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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