Exercise 3.  Find the constants a and b such that    is differentiable for all  z.  

Solution 3.

See text and/or instructor's solution manual.

Answer.   .  

Solution.  ,    and  

                     ,           so that

,     ,     ,     .   

The  Cauchy Riemann equations are  

        ,  

        ,

all to hold for  z.  Hence we must have    which implies that   .  

We are done.   

Aside.  With the above substitutions the function is    .

Recall the identities    and    that were used in Section 2.1.

They can be substituted in ,  and the result is

                      

This is an alternate way to obtain the formula  .  Then calculate  .  

This solution is complements of the authors.

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