Definition:Inverse Secant/Arcsecant
Definition
Real Numbers
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 $\closedint 0 \pi \setminus \dfrac \pi 2$.
Complex Plane
The principal branch of the complex inverse secant function is defined as:
- $\forall z \in \C_{\ne 0}: \map \arcsec z := \dfrac 1 i \, \map \Ln {\dfrac {1 + \sqrt {1 - z^2} } z}$
where:
- $\Ln$ denotes the principal branch of the complex natural logarithm
- $\sqrt {1 - z^2}$ denotes the principal square root of $1 - z^2$.
Terminology
There exists the popular but misleading notation $\sec^{-1} x$, which is supposed to denote the inverse secant function.
However, note that as $\sec x$ is not an injection, it does not have a well-defined inverse.
The $\arcsec$ 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 $\sec^{-1} x$, which (as pointed out) can be confusing and misleading.
Sometimes, $\operatorname {Sec^{-1} }$ (with a capital $\text S$) is taken to mean the same as $\arcsec$.
However, this can also be confusing due to the visual similarity between that and the lowercase $\text{s}$.
Some sources hyphenate: arc-secant.