Extra Example 2.  Use Poisson's Integral formula to find the harmonic function that is harmonic in the upper half-plane , that takes on the boundary values  

              

Explore Extra Solution 2.

Use Mathematica to make a contour plot of the solution.

Then use Mathematica to make a 3D plot of the solution.

Therefore, the function is harmonic in the upper half-plane  ,  and takes on the desired boundary values.

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