Definition:Hyperbolic Cotangent/Real

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

• Equivalence of Definitions of Real Hyperbolic Cotangent

• 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