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Hyperbolic analogues of trigonometric functions
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just
Hyperbolic_functions
Mathematical functions
mathematics, the inverse hyperbolic functions are inverses of the hyperbolic functions, analogous to the inverse circular functions. There are six in common
Inverse_hyperbolic_functions
Argument of the hyperbolic functions
{\sqrt {2}}} . Hyperbolic angle is used as the independent variable for the hyperbolic functions sinh, cosh, and tanh, because these functions may be premised
Hyperbolic_angle
Family of solutions to related differential equations
Bessel functions are a class of special functions that commonly appear in problems involving wave motion, heat conduction, and other physical phenomena
Bessel_function
Region of the Cartesian plane bounded by a hyperbola and two radii
words, the hyperbolic angle is the argument of hyperbolic functions in the same way that the circular angle is the argument of circular functions. When in
Hyperbolic_sector
Mathematical notation based on the Arabic script
second word of دالة زائدية "hyperbolic function") is added to the end of trigonometric functions to express hyperbolic functions. This is similar to the way
Modern Arabic mathematical notation
Modern_Arabic_mathematical_notation
Mathematical function relating circular and hyperbolic functions
circular and hyperbolic functions in 1830. The Gudermannian function and its inverse were used historically to construct tables of hyperbolic functions or to
Gudermannian_function
Algorithm for computing trigonometric, hyperbolic, logarithmic and exponential functions
simple and efficient algorithm to calculate trigonometric functions, hyperbolic functions, square roots, multiplications, divisions, exponentials, and
CORDIC
list of integrals (anti-derivative functions) of hyperbolic functions. For a complete list of integral functions, see list of integrals. In all formulas
List of integrals of hyperbolic functions
List_of_integrals_of_hyperbolic_functions
Mathematical approximation of a function
Euler numbers. The hyperbolic functions have Maclaurin series closely related to the series for the corresponding trigonometric functions: sinh x = ∑ n
Taylor_series
Plane curve: conic section
hyperbolic geometry (Lobachevsky's celebrated non-Euclidean geometry), hyperbolic functions (sinh, cosh, tanh, etc.), and gyrovector spaces (a geometry proposed
Hyperbola
Analytic function that does not satisfy a polynomial equation
contrast to an algebraic function. The most familiar transcendental functions are the exponential, trigonometric, and hyperbolic functions, and their inverses
Transcendental_function
Growth function exhibiting a singularity at a finite time
growth, hyperbolic growth is highly nonlinear, but differs in important respects. These functions can be confused, as exponential growth, hyperbolic growth
Hyperbolic_growth
Type of non-Euclidean geometry
the theory of hyperbolic functions and, indeed, our present notation for these functions. Ratcliffe, John (2006), Foundations of Hyperbolic Manifolds, Graduate
Hyperbolic_geometry
Transcendental functions are functions that are not algebraic. Exponential function: raises a fixed number to a variable power. Hyperbolic functions: formally
List of mathematical functions
List_of_mathematical_functions
Function equal to cos x + i sin x
is an acronym for "cos i sin". It connects trigonometric functions with exponential functions in the complex plane via Euler's formula. While the domain
Cis_(mathematics)
trigonometric functions List of integrals of hyperbolic functions List of integrals of inverse hyperbolic functions List of integrals of exponential functions List
Lists_of_integrals
Functions of an angle
trigonometric functions has a corresponding inverse function and has an analog among the hyperbolic functions. The oldest definitions of trigonometric functions, related
Trigonometric_functions
Topics referred to by the same term
plane in mathematics Hyperbolic geometry, a non-Euclidean geometry Hyperbolic functions, analogues of ordinary trigonometric functions, defined using the
Hyperbolic
Type of mathematical function
division provides the hyperbolic functions, while initial composition with i z {\displaystyle iz} instead provides the trigonometric functions. Examples of elementary
Elementary_function
integrals (antiderivatives) of expressions involving the inverse hyperbolic functions. For a complete list of integral formulas, see lists of integrals
