Exercise 16.  Find the image of the semi-infinite vertical strip      under the mapping   .

Solution 16.

Answer.   The image of the semi infinite vertical strip    under the mapping      

is the horizontal strip  .

Hint.   Extend the results in Example 10.13.  You can use the information in Figure 10.17.

Short Solution.   The transformation    maps the semi-infinite vertical strip      

onto the upper half plane  .   Then the transformation      maps the upper half plane  

   onto the horizontal strip   .

Therefore, the image of the semi infinite vertical strip    under the mapping      

is the horizontal strip  .

We might be done.   

Solution.   The mapping    can be written as a composition  

                     ,    

where   

                    ,    and   

                    .

        Using Equation (5-33), we write  

                    .  

If  ,  then the image of the vertical line    is the curve in the Z-plane given by the parametric equations  

                        and   ,    for    .  

The inequalities    imply that the image points lie in the upper half plane  .

Eliminate y from the parametric equations and the result is  .  

When the image curve approaches the ray  .  

When the image curve approaches the ray  .  

        Hence, the image of the semi-infinite strip      under the mapping     

the upper half plane  .

        The upper half plane    can be expressed as

                    .  

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, image of the semi-infinite strip      under the mapping     

is the horizontal strip  .

We are done.   

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

          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