PhD/EdD Comprehensive Exam Complex Analysis August 1995

NOTATION: Ω will denote an open subset of ℂ, the complex plane. H(Ω)= {holomorphic functions in Ω}, and D(z, r) = {w ∈ ℂ : |z − w| < r}, the open disc with center z and radius r.

Six complete solutions will constitute a pass.

1. Suppose f is entire (that is, f ∈ H(ℂ)) and |f(z) − f(w)| ≤ |z − w| for all z, w, ∈ ℂ. Show that f is a polynomial.

2. Suppose f_{n} ∈ H(D(0, 1)) for n = 1, 2, .... Suppose that f_{n} → f uniformly in D(0, 1) as n → ∞, and f has exactly one zero in D(0, 1/2). Show that there exists an N such that f_{n} has exactly one zero in D(0, 1/2) for all n > N.

3. True or False (with brief explanation): There exists a sequence of (holomorphic) polynomials P_{n} such that P_{n}(0) = 1 for all n, although P_{n}(z) → 0 for all z ∈ ℂ with z ≠ 0.

4. Suppose we are given real numbers a, b, c and d, and we define T : ℂ → ℂ by T (x + iy) = ax+by+i(cx+dy)(x, y real). For what values of a, b, c and d will T be holomorphic? (Explain briefly.)

5. Let Ω = D(0, 1) \ {0}. True or False: Any function f ∈ H(Ω) is the derivative of some other function g ∈ H(Ω). Explain.

6. Let Ω = {x + iy : x, y ∈ R, 0 < y < 1 + x^{2}}. Suppose that z, w ∈ Ω. Show that there exists a function f ∈ H(Ω) such that f is 1-1 in Ω, f(Ω) = Ω and f(z) = w. (In other words, show that Ω has a holomorphic automorphism mapping z to w.)

7. Suppose that Ω is connected and open, f, g ∈ H(Ω) are holomorphic in Ω and the product fg vanishes identically in Ω. Show that at least one of f and g must also vanish in all of Ω.

8. (a) Define “simply connected.”

(b) Recall the following definition: A set Ω ⊂ ℂ is convex if the line segment joining any two points of Ω lies in Ω. Now suppose that Ω is a convex open subset of ℂ and show that Ω is simply connected.

9. Suppose that {u_{n}}is a sequence of non-negative functions harmonic in D(0, 2). Suppose that u_{n}(0) → ∞ and show that u_{n}(1/2) → ∞.

(Note: You’re not allowed to use “Harnack’s Theorem” (or “Harnack’s Inequality”) in this one—that would amount to assuming what you are supposed to prove.)