Hyperbolic function

From Wikipedia, the free encyclopedia

A ray through the origin intercepts the hyperbola

x 2 - y 2 = 1

in the point

(cosh a,sinh a)

, where

a

is the area between the ray, its mirror image with respect to the

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 "arсsinh" 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. 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 hyperbolic functions take a real value for real argument called a hyperbolic angle. In complex analysis, they are simply algebraic functions of exponentials, and so are entire.

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

[edit] Standard algebraic expressions

sinh, cosh and tanh

csch, sech and coth

The hyperbolic functions are:

Hyperbolic sine, pronounced "shine" or "sinch":

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

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

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

Hyperbolic tangent, pronounced "than" or "tanch":

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

Hyperbolic cotangent, pronounced "coth" or "chot":

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

Hyperbolic secant, pronounced "sheck" or "sech":

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

Hyperbolic cosecant, pronounced "cosheck" or "cosech"

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

where $i$ is the imaginary unit.

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

[edit] 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.

[edit] Standard Integrals

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

$\int\sinh cx\,dx = \frac{1}{c}\cosh cx$

$\int\cosh cx\,dx = \frac{1}{c}\sinh cx$

$\int \tanh cx\,dx = \frac{1}{c}\ln|\cosh cx|$

$\int \coth cx\,dx = \frac{1}{c}\ln|\sinh cx|$

[edit] 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

[edit] Relationship to ordinary trigonometric functions

A point on the hyperbola x y = 1 with x > 1 determines an 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 periodic with complex period $2π i$ .

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, and cosh 0 = 1.

The function sinh x is an odd function, that is symmetric with respect to the origin, and sinh 0 = 0.

The hyperbolic functions satisfy many identities, all of them similar in form to the trigonometric identities. In fact, Osborne'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"

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

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

and the "half-angle formulas"

$\cosh^2\frac{x}{2} = \frac{1+\cosh x}{2}$ 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.

The graph of the function cosh x is the catenary curve.

[edit] 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.

[edit] 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.

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

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