Exercise 5.  Let  .  

Show that both  f(z)  and  f'(z)  are differentiable for all  z.  

Solution 5.

See text and/or instructor's solution manual.

Answer.    and    by Theorem 3.4.

Solution.  ,    and  

                 ,           so that  

,     ,     ,     .   

The  Cauchy Riemann equations are  

        ,  

        ,

which hold for all  z.

The partials are continuous everywhere, so   

,  

for all  z.

We are done.   

Remark. In Section 5.1 we will learn that    is the complex exponential function.  

Then we will be able to use the rule for differentiation  .  

Aside.  We can let Mathematica double check our work.

                    The analytic function    maps any orthogonal grid onto an orthogonal grid.

This solution is complements of the authors.

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