Definition:Inverse Hyperbolic Secant/Complex/Definition 2

Definition
The inverse hyperbolic secant is a multifunction $\operatorname{sech}^{-1}: \C \to \C$ defined as:


 * $\forall x \in \C: \operatorname{sech}^{-1} \left({x}\right) = \ln \left({\dfrac 1 x + \sqrt{\dfrac 1 {x^2} - 1} }\right)$

where $\ln$ is the complex natural logarithm function.

As $\ln$ is a multifunction it follows that $\operatorname{sech}^{-1}$ is likewise a multifunction.

Also see

 * Equivalence of Definitions of Inverse Hyperbolic Secant


 * Definition:Inverse Hyperbolic Sine
 * Definition:Inverse Hyperbolic Cosine
 * Definition:Inverse Hyperbolic Tangent
 * Definition:Inverse Hyperbolic Cotangent
 * Definition:Inverse Hyperbolic Cosecant