Exercise 16.  The complex form of the Cauchy-Riemann equations.

Recall that, for  ,  we have the substitutions     and  .  

16 (a).  Temporarily, think of    as dummy symbols for variables.  

With this perspective, x and y can be viewed as functions of z and .  

Use the chain rule for a function    of two variables to show that

                    .

Solution 16 (a).

See text and/or instructor's solution manual.

Solution.  For  ,      and   ,   we have  

                    ,    ,    ,   and   .   

Therefore,  

                    

This solution is complements of the authors.

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