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, similarly to how the derivatives of sin(t) and cos(t) are cos(t) and –sin(t) respectively, the derivatives of sinh(t) and cosh(t) are cosh(t) and +sinh(t) respectively.
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]
from which are derived: [4]
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 when applying the ordinary sine and cosine functions to an imaginary angle. 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. [12]
Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati and Johann Heinrich Lambert. [13] 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. [14] The abbreviations sh, ch, th, cth are also currently used, depending on personal preference.
There are various equivalent ways to define the hyperbolic functions.
In terms of the exponential function: [1] [4]
The hyperbolic functions may be defined as solutions of differential equations: The hyperbolic sine and cosine are the solution (s, c) of the system with the initial conditions The initial conditions make the solution unique; without them any pair of functions would be a solution.
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.
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).
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: [15]
The hyperbolic tangent is the (unique) solution to the differential equation f ′ = 1 − f 2, with f (0) = 0. [16] [17]
The hyperbolic functions satisfy many identities, all of them similar in form to the trigonometric identities. In fact, Osborn's rule [18] states that one can convert any trigonometric identity (up to but not including sinhs or implied sinhs of 4th degree) 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.
particularly
Also:
Also: [19]
where sgn is the sign function.
If x ≠ 0, then [20]
The following inequality is useful in statistics: [21]
It can be proved by comparing the Taylor series of the two functions term by term.
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 . [22]
The following integrals can be proved using hyperbolic substitution:
where C is the constant of integration.
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.
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:
The following expansions are valid in the whole complex plane:
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 regions together depict a hyperbolic sector with area corresponding to 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. [23]
The Gudermannian function gives a direct relationship between the circular functions and the hyperbolic functions 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.
The decomposition of the exponential function in its even and odd parts gives the identities and Combined with Euler's formula this gives for the general complex exponential function.
Additionally,
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).
The natural logarithm of a number is its logarithm to the base of the mathematical constant e, which is an irrational and transcendental number approximately equal to 2.718281828459. The natural logarithm of x is generally written as ln x, logex, or sometimes, if the base e is implicit, simply log x. Parentheses are sometimes added for clarity, giving ln(x), loge(x), or log(x). This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity.
Integration is the basic operation in integral calculus. While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful. This page lists some of the most common antiderivatives.
In mathematics, the Gudermannian function relates a hyperbolic angle measure to a circular angle measure called the gudermannian of and denoted . The Gudermannian function reveals a close relationship between the circular functions and hyperbolic functions. It was introduced in the 1760s by Johann Heinrich Lambert, and later named for Christoph Gudermann who also described the relationship between circular and hyperbolic functions in 1830. The gudermannian is sometimes called the hyperbolic amplitude as a limiting case of the Jacobi elliptic amplitude when parameter
In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions, under suitably restricted domains. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry.
In mathematics, a trigonometric substitution replaces a trigonometric function for another expression. In calculus, trigonometric substitutions are a technique for evaluating integrals. In this case, an expression involving a radical function is replaced with a trigonometric one. Trigonometric identities may help simplify the answer. Like other methods of integration by substitution, when evaluating a definite integral, it may be simpler to completely deduce the antiderivative before applying the boundaries of integration.
In trigonometry, tangent half-angle formulas relate the tangent of half of an angle to trigonometric functions of the entire angle.
In geometry, hyperbolic angle is a real number determined by the area of the corresponding hyperbolic sector of xy = 1 in Quadrant I of the Cartesian plane. The hyperbolic angle parametrizes the unit hyperbola, which has hyperbolic functions as coordinates. In mathematics, hyperbolic angle is an invariant measure as it is preserved under hyperbolic rotation.
In hyperbolic geometry, angle of parallelism is the angle at the non-right angle vertex of a right hyperbolic triangle having two asymptotic parallel sides. The angle depends on the segment length a between the right angle and the vertex of the angle of parallelism.
In mathematics, the inverse hyperbolic functions are inverses of the hyperbolic functions, analogous to the inverse circular functions. There are six in common use: inverse hyperbolic sine, inverse hyperbolic cosine, inverse hyperbolic tangent, inverse hyperbolic cosecant, inverse hyperbolic secant, and inverse hyperbolic cotangent. They are commonly denoted by the symbols for the hyperbolic functions, prefixed with arc- or ar-.
In special relativity, the classical concept of velocity is converted to rapidity to accommodate the limit determined by the speed of light. Velocities must be combined by Einstein's velocity-addition formula. For low speeds, rapidity and velocity are almost exactly proportional but, for higher velocities, rapidity takes a larger value, with the rapidity of light being infinite.
Toroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional bipolar coordinate system about the axis that separates its two foci. Thus, the two foci and in bipolar coordinates become a ring of radius in the plane of the toroidal coordinate system; the -axis is the axis of rotation. The focal ring is also known as the reference circle.
The integral of secant cubed is a frequent and challenging indefinite integral of elementary calculus:
Bending of plates, or plate bending, refers to the deflection of a plate perpendicular to the plane of the plate under the action of external forces and moments. The amount of deflection can be determined by solving the differential equations of an appropriate plate theory. The stresses in the plate can be calculated from these deflections. Once the stresses are known, failure theories can be used to determine whether a plate will fail under a given load.
In integral calculus, the tangent half-angle substitution is a change of variables used for evaluating integrals, which converts a rational function of trigonometric functions of into an ordinary rational function of by setting . This is the one-dimensional stereographic projection of the unit circle parametrized by angle measure onto the real line. The general transformation formula is:
In calculus, the integral of the secant function can be evaluated using a variety of methods and there are multiple ways of expressing the antiderivative, all of which can be shown to be equivalent via trigonometric identities,
In thermal quantum field theory, the Matsubara frequency summation is a technique used to simplify calculations involving Euclidean path integrals.
In the hyperbolic plane, as in the Euclidean plane, each point can be uniquely identified by two real numbers. Several qualitatively different ways of coordinatizing the plane in hyperbolic geometry are used.
George Osborn (1864–1932) was an English mathematician, known for Osborn’s rule that deals with hyperbolic trigonometric identities.