complement of the set {z| |f(z)|< M}.
6. Let Pdenote the set {z=reiθ |0< r < 1,−π/2< θ < π}. Explicitly describe a 1-1
conformal map of Ponto the unit disc ∆.