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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 .
There are six in common use: inverse hyperbolic sine, inverse hyperbolic cosine, inverse hyperbolic tangent, inverse hyperbolic cosecant, inverse hyperbolic secant, and inverse hyperbolic cotangent. They are commonly denoted by the symbols for the hyperbolic functions, prefixed with arc- or ar- , or with a superscript − 1 {\displaystyle {-1 ...
The following is a list of integrals (anti-derivative functions) of hyperbolic functions. For a complete list of integral functions, see list of integrals . In all formulas the constant a is assumed to be nonzero, and C denotes the constant of integration .
Just as the trigonometric functions are defined in terms of the unit circle, so also the hyperbolic functions are defined in terms of the unit hyperbola, as shown in this diagram. In a unit circle, the angle (in radians) is equal to twice the area of the circular sector which that angle subtends.
In mathematics, hyperbolic trigonometry can mean: The study of hyperbolic triangles in hyperbolic geometry (traditional trigonometry is the study of triangles in plane geometry ) The use of the hyperbolic functions
A wide variety of sigmoid functions including the logistic and hyperbolic tangent functions have been used as the activation function of artificial neurons. Sigmoid curves are also common in statistics as cumulative distribution functions (which go from 0 to 1), such as the integrals of the logistic density , the normal density , and Student's ...
Geometrically, the hyperbolic tangent function is the hyperbolic angle on the unit hyperbola =, which factors as (+) =, and thus has asymptotes the lines through the origin with slope and with slope , and vertex at (,) corresponding to the range and midpoint ( ) of tanh.
where is the hyperbolic tangent function, is the carrying capacity, and both and > jointly determine the growth rate. In addition, the parameter γ {\displaystyle \gamma } represents acceleration in the time course.