Definition:Hyperbolic Secant
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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
- 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