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# Hyperbolic function

In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. [1]

71 relations: Abramowitz and Stegun, Algebraic number, Angle, Argument of a function, Bernoulli number, Boundary value problem, Bulletin of the American Mathematical Society, Cartesian coordinate system, Catenary, Complex analysis, Complex number, Constant of integration, Derivative, Differential equation, Dover Publications, E (mathematical constant), Electromagnetism, Entire function, Equal incircles theorem, Eric W. Weisstein, Euler number, Euler's formula, Even and odd functions, Exponential function, Fluid dynamics, Google Books, Gudermannian function, Heat transfer, Holomorphic function, Hyperbola, Hyperbolic angle, Hyperbolic geometry, Hyperbolic sector, Imaginary unit, Invariant measure, Inverse hyperbolic functions, Java Web Start, Johann Heinrich Lambert, Laplace's equation, Lindemann–Weierstrass theorem, Linear combination, List of integrals of hyperbolic functions, List of trigonometric identities, Mathematics, MathWorld, Mellen Woodman Haskell, Meromorphic function, Nonlinear system, Periodic function, Physics, ... Expand index (21 more) »

## Abramowitz and Stegun

Abramowitz and Stegun (AS) is the informal name of a mathematical reference work edited by Milton Abramowitz and Irene Stegun of the United States National Bureau of Standards (NBS), now the National Institute of Standards and Technology (NIST).

## Algebraic number

An algebraic number is any complex number (including real numbers) that is a root of a non-zero polynomial (that is, a value which causes the polynomial to equal 0) in one variable with rational coefficients (or equivalently – by clearing denominators – with integer coefficients).

## Angle

In plane geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.

## Argument of a function

In mathematics, an argument of a function is a specific input in the function, also known as an independent variable.

## Bernoulli number

In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in number theory.

## Boundary value problem

In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions.

## Bulletin of the American Mathematical Society

The Bulletin of the American Mathematical Society is a quarterly mathematical journal published by the American Mathematical Society.

## Cartesian coordinate system

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length.

## Catenary

In physics and geometry, a catenary is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends.

## Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers.

## Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

## Constant of integration

In calculus, the indefinite integral of a given function (i.e., the set of all antiderivatives of the function) on a connected domain is only defined up to an additive constant, the constant of integration.

## Derivative

The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).

## Differential equation

A differential equation is a mathematical equation that relates some function with its derivatives.

## Dover Publications

Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward Cirker and his wife, Blanche.

## E (mathematical constant)

The number is a mathematical constant, approximately equal to 2.71828, which appears in many different settings throughout mathematics.

## Electromagnetism

Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electrically charged particles.

## Entire function

In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic at all finite points over the whole complex plane.

## Equal incircles theorem

In geometry, the equal incircles theorem derives from a Japanese Sangaku, and pertains to the following construction: a series of rays are drawn from a given point to a given line such that the inscribed circles of the triangles formed by adjacent rays and the base line are equal.

## Eric W. Weisstein

Eric Wolfgang Weisstein (born March 18, 1969) is an encyclopedist who created and maintains MathWorld and Eric Weisstein's World of Science (ScienceWorld).

## Euler number

In mathematics, the Euler numbers are a sequence En of integers defined by the Taylor series expansion where is the hyperbolic cosine.

## Euler's formula

Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function.

## Even and odd functions

In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses.

## Exponential function

In mathematics, an exponential function is a function of the form in which the argument occurs as an exponent.

## Fluid dynamics

In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids - liquids and gases.

Google Books (previously known as Google Book Search and Google Print and by its codename Project Ocean) is a service from Google Inc. that searches the full text of books and magazines that Google has scanned, converted to text using optical character recognition (OCR), and stored in its digital database.

## Gudermannian function

The Gudermannian function, named after Christoph Gudermann (1798–1852), relates the circular functions and hyperbolic functions without explicitly using complex numbers.

## Heat transfer

Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems.

## Holomorphic function

In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain.

## Hyperbola

In mathematics, a hyperbola (plural hyperbolas or hyperbolae) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set.

## Hyperbolic angle

In mathematics, a hyperbolic angle is a geometric figure that divides a hyperbola. The science of hyperbolic angle parallels the relation of an ordinary angle to a circle. The hyperbolic angle is first defined for a "standard position", and subsequently as a measure of an interval on a branch of a hyperbola. A hyperbolic angle in standard position is the angle at (0, 0) between the ray to (1, 1) and the ray to (x, 1/x) where x > 1. The magnitude of the hyperbolic angle is the area of the corresponding hyperbolic sector which is ln x. Note that unlike circular angle, hyperbolic angle is unbounded, as is the function ln x, a fact related to the unbounded nature of the harmonic series. The hyperbolic angle in standard position is considered to be negative when 0 a > 1 so that (a, b) and (c, d) determine an interval on the hyperbola xy.

## Hyperbolic geometry

In mathematics, hyperbolic geometry (also called Bolyai–Lobachevskian geometry or Lobachevskian geometry) is a non-Euclidean geometry.

## Hyperbolic sector

A hyperbolic sector is a region of the Cartesian plane bounded by rays from the origin to two points (a, 1/a) and (b, 1/b) and by the rectangular hyperbola xy.

## Imaginary unit

The imaginary unit or unit imaginary number is a solution to the quadratic equation.

## Invariant measure

In mathematics, an invariant measure is a measure that is preserved by some function.

## Inverse hyperbolic functions

In mathematics, the inverse hyperbolic functions are the inverse functions of the hyperbolic functions.

