Solution 3.

See text and/or instructor's solution manual.

Answer.   Both      and      are harmonic in the upper half plane and satisfy the boundary conditions   .

Also,   .

The Poisson integral formula defines a bounded function in the upper half plane, therefore the desired solution is   .  

Solution.   The Poisson's integral formula     defines a harmonic in the upper half plane and has the boundary values  

                    ,  

wherever is continuous.   Hence,  

                    ,

is harmonic in the upper half plane and has the boundary conditions .

There are two possible choices for  ,  both

                      
                    
are harmonic in the upper half plane and satisfy the boundary conditions .

Also,      all values of  x,  but      does not go to zero for all values of  x.  

The Poisson integral formula defines a bounded function in the upper half plane, therefore the desired solution is the choice  

                    .  

We are done.   

Aside.  We can graph the solution   .  

                     A contour graph of the function     

                     where      for   .  

                     A graph of the function   .   

We are really done.   

Aside.  We can let Mathematica 4.0 investigate this situation.

We are really really done.   

Aside.  We can construct an integral.

The special functions  CosIntegral  and  SinIntegral  are not part of our course.  

Aside.  We can construct another integral.

Aside.  If you use Mathematica 7.0 you will get a slightly different answer.

              

            

We can expand the integral and get   

The imaginary part is identically zero in the upper half-plane.  

The functions    and    take on the values

                    

Thus, for values of z in the upper half-plane the real part becomes

                    

Unfortunately, it's value is zero on the real axis.

The software Maple will construct an equally complicated expression for the integral   .  

Caveat.  Be careful when using computer algebra systems, often times they might use "special functions."

This solution is complements of the authors.

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