\begin{definition}[Definition:Real Hyperbolic Cosecant/Definition 1]
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.
\end{definition}