[rnhmjoj]

> My concern is that the word “elementary” in the title carries a much broader meaning in standard mathematical usage, and in this meaning, the paper’s title does not hold.

> Elementary functions typically include arbitrary polynomial roots, and EML terms cannot express them.

If you take a real analysis class, the elementary functions will be defined exactly as the author of the EML paper does.

I've actually just learnt that some consider roots of arbitrary polynomials being part of the elementary functions before, but I'm a physicist and only ever took some undergraduate mathematics classes. Nonetheless, calling these elementary feels a bit of stretch considering that the word literally means basic stuff, something that a beginner will learn first.

[burnished]

All I know is that when a class starts with 'elementary' or 'fundamentals of' you had best buckle up.

[quchen]

Algebraic too.

There's also the opposite in physics though, "modern" means from the 60s with square roots drawn in manually.

[TeMPOraL]

Introduction to ...

[Nevermark]

That's code for 101.

[TeMPOraL]

No. It's code for the thickest, densest book on the subject that you're ever gonna not read, as it actually assumes you're experienced in the subject and goes into everything except intro level topics.

See e.g. Petzold, et al.

[mr_mitm]

I'm getting flashbacks to Spivak, who wrote a 2000 page "introduction" to differential geometry.

[reikonomusha]

To be fair to Spivak, he did say it was comprehensive introduction. :)

[SideQuark]

> If you take a real analysis class, the elementary functions will be defined exactly as the author of the EML paper does.

I just looked through many of the best known real analysis texts, and not a single one defines them this way. This list included the texts by

Royden, Terence Tao, Rudin, Spivak, Bartle & Sherbert, Pugh, and a few others....

Can you cite a single text book that has this definition you claim is in every real analysis course? I find all evidence points to the opposite.

[rnhmjoj]

I guess you're right, I was probably mislead this whole time. I went through my old analysis class book [1] and there doesn't seem to be an explicit definition of elementary functions. The best I can find is this paragraph (I translate from italian):

> The elementary functions of analysis, that is powers, roots, exponentials, logarithms and their inverses, functions obtained from the former by arithmetic operations or composition, admit the limit f(p) for x → p, for any p in their set of definition. The study of such functions, which is not limited to the sole real functions of real variable, is carried out naturally in the setting of metric spaces.

That said, I'm relatively sure that a definition was given in class and it didn't include arbitrary roots: despite being notoriously difficult, the exam didn't require students to draw the graph of any elementary function including implicitly-defined algebraic roots.

I picked up another one of the old recommended books [2] and it seems to be similarly vague; while the book currently taught in my university [3], gives this definition:

> The following functions (from ℂ to ℂ) are called the elementary functions of the Analysis:

> 1) Rational functions (integral or fractional)

> 2) Algebraic functions (explicit or implicit)

> 3) The exponential function

> 4) The logarithm function

> 5) All those functions that can be obtained by combining a finite number of times the functions of kind 1)...4).

So, roots of arbitrary polynomials implicitly defined are indeed considered elementary. I never knew this.

[1]: https://search.worldcat.org/title/1261811544

[2]: https://search.worldcat.org/title/801297519

[3]: https://search.worldcat.org/title/935666878

[rnhmjoj]

So, I did a bit of research and I wasn't going crazy: there are apparently two competing definitions of "elementary" in use [1]:

> the class of functions [...] is what I would call exponential-logarithmic functions or EL functions; that is, they are the functions that can be expressed using some finite combination of constant functions, the identity function, exp, log, composition, and arithmetic operations (+−×÷). Some authors call this class of functions elementary functions, but that term is now more commonly used in a different sense, which includes algebraic functions.

Evidently my professor was in the exponential-logarithmic camp.

[1]: https://mathoverflow.net/a/442656

[reikonomusha]

The definition of "elementary function" typically includes functions which solve polynomials, like the Bring radical. The definition was developed and is most fitting in algebraic contexts where algebraic structure is meaningful, like Liouvillian structure theorems, algorithmic integration, and computer algebra. See e.g.

