Definition:Inverse Hyperbolic Cosine/Real/Principal Branch

Definition
Let $S$ denote the subset of the real numbers:
 * $S = \set {x \in \R: x \ge 1}$

The principal branch of the real inverse hyperbolic cosine function is defined as:
 * $\forall x \in S: \map \arcosh x := \map \ln {x + \sqrt {x^2 - 1} }$

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

That is, where $\map \arcosh x \ge 0$.

Also see

 * Derivation of Area Hyperbolic Cosine from Inverse Hyperbolic Cosine Multifunction


 * Definition:Real Area Hyperbolic Sine
 * Definition:Real Area Hyperbolic Tangent
 * Definition:Real Area Hyperbolic Cotangent
 * Definition:Real Area Hyperbolic Secant
 * Definition:Real Area Hyperbolic Cosecant