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{{Redirect|Closed formula|"closed formula" in the sense of a logic formula with no free variables|Sentence (mathematical logic)}}
{{Use American English|date = January 2019}}
{{Short description|Mathematical formula involving a given set of operations}}
{{more citations needed|date=June 2014}}

In [[mathematics]], an [[mathematical expression|expression]] or [[equation]] is in '''closed form''' if it is formed with [[Constant (mathematics)|constants]], [[Variable (mathematics)|variables]], and a [[set (mathematics)|set]] of [[function (mathematics)|functions]] considered as ''basic'' and connected by arithmetic operations ({{itco|{{math|+, −, ×, /}}}}, and [[exponentiation#Integer|integer powers]]) and [[function composition]]. Commonly, the basic functions that are allowed in closed forms are [[Nth root|''n''th root]], [[exponential function]], [[logarithm]], and [[trigonometric functions]].{{efn|[[Hyperbolic functions]], [[inverse trigonometric functions]] and [[inverse hyperbolic functions]] are also allowed, since they can be expressed in terms of the preceding ones.}} However, the set of basic functions depends on the context. For example, if one adds [[polynomial root]]s to the basic functions, the functions that have a closed form are called [[elementary function]]s.

The ''closed-form problem'' arises when new ways are introduced for specifying [[mathematical objects]], such as [[limit (mathematics)|limit]]s, [[series (mathematics)|series]], and [[integral]]s: given an object specified with such tools, a natural problem is to find, if possible, a ''closed-form expression'' of this object; that is, an expression of this object in terms of previous ways of specifying it.

== Example: roots of polynomials ==

The [[quadratic formula]]

<math display="block">x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}</math>

is a ''closed form'' of the solutions to the general [[quadratic equation]] <math>ax^2+bx+c=0.</math>

More generally, in the context of [[polynomial equation]]s, a closed form of a solution is a [[solution in radicals]]; that is, a closed-form expression for which the allowed functions are only {{mvar|n}}th-roots and field operations <math>(+, -, \times ,/).</math> In fact, [[field theory (mathematics)|field theory]] allows showing that if a solution of a polynomial equation has a closed form involving exponentials, logarithms or trigonometric functions, then it has also a closed form that does not involve these functions.{{cn|date=August 2023}}

There are expressions in radicals for all solutions of [[cubic equation]]s (degree 3) and [[quartic equation]]s (degree 4). The size of these expressions increases significantly with the degree, limiting their usefulness.

In higher degrees, the [[Abel–Ruffini theorem]] states that there are equations whose solutions cannot be expressed in radicals, and, thus, have no closed forms. A simple example is the equation <math>x^5-x-1=0.</math> [[Galois theory]] provides an [[algorithmic method]] for deciding whether a particular polynomial equation can be solved in radicals.

== Symbolic integration ==

[[Symbolic integration]] consists essentially of the search of closed forms for [[antiderivative]]s of functions that are specified by closed-form expressions. In this context, the basic functions used for defining closed forms are commonly [[logarithm]]s, [[exponential function]] and [[polynomial root]]s. Functions that have a closed form for these basic functions are called [[elementary function]]s and include [[trigonometric functions]], [[inverse trigonometric functions]], [[hyperbolic functions]], and [[inverse hyperbolic functions]].

The fundamental problem of symbolic integration is thus, given an elementary function specified by a closed-form expression, to decide whether its antiderivative is an elementary function, and, if it is, to find a closed-form expression for this antiderivative.

For [[rational function]]s; that is, for fractions of two [[polynomial function]]s; antiderivatives are not always rational fractions, but are always elementary functions that may involve logarithms and polynomial roots. This is usually proved with [[partial fraction decomposition]]. The need for logarithms and polynomial roots is illustrated by the formula 

<math display="block">\int\frac{f(x)}{g(x)}\,dx=\sum_{\alpha \in \operatorname{Roots}(g(x))} \frac{f(\alpha)}{g'(\alpha)}\ln(x-\alpha),</math>

which is valid if <math>f</math> and <math>g</math> are [[coprime polynomials]] such that <math>g</math> is [[squarefree polynomial|square free]] and <math>\deg f <\deg g.</math>

== Alternative definitions ==

Changing the basic functions to include additional functions can change the set of equations with closed-form solutions.  Many [[cumulative distribution function]]s cannot be expressed in closed form, unless one considers [[special functions]] such as the [[error function]] or [[gamma function]] to be basic.  It is possible to solve the quintic equation if general [[hypergeometric function]]s 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 basic since numerical implementations are widely available.

