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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, hyperbolic functions are analogues of the ordinary
trigonometric function In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
s, but defined using the
hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...
rather than the
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
. Just as the points form a circle with a unit radius, the points form the right half of the
unit hyperbola In geometry, the unit hyperbola is the set of points (''x'',''y'') in the Cartesian plane that satisfy the implicit equation x^2 - y^2 = 1 . In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an ''alternative ra ...
. Also, similarly to how the derivatives of and are and respectively, the derivatives of and are and respectively. Hyperbolic functions occur in the calculations of angles and distances in
hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P ...
. They also occur in the solutions of many linear
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
s (such as the equation defining a
catenary In physics and geometry, a catenary (, ) is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends in a uniform gravitational field. The catenary curve has a U-like shape, superfici ...
), cubic equations, and
Laplace's equation In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delta = \na ...
in
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
.
Laplace's equation In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delta = \na ...
s are important in many areas of
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, including electromagnetic theory,
heat transfer Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy ( heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conducti ...
,
fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) a ...
, and
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...
. The basic hyperbolic functions are: * hyperbolic sine "" (), * hyperbolic cosine "" (),''Collins Concise Dictionary'', p. 328 from which are derived: * hyperbolic tangent "" (), * hyperbolic cosecant "" or "" () * hyperbolic secant "" (), * hyperbolic cotangent "" (), corresponding to the derived trigonometric functions. The
inverse hyperbolic functions In mathematics, the inverse hyperbolic functions are the inverse functions of the hyperbolic functions. For a given value of a hyperbolic function, the corresponding inverse hyperbolic function provides the corresponding hyperbolic angle. T ...
are: * area hyperbolic sine "" (also denoted "", "" or sometimes "") * area hyperbolic cosine "" (also denoted "", "" or sometimes "") * and so on. The hyperbolic functions take a real argument called a
hyperbolic angle In geometry, hyperbolic angle is a real number determined by the area of the corresponding hyperbolic sector of ''xy'' = 1 in Quadrant I of the Cartesian plane. The hyperbolic angle parametrises the unit hyperbola, which has hyperbolic function ...
. The size of a hyperbolic angle is twice the area of its
hyperbolic sector A hyperbolic sector is a region of the Cartesian plane bounded by a hyperbola and two rays from the origin to it. For example, the two points and on the rectangular hyperbola , or the corresponding region when this hyperbola is re-scaled and ...
. The hyperbolic functions may be defined in terms of the legs of a right triangle covering this sector. In
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, the hyperbolic functions arise when applying the ordinary sine and cosine functions to an imaginary angle. The hyperbolic sine and the hyperbolic cosine are
entire function In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any fin ...
s. As a result, the other hyperbolic functions are meromorphic in the whole complex plane. By
Lindemann–Weierstrass theorem In transcendental number theory, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states the following: In other words, the extension field \mathbb(e^, \dots, e^) has transce ...
, the hyperbolic functions have a transcendental value for every non-zero algebraic value of the argument. Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati and
Johann Heinrich Lambert Johann Heinrich Lambert (, ''Jean-Henri Lambert'' in French; 26 or 28 August 1728 – 25 September 1777) was a polymath from the Republic of Mulhouse, generally referred to as either Swiss or French, who made important contributions to the subject ...
. Riccati used and (''sinus/cosinus circulare'') to refer to circular functions and and (''sinus/cosinus hyperbolico'') to refer to hyperbolic functions. Lambert adopted the names, but altered the abbreviations to those used today. The abbreviations , , , are also currently used, depending on personal preference.


Notation


Definitions

There are various equivalent ways to define the hyperbolic functions.


Exponential definitions

In terms of the
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
: * Hyperbolic sine: the odd part of the exponential function, that is, \sinh x = \frac = \frac = \frac . * Hyperbolic cosine: the even part of the exponential function, that is, \cosh x = \frac = \frac = \frac . * Hyperbolic tangent: \tanh x = \frac = \frac = \frac . * Hyperbolic cotangent: for , \coth x = \frac = \frac = \frac . * Hyperbolic secant: \operatorname x = \frac = \frac = \frac . * Hyperbolic cosecant: for , \operatorname x = \frac = \frac = \frac .


