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In Fluid dynamics, the closure form of the governing equations is called for the boundary layer approximation or the parabolic approximation.
I'd thought that closed-form solutions were possible for quintic equations, although there is no general formula for them? Could someone more knowledgeable edit this article?
- You can of course find closed form solutions to quintics such as (the five roots of one). There is even a general process to solve quintics by introducing a new radical (The Bring radical). --njh 12:12, 12 July 2006 (UTC)
This article title is not so good
Why the word "solution"?? Some cases of closed-form expressions in mathematics are in some sense "solutions", and others are not. This seems to me like one of those cases where someone picks a word like "equation" or "solution" as a sort of catch-all term to be used when they don't know the right nomenclature. This happens frequently in math, when, for example, lay persons promiscuously use the word "equation" to denote anything at all that is written in mathematical notation.
- I agree with your basic point. I have a general problem with the notion of "closed form", as I think it's a bit difficult to define "closed form" in the abstract, or to differentiate "expressed in closed form" from "expressed analytically". For example, one can often solve (in)equalities by using inverse functions, but whether or not the resulting expression would be said to be "in closed form" depends on what the basic inventory of functions and expressions is. I'm not sure if e.g. Lambert's W function would be considered part of that inventory, but it's obviously very convenient for expressing solutions of certain equations in a form that can easily be evaluated by numerical software. I have the feeling that large aspects of this notion of "closed form" are very much a remnant of a past time when only a small number of expressions could be conveniently evaluated by hand or looked up in a table. --MarkSweep (call me collect) 01:27, 26 November 2005 (UTC)
I am inclined to suspect that the concept does admit some precise definition, but I am skeptical of the claims even of some fairly sophisticated mathematicians to have done that definitively. But the fact that it is not yet fully precise doesn't mean there should be no article on it. Michael Hardy 22:33, 26 November 2005 (UTC)
- So, let's move it, shall we? Regarding the content, here are some external links, which may or may not shed further light on the meaning of "closed form":
- Also note that these articles all talk about solutions (though I agree with you now that that's too limited). --MarkSweep (call me collect) 10:58, 29 November 2005 (UTC)
- Are there any expressions that are not closed-form expressions? Can someone give examples?
- Are there indeed contexts in which people call certain expressions "closed-form expressions" in which these expressions are not the solutions of equations discussed in that context? Can someone give examples?
- The Google search term [closed-form-solution -wikipedia] gets almost twice the number of hits of [closed-form-expression -wikipedia]. In general the recommendation is to use the most common form for the article's title.
- --Lambiam 14:31, 26 December 2007 (UTC)
Opposite of closed form
Should the opposite of a closed form expression be defined on (or linked from) this page? It would seem to be a useful addition to the article. Alchemeleon 21:12, 7 June 2007 (UTC)
The Fibonacci numbers as an example
Perhaps the Fibonacci numbers provide a good example of a closed-form solution in contrast to a definition that does not use a closed-form expression. By the way, I don't think there is a single term with the opposite meaning. —22.214.171.124 (talk) 04:36, 15 February 2010 (UTC)
Bounded or finite?
