UMD Analysis Qualifying Exam/Aug08 Complex

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  ,


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 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