UMD Analysis Qualifying Exam/Jan10 Complex

Problem 2 edit

The function   has a convergent Taylor expansion  . Find  .

Solution edit

 . By using the definition  , we get that   if and only if  . It is not hard to show that this happens if and only if   and  . Therefore, the only zeros of   all occur on the real axis at integer distances away from 1/2. Therefore,   is analytic everywhere except at these points.

Our Taylor series   is centered at  . By simple geometry, the shortest distance from   to   or   (the closest poles of  ) is  . This is the radius of convergence of the Taylor series.

From calculus (root test), we know that  . Therefore,  .