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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 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 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 and the sine lemniscate and cosine lemniscate 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. 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 and related functions Hypergeometric functions: Versatile family of power series. Confluent hypergeometric function Associated Legendre functions Meijer G-function Iterated exponential and related functions 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 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 |
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