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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.

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Bronstein's "Symbolic Integration Tutorial" may help you (see '3.1 Differential algebra' and the remaining of the tutorial...) – Raymond Manzoni Jul 18 '12 at 16:54

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