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.

Contents

1 Elementary functions 1.1 Elementary periodic functions 1.2 Elementary transcendental functions 2 Special functions 2.1 Antiderivatives of elementary functions 2.2 Gamma and related functions 2.3 Elliptic and related functions 2.4 Bessel and related functions 2.5 Riemann zeta and related functions 2.6 Hypergeometric and related functions 2.7 Other standard special functions 3 Number theoretic functions 4 Miscellaneous

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

• Square wave
• Triangle wave

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:
  • 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.
• 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.
• 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.

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