Arcsinh

Arcsinh, or inverse hyperbolic sine, is the inverse of the hyperbolic sine function on the real numbers. It returns the unique real y with sinh(y) = x. A standard closed-form is arcsinh(x) = ln(x + sqrt(x^2 + 1)), valid for all real x since sqrt(x^2+1) > |x|.

Properties: arcsinh is an odd, strictly increasing function with derivative 1 / sqrt(x^2 + 1). As x → ∞, arcsinh(x) ~ ln(2x); as x → -∞, arcsinh(x) ~ -ln(-2x). For small x, arcsinh(x) ≈ x.

Relations: It is often denoted as asinh(x) in libraries and literature. It is the inverse of sinh, and can be related to other inverse hyperbolic functions via logarithmic expressions.

Applications: In calculus arcsinh appears in ∫ dx / sqrt(x^2+1) = arcsinh(x). In statistics, the arcsinh transform is used to stabilize variance and handle data with zero or negative values, behaving approximately linearly near zero and logarithmically for large values. In physics and geometry, arcsinh arises in problems involving catenaries and relativistic kinematics.

Computational notes: Many programming languages provide asinh or asinh(x); the equivalent formula ln(x + sqrt(x^2+1)) is a reliable evaluation method.