mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 natur ...

function. It is sometimes known as the incomplete Fermi–Dirac integral 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 Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...

and Γ(s,x) is the upper

incomplete gamma function In mathematics, the upper and lower incomplete gamma functions are types of special functions which arise as solutions to various mathematical problems such as certain integrals. Their respective names stem from their integral definitions, whic ...

. 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