A generalization of Liouville's theorem on integration in finite terms (Q2785252)

The paper is devoted to the theory of integration in finite terms. In it the authors enter a new class (to be exact two more of elementary integrals) of elementary extensions of differential fields, so-called Ei-Gamma extensions. Let \(F\) be a differential field of characteristic zero with an algebraically closed field of constants. As the main result of the paper it is proved, that any element from some Ei-Gamma extension of \(F\) that is primitive over \(F\) has a very specific representation of its derivative in \(\overline{F}\). Note that the use of the word Gamma in the name of such extensions is not absolutely successful. The truth of the matter is that extensions of Liouville type turn up due to the connection with solutions of elementary differential equations, and via this way to receive the classical Gamma function is impossible. Therefore it is clear that it is more natural to keep an adjective Gamma behind the extensions obtained due to the connection with solutions of elementary functional equations.