Exercise 5.   Find the electrostatic potential    in the domain  D  in the half-plane   ,  

that lies to the left of the hyperbola   ,   

and satisfies the following boundary values,  (shown in Figure 11.43)  

                      

Solution 5.

See text and/or instructor's solution manual.

Answer.   Map the given region onto a vertical strip with the mapping  ,  

then construct  .   

Solution.   The transformation    maps the domain  D  in the half-plane   ,  

that lies to the left of the hyperbola      onto the infinite strip   .

                      The mapping   .  

Now construct the intermediate solution    in the infinite strip in the w-plane that has the boundary values

                      

Applying the method in Example 11.1 in Section 11.1, the solution takes on constant values along the vertical lines and has the form  

                      

Substitute    and obtain the intermediate solution

                       

Now use      and      and construct   .  

Therefore,  

                    .   

We are done.   

        For computational purposes we can use the formulas for the real and imaginary parts of  ,  that were derived in Section 10.4.

                    .  
                    
In particular,

(10-26)          .  

Therefore,  

                    .   

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   ,   

                      

                     A graph of the function   ,   

                      

This solution is complements of the authors.

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