Exercise 18.  Show the following concerning the exponential map  .  

18 (b).  The image of the first quadrant    is the region  .

Solution 18 (b).

See text and/or instructor's solution manual.

Solution.    .  

Since    we have  .

If      then   ,      and we find that

the point      is mapped onto   .

Therefore the image of the first quadrant    is the region  .

We are done.   

Aside.  We can let Mathematica double check our work.

                                   The image the first quadrant   ,   is the region   ,
                                   this mapping is actually infinitely many-to-one  (this is not apparent when you look at the image set).

We are done.   

Reminder.  Since     is periodic with period ,  the mapping is infinitely many to one.

It does not take the entire first quadrant    to map onto the region  ,

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

                    The image the semi-infinite horizontal strip   ,   is the region   ,
                    Since      is periodic with period  ,  this mapping is actually one-to-one.

This solution is complements of the authors.

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