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

Solution 1.

Answer.   The image of   ,   under      is   .  

Short Solution.   The image of      under      is   ,

then the image of      under      is   .  

Solution.   The mapping    can be written as a composition  

                    ,    

where   

                    ,    and   

                    .

The mapping        rotates the complex plane    counterclockwise about the origin.  

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

                    .

        For the mapping  ,   recall that      

then       implies that   ,   thus   .   

Also, recall that      then       implies that   .  

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

                    .  

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

is the portion of the disk    that lies in the first quadrant  ,    i. e.   .  

We are done.   

          The mapping   ,   followed by the mapping   .

          Observe the points and their images    and    

                                                              

We are really done.   

Aside.   If it is important then we can include the following details.

          Here the value        can be calculated with a limit

                     .  

          Similarly, the value        can be calculated with a limit

                     .  

This solution is complements of the authors.

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