Definition:Inverse Hyperbolic Cosecant/Real/Definition 2

Definition
The inverse hyperbolic cosecant $\csch^{-1}: \R_{\ne 0} \to \R$ is a real function defined on the non-zero real numbers $\R_{\ne 0}$ as:


 * $\forall x \in \R_{\ne 0}: \map {\csch^{-1} } x := \map \ln {\dfrac 1 x + \dfrac {\sqrt {x^2 + 1} } {\size x} }$

where:
 * $\sqrt {x^2 + 1}$ denotes the positive square root of $x^2 + 1$
 * $\ln$ denotes the natural logarithm of a (strictly positive) real number.

Also known as
The inverse hyperbolic cosecant function is also known as the hyperbolic arccosecant function.

Also see

 * Equivalence of Definitions of Real Inverse Hyperbolic Cosecant