Hyperbolic function

A ray through the origin intercepts the hyperbola $\scriptstyle x^2\ -\ y^2\ =\ 1$ in the point $\scriptstyle (\cosh\,a,\,\sinh\,a)$ , where $\scriptstyle a$ is the area between the ray, its mirror image with respect to the $\scriptstyle x$ -axis, and the hyperbola (see animated version with comparison with the trigonometric (circular) functions).

In mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The basic hyperbolic functions are the hyperbolic sine "sinh", and the hyperbolic cosine "cosh", from which are derived the hyperbolic tangent "tanh", etc. , in analogy to the derived trigonometric functions. The inverse functions are the inverse hyperbolic sine "arsinh" (also called "arcsinh" or "asinh") and so on.

Just as the points (cos t, sin t) define a circle, the points (cosh t, sinh t) define the right half of the equilateral hyperbola. In Geometry, a hyperbola ( Greek, "over-thrown" has several equivalent definitions Hyperbolic functions are also useful because they occur in the solutions of some important linear differential equations, notably that defining the shape of a hanging cable, the catenary, and Laplace's equation (in Cartesian coordinates), which is important in many areas of physics including electromagnetic theory, heat transfer, fluid dynamics, and special relativity. A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the In Physics and Geometry, the catenary is the theoretical Shape of a hanging flexible Chain or Cable when supported at its ends and In Mathematics, Laplace's equation is a Partial differential equation named after Pierre-Simon Laplace who first studied its properties In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric In thermal physics, heat transfer is the passage of Thermal energy from a hot to a colder body Fluid dynamics is the sub-discipline of Fluid mechanics dealing with fluid flow: Fluids ( Liquids and Gases in motion Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial

The hyperbolic functions take real values for real argument called a hyperbolic angle. A hyperbolic angle in standard position is the Angle at (0 0 between the ray to (1 1 and the ray to ( x, 1/ x) where x > 1 In complex analysis, they are simply rational functions of exponentials, and so are meromorphic. In Mathematics, a rational function is any function which can be written as the Ratio of two Polynomial functions Definitions In The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) In Complex analysis, a meromorphic function on an open subset D of the Complex plane is a function that is holomorphic

1 Standard algebraic expressions
- 1.1 Useful relations
2 Derivatives
3 Standard Integrals
4 Taylor series expressions
5 Similarities to circular trigonometric functions
6 Relationship to the exponential function
7 Hyperbolic functions for complex numbers
8 References
9 See also
10 External links

Standard algebraic expressions

sinh, cosh and tanh

csch, sech and coth

The hyperbolic functions are:

Hyperbolic sine, often pronounced "sinch", or (especially in the U. K. ) "shine":

$\sinh x = \frac{e^x - e^{-x}}{2} = -i \sin ix \!$

Hyperbolic cosine, often pronounced "cosh", "co-sinch", or "co-shine":

$\cosh x = \frac{e^{x} + e^{-x}}{2} = \cos ix \!$

Hyperbolic tangent, often pronounced "tanch" (or "than"):

$\tanh x = \frac{\sinh x}{\cosh x} = \frac {\frac {e^x - e^{-x}} {2}} {\frac {e^x + e^{-x}} {2}} = \frac {e^x - e^{-x}} {e^x + e^{-x}} = \frac{e^{2x} - 1} {e^{2x} + 1} = -i \tan ix \!$

Hyperbolic cotangent, often pronounced "coth", "co-tanch", or "chot":

$\coth x = \frac{\cosh x}{\sinh x} = \frac {\frac {e^x + e^{-x}} {2}} {\frac {e^x - e^{-x}} {2}} = \frac {e^x + e^{-x}} {e^x - e^{-x}} = \frac{e^{2x} + 1} {e^{2x} - 1} = i \cot ix \!$

Hyperbolic secant, often pronounced "setch" or "sheck":

$\operatorname{sech} x = \frac{1}{\cosh x} = \frac {2} {e^x + e^{-x}} = \sec {ix} \!$

