Solution 1.

See text and/or instructor's solution manual.

Answer.   .  

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

              

Using techniques from calculus we have the integration formulas

            ,   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    ,   will convert Mathematica's solution into  

                    .  

We are really done.   

Aside.  For illustration purposes we can graph the function   .   

                     A contour graph of the function   

                        for   .  

                     A contour graph of the function   

                        for   .  

                     A graph of the function   

                       

We are really done.   

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

This solution is complements of the authors.

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