Exercise 8.   Consider the conformal mapping   .

8 (b).   Find the electrostatic potential    in the domain  D  that satisfies the boundary values,  (shown in Figure 11.46)  

                      

Solution 8 (b).

See text and/or instructor's solution manual.

Answer.   Use the results of part 8 (a).

Map the given region onto the annulus    with the mapping  ,  

then construct   .  

Solution.   The transformation    maps the domain  D  that is the portion of the disk   ,  

that lies outside the circle      onto the annulus   .  

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

                      

Applying Example 11.3 in Section 11.1 we know that the form of the intermediate solution is  

                    .  

Substitute      and get    

                      

Thus, the intermediate solution is  

                    .   

To find the solution in the z-plane make the substitution   .

Therefore,

                    ,  

                    .  

We are done.   

Aside.  For illustration purposes we can graph the function   .   

                     A contour graph of the function   

                     where      for   .  

We are really done.   

                     A graph of the function   ,   

                      

                     A graph of the function   ,  

                      

                     A graph of the function   ,  

                      

We are really really done.   

        It is possible to expand the quantity     as follows

                    .  

Then

                    .  

Therefore,

                    .  

Aside.  For illustration purposes we can graph the function   .   

                     A contour graph of the function   

                     where      for   .  

                     A graph of the function   ,  

                      

                     A graph of the function   ,  

                      

                     A graph of the function   .  

                     In Cartesian coordinates   ,  

                      

We are really really really done.   

Aside.   We can graph the intermediate solution   .  

                     A contour graph of the intermediate solution   

                     where      for   .  

                     A graph of the intermediate solution   ,   

                      

                     A graph of the intermediate solution   ,   

                      

                     A graph of the intermediate solution   .   

                     In Cartesian coordinates   ,   

                      

This solution is complements of the authors.

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