Complex exponential

Definition

The complex exponential is a holomorphic function from $\mathbb{C}$ to $\mathbb{C}$. 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:

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

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 $\mathrm{\infty }$.