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


In ], a '''closed-form expression''' is a ] that can be evaluated in a ] number of operations. It may contain ], ], certain "well-known" ] (e.g., + − × ÷), and ]s (e.g., ], ], ], ], and ]), but e.g. usually no ]. The set of operations and functions admitted in a closed-form expression may vary with author and context. In ], an ] or ] is in '''closed form''' if it is formed with ], ] and a ] of basic ] connected by arithmetic operations ({{itco|{{math|+, −, ×, /}}}}, and ]) and ]. Commonly, the allowed functions are ], ], ], and ].{{efn|], ] and ] are also allowed, since they can be expressed in terms of the preceding ones.}} However, the set of basic functions depends on the context.


The ''closed-form problem'' arises when new ways are introduced for specifying ], such as ]s, ] and ]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.
Problems are said to be '''tractable''' if they can be solved in terms of a closed-form expression.


== Example: roots of polynomials == == Example: roots of polynomials ==


The ]
The solutions of any ] with ] ] can be expressed in closed form in terms of ], ], ], ], and ] extraction, each of which is an elementary function. For example, the quadratic equation:
⚫ :<math>x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.</math>
is a ''closed form'' of the solutions to the general ] <math>ax^2+bx+c=0.</math>


More generally, in the context of ]s, a closed form of a solution is a ]; 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, ] 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}}
<div style="text-align: center;"><math>ax^2+bx+c=0,\,</math></div>


There are expressions in radicals for all solutions of ]s (degree 3) and ]s (degree 4). The size of these expressions increases significantly with the degree, limiting their usefulness.
is tractable since its solutions can be expressed as closed-form expression, i.e. in terms of elementary functions:


In higher degrees, the ] 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> ] provides an ] for deciding whether a particular polynomial equation can be solved in radicals.
⚫ <div style="text-align: center;"><math>x={-b\pm\sqrt{b^2-4ac} \over 2a}</math></div>


== Symbolic integration ==
Similarly solutions of cubic and quartic (third and fourth degree) equations can be expressed using arithmetic, square roots, and ]s, or alternatively using arithmetic and trigonometric functions. However, there are ]s without closed-form solutions using elementary functions, such as ''x''<sup>5</sup>&nbsp;−&nbsp;''x''&nbsp;+&nbsp;1&nbsp;=&nbsp;0.


] consists essentially of the search of closed forms for ]s of functions that are specified by closed-form expressions. In this context, the basic functions used for defining closed forms are commonly ]s, ] and ]s. Functions that have a closed form for these basic functions are called ]s and include ], ], ], and ].
An area of study in mathematics referred to broadly as ] involves proving that no closed-form expression exists in certain contexts, based on the central example of closed-form solutions to polynomials.

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 ]s; that is, for fractions of two ]s; antiderivatives are not always rational fractions, but are always elementary functions that may involve logarithms and polynomial roots. This is usually proved with ]. The need for logarithms and polynomial roots is illustrated by the formula
:<math>\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 ] such that <math>g</math> is ] and <math>\deg f <\deg g.</math>


== Alternative definitions == == Alternative definitions ==


Changing the definition of "well-known" to include additional functions can change the set of equations with closed-form solutions. Many ]s cannot be expressed in closed form, unless one considers ] such as the ] or ] to be well known. It is possible to solve the quintic equation if general ]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 well-known, since numerical implementations are widely available. Changing the definition of "well known" to include additional functions can change the set of equations with closed-form solutions. Many ]s cannot be expressed in closed form, unless one considers ] such as the ] or ] to be well known. It is possible to solve the quintic equation if general ]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 well known since numerical implementations are widely available.


== Analytic expression == == Analytic expression ==
{{see also| Analytic function}}


An '''analytic expression''' (or '''expression in analytic form''') is a ] constructed using well-known operations that lend themselves readily to calculation. Similar to ]s, the set of well-known functions allowed can vary according to context but always includes the ] (addition, subtraction, multiplication, and division), exponentiation to a real exponent (which includes extraction of the ]), logarithms, and trigonometric functions. An '''analytic expression''' (also known as '''expression in analytic form''' or '''analytic formula''') is a ] constructed using well-known operations that lend themselves readily to calculation.{{vague|date=February 2021}}{{cn|date=February 2021}} Similar to closed-form expressions, the set of well-known functions allowed can vary according to context but always includes the ] (addition, subtraction, multiplication, and division), exponentiation to a real exponent (which includes extraction of the ]), logarithms, and trigonometric functions.


