Exercise 9.  Find the image of the upper half-plane      under   .  

Solution 9.

Remark.  Compare this exercise with Exercise 10.

Answer.   The image of    under    is the horizontal strip  .  

Short Solution.   The image of     under      is the upper half-plane   ,  

then the image of    ,  under       is the horizontal strip  .  

Solution.   The mapping    can be written as a composition  

                    ,    

where   

                    ,    and   

                    .

        The boundary of the upper half-plane      is the real axis   ,  

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

Then      maps        onto    ,  

which is a positive orientation for the real axis   .

        Hence, the image of the upper half-plane      under      is the upper half-plane   .
        
Furthermore, as a double-check we can choose the point    in the upper half plane  ,  

then    lies in the upper half-plane   ,

which leads us to conclude that the image region lies above the real axis  .

        The upper half-plane      can be written in the form   .

Recall that      which can be written as  

                       and   .

Then,      implies that      which in turn implies that   .   

Also,      implies that   .   

        Hence, the image of the upper half-plane   ,  under the mapping  

  is the horizontal strip  .

        Therefore, the image of the upper half-plane   ,   

under the mapping      is the horizontal strip   .

We are done.   

          The mapping   ,   followed by the mapping   .

          Observe the points and their images    and    

                                                              

We are really done.   

Aside.   If it is important then we can include the following details.

          Here the values    ,    ,    and        are calculated as limits.

                    ,  

                    ,  

                    .  

          Similarly, the values    ,    ,    and        are calculated as limits.

                    ,  

                    ,  

                    .  

This solution is complements of the authors.

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