Class polylog (o2scl)

O2scl : Class List

class o2scl::polylog

Class to compute the polylogarithm function.

This class uses long double arithmetic and integral representations to compute the polylog functions.

Note

experimental

The relationship between the polylogarithm and the Fermi-Dirac distribution is:

\[ \mathrm{Li}_{1+s}(-e^{\mu}) = - \frac{1}{\Gamma(s+1)} \int_0^{\infty} \frac{k^{s}}{e^{k-\mu}+1} \]
or
\[ \mathrm{Li}_{s}(z) = - \frac{1}{\Gamma(s)} \int_0^{\infty} \frac{k^{s-1}}{e^{k-\ln(-z)}+1} \]
this representation works for negative values of \( z \).

The relationship between the polylogarithm and the Bose-Einstein distribution is:

\[ \mathrm{Li}_{1+s}(e^{\mu}) = \frac{1}{\Gamma(s+1)} \int_0^{\infty} \frac{k^{s}}{e^{k-\mu}-1} \]
or
\[ \mathrm{Li}_{s}(z) = \frac{1}{\Gamma(s)} \int_0^{\infty} \frac{k^{s-1}}{e^{k-\ln(z)}-1} \]
this representation works for positive values of \( z \).

Public Functions

polylog()
void set_tol(double tol)
double calc(double s, double y)

Polylogarithm function.

Protected Attributes

fermi_dirac_integ_tl<o2scl::inte_exp_sinh_boost<funct_ld, 15, long double>, long double> it

The integrator for negative arguments.

bose_einstein_integ_tl<o2scl::inte_exp_sinh_boost<funct_ld, 15, long double>, long double> it2

The integrator for positive arguments.