Wikipedia:Reference desk/Archives/Mathematics/2023 June 27

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June 27

Characteristic of closed expressions

A property that, I think, closed-form expressions have in common – provided they are variable-free – is that it is generally feasible to evaluate them to a high precision. For example, the first 100 digits in the decimal expansion of $\pi ^{e}$ are $22.45915771836104547342715220454373502758931513399669224920300255406692604039911791231851975272714303\,.$ In contrast, it is not easy to give even the first 10 digits of $\textstyle \sum _{i=1}^{\infty }{\frac {\sin i}{i}}$ ... until you realize this is the same as $\textstyle {\arctan {\frac {\sin 1}{1-\cos 1}}},$ and then it is suddenly easy.

I'd like to add this observation to our article Closed-form expression, because I think it is illuminating, but as of now this is just my OR, and the article is already tagged with {{more citations needed}}. Is anyone aware of an RS that can be used as source?  --Lambiam 21:13, 27 June 2023 (UTC)

Another piece of evidence that I thought was particularly relevant and may aid you on your search for a source: the Kempner series is so slowly convergent that its partial sum after $10^{24}$ terms is still off by a remainder of around $1$. At the same time, I don't think the Kempner series' slow convergence is indicative of closed-form expressions being fast per se, but rather this particular method of partial summation being slow. After all, even algorithms that calculate elementary functions that form closed-form expressions can have differing levels of efficiency. Baillie's improved algorithm allowed for a much quicker evaluation of the Kempner series than just brute-force summation, and using something like CORDIC to evaluate trigonometric functions is going to be a lot quicker than, say, taking partial sums of the Taylor series. GalacticShoe (talk) 23:48, 27 June 2023 (UTC)

Obviously, some explicit expressions that are not in closed-form can be evaluated to high precision with relatively little effort, but in general, if possible at all, this may require mathematical ingenuity. But, given an arbitrary-precision package for the elementary functions, it is a routine operation for closed-form expressions.  --Lambiam 07:19, 28 June 2023 (UTC)

Oh certainly, I just figure that the only reason why closed-form expressions may in general be more readily-evaluated is because more time has been dedicated to creating fast and precise algorithms for constants and standard functions/operations. Even evaluating constants (perhaps the most elementary building block of a closed-form expression) to arbitrary precision can require a lot of machinery. The Bailey–Borwein–Plouffe formula can calculate digits of $\pi$ to arbitrary $n$-th place precision in relatively quick $n\log n$ time, but it took until 1995 to actually create the formula. The Chudnovsky algorithm, though asymptotically slower at $n(\log n)^{3}$ time, is the modern go-to for calculating digits of $\pi$. With its use of a rapidly-convergent generalized hypergeometric series, it wasn't developed until 1988.

Ultimately, it may just be a matter of semantics; evaluating "standard" functions/operations and constants up to certain levels of precision must often be done through large amounts of even more elementary standard operations. The only reasons why we regard such functions as standard for the purposes of a closed-form function are probably because of commonality of usage, and ease of evaluation. The Lambert W function cannot be expressed in terms of elementary functions, but if one uses it enough to regard it as a standard operation, then given that it can be fairly quickly computed to arbitrary precision, one can use it to make a closed-form expression nonetheless. GalacticShoe (talk) 17:53, 28 June 2023 (UTC)

This claim seems much too vague: either untrue or tautological depending on how you make it precise. It seems tautologically true that "A number specified by the combination of a manageable number of operations each of which we have researched extensively and know how to compute to high precision quickly is overall faster to compute than a number specified by the combination of an infinite number of operations or by operations which are individually very expensive, where nobody has yet investigated possible shortcuts." –jacobolus (t) 18:45, 28 June 2023 (UTC)

The claim is not that computing them to high precision quickly is overall faster than for other ground terms. Obviously, one does not need to be a mathematical genius to compute the Champernowne constant to 100 decimals in just a few minutes. A counterexample to the claim requires a case where a researcher found a closed-form solution for some problem that is difficult to evaluate. (I know that is not easy to evaluate $\sin((17!)!),$ but this is due to the excessive size of an intermediate result, and this would not naturally arise as a solution for an actual problem.)  --Lambiam 21:05, 28 June 2023 (UTC)

As I see it, "closed-form expression" is one of those bits of jargon that tend to be used without ever being isolated as abstract concepts in and of themselves, or indeed even precisely defined. That makes it a problematic topic for a mathematical encyclopedia article. These things should really be treated in a glossary somewhere rather than made the subjects of articles.