List of integrals of inverse hyperbolic functions
List_of_integrals_of_inverse_hyperbolic_functions
Geometric mean and hyperbolic angle as coordinates in quadrant I
stock splits versus stock buy-back. The hyperbolic functions sinh, cosh, and tanh can be illustrated with hyperbolic coordinates. Let A = ( e − u , e u )
Hyperbolic_coordinates
Change of variable for integrals involving trigonometric functions
functions we have a function from angles to slopes. As with other properties shared between the trigonometric functions and the hyperbolic functions,
Tangent half-angle substitution
Tangent_half-angle_substitution
Mathematical formula involving a given set of operations
and include trigonometric functions, inverse trigonometric functions, hyperbolic functions, and inverse hyperbolic functions. The fundamental problem of
Closed-form_expression
Linear map that preserves areas
transformation groups. The hyperbolic functions, which take hyperbolic angle as argument, perform the role that circular functions play with the circular
Squeeze_mapping
Mathematical technique in thermal field theory
&{\text{if }}\eta =-1.\end{cases}}} The Bose distribution function is related to hyperbolic cotangent function by n B ( ξ ) = 1 2 ( coth β ξ 2 − 1 ) . {\displaystyle
Matsubara_summation
Rules for computing derivatives of functions
rules for computing the derivative of a function in calculus. Unless otherwise stated, all functions are functions of real numbers ( R {\textstyle \mathbb
Differentiation_rules
Inverse functions of sin, cos, tan, etc.
trigonometric functions (occasionally also called antitrigonometric, cyclometric, or arcus functions) are the inverse functions of the trigonometric functions, under
Inverse trigonometric functions
Inverse_trigonometric_functions
Italian mathematician and physicist (1707–1775)
priest, mathematician, and physicist. He is credited with introducing hyperbolic functions. Vincenzo Riccati was born in 1707 in Castelfranco Veneto, a small
Vincenzo_Riccati
Special function defined by an integral
{\displaystyle \operatorname {Ci} (x)=\gamma +\ln x-\operatorname {Cin} (x)~.} The hyperbolic sine integral is defined as Shi ( x ) = ∫ 0 x sinh ( t ) t d t . {\displaystyle
Trigonometric_integral
Mathematical concept
function is typically called the arcsine function, written as arcsin(x). Similarly, the inverse of a hyperbolic function is indicated by the prefix "ar" (for
Inverse_function
functions and their inverses in terms of the exponential function and the complex logarithm. Trigonometric functions may be deduced from hyperbolic functions
List of trigonometric identities
List_of_trigonometric_identities
Equation that is satisfied for all values of the variables
even number of hyperbolic sines. The Gudermannian function gives a direct relationship between the trigonometric functions and the hyperbolic ones that does
Identity_(mathematics)
Triangle in hyperbolic geometry
In hyperbolic geometry, a hyperbolic triangle is a triangle in the hyperbolic plane. It consists of three line segments called sides or edges and three
Hyperbolic_triangle
Economics concept
In economics, hyperbolic discounting is a time-inconsistent model of delay discounting. It is one of the cornerstones of behavioral economics and its brain-basis
Hyperbolic_discounting
Mathematical identity
}}}}}}}}.} The inverse hyperbolic functions are related to the inverse trigonometric functions similar to how the hyperbolic functions are related to the
Euler's continued fraction formula
Euler's_continued_fraction_formula
Number with a real and an imaginary part
The value of a trigonometric or hyperbolic function of a complex number can be expressed in terms of those functions evaluated on real numbers, via angle-addition
Complex_number
Mathematical function
In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum, as
Jacobi_elliptic_functions
Family of linear transformations
that resembles circular rotations in 3-dimensional space using the hyperbolic functions. For the boost in the x direction, the results are Lorentz boost
Lorentz_transformation
Commonly encountered and tricky integral
simplest case. The other cases are done in the same way. The utility of hyperbolic functions in integration can be demonstrated in cases of odd powers of secant
Integral_of_secant_cubed
Mathematical functions
In mathematics, the lemniscate elliptic functions are elliptic functions related to the arc length of the lemniscate of Bernoulli. They were first studied
Lemniscate_elliptic_functions
