List of mathematical functions

List of mathematical functions

In mathematics, several functions or groups of functions are important enough to deserve their own names. This is a listing of pointers to those articles which explain 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 spaces, 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 representations.

See also List of types of functions

Elementary functions

Elementary functions are functions built from basic operations (e.g. addition, exponentials, logarithms...)

Algebraic functions

Algebraic functions are functions that can be expressed as the solution of a polynomial equation with integer coefficients.
* Polynomials: Can be generated by addition and multiplication alone.
** Linear function: First degree polynomial, graph is a straight line.
** Quadratic function: Second degree polynomial, graph is a parabola.
** Cubic function: Third degree polynomial.
** Quartic function: Fourth degree polynomial.
** Quintic function: Fifth degree polynomial.
* Rational functions: A ratio of two polynomials.
* Power functions (with a rational power): A function of the form x^{frac{m}{n ! .
** Square root: Yields a number whose square is the given one x^{frac{1}{2 ! .

Elementary transcendental functions

Transcendental functions are functions that are not algebraic.
* Exponential function: raises a fixed number to a variable power.
* Hyperbolic functions: formally similar to the trigonometric functions.
* Logarithms: the inverses of exponential functions; useful to solve equations involving exponentials.
* Power functions: raise a variable number to a fixed power; also known as Allometric functions; note: if the power is a rational number it is not strictly a transcendental function.
* Periodic functions
** Trigonometric functions: sine, cosine, tangent, etc.; used in geometry and to describe periodic phenomena. See also Gudermannian function.
** Sawtooth wave
** Square wave
** Triangle wave

Special functions

Basic special functions

* Indicator function: maps "x" to either 1 or 0, depending on whether or not "x" belongs to some subset.
* Step function: A finite linear combination of indicator functions of half-open intervals.
** Floor function: Largest integer less than or equal to a given number.
** Heaviside step function: 0 for negative arguments and 1 for positive arguments. The integral of the Dirac delta function.
** Sign function: Returns only the sign of a number, as +1 or −1.
* Absolute value: distance to the origin (zero point)

Number theoretic functions

* Sigma function: Sums of powers of divisors of a given natural number.
* Euler's totient function: Number of numbers coprime to (and not bigger than) a given one.
* Prime-counting function: Number of primes less than or equal to a given number.
* Partition function: Order-independent count of ways to write a given positive integer as a sum of positive integers.

Antiderivatives of elementary functions

* Logarithmic integral function: Integral of the reciprocal of the logarithm, important in the prime number theorem.
* Exponential integral
* Error function: An integral important for normal random variables.
** Fresnel integral: related to the error function; used in optics.
** Dawson function: occurs in probability.

Gamma and related functions

* Gamma function: A generalization of the factorial function.
* Barnes G-function
* Beta function: Corresponding binomial coefficient analogue.
* Digamma function, Polygamma function
* Incomplete beta function
* Incomplete gamma function
* K-function
* Multivariate gamma function: A generalization of the Gamma function useful in multivariate statistics.
* Student's t-distribution

Elliptic and related functions

* Elliptic integrals: Arising from the path length of ellipses; important in many applications. Related functions are the quarter period and the nome. Alternate notations include:
** Carlson symmetric form
** Legendre form
* Elliptic functions: The inverses of elliptic integrals; used to model double-periodic phenomena. Particular types are Weierstrass's elliptic functions and Jacobi's elliptic functions.
* Theta function
* Closely related are the modular forms, which include
** J-invariant
** Dedekind eta function

Bessel and related functions

* Airy function
* Bessel functions: Defined by a differential equation; useful in astronomy, electromagnetism, and mechanics.
* Bessel-Clifford function
* Legendre function: From the theory of spherical harmonics.
* Scorer's function
* Sinc function
* Hermite polynomials
* Chebyshev polynomials

Riemann zeta and related functions

* Riemann zeta function: A special case of Dirichlet series.
* Dirichlet eta function: An allied function.
* Hurwitz zeta function
* Legendre chi function
* Lerch transcendent
* Polylogarithm and related functions:
** Incomplete polylogarithm
** Clausen function
** Complete Fermi–Dirac integral, an alternate form of the polylogarithm.
** Incomplete Fermi–Dirac integral
** Kummer's function
** Spence's function
* Riesz function

Hypergeometric and related functions

* Hypergeometric functions: Versatile family of power series.
* Confluent hypergeometric function
* Associated Legendre polynomials
* Meijer G-Function

Iterated exponential and related functions

* Hyper operators
* Iterated logarithm
* Super-logarithms
* Tetration
* Lambert W function: Inverse of "f"("w") = "w" exp("w").
* Ultra exponential function
* Infra logarithm function

Other standard special functions

* Lambda function
* Lamé function
* Mittag-Leffler function
* Painleve transcendents
* Parabolic cylinder function
* Synchrotron function

Miscellaneous functions

* Ackermann function: in the theory of computation, a computable function that is not primitive recursive.
* Dirac delta function: everywhere zero except for "x" = 0; total integral is 1. Not a function but a 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.
* 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

External links

* [ Special functions] : A programmable special functions calculator.
* [ Special functions] at EqWorld: The World of Mathematical Equations.

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • List of mathematical examples — This page will attempt to list examples in mathematics. To qualify for inclusion, an article should be about a mathematical object with a fair amount of concreteness. Usually a definition of an abstract concept, a theorem, or a proof would not be …   Wikipedia

  • List of mathematical logic topics — Clicking on related changes shows a list of most recent edits of articles to which this page links. This page links to itself in order that recent changes to this page will also be included in related changes. This is a list of mathematical logic …   Wikipedia

  • List of mathematical series — This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums.ums of powers*sum {i=1}^n i = frac{n(n+1)}{2} *sum {i=1}^n i^2 = frac{n(n+1)(2n+1)}{6} =… …   Wikipedia

  • List of mathematical jargon — The language of mathematics has a vast vocabulary of specialist and technical terms. It also has a certain amount of jargon: commonly used phrases which are part of the culture of mathematics, rather than of the subject. Jargon often appears in… …   Wikipedia

  • List of mathematical proofs — A list of articles with mathematical proofs:Theorems of which articles are primarily devoted to proving them: See also: *Bertrand s postulate and a proof *Estimation of covariance matrices *Fermat s little theorem and some proofs *Gödel s… …   Wikipedia

  • List of special functions and eponyms — This is a list of special function eponyms in mathematics, to cover the theory of special functions, the differential equations they satisfy, named differential operators of the theory (but not intended to include every mathematical eponym).… …   Wikipedia

  • List of mathematical identities — This page lists identities in the sense of mathematics, that is, identically true relations holding in algebra or between special functions.* Cassini s identity * Difference of two squares * Bézout s identity * Euler s identity * Vandermonde s… …   Wikipedia

  • List of mathematics articles (L) — NOTOC L L (complexity) L BFGS L² cohomology L function L game L notation L system L theory L Analyse des Infiniment Petits pour l Intelligence des Lignes Courbes L Hôpital s rule L(R) La Géométrie Labeled graph Labelled enumeration theorem Lack… …   Wikipedia

  • List of mathematics reference tables — See also: List of reference tables Mathematics *List of mathematical topics *List of statistical topics *List of mathematical functions *List of mathematical theorems *List of mathematical proofs *List of matrices *List of numbers *List of… …   Wikipedia

  • Mathematical logic — (also known as symbolic logic) is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic.[1] The field includes both the mathematical study of logic and the… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”