Definition:Inverse Hyperbolic Sine/Complex/Definition 2

Definition
The inverse hyperbolic sine is a multifunction defined as:


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

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
In expositions of the inverse hyperbolic functions, it is frequently the case that the $2 k \pi i$ constant is ignored, in order to simplify the presentation.

It is also commonplace to gloss over the multifunctional nature of the complex square root, and report this definition as:
 * $\forall z \in \C: \map {\sinh^{-1} } z := \map \ln {z + \sqrt {z^2 + 1} }$

Also see

 * Equivalence of Definitions of Complex Inverse Hyperbolic Sine


 * Definition:Inverse Sine/Complex/Definition 2