Differential equation definitions

The hyperbolic functions may be defined as solutions of
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
s: The hyperbolic sine and cosine are the solution of the system \begin c'(x)&=s(x),\\ s'(x)&=c(x),\\ \end with the initial conditions s(0) = 0, c(0) = 1. The initial conditions make the solution unique; without them any pair of functions (a e^x + b e^, a e^x - b e^) would be a solution. and are also the unique solution of the equation , such that , for the hyperbolic cosine, and , for the hyperbolic sine.


Complex trigonometric definitions

Hyperbolic functions may also be deduced from
trigonometric function In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
s with complex arguments: * Hyperbolic sine: \sinh x = -i \sin (i x). * Hyperbolic cosine: \cosh x = \cos (i x). * Hyperbolic tangent: \tanh x = -i \tan (i x). * Hyperbolic cotangent: \coth x = i \cot (i x). * Hyperbolic secant: \operatorname x = \sec (i x). * Hyperbolic cosecant:\operatorname x = i \csc (i x). where is the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
with . The above definitions are related to the exponential definitions via
Euler's formula 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. Euler's formula states that ...
(See below).


Characterizing properties


Hyperbolic cosine

It can be shown that the area under the curve of the hyperbolic cosine (over a finite interval) is always equal to the arc length corresponding to that interval: \text = \int_a^b \cosh x \,dx = \int_a^b \sqrt \,dx = \text


Hyperbolic tangent

The hyperbolic tangent is the (unique) solution to the
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
, with .


Useful relations

The hyperbolic functions satisfy many identities, all of them similar in form to the trigonometric identities. In fact, Osborn's rule states that one can convert any trigonometric identity for \theta, 2\theta, 3\theta or \theta and \varphi into a hyperbolic identity, by expanding it completely in terms of integral powers of sines and cosines, changing sine to sinh and cosine to cosh, and switching the sign of every term containing a product of two sinhs. Odd and even functions: \begin \sinh (-x) &= -\sinh x \\ \cosh (-x) &= \cosh x \end Hence: \begin \tanh (-x) &= -\tanh x \\ \coth (-x) &= -\coth x \\ \operatorname (-x) &= \operatorname x \\ \operatorname (-x) &= -\operatorname x \end Thus, and are
even function In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power se ...
s; the others are odd functions. \begin \operatorname x &= \operatorname \left(\frac\right) \\ \operatorname x &= \operatorname \left(\frac\right) \\ \operatorname x &= \operatorname \left(\frac\right) \end Hyperbolic sine and cosine satisfy: \begin \cosh x + \sinh x &= e^x \\ \cosh x - \sinh x &= e^ \\ \cosh^2 x - \sinh^2 x &= 1 \end the last of which is similar to the
Pythagorean trigonometric identity The Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations b ...
. One also has \begin \operatorname ^ x &= 1 - \tanh^ x \\ \operatorname ^ x &= \coth^ x - 1 \end for the other functions.


Sums of arguments

\begin \sinh(x + y) &= \sinh x \cosh y + \cosh x \sinh y \\ \cosh(x + y) &= \cosh x \cosh y + \sinh x \sinh y \\ px \tanh(x + y) &= \frac \\ \end particularly \begin \cosh (2x) &= \sinh^2 + \cosh^2 = 2\sinh^2 x + 1 = 2\cosh^2 x - 1\\ \sinh (2x) &= 2\sinh x \cosh x \\ \tanh (2x) &= \frac \\ \end Also: \begin \sinh x + \sinh y &= 2 \sinh \left(\frac\right) \cosh \left(\frac\right)\\ \cosh x + \cosh y &= 2 \cosh \left(\frac\right) \cosh \left(\frac\right)\\ \end


Subtraction formulas

\begin \sinh(x - y) &= \sinh x \cosh y - \cosh x \sinh y \\ \cosh(x - y) &= \cosh x \cosh y - \sinh x \sinh y \\ \tanh(x - y) &= \frac \\ \end Also: \begin \sinh x - \sinh y &= 2 \cosh \left(\frac\right) \sinh \left(\frac\right)\\ \cosh x - \cosh y &= 2 \sinh \left(\frac\right) \sinh \left(\frac\right)\\ \end


