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?
2026-04-05 11:03:37.1775387017
Proving that the Gamma function is a special function
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