Exercise 8.  Find the function    that is harmonic in the unit disk    and has the boundary values  

Solution 8.

See text and/or instructor's solution manual.

Answer.   .  

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  .        

Apply Theorem 11.2 to construct a Dirichlet solution in the upper half-plane

use formula (11-5)      with   ,  

                    .  

Now we substitute      and      to obtain  

                    .  

        Since   ,   the solution    in the z-plane will be

                    .  

Therefore,   

                    .  

We are done.   

Aside.  We can let Mathematica double check our work.

Enter the boundary values and construct the Dirichlet form of the solution.

We are really done.   

Remark.   For some situations it might be useful to use the trigonometric identity  .

This gives

                    

which is a version that plots the boundary values better.  

Therefore,   

                    .  

We are really really done.   

Aside.  For illustration purposes we can graph the function   .   

                     A contour graph of the function   

                     where      for   .  

We are really 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   ,  

                       

This solution is complements of the authors.

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