Theorem (Milne-Thomson Method for constructing a harmonic conjugate.)  

Formula (i)  Given the harmonic function    then construct   

                     .  

Show that under the proper conditions,    is a harmonic conjugate of  ,  and

                       is an analytic function.  

Proof of (i).

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  .

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