Exercise 8.  Find     using two methods.   (i) Partial fractions and Table 9.1.   (ii) Using residues.  

8 (a).  .     Hint.  .  

Solution 8 (a).

See text and/or instructor's solution manual.

Answer.   .  

Solution (i).   Using Table 9.1 of z-transforms we get

                      

Remark.  The details for the partial fraction expansion are at the bottom of the page.

We are done.   

Solution (ii).   Using residues we get

                      

                    and

                      

Hence,  

                      

Therefore,

                    .  

We are done.   

Aside.  We can let Mathematica double check our work.

          The Maple code is similar

  

                                                            


  

                                                            


  

                                                            

  

                                                            

          The Maple code using limits is similar

  

                                                            


  

                                                            


  

                                                            

          The Maple code using residues is similar

  

                                                            


  

                                                            


  

                                                            

We are really done.   

Aside.  We can graph some of the terms in the sequence.

                                                                                                                        The sequence   .   

We are really really done.   

The Details for the Partial Fractions.   

Aside.  How can we expand      into the proper partial fractions?

It is natural to try the command:

But this is not the desired form for using Table 9.1 of z-transforms.

Method (i).   Use the following algebra steps  

                      

Method (ii).   Find the linear combination of   ,  

                    .  

Equate the numerators   ,  

and solve the linear system  

                    

and get   .   

Therefore, the desired form is  

                    .  

Aside.   The Mathematica commands for Method (ii)  are

Method (iii).  (For distinct real roots)   First make the substitution      in      and get  

                    .  

Then use the standard procedure for expanding in partial fractions   

                    .  

Then make the substitution      in      and get  

                      


Therefore, the desired form is  

                    .  

Aside.   The Mathematica commands for Method (iii)  are

Now we have the desired form:

                    .  

This solution is complements of the authors.

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