Ph.D. Preliminary Examination Complex Analysis September 3, 1993

\underline{Notations}.

(a) Δ = {z ∈ C : |z| < 1} is the open unit disk.

(b) \overline{Δ} = \{z ∈ C : |z| ≤ 1\} is the closed unit disk.

(c) \overline{C} = C ∪ \{∞\} is the extended complex plane.

1. Let f : Δ → Δ be analytic. Suppose there exists z_{0} ∈ Δ with f(z_{0}) = z_{0} and f′(z_{0}) = 1. Prove that f(z) ≡ z.

2. Consider the elementary functions:

(1) cos : C ↦ C;

(2) tan : C ↦ \overline{C}. A. Is tan continuous on C? B. Determine

(a) the range of cos z;

(b) the range of tan;

(c) cos^{−1}\{\frac{5}{4}\};

(d) tan^{−1}{∞}.

3. Let L denote the line segment joining −i and i, and let Ω = \overline{C}\L.

A. Map Ω conformally onto Δ. B. Deduce that if f is an entire function such that f(C)∩L = ϕ then f(z) is constant.

4. Let z_{0} be a simple pole of a function f(z).

A. Prove

lim_{r→0^{+}} ∫_{γ_{r}} f(z)dz = (b − a)i Res[f, z_{0}] ,

where

γ_{r} = {z_{0} + re^{it} : a ≤ t ≤ b} .

B. Use this result to evaluate ∫_{−∞}^{∞} \frac{sin x}{x}dx.

5. Suppose f is a nonconstant analytic function in Δ. Show that the function

M(r) = max_{|z|=r} |f(z)|, 0 < r < 1 ,

is strictly increasing.

6. Let f(z) = exp(−\frac{1 + z}{1 − z}).

A. Show that f is bounded on Δ. B. Determine the range of f on Δ.

7. How many zeros has

P (z) = 1 + 2z^{4} + \frac{7}{10}z^{10}

in Δ? What are the multiplicities of these zeros?

8. Suppose that the functions f_{n} are holomorphic in a domain D, none of the functions f_{n} has a zero in D, and {f_{n}} converges to f uniformly on compact subsets of D. Show that either f has no zeros in D or f ≡ 0 in D.