Closed-form expression
Encyclopedia
In mathematics
, an expression
is said to be a closed-form expression if it can be expressed analytically in terms of a bounded
number of certain "well-known" function
s. Typically, these well-known functions are defined to be elementary functions
—constants, one variable x, elementary operations of arithmetic
(+ − × ÷), nth roots, exponent and logarithm (which thus also include trigonometric functions and inverse trigonometric functions).
By contrast, infinite series, integral
s, limits
, and infinite continued fraction
s are not permitted. Indeed, by the Stone–Weierstrass theorem, any continuous function on the unit interval can be expressed as a limit of polynomials, so any class of functions containing the polynomials and closed under limits will necessarily include all continuous functions.
Similarly, an equation
or system of equations is said to have a closed-form solution if, and only if, at least one solution
can be expressed as a closed-form expression. There is a subtle distinction between a "closed-form function" and a "closed-form number" in the discussion of a "closed-form solution", discussed in and below.
An area of study in mathematics is proving that no closed-form expression exists, which is referred to broadly as "Galois theory
", based on the central example of closed-form solutions to polynomials.
In physics
and engineering
the usual terminology is "analytical solution", a solution found by evaluating functions and solving equations; systems too complex for analytical solutions can often be analysed by mathematical model
ling and computer simulation
.
with complex
coefficients can be expressed in closed form in terms of addition
, subtraction
, multiplication
, division
, and square root
extraction, all elementary functions. Similarly roots of cubic and quartic (degrees three and four) can be expressed using arithmetic and roots. However, there are quintic equation
s without closed-form solutions using elementary functions, such as x5 − x + 1 = 0; see Galois theory
.
, by analogy with algebraic Galois theory.
The basic theorem of differential Galois theory is due to Joseph Liouville
in the 1830s and 1840s and hence referred to as Liouville's theorem
.
A standard example of an elementary function whose antiderivative does not have a closed-form expression is e−x2, whose antiderivative is (up to constants) the error function
.
s cannot be expressed in closed form, unless one considers special functions
such as the error function
or gamma function
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.
Whether a number is a closed-form number is related to whether a number is transcendental
. Formally, Liouville numbers and elementary numbers contain the algebraic number
s, and hence include some but not all transcendental numbers. Conversely, EL numbers do not contain all algebraic numbers, but do include some transcendental numbers. Closed-form numbers can be studied via transcendence theory
, in which a major result is the Gelfond–Schneider theorem
, and a major open question is Schanuel's conjecture
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an expression
Expression (mathematics)
In mathematics, an expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Symbols can designate numbers , variables, operations, functions, and other mathematical symbols, as well as punctuation, symbols of grouping, and other syntactic...
is said to be a closed-form expression if it can be expressed analytically in terms of a bounded
Bounded set
In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. Conversely, a set which is not bounded is called unbounded...
number of certain "well-known" function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
s. Typically, these well-known functions are defined to be elementary functions
Elementary function (differential algebra)
In mathematics, an elementary function is a function of one variable built from a finite number of exponentials, logarithms, constants, and nth roots through composition and combinations using the four elementary operations...
—constants, one variable x, elementary operations of arithmetic
Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...
(+ − × ÷), nth roots, exponent and logarithm (which thus also include trigonometric functions and inverse trigonometric functions).
By contrast, infinite series, integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
s, limits
Limit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
, and infinite continued fraction
Continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...
s are not permitted. Indeed, by the Stone–Weierstrass theorem, any continuous function on the unit interval can be expressed as a limit of polynomials, so any class of functions containing the polynomials and closed under limits will necessarily include all continuous functions.
Similarly, an equation
Equation
An equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...
or system of equations is said to have a closed-form solution if, and only if, at least one solution
Equation solving
In mathematics, to solve an equation is to find what values fulfill a condition stated in the form of an equation . These expressions contain one or more unknowns, which are free variables for which values are sought that cause the condition to be fulfilled...
can be expressed as a closed-form expression. There is a subtle distinction between a "closed-form function" and a "closed-form number" in the discussion of a "closed-form solution", discussed in and below.
An area of study in mathematics is proving that no closed-form expression exists, which is referred to broadly as "Galois theory
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
", based on the central example of closed-form solutions to polynomials.
In physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
and engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...
the usual terminology is "analytical solution", a solution found by evaluating functions and solving equations; systems too complex for analytical solutions can often be analysed by mathematical model
Mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...
ling and 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 simulate an abstract model of a particular system...
.
Roots of polynomials
For example, the roots of any quadratic equationQuadratic equation
In mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the formax^2+bx+c=0,\,...
with complex
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
coefficients can be expressed in closed form in terms of addition
Addition
Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . 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....
, subtraction
Subtraction
In arithmetic, subtraction is one of the four basic binary 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 mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic ....
, division
Division (mathematics)
right|thumb|200px|20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...
