UMD Analysis Qualifying Exam/Aug06 Complex

Problem 2 edit

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 2 edit

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.


Problem 4 edit

Let   have boundary  . For   define


(a) Show that   if and only if  .

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

Solution 4 edit

4a edit

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  .

4b edit

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 6 edit

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 6 edit