In mathematics, the incomplete polylogarithm function is related to the polylogarithm function. It is sometimes known as the incomplete FermiâÂÂDirac integral or the incomplete BoseâÂÂEinstein integral. It may be defined by:
Expanding about z=0 and integrating gives a series representation:
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:
where Li<sub>s</sub>(.) is the polylogarithm function.