Definition:Real Hyperbolic Cotangent/Definition 1

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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.

Also see