Exercise 4.   Find the electrostatic potential    in the semi-infinite strip   ,  

that has the boundary values,  (shown in Figure 11.42)    

                      

Solution 4.

See text and/or instructor's solution manual.

Answer.   Map the semi-infinite strip onto the upper half plane with the mapping  ,  

then construct  .  

Solution.   The transformation    maps the semi-infinite strip   onto the upper half-plane  .     

                      The mapping   .   

Now construct the intermediate solution    in the upper half w-plane that has the boundary values  

                      

This can be solved using the methods in Section 11.2 for the n-value Dirichlet problem.  

Use the formula  

and the values ,  ,  ,  ,  .  

Substitute and get    the intermediate solution

                     .  

Now substitute    and get the solution in the given domain is  

                     ,    

                     

We are done.   

Aside.  We can let Mathematica double check our work.

Enter the boundary values and construct the  Dirichlet sum.

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 graph of the function   ,  

                      

We are really really really done.   

Aside.   We can explore the intermediate solution   .  

                     A contour graph of the intermediate solution   

                     where      for   .  

                     A graph of the intermediate solution   ,   

                      

This solution is complements of the authors.

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