A Note on Symbolic Integration with Polylogarithms

Abstract

We generalize partially Liouville's theorem on integration in finite terms to allow polylogarithms of any order to occur in the integral in addition to elementary functions. The result is a partial generalization of a theorem proved by the author for the dilogarithm. It is also a partial proof of a conjecture postulated by the author in 1994 [1]. The basic conclusion is that an associated function to the nth polylogarithm appears linearly with logarithms appearing possibly in a polynomial way with non-constant coefficients. Mathematics Subject Classification (2010). 99Z99; 12H05; 33F10.

Integration in terms of polylogarithm

“…We give partial result for polylogarithm (Theorem 3.1). It seems that we give first proof of nontrivial symbolic integration result for polylogarithms of arbitrary integer order (Baddoura in [3] gives a useful result, but leaves main difficulty unresolved).…”