Definition:Inverse Hyperbolic Cosecant/Complex/Definition 2

Definition

The inverse hyperbolic cosecant is a multifunction defined as:

$\forall z \in \C_{\ne 0}: \map {\csch^{-1} } z := \set {\map \ln {\dfrac {1 + \sqrt {\size {z^2 + 1} } e^{\paren {i / 2} \map \arg {z^2 + 1} } } z} + 2 k \pi i: k \in \Z}$

where:

$\sqrt {\size {z^2 + 1} }$ denotes the positive square root of the complex modulus of $z^2 + 1$

$\map \arg {z^2 + 1}$ denotes the argument of $z^2 + 1$

$\ln$ denotes the complex natural logarithm considered as a multifunction.

Also defined as

This concept is also reported as:

$\map {\csch^{-1} } z := \set {\map \ln {\dfrac 1 z + \sqrt {\dfrac 1 {z^2} + 1} } }$

In the above, the complication arising from the multifunctional nature of the complex square root has been omitted for the purpose of simplification.

Also known as

The principal branch of the inverse hyperbolic cosecant is also known as the area hyperbolic cosecant, as it can be used, among other things, for evaluating areas of regions bounded by hyperbolas.

Some sources refer to it as hyperbolic arccosecant, but this is strictly a misnomer, as there is nothing arc related about an inverse hyperbolic cosecant.

Also see

• Equivalence of Definitions of Complex Inverse Hyperbolic Cosecant

• Definition:Inverse Cosecant/Complex/Definition 2

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.60$: Inverse Hyperbolic Functions

• Weisstein, Eric W. "Inverse Hyperbolic Cosecant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/InverseHyperbolicCosecant.html