How can we prove that the Gamma function $\Gamma (x)$ is non-elementary? I know that the Liouville theorem proves whether or not an indefinite integral is non-elementary. So, we need a form of the Gamma function that can be expressed in terms of an indefinite integral. In a similar manner, how can we prove other special functions without an indefinite integral representation to be non-elementary (e.g: Riemann Zeta function, Hypergeometric functions, Bessel functions, etc), are not a combination of elementary functions?
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3there is a unique analytic continuation to the gamma function. The continuation has infinitely many poles and this is the problem. – Adelafif Sep 23 '15 at 08:43
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2This was proved by Hölder, see this answer. Concerning Liouvillian extensions see this one. – Raymond Manzoni Sep 23 '15 at 08:47
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Euler proved that:
$$\Gamma (x)=\int_0^1(-\ln t)^xdt$$
Wikipedia shows that:
$$\zeta (s)=\frac{1}{\Gamma (s)}\int_0^\infty\frac{x^{s-1}}{e^x-1}dx$$
Wikipedia also shows that:
$$J_n(x)=\frac{1}{\pi}\int_0^\pi\cos (nt-x\sin t)dt$$
And once again, Wikipedia shows a recursive integral formula for the Hypergeometric function, however, some forms of the Hypergeometric function are indeed elementary.
The Liouville theorem should be able to take it from there.
In terms of other functions without a known indefinite integral representation, I haven't but a clue at my disposal as to how to show whether or not they're elementary.
Vessel
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