Solution 1 (a).

See text and/or instructor's solution manual.

Answer.      valid for all  z.

Solution.   Given  ,  the derivatives are  ,  ,  , etc.  

In general the even derivatives are      for   ,  

and the odd derivatives are      for   .  

Now evaluate these derivatives at   and get:

                        for   ,  

                        for   .  

      By Theorem 7.4, the coefficients of the Maclaurin series are

                         for   ,  

                         for   .  

and the sequence of coefficients is

                      

Or if you prefer, you can write it as:

                      

The series is usually expressed by adding up the non-zero odd powers  

                      

                    

Or if you prefer the series can be written as

                      

We are done.   

Remark.   If this last series looks strange, then recall that   ,   

and that   .   Hence, we obtain

                      

We are really done.   

                    The images of the disk    under   ,  for  .  

                    The image of the disk    under the mapping   .  

We are really really 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