Exercise 8.  Show that    maps the horizontal strip      onto the unit disk   .

Solution 8.

Answer.   The image of the horizontal strip      under    is the unit disk   .  

Short Solution.   The image of      under      is   ,   

then the image of      under      is   .  

Solution.   The mapping      can be written as a composition  

                    ,    

where   

                    ,    and   

                    .

Recall that        and    .   

Then      implies that  .

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

The boundary of the right half-plane      is the imaginary axis   ,  

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      under      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 the horizontal strip    under    is the unit disk  .  

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.

                    ,  

                    ,  

                    ,  

                    .  

This solution is complements of the authors.

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