Solution 3 (a).

See text and/or instructor's solution manual.

Answer.      has a simple pole at the origin.

Solution.   Use a convenient method to determine where the denominator is zero.  

For example, use the series    and get

                      

Then     

                      

Now apply  Corollary 7.5 to conclude that   has a simple pole at the origin.

You do not need to find any other singularity for this exercise.

We are  done.   

Aside.  We can let Mathematica double check our work.

We are really done.   

                              The simple pole at the origin.  

                              A plot for   .  

Remark.  There does not seem to be an analytic way to determine the other poles of  .

For this exercise it will suffice to determine that there is a simple pole at the origin.  

However, the graphs below indicated that there are infinitely many poles.  

                                                                      A plot for   .  

Remark.  There does not seem to be an analytic way to determine solutions to  .  

However, the graph below indicated that there are solutions.  

It is possible to determine the location of other non-zero solutions, but it requires a numerical analysis computation.

                    A plot for   .  

This solution is complements of the authors.

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