Exercise 7.  Show that the function    maps the horizontal strip    onto the region  .

Solution 7.

Answer.   The image of      under      is   .  

Short Solution.   The image of     under     is    ,  

then the image of      under      is   .  

Solution.   The mapping      can be written as a composition  

                    ,    

where   

                    ,    and   

                    .

The horizontal strip can be expressed as   .

The mapping    can be written as  

                    ,

where we have   

                        and    .  

Then      implies that     implies that  

which in turn implies that   .    

Also,      implies that   .  

        Hence, the image of the horizontal strip      under the mapping      

is the  the lower half-plane   

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

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

Then      maps      onto   ,  

which is a clockwise orientation for the unit circle   .

        Hence, the image of the lower half-plane      under      is the region   .  

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

then      lies in the region   ,  

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

        Therefore, the image of the horizontal strip    

under the mapping      is the region   .  

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 with limits

                    ,    and

                    .  

This solution is complements of the authors.

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