|
|
| Line 1: |
Line 1: |
| {{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.
| | At the time a association struggle begins, you will see Some particular War Map, a good solid map of this war area area association conflicts booty place. Beneficial territories will consistently be very on the left, along with the adversary association just in the right. Every last boondocks anteroom on typically the war map represents some kind of war base.<br><br>If you are purchasing a game to your child, appear for a bed that allows several individuals to perform together. In the event you loved this post and you want to receive details concerning clash of clans cheat ([http://circuspartypanama.com Click at circuspartypanama.com]) generously visit our own web page. Gaming generally is a singular activity. Nonetheless, it's important to service your youngster to you should be societal, and multiplayer clash of clans trucos gaming can do that. They allow siblings and as a [http://wordpress.org/search/result+buddies result buddies] to all seated and laugh and play together.<br><br>There is a patch game title button that you will want to click after entering the desired values. when you check back high on the game after their late twenties seconds to a minute, you will already maintain the items. There is nothing wrong making use of special secrets. To hack is in fact the best way into enjoy clash of clans cheats. Make use of a new Resources that you have, and take advantage connected with this 2013 Clash for Clans download! Then why pay for coins or gems when you could well get the needed items with this tool! Hurry and get your trusty very own Clash of the Clans hack tool at present. The needed parts are just a few clicks away.<br><br>Games are fun, nonetheless typically also be costly. The costs of golf games and consoles can set you back more than many people would probably choose those to be, but this may easily be eliminated.<br><br>A blog is offering Clash created by Clans hack tool personal trainer to users who demand it. The website offering this tool remains safe and secure and it guarantees finest quality software. There will also other sites which provides you with the tool. But is a superb either incomplete or akin to bad quality. When users download these partially hack tools, instead to complete well they end in mid-air in trouble. So, players are advised to decide the tool from a website that offers complete computer software programs.Users who are finding it tough to cross-stitching the hurdles can choose a site that allows tourists to download the cheats. Most of the sites allow free download and some [http://www.google.co.uk/search?hl=en&gl=us&tbm=nws&q=websites+charge&gs_l=news websites charge] fees. Users can locate an internet site from where they acquire good quality software.<br><br>If you are playing a applying game, and you have no any experience with it, set the difficulty floor to rookie. This particular can help you pick all the way up on the unique makes use of of the game and learn your way around the field. Incase you set it higher than that, you are usually to get frustrated to not have any pleasure.<br><br>There is a "start" control to click on located in the wake of getting in the wanted traits. When you start shut off Clash of Clans crack hack cheats tool, wait around for a 50 % of moment, get to refresh and you should have the means families needed. There is nothing at all the wrong type of in working with thjis hack and cheats piece of equipment. Make utilization associated with the Means that you have, and exploit your 2013 Clash of Clans hack obtain! Why fork out for cashflow or gems when they can get the thoughttlikely things with this building! Sprint and have your proprietary Clash related to Clans hack software today. The required property are only a few of clicks absent. |
| | |
| 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]]
| |
At the time a association struggle begins, you will see Some particular War Map, a good solid map of this war area area association conflicts booty place. Beneficial territories will consistently be very on the left, along with the adversary association just in the right. Every last boondocks anteroom on typically the war map represents some kind of war base.
If you are purchasing a game to your child, appear for a bed that allows several individuals to perform together. In the event you loved this post and you want to receive details concerning clash of clans cheat (Click at circuspartypanama.com) generously visit our own web page. Gaming generally is a singular activity. Nonetheless, it's important to service your youngster to you should be societal, and multiplayer clash of clans trucos gaming can do that. They allow siblings and as a result buddies to all seated and laugh and play together.
There is a patch game title button that you will want to click after entering the desired values. when you check back high on the game after their late twenties seconds to a minute, you will already maintain the items. There is nothing wrong making use of special secrets. To hack is in fact the best way into enjoy clash of clans cheats. Make use of a new Resources that you have, and take advantage connected with this 2013 Clash for Clans download! Then why pay for coins or gems when you could well get the needed items with this tool! Hurry and get your trusty very own Clash of the Clans hack tool at present. The needed parts are just a few clicks away.
Games are fun, nonetheless typically also be costly. The costs of golf games and consoles can set you back more than many people would probably choose those to be, but this may easily be eliminated.
A blog is offering Clash created by Clans hack tool personal trainer to users who demand it. The website offering this tool remains safe and secure and it guarantees finest quality software. There will also other sites which provides you with the tool. But is a superb either incomplete or akin to bad quality. When users download these partially hack tools, instead to complete well they end in mid-air in trouble. So, players are advised to decide the tool from a website that offers complete computer software programs.Users who are finding it tough to cross-stitching the hurdles can choose a site that allows tourists to download the cheats. Most of the sites allow free download and some websites charge fees. Users can locate an internet site from where they acquire good quality software.
If you are playing a applying game, and you have no any experience with it, set the difficulty floor to rookie. This particular can help you pick all the way up on the unique makes use of of the game and learn your way around the field. Incase you set it higher than that, you are usually to get frustrated to not have any pleasure.
There is a "start" control to click on located in the wake of getting in the wanted traits. When you start shut off Clash of Clans crack hack cheats tool, wait around for a 50 % of moment, get to refresh and you should have the means families needed. There is nothing at all the wrong type of in working with thjis hack and cheats piece of equipment. Make utilization associated with the Means that you have, and exploit your 2013 Clash of Clans hack obtain! Why fork out for cashflow or gems when they can get the thoughttlikely things with this building! Sprint and have your proprietary Clash related to Clans hack software today. The required property are only a few of clicks absent.