Preliminary Exam Complex Analysis August 1999

1. Find an analytic function f(z) whose real part is (z = x + iy).

Re f(z) = xy − 10 .

Does such a function exist? Justify your answer.

2. Find the general form of an entire function f(z) satisfying

|f(z)| ≤ A + B|z|^{3/2}, where A and B are constants .

3. Find the general form of a function f(z) which is analytic inside the ellipse D (z =

x + iy)

\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1 ,

continuous in \overline{D}, and

Im f(z) = −5 (z ∈ ∂D)

4. Find a conformal mapping from C\{[0, +∞)} to the unit disk.

5. 1. Prove that for any polynomial p and any a ∈ Δ

p(a) = \frac{1}{2π} ∫_{0}^{2π} \frac{p(e^{iθ})}{1 − e^{−iθ}a}dθ

2. Deduce from 5.1 that

|p(a)| ≤ [\frac{1}{(1 − |a|^{2})} \frac{1}{2π} ∫_{0}^{2π} |p(e^{iθ})|^{2}dθ]^{1/2}

6. Let f be analytic in the unit disk and map the unit disk into itself given f(1/2) = 0. Prove that |f′(1/2)| ≤ \frac{4}{3}.

7. Let

f(z) = \frac{1}{z} · \frac{1 − 2z}{z − 2} · . . . · \frac{1 − 10z}{z − 10}

Find ∫_{|z|=100} f(z)dz.

8. Let f(z) ≢ 0 be a meromorphic function in C such that

|f(z)| = 1 (|z| = 1)

and

f (\frac{1}{2}) = 0 .

Can f be an entire function?