Definition:Inverse Hyperbolic Cosecant/Complex/Definition 2

Definition
The inverse hyperbolic cosecant is a multifunction defined as:


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

where:
 * $\sqrt {\size {z^2 + 1} }$ denotes the positive square root of the complex modulus of $z^2 + 1$
 * $\map \arg {z^2 + 1}$ denotes the argument of $z^2 + 1$
 * $\ln$ denotes the complex natural logarithm considered as a multifunction.

Also defined as
This concept is also reported as:
 * $\map {\csch^{-1} } z := \set {\map \ln {\dfrac 1 z + \sqrt {\dfrac 1 {z^2} + 1} } }$

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 Cosecant


 * Definition:Inverse Cosecant/Complex/Definition 2