Half argument formulas

\begin \sinh\left(\frac\right) &= \frac &&= \sgn x \, \sqrt \frac \\ px \cosh\left(\frac\right) &= \sqrt \frac\\ px \tanh\left(\frac\right) &= \frac &&= \sgn x \, \sqrt \frac = \frac \end where is the
sign function In mathematics, the sign function or signum function (from '' signum'', Latin for "sign") is an odd mathematical function that extracts the sign of a real number. In mathematical expressions the sign function is often represented as . To avo ...
. If , then \tanh\left(\frac\right) = \frac = \coth x - \operatorname x


Square formulas

\begin \sinh^2 x &= \tfrac(\cosh 2x - 1) \\ \cosh^2 x &= \tfrac(\cosh 2x + 1) \end


Inequalities

The following inequality is useful in statistics: \operatorname(t) \leq e^ It can be proved by comparing term by term the Taylor series of the two functions.


Inverse functions as logarithms

\begin \operatorname (x) &= \ln \left(x + \sqrt \right) \\ \operatorname (x) &= \ln \left(x + \sqrt \right) && x \geq 1 \\ \operatorname (x) &= \frac\ln \left( \frac \right) && , x , < 1 \\ \operatorname (x) &= \frac\ln \left( \frac \right) && , x, > 1 \\ \operatorname (x) &= \ln \left( \frac + \sqrt\right) = \ln \left( \frac \right) && 0 < x \leq 1 \\ \operatorname (x) &= \ln \left( \frac + \sqrt\right) && x \ne 0 \end


Derivatives

\begin \frac\sinh x &= \cosh x \\ \frac\cosh x &= \sinh x \\ \frac\tanh x &= 1 - \tanh^2 x = \operatorname^2 x = \frac \\ \frac\coth x &= 1 - \coth^2 x = -\operatorname^2 x = -\frac && x \neq 0 \\ \frac\operatorname x &= - \tanh x \operatorname x \\ \frac\operatorname x &= - \coth x \operatorname x && x \neq 0 \end \begin \frac\operatorname x &= \frac \\ \frac\operatorname x &= \frac && 1 < x \\ \frac\operatorname x &= \frac && , x, < 1 \\ \frac\operatorname x &= \frac && 1 < , x, \\ \frac\operatorname x &= -\frac && 0 < x < 1 \\ \frac\operatorname x &= -\frac && x \neq 0 \end


Second derivatives

Each of the functions and is equal to its
second derivative In calculus, the second derivative, or the second order derivative, of a function is the derivative of the derivative of . Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, ...
, that is: \frac\sinh x = \sinh x \frac\cosh x = \cosh x \, . All functions with this property are linear combinations of and , in particular the
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
s e^x and e^ .


Standard integrals

\begin \int \sinh (ax)\,dx &= a^ \cosh (ax) + C \\ \int \cosh (ax)\,dx &= a^ \sinh (ax) + C \\ \int \tanh (ax)\,dx &= a^ \ln (\cosh (ax)) + C \\ \int \coth (ax)\,dx &= a^ \ln \left, \sinh (ax)\ + C \\ \int \operatorname (ax)\,dx &= a^ \arctan (\sinh (ax)) + C \\ \int \operatorname (ax)\,dx &= a^ \ln \left, \tanh \left( \frac \right) \ + C = a^ \ln\left, \coth \left(ax\right) - \operatorname \left(ax\right)\ + C = -a^\operatorname \left(\cosh\left(ax\right)\right) +C \end The following integrals can be proved using hyperbolic substitution: \begin \int & = \operatorname \left( \frac \right) + C \\ \int &= \sgn \operatorname \left, \frac \ + C \\ \int \,du & = a^\operatorname \left( \frac \right) + C && u^2 < a^2 \\ \int \,du & = a^\operatorname \left( \frac \right) + C && u^2 > a^2 \\ \int & = -a^\operatorname\left, \frac \ + C \\ \int & = -a^\operatorname\left, \frac \ + C \end where ''C'' is the
constant of integration In calculus, the constant of integration, often denoted by C (or c), is a constant term added to an antiderivative of a function f(x) to indicate that the indefinite integral of f(x) (i.e., the set of all antiderivatives of f(x)), on a connecte ...
.


