Exercise 6.  Find the function    that is harmonic in the first quadrant    and has the boundary values  

Solution 6.

See text and/or instructor's solution manual.

Answer.   ,    which can be written as   

                 .

Solution.   The transformation      maps the first quadrant onto the upper half-plane  .   Furthermore,

the segment    is mapped onto the segment  ,  

the segment    is mapped onto the segment  ,  

the ray    is mapped onto the ray  ,  

and the ray    is mapped onto the ray  .  

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

                      

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

The solution in the w-plane is similar to Example 11.7.

                      

Here we have    and  ,  which we substitute into the above equation for    to obtain  

                       

Hence the the solution in the z-plane is   

                      

Therefore,   

                    .   

Remark.   For some situations it might be useful to write this solution in the following form
        
          

        .  

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.   

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     

                       

This solution is complements of the authors.

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