Exercise 9.  Show that the function      maps the rectangle      one-to-one and onto

the portion of the upper half plane      that lies inside the ellipse   .

Solution 9.

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

Solution.   Using Equation (5-33), we write  

                    .  

If  ,  then the image of the vertical line    is the curve in the w 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      under     the upper half-plane   .   

        Next, using Equation (5-33), we write  

                    .  

        If  ,  then the image of the horizontal segment    is

the curve in the w-plane given by the parametric equations  

                    ,

We rewrite them as  

                    .  

We now eliminate x from the equations by squaring and using the trigonometric identity  .  The result is the single equation

                    .

This curve is an ellipse in the w-plane that passes through the points    and   and has foci at the points .  

The inequality      implies that      and   ,   and we can conclude

that the ellipse      lies inside the bounding ellipse   .  

        Hence, the image of   ,  lies inside the bounding ellipse   .  

        Therefore, the image of the rectangle      under the transformation      is  

                    ,  

which is the portion of the upper half plane    that lies inside the ellipse   .  

We are done.   

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

                                        The mapping   .

                                        Observe the points and their images    

                                                    

This solution is complements of the authors.

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