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Why exponentiation and power function with non-integer power are considered elementary functions while some other functions like Bernoulli polynomials generalized to non-integer order, polylogarithm, Hurwitz zeta and polygamma are not?

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Definition of elementary function –  pedja Apr 7 '12 at 10:14
    
The definition only lists which functions considered elementary. –  Anixx Apr 7 '12 at 10:23
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The definition is rather arbitrary, and dictated by history. I think only a historian of mathematics can give a better answer than this, and it will probably come down to the same answer, just in more words. –  Harald Hanche-Olsen Apr 7 '12 at 11:44
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While I agree with Harald Hanche-Olsen that the definition of an elementary function is "rather arbitrary, and dictated by history", the inclusion of $x^y$ can be justified on the basis of

Axiom 1 of elementary functions: a function that is elementary on rational numbers stays elementary after passing to reals by continuity.

For positive integers $m,n$ the operation $m^n$ comes naturally after $m+n$ and $mn$, as repeated multiplication. Solving simple algebraic equations $x^2=16$, $x^3=27$, etc naturally leads to $m^{1/n}$ as the inverse function of $m^n$. The obvious rules of exponentiation now define $x^{y}$ for rational $x,y$: for example, $(36/25)^{3/2}=216/125$. We can begin to work with these even if we still struggle with the meaning of irrationals such as $2^{1/2}$. In contrast, generalization of Bernoulli polynomials to rational orders does not happen quite as naturally. To say nothing of polylogarithm.


Axiom 1 does not justify the inclusion of trigonometric functions among elementary. For them we need Euler's identity and

Axiom 2 of elementary functions: a function that is elementary on real numbers stays elementary after passing to complex numbers by analyticity.

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