List of mathematical functions: Difference between revisions

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{{shortShort description|Wikipedia list articlenone}}

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. >:V

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

** [[Linear function]]: First degree polynomial, graph is a 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]]

** [[Common logarithm]]

** [[Binary logarithm]]

* [[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]]==

===Basic special{{main|Special functions===}}

===Piecewise special functions===

{{columns-list|colwidth=20em|

* [[Indicator function]]: maps ''x'' to either 1 or 0, depending on whether or not ''x'' belongs to some subset.

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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]]

* [[SynchrotronRectangular 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|Number theoretic functions]]===

===Arithmetic functions===

{{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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* [[Partition function (number theory)|Partition function]]: Order-independent count of ways to write a given positive integer as a sum of positive integers.

* [[Möbius function|Möbius μ function]]: Sum of the nth primitive roots of unity, it depends on the prime factorization of n.

** [[Spence'sPrime 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>

* [[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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* [[Exponential integral]]

* [[Trigonometric integral]]: Including Sine Integral and Cosine Integral

* [[Inverse tangent integral]]

* [[Error function]]: An integral important for [[normal distribution|normal random variables]].

** [[Fresnel integral]]: related to the error function; used in [[optics]].

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* [[Multivariate gamma function]]: A generalization of the Gamma function useful in [[multivariate statistics]].

* [[Student's t-distribution]]

* [[Gamma function#Pi function|Pi function]] ∏<math>\Pi(z) = z*Γ \Gamma(z) = (z)!</math>

===Elliptic and related functions===

{{columns-list|colwidth=20em|

* [[Elliptic integral]]s: Arising from the path length of [[ellipse]]s; important in many applications. Related functions are the [[quarter period]] and the [[nome (mathematics)|nome]]. Alternate notations include:

** [[Carlson symmetric form]]

** [[Legendre form]]

* [[Nome (mathematics)|Nome]]

* [[Elliptic function]]s: The inverses of elliptic integrals; used to model double-periodic phenomena. Particular types are [[Weierstrass's elliptic functions]] and [[Jacobi's elliptic functions]] and the [[sine lemniscate]] and [[cosine lemniscate]] functions.

* [[ThetaQuarter functionperiod]]

* [[Elliptic function]]s: The inverses of elliptic integrals; used to model double-periodic phenomena.

**[[Jacobi's elliptic functions]]

**[[Weierstrass's elliptic functions]]

**[[Lemniscate elliptic functions]]

* [[Theta functions]]

* [[Neville theta functions]]

* [[Modular lambda function]]

* Closely related are the [[modular form]]s, which include

** [[J-invariant]]

** [[Dedekind eta function]]

}}

===Bessel and related functions===

{{columns-list|colwidth=20em|

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* [[Laguerre polynomials]]

* [[Chebyshev polynomials]]

* [[Synchrotron function]]

}}

===Riemann zeta and related functions===

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** [[Clausen function]]

** [[Complete Fermi–Dirac integral]], an alternate form of the polylogarithm.

** [[Dilogarithm]]

** [[Incomplete Fermi–Dirac integral]]

** [[Kummer's function]]

** [[Spence's function]]

* [[Riesz function]]

}}

===Hypergeometric and related functions===

* [[Hypergeometric function]]s: Versatile family of [[power series]].

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* [[Associated Legendre functions]]

* [[Meijer G-function]]

* [[Fox H-function]]

===Iterated exponential and related functions===

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* [[Pentation]]

* [[Super-logarithm]]s

* [[Super-root]]s

* [[Tetration]]

* [[Lambert W function]]: Inverse of ''f''(''w'') = ''w'' exp(''w'').

===Other standard special functions===

* Dirichlet[[Lambert lambdaW function,]]: Inverse of ''λf''(''sw'') = (1&nbsp;–&nbsp;2^−''sw'')ζ exp(''sw'') where ''ζ'' is the [[Riemann zeta function]].

* [[Liouville function]], λ(''n'') = (–1)^Ω(''n'')

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

* [[Modular lambda function]], λ(τ), a highly symmetric holomorphic function on the complex upper half-plane

* [[Lamé function]]

* [[Mathieu function]]

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* [[Painlevé transcendents]]

* [[Parabolic cylinder function]]

* [[Synchrotron function]]

* [[Arithmetic–geometric mean]]

===Miscellaneous functions===

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

* [[Böttcher's equation|Böttcher's function]]

* [[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.

* [[Minkowski's question mark function]]: Derivatives vanish on the rationals.

* [[Weierstrass function]]: is an example of [[continuous function]] that is nowhere [[Differentiable function|differentiable]]

== See also ==

* [[List of mathematicaltypes of abbreviationsfunctions]]

* [[Test functions for optimization]]

* [[List of mathematical abbreviations]]

* [[List of special functions and eponyms]]

== External links ==

* [https://archive.istoday/20130105073730/http://www.special-functions.com/ Special functions] : A programmable special functions calculator.

* [http://eqworld.ipmnet.ru/en/auxiliary/aux-specfunc.htm Special functions] at EqWorld: The World of Mathematical Equations.

[[Category:Calculus|Functions]]

[[Category:Mathematics-related lists|Functions]]

[[Category:Number theory|Functions]]

[[Category:Functions and mappings| ]]

[[pl:Funkcje elementarne]]