![sinh cosh tanh](https://upload.wikimedia.org/wikipedia/commons/2/29/Sinh%2Bcosh%2Btanh.svg)
In
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 con ...
. 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 radi ...
. 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, an ...
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
In algebra, a cubic equation in one variable is an equation of the form
:ax^3+bx^2+cx+d=0
in which is nonzero.
The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of th ...
, and
Laplace's equation in
Cartesian coordinates.
Laplace's equations 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 r ...
, including
electromagnetic theory
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
,
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 conduction, ...
,
fluid dynamics, 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 laws ...
.
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. Th ...
are:
* area hyperbolic sine "" (also denoted "", "" or sometimes "")
* area hyperbolic cosine "" (also denoted "", "" or sometimes "")
* and so on.
![Hyperbolic functions-2](https://upload.wikimedia.org/wikipedia/commons/b/bc/Hyperbolic_functions-2.svg)
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, 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 mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are poles of the function. The ...
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
Vincenzo Riccati (Castelfranco Veneto, 11 January 1707 – Treviso, 17 January 1775) was a Venetian mathematician and physicist.
Life
Vincenzo Riccati was the brother of Giordano Riccati, and the second son of Jacopo Riccati. He entered the ...
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
![csch sech coth](https://upload.wikimedia.org/wikipedia/commons/6/61/Csch_sech_coth.svg)
There are various equivalent ways to define the hyperbolic functions.
Exponential definitions
![Hyperbolic and exponential; cosh](https://upload.wikimedia.org/wikipedia/commons/e/ef/Hyperbolic_and_exponential%3B_cosh.svg)
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,
* Hyperbolic cosine: the
even part of the exponential function, that is,
* Hyperbolic tangent:
* Hyperbolic cotangent: for ,
* Hyperbolic secant:
* Hyperbolic cosecant: for ,
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, an ...
s: The hyperbolic sine and cosine are the solution of the system
with the initial conditions
The initial conditions make the solution unique; without them any pair of functions
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
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
arguments:
* Hyperbolic sine:
* Hyperbolic cosine:
* Hyperbolic tangent:
* Hyperbolic cotangent:
* Hyperbolic secant:
* Hyperbolic cosecant:
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 fo ...
(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:
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, an ...
, with .
Useful relations
The hyperbolic functions satisfy many identities, all of them similar in form to the
trigonometric identities
In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ...
. In fact, Osborn's rule
states that one can convert any trigonometric identity for
,
,
or
and
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:
Hence:
Thus, and are
even functions; the others are
odd functions
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 ser ...
.
Hyperbolic sine and cosine satisfy:
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
for the other functions.
Sums of arguments
particularly
Also:
Subtraction formulas
Also:
Half argument formulas
where is the
sign function.
If , then
Square formulas
Inequalities
The following inequality is useful in statistics:
It can be proved by comparing term by term the Taylor series of the two functions.
Inverse functions as logarithms
Derivatives
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:
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
and
.
Standard integrals
The following integrals can be proved using
hyperbolic substitution
In mathematics, trigonometric substitution is the replacement of trigonometric functions for other expressions. In calculus, trigonometric substitution is a technique for evaluating integrals. Moreover, one may use the trigonometric identitie ...
:
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 connected ...
.
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, if the function is not defined at zero) of the above functions.
This series is
convergent for every
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
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.
This series is
convergent for every
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
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 w ...
, 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
In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series ...
, where the series is convergent and its sum equals the function.
where:
*
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, ...
*
is the ''n''th
Euler number
In mathematics, the Euler numbers are a sequence ''En'' of integers defined by the Taylor series expansion
:\frac = \frac = \sum_^\infty \frac \cdot t^n,
where \cosh (t) is the hyperbolic cosine function. The Euler numbers are related to a ...
Infinite products and continued fractions
The following expansions are valid in the whole complex plane:
:
:
:
Comparison with circular functions
![Circular and hyperbolic angle](https://upload.wikimedia.org/wikipedia/commons/c/c6/Circular_and_hyperbolic_angle.svg)
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 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. Both types depend on an
argument, 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
In linear algebra, a squeeze mapping, also called a squeeze transformation, is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is ''not'' a rotation or shear mapping.
For a fixed positive real number , th ...
, 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. I ...
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 betwee ...
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
and
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 fo ...
this gives
for the general complex exponential function.
Additionally,
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
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
argument, we can also extend the definitions of the hyperbolic functions to complex arguments. The functions and are then holomorphic
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivati ...
.
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 fo ...
for complex numbers:
so:
Thus, hyperbolic functions are periodic with respect to the imaginary component, with period ( 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 function
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. Th ...
s
* List of integrals of hyperbolic functions
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 constan ...
* Poinsot's spirals
* 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 = \ ...
* 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 fun ...
* 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 al ...
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