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.
- Polynomials: can be generated by addition and multiplication alone.
- Square root: yields a number whose square is the given one.
- Exponential function: raise a fixed number to a variable power.
- Logarithm: the inverses of exponential functions; useful to solve equations involving exponentials.
- Trigonometric functions: sine, cosine, etc.; used in geometry and to describe periodic phenomena. See also Gudermannian function.
- Hyperbolic functions: formally similar to the trigonometric functions.
- Absolute value: drops the sign of a given number.
- Identity function: maps a given element to itself
- Constant function: a fixed value regardless of arguments, if any.
- Floor function: largest integer ≤ a given number.
- Signum function: returns only the sign of a number, as +1 or -1
- Gamma function: A generalization of the factorial function.
Riemann zeta function: A special case of Dirichlet series.
- Dirichlet eta function is an allied function.
- Elliptic integrals: Arising from the path length of ellipses; important in many applications.
- 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.
- Hypergeometric functions: Versatile family of power series
- Legendre function: From the theory of spherical harmonics
- Bessel functions: Defined by a differential equation; useful in astronomy, electromagnetism, and mechanics. See also Airy function.
- Logarithmic integral: Integral of the reciprocal of the logarithm, important in the prime number theorem.
- Lambert's W function: inverse of f(w) = w exp(w).
- Error function: an integral important for normal random variables.
Number theoretic functions
- Sigma function: Sums of powers of divisors of a given natural number.
- Euler's phi function: Number of numbers relatively prime 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.
Other standard special functions
- Carlson symmetric form
- Clausen function
- Dawson function
- Dedekind eta function
- Exponential integral
- Hurwitz zeta function
- Incomplete beta function
- Incomplete gamma function
- Lambda function
- Sinc function
- Synchrotron function
- 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.
- Heaviside step function: 0 for negative arguments and 1 for positive arguments. The integral of the Dirac delta distribution.
- Weierstrass function: Continuous, nowhere differentiable
Topics in mathematics related to change