Special function - Definition and Overview

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.

Elementary functions

• Constant function: a fixed value regardless of arguments.
• Identity function: maps a given element to itself.
• Linear function: Graph is a straight line.
• Exponential function: raises a fixed number to a variable power.
• Logarithm: the inverses of exponential functions; useful to solve equations involving exponentials.
• Square root: yields a number whose square is the given one.
• Power function: raises a variable number to a fixed power; also known as Allometric function.
• Polynomials: can be generated by addition and multiplication alone.
• Absolute value: leaves positive numbers alone, multiplies negative numbers by -1 to make them positive.
• Trigonometric functions: sine, cosine, tangent, etc.; used in geometry and to describe periodic phenomena. See also Gudermannian function.
• Hyperbolic functions: formally similar to the trigonometric functions.

Special functions

• Floor function: Largest integer less than or equal to a given number.
• Signum function: Returns only the sign of a number, as +1 or -1.
• Gamma function: A generalization of the factorial function.
  • Beta function: Corresponding binomial coefficient analogue.
  • Digamma function, Polygamma function
  • Multivariate gamma function: A generalization of the Gamma function useful in multivariate statistics.
• Riemann zeta function: A special case of Dirichlet series.
  • Dirichlet eta function: An allied function.
• Elliptic integrals: Arising from the path length of ellipses; important in many applications. Related functions are the quarter period and the nome.
• 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 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.

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
• Lerch Transcendent
• Polylogarithm
• Sinc function
• Synchrotron function

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.
• Heaviside step function: 0 for negative arguments and 1 for positive arguments. The integral of the Dirac delta distribution.
• Dirichlet function Nowhere continuous.
• Weierstrass function: Continuous, nowhere differentiable