Exercise 3.   Find the electrostatic potential    in the crescent-shaped region

that lies inside the disk      and outside the circle   ,   

that satisfies the following boundary values,  (shown in Figure 11.41)     

                      

Solution 3.

See text and/or instructor's solution manual.

Answer.   Map the crescent-shaped region onto a vertical strip with the mapping  ,  

then construct  .  

Solution.     First, find the image of the disk    under the mapping  .  

The circle     can be give a positive orientation by using the points ,  ,  and   .  

The image points are  ,  ,  and  ,  

and they give the right half-plane    a positive orientation.   

Next, consider region that lies exterior to the circle   .    

The boundary    can be give a positive orientation by using the points ,  ,  and  .  

The image points are  ,  ,  and    

and give the left half-plane    a positive orientation.   

        Therefore      maps the the crescent-shaped region that lies inside the disk      

and outside the circle   ,    onto the vertical strip  .  

Furthermore,  the circle      is mapped onto the line   ,  

and the  circle      is mapped onto the line   .   

                      The mapping   .   

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

                       for  ,  and  

                       for  .  

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.   

Aside.  We can let Mathematica double check our work.

Check out the boundary values   

.  

Check out the boundary values   

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 plot for    where  .  

We are really really done.   

                    A graph of   ,   

                      

                    A graph of   ,   

                      

                    A graph of   .   

                    In Cartesian coordinates   ,   

                      

We are really really really done.   

        We can use Mathematica to explore some of the computations.

        The function      maps  

onto   ,   respectively.  

Hence, the  circle      is mapped onto the line   .   

        The function      maps  

onto    ,   respectively.  

Hence, the circle      is mapped onto the line   ,

        Therefore      maps the the crescent-shaped region that lies inside the disk      

and outside the circle   ,    onto the vertical strip  .  

This solution is complements of the authors.

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