Collision in which kinetic energy is conserved
)\cosh(s_{4})-\sinh(s_{4})\sinh(s_{3}))} for non-zero mass, using the hyperbolic trigonometric identity cosh ( a − b ) = cosh ( a ) cosh ( b ) −
Elastic_collision
Extension of the factorial function
of two gamma functions. Starting from this formula, the exponential function as well as all the trigonometric and hyperbolic functions can be expressed
Gamma_function
Generalized mathematical function
derivative of a constant function is 0. Inverse hyperbolic functions over the complex domain are multiple-valued because hyperbolic functions are periodic along
Multivalued_function
Development of linear transformations forming the Lorentz group
metric, hyperboloid model and other models of hyperbolic geometry, computations of elliptic functions and integrals, transformation of indefinite quadratic
History of Lorentz transformations
History_of_Lorentz_transformations
Specific values of a multivalued function
several graphs of smooth functions, which are called branches of the multivalued functions. In the case of complex analytic functions, these branches can be
Principal_value
there is a corresponding formula in the list of integrals of inverse hyperbolic functions. ∫ arcsin ( x ) d x = x arcsin ( x ) + 1 − x 2 + C {\displaystyle
List of integrals of inverse trigonometric functions
List_of_integrals_of_inverse_trigonometric_functions
Sigmoid shape special function
simple approximation for real-valued arguments can be done through hyperbolic functions: erf ( x ) ≈ z ( x ) = tanh ( 2 π ( x + 11 123 x 3 ) ) {\displaystyle
Error_function
Probability distribution
{\displaystyle t=1} give the expected value of these basic trigonometric and hyperbolic functions over a Gaussian random variable X ∼ N ( μ , σ 2 ) {\displaystyle
Normal_distribution
Polynomial equation of degree 3
real root (and p ≠ 0), this root can be similarly represented using hyperbolic functions, as t 0 = − 2 | q | q − p 3 cosh [ 1 3 arcosh ( − 3 | q | 2 p
Cubic_equation
)^{4}}}x^{2n}\end{aligned}}} The Jacobi theta functions describe the world of the elliptic modular functions and they have these Taylor series: ϑ 00 ( x
List_of_mathematical_series
Analytic function in mathematics
Riemann zeta function, such as Dirichlet series, Dirichlet L-functions and L-functions, are known. The Riemann zeta function ζ(s) is a function of a complex
Riemann_zeta_function
Quadric surface with one axis of symmetry and no center of symmetry
plane parallel to the axis of symmetry is a parabola. The paraboloid is hyperbolic if every other plane section is either a hyperbola, or two crossing lines
Paraboloid
C standard library header file
operations are a group of functions in the standard library of the C programming language implementing basic mathematical functions. Different C standards
C_mathematical_functions
Measure of relativistic velocity
associated with distance and time coordinates. Using the inverse hyperbolic function artanh, the rapidity w corresponding to velocity v is: w = artanh
Rapidity
Type of partial differential equations
In mathematics, a hyperbolic partial differential equation of order n {\displaystyle n} is a partial differential equation (PDE) that, roughly speaking
Hyperbolic partial differential equation
Hyperbolic_partial_differential_equation
Mathematical functions having established names and notations
Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical
Special_functions
of hyperbolic functions List of integrals of exponential functions List of integrals of logarithmic functions List of integrals of area functions Partial
List_of_calculus_topics
Fundamental trigonometric functions
elliptic functions Euler's formula Generalized trigonometry Hyperbolic function Lemniscate elliptic functions Law of sines List of periodic functions List
Sine_and_cosine
a list of indefinite sums (also known as antidifferences) of various functions. An indefinite sum ∑ x f ( x ) {\textstyle \sum _{x}f(x)} is the inverse
List_of_indefinite_sums
Figure formed by two rays meeting at a common point
just alternating series forms of the hyperbolic functions. This comparison of the two series corresponding to functions of angles was described by Leonhard
Angle
Input to a mathematical function
argument of a circular function is an angle. The argument of a hyperbolic function is a hyperbolic angle. A mathematical function has one or more arguments
Argument_of_a_function
Topics referred to by the same term
