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

9 (e).  .

Solution 9 (e).

See text and/or instructor's solution manual.

Answer.   .  

Solution.   The root of the denominator is  .  

Calculating the residues we get

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

                       

Computing the residues using Theorem 8.2 we obtain:

                    



                    


                    
                    

We are done.   

Aside.  We can let Mathematica double check our work.

Maple can check our work too!

     > residue( (2*z^2-3*z-1)/(z-1)^3, z=1 );

               2

     > residue( (z-1)*(2*z^2-3*z-1)/(z-1)^3, z=1 );

               1

     > residue( (z-1)^2*(2*z^2-3*z-1)/(z-1)^3, z=1 );

               -2

We are really done.   

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

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

     > convert( (2*z^2-3*z-1)/(z-1)^3, parfrac, z);

                 

This solution is complements of the authors.

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