Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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
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

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.