PhD/EdD Comprehensive Exam Complex Analysis May 1995

NOTATION: Ω always denotes a connected open subset of ℂ (perhaps not simply connected) and H(Ω) is the class of all functions holomorphic (or “analytic”) in Ω.

1. (a) Define “isolated singularity.”

(b) Define “removable singularity,” “pole,” “essential singularity.”

2. Suppose that f is an entire function (that is, f is holomorphic in ℂ) and

∫_{0}^{2π} |f(re^{it})|dt ≤ 1

for all r > 0. Show that f is constant.

3. Give an example of a non-vanishing holomorphic function with no logarithm. (In other words, give an example of a connected open set Ω and a function f ∈ H(Ω) such that f has no zero in Ω but there does not exist g ∈ H(Ω) with f = e^{g}). Explain briefly.

4. Suppose f∈ H(Ω). Under what conditions does it follow that there exists F ∈ H(Ω) with f = F′? (We know that this happens for all f∈ H(Ω) if Ω is simply connected. Now what if Ω is not simply connected? Whether Ω is simply connected or not, we know that there exists F ∈ H(Ω) with f = F′ if and only if f and Ω satisfy exactly what condition? Explain briefly.)

5. (a) State Cauchy’s theorem for simply-connected domains.

(b) State the general form of Cauchy’s theorem.

6. (a) Define “normal family.”

(b) Suppose that f_{n}∈ H(Ω) for n = 1, 2, ..., |f_{n}(z)| ≤ 1 for all z ∈ Ω and the sequence (f_{n}(z)) converges to f(z) for all z ∈ Ω. Show that f∈ H(Ω). (NOTE: The result is false without the hypothesis |f_{n}(z)| ≤ 1.)

7. Let S be the open unit square, S = {x+iy : 0 < x < 1, 0 < y < 1}. Let D be the unit disc and suppose that f : S → D is holomorphic and bijective (the Riemann mapping theorem shows that such a function exists.) Explain briefly why f cannot have an analytic continuation to a neighborhood of the closure of S.