triangles in plane geometry) The use of the hyperbolic functions The use of gyrotrigonometry in hyperbolic geometry This disambiguation page lists mathematics
Hyperbolic_trigonometry
Aids the computation of indefinite integrals involving sines and cosines
) {\displaystyle \sin t,\tan t,\cos(2t),\tan(t/2)} ), in the case of hyperbolic sine and cosine, a good change of variable is u = cosh ( t ) {\displaystyle
Bioche's_rules
integrals of exponential functions List of integrals of hyperbolic functions List of integrals of inverse hyperbolic functions List of integrals of inverse
Lists_of_mathematics_topics
Scientific calculator manufactured by Casio and released in 1978
key. In the previous version, the hyperbolic functions had to be carried out by using the exponential (e^x) function key, employing the formulas: sinh
Casio_fx-39
Indefinite sum Gamma function List of limits Reynolds, Robert; Stauffer, Allan (2020). "Derivation of Logarithmic and Logarithmic Hyperbolic Tangent Integrals
List_of_definite_integrals
One-dimensional complex manifold
non-constant negative subharmonic functions on the surface and is otherwise called hyperbolic. This class of hyperbolic surfaces is further subdivided into
Riemann_surface
Study of triangles in other spaces than the Euclidean plane
trigonometric functions but differ from the plane triangle identities. Hyperbolic trigonometry: Study of hyperbolic triangles in hyperbolic geometry with
Generalized_trigonometry
Device used for calculations
simple graph-based calculator for solving line equations involving hyperbolic functions. This allowed electrical engineers to simplify calculations for inductance
Calculator
Continuous probability distribution
are proportional to the hyperbolic secant function. The hyperbolic secant function is equivalent to the reciprocal hyperbolic cosine, and thus this distribution
Hyperbolic secant distribution
Hyperbolic_secant_distribution
Special function occurring in problems possessing elliptic symmetry
In mathematics, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's differential equation d 2 y d x 2 + ( a − 2
Mathieu_function
Relaxation model
\alpha \pi /2+(\omega \tau )^{2(1-\alpha )}}}} Upon introduction of hyperbolic functions, the above expressions reduce to: ε ′ = ε ∞ + 1 2 ( ε 0 − ε ∞ ) [
Cole–Cole_equation
Topics referred to by the same term
disease Autosomal recessive inheritance ar-, a prefix of inverse hyperbolic functions Autoregressive model, concerning random processes in statistics Aqua
AR
On rays from a point to a line, with equal inscribed circles between adjacent rays
relate to a continuous scaling function which defines the spacing of the rays. In fact, this function is the hyperbolic sine. The theorem is a direct corollary
Equal_incircles_theorem
Function that takes one argument
unary functions, including the trigonometric functions, logarithm with a specified base, exponentiation to a particular power or base, and hyperbolic functions
Unary_function
Set of spacetime events, light-connected to a given event
angles, and calculated with trig functions. Space-time tilt is measured by rapidity, and calculated with hyperbolic functions.) In flat spacetime, the future
Light_cone
Topics referred to by the same term
equipped with at least one multivalued operation Hyperbolic functions, analogues of trigonometric functions defined using the hyperbola rather than the circle
Hyper
American mathematician
professor emeritus in 1933. Haskell provided a foundation for hyperbolic angle and hyperbolic functions with his article in Bulletin of American Mathematical
Mellen_Woodman_Haskell
Mechanical analog computer
exponential and logarithmic functions; the HP had trigonometric functions (sine, cosine, and tangent) and hyperbolic trigonometric functions as well. The HP used
Slide_rule
Numerical integration method
infinite derivatives exist at one or both endpoints. The method uses hyperbolic functions in the change of variables x = tanh ( 1 2 π sinh t ) {\displaystyle
Tanh-sinh_quadrature
Category of coordinate systems
constant Gaussian curvature of the plane is −1. Sinh, cosh and tanh are hyperbolic functions. The polar coordinate system is a two-dimensional coordinate system
Coordinate systems for the hyperbolic plane
Coordinate_systems_for_the_hyperbolic_plane
Relates the tangent of half of an angle to trigonometric functions of the entire angle
the Gudermannian function. The Gudermannian function gives a direct relationship between the circular functions and the hyperbolic ones that does not
Tangent_half-angle_formula
