Are the elementary functions chosen by convention? What leads me to this question is that many non-elementary functions have power series representations, allowing them to be computed numerically, just as the trig and exponential functions. The sine function and the error function both usually require numerical methods as you can't just "plug in" a number and perform the calculation (I.e. power series or other infinite series would be required). So then what differentiates elementary functions from non-elementary functions?
There is a technical definition used in differential algebra. Elementary functions are the smallest field containing complex constants and the variable $x$ and closed under the following operations:
- (Algebraic extensions): if $P(X)$ is a polynomial of degree $> 1$ with elementary function coefficients, then there is an elementary function $g$ such that $P(g) = 0$.
- (Exponential extensions): if $f$ is an elementary function, then there is an elementary function $g$ such that $g' = f' g$.
- (Logarithmic extensions): if $f$ is an elementary function (not identically $0$), then there is an elementary function $g$ such that $g' = f'/f$.