Talk:Closed-form expression

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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 ${\displaystyle \textstyle \sum _{i=1}^{n}i}$ of a triangular number to be a closed-form expression for it, while ${\displaystyle \textstyle {\frac {n^{2}+n}{2}}}$ 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)Reply

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 ${\displaystyle \textstyle \sum _{i\in \mathbb {N} }{\frac {x^{i}}{i!}}}$ 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)Reply

I tend to agree with your suggestion that the boundedness of a closed-form formula be required. This would exclude ${\displaystyle n!}$. However, it could also exclude ${\displaystyle k^{n}}$ as it is usually defined as ${\displaystyle \underbrace {k\times ...\times k} _{n}}$ so that its length does depend on n, just like the length of ${\displaystyle n!}$ = ${\displaystyle \underbrace {1\times ...\times n} _{n}}$. If, however, one considers products ${\displaystyle n!}$ and ${\displaystyle k^{n}}$ of de facto unbounded lengths closed forms, what would be the reason for excluding sums of unbounded lengths? For instance, in such a case, ${\displaystyle \prod _{i=0}^{n}{\frac {1}{2^{2^{i}}}}}$ will be considered a closed form so why ${\displaystyle \sum _{i=0}^{n}{\frac {1}{2^{2^{i}}}}}$ should not? After all, ${\displaystyle k\times n}$ = ${\displaystyle \underbrace {k+...+k} _{n}}$ so that products seem much less closed forms than sums are.172.88.206.28 (talk) 15:29, 22 September 2016 (UTC)Reply

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.). 172.88.206.28 (talk) 01:23, 21 September 2016 (UTC) — Preceding unsigned comment added by 172.88.206.28 (talk) 01:29, 20 September 2016 (UTC)Reply

Is this grammar of the Wikipedia article correct? "to numbers defined in explicitly or implicitly in terms of algebraic operations"— Preceding unsigned comment added by Wilkibur (talk • contribs) 18:35, 28 March 2015 (UTC)Reply

The phrase "analytic function" is a technical term with a precise meaning. The term "analytic" is defined twice in this article; one definition is possibly consistent with that meaning, but the other is not. It is tempting to assume that an analytic expression is the value of an analytic function. The article should mention that there are such things as analytic functions, and how they relate to the expressions.

2001:388:6080:109:D578:2477:98CA:8194 (talk) 04:34, 20 March 2017 (UTC)Reply

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@Jochen Burghardt: Hi. Why did you write in a comment that sin(x)+sin(y) is not an analytic expression?

Maybe the sentence should say, "Closed-form expressions are an important sub-class of analytic expressions, which contain a finite or infinite number of applications of well-known functions".

Please ping me when you answer. Eric Kvaalen (talk) 16:51, 21 February 2021 (UTC)Reply

To editor Eric Kvaalen: Both formulations are nonsensical as none of the involved terms are clearly defined. In particular, the difference between an analytic expression of a function and an analytic function is never defined. Difficult to clarify sech a mess. D.Lazard (talk) 21:34, 21 February 2021 (UTC)Reply

The article mentions that differentiation usually isn't allowed to take place in closed-form expressions. However, isn't it always possible to expressed the derivative of a closed-form expression as another close-form expression? If so, should perhaps differentiation be allowed in closed-form expressions? Or maybe it doesn't matter whether an expression is equivalent or not to another expression that is on a closed form? —Kri (talk) 15:43, 16 March 2023 (UTC)Reply

Good catch. I have edited the article for fixing it. Feel free to improve my wording if needed. D.Lazard (talk) 17:32, 16 March 2023 (UTC)Reply

The polynomial ${\displaystyle f(x)=2x^{5}-5x^{4}+5}$ is not solvable by radicals over Q. Hence, none of its five individual roots can be expressed exactly, in a finite number of symbols, by a formula using only integers, radicals, and the four basic arithmetic operations. However, the roots are algebraic numbers, since they are roots of a nonzero polynomial in one variable with integer coefficients. Are these roots considered expressible in closed form just because they are roots of the polynomial, ${\displaystyle f(x)}$, which can be expressed in closed form?—Anita5192 (talk) 17:02, 31 July 2023 (UTC)Reply

The concept of "closed form" is rarely accurately defined. In particular, the description of a function through an algorithm in pseudocode may fit the definition in the first sentence of the article, but it is rarely considered as a closed form. For example, the definition of the nth Fibonacci number as $${\displaystyle F(n)=={\text{ if }}n=0{\text{ or }}n=1{\text{ then }}1{\text{ else }}F(n-2)+F(n-1).}$$

In the context of polynomial equations, "closed-form solution" means generally solution in radicals. So, your example has no closed-form solution. In other contexts, the roots of this polynomial may occur in a closed-form expression. For example, $${\displaystyle \int {\frac {dx}{f(x)}}=\sum _{\alpha \in \operatorname {roots} (f)}{\frac {1}{f'(\alpha )}}\ln(x-\alpha )}$$ is a closed form of the antiderivative of 1/f, which is valid for every squarefree polynomial.

