Definition:Inverse Hyperbolic Secant/Real/Principal Branch

Definition
Let $S$ denote the subset of the real numbers:
 * $S := \hointl 0 1$

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

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

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

Also see

 * Derivation of Area Hyperbolic Secant from Inverse Hyperbolic Secant Multifunction


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