Exercise 9.  Explain why  

9 (a).     holds for all  z.  

Solution 9 (a).

See text and/or instructor's solution manual.

Solution  Method I.  Use formula  (5-1)  .   

                       

We are done.   

Solution  Method II.  Use the Definition 5.1:   .  

And substitute into this and get

                    .  

Recall Exercise 6 in Section 4.1 where we proved:   If    then   .   

Setting    and    and applying this result yields   

                    .

Combining the above finishes our proof

                    .

Detailed solution for  Solution.  Method II.   Let   ,   then   .   

It follows that  

                       

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