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

18 (c).  If  [Graphics:Images/ComplexFunExponentialModHome_gr_572.gif]  is a real constant, the horizontal strip  [Graphics:Images/ComplexFunExponentialModHome_gr_573.gif]  is mapped one-to-one and onto the nonzero complex numbers.

Solution 18 (c).

See text and/or instructor's solution manual.

Solution.   Suppose  [Graphics:../Images/ComplexFunExponentialModHome_gr_574.gif].  If we write w in its exponential form as  [Graphics:../Images/ComplexFunExponentialModHome_gr_575.gif],   identity (5-1) gives
    
            [Graphics:../Images/ComplexFunExponentialModHome_gr_576.gif].  

Using property (1-39) of Section 1.5 we have  [Graphics:../Images/ComplexFunExponentialModHome_gr_577.gif] and [Graphics:../Images/ComplexFunExponentialModHome_gr_578.gif], where n is an integer.  

Furthermore, we can find a value of y that lies in the interval  [Graphics:../Images/ComplexFunExponentialModHome_gr_579.gif],  hence

            [Graphics:../Images/ComplexFunExponentialModHome_gr_580.gif]   and   [Graphics:../Images/ComplexFunExponentialModHome_gr_581.gif],     and   [Graphics:../Images/ComplexFunExponentialModHome_gr_582.gif].  

Thus we have  [Graphics:../Images/ComplexFunExponentialModHome_gr_583.gif].

Therefore,  [Graphics:../Images/ComplexFunExponentialModHome_gr_584.gif]  maps the horizontal strip  [Graphics:../Images/ComplexFunExponentialModHome_gr_585.gif]  one-to-one and onto the set of nonzero complex numbers   [Graphics:../Images/ComplexFunExponentialModHome_gr_586.gif].

We are done.   

Aside.  We can let Mathematica double check our work.

                              [Graphics:../Images/ComplexFunExponentialModHome_gr_587.gif]          [Graphics:../Images/ComplexFunExponentialModHome_gr_588.gif]

                                   The image the horizontal strip  [Graphics:../Images/ComplexFunExponentialModHome_gr_589.gif],  is the set of all nonzero complex numbers   [Graphics:../Images/ComplexFunExponentialModHome_gr_590.gif].

 

 

                              [Graphics:../Images/ComplexFunExponentialModHome_gr_591.gif]          [Graphics:../Images/ComplexFunExponentialModHome_gr_592.gif]

                    The image the semi-infinite horizontal strip  [Graphics:../Images/ComplexFunExponentialModHome_gr_593.gif],  is the region   [Graphics:../Images/ComplexFunExponentialModHome_gr_594.gif].

 

 

                              [Graphics:../Images/ComplexFunExponentialModHome_gr_595.gif]          [Graphics:../Images/ComplexFunExponentialModHome_gr_596.gif]

                              The image the semi-infinite horizontal strip  [Graphics:../Images/ComplexFunExponentialModHome_gr_597.gif],  is the region   [Graphics:../Images/ComplexFunExponentialModHome_gr_598.gif].

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This solution is complements of the authors.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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