Exercise 7.  Find the function    that is harmonic in the unit disk    and has the boundary values  

Solution 7.

See text and/or instructor's solution manual.

Answer.   Use the result of Example 11.8 and get   .  

Alternative Answer.       .

Solution.   Use the result of Example 11.8 and the function   ,   where

                    

Then it is easy to see that  .  

Therefore,   

                    .  

A More Detailed Solution.   Example 10.3 in Section 10.2 showed that the transformation  

   maps the unit disk onto the upper half-plane .   

Furthermore, the upper semi-circle is mapped onto the positive u-axis, i.e.  ,

and the lower semi-circle is mapped onto the negative u-axis, i.e.  .  

This makes a new boundary value problem in the w-plane   

                    ,   for  ,   and      
            
                    ,   for  .        

Now let   apply the result in Example 11.5.  Then for the boundary value problem

                    ,   for  ,   and      
            
                    ,   for  ,          

has the solution   

                    .  

        Since   ,   the solution    in the z-plane will be

                    .  

Therefore,   

                    .  

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   .  

                    

                    A graph of the function   .  

                    

                      A graph of the function   .  

                    

We are really really really done.   

Alternative Solution.  If the trigonometric identity    is used then we get:  

                       

                     A contour graph of the function   

                     where      for   .  

                    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