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Caveat: this question has already been asked on this site more than once, but the question has not been addressed completely.

The question I want to ask is: there are two common definitions of a "transcendental function", both of which are readily found in the literature, and both of which are inconsistent with each other; which is "correct"?

This is the first:

  • A transcendental function is an analytic function which cannot be expressed in terms of finite polynomials. That is, one which is not an algebraic function.

See for example Wikipedia, and also the Penguin Dictionary of Mathematics (2nd and 4th editions, 1998 and 2008, I have both immediately to hand).

Hence under this definition, the trigonometric, logarithmic and exponential functions are classed as transcendental, which is what you would expect.

This is the second definition:

  • A transcendental function is "a function which cannot be defined in a finite number of steps from the elementary functions, and their inverses, such as $\sin x$."

See, for example, the Collins Dictionary of Mathematics (1989).

The elementary functions seem to be conventionally defined as: polynomial functions, rational functions, exponential, logarithmic and trig functions and their composites.

So on the one hand you have "not an algebraic function", hence including log, exp and trig functions.

On the other hand you have "not an elementary function", hence not including log, exp and trig.

The question now is: which of these definitions is considered canonical nowadays? Or is it generally understood that there are two definitions, and either one is valid, nobody really cares as long as you define which you mean when you use it? Or is it even that there are two warring camps which know that their definition is the correct one and anyone using the other definition is a heretic?

In the interest of creating a "definitive" definition of "transcendental function", it would be useful to know the current school of thought on the subject: do different branches of mathematics use different definitions? Is one more for advanced (PhD+) mathematics and the other a general convenient definition for less advanced (BSc-) mathematics? Or what?

I understand there is a lot of room here for personal opinions and/or professional bias as to which is correct, that's to be expected. But is there anyone out there with an objective view on this, so the definition can be nailed down (with whatever nuances necessary)?

Prime Mover
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    I'm not a fan of the Collins dictionary: too many errors, including this one. – Angina Seng Oct 06 '20 at 08:32
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    I would say that the first definition is used by anyone doing pure mathematics (as it goes in line with definition of transcendental and algebraic numbers), the second probably by applied mathematicians. Personally, I have a PhD in pure mathematics and only know the first definition. – sampleuser Oct 06 '20 at 08:34
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    And, just as an aside, wikipedia is not reliable source as anyone can edit it and oftentimes no adequate references are provided. – sampleuser Oct 06 '20 at 08:35
  • @AnginaSeng Yet it's not the only place you see this definition. Take "Applied Bessel Functions" by F.E. Relton (1946). Its opening sentences: "The functions that one encounters ... "the elementary functions". They include the trigonometrical functions ... Functions other than these are termed "transcendental functions"." For a start. – Prime Mover Oct 06 '20 at 08:35
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    @sampleuser So might it be accurate to suggest that "non-algebraic" would be a pure mathematician's definition, and "non-elementary" an applied mathematicisn's definition? I also see from Wikipedia the definition "transcendentally transcendental", which may be what is really meant by "non-elementary" (the difference being to do with solutions to algebraic diffeqs). Does that sound correct? – Prime Mover Oct 06 '20 at 08:44
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    @PrimeMover It makes sense. "transcendental" is of course not the same as "non-elementary" and $\sin(x)$ is for me definitely a transcendental function. – Peter Oct 06 '20 at 08:49
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    There is no standards body for mathematics so you have to accept that definitions may vary from author to author. – badjohn Oct 06 '20 at 08:49
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    @PrimeMover I'm definitely not an expert on this (as I wrote, I just learned about the second definition in your question), I just guess that the latter definition might be used by applied mathematicians. Usually, you don't try to build your functions from simpler ones in pure mathematics (well, except for Taylor approximation), thus the second definition seems to naturally fit into applied mathematics. On the other hand, non-algebraic seems to be what most pure mathematicians expect when they hear the word transcendental. Hopefully, someone with deeper knowledge on this can help out ;) – sampleuser Oct 06 '20 at 08:50
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    @Joppy Your definition of transcendental number is not correct. A transcendental number is a number that cannot be obtained as root of a polynomial with integer coefficients. For example, $\sqrt 2$ is an irrational real number but it is not transcendental as it is a root of $x^2-2$. – sampleuser Oct 06 '20 at 09:53
  • @sampleuser Oh yes that was a silly mistake, let me amend my suggestion: Let $A \subseteq \mathbb{R}$ be the set of algebraic numbers (numbers which are a root of a polynomial with integer or rational coefficients). Perhaps we could define a transcendental function $f \colon \mathbb{R} \to \mathbb{R}$ as one such that $f(A) \not \subset A$? – Joppy Oct 06 '20 at 09:54
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    As a tip for researching questions such as you've asked, don't look at mathematics dictionaries or popularizations. Wikipedia is better, but Wikipedia can still sometimes be problematic, since not all entries are written by experts in the field. What you need to do is look at the actual literature in the field, and not rely on distilled versions, at least if you're trying to find some kind of definitive answer (or determine whether an answer exists), rather than wanting a generalist overview of something. – Dave L. Renfro Oct 06 '20 at 10:17
  • See Precise definition of an “algebraic function” (especially the paper I cite in a comment to GEdgar's answer). Regarding notions that go beyond elementary functions (which themselves go well beyond the algebraic functions), see my answer to Expanded concept of elementary function?. – Dave L. Renfro Oct 06 '20 at 10:36
  • @DaveL.Renfro The problem is that I have limited access to literature: a few dictionaries, and a bunch of stuff on my shelves. And, as I say, some books say one thing, and other books say other things. I don't know which are definitive. Hence my question here. – Prime Mover Oct 06 '20 at 11:23
  • If you were to use the term, and the distinction actually mattered, you would have to explicitly state what you meant by it. As you can see, there is not sufficient agreement on what it means. – JonathanZ Oct 06 '20 at 15:26
  • @Joppy "Perhaps we could define a transcendental function $f:\mathbb{R}\rightarrow\mathbb{R}$ as one such that $f(A)\not\subset A$?" No, that's not going to behave well. There are lots of functions which are terrible (e.g. not definable in any reasonable sense) but which would be non-transcendental according to that definition. In particular, any map $g:\mathbb{Q}\rightarrow\mathbb{Q}$ induces such a function: let $f(x)=g(x)$ for rational $x$ and $f(x)=x$ for irrational $x$. And there are lots of terrible maps from $\mathbb{Q}$ to itself. – Noah Schweber Oct 06 '20 at 15:56
  • @NoahSchweber: For sure, this is a hopeless classification of all functions $\mathbb{R} \to \mathbb{R}$. But in the spirit of the original post, I think the class of functions we are considering is restricted: say to analytic or $C^\infty$ functions (or perhaps functions which are analytic or $C^\infty$ piecewise, for finitely many pieces): do you think the approach I suggested could be meaningful given this restriction? – Joppy Oct 06 '20 at 16:11
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    @Joppy I suspect it will still not match up with algebraicity in any good way, but I'm not sure anymore - and regardless it now sounds interesting on its own. It might be worth asking a question here about what sort of functions fit that description. – Noah Schweber Oct 06 '20 at 16:14

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I don't know that anyone really uses the term "transcendental function" one way or another, so I don't think this matters very much, but for what it's worth the first definition seems straightforwardly correct to me and the second definition seems like it means "non-elementary function" which is a different concept. I've never heard of the Collins Dictionary of Mathematics and I used to be a graduate student.

Qiaochu Yuan
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