In mathematics, the inverse hyperbolic functions are the inverse functions of the hyperbolic functions. For a given value of a hyperbolic function, the corresponding inverse hyperbolic function provides the corresponding hyperbolic angle. The size of the hyperbolic angle is equal to the area of the corresponding hyperbolic sector of the hyperbola , or twice the area of the corresponding sector of the unit hyperbola , just as a circular angle is twice the area of the circular sector of the unit circle. Some authors have called inverse hyperbolic functions "area functions" to realize the hyperbolic angles. 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. Notation The ISO 80000-2 standard abbreviations consist of ar- followed by the abbreviation of the corresponding hyperbolic function (e.g., arsinh, arcosh). The prefix arc- followed by the corresponding hyperbolic function (e.g., arcsinh, arccosh) is also commonly seen, by analogy with the nomenclature for inverse trigonometric functions. These are misnomers, since the prefix arc is the abbreviation for arcus, while the prefix ar stands for area; the hyperbolic functions are not directly related to arcs. Other authors prefer to use the notation argsinh, argcosh, argtanh, and so on, where the prefix arg is the abbreviation of the Latin argumentum. In computer science, this is often shortened to asinh. The notation , , etc., is also used, despite the fact that care must be taken to avoid misinterpretations of the superscript −1 as a power, as opposed to a shorthand to denote the inverse function (e.g., versus Definitions in terms of logarithms Since the hyperbolic functions are rational functions of whose numerator and denominator are of degree at most two, these functions may be solved in terms of , by using the quadratic formula; then, taking the natural logarithm gives the following expressions for the inverse hyperbolic functions. For complex arguments, the inverse hyperbolic functions, the square root and the logarithm are multi-valued functions, and the equalities of the next subsections may be viewed as equalities of multi-valued functions. For all inverse hyperbolic functions (save the inverse hyperbolic cotangent and the inverse hyperbolic cosecant), the domain of the