Exercise 13.  Use Identities (5-36) and (5-37) to help establish the inequality    and show that equality holds if and only if  z  is a real number.  

Solution 13.

See text and/or instructor's solution manual.

Solution.   Combining (5-36) and (5-37), and letting  ,  we get

                      

From this equation we obtain the following two important facts:

                        when    is a real number,  and

                       when  .

Aside.  We can let Mathematica double check our work.

                    A portion of the surface  .

This solution is complements of the authors.

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