# Definition:Hyperbolic Cosecant/Real

## Definition

### Definition 1

The real hyperbolic cosecant function is defined on the real numbers as:

$\csch: \R_{\ne 0} \to \R$:
$\forall x \in \R_{\ne 0}: \csch x := \dfrac 2 {e^x - e^{-x} }$

where it is noted that at $x = 0$:

$e^x - e^{-x} = 0$

and so $\csch x$ is not defined at that point.

### Definition 2

The real hyperbolic cosecant function is defined on the real numbers as:

$\csch: \R_{\ne 0} \to \C$:
$\forall x \in \R_{\ne 0}: \csch x := \dfrac 1 {\sinh x}$

where $\sinh$ is the real hyperbolic sine.

It is noted that at $x = 0$ we have that $\sinh x = 0$, and so $\csch x$ is not defined at that point.

## Also see

• Results about the hyperbolic cosecant function can be found here.