Exercise 15.  Let  f(z)  be a nonconstant analytic function in the domain D.  

Show that the function    is not analytic in D.  

Solution 15.

See text and/or instructor's solution manual.

Solution.  Proof by contradiction.  

      Assume that    is analytic and nonconstant.  

We have   ,     ;   so that

,     ,     ,     .   

The  Cauchy Riemann equations for      are  

(i)        ,  

(ii)        .  

Hence we obtain    and  .

Now use the Cauchy Riemann equation    for    and substitute it into (i)   and get

(iii)                    ,  

and use the Cauchy Riemann equation    for    and substitute it into (ii)   and get  

(iv)                    ,  

(iii) and (iv) imply that both   and    which in turn implies that  ,  where A is a constant.

Using similar reasoning it also follows that  ,  where B is a constant.   

But this contradicts our assumption.  Therefore    is not analytic in D.  

This solution is complements of the authors.

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