2016 Fall Qualifying Exam in Complex Analysis

Problem 1.

Evaluate

Proof.

Use the standard formula

Here and , so

Problem 2.

Suppose is analytic in the annulus

and there exists a sequence of polynomials converging to uniformly on the unit circle . Show that can be extended analytically to .

Proof.

Write the Laurent expansion of in the annulus:

For ,

Since uniformly on ,

But is holomorphic in the unit disk, so each integral is zero. Hence

for all .

Thus the Laurent expansion has no negative powers, and extends holomorphically to .

Problem 3.

Prove or disprove: there is a nonzero holomorphic function on such that

for all .

Proof.

No such nonzero function exists.

If the inequality holds, then at every zero of , the function must vanish. Consider

The zeros of are simple. Since has zeros of even order, has removable singularities at the zeros of , so is entire. The inequality gives

away from the zeros, and hence everywhere by removability.

By Liouville's theorem, is constant:

Thus

If , then the right side has simple zeros, while the left side has zeros of even order, impossible. Therefore , so .

Problem 4.

Let be meromorphic in such that

on . Prove that is rational.

Proof.

Define the reflected meromorphic function

On , the condition gives

By the identity theorem for meromorphic functions,

where both are defined.

This identity shows that the behavior of at infinity is the reflection of its behavior at . Since is meromorphic at , it is also meromorphic at infinity. Therefore is meromorphic on the Riemann sphere.

Every meromorphic function on the Riemann sphere is rational. Hence is rational.

Problem 5.

Find the radius of convergence of

Proof.

The sine function is entire, so the only singularities come from the poles of

These occur at

Their moduli are

The nearest singularity to is , so the radius of convergence is

Problem 6.

Prove or disprove: there is a holomorphic function on such that

Proof.

Such an exists if and only if the integral of the proposed derivative around a generator of the exterior domain is zero.

For ,

The rational function behaves like at infinity, so the sum of the finite residues is . Hence the integral is

Therefore the function has no primitive on . No such holomorphic exists.

Problem 7.

Let be analytic on the upper half-plane and satisfy . Suppose

Give an upper bound for and state which functions realize the extremum.

Proof.

Use the Schwarz-Pick lemma on the upper half-plane. The automorphism

maps the upper half-plane to the unit disk and sends to . Since , Schwarz's lemma applied to gives

Equality occurs exactly for rotations:

Problem 8.

Let be real-valued and harmonic in such that

Show that there is a harmonic function on such that

for all .

Proof.

An isolated singularity of a harmonic function has the form

where is harmonic near , unless the singularity is removable.

The hypothesis

forces and rules out every principal part term, since those grow faster than along suitable directions.

Thus the singular part vanishes, and has a removable singularity at . The resulting extension is harmonic on .