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# Incomplete polylogarithm

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:

:$Li_s\left(b,z\right) = frac\left\{1\right\}\left\{Gamma\left(s\right)\right\}int_b^infty frac\left\{x^\left\{s-1\left\{e^x/z-1\right\}~dx$

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

:$Li_s\left(b,z\right) = sum_\left\{k=1\right\}^infty frac\left\{z^k\right\}\left\{k^s\right\}~frac\left\{Gamma\left(s,kb\right)\right\}\left\{Gamma\left(s\right)\right\}$

where &Gamma;(s) is the gamma function and &Gamma;(s,x) is the incomplete gamma function.

References

* GNU Scientific Library - Reference Manual http://www.gnu.org/software/gsl/manual/gsl-ref.html#SEC117

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