Exercise 2.  Find the function    that is harmonic in the sector      and has the boundary values  

Solution 2.

See text and/or instructor's solution manual.

Answer.   .   

Solution.   Applying Example 11.2 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

                       

Therefore,   

                    .   

We are done.   

Aside.  We can let Mathematica double check our work.

Aside.   If you prefer to use the function  Arg,  then the solution can be written in the form   

                    .   

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 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