Exercise 6.  Show that     maps the portion of the right half-plane    

that lies to the right of the hyperbola      onto the unit disk   .

Solution 6.

Answer.   The image of   under    is  .

Short Solution.   The image of      under      is   ,   

then the image of      under      is the unit disk   .  

Solution.   The portion of the right half-plane      that lies to the right of the hyperbola    is the set

                    .  

The mapping    can be written as a composition  

                     ,    

where   

                    ,    and   

                    .

We can write  ,  as  

                          and    .

The mapping      is two to one and the image      is   .

and so is the image of   .

        Hence, the image of    under    is the right half plane  .  

        The boundary of the right half-plane      is the vertical line   ,  

and we can give the right half-plane      a left orientation by using the points   .

Then      maps        onto    ,  

which is a positive orientation for the unit circle  .

        Hence, the image of the right half-plane      is the unit disk   .  

Furthermore, as a double-check we can choose the point    in the right half-plane  ,  

then    lies in the unit disk  ,

which leads us to conclude that the image region lies inside the unit circle  .

        Therefore, the image of  

under the mapping   is the unit disk  .

We are done.   

          The mapping   ,   followed by the mapping   .

          Observe the points and their images    and    

                                                              

This solution is complements of the authors.

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