# Definition:Hyperbolic Secant

## Definition

### Definition 1

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

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

where:

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

### Definition 2

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

$\sech: X \to \C$:
$\forall z \in X: \sech z := \dfrac 1 {\cosh z}$

where:

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

## Also see

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