Taylor series expressions

It is possible to express explicitly the
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
at zero (or the
Laurent series In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion c ...
, if the function is not defined at zero) of the above functions. \sinh x = x + \frac + \frac + \frac + \cdots = \sum_^\infty \frac This series is convergent for every complex value of . Since the function is
odd Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric. Odd may also refer to: Acronym * ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...
, only odd exponents for occur in its Taylor series. \cosh x = 1 + \frac + \frac + \frac + \cdots = \sum_^\infty \frac This series is convergent for every complex value of . Since the function is
even Even may refer to: General * Even (given name), a Norwegian male personal name * Even (surname) * Even (people), an ethnic group from Siberia and Russian Far East **Even language, a language spoken by the Evens * Odd and Even, a solitaire game wh ...
, only even exponents for occur in its Taylor series. The sum of the sinh and cosh series is the
infinite series In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
expression of the
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
. The following series are followed by a description of a subset of their domain of convergence, where the series is convergent and its sum equals the function. \begin \tanh x &= x - \frac + \frac - \frac + \cdots = \sum_^\infty \frac, \qquad \left , x \right , < \frac \\ \coth x &= x^ + \frac - \frac + \frac + \cdots = \sum_^\infty \frac , \qquad 0 < \left , x \right , < \pi \\ \operatorname x &= 1 - \frac + \frac - \frac + \cdots = \sum_^\infty \frac , \qquad \left , x \right , < \frac \\ \operatorname x &= x^ - \frac +\frac -\frac + \cdots = \sum_^\infty \frac , \qquad 0 < \left , x \right , < \pi \end where: *B_n is the ''n''th
Bernoulli number In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, ...
*E_n is the ''n''th Euler number


Infinite products and continued fractions

The following expansions are valid in the whole complex plane: :\sinh x = x\prod_^\infty\left(1+\frac\right) = \cfrac :\cosh x = \prod_^\infty\left(1+\frac\right) = \cfrac :\tanh x = \cfrac


Comparison with circular functions

The hyperbolic functions represent an expansion of
trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...
beyond the circular functions. Both types depend on an
argument An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialecti ...
, either circular angle or
hyperbolic angle In geometry, hyperbolic angle is a real number determined by the area of the corresponding hyperbolic sector of ''xy'' = 1 in Quadrant I of the Cartesian plane. The hyperbolic angle parametrises the unit hyperbola, which has hyperbolic function ...
. Since the area of a circular sector with radius and angle (in radians) is , it will be equal to when . In the diagram, such a circle is tangent to the hyperbola ''xy'' = 1 at (1,1). The yellow sector depicts an area and angle magnitude. Similarly, the yellow and red sectors together depict an area and hyperbolic angle magnitude. The legs of the two
right triangle A right triangle (American English) or right-angled triangle ( British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right a ...
s with hypotenuse on the ray defining the angles are of length times the circular and hyperbolic functions. The hyperbolic angle is an
invariant measure In mathematics, an invariant measure is a measure that is preserved by some function. The function may be a geometric transformation. For examples, circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mapping ...
with respect to the squeeze mapping, just as the circular angle is invariant under rotation. Mellen W. Haskell, "On the introduction of the notion of hyperbolic functions",
Bulletin of the American Mathematical Society The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. ...
1:6:155–9
full text
/ref> The
Gudermannian function In mathematics, the Gudermannian function relates a hyperbolic angle measure \psi to a circular angle measure \phi called the ''gudermannian'' of \psi and denoted \operatorname\psi. The Gudermannian function reveals a close relationship betwe ...
gives a direct relationship between the circular functions and the hyperbolic functions that does not involve complex numbers. The graph of the function is the
catenary In physics and geometry, a catenary (, ) is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends in a uniform gravitational field. The catenary curve has a U-like shape, superfici ...
, the curve formed by a uniform flexible chain, hanging freely between two fixed points under uniform gravity.


