In mathematics, a function 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 are functions built from basic operations (e.g. addition, exponentials, logarithms...)
Algebraic functions are functions that can be expressed as the solution of a polynomial equation with integer coefficients.
Polynomials: Can be generated by addition, multiplication, and exponentiation alone.
Constant function: polynomial of degree zero, graph is a horizontal straight line
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.
Sextic function: Sixth degree polynomial.
Rational functions: A ratio of two polynomials.
nth root
Square root: Yields a number whose square is the given one.
Cube root: Yields a number whose cube is the given one.
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.
Natural logarithm
Common logarithm
Binary logarithm
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, cotangent, secant, cosecant, exsecant, excosecant, versine, coversine, vercosine, covercosine, haversine, hacoversine, havercosine, hacovercosine, etc.; used in geometry and to describe periodic phenomena. See also Gudermannian function.
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
Floor function: Largest integer less than or equal to a given number.
Sign function: Returns only the sign of a number, as +1 or −1.
Absolute value: distance to the origin (zero point)
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.
Logarithmic integral function: Integral of the reciprocal of the logarithm, important in the prime number theorem.
Exponential integral
Trigonometric integral: Including Sine Integral and Cosine 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 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 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 and the sine lemniscate and cosine lemniscate functions.
Theta function
Closely related are the modular forms, which include
J-invariant
Dedekind eta function
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
Laguerre polynomials
Chebyshev polynomials
Riemann zeta function: A special case of Dirichlet series.
Riemann Xi function
Dirichlet eta function: An allied 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.
Incomplete Fermi–Dirac integral
Kummer's function
Spence's function
Riesz function
Hypergeometric functions: Versatile family of power series.
Confluent hypergeometric function
Associated Legendre functions
Meijer G-function
Hyper operators
Iterated logarithm
Pentation
Super-logarithms
Super-roots
Tetration
Lambert W function: Inverse of f(w) = w exp(w).
Other standard special functions
Lambda function
Lamé function
Mittag-Leffler function
Painlevé transcendents
Parabolic cylinder function
Synchrotron function
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.
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.
Minkowski's question mark function: Derivatives vanish on the rationals.
Weierstrass function: is an example of continuous function that is nowhere differentiable
