Exercise 4.  Find and illustrate the images of the following sets under the mapping  .  

4 (a).  .  

Solution 4 (a).

See text and/or instructor's solution manual.

Answer.   .  

Solution.  Use polar coordinates    in the z-plane and    in the  w-plane.  The mapping    can be written as  

        ,   

the point    in the xy-plane is mapped to the point    in the uv-plane,

and we get the equations    and  .  

Using the equations    and  ,  we find that the image of   is

  which can be written as  .  

We are done.   

Aside.  We can let Mathematica double check our work.

                    The image of    under    is  .  

This solution is complements of the authors.

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