List of mathematical functions: Difference between revisions

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{{Short description|none}}

In [[mathematics]], some functions or groups of functions are important enough to deserve their own names. This is a listing of articles which explain some of these functions in more detail. There is a large theory of [[special functions]] which developed out of [[statistics]] and [[mathematical physics]]. A modern, abstract point of view contrasts large [[function space]]s, which are infinite-dimensional and within which most functions are '"anonymous'", with special functions picked out by properties such as [[symmetry]], or relationship to [[harmonic analysis]] and [[group representation]]s.

See also [[List of types of functions]]

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===Algebraic functions===

[[Algebraic function]]s are functions that can be expressed as the solution of a polynomial equation with integerpolynomial coefficients.

* [[Polynomial]]s: Can be generated solely by addition, multiplication, and raising to the power of a positive integer.

** [[Constant function]]: polynomial of degree zero, graph is a horizontal straight line

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** [[Cubic function]]: Third degree polynomial.

** [[Quartic function]]: Fourth degree polynomial.

** [[Quintic equation|Quintic function]]: Fifth degree polynomial.

** [[Sextic equation|Sextic function]]: Sixth degree polynomial.

* [[Rational function]]s: A ratio of two polynomials.

* [[nth root|''n''th root]]

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[[Transcendental function]]s are functions that are not algebraic.

* [[Exponential function]]: raises a fixed number to a variable power.

* [[Hyperbolic function]]s: formally similar to the [[trigonometric functions]].

** [[Inverse hyperbolic functions]]: [[Inverse function|inverses]] of the [[hyperbolic functions]], analogous to the [[Inverse trigonometric functions|inverse circular functions]].

* [[Logarithm]]s: the inverses of exponential functions; useful to solve equations involving exponentials.

** [[Natural logarithm]]

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* [[Exponentiation#Power functions|Power functions]]: raise a variable number to a fixed power; also known as [[Allometric function]]s; note: if the power is a rational number it is not strictly a transcendental function.

* [[Periodic function]]s

** [[Trigonometric function]]s: [[sine]], [[cosine]], [[tangent (trigonometry)|tangent]], [[cotangent]], [[secant (trigonometry)|secant]], [[cosecant]], [[exsecant]], [[excosecant]], [[versine]], [[coversine]], [[vercosine]], [[covercosine]], [[haversine]], [[hacoversine]], [[havercosine]], [[hacovercosine]], [[Inverse trigonometric functions]] etc.; used in [[geometry]] and to describe periodic phenomena. See also [[Gudermannian function]].

==[[Special functions]]==

{{main|Special functions}}

===Piecewise special functions===

{{columns-list|colwidth=20em|

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** [[Heaviside step function]]: 0 for negative arguments and 1 for positive arguments. The integral of the [[Dirac delta function]].

* [[Sawtooth wave]]

* [[Square wave (waveform)|Square wave]]

* [[Triangle wave]]

* [[Rectangular function]]

* [[Floor function]]: Largest integer less than or equal to a given number.

* [[Ceiling function]]: Smallest integer larger than or equal to a given number.

* [[Sign function]]: Returns only the sign of a number, as +1 or, −1 or 0.

* [[Absolute value]]: distance to the origin (zero point)

}}

===[[Arithmetic function]]sfunctions===

{{main|Arithmetic function}}

* [[divisor function|Sigma function]]: [[Summation|Sums]] of [[Exponentiation|power]]s of [[divisor]]s of a given [[natural number]].

* [[Euler's totient function]]: Number of numbers [[coprime]] to (and not bigger than) a given one.

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* [[Prime omega function]]s

* [[Chebyshev function]]s

* [[Liouville function]],: λ<math>\Lambda(''n'') = (–1-1)<sup>Ω^{\Omega(''n'')}</supmath>

* [[Von Mangoldt function]], Λ(''n'') = log&nbsp;''p'' if ''n'' is a positive power of the prime ''p''

* [[Carmichael function]]: <math>\lambda(n)=</math> The smallest integer <math>m</math> such that <math>a^m\equiv 1\pmod{n}</math> for all <math>a</math> coprime to <math>n</math>

* [[Carmichael function]]

* [[Radical of an integer|Radical function]]: The product of the distinct prime factors of a positive integer input.

===Antiderivatives of elementary functions===

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* [[Pentation]]

* [[Super-logarithm]]s

* [[Super-root]]s

* [[Tetration]]

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* [[Ackermann function]]: in the [[theory of computation]], a [[computable function]] that is not [[primitive recursive function|primitive recursive]].

* [[Dirac delta function]]: everywhere zero except for ''x'' = 0; total integral is 1. Not a function but a [[distribution (mathematics)|distribution]], but sometimes informally referred to as a function, particularly by physicists and engineers.

* [[Dirichlet function]]: is an [[indicator function]] that matches 1 to [[Rational number|rational numbers]] and 0 to [[Irrational number|irrationals]]. It is [[nowhere continuous]].

* [[Thomae's function]]: is a function that is continuous at all irrational numbers and discontinuous at all rational numbers. It is also a modification of Dirichlet function and sometimes called Riemann function.

* [[Kronecker delta function]]: is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise.