Complex Prelim, September 1998
Do any six problems
1. Suppose f is analytic in |z| < 1 and
f(
1
n2
) =
1
n
for all n > 2. Show that f is identically zero in |z| < 1.
2. Suppose f is entire and |f(z)| ≤ log(1 + |z|) for all z. Show that f is identically zero.
3. Show that the function
u(x, y) = arctan(y/x)
is harmonic in the (open) right half plane. Find its harmonic conjugate there.
4. Suppose {fn} is a sequence of analytic functions in |z| < 1. If
lim
n→∞
∫
|z|<1
|fn(z)| dA(z) = 0,
where dA is area measure on |z| < 1, show that
lim
n→∞
fn(z) = 0
for all |z| < 1.
5. Evaluate the integeral
I =
∫
C
sin ζ
ζ(π − 6ζ)2
dζ,
where C is the positively oriented unit circle.
6. Suppose f is entire and
|f(neiθ)| ≤ exp(n cos θ)
for all n ≥ 1 and θ ∈ [0, 2π]. Show that f(z) = cez for some constant c with |c| ≤ 1.
7. Find a conformal map from the unit disk |z| < 1 onto the region |argz| < π/3.