Extra Example 1.  Show that    is a harmonic function and find the harmonic conjugate  .

Explore Extra Solution 1.

Enter the function u[x,y], do a step by step construction.

We are done!

Aside.  We can also construct v[x,y] by the following alternative method.

Enter the function u[x,y], and determine if it is a harmonic function.  If so, proceed with the construction of the harmonic conjugate v[x,y].

Form the analytic function  f[z] = u[x,y]+ i v[x,y]  and verify that the Cauchy-Riemann equations hold.

Therefore   is the harmonic conjugate of  .  

Aside.  We can use Mathematica to get the standard form of an analytic function.  

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