Solution 3 (c).

See text and/or instructor's solution manual.

Answer.     has an essential singularity at the origin.  

Solution.   Use the fact that  ,  and express this function as a Laurent series.  

Use the substitution    and get  

                      

Then apply  Definition 7.5 to conclude that   has an essential singularity at the origin.

We are  done.   

Aside.  We can let Mathematica double check our work.

We are really done.   

                              The essential singularity at the origin.

                              A plot for   .                  A plot for   .  

                              Note. These are plots of the real and imaginary parts of the function.

This solution is complements of the authors.

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