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Is there a theorem or method to prove that we can't express the antiderivative of some functions with elementary functions ? For example Gamma or Zeta or $\dfrac{\sin x}{x!}$? thanks

Moo
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Gustave
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Yes, Liouville's theorem.

The use of such a result in order to prove the non-existence of an elementary primitive for some functions (among them, the famous $e^{-x^2}$) is explained in these notes by Brian Conrad.

Francesco Polizzi
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