Exercise 7.  Find the image of    under the following mappings.  

7 (a).  .

Solution 7 (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  .  

The image of  set    under the the mapping    satisfies

        ,  and  
        
          or  .  

Hence, the image set is  .

We are done.   

Aside.  We can let Mathematica double check our work.

                    The image of the portion under    is  .

This solution is complements of the authors.

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