Hyperbolic functions – Wikipedia

“Hyperbolic curve” redirects here. For the geometric curve, see Hyperbola.

In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than 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 unit hyperbola. Also, just as the derivatives of sin(t) and cos(t) are cos(t) and –sin(t), the derivatives of sinh(t) and cosh(t) are cosh(t) and +sinh(t).

Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry. They also occur in the solutions of many linear differential equations (such as the equation defining a catenary), cubic equations, and 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:^[1][2]

• hyperbolic sine “sinh” (),^[3]
• hyperbolic cosine “cosh” (),^[4]

from which are derived:^[5]

• hyperbolic tangent “tanh” (),^[6]
• hyperbolic cosecant “csch” or “cosech” (^[4])
• hyperbolic secant “sech” (),^[7]
• hyperbolic cotangent “coth” (),^[8][9]

corresponding to the derived trigonometric functions.

The inverse hyperbolic functions are:^[1]

• area hyperbolic sine “arsinh” (also denoted “sinh⁻¹“, “asinh” or sometimes “arcsinh”)^[10][11][12]
• area hyperbolic cosine “arcosh” (also denoted “cosh⁻¹“, “acosh” or sometimes “arccosh”)
• and so on.

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.

By Lindemann–Weierstrass theorem, the hyperbolic functions have a transcendental value for every non-zero algebraic value of the argument.^[13]

Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati and Johann Heinrich Lambert.^[14] 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 those used today.^[15] The abbreviations sh, ch, th, cth are also currently used, depending on personal preference.

Definitions[edit]

There are various equivalent ways to define the hyperbolic functions.

Exponential definitions[edit]

In terms of the exponential function:^[2][5]

Differential equation definitions[edit]

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.

(The initial conditions are necessary because every pair of functions of the form solves the two differential equations.)

Sinh(x) and cosh(x) 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.

Complex trigonometric definitions[edit]

Hyperbolic functions may also be deduced from trigonometric functions with complex arguments:

where i is the imaginary unit with i² = −1.

The above definitions are related to the exponential definitions via Euler’s formula (See § Hyperbolic functions for complex numbers below).

Characterizing properties[edit]

Hyperbolic cosine[edit]

It can be shown that the area under the curve of the hyperbolic cosine (over a finite interval) is always equal to the arc length corresponding to that interval:^[16]

Hyperbolic tangent[edit]

The hyperbolic tangent is the (unique) solution to the differential equation f ′ = 1 − f ², with f (0) = 0.

Useful relations[edit]

The hyperbolic functions satisfy many identities, all of them similar in form to the trigonometric identities. In fact, Osborn’s rule^[17] states that one can convert any trigonometric identity for , , or and 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 containing a product of two sinhs.

Odd and even functions:

Hence:

Thus, cosh x and sech x are even functions; the others are odd 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[edit]

particularly

Also:

Subtraction formulas[edit]

Also:^[18]

Half argument formulas[edit]

where sgn is the sign function.

If x ≠ 0, then^[19]

Square formulas[edit]

Inequalities[edit]

The following inequality is useful in statistics: ^[20]

It can be proved by comparing term by term the Taylor series of the two functions.

Inverse functions as logarithms[edit]

Derivatives[edit]

Second derivatives[edit]

Each of the functions sinh and cosh is equal to its second derivative, that is:

All functions with this property are linear combinations of sinh and cosh, in particular the exponential functions and .

Standard integrals[edit]

The following integrals can be proved using hyperbolic substitution:

where C is the constant of integration.

Taylor series expressions[edit]

It is possible to express explicitly the Taylor series at zero (or the Laurent series, if the function is not defined at zero) of the above functions.

This series is convergent for every complex value of x. Since the function sinh x is odd, only odd exponents for x occur in its Taylor series.

This series is convergent for every complex value of x. Since the function cosh x is even, only even exponents for x occur in its Taylor series.

The sum of the sinh and cosh series is the infinite series expression of the exponential function.

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The following series are followed by a description of a subset of their domain of convergence, where the series is convergent and its sum equals the function.

where:

is the nth Bernoulli number

is the nth Euler number

Comparison with circular functions[edit]

The hyperbolic functions represent an expansion of trigonometry beyond the circular functions. Both types depend on an argument, either circular angle or hyperbolic angle.

Since the area of a circular sector with radius r and angle u (in radians) is r²u/2, it will be equal to u when r = √2. 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 √2 times the circular and hyperbolic functions.

The hyperbolic angle is an invariant measure with respect to the squeeze mapping, just as the circular angle is invariant under rotation.^[21]

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[edit]

The decomposition of the exponential function in its even and odd parts gives the identities

and

The first one is analogous to Euler’s formula

Additionally,

Hyperbolic functions for complex numbers[edit]

Since the exponential function can be defined for any complex argument, we can also extend the definitions of the hyperbolic functions 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:

so:

Thus, hyperbolic functions are periodic with respect to the imaginary component, with period ( for hyperbolic tangent and cotangent).

Hyperbolic functions in the complex plane

See also[edit]

• e (mathematical constant)
• Equal incircles theorem, based on sinh
• Hyperbolic growth
• Inverse hyperbolic functions
• List of integrals of hyperbolic functions
• Poinsot’s spirals
• Sigmoid function
• Soboleva modified hyperbolic tangent
• Trigonometric functions

References[edit]

External links[edit]

• “Hyperbolic functions”, Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Hyperbolic functions on PlanetMath
• GonioLab: Visualization of the unit circle, trigonometric and hyperbolic functions (Java Web Start)
• Web-based calculator of hyperbolic functions