List of mathematical functions

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In mathematics, several functions or groups of functions are important enough to deserve their own names. 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 statistics and 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 Algebraic functions 1.2 Elementary transcendental functions 2 Special functions 2.1 Basic special functions 2.2 Number theoretic functions 2.3 Antiderivatives of elementary functions 2.4 Gamma and related functions 2.5 Elliptic and related functions 2.6 Bessel and related functions 2.7 Riemann zeta and related functions 2.8 Hypergeometric and related functions 2.9 Other standard special functions 2.10 Miscellaneous functions 3 Function classification properties 3.1 Relative to set theory 3.2 Relative to an operator (c.q. a group) 3.3 Relative to a topology 3.4 Relative to an ordering 3.5 Relative to the real/complex numbers 4 External links

[edit] Elementary functions

Elementary function are functions built from basic operations (e.g. addition, exponentials, logarithms...)

[edit] Algebraic functions

Algebraic functions are functions that can be expressed as the solution of a polynomial equation with integer coefficients.

• Polynomials: Can be generated by addition and multiplication alone.
  • 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.
• Rational functions: A ratio of two polynomials.
• Power functions (with a rational power): A function of the form x^m/n.
  • Square root: Yields a number whose square is the given one (x^1/2).

[edit] Elementary transcendental functions

Transcendental functions are functions that are not algebraic.

• Exponential function: raises a fixed number to a variable power.
• Hyperbolic functions: formally similar to the trigonometric functions.
• Logarithms: the inverses of exponential functions; useful to solve equations involving exponentials.
• Power functions: raise a variable number to a fixed power; also known as Allometric functions; note: if the power is a rational number it is not strictly a transcendental function.
• Periodic functions
  • Trigonometric functions: sine, cosine, tangent, etc.; used in geometry and to describe periodic phenomena. See also Gudermannian function.
  • Sawtooth wave
  • Square wave
  • Triangle wave

[edit] Special functions

[edit] Basic special functions

• Indicator function: maps x to either 1 or 0, depending on whether or not x belongs to some subset.
• Step function: A finite linear combination of indicator functions of half-open intervals.
  • 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.
  • Signum function: Returns only the sign of a number, as +1 or −1.
• Absolute value: distance to the origin (zero point)

[edit] 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.

[edit] 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.

[edit] 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

[edit] 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

[edit] Bessel and related functions

• Airy function
• Bessel functions: Defined by a differential equation; useful in astronomy, electromagnetism, and mechanics.
• Bessel-Clifford function
• Legendre function: From the theory of spherical harmonics.
• Scorer's function
• Sinc function
• Hermite polynomials
• Chebyshev polynomials

[edit] 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

[edit] Hypergeometric and related functions

• Hypergeometric functions: Versatile family of power series.
• Confluent hypergeometric function
• Associated Legendre polynomials
• Meijer G-Function

[edit] 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

[edit] Miscellaneous functions

• Ackermann function: in the theory of computation, a computable 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: is an indicator function that matches 1 to rational numbers and 0 to irrationals. It is nowhere continuous.
• Kronecker delta function: is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise.
• Minkowski's question mark function: Derivatives vanish on the rationals.
• Weierstrass function: is an example of continuous function that is nowhere differentiable

[edit] Function classification properties

Functions can be classified according to the properties they have. These properties describe the functions behaviour under certain conditions.

[edit] Relative to set theory

These properties concern the domain, the codomain and the range of functions.

• Bijective function: is both an injective and a surjection, and thus invertible.
• Composite function: is formed by the composition of two functions f and g, by mapping x to f(g(x)).
• Constant function: has a fixed value regardless of arguments.
• Empty function: whose domain equals the empty set.
• Inverse function: is declared by "doing the reverse" of a given function (e.g. arcsine is the inverse of sine).
• Injective function: has a distinct value for each distinct argument. Also called an injection or, sometimes, one-to-one function.
• Surjective function: has a preimage for every element of the codomain, i.e. the codomain equals the range. Also called a surjection or onto function.
• Identity function: maps any given element to itself.
• Piecewise function: is defined by different expressions at different intervals.

[edit] Relative to an operator (c.q. a group)

These properties concern how the function is affected by arithmetic operations on its operand.

• Additive function: preserves the addition operation: f(x+y) = f(x)+f(y).
• Even function: is symmetric with respect to the Y-axis. Formally, for each x: f(x) = f(−x).
• Odd function: is symmetric with respect to the origin. Formally, for each x: f(−x) = −f(x).
• Subadditive function: for which the value of f(x+y) is less than or equal to f(x)+f(y).
• Superadditive function: for which the value of f(x+y) is greater than or equal to f(x)+f(y).

[edit] Relative to a topology

• Continuous function: in which preimages of open sets are open.
• Nowhere continuous function: is not continuous at any point of its domain (e.g. Dirichlet function).
• Homeomorphism: is an injective function that is also continuous, whose inverse is continuous.

[edit] Relative to an ordering

• Monotonic function: does not reverse ordering of any pair.
• Strict Monotonic function: preserves the given order.

[edit] Relative to the real/complex numbers

• Analytic function: Can be defined locally by a convergent power series.
• Arithmetic function: A function from the positive integers into the complex numbers.
• Differentiable function: Has a derivative.
• Holomorphic function: Complex valued function of a complex variable which is differentiable at every point in its domain.
• Entire function: A holomorphic function whose domain is the entire complex plane.

[edit] External links

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