\begin{definition}[Definition:Real Hyperbolic Cotangent/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.
\end{definition}