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. Euclid, detail from The School of Athens by Raphael. ... Partial plot of a function f. ... In mathematics, there is a theory or theories of special functions, particular functions such as the trigonometric functions that have useful or attractive properties, and which occur in different applications often enough to warrant a name and attention of their own. ... A graph of a bell curve in a normal distribution showing statistics used in educational assessment, comparing various grading methods. ... Mathematical physics is the scientific discipline concerned with the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories1. ... In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications, it is a topological space or a vector space or both. ... Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ... Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... In mathematics, two polynomials f and g are orthogonal to each other with respect to a nonnegative weight function w precisely if In other words, if polynomials are treated as vectors and the inner product of two polynomials f(x) and g(x) is defined as then the orthogonal polynomials...

Elementary functions

Elementary functions are functions built from basic operations (e.g. addition, exponentials, logarithms...) In differential algebra, an elementary function is a function built from a finite number of exponentials, logarithms, constants, one variable, and roots of equations through composition and combinations using the four elementary operations (+ − × ÷). The trigonometric functions and their inverses are assumed to be included in the elementary functions by using...

Algebraic functions

Algebraic functions are functions that can be expressed as the solution of a polynomial equation. In mathematics, an algebraic function of indeterminates X1, X2, ..., Xn, is a function F that satisfies some non-trivial equation P(F, X1, X2, ..., Xn) = 0, with P a polynomial in n + 1 variables over a given field K. That is, F is an implicit function that solves an algebraic...

• 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).

In mathematics, a polynomial is an expression in which constants and powers of variables are combined using (only) addition, subtraction, and multiplication. ... A linear function is a mathematical function term of the form: f(x) = m x + c where c is a constant. ... f(x) = x2 - x - 2 In mathematics, a quadratic function is a polynomial function of the form , where is nonzero. ... A parabola The parabola (from the Greek: παραβολή) is a conic section generated by the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface. ... Polynomial of degree 3 In mathematics, a cubic function is a function of the form where a is nonzero; or in other words, a polynomial of degree three. ... Polynomial of degree 4: f(x) = (x+4)(x+1)(x-1)(x-3)/14+0. ... Polynomial of degree 5: f(x) = (x+4)(x+2)(x+1)(x-1)(x-3)/20+2 Quintic functions are polynomial functions in which the highest degree is five. ... In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ... In mathematics, exponentiation is a process generalized from repeated multiplication, in much the same way that multiplication is a process generalized from repeated addition. ... In mathematics, the principal square root of a non-negative real number is denoted and represents the non-negative real number whose square (the result of multiplying the number by itself) is For example, since This example suggests how square roots can arise when solving quadratic equations such as or...

Elementary trancendental functions

Transcendental functions are functions that are not algebraic. A transcendental function is a function which does not satisfy a polynomial equation whose coefficients are themselves polynomials. ...

• 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; 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

The exponential function is one of the most important functions in mathematics. ... In mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. ... Logarithms to various bases: is to base e, is to base 10, and is to base 1. ... In mathematics, exponentiation is a process generalized from repeated multiplication, in much the same way that multiplication is a process generalized from repeated addition. ... In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... Table of Geometry, from the 1728 Cyclopaedia. ... Gudermannian function with its asymptotes y = ±π/2 marked in gray. ... The sawtooth wave (or saw wave) is a kind of basic non-sinusoidal waveform. ... A square wave is a kind of basic waveform. ... A triangle wave is a waveform named for its triangular shape. ...

Special functions

In mathematics, there is a theory or theories of special functions, particular functions such as the trigonometric functions that have useful or attractive properties, and which occur in different applications often enough to warrant a name and attention of their own. ...

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)

In the mathematical subfield of set theory, the indicator function, or characteristic function, is a function defined on a set X which is used to indicate membership of an element in a subset A of X. Remark. ... A function on the reals is a step function if it can be written as a finite linear combination of semi-open intervals. ... In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ... In the mathematical subfield of set theory, the indicator function, or characteristic function, is a function defined on a set X which is used to indicate membership of an element in a subset A of X. Remark. ... In elementary algebra, an interval is a set that contains every real number between two indicated numbers, and possibly the two numbers themselves. ... In mathematics, the floor function is the function defined as follows: for a real number x, floor(x) is the largest integer less than or equal to x. ... The Heaviside step function, using the half-maximum convention The Heaviside step function, sometimes called the unit step function and named in honor of Oliver Heaviside, is a discontinuous function whose value is zero for negative argument and one for positive argument: The function is used in the mathematics of... The Dirac delta function, sometimes referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere. ... Signum function In mathematics and especially in computer science, the sign function is a logical function which extracts the sign of a real number. ... In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ...

