Preliminary Examination
Complex Analysis
January 1998
1. Suppose that f is an entire function and f(C) ∩ {w : Re w = 0} = ∅. Prove that f is
constant.
2. Evaluate
∫ π
0
dθ
2 + cos θ
.
3. Give an example of a function f which is holomorphic in C\{z0} for some z0
6= 0,
has an essential singularity at z0 and is continuous in {z : |z| ≤ |z0|}. Show that the
function given actually has these properties.
4. Find the maximum value of |g(z)| if g(z) =
z
4z2 − 1
and z varies over the region
{z : |z| ≥ 1}.
5. A. State carefully the Riemann Mapping Theorem.
B. Let D = {z : |z| < 1}, Ω = {z : Re z > 0} and fix α ∈ Ω. Find all conformal
maps g from Ω onto D such that g(α) = 0.
6. Suppose that f is a holomorphic function in an open disk D, f is continuous on D
and |f | is constant and nonzero on ∂D. Prove that f is a rational function.
7. Let P be a nonzero polynomial. Suppose that
∫
|z|=r
1
P (z)
dz
6= 0 whenever r > 0 and
the integral is defined. Show that degP = 1.
8. Suppose that f is an entire function, and for r > 0 let Mf (r) = sup{|f(z)| : |z| ≤ r}.
Assume that 0 < α < 1 and let
L(α) = lim
r→∞
Mf (αr)
Mf (r)
.
(a) Determine L(α) in the case f is a polynomial.
(b) Show that L(α) = 0 if f is not a polynomial.