Is there any standard nomenclature for the sets of rational, algebraic, and elementary functions? The rational functions of $X$ can be denoted $\mathbb{C}(X)$, i.e., quotients of polynomials. Is there a standard notation for the algebraic and elementary functions?
By the set of elementary functions here I mean (informally) the set of functions of X comprising all possible combinations of algebraic functions and $e^Z$ and $\log Z$ by way of the basic algebraic operations.
 A: No, there is no standard nomenclature for the algebraic functions and for the elementary functions. But there is a notation that could be used.
An overline often denotes the algebraic closure.
$\overline{\mathbb{Q}}$ denotes the algebraic numbers.
Let $K$ be a field.
$K(X_1,...X_n)$ is the set of all rational functions of $X_1,...,X_n$ over $K$.
Usually, one writes $k=K(X_1,...X_n)$, and $\overline{k}$ for the set of all algebraic functions of $X_1,...X_n$ over $K$.
If $\overline{K(X_1,...X_n)}$ denotes the algebraic closure of $K(X_1,...X_n)$, it denotes the set of all algebraic functions of $X_1,...X_n$ over $K$.
$\mathbb{C}(x)$ is the set of all rational functions of the variable $x$ over $\mathbb{C}$.
$\overline{\mathbb{C}(x)}$ could denote the set of all algebraic functions of the variable $x$ over $\mathbb{C}$.
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There is no standard nomenclature for the elementary functions.
The term "elementary function" is not defined or defined differently.
Let's take the definition of the elementary functions from differential algebra. The elementary functions are then the field $\overline{\mathbb{C}(x,\exp(x),\ln(x))}$ together with the composition.
