real function is connected. Inverse hyperbolic sine Inverse hyperbolic sine (a.k.a. area hyperbolic sine) (Latin: Area sinus hyperbolicus): The domain is the real numbers. Inverse hyperbolic cosine Inverse hyperbolic cosine (a.k.a. area hyperbolic cosine) (Latin: Area cosinus hyperbolicus): The domain is the closed interval . Inverse hyperbolic tangent Inverse hyperbolic tangent (a.k.a. area hyperbolic tangent) (Latin: Area tangens hyperbolicus): The domain is the open interval . Inverse hyperbolic cotangent Inverse hyperbolic cotangent (a.k.a., area hyperbolic cotangent) (Latin: Area cotangens hyperbolicus): The domain is the union of the open intervals and . Inverse hyperbolic secant Inverse hyperbolic secant (a.k.a., area hyperbolic secant) (Latin: Area secans hyperbolicus): The domain is the semi-open interval . Inverse hyperbolic cosecant Inverse hyperbolic cosecant (a.k.a., area hyperbolic cosecant) (Latin: Area cosecans hyperbolicus): The domain is all the real numbers except 0. Addition formulae Other identities Composition of hyperbolic and inverse hyperbolic functions Composition of inverse hyperbolic and trigonometric functions Conversions Derivatives For an example differentiation: let θ = arsinh x, so (where sinh2 θ = (sinh θ)2): Series expansions Expansion series can be obtained for the above functions: An asymptotic expansion for arsinh is given by Principal values in the complex plane As functions of a complex variable, inverse hyperbolic functions are multivalued functions that are analytic, except at a finite number of points. For such a function, it is common to define a principal value, which is a single valued analytic function which coincides with one specific branch of the multivalued function, over a domain consisting of the complex plane in which a finite number of arcs (usually half lines or line segments) have been removed. These arcs are called branch cuts. For specifying the branch, that is, defining which value of the multivalued function is considered at each point, one generally define it at a particular point, and deduce the value everywhere in the domain of definition of the principal value by analytic continuation. When possible, it is better to define the principal value directly—without referring to analytic continuation. For example, for the square root, the principal value is defined as the square root that has a positive real part. This defines a single valued analytic function, which is defined everywhere, except for non-positive real values of the variables (where the two square roots have a zero real part). This principal value of the square root function is denoted in what follows. Similarly, the