Exercise 9.  Use residues to find the partial fraction representations of  

9 (c).  .

Solution 9 (c).

See text and/or instructor's solution manual.

Answer.   .  

Solution.   The roots of the denominator are  .  

Applying the method outlined in  Example 8.7 and the remark following it yields  

                      

Computing the residues using Theorem 8.2 we obtain:

                    


                    
                    

Similarly  
                    
                    

We are done.   

Aside.  We can let Mathematica double check our work.

Maple can check our work too!

     > residue( (z^2-7*z+4)/(z^2*(z+4)), z=0 );

               -2

     > residue( z*(z^2-7*z+4)/(z^2*(z+4)), z=0 );

               1

     > residue( (z^2-7*z+4)/(z^2*(z+4)), z=-4 );

               3

We are really done.   

Aside.  We can use Mathematica built in command "Apart."

Or we can use Maple's built in command "convert."

     > convert( (z^2-7*z+4)/(z^2*(z+4)), parfrac, z);

               

This solution is complements of the authors.

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