Exercise 5.  Show that, for all z,  

5 (a).   .  

Solution 5 (a).

See text and/or instructor's solution manual.

Solution.  This follows immediately from the identity   .  

                       

Aside.  We can let Mathematica double check our work.

Alternate Solution.   Use the Identity (5-34)   and get:  

                      

We are done.   

Aside.  We can let Mathematica graph the compositions that are involved.

                    The composite mappings  ,  ,  ,   

                    where    and  .   

                    The equivalent mapping   .    

                    The composite mappings  ,  ,  ,   

                    where    and  .   

                    The equivalent mapping   .    

This solution is complements of the authors.

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