Exercise 5.  Show that the multivalued function    maps

the annulus    onto the vertical strip   .

Solution 5.

Answer.   The image of    under    is  .  

Solution.   The annulus can be written as  ,

where each point    has infinitely many representations   where  n  is an integer.

Use       and the formulas

                        and    .   

Then      implies that      implies that   ,

which in turn implies that    .   

Also,      implies that   .

        Therefore, the image of the annulus      under the mapping   

is the vertical strip   .  

We are done.   

          The multivalued mapping   .

          Observe the points and their images    

                                                  
          
          where  n  is an integer.  

This solution is complements of the authors.

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