arcsech

arcsech is the inverse function of the hyperbolic secant, sech(y) = 1 / cosh(y). Since cosh(y) ≥ 1 for all real y, sech(y) takes values in (0, 1], making arcsech defined on the real interval (0, 1], with arcsech(1) = 0 and arcsech(x) → ∞ as x → 0+.

For real x in (0, 1], arcsech x can be expressed as arcsech x = arcosh(1/x). A convenient closed form is arcsech x = ln((1 + sqrt(1 − x^2)) / x). Another equivalent form is arcsech x = −ln(x / (1 + sqrt(1 − x^2))). These expressions make the principal value explicit.

Derivative properties: on 0 < x < 1, d/dx arcsech x = −1 / (x sqrt(1 − x^2)). This derivative mirrors the inverse relationship with sech and arcosh and reflects the vertical tangent behavior as x approaches 0 or 1.

Domain, range, and behavior: the real domain of arcsech is 0 < x ≤ 1, with arcsech(1) = 0. The function is strictly decreasing on (0, 1], mapping onto [0, ∞). The value tends to infinity as x approaches 0 from the right and equals zero at x = 1.

Relationships and usage: arcsech is the inverse of sech when restricted to nonnegative arguments, so for y ≥ 0, arcsech(sech(y)) = y. It is used in contexts involving inverse hyperbolic functions and integrals where the hyperbolic secant appears.

Extensions to complex arguments exist and require branch choices, but the above describes the standard real-valued arcsech function.