Example 3.12. The function    is analytic for all values of z, hence it follows that  

              
is harmonic, and
              
        
is a harmonic conjugate of u(x,y).

Explore Solution 3.12.

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