hyperbolic function

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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 functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the unit hyperbola. 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 János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For an ...
. They also occur in the solutions of many linear
differential equation In mathematics, a differential equation is an functional equation, equation that relates one or more unknown function (mathematics), functions and their derivatives. In applications, the functions generally represent physical quantities, the der ...
s (such as the equation defining a catenary), cubic equations, 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 Elementary particle, fundamental constituents, its motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical science is that depar ...
, 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 In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic s ...
,
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) and ...
, and
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between Spacetime, space and time. In Albert Einstein's original treatment, the theory is based on two Postulates of ...
. 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 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. The size of a hyperbolic angle is twice the area of its hyperbolic sector. 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 functions. As a result, the other hyperbolic functions are meromorphic in the whole complex plane. By Lindemann–Weierstrass theorem, 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. 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.

# 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 (mathematics), function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponentiation, exponent). Unless otherwise specified, the term generally refers to the positiv ...
: * 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 functional equation, equation that relates one or more unknown function (mathematics), functions and their derivatives. In applications, the functions generally represent physical quantities, the der ...
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 $s\left(0\right) = 0, c\left(0\right) = 1.$ The initial conditions make the solution unique; without them any pair of functions $\left(a e^x + b e^, a e^x - b e^\right)$ 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 functions 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 with . The above definitions are related to the exponential definitions via Euler's formula (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 functional equation, equation that relates one or more unknown function (mathematics), functions and their derivatives. In applications, the functions generally represent physical quantities, the der ...
, 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 functions; 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. One also has $\begin \operatorname ^ x &= 1 - \tanh^ x \\ \operatorname ^ x &= \coth^ x - 1 \end$ for the other functions.

## Sums of arguments

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

where is the sign function. 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: $\operatorname\left(t\right) \leq e^$ 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, 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 (mathematics), function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponentiation, exponent). Unless otherwise specified, the term generally refers to the positiv ...
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.

# Taylor series expressions

It is possible to express explicitly the Taylor series at zero (or the Laurent series, 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, 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, only even exponents for occur in its Taylor series. The sum of the sinh and cosh series is the infinite series expression of the
exponential function The exponential function is a mathematical Function (mathematics), function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponentiation, exponent). Unless otherwise specified, the term generally refers to the positiv ...
. 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 *$E_n$ is the ''n''th Euler number

# Infinite products and continued fractions

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

# Comparison with circular functions

The hyperbolic functions represent an expansion of trigonometry beyond the
circular function 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 ...
s. 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 a conclusion. Arguments can be studied from three main perspectives: the logical, the dialect ...
, either circular angle or hyperbolic angle. 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 triangles with hypotenuse on the ray defining the angles are of length times the circular and hyperbolic functions. The hyperbolic angle is an invariant measure 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 1:6:155–9
full text
/ref> The Gudermannian function 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, 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 $e^ = \cos x + i\sin x,$ 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 (mathematics), function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponentiation, exponent). Unless otherwise specified, the term generally refers to the positiv ...
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 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).

* e (mathematical constant) * Equal incircles theorem, based on sinh * Hyperbolic growth * Inverse hyperbolic functions * List of integrals of hyperbolic functions * Poinsot's spirals * Sigmoid function * Soboleva modified hyperbolic tangent * Trigonometric functions