2008 Spring Qualifying Exam in Complex Analysis

Problem 1.

Find explicitly a conformal mapping of

onto the unit disk.

Proof.

The map

sends the exterior first-quadrant domain onto the quarter unit disk

Then maps this quarter disk onto the upper half unit disk. The map

maps the upper half unit disk onto the first quadrant. Squaring maps the first quadrant onto the upper half-plane, and

maps the upper half-plane onto the unit disk.

Thus one explicit conformal map is

Problem 2.

Let be entire and suppose

for all . Prove that

for some constant .

Proof.

At every zero of , the inequality forces to vanish at least to the same order. Hence

has removable singularities at the zeros of and extends to an entire function.

Away from the zeros of ,

By removability, this bound holds everywhere. Therefore is bounded and entire, so by Liouville's theorem is constant. Hence

Problem 3.

Show that there is a holomorphic function on

whose derivative is

Is there a holomorphic function on whose derivative is

Proof.

On the exterior domain , a holomorphic function has a primitive exactly when its integral around a large circle is .

For

we have

Thus there is no term in the Laurent expansion at infinity, so

Therefore has a primitive on .

For

we have

Hence

So has no primitive on .

Problem 4.

Let be the unit disk. Does there exist a holomorphic function such that

Proof.

No.

By the Schwarz-Pick derivative estimate,

Taking and , we get

But

Thus no such holomorphic map exists.

Problem 5.

Evaluate

Proof.

Although the denominator vanishes at , the real integrand has a removable singularity there because also vanishes.

Consider

and close the contour in the upper half-plane, indenting above the pole at . The upper half-plane pole is

The residue at is

The indented pole at contributes one half-residue, and

Thus the principal value of the complex exponential integral is

Since

the first term is real. Therefore the imaginary part is

Hence

Problem 6.

Prove that

converges uniformly on compact sets to an entire function.

Proof.

Write each factor as

where

On a compact set ,

and

Both

converge. Hence

converges.

Therefore the infinite product converges uniformly on compact sets. Its limit is holomorphic on every compact disk, hence entire.

Problem 7.

Let be a family of holomorphic functions on such that every satisfies

Prove that is a normal family on .

Proof.

Write

Then

and orthogonality gives

Also , so the hypothesis gives uniform bounds on and on

For ,

By Cauchy-Schwarz,

The second sum is finite for , and the first is uniformly bounded. Thus is locally uniformly bounded.

By Montel's theorem, is a normal family.

Problem 8.

Let be holomorphic on the upper half-plane and continuous on . Assume that

Find all such functions .

Proof.

Let

and set

Then is holomorphic on , continuous on , and

for .

By the Schwarz reflection principle, extends holomorphically across the interval by

in the reflected lower half-neighborhood. The extended function vanishes on the interval, which is now a set with accumulation points inside the extended domain. By the identity theorem, .

Thus

Problem 9.

Show that there is no holomorphic function on

satisfying

Proof.

The inequality implies

takes values in the disk , which does not contain . Hence has no zeros in the annulus.

Fix with and let . Since has no zeros, the winding number of about is an integer . Therefore the winding number of

around is

This is an odd integer.

But lies entirely in , a disk that does not wind around . Hence its winding number about must be .

This contradiction proves that no such holomorphic function exists.