Theory of interwoven space and time by Albert Einstein
Lorentz boosts represent hyperbolic rotations in Minkowski spacetime.[citation needed] The advantages of using hyperbolic functions are such that some textbooks
Special_relativity
Early scientific pocket calculator
and hyperbolic trig functions, which were found on very few calculators (including the HP-35 and HP-45) at the time. The user invoked the hyperbolic functions
TI_SR-50
Angle in certain right triangles in the hyperbolic plane
(a),} where sinh, cosh, tanh, sech and csch are hyperbolic functions and gd is the Gudermannian function. János Bolyai discovered a construction which gives
Angle_of_parallelism
geometry looking at hyperbolic space. hyperbolic trigonometry the study of hyperbolic triangles in hyperbolic geometry, or hyperbolic functions in Euclidean
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
functions Trigonometric functions Hyperbolic functions Logarithmic functions Inverse trigonometric functions Inverse hyperbolic functions Residue theory Isometries
List of complex analysis topics
List_of_complex_analysis_topics
S-shaped curve
boundaries. Cross fluid Hyperbolic growth Heaviside step function Hill equation (biochemistry) Hubbert curve List of mathematical functions STAR model Michaelis–Menten
Logistic_function
American electrical engineer and professor (1883–1959)
on hyperbolic functions, equivalent circuits, and graphical analysis of electric power systems. On February 8, 1926, she showed the use of hyperbolic functions
Edith_Clarke
Functions such that f(–x) equals f(x) or –f(x)
an odd integer. Even functions are those real functions whose graph is self-symmetric with respect to the y-axis, and odd functions are those whose graph
Even_and_odd_functions
relativity, and mathematical tools, spanning from elementary algebra and hyperbolic functions, to linear algebra and group theory. This article provides a few
Derivations of the Lorentz transformations
Derivations_of_the_Lorentz_transformations
\end{aligned}}} Similarly, by comparing with the expansion of the hyperbolic functions sinh and cosh we find for x < 0 : {\displaystyle ~x<0\ :} c 0
Stumpff_function
Topics referred to by the same term
formal grammars Continuum hypothesis, in set theory Hyperbolic cosine, in mathematics, a hyperbolic function, ch(x) = cosh(x) Curry–Howard correspondence, the
CH
Important functions in solving differential equations
Y\end{aligned}}} The tangent, as well as inverse trigonometric functions, hyperbolic and inverse hyperbolic functions have also been defined for matrices: arcsin X
Trigonometric functions of matrices
Trigonometric_functions_of_matrices
Triangle containing a 90-degree angle
For the expression of hyperbolic functions as ratio of the sides of a right triangle, see the hyperbolic triangle of a hyperbolic sector. The values of
Right_triangle
American electrical engineer and mathematician (1861–1939)
applied mathematics by communicating the theory of the hyperbolic angle and hyperbolic functions, first in a course at the University of London and then
Arthur_Edwin_Kennelly
Product of numbers from 1 to n
other Taylor series (in particular those of the trigonometric and hyperbolic functions), where they cancel factors of n ! {\displaystyle n!} coming from
Factorial
coordinates of nodes are sprinkled according to a probability density function into a hyperbolic space of constant negative curvature and (2) an edge between two
Hyperbolic_geometric_graph
Concept in mathematics
In mathematics, a hyperbolic metric space is a metric space satisfying certain metric relations (depending quantitatively on a nonnegative real number
Hyperbolic_metric_space
HYPERBOLIC FUNCTIONS
HYPERBOLIC FUNCTIONS
Surname or Lastname
English (chiefly Kent and Sussex)
English (chiefly Kent and Sussex) : occupational name for a designer or engineer, from a Middle English reduced form of Old French engineor ‘contriver’ (a derivative of engaigne ‘cunning’, ‘ingenuity’, ‘stratagem’, ‘device’). Engineers in the Middle Ages were primarily designers and builders of military machines, although in peacetime they might turn their hands to architecture and other more pacific functions.German : from the Latin personal name Januarius (see January 1). Jänner is a South German word for ‘January’, and so it is possible that this is one of the surnames acquired from words denoting months of the year, for example by converts who had been baptized in that month, people who were born or baptized in that month, or people whose taxes were due in January.
