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Wikipedia defines an elementary function as

a function of a single variable that is defined as taking sums, products, and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponential functions, including possibly their inverse functions

while Wolfram's definition doesn't claim it has to be a function of a single variable.

So, could there more than one variable in an elementary function?

JJJohn
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1 Answers1

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The sources I have seen use a definition consistent with a single variable. As for the wolfram link, I think the wolfram definition is suggestive of defining the term for a single variable since it refers to inverses and repeated compositions (though this phrasing technically allows for the possibility of self maps on $\mathbb{C}^n$).

The question then becomes: Why can't we call more functions "elementary"? This question is explored here. Of course, nothing is stopping us from calling other functions, such as polynomials in several variables, "elementary." The question is: How useful is it to do so?

For instance, our defined notion of "elementary functions" enjoys application in differential algebra. These functions are closed under differentiation, but not under antidifferentiation. Liouville's theorem gives an answer to which kind of elementary functions have elementary antiderivatives, and the Risch algorithm makes this theorem practicable.

Golden_Ratio
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  • Thank you. So, the ellipse area formula doesn't count as an elementary function, right? – JJJohn Jan 23 '22 at 01:17
  • @JJJohn What formula? You mean the function $f(a,b)=\pi a b$, the area of an ellipse? – Golden_Ratio Jan 23 '22 at 01:20
  • Yes. Is the guy $f(a,b)=\pi a b$ an elementary function or not? – JJJohn Jan 23 '22 at 01:29
  • @JJJohn Not sure why that in particular; it is one of infinitely many polynomials in two variables. But again, in keeping with the single variable characterization of an elementary function, I would not call that "elementary." – Golden_Ratio Jan 23 '22 at 01:36