UMD Analysis Qualifying Exam/Aug08 Complex

Problem 2

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Compute


 


Solution 2

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We will compute the general case:


 


Find Poles of f(z)

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

If   is a solution of  ,

then  

  and  

  , k=0,1,2,...,n-1.

Thus, the poles of   are of the form   with  

Choose Path of Contour Integral

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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}

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Bound Arc Portion (B) of Integral

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Hence as  ,  

Parametrize (C) in terms of (A)

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Let   where   is real number. Then  


 

Apply Cauchy Integral Formula

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From Cauchy Integral Formula, we have,


 


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


 


We then have,


 

Problem 4

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

Solution 4

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Lemma: Two fixed points imply identity

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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  .

Therefore,

 ,

which implies

 

Shift Points to Create Fixed Points

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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

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

Define Composition Function

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Let  . Then   maps   to  .


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

Problem 6

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Let   be the family of functions   analytic on   so that


 


Prove that   is a normal family on  

Solution 6

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Choose any compact set K in D

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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)|

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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)

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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

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Integrating the left hand side, we have


 


Hence,


 

Bound |f(z0)| by using hypothesis

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Apply Montel's Theorem

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