Solution 7.

See text and/or instructor's solution manual.

Solution.   Since    have poles of order  ,  respectively, at    we can write

                      

                       

        If it so happens that  ,  and    for  ,  then

                    

In this case the Laurent series for     reduces to a Taylor series

                    ,

making    a removable singularity.  

        If  ,  and    then   ,   and

                      

Therefore    has a pole of order  m  at  .  

Similarly,  if  ,  then  ,  and then and    has a pole of order  n  at  .  

This solution is complements of the authors.

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