Definition:Inverse Cosecant

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Definition

Real Numbers

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:

$\csc^{-1} \left({x}\right) := \left\{{y \in \R: \csc \left({y}\right) = x}\right\}$

where $\csc \left({y}\right)$ is the cosecant of $y$.


Complex Plane

Let $z \in \C_{\ne 0}$ be a non-zero complex number.

The inverse cosecant of $z$ is the multifunction defined as:

$\csc^{-1} \left({z}\right) := \left\{{w \in \C: \csc \left({w}\right) = z}\right\}$

where $\csc \left({w}\right)$ is the cosecant of $w$.


Arccosecant

Real Numbers

Arccosecant Function

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 $f: \closedint {-\dfrac \pi 2} {\dfrac \pi 2} \setminus 0 \to \R \setminus \openint {-1} 1$:

$\map f x = \begin{cases} \map g x & : -\dfrac \pi 2 \le x < 0 \\ \map h x & : 0 < x \le \dfrac \pi 2 \end{cases}$


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:

$\map {f^{-1} } x = \begin{cases} \map {g^{-1} } x & : x \le -1 \\ \map {h^{-1} } x & : x \ge 1 \end{cases}$


This function $\map {f^{-1} } x$ is called arccosecant of $x$ and is written $\arccsc x$.

Thus:

The domain of $\arccsc x$ is $\R \setminus \openint {-1} 1$
The image of $\arccsc x$ is $\closedint {-\dfrac \pi 2} {\dfrac \pi 2} \setminus 0$.


Complex Plane

The principal branch of the complex inverse cosecant function is defined as:

$\forall z \in \C_{\ne 0}: \map \arccsc z := \dfrac 1 i \, \map \Ln {\dfrac {i + \sqrt {z^2 - 1} } z}$

where:

$\Ln$ denotes the principal branch of the complex natural logarithm
$\sqrt {z^2 - 1}$ denotes the principal square root of $z^2 - 1$.


Also see