Exercise 18.  Show the following concerning the exponential map  .  

18 (c).  If    is a real constant, the horizontal strip    is mapped one-to-one and onto the nonzero complex numbers.

Solution 18 (c).

See text and/or instructor's solution manual.

Solution.   Suppose  .  If we write w in its exponential form as  ,   identity (5-1) gives
    
            .  

Using property (1-39) of Section 1.5 we have   and , where n is an integer.  

Furthermore, we can find a value of y that lies in the interval  ,  hence

               and   ,     and   .  

Thus we have  .

Therefore,    maps the horizontal strip    one-to-one and onto the set of nonzero complex numbers   .

We are done.   

Aside.  We can let Mathematica double check our work.

                                   The image the horizontal strip  ,  is the set of all nonzero complex numbers   .

                    The image the semi-infinite horizontal strip  ,  is the region   .

                              The image the semi-infinite horizontal strip  ,  is the region   .

This solution is complements of the authors.

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