Exercise 3.  Verify Equations (5-3) and (5-4).  

Solution 3.

See text and/or instructor's solution manual.

Solution.   First prove  (5-3)   [Graphics:../Images/ComplexFunExponentialModHome_gr_42.gif],  if and only if  [Graphics:../Images/ComplexFunExponentialModHome_gr_43.gif],  where  n  is an integer.  

Let n be an integer, and set  [Graphics:../Images/ComplexFunExponentialModHome_gr_44.gif].  

Then  [Graphics:../Images/ComplexFunExponentialModHome_gr_45.gif].   Conversely, suppose  [Graphics:../Images/ComplexFunExponentialModHome_gr_46.gif].  

Then  [Graphics:../Images/ComplexFunExponentialModHome_gr_47.gif].   This implies  [Graphics:../Images/ComplexFunExponentialModHome_gr_48.gif],  this means that  [Graphics:../Images/ComplexFunExponentialModHome_gr_49.gif]  for some integer  n.  

Since  [Graphics:../Images/ComplexFunExponentialModHome_gr_50.gif]  is always positive and  [Graphics:../Images/ComplexFunExponentialModHome_gr_51.gif],  this means that  [Graphics:../Images/ComplexFunExponentialModHome_gr_52.gif]  for some integer  n.  

And  [Graphics:../Images/ComplexFunExponentialModHome_gr_53.gif] implies [Graphics:../Images/ComplexFunExponentialModHome_gr_54.gif] and this forces  [Graphics:../Images/ComplexFunExponentialModHome_gr_55.gif],  so    [Graphics:../Images/ComplexFunExponentialModHome_gr_56.gif].  

This establishes Property  (5-3) .

          Second,  prove  (5-4)   [Graphics:../Images/ComplexFunExponentialModHome_gr_57.gif],  if and only if   [Graphics:../Images/ComplexFunExponentialModHome_gr_58.gif],   for some integer  n.  

Property  (5-4)  comes from observing that   [Graphics:../Images/ComplexFunExponentialModHome_gr_59.gif]  iff  [Graphics:../Images/ComplexFunExponentialModHome_gr_60.gif],   and appealing to Property (5-3)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This solution is complements of the authors. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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