Exercise 11.  Let   be analytic at the points  .  

If   ,   then show that      for  .  

Hint:  The solution is a little tricky.  You might need to use  .  

Solution 11.

See text and/or instructor's solution manual.

Solution.   By Theorem 8.2 we have  ,  where n is any integer.  

Since  ,  and because    is analytic at n,  we use L'Hôpital's rule to get  .  

Explain how this gives the result.

Caveat:  .   Unfortunately, Mathematica is in error and computes this limit to be zero.  

The computation using L'Hôpital's rule will give the correct answer  .

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell