Exercise 9.  Find the function    that is harmonic in the upper half-disk    and has the boundary values  

Solution 9.

See text and/or instructor's solution manual.

Answer.   Use the result of Example 11.9 and observe that       

will be zero on the semicircle   

and    on the diameter  .   

Hence the solution is  .  

Solution.   Use the result of Example 11.9 and observe that      

is harmonic in upper half-disk    and has the boundary values   

                      

Hence the solution is  

                        

Therefore,   

                    .  

We are done.   

A More Detailed Solution.   If more details are required, then recall Exercise 4 in Section 10.2 where we showed that that the transformation  

  maps the upper half-disk    onto the first quadrant  .   

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

and the diameter (or segment)    is mapped onto the positive v-axis, i.e.  .  

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

                    ,   for  ,   and      
            
                    ,   for  .        

Applying Example 11.2 in Section 11.1 we know that the form of the solution is  

                    .  

Use the values   ,  and     and write the system of equations  

                    

                    .  

Simplify them and obtain  

                     ,    and    .  

Solving we get      and the desired solution.

Thus,    .   

Since   ,   the solution    in the z-plane in the half-disk  H will be

                    .  

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 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