Definition:Hyperbolic Secant

Definition

The hyperbolic secant is one of the hyperbolic functions:

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

• Equivalence of Definitions of Hyperbolic Secant

• Definition:Hyperbolic Sine
• Definition:Hyperbolic Cosine
• Definition:Hyperbolic Tangent
• Definition:Hyperbolic Cotangent
• Definition:Hyperbolic Cosecant

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

Sources

• Weisstein, Eric W. "Hyperbolic Secant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HyperbolicSecant.html