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 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.

Elementary transcendental functions

Transcendental functions are functions that are not algebraic.

Special functions

Piecewise 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. • * Heaviside step function: 0 for negative arguments and 1 for positive arguments. The integral of the Dirac delta function. • Sawtooth wave • 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, −1 or 0. • Absolute value: distance to the origin (zero point)

Arithmetic functions

Antiderivatives of elementary functions

Gamma and related functions

Elliptic and related functions

• Elliptic integrals: Arising from the path length of ellipses; important in many applications. Alternate notations include: • * Carlson symmetric form • * Legendre form • Nome • Quarter period • Elliptic functions: 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 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 • Kelvin functions • Legendre function: From the theory of spherical harmonics. • Scorer's function • Sinc function • Hermite polynomials • Laguerre polynomials • Chebyshev polynomials • Synchrotron function

Riemann zeta and related functions

• Riemann zeta function: A special case of Dirichlet series. • Riemann Xi function • Dirichlet eta function: An allied function. • Dirichlet beta function • Dirichlet L-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. • * Dilogarithm • * Incomplete Fermi–Dirac integral • * Kummer's function • Riesz function

Hypergeometric and related functions

Iterated exponential and related functions

Other standard special functions

Miscellaneous functions