Solution 9.

See text and/or instructor's solution manual.

Answer.   Appeal to Theorem 7.12 and mimic the argument given in the solution to Exercise 5.

Solution Method I.   Appeal to Theorem 7.12 and express  f(z)  in the form

                    ,

where the function h(z) is analytic at the point .   Then

                    

where  ,  is analytic at the point ,  

and   .  

Therefore,    has a pole of order  k+1  at  .   

Solution Method II.   Since    has a pole of order  , at    we can write

                      

Differentiate the series termwise and obtain
                    
                      

Therefore,    has a pole of order  k+1  at  .   

This solution is complements of the authors.

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