Why exponentiation is considered elementary funtion? 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?
 A: 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.
