In mathematics, one of a set of functions related to the hyperbola in the same way the trigonometric functions relate to the circle. They are the hyperbolic sine, cosine, tangent, secant, cotangent, and cosecant (written “sinh,” “cosh,” etc.). The hyperbolic equivalent of the fundamental trigonometric identity is cosh²math.z − sinh²math.z = 1. The hyperbolic sine and cosine, particularly useful for finding special types of integrals, can be defined in terms of exponential functions: sinhmath.x = (math.e^math.x − math.e^−math.x) ÷ 2 and coshmath.x = (math.e^math.x + math.e^−math.x) ÷ 2

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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.

Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form 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.

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.

Standard algebraic expressions

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 $i^2=-1$.

The complex forms in the definitions above derive from Euler's formula.

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

Inverse Functions as Logarithms

$sinh\; ^\{-1\}x=ln\; left(x+sqrt\{x^\{2\}+1\}\; right)$

$cosh\; ^\{-1\}x=ln\; left(x+sqrt\{x^\{2\}-1\}\; right);xge\; 1$

$operatorname\{csch\}^\{-1\}x=ln\; left(frac\{1\}\{x\}+frac\{sqrt\{1+x^\{2\}\}\}\{left|\; x\; right|\}\; right)$

$coth\; ^\{-1\}x=frac\{1\}\{2\}ln\; left(frac\{x+1\}\{x-1\}\; right);left|\; x\; right|>1$

From the above relationships it is easy to show that: $operatorname\{csch\}^\{-1\}2=ln\; (Golden\; Ratio)$

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.

$operatorname\{sech\}^\{-1\}x=cosh\; ^\{-1\}left(frac\{1\}\{x\}\; right)$

$operatorname\{csch\}^\{-1\}x=sinh\; ^\{-1\}left(frac\{1\}\{x\}\; right)$

$coth\; ^\{-1\}x=tanh\; ^\{-1\}left(frac\{1\}\{x\}\; right)$

Derivatives

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

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

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

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

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

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

$frac\{d\}\{dx\}left(sinh\; ^\{-1\}u\; right)=frac\{1\}\{sqrt\{1+u^\{2\}\}\}cdot\; frac\{du\}\{dx\}$

$frac\{d\}\{dx\}left(cosh\; ^\{-1\}u\; right)=frac\{1\}\{sqrt\{u^\{2\}-1\}\}cdot\; frac\{du\}\{dx\}$

$frac\{d\}\{dx\}left(tanh\; ^\{-1\}u\; right)=frac\{1\}\{1-u^\{2\}\}cdot\; frac\{du\}\{dx\}$

$frac\{d\}\{dx\}left(operatorname\{csch\}^\{-1\}u\; right)=frac\{1\}\{left|\; u\; right|sqrt\{1+u^\{2\}\}\}cdot\; -frac\{du\}\{dx\}$

$frac\{d\}\{dx\}left(operatorname\{sech\}^\{-1\}u\; right)=frac\{1\}\{usqrt\{1-u^\{2\}\}\}cdot\; -frac\{du\}\{dx\}$

$frac\{d\}\{dx\}left(coth\; ^\{-1\}u\; right)=frac\{1\}\{1-u^\{2\}\}cdot\; frac\{du\}\{dx\}$

Standard Integrals

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

$intsinh\; ax,dx\; =\; frac\{1\}\{a\}cosh\; ax\; +\; C$

$intcosh\; ax,dx\; =\; frac\{1\}\{a\}sinh\; ax\; +\; C$

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

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

$int\{frac\{du\}\{sqrt\{a^\{2\}+u^\{2\}\}\}\}=sinh\; ^\{-1\}left(frac\{u\}\{a\}\; right)+C$

$int\{frac\{du\}\{sqrt\{u^\{2\}-a^\{2\}\}\}\}=cosh\; ^\{-1\}left(frac\{u\}\{a\}\; right)+C$

$int\{frac\{du\}\{a^\{2\}-u^\{2\}\}\}=frac\{1\}\{a\}coth\; ^\{-1\}left(frac\{u\}\{a\}\; right)+C;\; u^\{2\}>a^\{2\}$

$int\{frac\{du\}\{usqrt\{a^\{2\}-u^\{2\}\}\}\}=-frac\{1\}\{a\}operatorname\{sech\}^\{-1\}left(frac\{u\}\{a\}\; right)+c$

$int\{frac\{du\}\{usqrt\{a^\{2\}+u^\{2\}\}\}\}=-frac\{1\}\{a\}operatorname\{csch\}^\{-1\}left|\; frac\{u\}\{a\}\; right|+c$

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:

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

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

(Laurent series)

(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. 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.

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

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

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

The hyperbolic functions are periodic with complex period $2\; pi\; i$.

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.

The function cosh x is an even function, that is symmetric with respect to the y-axis.

The function sinh x is an odd function, that is -sinh x = sinh -x, and sinh 0 = 0.

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 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"

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

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

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.

$tanh\; ^\{2\}x=1-operatorname\{sech\}^\{2\}x$

$coth\; ^\{2\}x=1+operatorname\{csch\}^\{2\}x$

The derivative of sinh x is given by cosh x and the derivative of cosh x is sinh x.

The Gudermannian function gives a direct relationship between the circular functions and the hyperbolic ones that does not involve complex numbers.

The graph of the function cosh x is the catenary, the curve formed by a uniform flexible chain hanging freely under gravity.

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.

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.

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

See also

• Inverse hyperbolic function
• List of integrals of hyperbolic functions
• Sigmoid function
• Poinsot's spirals
• Catenary
• e (mathematical constant)

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