Exercise 8.   Consider the conformal mapping   .

8 (a).   Show that    maps the domain  D  that is the portion of the disk   ,  

that lies outside the circle      onto the annulus   .  

Solution 8 (a).

See text and/or instructor's solution manual.

Solution.   First, the boundary of the region   can be give a positive orientation by using the points ,  ,  and  .  

The image points are  ,  ,  (where  ),  

and    and give the disk   a positive orientation.  

Next, the boundary of the disk     can be give a positive orientation by using the points ,  ,  and  .  

The image points are  ,  ,  (where  ),  and    

and give the region   a positive orientation.  

        Therefore      maps  the portion of the disk   ,  

that lies outside the circle      onto the annulus   .  

Furthermore,  the circle      is mapped onto the unit circle   ,  

and the  circle      is mapped onto the circle   .    

We are done.   

Aside.  For illustration purposes we can graph the mapping   .   

                    The mapping   .  

We are really done.   

        We can use Mathematica to explore some of the computations.

        The function      maps  

onto   ,   respectively.  

Hence, the circle    is mapped onto the circle  .   

        The function      maps  

onto    ,   respectively.  

Hence, the circle      is mapped onto the circle  .   

        Therefore      maps  the portion of the disk   ,  

that lies outside the circle      onto the annulus   .  

This solution is complements of the authors.

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