However, the class of expressions considered to be analytic expressions tends to be wider than that for closed-form expressions. In particular, ] such as the ] and the ] are usually allowed, and often so are ] and ]s. On the other hand, ] in general, and ]s in particular, are typically excluded. However, the class of expressions considered to be analytic expressions tends to be wider than that for closed-form expressions. In particular, ] such as the ] and the ] are usually allowed, and often so are ] and ]s. On the other hand, ] in general, and ]s in particular, are typically excluded.{{citation needed|reason=This paragraph seems ]. In particular, here, the distinction between series and limits is completely irrelevant.|date=June 2018}}


If an analytic expression involves only the algebraic operations (addition, subtraction, multiplication, division and exponentiation to a rational exponent) and rational constants then it is more specifically referred to as an ]. If an analytic expression involves only the algebraic operations (addition, subtraction, multiplication, division, and exponentiation to a rational exponent) and rational constants then it is more specifically referred to as an ].


==<span id="Closed-form_vs._analytical_expressions"/> Comparison of different classes of expressions == ==<span id="Closed-form_vs._analytical_expressions"></span> Comparison of different classes of expressions ==


Closed-form expressions are an important sub-class of analytic expressions, which contain a bounded{{cn|reason=This would make the sub-class non-closed e.g. wrt. addition: if e.g. up to one application of 'sin' is allowed, both 'sin(x)' and 'sin(y)' would be a member, but their sum would not. I wonder if any author can be found for this.|date=October 2014}} or unbounded number of applications of well-known functions. Unlike the broader analytic expressions, the closed-form expressions do not include ] or ]s; neither includes ]s or ]. Indeed, by the ], any ] on the ] 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. Closed-form expressions are an important sub-class of analytic expressions, which contain a finite number of applications of well-known functions. Unlike the broader analytic expressions, the closed-form expressions do not include ] or ]s; neither includes ]s or ]. Indeed, by the ], any ] on the ] 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 ] or ] is said to have a '''closed-form solution''' if, and only if, at least one ] 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 "]" in the discussion of a "closed-form solution", discussed in {{Harv|Chow|1999}} and ]. A closed-form or analytic solution is sometimes referred to as an '''explicit solution'''. Similarly, an ] or ] is said to have a '''closed-form solution''' if, and only if, at least one ] 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 "]" in the discussion of a "closed-form solution", discussed in {{Harv|Chow|1999}} and ]. A closed-form or analytic solution is sometimes referred to as an '''explicit solution'''.
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The expression: 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 ] 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>
⚫ <div style="text-align: center;"><math>f(x) = \sum_{i=0}^\infty {x \over 2^i}</math></div>
<math display="block">f(x) = 2x.</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|accessdate=31 December 2012}}</ref>

<div style="text-align: center;"><math>f(x) = 2x</math></div>


=== Differential Galois theory === === Differential Galois theory ===
{{main|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 ], by analogy with algebraic Galois theory. 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 ], by analogy with algebraic Galois theory.
Line 59: Line 68:
The basic theorem of differential Galois theory is due to ] in the 1830s and 1840s and hence referred to as ''']'''. The basic theorem of differential Galois theory is due to ] in the 1830s and 1840s and hence referred to as ''']'''.


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

<div style="text-align: center;"><math>e^{-x^2}</math></div>

whose antiderivative is (up to constants) the ]:

⚫ <div style="text-align: center;"><math>\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}}\int_{0}^x e^{-t^2}\,\mathrm dt.</math></div>


=== Mathematical modelling and computer simulation === === Mathematical modelling and computer simulation ===


Equations or systems too complex for closed-form or analytic solutions can often be analysed by ]ling and ]. Equations or systems too complex for closed-form or analytic solutions can often be analysed by ]ling and ] (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 == == Closed-form number ==
{{confusing|section|reason=as the section is written, it seems that Liouvillian numbers and elementary numbers are exactly the same|date=October 2020}}
{{see also|Transcendence theory}} {{see also|Transcendental number theory}}


Three subfields of the complex numbers '''C''' have been suggested as encoding the notion of a "closed-form number"; in increasing order of generality, these are the ]s, ], and ]s. The '''Liouville numbers''', denoted '''L''' (not to be confused with ]s in the sense of rational approximation), form the smallest '']'' 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 {{Harv|Ritt|1948|loc=p. 60}}. '''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 {{Harv|Chow|1999|loc=pp. 441–442}}, 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". Three subfields of the ]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 ]s in the sense of rational approximation), EL numbers and ]s. The '''Liouvillian numbers''', denoted {{math|'''L'''}}, form the smallest '']'' 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 ]. Formally, Liouville numbers and elementary numbers contain the ]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 ], in which a major result is the ], and a major open question is ]. Whether a number is a closed-form number is related to whether a number is ]. Formally, Liouvillian numbers and elementary numbers contain the ]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 ], in which a major result is the ], and a major open question is ].


== Numerical computations == == 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. 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 ] or the ]. Therefore, the future states of these systems must be computed numerically.


