Definition:Hyperbolic Secant

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


Real Hyperbolic Secant

On the real numbers it is defined similarly.

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

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


Also see

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


Sources