Definition:Inverse Hyperbolic Sine/Complex/Definition 2

Definition
The inverse hyperbolic sine is a multifunction defined as:


 * $\forall z \in \C: \sinh^{-1} \left({z}\right) := \left\{{\ln \left({z + \sqrt{\left|{z^2 + 1}\right|} e^{\left({i / 2}\right) \arg \left({z^2 + 1}\right)} }\right) + 2 k \pi i: k \in \Z}\right\}$

where:
 * $\sqrt{\left|{z^2 + 1}\right|}$ denotes the positive square root of the complex modulus of $z^2 + 1$
 * $\arg \left({z^2 + 1}\right)$ 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:
 * $\forall z \in \C: \sinh^{-1} \left({z}\right) := \ln \left({z + \sqrt{z^2 + 1} }\right)$

Also see

 * Equivalence of Definitions of Complex Inverse Hyperbolic Sine


 * Definition:Inverse Sine/Complex/Definition 2