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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, an expression is said to be a closed-form expression if, and only if, it can be expressed analytically in terms of a bounded number of certain "well-known" function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
s. Typically, these well-known functions are defined to be elementary functions
Elementary function (differential algebra)

In mathematics, an elementary function is a function built from a finite number of exponential functions, logarithms, constants, one variable, and nth roots through function composition and combinations using the four arithmetic ....
; so infinite series
Series (mathematics)

In mathematics, given an infinite set sequence of numbers , a series is informally the result of adding all those terms together: . These can be written more compactly using the summation symbol ?....
, limits
Limit of a sequence

The limit of a sequence is one of the oldest concepts in mathematical analysis. It provides a rigorous definition of the idea of a sequence converging towards a point called the limit....
, and continued fraction
Continued fraction

In mathematics, a continued fraction is an expression such aswhere a0 is an integer and all the other numbers ai are positive integers....
s are not permitted.

Similarly, an equation
Equation

An equation is a mathematics Proposition, in table of mathematical symbols, that two things are exactly the same . Equations are written with an equal sign, as in...
 or system of equations is said to have a closed-form solution if, and only if, at least one solution
Equation solving

In mathematics, equation solving is the problem of finding what values fulfill a condition stated as an equality . Usually, this condition involves expressions with variables , which are to be substituted by values in order for the equality to hold....
 can be expressed as a closed-form expression.

For example, the roots of any quadratic equation
Quadratic equation

In mathematics, a quadratic equation is a polynomial equation of the second degree of a polynomial. The general form iswhere a ? 0. The letters a, b, and c are called coefficients: the quadratic coefficient a is the coefficient of x2, the linear coefficient b is the coefficient of x, and c i...
 with complex
Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
 coefficients can be expressed in closed form in terms of addition
Addition

Addition is the mathematics process of putting things together. The plus sign "+" means that numbers are added together. For example, in the picture on the right, there are 3 + 2 apples?meaning three apples and two other apples?which is the same as five apples, since 3 + 2 = 5....
, subtraction
Subtraction

Subtraction is one of the four basic arithmetic operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with....
, multiplication
Multiplication

Multiplication is the Operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic .Multiplication is defined for Natural number in terms of repeated addition; for example, 4 multiplied by 3 can be calculated by adding 3 copies of 4 together:...
, division
Division (mathematics)

In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication.Specifically, if c times b equals a, written:...
, and square root
Square root

In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x....
 extraction, all elementary functions.






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, an expression is said to be a closed-form expression if, and only if, it can be expressed analytically in terms of a bounded number of certain "well-known" function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
s. Typically, these well-known functions are defined to be elementary functions
Elementary function (differential algebra)

In mathematics, an elementary function is a function built from a finite number of exponential functions, logarithms, constants, one variable, and nth roots through function composition and combinations using the four arithmetic ....
; so infinite series
Series (mathematics)

In mathematics, given an infinite set sequence of numbers , a series is informally the result of adding all those terms together: . These can be written more compactly using the summation symbol ?....
, limits
Limit of a sequence

The limit of a sequence is one of the oldest concepts in mathematical analysis. It provides a rigorous definition of the idea of a sequence converging towards a point called the limit....
, and continued fraction
Continued fraction

In mathematics, a continued fraction is an expression such aswhere a0 is an integer and all the other numbers ai are positive integers....
s are not permitted.

Similarly, an equation
Equation

An equation is a mathematics Proposition, in table of mathematical symbols, that two things are exactly the same . Equations are written with an equal sign, as in...
 or system of equations is said to have a closed-form solution if, and only if, at least one solution
Equation solving

In mathematics, equation solving is the problem of finding what values fulfill a condition stated as an equality . Usually, this condition involves expressions with variables , which are to be substituted by values in order for the equality to hold....
 can be expressed as a closed-form expression.

For example, the roots of any quadratic equation
Quadratic equation

In mathematics, a quadratic equation is a polynomial equation of the second degree of a polynomial. The general form iswhere a ? 0. The letters a, b, and c are called coefficients: the quadratic coefficient a is the coefficient of x2, the linear coefficient b is the coefficient of x, and c i...
 with complex
Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
 coefficients can be expressed in closed form in terms of addition
Addition

Addition is the mathematics process of putting things together. The plus sign "+" means that numbers are added together. For example, in the picture on the right, there are 3 + 2 apples?meaning three apples and two other apples?which is the same as five apples, since 3 + 2 = 5....
, subtraction
Subtraction

Subtraction is one of the four basic arithmetic operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we added, we return to the number we started with....
, multiplication
Multiplication

Multiplication is the Operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic .Multiplication is defined for Natural number in terms of repeated addition; for example, 4 multiplied by 3 can be calculated by adding 3 copies of 4 together:...
, division
Division (mathematics)

In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication.Specifically, if c times b equals a, written:...
, and square root
Square root

In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x....
 extraction, all elementary functions. However, there are quintic equation
Quintic equation

In mathematics, a quintic equation is a polynomial equation of Degree of a polynomial five. It is of the form:where .......
s without closed-form solutions using elementary functions — see Galois theory
Galois theory

In mathematics, more specifically in abstract algebra, Galois theory, named after ?variste Galois, provides a connection between field theory and group theory....
.

Changing the definition of "well-known" to include additional functions can change the set of equations with closed-form solutions. Many cumulative distribution function
Cumulative distribution function

In probability theory and statistics, the cumulative distribution function or just distribution function, completely describes the probability distribution of a real-valued random variable X....
s cannot be expressed in closed form, unless one considers special functions
Special functions

Special functions are particular function s which have more or less established names and notations due to their importance in mathematical analysis, functional analysis, physics, or other applications....
 such as the error function
Error function

In mathematics, the error function is a special function which occurs in probability, statistics, materials science, and partial differential equations....
 or gamma function
Gamma function

In mathematics, the Gamma function is an extension of the factorial function to real number and complex number numbers. For a complex number z with positive real part the Gamma function is defined by...
 to be well-known. It is possible to solve the quintic equation if general hypergeometric functions are included, although the solution is far too complicated algebraically to be useful. For many practical computer applications, it is entirely reasonable to assume that the gamma function and other special functions are well-known, since numerical implementations are widely available.

See also

  • Analytic expression
  • Finitary operation
  • Numerical solution
  • Computer simulation
    Computer simulation

    A computer simulation, a computer model or a computational model is a computer program, or network of computers, that attempts to simulation an abstract model of a particular system....


Further reading

  • : provides closed form of arithmetic and linear recurrence sequences