Exercise 1.  Use the Cauchy Riemann conditions to determine where the following functions are differentiable, and evaluate the derivatives at those points where they exist.  

1 (a).  .

Solution 1 (a).

See text and/or instructor's solution manual.

Answer.  Given   ,  calculate  .

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.  This agrees with the rule for differentiation that were given in Section 3.1.  

Given   ,  calculate  .

Aside.  We can let Mathematica double check our work.

                      maps this orthogonal grid onto an orthogonal grid.

                    We will study this phenomenon in detail in Section 10.1 and Section 11.4.  

This solution is complements of the authors.

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