Exercise 6.  Prove that   [Graphics:Images/ComplexFunExponentialModHome_gr_274.gif]   for all  z.  Where does equality hold?  

Solution 6.

See text and/or instructor's solution manual.

Solution.   Set  [Graphics:../Images/ComplexFunExponentialModHome_gr_275.gif].  We have  [Graphics:../Images/ComplexFunExponentialModHome_gr_276.gif]  and  

                    [Graphics:../Images/ComplexFunExponentialModHome_gr_277.gif][Graphics:../Images/ComplexFunExponentialModHome_gr_278.gif],

also
                    [Graphics:../Images/ComplexFunExponentialModHome_gr_279.gif].  

Since  [Graphics:../Images/ComplexFunExponentialModHome_gr_280.gif]  and for real numbers t,  [Graphics:../Images/ComplexFunExponentialModHome_gr_281.gif]  is a non-negative increasing function, we have  [Graphics:../Images/ComplexFunExponentialModHome_gr_282.gif].

Therefore,   [Graphics:../Images/ComplexFunExponentialModHome_gr_283.gif].

We will have equality when    [Graphics:../Images/ComplexFunExponentialModHome_gr_284.gif]   iff    [Graphics:../Images/ComplexFunExponentialModHome_gr_285.gif]    iff    [Graphics:../Images/ComplexFunExponentialModHome_gr_286.gif].

Therefore,   [Graphics:../Images/ComplexFunExponentialModHome_gr_287.gif]  precisely when  [Graphics:../Images/ComplexFunExponentialModHome_gr_288.gif],  i.e., when z is a real number. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This solution is complements of the authors.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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