Exercise 20.  Many texts give an alternative definition for  [Graphics:Images/ComplexFunExponentialModHome_gr_643.gif],  starting with (5-1) as the definition for  [Graphics:Images/ComplexFunExponentialModHome_gr_644.gif].  

20 (f).  Identify the general solutions to part (e).

Then, given the initial conditions  [Graphics:Images/ComplexFunExponentialModHome_gr_703.gif],  

find the particular solutions and conclude that Identity (5-1) follows.

Solution 20 (f).

See text and/or instructor's solution manual.

Solution.   The general solution to   [Graphics:../Images/ComplexFunExponentialModHome_gr_704.gif]   is   

                         [Graphics:../Images/ComplexFunExponentialModHome_gr_705.gif],

and the general solution to   [Graphics:../Images/ComplexFunExponentialModHome_gr_706.gif]   is    

                         [Graphics:../Images/ComplexFunExponentialModHome_gr_707.gif],

       From property (3)   [Graphics:../Images/ComplexFunExponentialModHome_gr_708.gif],  we have  [Graphics:../Images/ComplexFunExponentialModHome_gr_709.gif],

and from part (c) we have   [Graphics:../Images/ComplexFunExponentialModHome_gr_710.gif],   and we get
                
[Graphics:../Images/ComplexFunExponentialModHome_gr_711.gif] ,   and we now have   [Graphics:../Images/ComplexFunExponentialModHome_gr_712.gif].   Thus,

                    [Graphics:../Images/ComplexFunExponentialModHome_gr_713.gif].  

       Also, from condition (3)   [Graphics:../Images/ComplexFunExponentialModHome_gr_714.gif],  we have  [Graphics:../Images/ComplexFunExponentialModHome_gr_715.gif],

and from part (c) we have   [Graphics:../Images/ComplexFunExponentialModHome_gr_716.gif],   and we get

[Graphics:../Images/ComplexFunExponentialModHome_gr_717.gif]   and we now have   [Graphics:../Images/ComplexFunExponentialModHome_gr_718.gif].   Thus,

                    [Graphics:../Images/ComplexFunExponentialModHome_gr_719.gif].  

But from part (d)  we have   [Graphics:../Images/ComplexFunExponentialModHome_gr_720.gif],  substitute  [Graphics:../Images/ComplexFunExponentialModHome_gr_721.gif]  and  [Graphics:../Images/ComplexFunExponentialModHome_gr_722.gif],  and get  

                    [Graphics:../Images/ComplexFunExponentialModHome_gr_723.gif].  

Setting  [Graphics:../Images/ComplexFunExponentialModHome_gr_724.gif]  yields  [Graphics:../Images/ComplexFunExponentialModHome_gr_725.gif]  and we get    [Graphics:../Images/ComplexFunExponentialModHome_gr_726.gif].
  
Setting  [Graphics:../Images/ComplexFunExponentialModHome_gr_727.gif]  yields  [Graphics:../Images/ComplexFunExponentialModHome_gr_728.gif]  and we get  [Graphics:../Images/ComplexFunExponentialModHome_gr_729.gif]  and we have  [Graphics:../Images/ComplexFunExponentialModHome_gr_730.gif].  

We have now shown that  [Graphics:../Images/ComplexFunExponentialModHome_gr_731.gif].

Therefore,

          [Graphics:../Images/ComplexFunExponentialModHome_gr_732.gif],   
          
and

          [Graphics:../Images/ComplexFunExponentialModHome_gr_733.gif].

We have now shown that  [Graphics:../Images/ComplexFunExponentialModHome_gr_734.gif].

Congratulations!

The construction of the complex exponential function is now complete.

                    [Graphics:../Images/ComplexFunExponentialModHome_gr_735.gif].  

It has been derived using only the three assumptions:

(1)   [Graphics:../Images/ComplexFunExponentialModHome_gr_736.gif]  is entire,

(2)   [Graphics:../Images/ComplexFunExponentialModHome_gr_737.gif]  for all  z,  and

(3)   [Graphics:../Images/ComplexFunExponentialModHome_gr_738.gif].  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This solution is complements of the authors.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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