Preliminary Examination in Complex Analysis
June 6, 1994
1. Show that every function that is meromorphic on the extended complex plane is
rational.
2. Show that if u is a real-valued harmonic function in a domain Ω ⊂ C such that u2 is
harmonic in Ω, then u is constant.
3. By using complex integration verify the formula
∫ 2π
0
1
a + sin θ
dθ =
2π
√
a2 − 1
where
a > 1.
4. Let f(z) = ez + z and for 0 ≤ θ < 2π let R(θ) = {z : z = reiθ, r ≥ 0}. Show that
lim
z→∞
z∈R(θ)
|f(z)| = ∞ for all θ. Does this imply that lim
z→∞
1
f(z)
= 0? Explain.
5. Suppose that f is an analytic function in H = {z : Im z > 0} and Im f(z) > 0 for
z ∈ H. Show that |f ′(z)| ≤ Im f(z)
Im z
for z ∈ H, and determine when equality holds in
this inequality.
6. Let ∆ denote the open unit disk in C and let A = {z : 3
4
< |z| < 1}. Show that the
function f(z) =
1
z − 12
cannot be approximated uniformly on compact subsets of A
by functions analytic in ∆.
7. For |z| < 1 let f(z) =
1
1− z
exp
[
−
1
1− z
]
, and for 0 ≤ θ < 2π let `θ = {z : z =
reiθ, 0 ≤ r < 1}. Show that f is bounded on each set `θ. Is f bounded in ∆? Explain.
8. Find all conformal automorphisms of the annulus {z : 1 < |z| < 2}.