Definition:Inverse Sine/Complex
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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 $2$
- $\map {\sin^{-1} } 2 = \dfrac {4 k + 1} 2 \pi - i \map \ln {2 \pm \sqrt 3}$
for $k \in \Z$.