Definition:Hyperbolic Secant

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Definition

Definition 1

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

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

where:

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


Definition 2

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

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

where:

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


Also see

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


Sources