- 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 ofspecial functions which developed out ofstatistics andmathematical physics . A modern, abstract point of view contrasts largefunction space s, which are infinite-dimensional and within which most functions are 'anonymous', with special functions picked out by properties such assymmetry , or relationship toharmonic analysis andgroup representation s.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 function s are functions that can be expressed as the solution of a polynomial equation with integer coefficients.

*Polynomial s: 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 aparabola .

**Cubic function : Third degree polynomial.

**Quartic function : Fourth degree polynomial.

**Quintic function : Fifth degree polynomial.

*Rational function s: A ratio of two polynomials.

*Power function s (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 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.

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

*Power function s: raise a variable number to a fixed power; also known asAllometric 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, etc.; used ingeometry and to describe periodic phenomena. See alsoGudermannian 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 finitelinear combination ofindicator function s ofhalf-open interval s.

**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 theDirac 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:

Sum s of powers ofdivisor s of a givennatural number .

*Euler's totient function : Number of numberscoprime 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 theprime number theorem .

*Exponential integral

*Error function : An integral important for normal random variables.

**Fresnel integral : related to the error function; used inoptics .

**Dawson function : occurs inprobability .**Gamma and related functions***

Gamma function : A generalization of thefactorial function.

*Barnes G-function

*Beta function : Correspondingbinomial coefficient analogue.

*Digamma function ,Polygamma function

*Incomplete beta function

*Incomplete gamma function

*K-function

*Multivariate gamma function : A generalization of the Gamma function useful inmultivariate statistics .

*Student's t-distribution **Elliptic and related functions***

Elliptic integral s: Arising from the path length ofellipse s; important in many applications. Related functions are thequarter period and the nome. Alternate notations include:

**Carlson symmetric form

**Legendre form

*Elliptic function s: The inverses of elliptic integrals; used to model double-periodic phenomena. Particular types areWeierstrass's elliptic functions andJacobi's elliptic functions .

*Theta function

* Closely related are themodular form s, which include

**J-invariant

**Dedekind eta function **Bessel and related functions***

Airy function

*Bessel function s: Defined by adifferential equation ; useful inastronomy ,electromagnetism , andmechanics .

*Bessel-Clifford function

*Legendre function : From the theory ofspherical harmonics .

*Scorer's function

*Sinc function

*Hermite polynomials

*Chebyshev polynomials **Riemann zeta and related functions***

Riemann zeta function : A special case ofDirichlet 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 function s: Versatile family ofpower series .

*Confluent hypergeometric function

*Associated Legendre polynomials

*Meijer G-Function **Iterated exponential and related functions***

Hyper operator s

*Iterated logarithm

*Super-logarithm s

*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 thetheory of computation , acomputable 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 anindicator function that matches 1 to rational numbers and 0 to irrationals. It isnowhere 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 ofcontinuous function that is nowheredifferentiable **External links*** [

*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.

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