asinhx

Asinh, short for inverse hyperbolic sine, denotes the inverse function of the hyperbolic sine function sinh. For a real number x, asinh(x) is the unique real y such that sinh(y) = x. The function is also called arsinh(x) in some references and is defined for all real x.

A convenient closed form is asinh(x) = ln(x + sqrt(x^2 + 1)). This expression is valid for every real x because sqrt(x^2 + 1) ≥ |x|, ensuring the argument of the logarithm is positive. The function is continuous and smooth on the entire real line.

Key properties include that asinh is an odd function (asinh(-x) = -asinh(x)) and strictly increasing. Its derivative is d/dx asinh(x) = 1 / sqrt(x^2 + 1), and its range is all real numbers. As x grows large, asinh(x) behaves like ln(2x); near zero, asinh(x) ~ x. The function is the inverse of sinh, so sinh(asinh(x)) = x for all x, and asinh(sinh(y)) = y for all real y.

Applications of the asinh function appear in solving equations involving sqrt(x^2 + 1), in integration, and in areas of hyperbolic geometry and physics where inverse hyperbolic measures arise. It is widely used in engineering and computer science as a convenient transform similar to the logarithm for certain nonlinear relationships.