Extra Example 1.  Find the function that is harmonic in the upper half-plane , which takes on the boundary values  

              

Explore Extra Solution 1.

Enter the function U[t] and use the Poisson integral to construct  .  

We can verify some of the boundary values by taking limits.

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