In mathematics, 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:
:<math>
\operatorname{Li}_s(b,z) = \frac{1}{\Gamma(s)}\int_b^\infty \frac{x^{s-1{e^{x}/z-1}~dx.
</math>
Expanding about z=0 and integrating gives a series representation:
:<math>
\operatorname{Li}_s(b,z) = \sum_{k=1}^\infty \frac{z^k}{k^s}~\frac{\Gamma(s,kb)}{\Gamma(s)}
</math>
where Γ(s) is the gamma function and Γ(s,x) is the upper incomplete gamma function. Since Γ(s,0)=Γ(s), it follows that:
:<math>
\operatorname{Li}_s(0,z) =\operatorname{Li}_s(z)
</math>
where Li<sub>s</sub>(.) is the polylogarithm function.
References
- GNU Scientific Library - Reference Manual https://www.gnu.org/software/gsl/manual/gsl-ref.html#SEC117