Definition:Hyperbolic Cotangent/Real
< Definition:Hyperbolic Cotangent(Redirected from Definition:Real Hyperbolic Cotangent)
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Definition
Definition 1
The real hyperbolic cotangent function is defined on the real numbers as:
- $\coth: \R_{\ne 0} \to \R$:
- $\forall x \in \R_{\ne 0}: \coth x := \dfrac {e^x + e^{-x} } {e^x - e^{-x} }$
where it is noted that at $x = 0$:
- $e^x - e^{-x} = 0$
and so $\coth x$ is not defined at that point.
Definition 2
The real hyperbolic cotangent function is defined on the real numbers as:
- $\coth: \R_{\ne 0} \to \R$:
- $\forall x \in \R_{\ne 0}: \coth x := \dfrac {\cosh x} {\sinh x}$
where:
- $\sinh$ is the real hyperbolic sine
- $\cosh$ is the real hyperbolic cosine
It is noted that at $x = 0$ we have that $\sinh x = 0$, and so $\coth x$ is not defined at that point.
Definition 3
The real hyperbolic cotangent function is defined on the real numbers as:
- $\coth: \R_{\ne 0} \to \R$:
- $\forall x \in \R_{\ne 0}: \coth x := \dfrac 1 {\tanh x}$
where $\tanh$ is the real hyperbolic tangent.
It is noted that at $x = 0$ we have that $\tanh x = 0$, and so $\coth x$ is not defined at that point.
Also denoted as
The notation $\operatorname {cth} z$ can also be found for hyperbolic cotangent.
Also see
- Definition:Real Hyperbolic Sine
- Definition:Real Hyperbolic Cosine
- Definition:Real Hyperbolic Tangent
- Definition:Real Hyperbolic Secant
- Definition:Real Hyperbolic Cosecant
- Results about the hyperbolic cotangent function can be found here.
Sources
- Weisstein, Eric W. "Hyperbolic Cotangent." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HyperbolicCotangent.html