The current lede says a closed form expression may contain a bounded number of operations. To me that is not clear, and I think it should be "finite" instead. Obviously infinite summations are not closed forms. But a summation whose number of terms varies with the argument of the expression would be bounded (for each argument), but not involve a finite number of operations because there is no bound in the expression itself. For instance I would not consider the definition of a triangular number to be a closed-form expression for it, while does give a closed-form expression. By the same token there would be no closed-form expression for factorials at all, unless we explicitly place them in our repertoire of "well-known"" functions. If my interpretation is agreed upon, I think "finite" would be the correct term to use. Marc van Leeuwen (talk) 10:33, 9 March 2010 (UTC)
Re-reading the intro, I think an even more radical change is in order: "[an expression] can be expressed analytically in terms of a bounded number of certain well-known functions" makes no sense: an expression cannot be expressed, it is already expressed. It would be silly to call an expression like a closed form just because it happens to be equivalent to (i.e., can be expressed as) a different one (guess) that is. Also an expression is always finite, although this might need stressing (in view of practices such as continued fractions and infinite summations that are written using ellipses; in fact these are improperly written expressions, corresponding in a well understood way to limit expressions over hopefully equally well understood (in spite of the ellipses) sequences). So an expression is a closed form if it only involves certain well-known operations and functions, where expressly are excluded summations (as opposed to additions which are allowed), products (again with a variable or infinite number of factors; multiplications are OK), limits, case distinctions, and maybe some more I've forgotten here. The point is one may add basic functions to the repertoire (provided this is clearly stated), but the operations excluded are always forbidden. If anybody disagrees, please explain here; otherwise I will make this change some day. Marc van Leeuwen (talk) 09:24, 16 March 2010 (UTC)
I tend to agree with your suggestion that the boundedness of a closed-form formula be required. This would exclude . However, it could also exclude as it is usually defined as so that its length does depend on n, just like the length of = . If, however, one considers products and of de facto unbounded lengths closed forms, what would be the reason for excluding sums of unbounded lengths? For instance, in such a case, will be considered a closed form so why should not? After all, = so that products seem much less closed forms than sums are.126.96.36.199 (talk) 15:29, 22 September 2016 (UTC)
Also, whether an expression is closed-form should be effectively verifiable (as a minimum, provable/disprovable) by means of finite number of obvious steps (just like whether a sequence of formulas is a proof is supposed to be), so a phrase "... can be expressed ..." seems inappropriate as "can" may turn out true but unprovable (or false but undisprovable) in any accepted system (PA, ZFC, etc.). 188.8.131.52 (talk) 01:23, 21 September 2016 (UTC) — Preceding unsigned comment added by 184.108.40.206 (talk) 01:29, 20 September 2016 (UTC)
The first paragraph of that section needs to be clarified:
- "...in increasing order of size, these are the EL numbers, Liouville numbers, and elementary numbers."
- "The first, denoted L for Liouville numbers..."
I might be able to work out the correct interpretation on my own, were it not for the fact that three sets are mentioned, and I see four labels in the paragraph: E, L, EL and C. —Preceding unsigned comment added by 220.127.116.11 (talk) 02:52, 16 March 2010 (UTC)
Proposed merger from Analytical expression
@CRGreathouse: Please try to be constructive here. I found the relevance of the sentence to what precedes it in the paragraph to be unclear, so I took a shot at stating what the point was. You reverted, saying the addition was invalid. Then I put in a clarification-needed tag, which you reverted, saying the point was already pretty clear and giving an informally worded version of the point in the edit box but not in the relevant place in the article. Then I tried to give a more carefully worded version of your edit-box explanation in the article, saying in my edit summary "If you don't like this version, put in something better." You responded by reverting without replacing, with the edit summary "I don't mind a clarification if it's correct, but this one is not". This is not constructive in the absence of a replacement clarification.
I'm trying to get this passage in the lede improved to a point where people who know enough math to read the article can understand the point of the sentence. If the point is not what I put in in either of my two edits, then its point is not clear to me, and if it's not clear to me then it's not clear to others as well.
The first paragraph has the following:
" Unlike the broader analytic expressions, the closed-form expressions do not include infinite series or continued fractions; neither includes integrals or limits."
"Neither" refers to infinite series and continued fractions, because 'neither' means something along the lines of "not the one nor the other of two people or things". But I am guessing here that the sentence was meant to say that closed-form expressions do also not include integrals or limits? (I don't know, I'm not a mathematician). If, OTOH, it is meant to say that neither infinite series or continued fractions include integrals or limits, then I wonder why this information is given, as it feels unrelated to the definition of closed-form expressions? maye (talk) 16:49, 26 September 2013 (UTC)
Is this grammar of the Wikipedia article correct?
Is this grammar of the Wikipedia article correct? "to numbers defined in explicitly or implicitly in terms of algebraic operations"