"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, 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.
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.
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.
Hyperbolic sine: the
odd part of the exponential function, that is,
Hyperbolic cosine: the
even part of the exponential function, that is,
Hyperbolic tangent:
Hyperbolic cotangent: for x ≠ 0,
Hyperbolic secant:
Hyperbolic cosecant: for x ≠ 0,
Differential equation definitions
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.
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 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.
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.
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
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.
Relationship to the exponential function
The decomposition of the exponential function in its
even and odd parts gives the identities
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).
^Robert E. Bradley, Lawrence A. D'Antonio, Charles Edward Sandifer. Euler at 300: an appreciation. Mathematical Association of America, 2007. Page 100.
^Georg F. Becker. Hyperbolic functions. Read Books, 1931. Page xlviii.
^Martin, George E. (1986). The foundations of geometry and the non-euclidean plane (1st corr. ed.). New York: Springer-Verlag. p. 416.
ISBN3-540-90694-0.
"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, 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.
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.
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.
Hyperbolic sine: the
odd part of the exponential function, that is,
Hyperbolic cosine: the
even part of the exponential function, that is,
Hyperbolic tangent:
Hyperbolic cotangent: for x ≠ 0,
Hyperbolic secant:
Hyperbolic cosecant: for x ≠ 0,
Differential equation definitions
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.
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 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.
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.
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
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.
Relationship to the exponential function
The decomposition of the exponential function in its
even and odd parts gives the identities
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).
^Robert E. Bradley, Lawrence A. D'Antonio, Charles Edward Sandifer. Euler at 300: an appreciation. Mathematical Association of America, 2007. Page 100.
^Georg F. Becker. Hyperbolic functions. Read Books, 1931. Page xlviii.
^Martin, George E. (1986). The foundations of geometry and the non-euclidean plane (1st corr. ed.). New York: Springer-Verlag. p. 416.
ISBN3-540-90694-0.