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:
\operatorname{Li}s(b,z)=
| 1 | |
| \Gamma(s) |
| infty | |
| \int | |
| b |
| xs-1 | |
| ex/z-1 |
~dx.
Expanding about z=0 and integrating gives a series representation:
\operatorname{Li}s(b,z)=
| infty | |
| \sum | |
| k=1 |
| zk | ~ | |
| ks |
| \Gamma(s,kb) | |
| \Gamma(s) |
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:
\operatorname{Li}s(0,z)=\operatorname{Li}s(z)
where Lis(.) is the polylogarithm function.