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 (g).  .  

Solution 1 (g).

See text and/or instructor's solution manual.

Answer.  ,   ,   .

The conditions necessary for Theorem 4.4 are satisfied if and only if  , which is the  x-axis.

At the points  , we have  .  

Solution.  ,     and      

                 ,           so that  

,     ,     ,     .   

The  Cauchy Riemann equations are  

        ,  

        ,

which implies that   ,  and this holds only when  .  

Therefore,    is differentiable only at points on the x-axis.

At the points    on the x-axis, we have

.  

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 not an orthogonal grid.

          An orthogonal grid mapped by    is not an orthogonal grid, because the function is not analytic.

This solution is complements of the authors.

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