Surname or Lastname
English
English : nickname from the animal, Middle English catte ‘cat’. The word is found in similar forms in most European languages from very early times (e.g. Gaelic cath, Slavic kotu). Domestic cats were unknown in Europe in classical times, when weasels fulfilled many of their functions, for example in hunting rodents. They seem to have come from Egypt, where they were regarded as sacred animals.English : from a medieval female personal name, a short form of Catherine.Variant spelling of German and Dutch Katt.
HYPERBOLIC FUNCTIONS
HYPERBOLIC FUNCTIONS
Girl/Female
Tamil
A flower
Girl/Female
Teutonic
Famous in war.
Boy/Male
French
Name of a count.
Surname or Lastname
English or Scottish
English or Scottish : unexplained. Possibly, as Black suggests, a reduced form of Langdon.French : from the old Germanic personal name element Lando (see Land), via the oblique case, Landonis.
Boy/Male
English
From the Cattle Crossing
Girl/Female
French, Indian, Tamil
Queen of Sky
Boy/Male
Gujarati, Hindu, Indian
Goddess Parvati / Lord Shiva
Female
Chinese
fragrant, incense.
Girl/Female
Tamil
Maralika | மாராலிகா
Small swan
Girl/Female
Latin American
Industrious; striving.
HYPERBOLIC FUNCTIONS
HYPERBOLIC FUNCTIONS
HYPERBOLIC FUNCTIONS
HYPERBOLIC FUNCTIONS
HYPERBOLIC FUNCTIONS
a.
Having some property that belongs to an hyperboloid or hyperbola.
a.
Having the form, or nearly the form, of an hyperbola.
n.
A surface of the second order, which is cut by certain planes in hyperbolas; also, the solid, bounded in part by such a surface.
a.
Relating to, containing, or of the nature of, hyperbole; exaggerating or diminishing beyond the fact; exceeding the truth; as, an hyperbolical expression.
adv.
In the form of an hyperbola.
v. t.
To state or represent hyperbolically.
v. i.
To speak or write with exaggeration.
imp. & p. p.
of Hyperbolize
n.
A figure of speech in which the expression is an evident exaggeration of the meaning intended to be conveyed, or by which things are represented as much greater or less, better or worse, than they really are; a statement exaggerated fancifully, through excitement, or for effect.
a.
Exaggerated; excessive; hyperbolical.
a.
Alt. of Hyperbolical
n.
Diminution; a species of hyperbole, representing a thing as being less than it really is.
a.
Belonging to the hyperbola; having the nature of the hyperbola.
n.
A curve formed by a section of a cone, when the cutting plane makes a greater angle with the base than the side of the cone makes. It is a plane curve such that the difference of the distances from any point of it to two fixed points, called foci, is equal to a given distance. See Focus. If the cutting plane be produced so as to cut the opposite cone, another curve will be formed, which is also an hyperbola. Both curves are regarded as branches of the same hyperbola. See Illust. of Conic section, and Focus.
p. pr. & vb. n.
of Hyperbolize
n.
One who uses hyperboles.
n.
A figure by which a grave and magnificent word is put for the proper word; amplification; hyperbole.
a.
Of or pertaining to an hyperbaton; transposed; inverted.
n.
The use of hyperbole.
n.
The act of exaggerating; the act of doing or representing in an excessive manner; a going beyond the bounds of truth reason, or justice; a hyperbolical representation; hyperbole; overstatement.