Asinh

Asinh, short for the inverse hyperbolic sine, is the inverse function of the hyperbolic sine, sinh. For real inputs it can be defined by asinh(x) = ln(x + sqrt(x^2 + 1)). This expression yields a real value for every real x, so asinh is defined on the whole real line.

As a function, asinh is strictly increasing, odd, and maps the real numbers onto themselves. Its derivative is d/dx asinh(x) = 1 / sqrt(x^2 + 1). Because asinh is the inverse of sinh, the compositions sinh(asinh(x)) = x and asinh(sinh(x)) = x hold for all real x.

Besides the closed form, asinh has a series expansion around 0: asinh(x) = x - x^3/6 + 3x^5/40 - 5x^7/112 + ... . For large |x|, asymptotically asinh(x) ~ sign(x) ln(2|x|).

A common integral involves asinh: ∫ asinh(x) dx = x asinh(x) - sqrt(x^2 + 1) + C. The function is widely used in calculus and applied mathematics, especially in integrals containing sqrt(x^2 + 1) and in problems involving hyperbolic geometry. It also serves as an explicit inverse in equations of the form sinh(y) = x. Notation varies between asinh(x) and arcsinh(x).