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

18 (a).  The image of the line  [Graphics:Images/ComplexFunExponentialModHome_gr_534.gif],  where  [Graphics:Images/ComplexFunExponentialModHome_gr_535.gif]  is a spiral.

Solution 18 (a).

See text and/or instructor's solution manual.

Solution.    [Graphics:../Images/ComplexFunExponentialModHome_gr_536.gif].  

Substitute  [Graphics:../Images/ComplexFunExponentialModHome_gr_537.gif]   into the equations  [Graphics:../Images/ComplexFunExponentialModHome_gr_538.gif]  and  [Graphics:../Images/ComplexFunExponentialModHome_gr_539.gif]  and obtain

                    [Graphics:../Images/ComplexFunExponentialModHome_gr_540.gif][Graphics:../Images/ComplexFunExponentialModHome_gr_541.gif]

This set is the parametric representation of a spiral.

We are done.   

Aside.  We can let Mathematica double check our work.

 

[Graphics:../Images/ComplexFunExponentialModHome_gr_542.gif]          [Graphics:../Images/ComplexFunExponentialModHome_gr_543.gif]

                                        The image of a portion of the line   [Graphics:../Images/ComplexFunExponentialModHome_gr_544.gif],
                                        is a portion of the spiral   [Graphics:../Images/ComplexFunExponentialModHome_gr_545.gif].

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This solution is complements of the authors.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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