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:

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

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 $\infty$.