Exercise 12.  Find the image of the right half plane    under the mapping   .  

Solution 12.

Answer.   The image of the right half-plane    under the mapping      

is the vertical strip  .  

Hint.   Extend the results in Example 10.12.

Short Solution.   The image of the right half-plane      under      is the lower half-plane   .  

Then the image of the lower half-plane   ,   under the mapping      is the horizontal strip  

.   Then the image of the horizontal strip   ,  

under the mapping      is the the vertical strip   .  

Therefore, the image of the right half-plane    under the mapping      

is the vertical strip  .  

We might be done.   

Solution.   The mapping      can be written as a composition  

                    ,    

where   

                    ,    

                    ,    and   

                    .  

        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 left orientation for the lower half-plane .  

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

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

then      lies in the lower half-plane .  

        The lower 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 lower half-plane   ,  under the mapping  

  is the horizontal strip  .

        Finally, the mapping    is a rotation of the plane about the origin by counter-clockwise and then shrinking by the factor .

        Hence, the image of the horizontal strip   ,   under the mapping   

is the the vertical strip   .  

        Therefore, the image of the right half-plane    under the mapping      

is the vertical strip  .  

We are done.   

Aside.  We can look at some graphs of the mapping  .

            The mapping   ,   followed by   ,   followed by   .

          The mapping   ,   followed by   ,   followed by   .

          Observe the points and their images  ,    and    

                                          

This solution is complements of the authors.

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