== Conversion from numerical forms == == 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 |accessdate = 30 April 2012 }}</ref> <tt>identify</tt> in ]<ref>{{cite web |title = identify |url = http://www.maplesoft.com/support/help/Maple/view.aspx?path=identify |work = Maple Online Help |publisher = Maplesoft |accessdate = 30 April 2012 }}</ref> and ],<ref>{{cite web |title = Number identification |url = http://docs.sympy.org/latest/modules/mpmath/identification.html |work = SymPy documentation }}</ref> Plouffe's Inverter,<ref>{{cite web |title = Plouffe's Inverter |url = http://pi.lacim.uqam.ca/eng/server_en.html |accessdate = 30 April 2012 }}</ref> and the ].<ref>{{cite web |title = Inverse Symbolic Calculator |url = http://oldweb.cecm.sfu.ca/projects/ISC/ |accessdate = 30 April 2012 }}</ref> 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 ]<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 ],<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 ].<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== ==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}}


==References== ==Notes==
{{Notelist}}


==References==
{{reflist}} {{reflist}}
{{refbegin}}
⚫ * {{ Citation | title = Integration in finite terms | last = Ritt | first = J. F. | authorlink = 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 = ] |date=May 1999 }}
{{refend}}


== External links == == 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 = ] |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 = ] | date = January 2013 | doi= 10.1090/noti936| doi-access = free }}


== External links ==
* {{MathWorld | urlname = Closed-FormSolution | title = Closed-Form Solution}} * {{MathWorld | urlname = Closed-FormSolution | title = Closed-Form Solution}}
*


{{DEFAULTSORT:Closed-Form Expression}} {{DEFAULTSORT:Closed-Form Expression}}
] ]
] ]
]
]

Latest revision as of 04:35, 19 August 2024

"Closed formula" redirects here. For "closed formula" in the sense of a logic formula with no free variables, see Sentence (mathematical logic).

Mathematical formula involving a given set of operations
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In mathematics, an expression or equation is in closed form if it is formed with constants, variables and a finite set of basic functions connected by arithmetic operations (+, −, ×, /, and integer powers) and function composition. Commonly, the allowed functions are nth root, exponential function, logarithm, and trigonometric functions. However, the set of basic functions depends on the context.

The closed-form problem arises when new ways are introduced for specifying mathematical objects, such as limits, series and integrals: 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

x = − b ± b 2 − 4 a c 2 a . {\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.}

is a closed form of the solutions to the general quadratic equation a x 2 + b x + c = 0. {\displaystyle ax^{2}+bx+c=0.}

More generally, in the context of polynomial equations, a closed form of a solution is a solution in radicals; that is, a closed-form expression for which the allowed functions are only nth-roots and field operations ( + , − , × , / ) . {\displaystyle (+,-,\times ,/).} In fact, 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.

There are expressions in radicals for all solutions of cubic equations (degree 3) and quartic equations (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 x 5 − x − 1 = 0. {\displaystyle x^{5}-x-1=0.} 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 antiderivatives of functions that are specified by closed-form expressions. In this context, the basic functions used for defining closed forms are commonly logarithms, exponential function and polynomial roots. Functions that have a closed form for these basic functions are called elementary functions 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 functions; that is, for fractions of two polynomial functions; 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

∫ f ( x ) g ( x ) d x = ∑ α ∈ Roots ⁡ ( g ( x ) ) f ( α ) g ′ ( α ) ln ⁡ ( x − α ) , {\displaystyle \int {\frac {f(x)}{g(x)}}\,dx=\sum _{\alpha \in \operatorname {Roots} (g(x))}{\frac {f(\alpha )}{g'(\alpha )}}\ln(x-\alpha ),}

which is valid if f {\displaystyle f} and g {\displaystyle g} are coprime polynomials such that g {\displaystyle g} is square free and deg ⁡ f < deg ⁡ g . {\displaystyle \deg f<\deg g.}

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 functions 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.

Analytic expression

See also: Analytic function

An analytic expression (also known as expression in analytic form or analytic formula) is a mathematical expression constructed using well-known operations that lend themselves readily to calculation. Similar to closed-form expressions, the set of well-known functions allowed can vary according to context but always includes the basic arithmetic operations (addition, subtraction, multiplication, and division), exponentiation to a real exponent (which includes extraction of the nth root), logarithms, and trigonometric functions.

However, the class of expressions considered to be analytic expressions tends to be wider than that for closed-form expressions. In particular, special functions such as the Bessel functions and the gamma function are usually allowed, and often so are infinite series and continued fractions. On the other hand, limits in general, and integrals in particular, are typically excluded.

