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 ( b , z ) = 1 Γ ( s ) ∫ b ∞ x s − 1 e x / z − 1 d x . {\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:
Li s ( b , z ) = ∑ k = 1 ∞ z k k s Γ ( s , k b ) Γ ( s ) {\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:
Li s ( 0 , z ) = Li s ( z ) {\displaystyle \operatorname {Li} _{s}(0,z)=\operatorname {Li} _{s}(z)}
where Lis(.) is the polylogarithm function.
References GNU Scientific Library - Reference Manual https://www.gnu.org/software/gsl/manual/gsl-ref.html#SEC117
