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

  (i).   ,  using Leibniz notation  .  

Demonstration for Theorem 5.1 (i).

Enter the D.E. with the given initial condition and solve it.
Then enter the function  ,  compute  .  

Construct the Maclaurin series for  ,  compute  .  

Determine the real and imaginary parts of  .  

Determine the real and imaginary parts of the series .  

Compare the above results with the two variable Taylor series expansions for  .  

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