Definition:Inverse Cosecant/Real/Arccosecant

Definition
From Shape of Cosecant Function, we have that $\csc x$ is continuous and strictly decreasing on the intervals $\left[{-\dfrac \pi 2 \,.\,.\, 0}\right)$ and $\left({0 \,.\,.\, \dfrac \pi 2}\right]$.

From the same source, we also have that:
 * $\csc x \to + \infty$ as $x \to 0^+$
 * $\csc x \to - \infty$ as $x \to 0^-$

Let $g: \left[{-\dfrac \pi 2 \,.\,.\, 0}\right) \to \left({-\infty \,.\,.\, -1}\right]$ be the restriction of $\csc x$ to $\left[{-\dfrac \pi 2 \,.\,.\, 0}\right)$.

Let $h: \left({0 \,.\,.\, \dfrac \pi 2}\right] \to \left[{1 \,.\,.\, \infty}\right)$ be the restriction of $\csc x$ to $\left({0 \,.\,.\, \dfrac \pi 2}\right]$.

Let $f: \left[{-\dfrac \pi 2 \,.\,.\, \dfrac \pi 2}\right] \setminus 0 \to \R \setminus \left({-1 \,.\,.\, 1}\right)$:


 * $f\left({x}\right) = \begin{cases}

g\left({x}\right) & : -\dfrac \pi 2 \le x < 0 \\ h\left({x}\right) & : 0 < x \le \dfrac \pi 2 \end{cases}$

From Inverse of Strictly Monotone Function, $g \left({x}\right)$ admits an inverse function, which will be continuous and strictly decreasing on $\left({-\infty \,.\,.\, -1}\right]$.

From Inverse of Strictly Monotone Function, $h \left({x}\right)$ admits an inverse function, which will be continuous and strictly decreasing on $\left[{1 \,.\,.\, \infty}\right)$.

As both the domain and range of $g$ and $h$ are disjoint, it follows that:


 * $f^{-1}\left({x}\right) = \begin{cases}

g^{-1}\left({x}\right) & : x \le -1 \\ h^{-1}\left({x}\right) & : x \ge 1 \end{cases}$

This function $f^{-1}\left({x}\right)$ is called arccosecant of $x$ and is written $\operatorname{arccsc} x$.

Thus:
 * The domain of $\operatorname{arccsc} x$ is $\R \setminus \left({-1 \,.\,.\, 1}\right)$
 * The image of $\operatorname{arccsc} x$ is $\left[{-\dfrac \pi 2 \,.\,.\, \dfrac \pi 2}\right] \setminus 0$.

Caution
There exists the a popular but misleading notation $\csc^{-1} x$, which is supposed to denote the inverse cosecant function.

However, note that as $\csc x$ is not an injection, it does not have an inverse.

The $\operatorname{arccsc}$ function as defined here has a well-specified image which (to a certain extent) is arbitrarily chosen for convenience.

Therefore it is preferred to the notation $\csc^{-1} x$, which (as pointed out) can be confusing and misleading.

Sometimes, $\operatorname{Csc}^{-1}$ (with a capital $\text{C}$) is taken to mean the same as $\operatorname{arccsc}$, although this can also be confusing due to the visual similarity between that and the lowercase $\text{c}$.