## Java Web Start

In computing, Java Web Start (also known as JavaWS, javaws or JAWS) is a framework developed by Sun Microsystems (now Oracle) that allows users to start application software for the Java Platform directly from the Internet using a web browser.

## Johann Heinrich Lambert

Johann Heinrich Lambert (Jean-Henri Lambert in French; 26 August 1728 – 25 September 1777) was a Swiss polymath who made important contributions to the subjects of mathematics, physics (particularly optics), philosophy, astronomy and map projections.

## Laplace's equation

In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties.

## Lindemann–Weierstrass theorem

In transcendental number theory, the Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers.

## Linear combination

In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants).

## List of integrals of hyperbolic functions

The following is a list of integrals (anti-derivative functions) of hyperbolic functions.

## List of trigonometric identities

In mathematics, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables where both sides of the equality are defined.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## MathWorld

MathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein.

Mellen Woodman Haskell (March 17, 1863 – January 15, 1948) was an American mathematician, specializing in geometry, group theory, and applications of group theory to geometry.

## Meromorphic function

In the mathematical field of complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all of D except for a discrete set of isolated points, which are poles of the function.

## Nonlinear system

In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input.

## Periodic function

In mathematics, a periodic function is a function that repeats its values in regular intervals or periods.

## Physics

Physics (from knowledge of nature, from φύσις phýsis "nature") is the natural science that studies matterAt the start of The Feynman Lectures on Physics, Richard Feynman offers the atomic hypothesis as the single most prolific scientific concept: "If, in some cataclysm, all scientific knowledge were to be destroyed one sentence what statement would contain the most information in the fewest words? I believe it is that all things are made up of atoms – little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another..." and its motion and behavior through space and time and that studies the related entities of energy and force."Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves."Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics. (...) You will come to see physics as a towering achievement of the human intellect in its quest to understand our world and ourselves."Physics is an experimental science. Physicists observe the phenomena of nature and try to find patterns that relate these phenomena.""Physics is the study of your world and the world and universe around you." Physics is one of the oldest academic disciplines and, through its inclusion of astronomy, perhaps the oldest. Over the last two millennia, physics, chemistry, biology, and certain branches of mathematics were a part of natural philosophy, but during the scientific revolution in the 17th century, these natural sciences emerged as unique research endeavors in their own right. Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the fundamental mechanisms studied by other sciences and suggest new avenues of research in academic disciplines such as mathematics and philosophy. Advances in physics often enable advances in new technologies. For example, advances in the understanding of electromagnetism and nuclear physics led directly to the development of new products that have dramatically transformed modern-day society, such as television, computers, domestic appliances, and nuclear weapons; advances in thermodynamics led to the development of industrialization; and advances in mechanics inspired the development of calculus.

## PlanetMath

PlanetMath is a free, collaborative, online mathematics encyclopedia.

## Poinsot's spirals

In mathematics, Poinsot's spirals are two spirals represented by the polar equations where csch is the hyperbolic cosecant, and sech is the hyperbolic secant.

## Pythagorean trigonometric identity

The Pythagorean trigonometric identity is a trigonometric identity expressing the Pythagorean theorem in terms of trigonometric functions.

## Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

## Right triangle

A right triangle (American English) or right-angled triangle (British English) is a triangle in which one angle is a right angle (that is, a 90-degree angle).

## Second derivative

In calculus, the second derivative, or the second order derivative, of a function is the derivative of the derivative of.

## Series (mathematics)

In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity.

## Sigmoid function

A sigmoid function is a mathematical function having a characteristic "S"-shaped curve or sigmoid curve.

## Sign function

In mathematics, the sign function or signum function (from signum, Latin for "sign") is an odd mathematical function that extracts the sign of a real number.

## Special relativity

In physics, special relativity (SR, also known as the special theory of relativity or STR) is the generally accepted and experimentally well-confirmed physical theory regarding the relationship between space and time.

## Squeeze mapping

In linear algebra, a squeeze mapping is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is not a rotation or shear mapping.

## Stack Exchange

Stack Exchange is a network of question-and-answer (Q&A) websites on topics in varied fields, each site covering a specific topic, where questions, answers, and users are subject to a reputation award process.

## Taylor series

In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.

## The Mathematical Gazette

The Mathematical Gazette is an academic journal of mathematics education, published three times yearly, that publishes "articles about the teaching and learning of mathematics with a focus on the 15–20 age range and expositions of attractive areas of mathematics." It was established in 1894 by Edward Mann Langley as the successor to the Reports of the Association for the Improvement of Geometrical Teaching.

## Transcendental number

In mathematics, a transcendental number is a real or complex number that is not algebraic—that is, it is not a root of a nonzero polynomial equation with integer (or, equivalently, rational) coefficients.

## Trigonometric functions

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are functions of an angle.

## Trigonometric substitution

In mathematics, Trigonometric substitution is the substitution of trigonometric functions for other expressions.

## Trigonometry

Trigonometry (from Greek trigōnon, "triangle" and metron, "measure") is a branch of mathematics that studies relationships involving lengths and angles of triangles.

## Unit circle

In mathematics, a unit circle is a circle with a radius of one.

## Vincenzo Riccati

Vincenzo Riccati (Castelfranco Veneto, 11 January 1707 &ndash; Treviso, 17 January 1775) was a Venetian mathematician and physicist.

## 0

0 (zero) is both a number and the numerical digit used to represent that number in numerals.

## References

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