The closest precise notion with a standard definition is that of a closed term; that is, a term with no variables, but only constant symbols and function symbols. There is no reason that the interpretation of such a term needs to be easy to calculate. --Trovatore (talk) 20:35, 28 June 2023 (UTC)

What about this definition: an expression using only the natural number $1$ and a finite number of applications of the elementary arithmetic operationss and other elementary functions? Then one can bicker whether the factorial function and the floor function are allowed, but these are fringe issues. It is much more restrictive than just a ground term, which includes $\Sigma (9),$ in which $\Sigma$ stands for the busy beaver function.  --Lambiam 21:16, 28 June 2023 (UTC)

You're totally free to use that definition in any paper you write, ideally stating it explicitly so your readers are not confused. What it is not is a standard definition. That's why it's problematic to write a Wikipedia article about it. --Trovatore (talk) 21:55, 28 June 2023 (UTC)

Note that Jonathan Borwein and Richard Crandall co-wrote a 16-page article under the title "Closed Forms: What They Are and Why We Care", which appeared in the Notices of the AMS.^[1] Borwein considered this important enough to include the article as the final chapter of his book 'Experimental and Computational Mathematics: Selected Writings.^[2] The criterion for whether a topic should be covered in Wikipedia is not our personal opinion of the topic, but whether it has been the subject of significant in-depth coverage.  --Lambiam 22:52, 28 June 2023 (UTC)

At a glance that looks like a great survey. The article closed-form expression should probably be edited in light of whatever those authors discovered in their search for a definition. –jacobolus (t) 23:31, 28 June 2023 (UTC)

Actually I would go further than that. This is exactly the sort of source you need to treat "closed-form expression" as a distinct topic. It's a bit unfortunate that it's only one source, but it's a very good start.

Note that the Notices article exactly recapitulates my concern that there is no standard definition of "closed-form expression". That's a serious problem if you have an article that picks one particular definition and presents it as the definition, even if sourced. Our article at least does not do that.

But it's still a problem if editors are taking definitions from different sources and synthesizing an underlying concept. You need sources that do the synthesis. This seems to be one.

Now, as to the Notices article substantively, it's very interesting, but has a couple of peculiarities I would call out:

• Their "current preferred approach", the "sixth approach" where they isolate a particular subring of the complex numbers, seems very idiosyncratic. These "generalized hypergeometric functions" would not, I think, be on most mathematicians' intuitive list of "closed forms".
• Some of their examples talk about closed forms for functions, but their final "preferred approach" concentrates on individual complex numbers. They don't seem to make a big deal of this distinction, though I haven't read the whole article in detail and I could have missed where they mentioned it. This is a pretty important distinction; usually closed forms for functions are much more important than closed forms for individual numbers. I missed this point myself, above, when I talked about interpretations of terms having no variables — if you want a closed form for a function, of course you need variables.

Anyway, yes, it's a nice find and this is the sort of source one needs. --Trovatore (talk) 17:42, 29 June 2023 (UTC)

As an aside, I have some concerns about the ground expression article to which you linked, which I hadn't seen before. One of the refs is to MathWorld, which is unreliable for terminology, and another is to some professor's class notes, which may or may not represent that professor's personal usage. The remaining one is Hodges, which I would accept as sufficient, but I don't have it and can't easily check whether it in fact supports the text. It seems very peculiar to treat terms and formulas in a common category just because they have no variables; I don't know why you would want to do that. It seems like a candidate to merge somewhere. --Trovatore (talk) 22:05, 28 June 2023 (UTC)

For instance, the Liouville constant $L:=\sum _{k\geq 1}10^{-k!}$ ha no closed form in the sense of the linked article, though it can clearly evaluated with any precision faster than most elementary functions. On the other hand, certainly there are closed-form expressions, even in the very restrictive meaning of that article, whose direct computation is very ill-conditioned. pma 00:57, 5 July 2023 (UTC)

Above I wrote, "A counterexample to the claim requires a case where a researcher found a closed-form solution for some problem that is difficult to evaluate." I know one can construct examples that are difficult to evaluate to a high precision, but do such cases appear in mathematics as she is practiced?  --Lambiam 08:16, 5 July 2023 (UTC)