Solution 17 (i).

See text and/or instructor's solution manual.

Solution.  To prove part (i) consider the analytic function    and it's conjugate  ,  then

                    .  

Let us observe that     and then define the function  as follows:

                     ,       and notice here that
                     
                         and    ).

Then we can express    in the form  

                    .

Now consider this as an identity in the variables  :  

                    ,  

and make the substitutions  ,  and get  

                        

Since    we can now conclude that

                        

is a harmonic conjugate of  .

This solution is complements of the authors.

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