Exercise 10.  Find the image of the vertical strip      under the mapping   .

Solution 10.

Answer.   The image of the vertical strip      under the mapping     

is the right 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.  The image of the vertical strip    under    is the vertical strip  

.   Then the image of the vertical strip      under the mapping   

  is the right half-plane      slit along the ray   .   

Therefore, the image of the vertical strip      under the mapping      is

the right 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   

                    .

        The mapping      translates points in the plane to the right by the amount  .  

        Hence, the image of the vertical strip    under    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 right half-plane  .  

Eliminate Y from the parametric equations and the result is  .  

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 right half-plane    slit along the ray  .

        Therefore, the image of the vertical strip      under the mapping   

is the right half-plane      slit along the ray   .

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