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

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