asech(): The Inverse Hyperbolic Secant

asech() reverses sech(), the reciprocal of hyperbolic cosine. It occurs in symbolic formulas, profiles that decay away from a center, and models involving hyperbolic functions.

Real input range

For real arithmetic, asech(x) accepts values from 0 through 1, with zero representing a limiting infinite result rather than an ordinary finite value. Values greater than 1 or negative values do not have real inverse-hyperbolic-secant results. Treat boundary data carefully.

asech(x) can be understood as acosh(1/x), linking it to acosh(). The reciprocal reveals two caveats: zero cannot be divided into, and tiny positive inputs can yield large values. Validate measured data with larger() and smallerEq() before evaluation.

Useful companions

To check a finite result, compute sech(asech(x)). Related hyperbolic inverses are asinh() and atanh(); each has its own domain. Do not confuse asech with asec, which belongs to circular trigonometry.

Writing asech directly communicates both the reciprocal form and the intended branch, making formulas easier to review than a manually expanded logarithmic expression.