polylog

Polylogarithms.

SYNOPSIS:

double x, y, polylog();
int n;
y = polylog(n, x);

The polylogarithm of order n is defined by the series:

               inf   k
                -   x
   Li (x) =    >   --- .
     n          -     n
               k=1   k

For x = 1,

                inf
                 -    1
    Li (1) =    >   ---   = Riemann zeta function (n).
      n          -     n
                k=1   k

When n = 2, the function is the dilogarithm, related to Spence's integral:

                  x                      1-x
                  -                        -
                 | | -ln(1-t)            | | ln t
    Li (x) =    |    -------- dt    =    |    ------ dt    =   spence(1-x) .
      2        | |       t              | |    1 - t
                -                        -
                 0                        1
References:

   Lewin, L., Polylogarithms and Associated Functions,
   North Holland, 1981.

   Lewin, L., ed., Structural Properties of Polylogarithms,
   American Mathematical Society, 1991.

ACCURACY:
                       Relative error:
arithmetic   domain   n   # trials      peak         rms
     IEEE      0, 1     2     50000      6.2e-16     8.0e-17
     IEEE      0, 1     3    100000      2.5e-16     6.6e-17
     IEEE      0, 1     4     30000      1.7e-16     4.9e-17
     IEEE      0, 1     5     30000      5.1e-16     7.8e-17