Complex Analysis Prelim January, 2005

Throughout let D = {z : |z| < 1} and C = complex plane.

1. Let u(x, y) = \frac{y}{x^{2} + y^{2}} . Show that u is harmonic in the punctured plane C\\{0\} in 2 ways: A. From the definition of harmonic. B. By finding a function f, analytic in C\{0} with u = Ref.

2. Construct a conformal map of the region D\\{z : |z + \frac{1}{2}| ≤ \frac{1}{2}\} onto the region C\{z : Re z ≤ 0}.

3. Let α be a real number and consider the integral ∫_{−∞}^{∞} \frac{dx}{x^{2} − 2x + α}. Determine for what α the integral converges and, in those cases, determine its value. Include the details of your contour argument.

4. Let f(z) = cos(i z^{3}). Determine Z(f) = {z : f(z) = 0}. Indicate with a picture where the solutions lie in C.

5. Let p(z) = 3z^{15} + 4z^{8} + 6z^{5} + 19z^{4} + 3z + 1. Show that p(z) has 4 zeros for |z| < 1 and 11 zeros for 1 < |z| < 2.

6. Let f : D → D be analytic and satisfy f(\frac{1}{2}) = \frac{1}{2} and f′(\frac{1}{2}) = −1. Find an explicit formula for f.

7. Let f and g be analytic in a nonempty connected open set U and satisfy |f| = |g| there. What else can you deduce about the relationship between f and g. Justify your answer.

8. Let f be analytic in D and satisfy |f(z)| ≤ \frac{1}{1 − |z|} there. Show that |f′(0)| ≤ 4.

9. Let \{b_{n}\} be a sequence of complex numbers such that lim sup |b_{n}|^{\frac{1}{n}} = 1. Let F be the family of function f(z) = ∑_{n=0}^{∞} a_{n}z^{n} which are analytic in D and satisfy |a_{n}| ≤ |b_{n}|, n = 0, 1, 2, … Prove that F is a compact family in the topology of uniform convergence on compact sets in D.

10. A. State carefully the Riemann Mapping Theorem.

B. Let f be a conformal map from D onto D satisfying f(0) = 0 and |f′(0)| = 1. Using only the Riemann Mapping Theorem show that f(e^{iθ}z) = e^{iθ}f(z) for every real number θ.

C. Deduce that there is a real number θ_{0} such that f(z) = e^{iθ_{0}}z.