Polylogarithm Function

The polylogarithm function is defined by successive integrations of the function \(x\mapsto \ln (1-x).\) It is a special case of hypergeometric function. However, it is simpler to study it directly, without using the previous chapters. Its first properties are given in Sect. 8.1, as well as the two most important cases (dilogarithm and trilogarithm functions). In Sect. 8.2, we show how to extend the polylogarithm function (and in particular the logarithm function) to complex values of the variable. Finally, we use all the results we have obtained for computing integrals and sums of series, both in the real frame (Sect. 8.3) and in the complex frame (BBP-type formulas, Sect. 8.4).