2009 Fall Qualifying Exam in Complex Analysis

Problem 1.

For , find the radius of convergence of

Proof.

Write the coefficient as

where is the rising factorial.

Then

As ,

Therefore, by the ratio test, the power series has radius of convergence

Problem 2.

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

Proof.

No such holomorphic function exists.

For each , the zero set of is either discrete in or all of .

If no derivative is identically zero, then the set

is a countable union of discrete sets. In particular, it cannot contain the whole interval , which has accumulation points inside the disk.

If some derivative is identically zero, then all higher derivatives are also identically zero. Since the problem allows nonnegative integers , this would make every point of belong to the displayed set, not just the interval .

Both cases are impossible. Hence no such holomorphic function exists.

Problem 3.

Let

Assume is entire and that for every , one has . If

find .

Proof.

Set

Then is entire, and is real-valued on the line .

Reflection across the line is

By the Schwarz reflection principle,

Taking , we get

Since

we have

Therefore

Thus

Problem 4.

Let be a family of holomorphic functions on such that for every and every ,

Prove that is a normal family.

Proof.

The forbidden set

contains at least two distinct complex numbers, for example and .

Thus every function omits the two values

By Montel's theorem for families omitting two fixed complex values, the family is normal in the sense that every sequence has a subsequence converging locally uniformly to a holomorphic function or to .

Problem 5.

Let

be holomorphic. Prove that is constant.

Proof.

Since takes values outside the unit disk, never vanishes and

Therefore

is holomorphic on and satisfies

The singularities of at and are removable, because is bounded near each of them. Hence extends to a bounded entire function on .

By Liouville's theorem, is constant. Therefore is constant.

Problem 6.

Suppose is a polynomial such that all of its zeros are inside the unit disk. Prove that all zeros of are also inside the unit disk.

Proof.

By the Gauss-Lucas theorem, every zero of lies in the convex hull of the zeros of .

The unit disk is convex. Since all zeros of lie inside the unit disk, their convex hull is also contained inside the unit disk. Therefore every zero of lies inside the unit disk.

Problem 7.

Find

Proof.

Integrate by parts. Let

Then

The boundary term

vanishes at both and . Hence

Using

we get

Therefore

Problem 8.

Does there exist a sequence of holomorphic functions on such that

uniformly on

Proof.

No.

If such a sequence existed, then uniform convergence on the circle would imply

But each is holomorphic in the disk, so by Cauchy's theorem,

On the other hand,

This is a contradiction. Therefore no such sequence exists.