arcsinhx

Arcsinh, written as arcsinh(x) or asinh(x), is the inverse hyperbolic sine function. It is the inverse of the hyperbolic sine, sinh, and maps real numbers to real numbers. A common closed form is arcsinh(x) = ln(x + sqrt(x^2 + 1)), which derives from solving x = sinh(y) for y.

Properties and values: arcsinh is an odd function, satisfying arcsinh(-x) = -arcsinh(x). It is strictly increasing on the real line. Its derivative is d/dx arcsinh(x) = 1 / sqrt(x^2 + 1), and its second derivative is -x / (x^2 + 1)^(3/2). The domain is all real x and the range is all real numbers. As x grows large, arcsinh(x) behaves like ln(2x); as x declines, it behaves like -ln(-2x). Since sinh is a bijection from R to R, arcsinh is its inverse, giving arcsinh(sinh(x)) = x and sinh(arcsinh(x)) = x for all real x.

Representations and related forms: besides the logarithmic form, arcsinh can be interpreted as the area hyperbolic sine and appears in integrals involving sqrt(x^2 + 1) and in solving equations with hyperbolic functions. It is related to other inverse hyperbolic functions such as arccosh and arctanh, but is distinct in its argument and range.

Examples: arcsinh(0) = 0; arcsinh(1) = ln(1 + sqrt(2)) ≈ 0.881373; arcsinh(-2) ≈ -1.443635.