2018 Spring Qualifying Exam in Complex Analysis

Problem 1.

Determine the number of roots, counted with multiplicity, of

inside the annulus

Proof.

Let

On ,

By Rouche's theorem, and have the same number of zeros in . Thus has zeros in .

On , compare

with . The polynomial has all zeros in , since the three roots of have modulus . On ,

while

Equality can occur only at , and there the two terms point in the same direction, so the homotopy

has no zero on . Hence the number of zeros inside is unchanged and equals .

Therefore the number of zeros in is

Problem 2.

Let and be analytic on

Suppose

for all . Prove that is constant on .

Proof.

Let

Then is analytic on and

for all .

The set is connected. Since on a connected domain, is constant. Therefore there exists such that

for all .

Problem 3.

Let

Find an explicit conformal map from to the unit disk.

Proof.

First square:

The first quadrant maps to the upper half-plane, and the ray maps to the vertical ray

Now use

This maps the upper half-plane to the unit disk and sends the slit to the interval .

Taking a square root opens the slit. Finally apply a Mobius map from the resulting half-disk to the unit disk. One explicit choice is

where the branch of the square root is chosen on the slit disk so that the composition is single-valued. This is a composition of conformal maps, hence is conformal from onto the unit disk.

Problem 4.

Suppose

is holomorphic. Show that there exist and such that

Proof.

The imaginary part of depends only on , so write it as . The Cauchy-Riemann equations give

Thus depends only on . Since , the left side depends only on and the right side depends only on , so both are constant. Let this constant be .

Then

Therefore

where .

Problem 5.

Suppose is a polynomial of degree with only simple zeros . Prove that

Proof.

Since all zeros are simple, the partial fraction decomposition of is

For large ,

Therefore the coefficient of in the expansion of the right-hand side is

But since ,

so there is no term. Hence

Problem 6.

Evaluate

Proof.

Write

For

close the contour in the upper half-plane. There is no pole there, so

For

close in the lower half-plane. The pole at is enclosed, with clockwise orientation. Its residue is

Thus

Therefore

Problem 7.

Prove that the range of

is the whole complex plane.

Proof.

Let

Then

The cosine function is surjective from onto . Also, is a nonconstant polynomial, hence by the fundamental theorem of algebra.

Therefore

Problem 8.

A holomorphic function on the unit disk is called good if, for some , the function does not take values on the ray

Prove that the collection of all good functions is normal.

Proof.

Fix one of the finitely many rays

The domain

is conformally equivalent to a half-plane by a suitable rotation followed by a square root. Hence the family of holomorphic maps from into is normal by Montel's theorem.

The good functions are the union of the finitely many normal families corresponding to . A finite union of normal families is normal: every sequence has a subsequence lying in one of the finitely many families, and that subsequence has a normally convergent subsequence.

Therefore the collection of all good functions is normal.