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

18 (b).  The image of the first quadrant  [Graphics:Images/ComplexFunExponentialModHome_gr_546.gif]  is the region  [Graphics:Images/ComplexFunExponentialModHome_gr_547.gif].

Solution 18 (b).

See text and/or instructor's solution manual.

Solution.    [Graphics:../Images/ComplexFunExponentialModHome_gr_548.gif].  

Since  [Graphics:../Images/ComplexFunExponentialModHome_gr_549.gif]  we have  [Graphics:../Images/ComplexFunExponentialModHome_gr_550.gif].

If   [Graphics:../Images/ComplexFunExponentialModHome_gr_551.gif]   then   [Graphics:../Images/ComplexFunExponentialModHome_gr_552.gif],   [Graphics:../Images/ComplexFunExponentialModHome_gr_553.gif]   and we find that

the point   [Graphics:../Images/ComplexFunExponentialModHome_gr_554.gif]   is mapped onto   [Graphics:../Images/ComplexFunExponentialModHome_gr_555.gif].

Therefore the image of the first quadrant  [Graphics:../Images/ComplexFunExponentialModHome_gr_556.gif]  is the region  [Graphics:../Images/ComplexFunExponentialModHome_gr_557.gif].

We are done.   

Aside.  We can let Mathematica double check our work.

                              [Graphics:../Images/ComplexFunExponentialModHome_gr_558.gif]          [Graphics:../Images/ComplexFunExponentialModHome_gr_559.gif]

                                   The image the first quadrant   [Graphics:../Images/ComplexFunExponentialModHome_gr_560.gif],   is the region   [Graphics:../Images/ComplexFunExponentialModHome_gr_561.gif],
                                   this mapping is actually infinitely many-to-one  (this is not apparent when you look at the image set).

We are done.   

Reminder.  Since   [Graphics:../Images/ComplexFunExponentialModHome_gr_562.gif]  is periodic with period [Graphics:../Images/ComplexFunExponentialModHome_gr_563.gif],  the mapping is infinitely many to one.

It does not take the entire first quadrant  [Graphics:../Images/ComplexFunExponentialModHome_gr_564.gif]  to map onto the region  [Graphics:../Images/ComplexFunExponentialModHome_gr_565.gif],

and you can choose a portion where is one-to-one, like

                    [Graphics:../Images/ComplexFunExponentialModHome_gr_566.gif]          [Graphics:../Images/ComplexFunExponentialModHome_gr_567.gif]

                    The image the semi-infinite horizontal strip   [Graphics:../Images/ComplexFunExponentialModHome_gr_568.gif],   is the region   [Graphics:../Images/ComplexFunExponentialModHome_gr_569.gif],
                    Since   [Graphics:../Images/ComplexFunExponentialModHome_gr_570.gif]   is periodic with period  [Graphics:../Images/ComplexFunExponentialModHome_gr_571.gif],  this mapping is actually one-to-one.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This solution is complements of the authors.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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