Hyperbolic cosecant, often pronounced "cosetch" or "cosheck"

$\operatorname{csch} x = \frac{1}{\sinh x} = \frac {2} {e^x - e^{-x}} = i\,\csc\,ix \!$

where $i$ is the imaginary unit defined as $i 2 = - 1$ . Definition By definition the imaginary unit i is one solution (of two of the Quadratic equation

The complex forms in the definitions above derive from Euler's formula. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted This article is about Euler's formula in Complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic

Note that, by convention, $sinh 2 x$ means $(sinh x) 2$ , not $sinh(sinh x)$ ; similarly for the other hyperbolic functions and positive exponents.

Useful relations

$\sinh(-x) = -\sinh x\,\!$

$\cosh(-x) = \cosh x\,\!$

Hence:

$\tanh(-x) = -\tanh x\,\!$

$\coth(-x) = -\coth x\,\!$

$\operatorname{sech}(-x) = \operatorname{sech}\, x\,\!$

$\operatorname{csch}(-x) = -\operatorname{csch}\, x\,\!$

It can be seen that both cosh x and sech x are even functions, others are odd functions. In Mathematics, even functions and odd functions are functions which satisfy particular Symmetry relations with respect to taking Additive In Mathematics, even functions and odd functions are functions which satisfy particular Symmetry relations with respect to taking Additive

Derivatives

$\frac{d}{dx}\sinh(x) = \cosh(x) \,$

$\frac{d}{dx}\cosh(x) = \sinh(x) \,$

$\frac{d}{dx}\tanh(x) = 1 - \tanh^2(x) = \hbox{sech}^2(x) = 1/\cosh^2(x) \,$

$\frac{d}{dx}\coth(x) = 1 - \coth^2(x) = -\hbox{csch}^2(x) = -1/\sinh^2(x) \,$

$\frac{d}{dx}\ \hbox{csch(x)} = - \coth(x)\ \hbox{csch(x)}\,$

$\frac{d}{dx}\ \hbox{sech(x)} = - \tanh(x)\ \hbox{sech(x)}\,$

Standard Integrals

For a full list of integrals of hyperbolic functions, see list of integrals of hyperbolic functions

$\int\sinh ax\,dx = \frac{1}{a}\cosh ax + C$

$\int\cosh ax\,dx = \frac{1}{a}\sinh ax + C$

$\int \tanh ax\,dx = \frac{1}{a}\ln|\cosh ax| + C$

$\int \coth ax\,dx = \frac{1}{a}\ln|\sinh ax| + C$

In the above expressions, C is called the constant of integration. The following is a list of Integrals ( Antiderivative functions of Hyperbolic functions For a complete list of Integral functions see List of integrals. In Calculus, the Indefinite integral of a given function (ie the set of all Antiderivatives of the function is always written with a constant the constant

Taylor series expressions

It is possible to express the above functions as Taylor series:

$\sinh x = x + \frac {x^3} {3!} + \frac {x^5} {5!} + \frac {x^7} {7!} +\cdots = \sum_{n=0}^\infty \frac{x^{2n+1}}{(2n+1)!}$

$\cosh x = 1 + \frac {x^2} {2!} + \frac {x^4} {4!} + \frac {x^6} {6!} + \cdots = \sum_{n=0}^\infty \frac{x^{2n}}{(2n)!}$

$\tanh x = x - \frac {x^3} {3} + \frac {2x^5} {15} - \frac {17x^7} {315} + \cdots = \sum_{n=1}^\infty \frac{2^{2n}(2^{2n}-1)B_{2n} x^{2n-1}}{(2n)!}, \left |x \right | < \frac {\pi} {2}$

$\coth x = \frac {1} {x} + \frac {x} {3} - \frac {x^3} {45} + \frac {2x^5} {945} + \cdots = \frac {1} {x} + \sum_{n=1}^\infty \frac{2^{2n} B_{2n} x^{2n-1}} {(2n)!}, 0 < \left |x \right | < \pi$ (Laurent series)

