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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)?

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    $\begingroup$ I'm not a fan of the Collins dictionary: too many errors, including this one. $\endgroup$
    – Angina Seng
    Oct 6, 2020 at 8:32
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    $\begingroup$ 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. $\endgroup$
    – sampleuser
    Oct 6, 2020 at 8:34
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    $\begingroup$ And, just as an aside, wikipedia is not reliable source as anyone can edit it and oftentimes no adequate references are provided. $\endgroup$
    – sampleuser
    Oct 6, 2020 at 8:35
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    $\begingroup$ There is no standards body for mathematics so you have to accept that definitions may vary from author to author. $\endgroup$
    – badjohn
    Oct 6, 2020 at 8:49
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    $\begingroup$ @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 $x^2-2$. $\endgroup$
    – sampleuser
    Oct 6, 2020 at 9:53

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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.

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