Complex Variables I & II August 1992

Show all work and justify your answers.

1. Let z_{0} be a point in the extended complex plane ℂ^{∗}.

a. Define the three types of isolated singularities of a function f analytic in {z : r > |z − z_{0}| > 0} for some r > 0.

b. Suppose that r(z) is a rational function, i.e., r(z) = \frac{p(z)}{q(z)} where p(z) and

q(z) are polynomials. Is it possible for r(z) to have an essential singularity at z_{0} = ∞. Why or why not?

c. Assume that ∞ is not an essential singularity of the rational function r(z).

How can one determine the type of singularity at ∞? d. Sketch the proof that if f(z) is analytic in a disk {z : |z − z_{0}| < r} for some

r > 0, then the Taylor’s series for f about z_{0} converges at least in this same disk.

e. Suppose that the rational function r(z) is defined at z_{0}. Explain how one could find the exact radius of convergence of the power series for r(z) with expansion point z_{0} without determining the coefficients of the power series.

2. Suppose that f(z) is non-constant and analytic in some region R and that z_{0} ∈ R. a. Define the winding number of a closed rectifiable curve γ in R about z_{0}.

b. Show that lim_{r→0^{+}}(2πi)^{−1} ∫_{γ_{r}} \frac{f′(z)}{f(z)}dz exists where γ_{r} is circle of radius r with

center at z_{0} traversed once in the counterclockwise direction.

c. For w ∈ R define g(w) = lim_{r→0^{+}}(2πi)^{−1} ∫_{γ_{r}} \frac{f′(z)}{f(z)}dz. What condition on f is

necessary for g to be analytic in R?

3. Suppose that f is analytic in the open unit disk D and that |f(z)| ≤ |z| for all z ∈ D. Prove that if f(0) = f′(0) = 0 and f(1/2) = i/4 then f(z) = iz^{2}.

4. Suppose that G is a region in ℂ and that F is a family of complex valued functions on G. a. Define “F is a normal family”. b. Prove that if F is a normal family of analytic functions on G, then

F′ = {f′ : f ∈ F} is also a normal family.

5. Let (z_{n}) be a sequence in ℂ.

a. Define ∏ z_{n} converges. b. Suppose that z_{n} = 2^{n}. Show that there is an entire function g with a zero of

order one at z_{n} and no other zeroes. Explicitly define such a function.

Typeset by AMS-TEX