Definition:Hyperbolic Cosecant
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Definition
The hyperbolic cosecant is one of the hyperbolic functions:
Definition 1
The hyperbolic cosecant function is defined on the complex numbers as:
- $\csch: X \to \C$:
- $\forall z \in X: \csch z := \dfrac 2 {e^z - e^{-z} }$
where:
- $X = \set {z: z \in \C, \ e^z - e^{-z} \ne 0}$
Definition 2
The hyperbolic cosecant function is defined on the complex numbers as:
- $\csch: X \to \C$:
- $\forall z \in X: \csch z := \dfrac 1 {\sinh z}$
where:
- $\sinh$ is the hyperbolic sine
- $X = \set {z: z \in \C, \ \sinh z \ne 0}$
Real Hyperbolic Cosecant
On the real numbers it is defined similarly.
The real hyperbolic cosecant function is defined on the real numbers as:
- $\csch: \R_{\ne 0} \to \R$:
- $\forall x \in \R_{\ne 0}: \csch x := \dfrac 2 {e^x - e^{-x} }$
where it is noted that at $x = 0$:
- $e^x - e^{-x} = 0$
and so $\csch x$ is not defined at that point.
Also see
- Definition:Hyperbolic Sine
- Definition:Hyperbolic Cosine
- Definition:Hyperbolic Tangent
- Definition:Hyperbolic Cotangent
- Definition:Hyperbolic Secant
- Results about the hyperbolic cosecant function can be found here.
Sources
- Weisstein, Eric W. "Hyperbolic Cosecant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HyperbolicCosecant.html