arccosh

Arccosh, denoted arccosh x, is the inverse of the hyperbolic cosine on the domain where cosh is increasing. It returns the unique nonnegative y such that cosh y = x. Accordingly, arccosh is defined for real x ≥ 1, and its range is y ≥ 0.

For x ≥ 1, arccosh x can be expressed as arccosh x = ln( x + sqrt(x^2 - 1) ). In particular, arccosh 1 = 0 and arccosh 2 = ln(2 + sqrt(3)). The derivative is d/dx arccosh x = 1 / sqrt(x^2 - 1) for x > 1, making arccosh an increasing function on [1, ∞).

As the inverse function, cosh(arccosh x) = x for x ≥ 1. Conversely, arccosh(cosh y) = |y| for all real y, reflecting that cosh is even and arccosh returns the principal nonnegative value. The expression arccosh x = ln( x + sqrt(x^2 - 1) ) also yields arccosh x = asinh( sqrt(x^2 - 1) ).

In addition to its use in solving equations involving the hyperbolic cosine, arccosh appears in integrations, in the geometry of the catenary, and in various areas of physics and engineering where hyperbolic functions arise. Extensions to complex values provide a broader, multivalued context, but the real-valued arccosh is standard for x ≥ 1.