Talk:Elementary function

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I suggest moving this to elementary function and deleting the disambiguation page. "an elementary function is a function built from a finite number of exponentials, logarithms, constants, one variable, and roots of equations through composition and combinations using the four elementary operations (+ − × ÷)" is the conventional, correct definition. Nothing useful comes out of pretending that there also is a class of "elementary functions" characterized by "not being complicated". Fredrik Johansson - talk - contribs 03:07, 14 February 2006 (UTC)[reply]

I agree. It's idiotic to require people to go through another page because "elementary" means "not complicated". That link should be merged into this article, and the disambiguation page removed. --Leapfrog314 03:24, 26 March 2006 (UTC)[reply]

I agree. When I added some content to elementary function (differential algebra) there was a reason why I had to add the (differential algebra), but from the history I cannot recreate it. I think there was a page that said that elementary functions are the simple functions we learn in a calculus course. — XaosBits 15:25, 26 March 2006 (UTC)[reply]

Definition

This definition could be much more concise and elegant. I have a precise definition that was presented to me by a classical mathematician. It's better than anything I've seen published in various books or on the web. However, it would require a major rewrite. Discuss. Tparameter 22:43, 10 November 2006 (UTC)[reply]

It is difficult to discuss your proposed definition when you haven't said what it is. Fredrik Johansson 23:10, 10 November 2006 (UTC)[reply]

Tparameter, please put your definition on this talk page, so we have something to discuss. --MarSch 13:01, 14 February 2007 (UTC)[reply]

trigonometric functions

There is no finite way of constructing sine, even with the use of i if you only use addition and multiplication and their inverses, is there? --MarSch 13:04, 14 February 2007 (UTC)[reply]

No, as far as I know at least. Nor is there a finite way to construct an exponential (again, AFAIK) but you can express your sine in terms of

exponentials (sina = ( e^(ia)-e^(-ia))/(2i) ) so it is an elementary function, as it is expressible in terms of exponentials.163.1.99.179 12:37, 25 May 2007 (UTC)[reply]

Disambiguation issue

In Kleene 1952, I discovered another use of "elementary function":

"EXAMPLE 1. "Elementary" is used by Kalmár 1943 in another sense, equivalent to the following. A function is "elementary", if it can be expressed explicitly in terms of variable natural numbers, the constant 1, the functions +, *, and [a/b], and the operations Σ_y<z and Π_y<z." (Kleene 1952:285)

Kleene then goes on to state that "Kalmár (1943, 1948, 1950, 1950a) uses his elementary functions in presenting Godel's theorem and other results which are presented in this book using primitive recursive functions (p. 287). BUT... Kleen produces examples from the literature to show that "there are non-elementary primitive recursive predicates" in a manner similar to Skolem 1944's demonstration that there exist double recursions that are not primitive recursive (cf p. 273, ibid).

In Kleene's Bibliography Kleene notes that Kalmar "takes, as his basis for elementary functions, the variables, 1, +, *, |a-b|, [a/b], Σ_y=z and Π_y=z, but remarks that then * and [a/b] are redundant." (p. 526)

Observe in the above the subtle subscript changes. (The following is likely O.R. until I can find reference(s) to the effect, but interesting none the less: In the context of tiny models of computation e.g. counter machines (aka register machines), this basis of computation has more than a passing similarity to, in particular, the model of Melzak 1961 and discussions in Shepherdson-Sturgis 1961 about the work post-WWII re models of computation in Europe (Ershov 1958, Hermes 1955, Hermes 1954, Kaphegst 1959, Obershelp 1958, Péter 1958)). wvbaileyWvbailey 18:13, 9 October 2007 (UTC)[reply]

The articles discussed here are not cited in the text. can you please offer more information about where these can be found so that they can be used as sources for the article? Thanks, Cliff (talk) 20:40, 1 April 2011 (UTC)[reply]

See counter machine and etc. for references. The only good reference I have re Kalmar is Kleene 1952 as noted above and the article about Kalmar that I believe includes this meager reference. I have no idea whatever the relationship of "elementary functions" to Presburger arithmetic and other restricted forms of recursion. In general one attempts to prove that, given an instruction-set, that a Turing-equivalent machine can be created. We know that { clear counter x to 0, add 1 to counter x, compare counts in x1 and x2, halt } is Turing-equivalent, but I think that the finiteness-restrictions on the for-all and existential operators reduce Kalmar's functions to only the restricted primitive-recursive and not the unrestricted mu-recursive functions. Bill Wvbailey (talk) 23:02, 1 April 2011 (UTC)[reply]

For more modern references, there's Subrecursion: Functions and Hierarchies by H E Rose, or Odifreddi's Classical recursion theory, Volume II. Relevant wiki pages are ELEMENTARY and Elementary function arithmetic. Ben Standeven (talk) 22:06, 10 June 2011 (UTC)[reply]

Definition revisited

I have a problem with this definition. It mentions "constants", which I assume means real numbers, but then it says, "roots of equations through composition and combinations using the four elementary operations", which sounds a whole lot like algebraic numbers. And then it says, "The trigonometric functions and their inverses are assumed to be included in the elementary functions by using complex variables and the relations between the trigonometric functions and the exponential and logarithm functions." Now, wait a minute, that brings in imaginary numbers, too ( $e^{i\theta }$ and all that)!

So which, if any, of the following would not be considered elementary?

