UMD Analysis Qualifying Exam/Aug06 Complex

Problem 2Edit

For real   consider the integral

 

(a) Compute the Cauchy Principal Value of the integral (when it exists)

(b) For which values of   is the integral convergent?

Solution 2Edit

Consider the complex function  . This function has a pole at  . We can calculate  .

Consider the contour   composed of the upper half circle   centered at the origin with radius   traversed counter-clockwise and the other part being the interval   on the real axis.

That is,

 

Let us estimate the integral of   along the half circle  . We parametrize   by the path  ,   for  . This gives

 

Break up the interval   into   for some  . This gives  .

Let us evaluate the first of the two integrals on the right-hand side.

  which tends to 0 as  . NOTE: This argument only works if we assume  . If we try this argument for  , we bound the integrand by   instead of  , but this will diverge as we send   (which implies that   must also diverge as  . This answers part b).

As for the other integral,   which tends to   as  .

Therefore, we've shown that  . But   was arbitrary, hence we can say that the integral vanishes.

Therefore,  

Problem 4Edit

Let   have boundary  . For   define

 .

(a) Show that   if and only if  .

(b) Show that   has at least one fixed point  .

Solution 4Edit

4aEdit

Consider   and  . Then  . We know that   is a conformal map from   to   and moreover,   if an only if  . The same is true for  , that is,   if any only if  . Therefore,   if and only if  .

4bEdit

If   is a fixed point of  , then  . Rearranging gives   By the fundamental theorem of algebra, we are guaranteed 3 solutions to this equation in the complex plane. All that we need to show is that at least on of these solutions lie on the circle n the circle  .

Problem 6Edit

Let   be a family of entire functions. For   define the domains

 .

If   is normal (i.e. convergence to   is allowed) on each   show that   is normal on  .

Solution 6Edit