== Analytic expression ==


This is a term that is sometimes understood as a synonym for closed-form (see {{cite web |url=https://mathworld.wolfram.com/Analytic.html |title=Wolfram Mathworld}}) but this usage is contested (see {{cite web |url=https://math.stackexchange.com/questions/1750296/what-is-the-difference-between-closed-form-expression-and-analytic-expression|title=Math Stackexchange}}).  It is unclear the extent to which this term is genuinely in use as opposed to the result of uncited earlier versions of this page.

==Comparison of different classes of expressions ==

The closed-form expressions do not include [[infinite series]] or [[continued fraction]]s; neither includes [[integral]]s or [[limit of a sequence|limits]]. 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 [[equation solving|solution]] can be expressed as a closed-form expression; and it is said to have an '''analytic solution''' if and only if at least one solution can be expressed as an analytic expression. There is a subtle distinction between a "closed-form ''function''" and a "[[#Closed-form number|closed-form ''number'']]" in the discussion of a "closed-form solution", discussed in {{Harv|Chow|1999}} and [[#Closed-form number|below]]. A closed-form or analytic solution is sometimes referred to as an '''explicit solution'''.

{{Mathematical expressions}}

== Dealing with non-closed-form expressions ==

=== Transformation into closed-form expressions ===

The expression:
<math display="block">f(x) = \sum_{n=0}^\infty \frac{x}{2^n}</math>
is not in closed form because the summation entails an infinite number of elementary operations. However, by summing a [[geometric series]] this expression can be expressed in the closed form:<ref>{{cite web | last=Holton | first=Glyn | title = Numerical Solution, Closed-Form Solution | url = http://www.riskglossary.com/link/closed_form_solution.htm |website=riskglossary.com | access-date = 31 December 2012 |url-status = dead | archive-url = https://web.archive.org/web/20120204082706/http://www.riskglossary.com/link/closed_form_solution.htm |archive-date = 4 February 2012 }}</ref>
<math display="block">f(x) = 2x.</math>

=== Differential Galois theory ===
{{main|Differential Galois theory}}
{{See also|Nonelementary integral}}

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 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 (differential algebra)|Liouville's theorem]]'''.

A standard example of an elementary function whose antiderivative does not have a closed-form expression is: <math display="block">e^{-x^2},</math> whose one antiderivative is ([[up to]] a multiplicative constant) the [[error function]]:
<math display="block">\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}} \int_{0}^x e^{-t^2} \, dt.</math>

=== Mathematical modelling and computer simulation ===

Equations or systems too complex for closed-form or analytic solutions can often be analysed by [[mathematical model]]ling and [[computer simulation]] (for an example in physics, see<ref>{{Cite journal |last=Barsan |first=Victor |date=2018 |title=Siewert solutions of transcendental equations, generalized Lambert functions and physical applications |publisher=De Gruyter |doi=10.1515/phys-2018-0034 |doi-access=free |journal=Open Physics|volume=16 |issue=1 |pages=232–242 |bibcode=2018OPhy...16...34B |arxiv=1703.10052 }}</ref>).

== Closed-form number ==
{{see also|Transcendental number theory}}
{{confusing|section|reason=as the section is written, it seems that Liouvillian numbers and elementary numbers are exactly the same|date=October 2020}}

Three subfields of the [[complex number]]s {{math|'''C'''}} have been suggested as encoding the notion of a "closed-form number"; in increasing order of generality, these are the Liouvillian numbers (not to be confused with [[Liouville number]]s in the sense of rational approximation), EL numbers and [[elementary number]]s. The '''Liouvillian numbers''', denoted {{math|'''L'''}}, form the smallest ''[[algebraically closed]]'' subfield of {{math|'''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 {{Harv|Ritt|1948|loc=p. 60}}. {{math|'''L'''}} was originally referred to as '''elementary numbers''', but this term is now used more broadly to refer to numbers defined explicitly or implicitly in terms of algebraic operations, exponentials, and logarithms. A narrower definition proposed in {{Harv|Chow|1999|loc=pp. 441–442}}, denoted {{math|'''E'''}}, and referred to as '''EL numbers''', is the smallest subfield of {{math|'''C'''}} closed under exponentiation and logarithm—this need not be algebraically closed, and corresponds 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 number|transcendental]]. Formally, Liouvillian numbers and elementary numbers contain the [[algebraic number]]s, and they include some but not all transcendental numbers. In contrast, EL numbers do not contain all algebraic numbers, but do include some transcendental numbers. Closed-form numbers can be studied via [[transcendental number theory]], in which a major result is the [[Gelfond–Schneider theorem]], and a major open question is [[Schanuel's conjecture]].

