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

9 (a).  .

Solution 9 (a).

See text and/or instructor's solution manual.

Answer.   .  

Solution.   Factoring the denominator, we obtain

                      

The roots are  .  

Applying the method outlined in  Example 8.7 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( 1/(z^2+3*z+2), z=-1 );

               1

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

               -1

We are really done.   

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

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

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

               

This solution is complements of the authors.

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