List of mathematical functions: Difference between revisions

Revision as of 01:13, 29 November 2021

Revision as of 23:22, 25 February 2022

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==[[Special functions]]==

===BasicPiecewise special functions===

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

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* [[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.

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

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===[[Arithmetic_function|Number theoreticArithmetic functions]]===

* [[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.

* [[Prime omega function|Prime omega functions]]

* [[Chebyshev function|Chebyshev functions]]

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

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

* [[Carmichael function]]

===Antiderivatives of elementary functions===

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===Elliptic and related functions===

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* [[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]]

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* [[Laguerre polynomials]]

* [[Chebyshev polynomials]]

* [[Synchrotron function]]

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===Riemann zeta and related functions===

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* [[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 – 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 ''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 numbers and 0 to irrationals. It is [[nowhere continuous]].

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* [[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 mathematical abbreviations]]