Solution 3.

See text and/or instructor's solution manual.

Answer.   .

Solution.  For  ,  we have

                    ,   and  ,  

                    ,   and  ,  

                    .  

In order for this to hold for all z  we must have  .  

We are done.

Aside.  The choice   .  determines the form of  ,  which must be  

                    .

The terms    are the real and imaginary parts of  .  It doesn't take long to discover that

                    

is the analytic function for which  .  

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