Definition:Inverse Cosecant/Complex

Definition

Definition 1

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 $w$.

Definition 2

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

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

$\csc^{-1} \left({z}\right) := \left\{{\dfrac 1 i \ln \left({\dfrac {i + \sqrt{\left|{z^2 - 1}\right|} e^{\left({i / 2}\right) \arg \left({z^2 - 1}\right)}} z}\right) + 2 k \pi: k \in \Z}\right\}$

where:

$\sqrt{\left|{z^2 - 1}\right|}$ denotes the positive square root of the complex modulus of $z^2 - 1$
$\arg \left({z^2 - 1}\right)$ denotes the argument of $z^2 - 1$
$\ln$ denotes the complex natural logarithm considered as a multifunction.

Arccosecant

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 $z^2 - 1$.