Relationship to the exponential function

The decomposition of the exponential function in its even and odd parts gives the identities e^x = \cosh x + \sinh x, and e^ = \cosh x - \sinh x. Combined with
Euler's formula 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. Euler's formula states that ...
e^ = \cos x + i\sin x, this gives e^=(\cosh x+\sinh x)(\cos y+i\sin y) for the general complex exponential function. Additionally, e^x = \sqrt = \frac


Hyperbolic functions for complex numbers

Since the
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, ...
can be defined for any complex argument, we can also extend the definitions of the hyperbolic functions to complex arguments. The functions and are then holomorphic. Relationships to ordinary trigonometric functions are given by
Euler's formula 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. Euler's formula states that ...
for complex numbers: \begin e^ &= \cos x + i \sin x \\ e^ &= \cos x - i \sin x \end so: \begin \cosh(ix) &= \frac \left(e^ + e^\right) = \cos x \\ \sinh(ix) &= \frac \left(e^ - e^\right) = i \sin x \\ \cosh(x+iy) &= \cosh(x) \cos(y) + i \sinh(x) \sin(y) \\ \sinh(x+iy) &= \sinh(x) \cos(y) + i \cosh(x) \sin(y) \\ \tanh(ix) &= i \tan x \\ \cosh x &= \cos(ix) \\ \sinh x &= - i \sin(ix) \\ \tanh x &= - i \tan(ix) \end Thus, hyperbolic functions are periodic with respect to the imaginary component, with period 2 \pi i (\pi i for hyperbolic tangent and cotangent).


See also

*
e (mathematical constant) The number , also known as Euler's number, is a mathematical constant approximately equal to 2.71828 that can be characterized in many ways. It is the base of the natural logarithms. It is the limit of as approaches infinity, an expressi ...
*
Equal incircles theorem In geometry, the equal incircles theorem derives from a Japanese Sangaku, and pertains to the following construction: a series of rays are drawn from a given point to a given line such that the inscribed circles of the triangles formed by adjacent ...
, based on sinh *
Hyperbolic growth When a quantity grows towards a singularity under a finite variation (a "finite-time singularity") it is said to undergo hyperbolic growth. More precisely, the reciprocal function 1/x has a hyperbola as a graph, and has a singularity at 0, meani ...
* Inverse hyperbolic functions * List of integrals of hyperbolic functions *
Poinsot's spirals In mathematics, Poinsot's spirals are two spirals represented by the polar equations : r = a\ \operatorname (n\theta) : r = a\ \operatorname (n\theta) where csch is the hyperbolic cosecant, and sech is the hyperbolic secant. They are named after ...
*
Sigmoid function A sigmoid function is a mathematical function having a characteristic "S"-shaped curve or sigmoid curve. A common example of a sigmoid function is the logistic function shown in the first figure and defined by the formula: :S(x) = \frac = \f ...
*
Soboleva modified hyperbolic tangent The Soboleva modified hyperbolic tangent, also known as (parametric) Soboleva modified hyperbolic tangent activation function ( MHTAF), is a special S-shaped function based on the hyperbolic tangent, given by :\operatornamex = \frac . This fu ...
*
Trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...


References


External links

*
Hyperbolic functions
on
PlanetMath PlanetMath is a free, collaborative, mathematics online encyclopedia. The emphasis is on rigour, openness, pedagogy, real-time content, interlinked content, and also community of about 24,000 people with various maths interests. Intended to be c ...

GonioLab
Visualization of the unit circle, trigonometric and hyperbolic functions (
Java Web Start In computing, Java Web Start (also known as JavaWS, javaws or JAWS) is a deprecated framework developed by Sun Microsystems (now Oracle) that allows users to start application software for the Java Platform directly from the Internet using a web b ...
)
Web-based calculator of hyperbolic functions
{{DEFAULTSORT:Hyperbolic Function Exponentials Hyperbolic geometry Analytic functions