Exercise 16.  Generalize Example 5.2, where the condition  [Graphics:Images/ComplexFunExponentialModHome_gr_494.gif]  is replaced by  [Graphics:Images/ComplexFunExponentialModHome_gr_495.gif].  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  [Graphics:../Images/ComplexFunExponentialModHome_gr_496.gif].

Solution.   Consider a rectangle  [Graphics:../Images/ComplexFunExponentialModHome_gr_497.gif], where  [Graphics:../Images/ComplexFunExponentialModHome_gr_498.gif].  

Show that the transformation  [Graphics:../Images/ComplexFunExponentialModHome_gr_499.gif]  maps the rectangle [Graphics:../Images/ComplexFunExponentialModHome_gr_500.gif] 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:  

            [Graphics:../Images/ComplexFunExponentialModHome_gr_501.gif],   and

            [Graphics:../Images/ComplexFunExponentialModHome_gr_502.gif],  

which is a portion of the annulus  [Graphics:../Images/ComplexFunExponentialModHome_gr_503.gif]  in the w plane subtended by the rays  [Graphics:../Images/ComplexFunExponentialModHome_gr_504.gif].   

As an illustration, consider the rectangle   [Graphics:../Images/ComplexFunExponentialModHome_gr_505.gif].  

                    [Graphics:../Images/ComplexFunExponentialModHome_gr_506.gif]          [Graphics:../Images/ComplexFunExponentialModHome_gr_507.gif]

               The rectangle  [Graphics:../Images/ComplexFunExponentialModHome_gr_508.gif],  and its image  [Graphics:../Images/ComplexFunExponentialModHome_gr_509.gif].

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This solution is complements of the authors.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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