Symbols:A/Arccosecant/arccsc
Arccosecant
- $\arccsc$
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 $\closedint {-\dfrac \pi 2} {\dfrac \pi 2} \setminus \set 0$.
The usual symbol used on $\mathsf{Pr} \infty \mathsf{fWiki}$ to denote the arccosecant function is $\arccsc$.
The $\LaTeX$ code for \(\arccsc\) is \arccsc .
Also denoted as
arccosec
- $\operatorname {arccosec}$
A variant symbol used to denote the arccosecant function is $\operatorname {arccosec}$.
The $\LaTeX$ code for \(\operatorname {arccosec}\) is \operatorname {arccosec} .
acosec
- $\operatorname {acosec}$
A variant symbol used to denote the arccosecant function is $\operatorname {acosec}$.
The $\LaTeX$ code for \(\operatorname {acosec}\) is \operatorname {acosec} .
acsc
- $\operatorname {acsc}$
A variant symbol used to denote the arccosecant function is $\operatorname {acsc}$.
The $\LaTeX$ code for \(\operatorname {acsc}\) is \operatorname {acsc} .