hyperbolic function

Any of a set of six functions related, for a real or complex variable x, to the hyperbola in a manner analogous to the relationship of the trigonometric functions to a circle, including:

a. The hyperbolic sine, defined by the equation sinh x = ¹/₂ (e^x - e^-^x).

b. The hyperbolic cosine, defined by the equation cosh x = ¹/₂ (e^x + e^-^x).

c. The hyperbolic tangent, defined by the equation tanh x = sinh x/cosh x.

d. The hyperbolic cotangent, defined by the equation coth x = cosh x/sinh x.

e. The hyperbolic secant, defined by the equation sech x = 1/cosh x.

f. The hyperbolic cosecant, defined by the equation csch x = 1/sinh x.

American Heritage® Dictionary of the English Language, Fifth Edition. Copyright © 2016 by Houghton Mifflin Harcourt Publishing Company. Published by Houghton Mifflin Harcourt Publishing Company. All rights reserved.

hyperbolic function

(Mathematics) any of a group of functions of an angle expressed as a relationship between the distances of a point on a hyperbola to the origin and to the coordinate axes. The group includes sinh (hyperbolic sine), cosh (hyperbolic cosine), tanh (hyperbolic tangent), sech (hyperbolic secant), cosech (hyperbolic cosecant), and coth (hyperbolic cotangent)

Collins English Dictionary – Complete and Unabridged, 12th Edition 2014 © HarperCollins Publishers 1991, 1994, 1998, 2000, 2003, 2006, 2007, 2009, 2011, 2014

hy′perbol′ic func′tion

a function of an angle expressed as a relationship between the distances from a point on a hyperbola to the origin and to the coordinate axes, as hyperbolic sine or hyperbolic cosine.

[1885–90]

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References in periodicals archive

The hyperbolic-type control schemes work as follows: for large errors [[??].sub.i], the hyperbolic function cosh [mathematical expression not reproducible] returns a bounded control output avoiding input saturation in the servo amplifier; when the errors [[??].sub.i] are small, that hyperbolic function is approximately linear; then it works like proportional controller.

A Family of Hyperbolic-Type Explicit Force Regulators with Active Velocity Damping for Robot Manipulators

Hyperbolic function solution, trigonometric function solution, and rational solution with general parameters are obtained.

Theoretical and Computational Advances in Nonlinear Dynamical Systems 2018

Caption: FIGURE 7: Prediction model for the settlement of the road shoulder: (a) hyperbolic function; (b) logarithmic function; (c) power function; (d) exponential function.

In Situ Monitoring of the Long-Term Settlement of High-Fill Subgrade

However, Mazur's hyperbolic function and the AUC are arguably the two most commonly used measures of DD.

A comparison of Mazur's k and area under the curve for describing steep discounters

Case 2 (trigonometric and hyperbolic function solution).

The Improved Generalized tanh-coth Method Applied to Sixth-Order Solitary Wave Equation

The Linear Hyperbolic function (LHF) can be used to describe milk yield and the percentages of fat and protein curves between different ages and calving seasons.

Milk production and percentages of fat and protein in Holstein breed cows raised in Rio Grande do Sul, Brazil/Producao de leite e porcentagens de gordura e proteina de bovinos da raca Holandesa criados no Rio Grande do Sul, Brasil

of Name of Type of equation empirical model mathematical model 1 Reeve (1957) Hyperbolic function 2 Dieleman (1963) Exponential function 3 Leffelaar & Shama (1977) Hyperbolic function 4 Hoffman (1980) Hyperbolic function 5 Pazira & Kawachi (1998) Hyperbolic function 6 Verma and Gupta (1989) Power function 7 Power function 8 Pazira & Keshavarz (1998) Power function 9 Rajabzadeh (2009) Exponential function 10 Asadi (2011) Logarithmic function 11 Asadi (2012) Logarithmic function 12 Mohammadzadeh (2013) Power function No.

Leaching mathematical modeling for salt leaching of saline-sodic soils of Ahvaz North plains of Khuzestan province

Consequently, as before, when [[lambda].sup.2] - 4[mu] > 0, we obtain the hyperbolic function solutions:

Conservation laws and traveling wave solutions of a generalized nonlinear ZK-BBM equation

The tangent hyperbolic function can be arranged in a vector in the following way:

Adaptive neural network motion control of manipulators with experimental evaluations

Let function [[eta].sub.1] be a hyperbolic function of time t; that is, [[eta].sub.1] = cosh([[OMEGA].sub.3]t), and then we have

Dynamics and matter-wave solitons in bose-einstein condensates with two- and three-body interactions