Polylogarithm

The Polylogarithm package provides C, C++ and Fortran implementations of various polylogarithms, including the real and complex dilogarithm, trilogarithm, and (Standard and Glaisher) Clausen functions. The implementations have been fully tested against the literature and many other implementations and are highly optimized for fast numerical evaluation.

The package has no external dependencies, except for the C/C++/Fortran standard libraries. The implementations of the individual polylogarithm functions are distributed among different source code files, so individual source code files can be easily extracted and incorporated into existing projects.

Example in C++

#include "Li.hpp"
#include "Li2.hpp"
#include "Li3.hpp"
#include "Li4.hpp"
#include "Li5.hpp"
#include "Li6.hpp"
#include <iostream>

int main() {
   using namespace polylogarithm;

   const double x = 1.0;
   const std::complex<double> z(1.0, 1.0);

   // real polylogarithms for real arguments
   std::cout
      << "Li_2(" << x << ") = " << Li2(x) << '\n'
      << "Li_3(" << x << ") = " << Li3(x) << '\n'
      << "Li_4(" << x << ") = " << Li4(x) << '\n';

   // complex polylogarithms for complex arguments
   std::cout
      << "Li_2(" << z << ") = " << Li2(z) << '\n'
      << "Li_3(" << z << ") = " << Li3(z) << '\n'
      << "Li_4(" << z << ") = " << Li4(z) << '\n'
      << "Li_5(" << z << ") = " << Li5(z) << '\n'
      << "Li_6(" << z << ") = " << Li6(z) << '\n'
      << "Li_10(" << z << ") = " << Li(10,z) << '\n';
}

Output:

Li_2(1) = 1.64493
Li_3(1) = 1.20206
Li_4(1) = 1.08232
Li_2((1,1)) = (0.61685,1.46036)
Li_3((1,1)) = (0.871159,1.26708)
Li_4((1,1)) = (0.959319,1.13804)
Li_5((1,1)) = (0.987467,1.06844)
Li_6((1,1)) = (0.99615,1.03355)
Li_10((1,1)) = (0.999962,1.00199)

Notes

The implementation of the real dilogarithm is an adaptation of [arXiv:2201.01678].

The implementation of the complex dilogarithm is inspired by the implementation in SPheno.

The implementation of the real trilogarithm is an adaptation of [arXiv:2308.11619].

The implementation of the general n-th order complex polylogarithm an adaptation of [arXiv:2010.09860].

Copying

Polylogarithm is licenced under the MIT License.