Definition:Inverse Hyperbolic Sine/Complex/Definition 2

Definition
Let $\sinh: \C \to \C$ denote the hyperbolic sine as defined on the set of complex numbers.

The inverse hyperbolic sine is a multifunction $\sinh^{-1}: \C \to \C$ defined as:


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

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

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

Also see

 * Equivalence of Definitions of Inverse Hyperbolic Sine


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