Exercise 16.  Generalize Example 5.2, where the condition    is replaced by  .  Illustrate what this means.  

Solution 16.

See text and/or instructor's solution manual.

Answer.   The image region is basically the same as in Example 5.2 in Section 5.1, as the complex exponential is periodic, with period  .

Solution.   Consider a rectangle  , where  .  

Show that the transformation    maps the rectangle onto a portion of an annular region bounded by two rays.

The image points in the w plane satisfy the following relationships involving the modulus and argument of w:  

            ,   and

            ,  

which is a portion of the annulus    in the w plane subtended by the rays  .   

As an illustration, consider the rectangle   .  

               The rectangle  ,  and its image  .

This solution is complements of the authors.

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