asech

asech, short for inverse hyperbolic secant, is the inverse function of the hyperbolic secant, sech(x) = 1/cosh(x). It is sometimes denoted arcsech or arsech. The function returns, for a given y, the nonnegative x such that sech(x) = y.

Domain and range: The function is defined for 0 < y ≤ 1. Its range is x ≥ 0, since cosh(x) ≥ 1 for all real x and y = 1/cosh(x) decreases from 1 to 0 as x increases from 0 to infinity. The function asech is strictly decreasing on its domain: as y increases toward 1, asech(y) decreases toward 0; as y approaches 0 from the right, asech(y) grows without bound.

Formulas: asech(y) can be expressed as arcosh(1/y), where arcosh is the inverse hyperbolic cosine. Equivalently, asech(y) = ln((1 + sqrt(1 − y^2)) / y) for 0 < y ≤ 1.

Examples: asech(1) = 0, since sech(0) = 1. For y = 1/2, asech(1/2) = arcosh(2) = ln(2 + sqrt(3)) ≈ 1.31696.

Relationships: asech(y) is related to other inverse hyperbolic functions by asech(y) = arcosh(1/y). It shares properties with inverse functions of hyperbolic trigonometric functions and is used in solving equations involving sech.

See also: sech, arcsech, arcosh.