Solution 5.

See text and/or instructor's solution manual.

Solution.   .   Expand the expression in brackets by replacing z with    in
the geometric series (valid, therefore, for ,  or  .

                

Therefore,   ,   is valid for  .  

We are done.   

                    The images of the disk    under   ,  for  .  

                    The image of the disk    under the mapping   .  

Aside.  In Section 10.2 we will learn that the image of the disk    is the disk  .

We are 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