If an analytic expression involves only the algebraic operations (addition, subtraction, multiplication, division, and exponentiation to a rational exponent) and rational constants then it is more specifically referred to as an algebraic expression.

Comparison of different classes of expressions

Closed-form expressions are an important sub-class of analytic expressions, which contain a finite number of applications of well-known functions. Unlike the broader analytic expressions, the closed-form expressions do not include infinite series or continued fractions; neither includes integrals or 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 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" in the discussion of a "closed-form solution", discussed in (Chow 1999) and below. A closed-form or analytic solution is sometimes referred to as an explicit solution.

Arithmetic expressions Polynomial expressions Algebraic expressions Closed-form expressions Analytic expressions Mathematical expressions
Constant Yes Yes Yes Yes Yes Yes
Elementary arithmetic operation Yes Addition, subtraction, and multiplication only Yes Yes Yes Yes
Finite sum Yes Yes Yes Yes Yes Yes
Finite product Yes Yes Yes Yes Yes Yes
Finite continued fraction Yes No Yes Yes Yes Yes
Variable No Yes Yes Yes Yes Yes
Integer exponent No Yes Yes Yes Yes Yes
Integer nth root No No Yes Yes Yes Yes
Rational exponent No No Yes Yes Yes Yes
Integer factorial No No Yes Yes Yes Yes
Irrational exponent No No No Yes Yes Yes
Exponential function No No No Yes Yes Yes
Logarithm No No No Yes Yes Yes
Trigonometric function No No No Yes Yes Yes
Inverse trigonometric function No No No Yes Yes Yes
Hyperbolic function No No No Yes Yes Yes
Inverse hyperbolic function No No No Yes Yes Yes
Root of a polynomial that is not an algebraic solution No No No No Yes Yes
Gamma function and factorial of a non-integer No No No No Yes Yes
Bessel function No No No No Yes Yes
Special function No No No No Yes Yes
Infinite sum (series) (including power series) No No No No Convergent only Yes
Infinite product No No No No Convergent only Yes
Infinite continued fraction No No No No Convergent only Yes
Limit No No No No No Yes
Derivative No No No No No Yes
Integral No No No No No Yes

Dealing with non-closed-form expressions

Transformation into closed-form expressions

The expression: f ( x ) = ∑ n = 0 ∞ x 2 n {\displaystyle f(x)=\sum _{n=0}^{\infty }{\frac {x}{2^{n}}}} 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: f ( x ) = 2 x . {\displaystyle f(x)=2x.}

Differential Galois theory

Main article: 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.

A standard example of an elementary function whose antiderivative does not have a closed-form expression is: e − x 2 , {\displaystyle e^{-x^{2}},} whose one antiderivative is (up to a multiplicative constant) the error function: erf ⁡ ( x ) = 2 π ∫ 0 x e − t 2 d t . {\displaystyle \operatorname {erf} (x)={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{2}}\,dt.}

Mathematical modelling and computer simulation

Equations or systems too complex for closed-form or analytic solutions can often be analysed by mathematical modelling and computer simulation (for an example in physics, see).

Closed-form number

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See also: Transcendental number theory

Three subfields of the complex numbers 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 numbers in the sense of rational approximation), EL numbers and elementary numbers. The Liouvillian numbers, denoted L, form 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 (Ritt 1948, p. 60). 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 (Chow 1999, pp. 441–442), 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 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. Formally, Liouvillian numbers and elementary numbers contain the algebraic numbers, 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, identify in Maple and SymPy, Plouffe's Inverter, and the Inverse Symbolic Calculator.

See also

Notes

  1. Hyperbolic functions, inverse trigonometric functions and inverse hyperbolic functions are also allowed, since they can be expressed in terms of the preceding ones.

References

  1. Holton, Glyn. "Numerical Solution, Closed-Form Solution". riskglossary.com. Archived from the original on 4 February 2012. Retrieved 31 December 2012.
  2. Barsan, Victor (2018). "Siewert solutions of transcendental equations, generalized Lambert functions and physical applications". Open Physics. 16 (1). De Gruyter: 232–242. arXiv:1703.10052. Bibcode:2018OPhy...16...34B. doi:10.1515/phys-2018-0034.
  3. Munafo, Robert. "RIES - Find Algebraic Equations, Given Their Solution". MROB. Retrieved 30 April 2012.
  4. "identify". Maple Online Help. Maplesoft. Retrieved 30 April 2012.
  5. "Number identification". SymPy documentation. Archived from the original on 2018-07-06. Retrieved 2016-12-01.
  6. "Plouffe's Inverter". Archived from the original on 19 April 2012. Retrieved 30 April 2012.
  7. "Inverse Symbolic Calculator". Archived from the original on 29 March 2012. Retrieved 30 April 2012.

Further reading

External links

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