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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.
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Graphs of the inverse hyperbolic functions The hyperbolic functions sinh, cosh, and tanh with respect to a unit hyperbola are analogous to circular functions sin, cos, tan with respect to a unit circle. The argument to the hyperbolic functions is a hyperbolic angle measure.
Tanh-sinh quadrature is a method for numerical integration introduced by Hidetoshi Takahashi and Masatake Mori in 1974. [1] It is especially applied where singularities or infinite derivatives exist at one or both endpoints.
In the integral , we may use = , = , = . Then, = = () = = = + = +. The above step requires that > and > We can choose to be the principal root of , and impose the restriction / < < / by using the inverse sine function.
Sinh may refer to: Hyperbolic sine, abbreviated as sinh, a mathematical function; Sinh (clothing), a traditional women's garment from Southeast Asia;
Si(x) (blue) and Ci(x) (green) shown on the same plot.Integral sine in the complex plane, plotted with a variant of domain coloring. Integral cosine in the complex plane. Note the branch cut along the negative real a
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function.