UMD Analysis Qualifying Exam/Aug08 Complex

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Problem 2[edit]


Solution 2[edit]

We will compute the general case:

Find Poles of f(z)[edit]

The poles of are just the zeros of , so we can compute them in the following manner:

If is a solution of ,



, k=0,1,2,...,n-1.

Thus, the poles of are of the form with

Choose Path of Contour Integral[edit]

In order to get obtain the integral of from 0 to , let us consider the path consisting in a line going from 0 to , then the arc of radius from the angle 0 to and then the line joining the end point of and the initial point of ,

Pi n contour.jpg

where is a fixed positive number such that

the pole is inside the curve . Then , we need to estimate the integral

Compute Residues of f at z0= exp{i\pi /n}[edit]

Bound Arc Portion (B) of Integral[edit]

Hence as ,

Parametrize (C) in terms of (A)[edit]

Let where is real number. Then

Apply Cauchy Integral Formula[edit]

From Cauchy Integral Formula, we have,

As , . Also can be written in terms of . Hence

We then have,

Problem 4[edit]

Suppose and there is an entire function with . If and , prove that

Solution 4[edit]

Lemma: Two fixed points imply identity[edit]

Lemma. Let be analytic on the unit , and assume that on the disc. Prove that if there exist two distinct points and in the disc which are fixed points, that is, and , then .

Proof Let be the automorphism defined as

Consider now . Then, F has two fixed points, namely


Since ,

(since is different to ), and


by Schwarz Lemma,


But, replacing into the last formula, we get .



which implies

Shift Points to Create Fixed Points[edit]

Let . Then and .

Notice that is an infinite horizontal strip centered around the real axis with height . Since is a unit horizontal shift left, .

Use Riemann Mapping Theorem[edit]

From the Riemann mapping theorem, there exists a biholomorphic (bijective and holomorphic) mapping , from the open unit disk to .

Define Composition Function[edit]

Let . Then maps to .

From the lemma, since has two fixed points, which implies which implies .

Problem 6[edit]

Let be the family of functions analytic on so that

Prove that is a normal family on

Solution 6[edit]

Choose any compact set K in D[edit]

Choose any compact set in the open unit disk . Since is compact, it is also closed and bounded.

We want to show that for all and all , is bounded i.e.

where is some constant dependent on the the choice of .

Apply Maximum Modulus Principle to find |f(z0)|[edit]

Choose that is the shortest distance from the boundary of the unit disk . From the maximum modulus principle, .

Note that is independent of the choice of .

Apply Cauchy's Integral Formula to f^2(z0)[edit]

We will apply Cauchy's Integral formula to (instead of ) to take advantage of the hypothesis.

Choose sufficiently small so that

Integrate with respect to r[edit]

Integrating the left hand side, we have


Bound |f(z0)| by using hypothesis[edit]

Apply Montel's Theorem[edit]

Then, since any is uniformly bounded in every compact set, by Montel's Theorem, it follows that is normal