Symbols:A/Arcsecant/arcsec
Arcsecant
- $\arcsec$
From Shape of Secant Function, we have that $\sec x$ is continuous and strictly increasing on the intervals $\hointr 0 {\dfrac \pi 2}$ and $\hointl {\dfrac \pi 2} \pi$.
From the same source, we also have that:
- $\sec x \to + \infty$ as $x \to \dfrac \pi 2^-$
- $\sec x \to - \infty$ as $x \to \dfrac \pi 2^+$
Let $g: \hointr 0 {\dfrac \pi 2} \to \hointr 1 \to$ be the restriction of $\sec x$ to $\hointr 0 {\dfrac \pi 2}$.
Let $h: \hointl {\dfrac \pi 2} \pi \to \hointl \gets {-1}$ be the restriction of $\sec x$ to $\hointl {\dfrac \pi 2} \pi$.
Let $f: \closedint 0 \pi \setminus \dfrac \pi 2 \to \R \setminus \openint {-1} 1$:
- $\map f x = \begin{cases} \map g x & : 0 \le x < \dfrac \pi 2 \\ \map h x & : \dfrac \pi 2 < x \le \pi \end{cases}$
From Inverse of Strictly Monotone Function, $\map g x$ admits an inverse function, which will be continuous and strictly increasing on $\hointr 1 \to$.
From Inverse of Strictly Monotone Function, $\map h x$ admits an inverse function, which will be continuous and strictly increasing on $\hointl \gets {-1}$.
As both the domain and range of $g$ and $h$ are disjoint, it follows that:
- $\inv f x = \begin {cases} \inv g x & : x \ge 1 \\ \inv h x & : x \le -1 \end {cases}$
This function $\inv f x$ is called the arcsecant of $x$.
Thus:
- The domain of the arcsecant is $\R \setminus \openint {-1} 1$
- The image of the arcsecant is $\closedint 0 \pi \setminus \dfrac \pi 2$.
The usual symbol used on $\mathsf{Pr} \infty \mathsf{fWiki}$ to denote the arcsecant function is $\arcsec$.
The $\LaTeX$ code for \(\arcsec\) is \arcsec
.
Also denoted as
asec
- $\operatorname {asec}$
A variant symbol used to denote the arcsecant function is $\operatorname {asec}$.
The $\LaTeX$ code for \(\operatorname {asec}\) is \operatorname {asec}
.