Exercise 6.  Find the image of the semi-infinite strip    under the mapping   .

Solution 6.

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

is the second quadrant  .  

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  ,  and    imply that

the image points lie in the second quadrant  .  

Eliminate y from the parametric equations and the result is  .  

When the image curve approaches the positive v axis where  .  

When the image curve approaches the ray  .  

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

is the second quadrant  .  

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