Example 3.11.  If  ,  then  ;   hence u is a harmonic function for all z.  We find that is also a harmonic function and that

            ,   and   

            .

Therefore v is a harmonic conjugate of u, and the function f given by  

              

is an analytic function.

Explore Solution 3.11.

Enter the functions u[x,y] and v[x,y] and show that the Cauchy-Riemann equations hold.

The Cauchy-Riemann equations hold, therefore    is analytic.  It follows that both    and    are harmonic functions.

Aside.  The underlying complex function is  .  

(c) 2006 John H. Mathews, Russell W. Howell