Solution 3 (b).

Answer.   No.  

Solution.   No.  The function    is not analytic at    because it has an essential singularity at  .

So this does not contradict Corollary 7.9 because it requires that    should be an analytic function in a neighborhood of   .

Aside.   It is easy to show that    has an essential singularity at the origin.  

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.

This solution is complements of the authors.

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