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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