COMPLEX ANALYSIS PRELIM. (AUGUST 25, 2005)

1. Find an analytic function f(z) whose real part is

Re(f(z)) = xy + 3, (z = x + iy).

Does such a function exist? Justify your answer.

2. Let u be a real-valued harmonic function. For what functions f is the function f(u) harmonic?

3. Let f be analytic in a domain Ω and Re(f) be a constant on Ω. Show that f is a constant.

4. Construct a conformal mapping of ℂ \ ([−1, 0] ∪ [−i, i]) onto the init disk.

5. State and prove Morera’s Theorem.

6. Compute the integral

∫_{|z|=1/2} \frac{dz}{(2z − z̄)^{8}} .

7. How many roots of the equation z^{4} − 6z + 3 = 0 have their modulus between 1 and2?

8. Let f be an entire function whose modulus is constant on some circle. Prove that f(z) = C(z − z_{0})^{n}.

9. By Picard’s Theorem every meromorphic function has at most 2 exceptional values (that is there are at most two complex numbers which are not in the range). How many exceptional values does tan z have? Find them.

10. Prove that there exists a constant C such that for every polynomial P |∫_{−1/2}^{1/2} P (x)dx| ≤ C ∫_{|z|=1} |P (z)||dz|.