Exercise 10.  Find the function    that is harmonic in the portion of the upper half-plane    

that lies outside the circle     and has the boundary values

Solution 10.

See text and/or instructor's solution manual.

Answer.   .  

Solution.   Use the result of Example 11.9,  and the function  

                    ,    which has the boundary values

                      
                    
                    

Then it is easy to see that  

                        has the boundary values

                      
                    
                    

        Apply the mapping    and find the image the region   .    

                      The image the region      under the mapping   

                    is the upper half disk   .  

Observe that the upper semi-circle    is mapped onto itself.  This can be seen by considering the curve

                       for    

which parameterizes the upper semi-circle starting at  ,  passing through   ,   and ending up at  .     

Furthermore, the ray    is mapped onto the segment    ,

and the ray    is mapped onto the segment    .  

        Substituting     in     produces the desired function.  

Therefore,    has the boundary values

                    

We can manipulate the quantity as follows:

                        

Therefore,    

                    .  

Use the trigonometric identity    and write

                      

Therefore,    

                    .  

We are done.   

Aside.  We can let Mathematica double check our work.

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 graph of the function   .   

                    

                    A graph of the function   .   

                    

                    A graph of the function   .  

                    

This solution is complements of the authors.

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