List of mathematical functions explained

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.

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 solely by addition, multiplication, and raising to the power of a positive integer.
- 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.
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.

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.
- Inverse hyperbolic functions: inverses of the hyperbolic functions, analogous to the inverse circular 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, cotangent, secant, cosecant, exsecant, excosecant, versine, coversine, vercosine, covercosine, haversine, hacoversine, havercosine, hacovercosine, Inverse trigonometric functions etc.; used in geometry and to describe periodic phenomena. See also Gudermannian function.

Special functions

See main article: Special functions.

Arithmetic functions

See main article: Arithmetic function.

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.
Möbius μ function: Sum of the nth primitive roots of unity, it depends on the prime factorization of n.
Prime omega functions
Chebyshev functions
Liouville function, λ(n) = (–1)^Ω(n)
Von Mangoldt function, Λ(n) = log p if n is a positive power of the prime p
Carmichael function

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
Inverse tangent 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.
- Faddeeva function

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

\Pi(z)=z\Gamma(z)=(z)!

Hypergeometric and related functions

Hypergeometric functions: Versatile family of power series.
Confluent hypergeometric function
Associated Legendre functions
Meijer G-function
Fox H-function

Iterated exponential and related functions

Other standard special functions

Lambert W function: Inverse of f(w) = w exp(w).
Lamé function
Mathieu function
Mittag-Leffler function
Painlevé transcendents
Parabolic cylinder function
Arithmetic–geometric mean

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.
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

External links

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