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.

Classes of functions

Functions can classified according to the properties they have.

• Additive function: The value of a product equals the sum of the values of the factors.
• Analytic function: Can be defined locally by a convergent power series.
• Arithmetic function: A function from the positive integers into the complex numbers.
• Bijection: Is both an injective and a surjection.
• Composition function: For two functions f and g, maps x to f(g(x)).
• Continuous function: Preimages of open sets are open.
• Differentiable function: Has a derivative.
• Entire function: A holomorphic function whose domain is the entire complex plane.
• Even function: f(x) = f(−x). Is symmetric with respect to the Y-axis.
• Holomorphic function: Complex valued function of a complex variable which is complex differentiable at every point in its domain.
• Homeomorphism: One-to-one function continuous function, whose inverse is continuous.
• Injection, injective function: Distinct arguments have distinct values. Also called a one-to-one function.
• Inverse function: "Does the reverse" of a given function.
• Monotonic function: Order preserving (or reversing).
• Odd function: f(x) = −f(x). Is symmetric with respect to the origin.
• One-to-one function: Distinct arguments have distinct values. Also called an injection or injective function.
• Onto function: Every element of the codomain has a preimage. Also called a surjection or surjective function.
• Subadditive function: The value of a sum is less than or equal to the sum of the values of the summands.
• Superadditive function: The value of a sum is greater than or equal to the sum of the values of the summands.
• Surjection, surjective function: Every element of the codomain has a preimage. Also called an onto function.

Elementary functions

• Absolute value: Leaves positive numbers alone, multiplies negative numbers by −1 to make them positive.
• Empty function: Domain equals the empty set.
• 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.
• Indicator function: Maps x to either 1 or 0, depending on whether, x is or is not in some set.
• Signum function: Returns only the sign of a number, as +1 or −1.
• Step function: A finite linear combination of indicator functions of half-open intervals.

Polynomials

Polynomials: can be generated by addition and multiplication alone.

• Constant function: Zero degree polynomial, fixed value regardless of arguments.
• 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.

Elementary periodic functions

• Sawtooth wave
• 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.
• 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 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

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 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.

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.
• Question mark function: Derivatives vanish on the rationals.
• Weierstrass function: Continuous, nowhere differentiable

External links

• Special functions at EqWorld: The World of Mathematical Equations.