Definition:Inverse Secant/Complex

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Definition

Definition 1

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

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

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

where $\sec \left({w}\right)$ is the secant of $w$.


Definition 2

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

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

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

where:

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


Arcsecant

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 $1 - z^2$.


Also see