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.

In mathematics, and specifically in number theory, a divisor function is an arithmetical function related to the divisors of an integer. ... Addition is one of the basic operations of arithmetic. ... In mathematics, exponentiation is a process generalized from repeated (or iterated) multiplication, in much the same way that multiplication is a process generalized from repeated addition. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... A natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory and computer science. ... In number theory, the totient φ(n) of a positive integer n is defined to be the number of positive integers less or equal than n and coprime to n. ... Coprime - Wikipedia /**/ @import /skins-1. ... In number theory, the prime number theorem (PNT) describes the approximate, asymptotic distribution of the prime numbers. ... A prime number (or a prime) is a natural number that only has trivial divisors. ... It has been suggested that this article or section be merged with Integer partition. ...

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.

In mathematics, the logarithmic integral function or integral logarithm li(x) is a non-elementary function defined for all positive real numbers x≠ 1 by the definite integral: Here, ln denotes the natural logarithm. ... In number theory, the prime number theorem (PNT) describes the approximate, asymptotic distribution of the prime numbers. ... In mathematics, the exponential integral Ei(x) is defined as Since 1/t diverges at t=0, the above integral has to be understood in terms of the Cauchy principal value. ... In mathematics, the error function (also called the Gauss error function) is a non-elementary function which occurs in probability, statistics and partial differential equations. ... The normal distribution, also called Gaussian distribution, is an extremely important probability distribution in many fields. ... In mathematics and optics, the two Fresnel integrals, S(x) and C(x), arise in the description of near field Fresnel diffraction phenomena, and are the integrals defined as follows: . Some may use Ï€ t2/2 instead of t2, in which case the S(x) and C(x) above should be... Table of Opticks, 1728 Cyclopaedia Optics (appearance or look in ancient Greek) is a branch of physics that describes the behavior and properties of light and the interaction of light with matter. ... In mathematics, the Dawson function is The notation D(x) is also in use. ... The word probability derives from the Latin probare (to prove, or to test). ...

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

The Gamma function along part of the real axis In mathematics, the Gamma function extends the factorial function to complex and non integer numbers (it is already defined on the naturals, and has simple poles at the negative integers). ... In mathematics, the factorial of a natural number n is the product of all positive integers less than and equal to n. ... In mathematics, the Barnes G-function (typically denoted G(z)) is a function that is an extension of superfactorials to the complex numbers. ... A separate article treats the beta-function (written with a hyphen) of physics. ... In mathematics, particularly in combinatorics, the binomial coefficient of the natural number n and the integer k is defined to be the natural number and (Here, for a natural number m, m! denotes the factorial of m. ... In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: It is the first of the polygamma functions. ... In mathematics, the polygamma function of order m is defined as the m+1 th derivative of the logarithm of the gamma function: Here is the digamma function and is the gamma function. ... In mathematics, the incomplete beta function is a generalization of the beta function that replaces the definite integral of the beta function with an indefinite integral. ... In mathematics, the gamma function is defined by a definite integral. ... In mathematics, the K-function, typically denoted K(z), is a generalization of the hyperfactorial to complex numbers, similar to the generalization of the factorial to the Gamma function. ... In mathematics, the multivariate gamma distribution , , is a generalization of the gamma function. ... Multivariate statistics or multivariate statistical analysis in statistics describes a collection of procedures which involve observation and analysis of more than one statistical variable at a time. ... In probability and statistics, the t-distribution or Students t-distribution is a probability distribution that arises in the problem of estimating the mean of a normally distributed population when the sample size is small. ...

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

In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse and were first studied by Giulio Fagnano and Leonhard Euler. ... The ellipse and some of its mathematical properties. ... In mathematics, the quarter periods K(m) and iK′(m) are special functions that appear in the theory of elliptic functions. ... In mathematics, specifically the theory of elliptic functions, the nome is a special function and is given by where K and iK are the quarter periods, and and are the fundamental pair of periods. ... In mathematics, the Carlson symmetric forms of elliptic integrals, RC(x,y), RD(x,y,z), RF(x,y,z) and RJ(x,y,z,p) are defined by Categories: Special functions | Elliptic functions | Math stubs ... In mathematics, the Legendre forms of elliptic integrals, F(φ,k), E(φ,k) and P(φ,k,n) are defined by and Categories: Math stubs ... In complex analysis, an elliptic function is, roughly speaking , a function defined on the complex plane which is periodic in two directions. ... In mathematics, Weierstrass introduced some particular elliptic functions that have become the basis for the most standard notations used. ... In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions, and auxiliary theta functions, that have historical importance with also many features that show up important structure, and have direct relevance to some applications (e. ... In mathematics, theta functions are special functions of several complex variables. ... Modular form - Wikipedia /**/ @import /skins-1. ... Real part of the j-invariant as a function of the nome q on the unit disk In mathematics, Kleins j-invariant, regarded as a function of a complex variable Ï„, is a modular function defined on the upper half-plane of complex numbers. ... The Dedekind eta function is a function defined on the upper half plane of complex numbers whose imaginary part is positive. ...

