Inverse hyperbolic functions

In mathematics, the inverse hyperbolic functions are inverses of the hyperbolic functions, analogous to the inverse circular functions. There are six in common use: inverse hyperbolic sine, inverse hyperbolic cosine, inverse hyperbolic tangent, inverse hyperbolic cosecant, inverse hyperbolic secant, and inverse hyperbolic cotangent. They are commonly denoted by the symbols for the hyperbolic functions, prefixed with arc- or ar-.

For a given value of a hyperbolic function, the inverse hyperbolic function provides the corresponding hyperbolic angle measure, for example ${\displaystyle \operatorname {arsinh} (\sinh a)=a}$ and ${\displaystyle \sinh(\operatorname {arsinh} x)=x.}$ Hyperbolic angle measure is the length of an arc of a unit hyperbola ${\displaystyle x^{2}-y^{2}=1}$ as measured in the Lorentzian plane (not the length of a hyperbolic arc in the Euclidean plane), and twice the area of the corresponding hyperbolic sector. This is analogous to the way circular angle measure is the arc length of an arc of the unit circle in the Euclidean plane or twice the area of the corresponding circular sector. Alternately hyperbolic angle is the area of a sector of the hyperbola ${\displaystyle xy=1.}$ Some authors call the inverse hyperbolic functions hyperbolic area functions.^[1]

Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry. It also occurs in the solutions of many linear differential equations (such as the equation defining a catenary), cubic equations, and Laplace's equation in Cartesian coordinates. Laplace's equations are important in many areas of physics, including electromagnetic theory, heat transfer, fluid dynamics, and special relativity.