Exercise 11.  Find the image of the horizontal strip    under the mapping   .  

Solution 11.

Answer.   The image of the horizontal strip      under     

is the upper half-plane      slit along the ray   .

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

Short Solution.   Use the trigonometric identity  .   

The image of the horizontal strip    under the mapping    is the vertical strip  

.   Then the image of the vertical strip      under the mapping     

is the left half-plane      slit along the ray   .   Then the image of the left half-plane   

   slit along the ray      under the mapping      

is the upper half-plane      slit along the ray   .

Therefore, the image of the horizontal strip      under     

is the upper half-plane      slit along the ray   .

We might be done.   

Solution.   Use the trigonometric identity  .   The mapping      can be written as a composition  

                    ,    

where   

                    ,    

                    ,    and   

                    .  

        Recall that the product    rotates the plane counterclockwise about the origin.

        Hence, the image of the horizontal strip    under the mapping    is the vertical strip  .  

        Next we can use Equation (5-33) to 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 left half-plane  .  

Eliminate Y from the parametric equations and the result is a left branch of the hyperbola   .  

When the image curve approaches the V-axis where  .  

When the image curve approaches the ray  .  

        Hence, the image of the vertical strip    under the mapping    

is the left half-plane    slit along the ray   .

        Recall that the product    rotates the plane clockwise about the origin.

        Hence, the image of the left half-plane     slit along the ray   
        
under the mapping      is the upper half-plane      slit along the ray   .

        Therefore, the image of the horizontal strip      under the mapping     

is the upper half-plane      slit along the ray   .

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