# Definition:Hyperbolic Cosecant

## Definition

### Definition 1

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

$\csch: X \to \C$:
$\forall z \in X: \csch z := \dfrac 2 {e^z - e^{-z} }$

where:

$X = \set {z: z \in \C, \ e^z - e^{-z} \ne 0}$

### Definition 2

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

$\csch: X \to \C$:
$\forall z \in X: \csch z := \dfrac 1 {\sinh z}$

where:

$\sinh$ is the hyperbolic sine
$X = \set {z: z \in \C, \ \sinh z \ne 0}$

### Real Hyperbolic Cosecant

On the real numbers it is defined similarly.

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.

## Also see

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