2019 Spring Qualifying Exam in Complex Analysis

Problem 1.

Let be analytic in the punctured disk and suppose

Show that is a removable singularity of .

Proof.

Write the Laurent expansion of at :

If the principal part were nonzero, let be the largest integer such that . Then near ,

Along suitable rays approaching , the real part of becomes arbitrarily large and positive. Hence would be unbounded above near , contradicting

Thus the principal part vanishes, so is removable.

Problem 2.

Let be a nonconstant real harmonic function. Show that there exists a sequence such that

Proof.

Suppose not. Then is bounded below on . Since is simply connected, there is an entire function such that

If is bounded below, then

is a bounded entire function. By Liouville's theorem, is constant, hence is constant, so is constant. This contradicts the hypothesis.

Therefore is not bounded below, and there exists a sequence such that

Problem 3.

(a) State the Schwarz-Pick lemma.

(b) Suppose is holomorphic and

Give an upper bound for , and characterize the functions for which equality holds.

Proof.

(a) Schwarz-Pick says that if is holomorphic, then

for all . Equivalently,

(b) At ,

Equality holds exactly when is a disk automorphism. Since , these functions are

where

Problem 4.

Let

(a) Find and classify all singularities of in the extended complex plane.

(b) Evaluate

where

(c) Does there exist a holomorphic function on such that on ?

Proof.

Since

the finite singularities are at

At , the factor has an essential singularity, so has an essential singularity. At , the denominator has simple zeros, so has simple poles.

At infinity,

Thus as , so infinity is a pole of order .

For the integral, expand at infinity:

The coefficient of is

Therefore

Since encloses all finite singularities,

For part (c), a primitive on the exterior domain would force the integral around a large positively oriented circle to vanish. But this integral is

Therefore no such holomorphic primitive exists.

Problem 5.

Assume

is holomorphic for and

Show that is one-to-one in .

Proof.

Let with . Then

For ,

Thus

Therefore

Hence whenever , so is one-to-one.

Problem 6.

Let

be entire and suppose

for all . Prove that

for all .

Proof.

By Cauchy's estimate, for any ,

Choose . Then

Problem 7.

Construct a conformal map from

onto

Proof.

First map the vertical strip to the upper half-plane by

Indeed, if and , then

so lies in the upper half-plane.

The map

maps the upper half-plane conformally onto the exterior of the unit disk. Therefore

is a conformal map from onto .

Problem 8.

Let be harmonic in and suppose

Prove that is harmonic on .

Proof.

An isolated singularity of a harmonic function has the form

where is harmonic near , unless the singularity is removable.

The condition

forces . It also rules out every principal part term , because such terms grow faster than along suitable paths.

Therefore the singular part vanishes, so has a removable singularity at . Hence extends harmonically to .