Definition:Inverse Cosecant/Real
Definition
Let $x \in \R$ be a real number such that $x \le -1$ or $x \ge 1$.
The inverse cosecant of $x$ is the multifunction defined as:
- $\inv \csc x := \set {y \in \R: \map \csc y = x}$
where $\map \csc y$ is the cosecant of $y$.
Arccosecant
From Shape of Cosecant Function, we have that $\csc x$ is continuous and strictly decreasing on the intervals $\hointr {-\dfrac \pi 2} 0$ and $\hointl 0 {\dfrac \pi 2}$.
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 $g: \hointr {-\dfrac \pi 2} 0 \to \hointl {-\infty} {-1}$ be the restriction of $\csc x$ to $\hointr {-\dfrac \pi 2} 0$.
Let $h: \hointl 0 {\dfrac \pi 2} \to \hointr 1 \infty$ be the restriction of $\csc x$ to $\hointl 0 {\dfrac \pi 2}$.
Let $f: \closedint {-\dfrac \pi 2} {\dfrac \pi 2} \setminus \set 0 \to \R \setminus \openint {-1} 1$:
- $\map f x = \begin{cases} \map g x & : -\dfrac \pi 2 \le x < 0 \\ \map h x & : 0 < x \le \dfrac \pi 2 \end{cases}$
From Inverse of Strictly Monotone Function, $\map g x$ admits an inverse function, which will be continuous and strictly decreasing on $\hointl {-\infty} {-1}$.
From Inverse of Strictly Monotone Function, $\map h x$ admits an inverse function, which will be continuous and strictly decreasing on $\hointr 1 \infty$.
As both the domain and range of $g$ and $h$ are disjoint, it follows that:
- $\inv f x = \begin {cases} \inv g x & : x \le -1 \\ \inv h x & : x \ge 1 \end {cases}$
This function $\map {f^{-1} } x$ is called the arccosecant of $x$.
Thus:
- The domain of the arccosecant is $\R \setminus \openint {-1} 1$
- The image of the arccosecant is $\closedint {-\dfrac \pi 2} {\dfrac \pi 2} \setminus \set 0$.
Terminology
There exists the 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 a well-defined inverse.
The $\arccsc$ 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 $\arccsc$.
However this can also be confusing due to the visual similarity between that and the lowercase $\text {c}$.
Some sources hyphenate: arc-cosecant.
Also see
- Definition:Real Inverse Sine
- Definition:Real Inverse Cosine
- Definition:Real Inverse Tangent
- Definition:Real Inverse Cotangent
- Definition:Real Inverse Secant
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): arc-cosecant