2011 Fall Qualifying Exam in Complex Analysis

Problem 1.

Describe all entire holomorphic functions and such that:

(a) for all positive integers .

(b) for all positive integers .

Proof.

(a) The function

has the required values. If another entire function has the same values, then vanishes at for every positive integer . These zeros accumulate at , so by the identity theorem,

Thus the only possibility is

(b) No such entire function exists.

Indeed, the condition

for all implies by the identity theorem that is even:

Hence there is an entire function such that

Then

Let . Then and

This gives and

so . Hence near . But

contradicting . Therefore no entire satisfies the stated condition.

Problem 2.

Let be an entire holomorphic function such that

Prove that is constant.

Proof.

Write

Fix . For all sufficiently large ,

on . By Cauchy's estimate,

For , this gives , so . For , letting gives .

Thus all nonconstant Taylor coefficients vanish, and is constant.

Problem 3.

Evaluate

Proof.

Use the standard formula

Here

so . Therefore

Thus

Problem 4.

(a) Show that there is an analytic function defined in

whose derivative is

(b) Does there exist an analytic function in with derivative

Proof.

On the exterior domain , a holomorphic function has a primitive if and only if its integral around one large positively oriented circle is . Equivalently, the coefficient of in its Laurent expansion at infinity must be .

(a) For

we have

Thus the Laurent expansion at infinity has no term. Therefore the integral of around every closed curve in is , and has a primitive in .

(b) For

we have

Therefore the coefficient of is , so

for . Hence does not have a primitive on .

So the answer to part (b) is no.

Problem 5.

Let be continuous. Define

Prove that is holomorphic on .

Proof.

Let be a compact subset of . Then there is such that

for all and .

For each fixed , the function

is holomorphic on . On , its derivative with respect to is

and this is uniformly bounded by

Thus differentiation under the integral sign is justified on compact subsets of .

Therefore

and is holomorphic on .

Problem 6.

Prove the Schwarz-Pick lemma: if is holomorphic, then

Proof.

For , define the disk automorphism

Also define

Then the function

maps holomorphically into itself and satisfies

By the Schwarz lemma,

for all .

Now take

Then

Thus

which is exactly

Problem 7.

Let be holomorphic in and let

Prove that is an increasing convex function of on .

Proof.

Write the Taylor expansion

For , Parseval's identity gives

Each function is increasing on , and for it is convex there. The term is constant.

Since the series has nonnegative coefficients and converges locally uniformly in , the sum

is increasing and convex on .

Problem 8.

Let be meromorphic in such that

Prove that is a removable singularity of .

Proof.

Since is meromorphic near , the isolated singularity is either removable or a pole. Suppose it is a pole of order . Then near ,

for some . Hence

In polar coordinates,

has the same convergence behavior as

This integral diverges for every integer .

Therefore cannot be a pole. The only remaining possibility is that is removable.