hyperboliske

Hyperboliske is the Danish term for hyperbolic, an adjective used in mathematics and related disciplines to describe concepts associated with hyperbolas, negative curvature, or exponential growth. The most common areas are hyperbolic functions, hyperbolic geometry and hyperbolic space.

Hyperbolic functions, including sine hyperbolic (sinh), cosine hyperbolic (cosh) and tangent hyperbolic (tanh), are defined by exponentials: sinh x = (e^x - e^-x)/2, cosh x = (e^x + e^-x)/2. They satisfy cosh^2 x - sinh^2 x = 1 and have derivatives d/dx sinh x = cosh x and d/dx cosh x = sinh x. They arise in solutions to certain differential equations and appear in physics, such as special relativity and wave propagation.

Hyperbolic geometry is a non-Euclidean geometry of constant negative curvature, with models such as the Poincaré disk and the Poincaré half-plane. Geodesics are circles or lines orthogonal to the model boundary; the triangle angle sum is less than 180 degrees, and area relates to angle deficit. Hyperbolic space H^n generalizes these ideas to higher dimensions and is a central object in differential geometry and geometric topology.

Etymology: the term derives from Greek hyperbolē meaning excess; in Danish, hyperboliske functions and spaces are described using the same root as their English counterparts. The concept appears across mathematics, theoretical physics and computer science when modelling phenomena with exponential behaviour or negative curvature.

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