A conjecture on integration in finite terms with elementary functions and polylogarithms

Abstract

In this abstract, we report on a conjecture that gives the form of an integral if it can be expressed using elementary functions and polylogarithms. The conjecture is proved by the author in the cases of the dilogarithm and the trilogarithm [3] and consists of a generalization of Liouville's theorem on integration in finite terms with elementary functions. Those last structure theorems, for the dilogarithm and the trilogarithm, are the first case of structure theorems where logarithms can appear with non-constant coefficients. In order to prove the conjecture for higher polylogarithms we need to find the functional identities, for the polylogarithms that we are using, that characterize all the possible algebraic relations among the considered polylogarithms of functions that are built up from the rational functions by taking the considered polylogarithms, exponentials, logarithms and algebraics. The task of finding those functional identities seems to be a difficult one and is an unsolved problem for the most part to this date.