Complex exponential

This article is about a particular entire function: a holomorphic function defined on the whole of $\mathbb {C}$
View a complete list of entire functions

Definition

The complex exponential is a holomorphic function from $\mathbb {C}$ to $\mathbb {C}$ (i.e. an entire function). The complex exponential of $z$ is denoted as $\exp(z)$ and also as $e^{z}$. It is defined as the sum of the following power series:

$\exp(z)=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}$

Alternatively, it can be defined in terms of the real exponential, real sine function, and real cosine function, as follows:

$\exp(x+iy)=e^{x}(\cos y+i\sin y)$

The complex exponential does not extend to a meromorphic function on the Riemann sphere: it has an essential singularity at $\infty$.