$\sin x$ — explicitly allowed by above
$\sin \pi x$ — contains non-algebraic number (inside function)
$\pi \sin x$ — contains non-algebraic constant
$i\sin x$ — contains imaginary number
$2^{x}$ — explicitly allowed
$\log _{2}x$ — explicitly allowed
$x\log _{2}3$ — contains non-algebraic constant
$e^{x}$ — non-algebraic base
$e2^{x}$ — contains non-algebraic constant

If all of these are elementary, why say anything about "roots..."? If not, where's the line drawn? - dcljr (talk) 20:28, 25 November 2008 (UTC)[reply]

As far as I can tell, constants can be any complex number - the "roots of equations" refers to functions (and I changed it to "nth roots", since I've never heard anyone claim that e.g. a root of an arbitrary quintic is an elementary function). Akriasas (talk) 12:39, 21 December 2008 (UTC)[reply]

Sources

The sourcing in this article is of very poor quality. In the introduction, for example, says that Joseph Liouville introduced the topic. There is no citation to verify this. Further, the series of papers that were published by Liouville (the definitive source I'd guess) are not included as sources. If anybody has these and other sources, please place them accordingly. Cliff (talk) 20:38, 1 April 2011 (UTC)[reply]

Strange sentence

The roots of equations are the functions implicitly defined as solving a polynomial equation with constant coefficients -- I tried hard to understand what this sentence should mean. I cannot make out its syntactic structure at all. I reduced it to: The roots are the functions defined as solving an equation. Does this make sense to anyone? Maybe I just do not understand, but to me, the sentence seems broken. I would be grateful for either an explanation or a correction. Thanks! --195.81.5.154 (talk) 09:25, 19 March 2012 (UTC)[reply]

Odd Notation F[u] for Differential Extension

In the Differential Algebra section, why is the extended field called F[u] and not something like F[exp,log]? It's being extended by adding the operations exp and log, not by adding "u", which is just an example element in that section. Stephen J. Brooks (talk) 15:18, 9 May 2013 (UTC)[reply]

Possible contradiction in opening section

The opening sections states that "It follows directly from the definition that the set of elementary functions is closed under arithmetic operations and composition", but also goes on to claim that "Elementary functions are analytic at all but a finite number of points." I don't see how these can both be true, since the function (cos(x))^-1 is a composition of two elementary functions, but is non-analytic at an infinite number of points, namely x = π(1+n), n∈ℤ. — Preceding unsigned comment added by 129.13.72.198 (talk) 11:53, 13 January 2016‎

Analytic

The article currently says "Elementary functions are analytic at all but a finite number of points."

That is obviously false, take for example $f(x)=\exp \left({\frac {-1}{\sin ^{2}x}}\right)$ .

Perhaps the correct statement is with "countable"? But I'm not sure that's correct either, there should be a citation for this.

Anyway, I went ahead and removed that line. -- Meni Rosenfeld (talk) 20:58, 10 September 2016 (UTC)[reply]

Is abs(x) an elementary function?

There seems to be controversy about whether abs(x) is an elementary function. Maybe it's good to clarify this somewhere in the article? MaigoAkisame (talk) 02:52, 21 March 2018 (UTC)[reply]

I also wonder. It never was considered elementary.--Reciprocist (talk) 20:07, 17 February 2021 (UTC)[reply]

Is factorial elementary?

The current #Basic examples section says the following:

All functions obtained by adding, subtracting, multiplying or dividing a finite number of any of the previous functions

The factorial function, $\textstyle n!=\prod _{k=1}^{n}k=\underbrace {n\cdot (n-1)\cdot (n-2)\cdots 1} _{n{\text{ times}}}$ , can be obtained by multiplying polynomial functions a "finite" number of times. However, this does not seem to be part of elementary functions. Shouldn't it be "finite and fixed number of" to be precise? Likewise, tetration would not be included in elementary functions. Also, should we include these in the examples of #Non-elementary functions? --126.236.164.157 (talk) 11:15, 14 December 2021 (UTC)[reply]

Root extraction

Currently it says "All functions obtained by root extraction of a polynomial with coefficients in elementary functions" with a ref to https://mathworld.wolfram.com/ElementaryFunction.html which includes "... root extractions ..." linked to https://mathworld.wolfram.com/RootExtraction.html which in turn says "The operation of taking an nth root of a number." That final operation is much narrower than the root of a polynomial (especially a polynomial of degree 5 or more). 2A00:23C6:148A:9B01:691F:F623:CF1E:F7D3 (talk) 10:27, 14 March 2023 (UTC)[reply]

Absolute value

Most mathematicians do not consider the absolute value function to be elementary, and the "definition" abs(x)=sqrt(x^2) is invalid, since that depends on the sqrt being positive, which is a common convention in numerical calculation and in engineering, but in math, the sqrt function is multi-valued. --Macrakis (talk) 20:01, 7 June 2024 (UTC)[reply]

When working with real-valued functions, sqrt(x) is not multi-valued. It is an elementary function.

The statement "Most mathematicians exclude ... the absolute value function" will need defending. Of the two citations at the end of that sentence, one is discussing only complex-valued functions and the other explicitly allows absolute values." Neither makes a claim for "most mathematicians." --seberle (talk) 13:12, 9 June 2024 (UTC)[reply]

Perhaps the "real elementary functions" are different from the "complex elementary functions"? --Macrakis (talk) 14:39, 10 June 2024 (UTC)[reply]

@Macrakis But in general, when we refer to "The square root of x," we mean the principal square root. Otherwise we would be talking about the square roots or using the ± square root. 2A02:3100:7D53:8D00:3867:8C97:4AFE:3A1E (talk) 09:57, 4 October 2024 (UTC)[reply]

Is it relevant that if we allow |x| as an elementary function, then the elementary functions cannot be a differential field? The polynomial f^(2)=x^2 would have more than two roots. Or for an example in direct terms of the field axioms, (x+|x|)(x-|x|)=0 shows we would have zero-divisors. At a minimum the article should make clear this may or may not be included depending on the definition and context, since it seems to be excluded in the context of differential algebra at least.136.25.107.203 (talk) 16:20, 31 October 2024 (UTC)[reply]