Cosine function

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 cosine function is a holomorphic function from $\mathbb{C}$ to $\mathbb{C}$ (i.e., an entire function). It is defined as:

$\cos z:=\sum _{n=0}^{\mathrm{\infty }}(-1{)}^n\frac{z^{2}n}{(2n)!}$

Alternatively, it can be defined as:

$\cos z:=\frac{\exp (iz)+\exp (-iz)}{2}$

For $z\in \mathbb{R}$, it coincides with the real cosine function.

The cosine function does not extend to a meromorphic function on the Riemann sphere: it has an essential singularity at $\mathrm{\infty }$.