5.1 The Complex Exponential Function

    Recall that the real exponential function can be represented by the power series  .  Thus it is only natural to define the complex exponential  ,  also written as , in the following way:

Definition 5.1 (Exponential Function).  The definition of  exp(z) is  

            .  

Demonstration for Definition 5.1.  

We can use Mathematica to sum a power series.

The ratio test can be used to show that the infinite series converges for all values of   z.

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