# Incomplete polylogarithm

In

mathematics , the**Incomplete Polylogarithm**function is related to thepolylogarithm function. It is sometimes known as theincomplete Fermi–Dirac integral or the incompleteBose–Einstein integral. It may be defined by::$Li\_s(b,z)\; =\; frac\{1\}\{Gamma(s)\}int\_b^infty\; frac\{x^\{s-1\{e^x/z-1\}~dx$

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

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

where Γ(s) is the

gamma function and Γ(s,x) is theincomplete gamma function .**References*** GNU Scientific Library - Reference Manual http://www.gnu.org/software/gsl/manual/gsl-ref.html#SEC117

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