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Owen Reich (talk | contribs) m Added the rad function (as in the ABC Conjecture), and added a definition for the Carmichael function. |
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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
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
* [[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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** [[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
* [[Absolute value]]: distance to the origin (zero point)
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* [[Prime omega function]]s
* [[Chebyshev function]]s
* [[Liouville function]]
* [[Von Mangoldt function]], Λ(''n'') = log ''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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* [[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.
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