IMO, "closed form" is a concept that can be explained, but it is too vague for being formally defined. This should be clarified in the lead, but I do not know how to do that in an encyclopedic style. D.Lazard (talk) 10:48, 1 August 2023 (UTC)Reply

Thank you! This explains a lot. —Anita5192 (talk) 13:08, 1 August 2023 (UTC)Reply

Two editors are trying to re-introduce in the article talking about "size" of an expression. One of them linking to a Quora post, that they themselves wrote, that doesn't even support the use of that terminology. The other, based on nothing. Size, is not a meaningful measure of anything. It could mean anything, from number of operations, operands, font, number of lines, etc. The article should instead mention at least one actual concrete reason that makes the expressions in radicals for quartics, cubic (and sometimes even quadratics) less useful as the degree increases. An ambiguous notion of size is irrelevant to usefulness. It is not uncommon to have expressions, procedures, algorithms, that are more sizable than others, longer to write, but are more useful for performing better than others that are less sizable, simpler to describe. The article should mention objective notions that make the expressions less useful: Number or operations, numerical instability, complexity of the symmetries (Galois group), complexity when testing the identity problem, complexity of rationalization of radicals, any, some, or all, but not a bogus reference to "size", which does not imply any lack of usefulness. Thatwhichislearnt (talk) 11:23, 27 March 2024 (UTC)Reply

Mathematics does not exclude common sense; here, "size" has its common meaning. It suffices to look at the solution of a general quartic equation to understand that it is too large to be used, and even to be read by humans. If you want for such a huge formula, ask your preferred computer algebra system. If you would try to verify the solution by putting it in the equation, I guess that you would never succeed to simplify the result to zero. D.Lazard (talk) 12:08, 27 March 2024 (UTC)Reply

Utter nonsense, and you know it. Size and usefulness are unrelated. Also, using intuition as a way of judging usefulness is also nonsense. Multiple examples, including your favorite Sturm vs Vincent's theorem, show how the size of a description is unrelated to usefulness. There are concrete mathematical reasons that make these formulas less useful as the degree increases. Those should be given as reasons, not size. Thatwhichislearnt (talk) 12:17, 27 March 2024 (UTC)Reply

It is impressive how you are always behind efforts to make Mathematical articles worse, less precise, less mathematical. What's up with that? Is it a quest of yours? 12:19, 27 March 2024 (UTC) Thatwhichislearnt (talk) 12:19, 27 March 2024 (UTC)Reply

@Thatwhichislearnt:Be careful not to make personal attacks against other editors like D.Lazard. If you disagree with what an editor claims, then attack the claim or the material—not the editor. If you continue, you may be blocked. And do not edit war. Instead, continue discussing the issue here on the talk page until consensus is reached.—Anita5192 (talk) 13:31, 27 March 2024 (UTC)Reply

Come on!! There is no definition of 'Well known' (in this article). The "Alternative Definitions" section is the only place (according to my browser's edit|search function) it even occurs. There it first occurs as "Changing the definition of "well known" to..." is absurd. The term must first be explicitly or by reference defined before that definition is changed. Fix this absurdity, please. I suggest removing the entire (useless) section - it appears to have been left in place as a compromise to some editor's stubbornness. Also, the lead is confused. And confusing. The exponential function is "basic"??? Since it's generally defined as a derivative (the function f(x) such that d(f(x))/dx = f(x)) and since the derivative is not "basic", how could anyone think it's coherent to claim it's basic if derivatives are not? It is notable that the lead contains no references. Closed form expressions are those which are expressed in terms of previously expressed forms?? LMFAO! OK, sure. But if that is true, then the term is semantic, not mathematical. (And it means exactly what the author(s) want it to mean, neither more nor less, LOL!) ...So, is it or is it not true that a closed form expression is one which, in the specific context it's being used in, is well defined (as opposed to "previous ways of specifying it")? (It's hard to believe any editor thinks "previous ways" is reasonable here. This article stinks. At the very least the lack of clear definition should be mentioned. 98.19.179.27 (talk) 15:10, 23 April 2025 (UTC)Reply

Good point. Apparently an editor has improved the lead without checking wether the body of the article needed to be upgraded. I edited the article to get rid of "well known", and to clarify that the functions that are considered as basic depend on the class of closed form that has considered. D.Lazard (talk) 16:36, 23 April 2025 (UTC)Reply