Hyperbolic functions | EPFL Graph Search

In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the unit hyperbola. Also, similarly to how the derivatives of sin(t) and cos(t) are cos(t) and –sin(t) respectively, the derivatives of sinh(t) and cosh(t) are cosh(t) and +sinh(t) respectively. Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry. They also occur 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. The basic hyperbolic functions are: hyperbolic sine "sinh" (ˈsɪŋ,_ˈsɪntʃ,_ˈʃaɪn), hyperbolic cosine "cosh" (ˈkɒʃ,_ˈkoʊʃ), from which are derived: hyperbolic tangent "tanh" (ˈtæŋ,_ˈtæntʃ,_ˈθæn), hyperbolic cosecant "csch" or "cosech" (ˈkoʊsɛtʃ,_ˈkoʊʃɛk) hyperbolic secant "sech" (ˈsɛtʃ,_ˈʃɛk), hyperbolic cotangent "coth" (ˈkɒθ,_ˈkoʊθ), corresponding to the derived trigonometric functions. The inverse hyperbolic functions are: area hyperbolic sine "arsinh" (also denoted "sinh−1", "asinh" or sometimes "arcsinh") area hyperbolic cosine "arcosh" (also denoted "cosh−1", "acosh" or sometimes "arccosh") and so on. The hyperbolic functions take a real argument called a hyperbolic angle. The size of a hyperbolic angle is twice the area of its hyperbolic sector. The hyperbolic functions may be defined in terms of the legs of a right triangle covering this sector. In complex analysis, the hyperbolic functions arise when applying the ordinary sine and cosine functions to an imaginary angle. The hyperbolic sine and the hyperbolic cosine are entire functions. As a result, the other hyperbolic functions are meromorphic in the whole complex plane.

Efficient Continual Finite-Sum Minimization

Volkan Cevher, Efstratios Panteleimon Skoulakis, Leello Tadesse Dadi

Given a sequence of functions

f_1,\ldots,f_n

with

f_i:\mathcal{D}\mapsto \mathbb{R}

, finite-sum minimization seeks a point

{x}^\star \in \mathcal{D}

minimizing

\sum_{j=1}^nf_j(x)/n

. In this work, we propose a key twist into the finite-sum minimizat ...

2024