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:
 * $\sec^{-1} z := \set {\dfrac 1 i \map \ln {\dfrac {1 + \sqrt {\size {1 - z^2} } e^{\paren {i / 2} \map \arg {1 - z^2} } } z} + 2 k \pi: k \in \Z}$

where:
 * $\sqrt {\size {1 - z^2} }$ denotes the positive square root of the complex modulus of $1 - z^2$
 * $\map \arg {1 - z^2}$ 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