Exercise 5.  Find the function    that is harmonic in the upper half-plane      and has the boundary values  

Solution 5.

See text and/or instructor's solution manual.

Answer.   ,   which can be written as   

                 .  

Solution.   Apply Theorem 11.2 to construct a Dirichlet solution in the upper half-plane

use formula (11-5)      with   ,  

                    .  

Now we substitute      and      to obtain  

                       

Therefore,   

                    .

We are done.   

Aside.  We can let Mathematica double check our work.

Enter the boundary values and construct the Dirichlet form of the solution.

We are really done.   

Aside.  For illustration purposes we can graph the function   .   

                     A contour graph of the function     

                     where     for   .   

We are really really done.   

                     A contour graph of the function     

                     where      for   .   

                     A contour graph of the function     

                     where      for   .   

                    A graph of the function     

                       

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell