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 (c).  .

Solution 1 (c).

See text and/or instructor's solution manual.

Answer.  ,  and  .

The partials are continuous everywhere, so     for all  z.

Solution.  ,     and      

                 ,            so that  

,     ,     ,     .   

The  Cauchy Riemann equations are  

        ,  

        ,

and equality holds 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.  

It is easily seen that if we set  ,  then we have  

          .

Hence, by the differentiation rules for    we get  

          .

which can be written as  

          .

As we anticipated, this agrees with the Cauchy-Riemann method.

We are really done.   

Aside.  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  .  

We are really really done.   

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