== 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. Some equations have no closed form solution, such as those that represent the [[Three-body problem]] or the [[Hodgkin–Huxley model]]. Therefore, the future states of these systems must be computed numerically.

== Conversion from numerical forms ==

There is software that attempts to find closed-form expressions for numerical values, including RIES,<ref>{{cite web |last = Munafo |first = Robert |title = RIES - Find Algebraic Equations, Given Their Solution |url = http://mrob.com/pub/ries/index.html |website=MROB |access-date = 30 April 2012 }}</ref> {{mono|identify}} in [[Maple (software)|Maple]]<ref>{{cite web |title = identify |url = http://www.maplesoft.com/support/help/Maple/view.aspx?path=identify |work = Maple Online Help |publisher = Maplesoft |access-date = 30 April 2012 }}</ref> and [[SymPy]],<ref>{{cite web |title = Number identification |url = http://docs.sympy.org/0.7.1/modules/mpmath/identification.html |work = SymPy documentation |access-date = 2016-12-01 |archive-date = 2018-07-06 |archive-url = https://web.archive.org/web/20180706114117/http://docs.sympy.org/0.7.1/modules/mpmath/identification.html |url-status = dead }}</ref> Plouffe's Inverter,<ref>{{cite web |title = Plouffe's Inverter |url = http://pi.lacim.uqam.ca/eng/server_en.html |access-date = 30 April 2012 |archive-url = https://web.archive.org/web/20120419132713/http://pi.lacim.uqam.ca/eng/server_en.html |archive-date = 19 April 2012 |url-status = dead }}</ref> and the [[Inverse Symbolic Calculator]].<ref>{{cite web |title = Inverse Symbolic Calculator |url = http://oldweb.cecm.sfu.ca/projects/ISC/ |access-date = 30 April 2012 |url-status = dead |archive-url = https://web.archive.org/web/20120329145758/http://oldweb.cecm.sfu.ca/projects/ISC/ |archive-date = 29 March 2012 }}</ref>

==See also==

* {{annotated link|Algebraic solution}}
* {{annotated link|Computer simulation}}
* {{annotated link|Elementary function}}
* {{annotated link|Finitary operation}}
* {{annotated link|Numerical solution}}
* {{annotated link|Liouvillian function}}
* {{annotated link|Symbolic regression}}
* {{annotated link|Tarski's high school algebra problem}}
* {{annotated link|Term (logic)}}
* {{annotated link|Tupper's self-referential formula}}

==Notes==
{{Notelist}}

==References==
{{reflist}}

== Further reading ==
* {{ Citation | title = Integration in finite terms | last = Ritt | first = J. F. | author-link = Joseph Ritt | year = 1948 }}
* {{Citation | title = What is a Closed-Form Number? | first = Timothy Y. | last = Chow | volume = 106 | number = 5 | pages = 440–448 | jstor = 2589148 | journal = [[American Mathematical Monthly]] |date=May 1999 | doi=10.2307/2589148| arxiv = math/9805045 }}
* {{Citation | title = Closed Forms: What They Are and Why We Care | author = Jonathan M. Borwein and Richard E. Crandall | volume = 60 | number = 1 | pages = 50–65 | journal = [[Notices of the American Mathematical Society]] | date = January 2013 | doi= 10.1090/noti936| doi-access = free }}

== External links ==
* {{MathWorld | urlname = Closed-FormSolution | title = Closed-Form Solution}}
* [https://www.nature.com/articles/s42256-022-00556-7 Closed-form continuous-time neural networks]

{{DEFAULTSORT:Closed-Form Expression}}
[[Category:Algebra]]
[[Category:Special functions]]