Incomplete polylogarithm

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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) = \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:


\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.

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