# 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
$X = \set {z: z \in \C, \ \sinh z \ne 0}$

## Also see

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