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:

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

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

${\displaystyle \operatorname {Li} _{s}(b,z)=\sum _{k=1}^{\infty }{\frac {z^{k)){k^{s))}~{\frac {\Gamma (s,kb)}{\Gamma (s)))}$

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:

${\displaystyle \operatorname {Li} _{s}(0,z)=\operatorname {Li} _{s}(z)}$

where Li_s(.) is the polylogarithm function.