2021 Winter Qualifying Exam in Complex Analysis

Problem 1.

For every , find all solutions of

Proof.

Write . Then

and

Separating real and imaginary parts gives

and

Thus

Substitute into the real equation:

Equivalently,

Therefore solutions exist exactly when

that is,

In that case,

If , there are no solutions.

Problem 2.

Evaluate

where

Proof.

The poles are at

The winding number of around a point is the number of zeros in of

where .

For , on ,

By Rouche's theorem, has one zero in . Hence

For , on ,

Thus has no zeros in , so

Only the pole at contributes. Its residue is

Therefore

Problem 3.

Let the sequence be defined by

Find

Proof.

The left-hand side is

The coefficients are the Taylor coefficients of this function centered at .

The radius of convergence is the distance from to the nearest singularity of

The singularities are the cube roots of unity:

The distances from are

and

Thus the radius of convergence is

Therefore

Problem 4.

(a) Suppose is a function such that both and are entire functions. Prove that is entire.

(b) Suppose is a function such that both and are entire functions. Does it imply that is entire? Explain.

Proof.

(a) Let

Where ,

Thus is holomorphic away from the zeros of .

At a zero of , the identity

implies that the vanishing orders are compatible. If

then

is divisible by . Since , is divisible by . Hence has a removable singularity at .

Therefore extends holomorphically across all zeros of , so is entire.

(b) No. Since , the powers only determine , not necessarily .

For example, choose any function satisfying

for every , with some arbitrary choice of square root at each point. Such an cannot be continuous, hence cannot be entire. But

which are entire functions.

Problem 5.

Let be analytic in , have a simple pole with residue at , and a simple pole with residue at . Prove that for any ,

Proof.

Since the residue at is and the residue at is , the function

is entire.

Thus

Compute derivatives at :

and

Therefore

Since is entire, Cauchy's estimates on the circle give

Given , choose . Then

This proves the claim.

Problem 6.

How many roots does

have in the first quadrant?

Proof.

There are no zeros on the positive real axis, since

for .

There are also no zeros on the positive imaginary axis. If , , then

and

Let . Then

Now apply the argument principle to a large quarter-circle in the first quadrant. On the circular arc, the term dominates, so the change of argument tends to

On the two axis segments, the image stays on the positive real axis and contributes no change of argument.

Thus the number of zeros in the first quadrant is

Problem 7.

Find explicitly a conformal map from

to the unit disk.

Proof.

Set

This maps onto the upper half of the unit disk:

Then

maps the upper half-disk onto the first quadrant. Squaring maps the first quadrant to the upper half-plane:

Finally,

maps the upper half-plane to the unit disk.

Therefore one explicit map is

Problem 8.

Let be entire, and set

where is the greatest integer less than or equal to . Prove that

is a normal family on .

Proof.

By Montel's theorem, it is enough to prove local boundedness.

Fix a compact set

For , the points

remain in some fixed compact disk depending only on .

For , we have

for all , so

Thus, for all and all , the arguments of lie in a single compact subset of depending only on .

Since is entire, it is bounded on that compact subset. Hence is uniformly bounded on . Therefore the family is locally bounded, and by Montel's theorem it is normal.