$\operatorname {sech}\, x = 1 - \frac {x^2} {2} + \frac {5x^4} {24} - \frac {61x^6} {720} + \cdots = \sum_{n=0}^\infty \frac{E_{2 n} x^{2n}}{(2n)!} , \left |x \right | < \frac {\pi} {2}$

$\operatorname {csch}\, x = \frac {1} {x} - \frac {x} {6} +\frac {7x^3} {360} -\frac {31x^5} {15120} + \cdots = \frac {1} {x} + \sum_{n=1}^\infty \frac{ 2 (1-2^{2n-1}) B_{2n} x^{2n-1}}{(2n)!} , 0 < \left |x \right | < \pi$ (Laurent series)

where

$B_n \,$ is the nth Bernoulli number

$E_n \,$ is the nth Euler number

Similarities to circular trigonometric functions

A point on the hyperbola x y = 1 with x > 1 determines a hyperbolic triangle in which the side adjacent to the hyperbolic angle is associated with cosh while the side opposite is associated with sinh. In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives In Mathematics, the Laurent series of a complex function f ( z) is a representation of that function as a Power series which includes terms In Mathematics, the Laurent series of a complex function f ( z) is a representation of that function as a Power series which includes terms In Mathematics, the Bernoulli numbers are a Sequence of Rational numbers with deep connections to Number theory. For other uses see Euler number (topology and Eulerian number. In Mathematics, the term hyperbolic triangle has more than one meaning However, since the point (1,1) on this hyperbola is a distance √2 from the origin, the normalization constant 1/√2 is necessary to define cosh and sinh by the lengths of the sides of the hyperbolic triangle. The concept of a normalizing constant arises in Probability theory and a variety of other areas of Mathematics.

Just as the points (cos t, sin t) define a circle, the points (cosh t, sinh t) define the right half of the equilateral hyperbola x² - y² = 1. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the In Geometry, a hyperbola ( Greek, "over-thrown" has several equivalent definitions This is based on the easily verified identity

$\cosh^2 t - \sinh^2 t = 1 \,$

and the property that cosh t > 0 for all t.

The hyperbolic functions are periodic with complex period $2π i$ . In Mathematics, a periodic function is a function that repeats its values after some definite period has been added to its Independent variable

The parameter t is not a circular angle, but rather a hyperbolic angle which represents twice the area between the x-axis, the hyperbola and the straight line which links the origin with the point (cosh t, sinh t) on the hyperbola. In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called A hyperbolic angle in standard position is the Angle at (0 0 between the ray to (1 1 and the ray to ( x, 1/ x) where x > 1

The function cosh x is an even function, that is symmetric with respect to the y-axis. In Mathematics, even functions and odd functions are functions which satisfy particular Symmetry relations with respect to taking Additive

The function sinh x is an odd function, that is -sinh x = sinh -x, and sinh 0 = 0. In Mathematics, even functions and odd functions are functions which satisfy particular Symmetry relations with respect to taking Additive

The hyperbolic functions satisfy many identities, all of them similar in form to the trigonometric identities. In Mathematics, trigonometric identities are equalities that involve Trigonometric functions that are true for every single value of the occurring variables In fact, Osborn's rule ^[1] states that one can convert any trigonometric identity 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 which contains a product of two sinh's. This yields for example the addition theorems

$\sinh(x+y) = \sinh x \cosh y + \cosh x \sinh y \,$

$\cosh(x+y) = \cosh x \cosh y + \sinh x \sinh y \,$

$\tanh(x+y) = \frac{\tanh x + \tanh y}{1 + \tanh x \tanh y} \,$

the "double angle formulas"

$\sinh 2x\ = 2\sinh x \cosh x \,$

$\cosh 2x\ = \cosh^2 x + \sinh^2 x = 2\cosh^2 x - 1 = 2\sinh^2 x + 1 \,$

and the "half-angle formulas"

$\cosh^2\frac{x}{2} = \frac{\cosh x + 1}{2}$ Note: This corresponds to its circular counterpart.

