Exercise 1.  Find the image of the semi-infinite strip      under the mapping   .  

Solution 1.

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

,   which is the portion of the disk    that lies in the second quadrant  .  

Hint.   Extend the results in Example 10.12.

Short Solution.   The image of the semi-infinite strip      under the mapping      is

.   

Then, the image of   ,   under      is  ,   

which is the portion of the disk    that lies in the second quadrant  .  

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

,   which is the portion of the disk    that lies in the second quadrant  .  

We might be done.   

Solution. Method I.   In Example 10.12 we showed that the mapping    can be written as the composition  

                     ,    

where   

                    ,    and   

                    .

        Recall that the product    rotates the plane and magnifies by a factor of  .  

For the mapping  ,   also recall that   

                    .  

Then      implies that      implies that      implies that   .   

Also, recall that   .

Then      implies that      implies that      implies that   .

        Hence, the image of   ,   under      is

                    ,

which is the portion of the disk    that lies in the fourth quadrant  .  

        Now find the inverse transformation for   .

Apply equations (10-13)  and  (10-14)  in the form

(10-13)             ,

(10-14)             .

Here we have    and   .  

Then  

                        

Thus, the inverse transformation is   .  

Then get  

                       

Here we have

                        and    .  

Then      implies      implies

                       

Then get

                    

                    

                    

                    

                    

                    

Thus      implies that   .  

Also,      implies       implies    implies     implies   

Also,      implies       implies      implies   .

        Hence, the image of   ,   under      is

,   which is the portion of the disk    that lies in the second quadrant  .  

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

,   which is the portion of the disk    that lies in the second quadrant  .  

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    

                                                              

We are really done.  

Solution. Method II.   In Example 10.12 we showed that the mapping    can be written as the composition  

                     ,    

where   

                    ,    and   

                    .

        Recall that the product    rotates the plane and magnifies by a factor of  .  

For the mapping  ,   also recall that   

                    .  

Then      implies that      implies that      implies that   .   

Also, recall that   .

Then      implies that      implies that      implies that   .

        Hence, the image of   ,   under      is

                    ,

which is the portion of the disk    that lies in the fourth quadrant  .  

        Next we will find the image of      under   . This will require that we  

find the images of the right half-plane  ,  the unit disk  ,  and the lower half-plane  .

(i).    The right half-plane    can be given a left orientation with the points   .

Then      maps        onto    ,  

which is a left orientation for the unit disk  .

(ii).   The unit disk    can be given a left orientation with the points  .  

Then      maps        onto    

which is a left orientation of the upper half-plane  .

(iii).  The lower half-plane    can be given a left orientation with the points  .  

Then      maps      onto   

which is a left orientation for the left half-plane  .   

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

the intersection of the three image sets in (i), (ii), and (iii), which is  .

which is the portion of the disk    that lies in the second quadrant  .  

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

,   which is the portion of the disk    that lies in the second quadrant  .  

We are really really 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