What kind of function is $a(x)^{b(x)}$, where $a(x), b(x) \in \mathbb R[x]$, and what field of functions does it "live" in? What if $a(x), b(x) \in \mathbb R(x)$?
- Certainly in neither case do we have a rational function, i.e. $a(x)^{b(x)}$ is not an element of $\mathbb R(x)$.
- It is also (I think?) not an algebraic function, i.e. it does not live in the algebraic closure of $\mathbb R(x)$.
Do such things have a name? And is there a name for the field of functions "like" this (whatever that means)?
(I realize that last question is fairly vague, as I'm not sure what exactly I mean by "like" this. I guess I'm looking for a name for a class of functions and partial functions that continues the sequence $\textbf{Polynomial}, \textbf{Rational}, \textbf{Algebraic}, \dots...$)