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
• Hermite polynomials
• Chebyshev polynomials

In mathematics, the Airy function Ai(x) is a special function, i. ... In mathematics, Bessel functions, first defined by the Swiss mathematician Daniel Bernoulli and named after Friedrich Bessel, are canonical solutions y(x) of Bessels differential equation: for an arbitrary real number α (the order). ... Graph of a differential equation In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... Lunar astronomy: the large crater is Daedalus, photographed by the crew of Apollo 11 as they circled the Moon in 1969. ... Electromagnetism is the physics of the electromagnetic field: a field, encompassing all of space, which exerts a force on those particles that possess the property of electric charge, and is in turn affected by the presence and motion of such particles. ... Mechanics refers to: a craft relating to machinery (from the Latin mechanicus, from the Greek mechanikos, meaning one skilled in machines), or a range of disciplines in science and engineering. ... Note: The term Legendre polynomials is sometimes used (wrongly) to indicate the associated Legendre polynomials. ... In mathematics, the spherical harmonics are an orthogonal set of solutions to Laplaces equation represented in a system of spherical coordinates. ... In mathematics, the Scorers functions are special functions denoted Gi(x) and Hi(x). ... The sinc function sinc(x) from x = âˆ’8Ï€ to 8Ï€. In mathematics, the sinc function (for sinus cardinalis), also known as the interpolation function, filtering function or the first spherical Bessel function , is the product of a sine function and a monotonically decreasing function. ... In mathematics, the Hermite polynomials, named in honor of Charles Hermite (Hermite is pronounced air MEET), are a polynomial sequence defined either by (the probabilists Hermite polynomials), or sometimes by (the physicists Hermite polynomials). These two definitions are not exactly equivalent; either is a trivial rescaling of the other. ... In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev (Пафнутий Чебышёв), are a sequence of orthogonal polynomials which are related to de Moivres formula and which are easily defined recursively, like Fibonacci or Lucas numbers. ...

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

There are a number of mathematical functions with the name zeta function, named after the Greek letter ζ. ... In mathematics, the Riemann zeta-function, named after Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ... In mathematics, a Dirichlet series, one of a number of concepts named in honor of Johann Peter Gustav Lejeune Dirichlet, is a series of the form The most famous of Dirichlet series is which is the Riemann zeta function. ... The Dirichlet eta function can be defined as where ζ is Riemanns zeta function. ... In mathematics, the Hurwitz zeta function is one of the many zeta functions. ... In mathematics, the Legendre chi function is defined as The discrete fourier transform of the Legendre chi function with respect to the order n is the Hurwitz zeta function (Cvijovic). ... In mathematics, the Lerch zeta function is a special function that generalizes the Hurwitz zeta function and the polylogarithm. ... The polylogarithm (also known as Jonquiéres function) is a special function that is defined for all complex numbers s and z where |z| < 1 by: The special cases s = 2 and s = 3 are called the dilogarithm (also referred to as Spences function) and trilogarithm respectively. ... In mathematics, the Incomplete Polylogarithm function is related to the polylogarithm function. ... In mathematics, the Clausen function is defined by the following integral: More generally, one defines . It is related to the polylogarithm by . Ernst Kummer and Rogers give the relation valid for . ... In mathematics, the complete Fermi-Dirac integral for an index j  is given by This is an alternate definition of the polylogarithm function. ... In mathematics, the incomplete Fermi-Dirac integral for an index j is given by This is an alternate definition of the incomplete polylogarithm. ... In mathematics, there are several functions known as Kummers function. ... In mathematics, the Riesz function is an entire function defined by Marcel Riesz in connection with the Riemann hypothesis, by means of the power series If we set we may define it in terms of the coefficients of the Laurent series development of the hyperbolic (or equivalently, the ordinary) cotangent...

