# Definition:Inverse Sine/Complex

## Definition

### Definition 1

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

The inverse sine of $z$ is the multifunction defined as:

$\sin^{-1} \paren z := \set {w \in \C: \sin \paren w = z}$

where $\sin \paren w$ is the sine of $w$.

### Definition 2

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

The inverse sine of $z$ is the multifunction defined as:

$\sin^{-1} \paren z := \set {\dfrac 1 i \ln \paren {i z + \sqrt {\cmod {1 - z^2} } \exp \paren {\dfrac i 2 \arg \paren {1 - z^2} } } + 2 k \pi: k \in \Z}$

where:

$\sqrt {\cmod {1 - z^2} }$ denotes the positive square root of the complex modulus of $1 - z^2$
$\arg \paren {1 - z^2}$ denotes the argument of $1 - z^2$
$\ln$ is the complex natural logarithm considered as a multifunction.

## Arcsine

The principal branch of the complex inverse sine function is defined as:

$\map \arcsin z = \dfrac 1 i \, \map \Ln {i z + \sqrt {1 - z^2} }$

where:

$\Ln$ denotes the principal branch of the complex natural logarithm
$\sqrt {1 - z^2}$ denotes the principal square root of $1 - z^2$.

## Examples

### Inverse Sine of $2$

$\sin^{-1} \paren 2 = \dfrac {4 k + 1} 2 \pi - i \ln \paren {2 \pm \sqrt 3}$

for $k \in \Z$.