Definition:Inverse Sine/Complex/Definition 2

Definition
Let $z \in \C$ be a complex number.

The inverse sine of $z$ is the multifunction defined as:
 * $\sin^{-1} \left({z}\right) := \left\{{\dfrac 1 i \ln \left({i z + \sqrt{\left|{1 - z^2}\right|} e^{\left({i / 2}\right) \arg \left({1 - z^2}\right)} }\right) + 2 k \pi: k \in \Z}\right\}$

where:
 * $\sqrt{\left|{1 - z^2}\right|}$ denotes the positive square root of the complex modulus of $1 - z^2$
 * $\arg \left({1 - z^2}\right)$ denotes the argument of $1 - z^2$
 * $\ln$ is the complex natural logarithm considered as a multifunction.

Also defined as
In expositions of the inverse trigonometric functions, it is frequently the case that the $2 k \pi$ 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:
 * $\sin^{-1} \left({z}\right) := \dfrac 1 i \ln \left({i z + \sqrt{1 - z^2} }\right)$

Also see

 * Equivalence of Definitions of Complex Inverse Sine Function


 * Definition:Complex Arcsine


 * Definition:Inverse Hyperbolic Sine/Complex/Definition 2