Exercise 3.  Find the function    that is harmonic in the annulus    and has the boundary values  

Solution 3.

See text and/or instructor's solution manual.

Answer.   .  

Solution.   Applying Example 11.3 in we know that the form of the solution is  

                    .  

Use the values   ,   and     and     and write the system of equations  

                    .  

                    .  

Solving we get      and the desired solution   

                      

Therefore,   

                    .   

We are done.   

Aside.  We can let Mathematica double check our work.

Aside.   If polar coordinates      are used, then the polar form of the solution is     

                    .   

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