Example 11.11.  Find the function that is harmonic in the upper half-plane , which takes on the boundary values  

              

Explore Solution 11.11.

The solution is similar to Example 11.7, but the method of solution is different.

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

Using the trigonometric identity  , the above result can be written as  .  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