In mathematics, the Incomplete Polylogarithm function is related to the

polylogarithm In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function of order and argument . Only for special values of does the polylogarithm reduce to an elementary function such as the natu ...

function. It is sometimes known as the

incomplete Fermi–Dirac integral In mathematics, the incomplete Fermi– Dirac integral for an index ''j'' is given by :F_j(x,b) = \frac \int_b^\infty \frac\,dt. This is an alternate definition of the incomplete polylogarithm. See also * Complete Fermi–Dirac integral E ...

or the incomplete Bose–Einstein integral. It may be defined by: :

\operatorname_s(b,z) = \frac\int_b^\infty \frac~dx.

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

\operatorname_s(b,z) = \sum_^\infty \frac~\frac

where Γ(s) is the

gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except th ...

and Γ(s,x) is the upper incomplete gamma function. Since Γ(s,0)=Γ(s), it follows that: :

\operatorname_s(0,z) =\operatorname{Li}_s(z)

where Li_s(.) is the polylogarithm function.

References

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