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 (e).  .  

Solution 1 (e).

See text and/or instructor's solution manual.

Answer.    is differentiable only at the point  ,  and  .  

Solution.  ,     and      

                 ,           so that  

,     ,     ,     .   

The  Cauchy Riemann relations are  

        ,  

        ,

which hold when ,  which occurs only at the point  .   

The partials are continuous everywhere, so at the point    we find that    

                    .  

Caveat.  It would be false to use the formula    for values other than  .    

We are done.   

Aside.  A reason why f(z) is not analytic is given in Exercise 16.  

     Loosely speaking, if f(z) is analytic then there cannot be any occurrence of the variable  .

Recall the identities    and    that were used in Section 2.1.

They can be substituted in ,  and the result is

                      

So technically there are "hidden" terms involving    in the formula for  f(z),  and this precludes the possibility that  f(z)  can be analytic.

Indeed, the complex form of the Cauchy-Riemann equations fails to hold because  .  

We are really done.   

Aside.  We can let Mathematica double check our work.

                    The image of this orthogonal grid under    is by luck an orthogonal grid.

This solution is complements of the authors.

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