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Encyclopedia > Hyperbolic function
A ray through the origin intercepts the hyperbola in the point , where is the area between the ray, its mirror image with respect to the -axis, and the hyperbola (see animated version with comparison with the trigonometric (circular) functions).
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. 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. Image File history File links Hyperbolic_functions. ... Image File history File links Hyperbolic_functions. ... Image File history File links No higher resolution available. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Sine redirects here. ...


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. 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. In mathematics, a hyperbola (Greek literally overshooting or excess) is a type of conic section defined as the intersection between a right circular conical surface and a plane which cuts through both halves of the cone. ... Visualization of airflow into a duct modelled using the Navier-Stokes equations, a set of partial differential equations. ... For the railroad term see Overhead lines For its use in ring theory, see Catenary ring. ... In mathematics, Laplaces equation is a partial differential equation named after its discoverer, Pierre-Simon Laplace. ... Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... Maxwells equations are the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter. ... In thermal physics, heat transfer is the passage of thermal energy from a hot to a cold body. ... Fluid dynamics is the sub-discipline of fluid mechanics dealing with fluids (liquids and gases) in motion. ... For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ...


The hyperbolic functions take real values for real argument called a hyperbolic angle. In complex analysis, they are simply rational functions of exponentials, and so are meromorphic. 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 mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ... The exponential function is one of the most important functions in mathematics. ... In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function. ...

Contents

Standard algebraic expressions

sinh, cosh and tanh
sinh, cosh and tanh
csch, sech and coth
csch, sech and coth

The hyperbolic functions are: Image File history File links Sinh_cosh_tanh. ... Image File history File links Sinh_cosh_tanh. ... Image File history File links Csch_sech_coth. ... Image File history File links Csch_sech_coth. ...

  • Hyperbolic sine, often pronounced "sinch", and less commonly "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 {e^x - e^{-x}} {e^x + e^{-x}} = frac{e^{2x} - 1} {e^{2x} + 1} = -i tan ix !
  • Hyperbolic cotangent, often pronounced "coth", "chot", or "co-tanch":
coth x = frac{cosh x}{sinh x} = frac {e^x + e^{-x}} {e^x - e^{-x}} = frac{e^{2x} + 1} {e^{2x} - 1} = i cot ix !
  • Hyperbolic secant, often pronounced "sech" or "sheck":
operatorname{sech} x = frac{1}{cosh x} = frac {2} {e^x + e^{-x}} = sec {ix} !
  • Hyperbolic cosecant, often pronounced "cosech" 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 i2 = − 1. In mathematics, the imaginary unit (or sometimes the Latin or the Greek iota, see below) allows the real number system to be extended to the complex number system . ...


The complex forms in the definitions above derive from Euler's formula. In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ...


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 negatives. ... In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking negatives. ...


Standard Integrals

For a full list of integrals of hyperbolic functions, see list of integrals of hyperbolic functions The following is a list of integrals (antiderivative functions) of hyperbolic functions. ...

intsinh cx,dx = frac{1}{c}cosh cx + C
intcosh cx,dx = frac{1}{c}sinh cx + C
int tanh cx,dx = frac{1}{c}ln|cosh cx| + C
int coth cx,dx = frac{1}{c}ln|sinh cx| + C

In the above expressions, C is called the constant of integration.


Taylor series expressions

It is possible to express the above functions as Taylor series: As the degree of the Taylor series rises, it approaches the correct function. ...

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 A Laurent series is defined with respect to a particular point c and a path of integration γ. The path of integration must lie in an annulus (shown here in red) inside of which f(z) is holomorphic. ... A Laurent series is defined with respect to a particular point c and a path of integration γ. The path of integration must lie in an annulus (shown here in red) inside of which f(z) is holomorphic. ...

B_n , is the nth Bernoulli number
E_n , is the nth Euler number

In mathematics, the Bernoulli numbers are a sequence of rational numbers with deep connections in number theory. ... In mathematics, in the area of number theory, the Euler numbers are a sequence En of integers defined by the following Taylor series expansion: where cosh t is the hyperbolic cosine. ...

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. 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. In mathematics, the term hyperbolic triangle has more than one meaning. ... 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. This is based on the easily verified identity Circle illustration This article is about the shape and mathematical concept of circle. ... In mathematics, a hyperbola (Greek literally overshooting or excess) is a type of conic section defined as the intersection between a right circular conical surface and a plane which cuts through both halves of the cone. ...

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 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. This article is about angles in geometry. ... 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 negatives. ...


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 negatives. ...


The hyperbolic functions satisfy many identities, all of them similar in form to the trigonometric identities. 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 In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ...

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

and the "half-angle formulas"

cosh^2frac{x}{2} = frac{cosh x + 1}{2} Note: This corresponds to its circular counterpart.
sinh^2frac{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. This article is about derivatives and differentiation in mathematical calculus. ...


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 - 1852), relates the circular and hyperbolic trigonometric functions without resorting to complex numbers. ...


The graph of the function cosh x is the catenary, the curve formed by a uniform flexible chain hanging freely under gravity. For the railroad term see Overhead lines For its use in ring theory, see Catenary ring. ...


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. Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ...


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 functions sinh z and cosh z are then holomorphic; their Taylor series expansions are given in the Taylor series article. The exponential function is one of the most important functions in mathematics. ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... As the degree of the Taylor series rises, it approaches the correct function. ...


Relationships to ordinary trigonometric functions are given by Euler's formula for complex numbers: Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ...

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 ,
sinh x = -i sin ix ,
cosh x = cos ix ,
tanh x = -i tan ix ,

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)},

See also

The function artanh. ... The following is a list of integrals (antiderivative functions) of hyperbolic functions. ... The logistic curve A sigmoid function is a mathematical function that produces a sigmoid curve — a curve having an S shape. ... The Poinsot spiral r=csch(θ/3). ... For the railroad term see Overhead lines For its use in ring theory, see Catenary ring. ... e is the unique number such that the value of the derivative of f (x) = ex (blue curve) at the point x = 0 is exactly 1. ...

External links

PlanetMath is a free, collaborative, online mathematics encyclopedia. ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ... Java Web Start, introduced in Java 2, allows provisioning applications over the Web by clicking a desktop icon or a link on a website. ...

External links

  1. ^ G. Osborn, Mnemonic for hyperbolic formulae, The Mathematical Gazette, p. 189, volume 2, issue 34, July 1902

  Results from FactBites:
 
Hyperbolic function - Wikipedia, the free encyclopedia (707 words)
In mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions.
Hyperbolic functions are also useful because they occur in the solutions of some simple linear differential equations, notably that defining the shape of a hanging cable, the catenary.
The inverses of the hyperbolic functions are the area hyperbolic functions.
  More results at FactBites »

 
 

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