Hypergeometric and related functions

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

In mathematics, a hypergeometric series could in principle be any formal power series in which the ratio of successive coefficients an/an-1 is a rational function of n. ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ... In mathematics, the confluent hypergeometric function is formed from hypergeometric series. ... In mathematics, the associated Legendre polynomials, named after Adrien-Marie Legendre, are defined by: These differ from the 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

In mathematics, the Dawson function is The notation D(x) is also in use. ... This article is about the mathematical lambda function. ... In mathematics, Lamberts W function, named after Johann Heinrich Lambert, also called the Omega function or product log, is the inverse function of where ew is the exponential function and w is any complex number. ... In mathematics, the Mittag-Leffler function is special function, a complex function which depends on two complex parameters and . ... In mathematics, the parabolic cylinder functions are special functions defined as solutions to the differential equation This can be brought into two distinct forms (A) and (B) by change of variable, to (A) and (B) If f(a,z) is a solution, then so are f(a,−z), f(−a... In mathematics the synchrotron functions are defined as follows (for x ≥ 0): First synchrotron function Second synchrotron function where Kj is the modified Bessel function of the second kind. ...

Miscellaneous functions

• 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: is an indicator function that matches 1 to rational numbers and 0 to irrationals. It is nowhere continuous.
• Minkowski question mark function: Derivatives vanish on the rationals.
• Weierstrass function: is an example of continuous function that is nowhere differentiable

In the theory of computation, the Ackermann function or Ackermann-Peter function is a simple example of a recursive function that is not primitively recursive. ... Computation can be defined as finding a solution to a problem from given inputs by means of an algorithm. ... It has been suggested that this article or section be merged into computable function. ... In computability theory, primitive recursive functions are a class of functions which form an important building block on the way to a full formalization of computability. ... The Dirac delta function, sometimes referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere. ... In mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and probability distributions. ... In mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain. ... In the mathematical subfield of set theory, the indicator function, or characteristic function, is a function defined on a set X which is used to indicate membership of an element in a subset A of X. Remark. ... In mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain. ... Minkowski question mark function In mathematics, the Minkowski question mark function, sometimes called the slippery devils staircase, is a function, denoted ?(x), possessing various unusual fractal properties. ... In mathematics, the Weierstrass function was the first example found of a kind of function with the property that it is continuous everywhere but differentiable nowhere. ... In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ...

Function classification properties

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

Relative to set theory

These properties concern the domain, the codomain and the range of functions. Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In mathematics, the domain of a function is the set of all input values to the function. ... A codomain in mathematics is the set of output values associated with (or mapped to) the domain of inputs in a function. ... In mathematics, the range of a function is the set of all output values produced by that function. ...

• 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.

In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ... In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... A surjective function. ... In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ... In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ... In mathematics a constant function is a function whose values do not vary and thus are constant. ... In the case where the domain of the function is the empty set {}, there is only one function with that domain (given any codomain), the empty function, and any formula can be used to define the empty function, since the formula wont apply to anything and will therefore never... In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ... In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ... In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ... A codomain in mathematics is the set of output values associated with (or mapped to) the domain of inputs in a function. ... In mathematics, injections, surjections and bijections are classes of functions distinguished by the manner in which arguments (input expressions) and images (output expressions) are related. ... An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ...

Relative to an operator (c.q. a group)

These properties concern how the function is affected by arithmetic operations on it's operand. Arithmetic or arithmetics (from the Greek word αριθμός = number) in common usage is a branch of (or the forerunner of) mathematics which records elementary properties of certain operations on numerals, though professional mathematicians often treat arithmetic as a synonym for number theory. ...

• 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).

In number theory, an additive function is an arithmetic function f(n) of the positive integer n such that whenever a and b are coprime we have: f(ab) = f(a) + f(b). ... In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking negatives. ... In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking negatives. ... In mathematics, the origin of a coordinate system is the point where the axes of the system intersect. ... A sequence { an }, n ≥ 1, is called subadditive if it satisfies the inequality for all m and n. ... A sequence { an }, n ≥ 1, is called superadditive if it satisfies the inequality for all m and n. ...

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.

In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain. ... In mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ... In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ...

Relative to an ordering

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

In mathematics, functions between ordered sets are monotonic (or monotone) if they preserve the given order. ... In mathematics, functions between ordered sets are monotonic (or monotone) if they preserve the given order. ...

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.
• Entire function: A holomorphic function whose domain is the entire complex plane.
• Holomorphic function: Complex valued function of a complex variable which is differentiable at every point in its domain.

In mathematics, an analytic function is a function that is locally given by a convergent power series. ... For a discussion of convergence and convergent series, see limit (mathematics). ... In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ... In number theory, an arithmetic function (or number-theoretic function) f(n) is a function defined for all positive integers and having values in the complex numbers. ... The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = −1. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... In mathematics, the derivative is defined to be the instantaneous rate of change of a function. ... In complex analysis, an entire function is a function that is holomorphic everywhere on the whole complex plane. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = −1. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = −1. ...

External links

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