In mathematics, hyperbolic functions are analogs of the ordinary trigonometric functions defined for the hyperbola rather than on the circle: 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 occur in the solutions of many linear differential equations (for example, the equation defining a catenary), of some cubic equations, in calculations of angles and distances in hyperbolic geometry, and of Laplace's equation in Cartesian coordinates. Laplace's equations are important in many areas of physics, including electromagnetic theory, heat transfer, fluid dynamics, and special relativity.
The basic hyperbolic functions are:
from which are derived:
- hyperbolic tangent "tanh" (/ /,),
- hyperbolic cosecant "csch" or "cosech" (// or //)
- hyperbolic secant "sech" (/ /,),
- hyperbolic cotangent "coth" (/ /,),
corresponding to the derived trigonometric functions.
The inverse hyperbolic functions are:
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, the hyperbolic functions arise as the imaginary parts of sine and cosine. The hyperbolic sine and the hyperbolic cosine are entire functions. As a result, the other hyperbolic functions are meromorphic in the whole complex plane.
Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati and Johann Heinrich Lambert. Riccati used Sc. and Cc. (sinus/cosinus circulare) to refer to circular functions and Sh. and Ch. (sinus/cosinus hyperbolico) to refer to hyperbolic functions. Lambert adopted the names but altered the abbreviations to what they are today. The abbreviations sh, ch, th, cth are also at disposition, their use depending more on personal preference of mathematics of influence than on the local language.
There are various equivalent ways for defining the hyperbolic functions.
They may be defined in terms of the exponential function:
The hyperbolic functions may be defined as solutions of differential equations: The hyperbolic sine and cosine are the unique solution (s, c) of the system
such that s(0) = 0 and c(0) = 1.
They are also the unique solution of the equation f ″(x) = f (x), such that f (0) = 1, f ′(0) = 0 for the hyperbolic cosine, and f (0) = 0, f ′(0) = 1 for the hyperbolic sine.
- Hyperbolic sine:
- Hyperbolic cosine:
- Hyperbolic tangent:
- Hyperbolic cotangent:
- Hyperbolic secant:
- Hyperbolic cosecant:
where i is the imaginary unit with the property that i2 = −1.
Odd and even functions:
Hyperbolic sine and cosine satisfy:
the last of which is similar to the Pythagorean trigonometric identity.
One also has
for the other functions.
Sums of arguments
Half argument formulas
where sgn is the sign function.
Inverse functions as logarithms
Sinh and cosh are both equal to their second derivative, that is:
The following integrals can be proved using hyperbolic substitution:
where C is the constant of integration.
Taylor series expressions
It is possible to express the above functions as Taylor series:
The function sinh x has a Taylor series expression with only odd exponents for x. Thus it is an odd function, that is, −sinh x = sinh(−x), and sinh 0 = 0.
The function cosh x has a Taylor series expression with only even exponents for x. Thus it is an even function, that is, symmetric with respect to the y-axis. The sum of the sinh and cosh series is the infinite series expression of the exponential function.
Comparison with circular functions
Since the area of a circular sector with radius r and angle u is r2u/2, it will be equal to u when r = √. 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 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 2, 6, 10, 14, ... sinhs. This yields for example the addition theorems
the "double argument formulas"
- (This is equivalent to its circular counterpart multiplied by −1.)
- (This corresponds to its circular counterpart.)
The derivative of sinh x is cosh x and the derivative of cosh x is sinh x; this is similar to trigonometric functions, albeit the sign is different (i.e., the derivative of cos x is −sin 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 a cosh(x/a) 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
The first one is analogous to Euler's formula
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:
Thus, hyperbolic functions are periodic with respect to the imaginary component, with period ( for hyperbolic tangent and cotangent).
|Wikimedia Commons has media related to Hyperbolic functions.|
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