Example 11.13.  Use Poisson's Integral formula to find the harmonic function that is harmonic in the upper half-plane , that takes on the boundary values  

              

Figure 11.15  The graph of .

Explore Solution 11.13.

This is similar to Example 11.17, but the method of solution is different.

Using techniques from Section 11.2, the function    is harmonic in the upper half plane and takes on the boundary values


Thus, we should add it to the solution    in Example 11.12 to obtain the desired result.  However, with Mathematica  we need to use  .

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

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