Exercise 19.  Explain how the complex function  [Graphics:Images/ComplexFunExponentialModHome_gr_626.gif]  and the real function  [Graphics:Images/ComplexFunExponentialModHome_gr_627.gif]  are different.  How are they similar?  

Solution 19.

See text and/or instructor's solution manual.

Solution.   Let   [Graphics:../Images/ComplexFunExponentialModHome_gr_628.gif]   and   [Graphics:../Images/ComplexFunExponentialModHome_gr_629.gif].  

They are similar in the following:

          [Graphics:../Images/ComplexFunExponentialModHome_gr_630.gif]   and   [Graphics:../Images/ComplexFunExponentialModHome_gr_631.gif].  

          [Graphics:../Images/ComplexFunExponentialModHome_gr_632.gif]   and   [Graphics:../Images/ComplexFunExponentialModHome_gr_633.gif].  

          [Graphics:../Images/ComplexFunExponentialModHome_gr_634.gif]   and   [Graphics:../Images/ComplexFunExponentialModHome_gr_635.gif].

          [Graphics:../Images/ComplexFunExponentialModHome_gr_636.gif]   and   [Graphics:../Images/ComplexFunExponentialModHome_gr_637.gif]

          [Graphics:../Images/ComplexFunExponentialModHome_gr_638.gif]   and   [Graphics:../Images/ComplexFunExponentialModHome_gr_639.gif]

          The rules for exponents apply to both functions.

They are different in the following regard:

          [Graphics:../Images/ComplexFunExponentialModHome_gr_640.gif]  is periodic with period  [Graphics:../Images/ComplexFunExponentialModHome_gr_641.gif].

          [Graphics:../Images/ComplexFunExponentialModHome_gr_642.gif]  maps the two dimensional complex z-plane onto the two dimensional w-plane.

Can you think of any more?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This solution is complements of the authors.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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