Definition:Inverse Secant/Complex/Definition 2

Definition
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.

Also see

 * Equivalence of Definitions of Complex Inverse Secant Function


 * Definition:Inverse Hyperbolic Secant/Complex/Definition 2


 * Definition:Complex Arcsecant