, 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. Similarly roots of cubic and quartic (degrees three and four) can be expressed using arithmetic and roots. However, there are quintic equation
Quintic equation
In mathematics, a quintic function is a function of the formg=ax^5+bx^4+cx^3+dx^2+ex+f,\,where a, b, c, d, e and f are members of a field, typically the rational numbers, the real numbers or the complex numbers, and a is nonzero...
s without closed-form solutions using elementary functions, such as x5 − x + 1 = 0; see Galois theory
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
.
Integrals
The integral of a closed-form expression may or may not itself be expressible as a closed-form expression. This study is referred to as differential Galois theoryDifferential Galois theory
In mathematics, differential Galois theory studies the Galois groups of differential equations.Whereas algebraic Galois theory studies extensions of algebraic fields, differential Galois theory studies extensions of differential fields, i.e. fields that are equipped with a derivation, D. Much of...
, by analogy with algebraic Galois theory.
The basic theorem of differential Galois theory is due to Joseph Liouville
Joseph Liouville
- Life and work :Liouville graduated from the École Polytechnique in 1827. After some years as an assistant at various institutions including the Ecole Centrale Paris, he was appointed as professor at the École Polytechnique in 1838...
in the 1830s and 1840s and hence referred to as Liouville's theorem
Liouville's theorem (differential algebra)
In mathematics, Liouville's theorem, originally formulated by Joseph Liouville in the 1830s and 1840s, places an important restriction on antiderivatives that can be expressed as elementary functions....
.
A standard example of an elementary function whose antiderivative does not have a closed-form expression is e−x2, whose antiderivative is (up to constants) the error function
Error function
In mathematics, the error function is a special function of sigmoid shape which occurs in probability, statistics and partial differential equations...
.
Alternative definitions
Changing the definition of "well-known" to include additional functions can change the set of equations with closed-form solutions. Many cumulative distribution functionCumulative distribution function
In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...
s cannot be expressed in closed form, unless one considers special functions
Special functions
Special functions are particular mathematical functions 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 of sigmoid shape which occurs in probability, statistics and partial differential equations...
or gamma function
Gamma function
In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
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.
Closed-form number
Three subfields of the complex numbers C have been suggested as encoding the notion of a "closed-form number"; in increasing order of size, these are the EL numbers, Liouville numbers, and elementary numbers. The first, denoted L for Liouville numbers (not to be confused with Liouville numbers in the sense of rational approximation), is the smallest algebraically closed subfield of C closed under exponentiation and logarithm (formally, intersection of all such subfields)—that is, numbers which involve explicit exponentiation and logarithms, but allow explicit and implicit polynomials (roots of polynomials); this is defined in . L was originally referred to as elementary numbers, but this term is now used more broadly to refer to numbers defined in explicitly or implicitly in terms of algebraic operations, exponentials, and logarithms. A narrower definition, proposed in , denoted E, and referred to as EL numbers, is the smallest subfield of C closed under exponentiation and logarithm—this need not be algebraically closed, and correspond to explicit algebraic, exponential, and logarithmic operations. "EL" stands both for "Exponential-Logarithmic" and as an abbreviation for "elementary".Whether a number is a closed-form number is related to whether a number is transcendental
Transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...
. Formally, Liouville numbers and elementary numbers contain the algebraic number
Algebraic number
In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients. Numbers such as π that are not algebraic are said to be transcendental; almost all real numbers are transcendental...
s, and hence include some but not all transcendental numbers. Conversely, EL numbers do not contain all algebraic numbers, but do include some transcendental numbers. Closed-form numbers can be studied via transcendence theory
Transcendence theory
Transcendence theory is a branch of number theory that investigates transcendental numbers, in both qualitative and quantitative ways.-Transcendence:...
, in which a major result is the Gelfond–Schneider theorem
Gelfond–Schneider theorem
In mathematics, the Gelfond–Schneider theorem establishes the transcendence of a large class of numbers. It was originally proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider...
, and a major open question is Schanuel's conjecture
Schanuel's conjecture
In mathematics, specifically transcendence theory, Schanuel's conjecture is a conjecture made by Stephen Schanuel in the 1960s concerning the transcendence degree of certain field extensions of the rational numbers.-Statement:The conjecture is as follows:...
.
Numerical computations
For purposes of numeric computations, being in closed form is not in general necessary, as many limits and integrals can be efficiently computed.See also
- Algebraic solutionAlgebraic solutionAn algebraic solution is a closed form expression that is the solution of an algebraic equation in terms of the coefficients, relying only on addition, subtraction, multiplication, division, and the extraction of roots ....
- Analytic expression
- Finitary operation
- Numerical solution
- Computer simulationComputer simulationA computer simulation, a computer model, or a computational model is a computer program, or network of computers, that attempts to simulate an abstract model of a particular system...