Exercise 18.  Show the following concerning the exponential map  [Graphics:Images/ComplexFunExponentialModHome_gr_533.gif].  

18 (e).  The image of the horizontal ray  [Graphics:Images/ComplexFunExponentialModHome_gr_615.gif]  is a ray.

Solution 18 (e).

See text and/or instructor's solution manual.

Solution.    [Graphics:../Images/ComplexFunExponentialModHome_gr_616.gif].  

Substitute  [Graphics:../Images/ComplexFunExponentialModHome_gr_617.gif]   into the equations  [Graphics:../Images/ComplexFunExponentialModHome_gr_618.gif]  and  [Graphics:../Images/ComplexFunExponentialModHome_gr_619.gif]  and obtain

                    [Graphics:../Images/ComplexFunExponentialModHome_gr_620.gif].

This set is the parametric representation the ray  [Graphics:../Images/ComplexFunExponentialModHome_gr_621.gif].

We are done.   

Aside.  We can let Mathematica double check our work.

                              [Graphics:../Images/ComplexFunExponentialModHome_gr_622.gif]          [Graphics:../Images/ComplexFunExponentialModHome_gr_623.gif]

                              The image the horizontal ray  [Graphics:../Images/ComplexFunExponentialModHome_gr_624.gif],  is the ray   [Graphics:../Images/ComplexFunExponentialModHome_gr_625.gif].

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This solution is complements of the authors.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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