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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.
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.
Trigonometric functions and their reciprocals on the unit circle. All of the right-angled triangles are similar, i.e. the ratios between their corresponding sides are the same.
The sinc function as audio, at 2000 Hz (±1.5 seconds around zero) In mathematics, the historical unnormalized sinc function is defined for x ≠ 0 by = .. Alternatively, the unnormalized sinc function is often called the sampling function, indicated as Sa(x).
Since cosh x + sinh x = e x, an analog to de Moivre's formula also applies to the hyperbolic trigonometry. For all integers n, ( + ) = + . If n is a rational number (but not necessarily an integer), then cosh nx + sinh nx will be one of the values of (cosh x + sinh x) n. [4]
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.
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Prolate spheroidal coordinates μ and ν for a = 1.The lines of equal values of μ and ν are shown on the xz-plane, i.e. for φ = 0.The surfaces of constant μ and ν are obtained by rotation about the z-axis, so that the diagram is valid for any plane containing the z-axis: i.e. for any φ.