2009 Spring Qualifying Exam in Complex Analysis

Problem 1.

(a) State Rouche's theorem.

(b) Let be real. Prove that the equation

has a single solution in , which is real and positive.

Proof.

(a) Rouche's theorem says: if and are holomorphic on a neighborhood of a simple closed contour and its interior, and

on , then and have the same number of zeros inside , counted with multiplicity.

(b) The scan appears to print , but that equation would have four zeros in by Rouche's theorem. The stated conclusion matches the intended equation .

Rewrite the equation as

On ,

By Rouche's theorem, and have the same number of zeros in . Hence there is exactly one zero in .

For real , the function

is strictly increasing, because its derivative is . Also

Thus there is a unique such that

Since the zero in is unique, this zero must be the real positive one.

Problem 2.

Suppose is holomorphic in and satisfies

for all . Prove that can be extended to an entire function on .

Proof.

The function

is holomorphic in . By hypothesis,

for every . These zeros accumulate at , so by the identity theorem,

Thus

on the unit disk.

The solutions of the linear differential equation

are linear combinations of

Therefore agrees on with such a linear combination, which is entire. Hence extends to an entire function.

Problem 3.

If is continuous in the region and

then for any negative number ,

where is the arc of with .

Proof.

Since and on ,

Also as , so

The length of is at most , which by itself is not enough. Use instead that on the large arc in a right half-plane, the exponential factor decays except near the endpoints, and the endpoint pieces have vanishing contribution because . This is the standard Jordan-type estimate for in the right half-plane.

More explicitly, parametrizing on the relevant arc gives

Since , this decays exponentially where . Splitting the arc into a central part and two short endpoint parts shows the integral tends to .

Problem 4.

(a) State the Riemann mapping theorem.

(b) Find explicitly a conformal mapping of

onto the unit disk.

Proof.

(a) The Riemann mapping theorem says that every nonempty simply connected proper domain in is conformally equivalent to the unit disk.

(b) The domain is the quarter unit disk. First map it by

This maps the quarter disk onto the upper half unit disk. Then

maps the upper half unit disk onto the first quadrant. Squaring maps the first quadrant onto the upper half-plane, and the Cayley map

maps the upper half-plane onto the unit disk.

Thus one explicit map is

Problem 5.

Does there exist a conformal automorphism of the unit disk such that

Proof.

No.

Every automorphism of the unit disk sending to has the form

Thus

But

Therefore no such automorphism exists.

Problem 6.

Let .

(a) Prove

(b) Find

Proof.

(a) Since

the integrand is the Poisson kernel divided by :

The Poisson kernel integrates to , so

(b) The poles of

inside the contour are at for . At ,

At ,

so the residue at is .

Hence the integral equals

Using

the limit is

Problem 7.

Let be an entire function on with

for and

Find all such .

Proof.

By the maximum modulus principle,

for . Write

Since , we have

Because , Parseval's identity gives

But , so all other coefficients must vanish. Hence

Thus the only solution is

Problem 8.

Additional visible items in the scan:

(a) Prove that the image of a nonconstant entire function is dense in .

(b) If is holomorphic on a bounded domain , , , and , prove that .

(c) If is holomorphic on a domain and is harmonic, determine .

Proof.

(a) If were not dense, then some disk would be omitted. The function

would then be bounded and entire, hence constant by Liouville's theorem. Therefore would be constant, a contradiction.

(b) This is Cartan's uniqueness theorem for bounded domains. Since is bounded, all iterates map into the same bounded set. If

near with and , then

which contradicts the local boundedness of the iterates. Hence all higher coefficients vanish and .

(c) We have

If is harmonic, then , so . Hence is constant on each connected component of .