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Liouville's theorem gives such a proof for antiderivatives of functions like $e^x/x$ or $e^{x^2}$, and differential Galois thory extends that to Bessel functions, say. But what tools exist for implicit functions like Lambert's W?

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  • $\begingroup$ Well, $W(z)$ does satisfy a differential equation, $z W'(z) = \frac{W(z)}{1+W(z)}$. So one could still pose this as a question in differential Galois theory. Unfortunately, this equation is non-linear, which makes it rather complicated. $\endgroup$ – Igor Khavkine Jul 6 '13 at 9:44
  • $\begingroup$ I would try to show elementary functions cannot have a branch point with such properties as $W(z)$ $\endgroup$ – reuns Feb 25 '17 at 22:22
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It seems that the non-linear differential equation satisfied by the Lambert $W$ function is simple enough for this question to have already been answered. This paper proves that the Lambert $W$ is non-elementary by appealing to a result of Rosenlicht (1969):

Bronstein, M., Corless, R. M., Davenport, J. H. and Jeffrey, D. J. (2008) Algebraic properties of the Lambert W Function from a result of Rosenlicht and of Liouville. Integral Transforms and Special Functions, 19 (10). pp. 709-712. http://dx.doi.org/10.1080/10652460802332342

The argument runs roughly as follows. First note that $W(x)$ satisfies a two-variable algebraic-differential system, where the differential part has a special form:

\begin{equation*} W e^W = x \implies W'/W + W' = 1/x \implies \{W Y - x=0, \frac{Y'}{Y} = W'\} . \end{equation*}

Now, consider the differential field $\mathbb{C}(x)(Y,W)$ where $x'=1$, $Y'/Y = W'$ and $f(x,Y,W)=0$, with $f$ polynomial in $Y$ and $W$ over $\mathbb{C}(x)$. According to a more general result of Rosenlicht, the field $\mathbb{C}(x)(Y,W)$ is Liouvillian only if $Y$ and $W$ are algebraic over $\mathbb{C}(x)$. So, if $W(x)$ were elementary (elementary is a special case of Liouvillian) then it is algebraic.

It remains to check that $W(x)$ is not algebraic. The above article shows that as well. I'll just leave it at saying that it follows from the fact that it satisfies a transcendental equation.

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The branches of Lambert W are the local inverses of the Elementary function $f$ with $f(z)=ze^z$, $z \in \mathbb{C}$.

The incomprehensibly unfortunately hardly noticed theorem of Joseph Fels Ritt in Ritt, J. F.: Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925) (1) 68-90 answers which kinds of Elementary functions can have an inverse which is an Elementary function.

And Ritt's theorem shows that no antiderivatives, no differentiation and no differential fields are needed for defining the Elementary functions.

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