Definition:Inverse Secant/Complex
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Definition
Definition 1
Let $z \in \C_{\ne 0}$ be a non-zero complex number.
The inverse secant of $z$ is the multifunction defined as:
- $\map {\sec^{-1} } z := \set {w \in \C: \map \sec w = z}$
where $\map \sec w$ is the secant of $w$.
Definition 2
Let $z \in \C_{\ne 0}$ be a non-zero complex number.
The inverse secant of $z$ is the multifunction defined as:
- $\sec^{-1} z := \set {\dfrac 1 i \map \ln {\dfrac {1 + \sqrt {\size {1 - z^2} } e^{\paren {i / 2} \map \arg {1 - z^2} } } z} + 2 k \pi: k \in \Z}$
where:
- $\sqrt {\size {1 - z^2} }$ denotes the positive square root of the complex modulus of $1 - z^2$
- $\map \arg {1 - z^2}$ denotes the argument of $1 - z^2$
- $\ln$ denotes the complex natural logarithm as a multifunction.
Arcsecant
The principal branch of the complex inverse secant function is defined as:
- $\forall z \in \C_{\ne 0}: \map \arcsec z := \dfrac 1 i \, \map \Ln {\dfrac {1 + \sqrt {1 - z^2} } z}$
where:
- $\Ln$ denotes the principal branch of the complex natural logarithm
- $\sqrt {1 - z^2}$ denotes the principal square root of $1 - z^2$.