Exercise 9.  Explain why  

9 (b).     is nowhere analytic.  

Solution 9 (b).

See text and/or instructor's solution manual.

Solution.   

                        


          ,           so that

,   ,   ,   .   

The  Cauchy Riemann equations are  

        ,  

        .  

Hence we obtain    and  ,  which hold if and only if both    and  ,  which is impossible.

Therefore,    is nowhere differentiable.  

We are done.   

Aside.  Another reason why f(z) is not analytic is given in Exercise 16 in Section 3.2.   

The complex form of the Cauchy-Riemann equations is  .

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

This solution is complements of the authors.

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