Exercise 20.  Many texts give an alternative definition for  ,  starting with (5-1) as the definition for  .  

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

Then, given the initial conditions  ,  

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      is   

                         ,

and the general solution to      is    

                         ,

       From property (3)   ,  we have  ,

and from part (c) we have   ,   and we get
                
,   and we now have   .   Thus,

                    .  

       Also, from condition (3)   ,  we have  ,

and from part (c) we have   ,   and we get

   and we now have   .   Thus,

                    .  

But from part (d)  we have   ,  substitute    and  ,  and get  

                    .  

Setting    yields    and we get    .
  
Setting    yields    and we get    and we have  .  

We have now shown that  .

Therefore,

          ,   
          
and

          .

We have now shown that  .

Congratulations!

The construction of the complex exponential function is now complete.

                    .  

It has been derived using only the three assumptions:

(1)     is entire,

(2)     for all  z,  and

(3)   .  

This solution is complements of the authors.

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