Exercise 7.  Consider the conformal mapping   .  

7 (a).   Show that    maps the domain  D  that is the portion of the right half-plane   ,  

that lies exterior to the circle      onto the annulus   .  

Solution 7 (a).

See text and/or instructor's solution manual.

Solution.   First, the boundary of the right half-plane    can be give a positive orientation by using the points ,  ,  and  .  

The image points are  ,  ,  and  ,  

and give the disk   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  ,  ,  (where  ),  and    

and give the region   a positive orientation.  

        Therefore      maps the portion of the right half-plane     

that lies exterior to the circle       onto the annulus  .  

Furthermore,  the imaginary axis      is mapped onto the circle   ,  

and the  circle      is mapped onto the unit 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 right half-plane    is mapped onto the circle  .   

        The function      maps  

onto    ,   respectively.  

Hence, the circle      is mapped onto the circle  .   

        Therefore      maps the portion of the right half-plane     

that lies exterior to the circle       onto the annulus  .  

This solution is complements of the authors.

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