Solution 2.

See text and/or instructor's solution manual.

Answer.   .  

Alternative Answer.   .  

Solution.   Use the Poisson's integral formula, Equation (11-12)   ,   and obtain  

              

Using techniques from calculus we have the integration formulas

                    ,    and  
    
                    ,    and   
    
                    .   

We obtain the solution as follows

              

The function is continuous in the upper half-plane, and on the boundary , except at the discontinuities and    on the real axis.  

We are done.   

Aside.  We can let Mathematica double check our work.

Using the identities    ,     ,       

and    Log,    will convert Mathematica's solution into the desired form:  

                    .  

We are really done.   

Aside.  For illustration purposes we can graph the function   

                    .   

                     A contour graph of   

                        for   .  

                     A contour graph of   

                        for   .  

                    A contour graph of   

                    

We are really really done.   

Aside.  We can let Mathematica check out some boundary values.

We are really really really done.   

The Alternative Solution.

        From Example 11.12, the function  

                      

is harmonic in the upper half-plane  ,  which takes on the boundary values  

                      

                    A graph of the terms   .  

Using techniques from Section 11.2, we find that the function  

                      

is harmonic in the upper half-plane and has the boundary values  ,    ,   and     

                    A graph of the terms   .  

This function can be added to the function   in Example 11.12 to get

                       

                    

                    A graph of the intermediate function   .  

Adjust the variables in   and rescale the output to obtain the desired solution  

                       

                    

Graph Of The Alternative Solution.   For illustration purposes we can graph the alternative solution     

                    .   

                     A contour graph of the alternative solution      

                        for   .  

                     A contour graph of the alternative solution      

                        for   .  

                    A graph of the alternative solution   

                    

This solution is complements of the authors.

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