List of mathematical functions
In mathematics, several functions are important enough to deserve their own name. This is a listing of pointers to those articles which explain these functions in more detail. There is a large theory of special functions which developed out of trigonometry, and then the needs of 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 orthogonal polynomial.
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Elementary functions
- Absolute value: leaves positive numbers alone, multiplies negative numbers by −1 to make them positive.
- Constant function: a fixed value regardless of arguments.
- 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.
- Identity function: maps a given element to itself.
- Linear function: Graph is a straight line.
- Signum function: Returns only the sign of a number, as +1 or −1.
Elementary periodic functions
Elementary transcendental functions
- Exponential function: raises a fixed number to a variable power.
- Hyperbolic functions: formally similar to the trigonometric functions.
- Logarithm: the inverses of exponential functions; useful to solve equations involving exponentials.
- Polynomials: can be generated by addition and multiplication alone.
- Power function: raises a variable number to a fixed power; also known as Allometric function.
- Square root: yields a number whose square is the given one.
- Trigonometric functions: sine, cosine, tangent, etc.; used in geometry and to describe periodic phenomena. See also Gudermannian function.
Special functions
Antiderivatives of elementary functions
- Logarithmic integral function: Integral of the reciprocal of the logarithm, important in the prime number theorem.
- Exponential 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 function
- Beta function: Corresponding binomial coefficient analogue.
- Digamma function, Polygamma function
- Incomplete beta function
- Incomplete gamma 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:
- 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.
- Theta function
- Closely related are the modular forms, which include
Bessel and related functions
- Airy function
- Bessel functions: Defined by a differential equation; useful in astronomy, electromagnetism, and mechanics.
- Legendre function: From the theory of spherical harmonics.
- Scorer's function
- Sinc function
Riemann zeta and related functions
- Riemann zeta function: A special case of Dirichlet 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
- Riesz function
Hypergeometric and related functions
- Hypergeometric functions: Versatile family of power series.
- Confluent hypergeometric function
- Associated Legendre polynomials
Other standard special functions
- Dawson function
- Lambda function
- Lambert's W function: Inverse of f(w) = w exp(w).
- Lame function
- Mittag-Leffler function
- Parabolic cylinder function
- Synchrotron function
Number theoretic functions
- Sigma function: Sums of powers of divisors of a given natural number.
- Euler's phi 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.
External links
- Special functions at EqWorld: The World of Mathematical Equations.
Miscellaneous
- Ackermann function: in the theory of computation, a recursive 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 Nowhere continuous.
- Weierstrass function: Continuous, nowhere differentiable