PhD/EdD Comprehensive Exam Complex Analysis January 1997

Do any six of the following problems. Be sure to give complete proofs and explanations. A domain is always a non-empty open subset of ℂ (or the Riemann sphere S^{2}) while a region is a non-empty, open and connected.

1. (a) Show that the real part of an analytic function f(z) on a domain Ω is harmonic.

(b) Is u(x, y) = log(x^{2} + y^{2}) harmonic in Ω = ℂ^{×}? Is it the real part of an analytic function on ℂ^{×}? (Here ℂ^{×} denotes the complex plane with the origin removed.)

2. Let ℌ = {z = x + iy|y > 0} be the upper half plane in ℂ. Show that every one-to-one and onto analytic map f : ℌ → ℌ is a linear fractional transformation with real coefficients.

3. Let S^{2} denote the Riemann sphere (or extended complex plane). Prove that the only functions that are meromorphic on all of S^{2} are the rational functions.

4. Let Ω be a region in ℂ and f an analytic function on Ω. Show that the following two statements are equivalent.

(a) f(z) is a polynomial.

(b) There is a point a ∈ Ω such that all derivatives f^{(n)}(a) = 0 for all but finitely many n.

5. (Euler, 1743) Evaluate ∫_{0}^{∞} \frac{dz}{1 + x^{n}} for n ≥ 2.

6. Show that the family of functions F = {f ∈ H(Ω)| Re(f) > 0} on a region Ω is a normal family. (Here H(Ω) is the set of all analytic functions on Ω.)

7. Let {a_{n}} be a discrete set of points in ℂ satisfying

∑_{n} \frac{1}{|a_{n}|^{h+1}} < ∞

for some integer h ≥ 0. Show that there exists an entire function of finite order having zeros precicely at the a_{n}. What is the general form of such functions?

8. Let S be the open sector defined by S = \{z ∈ ℂ|0 < |z| < 1 and 0 < arg(z) < \frac{π}{2} \}. Is this set conformally equivalent to the unit disc Δ = {z||z| < 1}? If so, explicitly determine a conformal equivalence and determine the image of the intersection of S with the ray R = \{z|arg(z) = \frac{π}{4} \}. If not, prove why not.

9. (a) Show that any continuous real valued function u(x, y) which satisfies the Mean Value Property in a region Ω is harmonic in Ω.

(b) Use (a) to prove the Schwartz Reflection Principle for harmonic functions is the following simple form: If u(x, y) is a continuous real valued function on the half disc {z||z| < 1 and Im(z) ≥ 0} such that u is harmonic on {z||z| < 1 and Im(z) > 0} and u(x, y) ≡ 0 on {z||z| < 1 and Im(z) = 0} then u(x, y) has a harmonic extension to all of Δ = {z||z| < 1}.