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

Formula (ii)  Given the harmonic function    then construct   

                     .  

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

                       is an analytic function.  

Proof of (ii).

To prove part (ii) consider the analytic function  .  

Then use the construction in (i) applied to   .  

                   ,
                   
                   ,
                   
                   ,
                   
                   .  

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