# Definition:Exponential Function/Complex/Sum of Series

## Definition

Let $\exp: \C \to \C \setminus \set 0$ denote the (complex) exponential function.

The exponential function can be defined as a (complex) power series:

 $\displaystyle \exp z$ $=$ $\displaystyle \sum_{n \mathop = 0}^\infty \frac {z^n} {n!}$ $\displaystyle$ $=$ $\displaystyle 1 + \frac z {1!} + \frac {z^2} {2!} + \frac {z^3} {3!} + \cdots + \frac {z^n} {n!} + \cdots$

The complex number $\exp z$ is called the exponential of $z$.