- Page 2 and the following example of https://billcookmath.com/courses/math4010-spring2016/math401... (2016)

- Ritt's Integration in Finite Terms: Liouville's Theory of Elementary Methods (1948)

It's not frequent that analysis books will define the class of elementary functions rigorously, but instead refer to examples of them informally.

[thaumasiotes]

> See e.g. Page 2 and the following example of https://billcookmath.com/courses/math4010-spring2016/math401... (2016)

There appears to be a typo in that example; I assume "Essentially elementary functions are the functions that can be built from ℂ and f(x) = x" should say something more like "the functions that can be built from ℂ and f(x) = y".

[reikonomusha]

Not a typo! Think of f(x) = x as a seed function that can be used to build other functions. It's one way to avoid talking about "variables" as a "data type" and just keep everything about functions. We can make a function like x + x*exp(log(x)) by "formally" writing

    f + f*(exp∘log)

where + and * are understood to produce new functions. Sort of Haskell-y.

[Joker_vD]

> The definition of "elementary function" typically includes functions which solve polynomials, like the Bring radical.

What. Does that "typical definition" of elementary function includes elliptic functions as well, by any chance?

[reikonomusha]

Not that I've seen.

[chii]

jargon are words being used that don't carry the typical laymen definition, but a specific one from the domain of said jargon.

If a written piece is intended for an audience who knows the jargon, then it's fine to use jargon - in fact it's appropriate and succinct. If it was intended for the laymen, then jargon is inappropriate.

But it seems you're lamenting that this jargon is wrong and that it shouldn't be jargon!?

[mcmoor]

I don't know if I read this right, but I thought it's proven that "elementary functions" can't solve 5th degree or higher polynomial, so I'm confused how it's interpreted if elementary functions also include arbitrary polynomial roots. Or is it different elementary functions?

[adrian_b]

That theorem is not formulated about "elementary functions".

It says that polynomial equations of the 5th degrees or higher cannot, in general, be solved using "radicals".

While something like "polynomials" or "radicals" has a clear meaning, which are the "elementary functions" is a matter of convention.

The usual convention is to include all algebraic functions and a few selected transcendental functions.

In "all algebraic functions", are included the rational functions, the radicals and the functions that compute solutions of arbitrary polynomial equations.

Some conventions used for "elementary functions" describe the expressions that you can use to write such "elementary functions", in which case not all algebraic functions are included, but only those written by combining rational functions with radicals.

For an algebraic function that computes a solution of a general polynomial equation, which cannot be expressed with radicals, you cannot write an explicit formula, but you can write the function only implicitly, by writing the corresponding polynomial equation.

So the difference between the 2 kinds of conventions about which are "the elementary functions" is usually based on whether only explicitly-written functions are considered, or also implicit functions.

[freehorse]

So the argument of the post is basically “this definition of elementary functions includes functions without closed form expression, and thus we cannot express these elementary functions with eml”, or sth more (that there exist elementary functions with closed form expressions that cannot be expressed by eml)?

FWIW I never thought that functions without closed form expressions were considered elementary functions, but i guess one could choose to allow this if they wanted

[eru]

The term 'elementary function' doesn't really have a single universally agreed on strict definition.

Definitions are either a bit fuzzy, or not universally agreed on.

Though interestingly https://en.wikipedia.org/wiki/Elementary_function says "More generally, in modern mathematics, elementary functions comprise the set of [...]". Though at least Wikipedia thinks that 'modern mathematics' has a consensus; of course, there's no guarantee that whoever you are talking to uses the 'modern mathematics' definition that Wikipedia brings up.

[fnordpiglet]

In math elementary usually means fundamental or foundational not elementary school. The root word is element and the relationship to “simple subject” is tangential and more related to its teaching the elemental topics for a lifetime education than definitionally cross discipline.