Exercise 11.  Determine where the following functions are differentiable and where they are analytic.  Explain!  

11 (a).  .

Solution 11 (a).

See text and/or instructor's solution manual.

Answer.  f(z)  is differentiable only at points on the coordinate axes.    f(z)  is nowhere analytic.

Solution.  ,   and  

                 ,     and         so that  

,     ,     ,     .   

The  Cauchy Riemann equations are  

        ,  

        ,

which hold only for the points that satisfy  ,  which implies that  ,  and this happens when either  .

Hence, f(z) is differentiable only at points on the coordinate axes.    

However, at each point on the x-axis or y-axis there fails to be a neighborhood in which f(z) is differentiable.  

Therefore  f(z)  is nowhere analytic.

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 not an orthogonal grid.

This solution is complements of the authors.

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