9 (a).  Use  ,  and show that the residue at   is  .  

Solution 9 (a).

See text and/or instructor's solution manual.

The residue at    is calculated by using Theorem 8.1 (Cauchy's Residue Theorem).  

Here the denominator of  f(z) has a factor of the form  ,  and  .

                      

We are done.   

Aside.  We can let Mathematica double check our work.

Maple can check our work too!

     > limit( (z+1)*z^(1/3)/(z^3*(z+1)), z=-1);

               

This solution is complements of the authors.

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