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{{Redirect|Closed formula|"closed formula" in the sense of a logic formula with no free variables|Sentence (mathematical logic)}} | |||
{{Merge from|Analytical expression|discuss=Talk:Closed-form expression#Proposed merger from Analytical expression|date=May 2011}} | |||
In [[mathematics]], an [[expression (mathematics)|expression]] is said to be a '''closed-form expression''' if it can be expressed analytically in terms of a [[finite set|finite]] number of certain "well-known" [[function (mathematics)|function]]s. Typically, these well-known functions are defined to be [[Elementary function (differential algebra)|elementary functions]]—constants, one variable ''x'', elementary operations of [[arithmetic]] (+ − × ÷), ''n''th roots, exponent and logarithm (which thus also include trigonometric functions and inverse trigonometric functions). Often problems are said to be '''tractable''' if they can be solved in terms of a closed-form expression. | |||
Closed–form expressions are an important sub-class of [[analytic expression]]s, which contain a bounded or unbounded number of applications of well-known functions. Unlike the broader analytic expressions, the closed-form expressions do not include [[Series (mathematics)#Infinite series|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'''. | |||
An area of study in mathematics referred to broadly as [[Galois theory]] involves proving that no closed-form expression exists in certain contexts, based on the central example of closed-form solutions to polynomials. | |||
Equations or systems too complex for closed-form or analytical solutions can often be analysed by [[mathematical model]]ling and [[computer simulation]]. | |||
== Examples == | |||
=== Roots of polynomials === | |||
The solutions of any [[quadratic equation]] with [[complex number|complex]] [[coefficients]] can be expressed in closed form in terms of [[addition]], [[subtraction]], [[multiplication]], [[Division (mathematics)|division]], and [[square root]] extraction, each of which is an elementary function. For example, the quadratic equation: | |||
<div style="text-align: center;"><math>ax^2+bx+c=0,\,</math></div> | |||
is closed-form since its solutions can be expressed in terms of elementary functions: | |||
<div style="text-align: center;"><math>x={-b\pm\sqrt{b^2-4ac} \over 2a}</math></div> | |||
Similarly solutions of cubic and quartic (third and fourth degree) equations can be expressed using arithmetic, square roots, and [[cube root]]s, or alternatively using arithmetic and trigonometric functions. However, there are [[quintic equation]]s without closed-form solutions using elementary functions, such as ''x''<sup>5</sup> − ''x'' + 1 = 0; see [[Galois theory]]. | |||
=== Non-closed-form expressions with closed-form solutions === | |||
The expression: | |||
<div style="text-align: center;"><math>f(x) = \sum_{i=0}^\infty {x \over 2^i}</math></div> | |||
is not 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> | |||
=== Integrals === | |||
{{main|Differential 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 [[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: | |||
<div style="text-align: center;"><math>e^{-x^2}</math></div> | |||
whose antiderivative is (up to constants) the [[error function]]: | |||
<div style="text-align: center;"><math>\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}}\int_{0}^x e^{-t^2}\,\mathrm dt.</math></div> | |||
== 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 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 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 well-known, since numerical implementations are widely available. | |||
== Closed-form number == | |||
{{see also|Transcendence 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 [[EL number]]s, [[Liouville number (closed form expression)|Liouville numbers]], and [[elementary number]]s. The '''Liouville numbers''', denoted '''L''' (not to be confused with [[Liouville number]]s in the sense of rational approximation), 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 {{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". | |||
Whether a number is a closed-form number is related to whether a number is [[transcendental number|transcendental]]. Formally, Liouville 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 [[transcendence 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. | |||
== 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 [[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|accessdate=30 April 2012}}</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 [[Inverse Symbolic Calculator]].<ref>{{cite web|title=Inverse Symbolic Calculator|url=http://oldweb.cecm.sfu.ca/projects/ISC/|accessdate=30 April 2012}}</ref> | |||
==See also== | |||
* [[Algebraic solution]] | |||
* [[Analytic expression]] | |||
* [[Finitary operation]] | |||
* [[Numerical solution]] | |||
* [[Computer simulation]] | |||
==References== | |||
{{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 = [[American Mathematical Monthly]] |date=May 1999 }} | |||
{{refend}} | |||
== External links == | |||
* {{MathWorld | urlname = Closed-FormSolution | title = Closed-Form Solution}} | |||
{{DEFAULTSORT:Closed-Form Expression}} | |||
[[Category:Algebra]] | |||
[[Category:Special functions]] | |||
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In mathematics, an expression is said to be a closed-form expression if it can be expressed analytically in terms of a finite number of certain "well-known" functions. 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). Often problems are said to be tractable if they can be solved in terms of a closed-form expression.
Closed–form expressions are an important sub-class of analytic expressions, which contain a bounded or unbounded 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 Template:Harv and below. A closed-form or analytic solution is sometimes referred to as an explicit solution.
An area of study in mathematics referred to broadly as Galois theory involves proving that no closed-form expression exists in certain contexts, based on the central example of closed-form solutions to polynomials.
Equations or systems too complex for closed-form or analytical solutions can often be analysed by mathematical modelling and computer simulation.
Examples
Roots of polynomials
The solutions of any quadratic equation with complex coefficients can be expressed in closed form in terms of addition, subtraction, multiplication, division, and square root extraction, each of which is an elementary function. For example, the quadratic equation:
is closed-form since its solutions can be expressed in terms of elementary functions:
Similarly solutions of cubic and quartic (third and fourth degree) equations can be expressed using arithmetic, square roots, and cube roots, or alternatively using arithmetic and trigonometric functions. However, there are quintic equations without closed-form solutions using elementary functions, such as x5 − x + 1 = 0; see Galois theory.
Non-closed-form expressions with closed-form solutions
The expression:
is not 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:[1]
Integrals
Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. 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:
whose antiderivative is (up to constants) the error function:
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.
Closed-form number
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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 EL numbers, Liouville numbers, and elementary numbers. The Liouville numbers, denoted L (not to be confused with Liouville numbers in the sense of rational approximation), 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 Template:Harv. 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 Template:Harv, 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. Formally, Liouville 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 transcendence 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.
Conversion from numerical forms
There is software that attempts to find closed-form expressions for numerical values, including RIES,[2] identify in Maple,[3] Plouffe's Inverter,[4] and the Inverse Symbolic Calculator.[5]
See also
References
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In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
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External links
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