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Stemming from a comment thread in another question I got curious about why exponential and trig functions are considered elementary but there are so very many other non-algebraic functions which are not. Are there any particular motivations or is it something that becomes obvious when one has studied enough analysis? Is it the exponential functions relation to being eigenfunction to differentiation that is central to this choice or something else?

mathreadler
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    One doesn't necessarily have to assume trig functions are elementary, rather, it follows that they are due to Euler's formula. – Simply Beautiful Art Jul 13 '17 at 22:26
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    @DonaldSplutterwit: thank you. – mathreadler Jul 13 '17 at 22:26
  • @SimplyBeautifulArt: yes so then the question is why the choice of particularly the exponential function. – mathreadler Jul 13 '17 at 22:27
  • I'd imagine exponential functions are usually taken to be elementary because they have relatively simple properties, are well known and used, and appear in many areas of mathematics. – Simply Beautiful Art Jul 13 '17 at 22:27
  • "What we choose to call an elementary function has very little significance. The only place it really matters is for mathematics education. Lambert W and friends don't really have that many applications below the research level to warrant pushing it into the curriculum early on. It just takes away time from other useful stuff we would like to teach our students. (cont.) – Simply Beautiful Art Jul 13 '17 at 22:31
  • (cont.) The only function I can think of that might come close to deserving being called elementary (just on the merit of it being so useful is so many areas) would be the error function. As a math conservative I hope it stays the way it is:)" - Winther – Simply Beautiful Art Jul 13 '17 at 22:31
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    this answer explains how we choose elementary functions well – Dando18 Jul 13 '17 at 22:32
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    My opinion is that $\Gamma(z), B(a,b)$ and $J_n(z)$ should be considered elementary functions too, but I am not a math conservative at all :D I am not sure about polylogarithms. – Jack D'Aurizio Jul 13 '17 at 22:33
  • So what would you make of $\sum_{n=1}^{\infty} \frac{H_n}{(n+1)^2} = \zeta(3)$ ? – Donald Splutterwit Jul 13 '17 at 22:33
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    @DonaldSplutterwit Not a function. – Simply Beautiful Art Jul 13 '17 at 22:33
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    @JackD'Aurizio For you, you might consider Liouvillian functions and solutions to algebraic differential equations (the former including the error function and sine integrals, the latter including Bessel functions and hypergeometric functions) to be elementary. Indeed, it much depends on who you are talking to I would suppose. – Simply Beautiful Art Jul 13 '17 at 22:36
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    The $\zeta$ function is probably the queen of functions, but I find it difficult to call her an elementary function, since we know so few about its behaviour in the critical strip... About solutions of "simple" differential equations, I share the viewpoint @SimplyBeautifulArt has just proposed. – Jack D'Aurizio Jul 13 '17 at 22:36
  • Say, when did the Wikipedia get a section concerning the fractional derivatives of the Riemann zeta function? (a bit off-topic) – Simply Beautiful Art Jul 13 '17 at 22:40
  • On the other hand, I suppose one could define "elementary function" as a finite combination of addition, multiplication, or exponentiation (basic arithmetic operations) and their inverses, which directly explains why we'd want exponential functions to be in the circle. – Simply Beautiful Art Jul 13 '17 at 22:43
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    from what I know it is just a convention. An idea of mine about this concept is consider any function that can be "approximated well" in all it domain as elementary. But we will need to define formally "approximated well". – Masacroso Jul 13 '17 at 23:09
  • @Masacroso That's very loose. Approximated in terms of what? Certainly not elementary functions, but then what qualifies? Probably way too broad. – Simply Beautiful Art Jul 15 '17 at 13:34
  • @Simply by example linear functions... Idk, it is just a non formalized idea. – Masacroso Jul 15 '17 at 14:18
  • I have the feeling that such a definition along those lines would be too broad, as every analytic function can be linearly approximated, but I'm certain we can agree not all analytic functions are elementary. – Simply Beautiful Art Jul 15 '17 at 14:19

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To qualify as being elementary, a function must be (1) well known to mathematicians for centuries, (2) taught in mathematics courses in most secondary schools, and (3) indispensable in a wide range of disciplines that use mathematics. The elementary functions pretty much coincide with the functions that feature on the more basic and popular models of "scientific" calculator.

The above "definition" might seem rather arbitrary, and unattractive to those who would prefer a more conceptual definition. However, the concept of elementary, as applied to functions, is not a mathematical one, and lies within the realm of common (albeit specialist) language, which is always subject to evolution and varying opinion.

John Bentin
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