Theorem 5.1 (The exponential function).  The function    is an entire function satisfying the following conditions:

  (i).   ,  using Leibniz notation  .  

(ii).   ,   i.e.   .  

(iii).   If    is a real number, then   .  

The exponential function is a solution to the differential equation    with the initial condition  .  

Proof of Theorem 5.1.

    By the ratio test, the series in Definition 5.1 has an infinite radius of convergence, so    is an function.  Using termwise differentiation, we get  

            ,  

which gives us part (i) of Theorem 5.1.

To prove part (ii), we let be an arbitrary complex number and define to be

            .

Using the product rule, chain rule, and part (i), we have

            .

Hence the function must be a constant function.  Thus, for all z,  .   Since    (for the reader to verify!), we deduce

             .

Hence, for all z,

            .

Setting    and letting  ,  we get

            .

which simplifies to be (ii) which is our desired result, i.e.  .

To prove part (iii), we let    be a real number.  By Definition 5.1,  

              

(c) 2006 John H. Mathews, Russell W. Howell