2008 Fall Qualifying Exam in Complex Analysis

Problem 1.

Compute the area of the image of the unit disk under

Proof.

The map is univalent on the unit disk; indeed has no zero in . Hence the area of the image is

Since

we get

The integrals of and over the disk are , so

Thus the area is

Problem 2.

Find all entire functions satisfying

for all .

Proof.

The function

is entire and vanishes at for every . Since these points accumulate at , the identity theorem gives

Thus

The entire solutions of this differential equation are

Problem 3.

Let

Assume is entire and maps into . If

prove that

Proof.

Reflection across the line is

Since maps into itself, the Schwarz reflection principle gives

Taking ,

Therefore

Problem 4.

Find the largest disk centered at in which the Taylor series

converges.

Proof.

The function

has singularities at

The radius of convergence of the Taylor series centered at is the distance from to the nearest singularity.

Both distances are

Thus the largest disk is

Problem 5.

Evaluate

Proof.

Use the standard identity

Taking gives

Problem 6.

Suppose , where , is holomorphic and

Show that cannot have a zero in the disk

Proof.

Suppose . By the Schwarz-Pick lemma,

Since and , this becomes

Therefore every zero of satisfies . Hence has no zero in .

Problem 7.

Let be harmonic on and suppose for all . Show that is constant.

Proof.

Since is continuous and never zero on the connected set , it has one fixed sign. Thus either everywhere or everywhere.

A harmonic function on the whole plane that is bounded below or bounded above is constant. Indeed, apply Harnack's inequality to the positive harmonic function or and let the radius tend to infinity.

Therefore is constant.

Problem 8.

How many zeros does

have in the unit disk? What are the multiplicities of the zeros?

Proof.

On ,

By Rouche's theorem, and have the same number of zeros in the unit disk, counted with multiplicity. Hence has zeros in the unit disk.

They are all simple. If were a multiple zero, then

and

Dividing the second equation by the first gives

so , which is not in the unit disk. Therefore no zero in the unit disk is multiple.

Thus has

zeros in the unit disk, all of multiplicity .