Definition:Inverse Hyperbolic Secant/Complex/Definition 2
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Definition
The inverse hyperbolic secant is a multifunction defined as:
- $\forall z \in \C_{\ne 0}: \map {\sech^{-1} } z := \set {\map \ln {\dfrac {1 + \sqrt {\size {1 - z^2} } e^{\paren {i / 2} \map \arg {1 - z^2} } } z} + 2 k \pi i: k \in \Z}$
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.
As $\ln$ is a multifunction it follows that ${\sech^{-1} }$ is likewise a multifunction.
Also known as
The principal branch of the inverse hyperbolic secant is also known as the area hyperbolic secant, as it can be used, among other things, for evaluating areas of regions bounded by hyperbolas.
Some sources refer to it as hyperbolic arcsecant, but this is strictly a misnomer, as there is nothing arc related about an inverse hyperbolic secant.
Also defined as
This concept is also reported as:
- $\map {\sech^{-1} } z := \set {\map \ln {\dfrac 1 z + \sqrt {\dfrac 1 {z^2} - 1} } + 2 k \pi i: k \in \Z}$
or:
- $\map {\sech^{-1} } z := \set {\map \ln {\dfrac 1 z + \sqrt {\paren {\dfrac 1 z + 1} } \sqrt {\paren {\dfrac 1 z - 1} } } + 2 k \pi i: k \in \Z}$
In the above, the complication arising from the multifunctional nature of the complex square root has been omitted for the purpose of simplification.
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.59$: Inverse Hyperbolic Functions
- Weisstein, Eric W. "Inverse Hyperbolic Secant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/InverseHyperbolicSecant.html