Definition:Inverse Hyperbolic Secant/Complex/Definition 2

Definition
The inverse hyperbolic secant is a multifunction defined as:


 * $\forall z \in \C_{\ne 0}: \map {\sech^{-1} } z := \set {\map \ln {\dfrac {1 + \sqrt {\size {1 - z^2} } e^{\paren {i / 2} \map \arg {1 - z^2} } } z} + 2 k \pi i: k \in \Z}$

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.

As $\ln$ is a multifunction it follows that $\sech^{-1}$ is likewise a multifunction.

Also defined as
This concept is also reported as:
 * $\map {\sech^{-1} } z := \set {\map \ln {\dfrac 1 z + \sqrt {\dfrac 1 {z^2} - 1} } + 2 k \pi i: k \in \Z}$

or:
 * $\map {\sech^{-1} } z := \set {\map \ln {\dfrac 1 z + \sqrt {\paren {\dfrac 1 z + 1} } \sqrt {\paren {\dfrac 1 z - 1} } } + 2 k \pi i: k \in \Z}$

In the above, the complication arising from the multifunctional nature of the complex square root has been omitted for the purpose of simplification.

Also see

 * Equivalence of Definitions of Complex Inverse Hyperbolic Secant


 * Definition:Inverse Secant/Complex/Definition 2