Complex Prelim, September 1998 Do any six problems

1. Suppose f is analytic in |z| < 1 and

f(\frac{1}{n^{2}} ) = \frac{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 {f_{n}} is a sequence of analytic functions in |z| < 1. If

lim_{n→∞} ∫_{|z|<1} |f_{n}(z)| dA(z) = 0,

where dA is area measure on |z| < 1, show that

lim_{n→∞} f_{n}(z) = 0

for all |z| < 1.

5. Evaluate the integeral

I = ∫_{C} \frac{sin ζ}{ζ(π − 6ζ)^{2}} dζ,

where C is the positively oriented unit circle.

6. Suppose f is entire and

|f(ne^{iθ})| ≤ exp(n cos θ)

for all n ≥ 1 and θ ∈ [0, 2π]. Show that f(z) = ce^{z} for some constant c with |c| ≤ 1.

7. Find a conformal map from the unit disk |z| < 1 onto the region |argz| < π/3.