$\sinh^2\frac{x}{2} = \frac{\cosh x - 1}{2}$ Note: This is equivalent to its circular counterpart multiplied by -1.

The derivative of sinh x is given by cosh x and the derivative of cosh x is sinh x. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change

The Gudermannian function gives a direct relationship between the circular functions and the hyperbolic ones that does not involve complex numbers. The Gudermannian function, named after Christoph Gudermann (1798 &ndash 1852 relates the circular and Hyperbolic trigonometric functions without using

The graph of the function cosh x is the catenary, the curve formed by a uniform flexible chain hanging freely under gravity. In Physics and Geometry, the catenary is the theoretical Shape of a hanging flexible Chain or Cable when supported at its ends and

Relationship to the exponential function

From the definitions of the hyperbolic sine and cosine, we can derive the following identities:

$e^x = \cosh x + \sinh x\!$

and

$e^{-x} = \cosh x - \sinh x.\!$

These expressions are analogous to the expressions for sine and cosine, based on Euler's formula, as sums of complex exponentials. This article is about Euler's formula in Complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic

Hyperbolic functions for complex numbers

Since the exponential function can be defined for any complex argument, we can extend the definitions of the hyperbolic functions also to complex arguments. The exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted The functions sinh z and cosh z are then holomorphic; their Taylor series expansions are given in the Taylor series article. Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane In Mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its Derivatives

Relationships to ordinary trigonometric functions are given by Euler's formula for complex numbers:

$e^{i x} = \cos x + i \;\sin x$

$e^{-i x} = \cos x - i \;\sin x$

so:

$\cosh ix = \frac{e^{i x} + e^{-i x}}{2} = \cos x$

$\sinh ix = \frac{e^{i x} - e^{-i x}}{2} = i \sin x$

$\tanh ix = i \tan x \,$

$\cosh x = \cos ix \,$

$\sinh x = -i \sin ix \,$

$\tanh x = -i \tan ix \,$

**Hyperbolic functions in the complex plane**

$\operatorname{sinh}(z)$	$\operatorname{cosh}(z)$	$\operatorname{tanh}(z)$	$\operatorname{coth}(z)$	$\operatorname{sech}(z)$	$\operatorname{csch}(z)$

References

^ G. This article is about Euler's formula in Complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic Osborn, Mnemonic for hyperbolic formulae, The Mathematical Gazette, p. 189, volume 2, issue 34, July 1902

External links

Hyperbolic functions on PlanetMath
Hyperbolic functions entry at MathWorld
GonioLab: Visualization of the unit circle, trigonometric and hyperbolic functions (Java Web Start)
Web-based calculator of hyperbolic functions

The inverses of the hyperbolic functions are the area hyperbolic functions. The following is a list of Integrals ( Antiderivative functions of Hyperbolic functions For a complete list of Integral functions see List of integrals. A sigmoid function is a Mathematical function that produces a sigmoid curve &mdash a curve having an "S" shape In Mathematics, Poinsot's spirals are two Spirals represented by the Polar equations:r=a\\operatorname{csch}(n\theta r=a\\operatorname{sech}(n\theta In Physics and Geometry, the catenary is the theoretical Shape of a hanging flexible Chain or Cable when supported at its ends and The Mathematical constant e is the unique Real number such that the function e x has the same value as the slope of the tangent line PlanetMath is a free, collaborative online Mathematics Encyclopedia. MathWorld is an online Mathematics reference work created and largely written by Eric W Java Web Start is a framework developed by Sun Microsystems which allows Application software for the Java Platform

Dictionary

-noun

(mathematics) A function that is derived from some arithmetic operations on the exponential function with base e and the inverse function, and was named after the corresponding similar trigonometric function.

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