Incomplete polylogarithm

The Incomplete Polylogarithm function is related to the polylogarithm function. It is sometimes known as the incomplete Fermi-Dirac integral or the incomplete Bose-Einstein integral. It may be defined by:

$Li_{s}(b,x)={\frac {1}{\Gamma (s)}}\int _{b}^{\infty }{\frac {x^{s-1}}{e^{x}/z-1}}~dx$

Expanding about z=0 and integrating gives a series representation:

$Li_{s}(b,x)=\sum _{k=1}^{\infty }{\frac {z^{k}}{k^{s}}}~{\frac {\Gamma (s,kx)}{\Gamma (s)}}$

where Γ(s) is the gamma function and Γ(s,x) is the incomplete gamma function.

References

• GNU Scientific Library - Reference Manual http://www.gnu.org/software/gsl/manual/gsl-ref.html#SEC117