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

Wikimedia Foundation. 2010.

Look at other dictionaries:

• Polylogarithm — Not to be confused with polylogarithmic. In mathematics, the polylogarithm (also known as Jonquière s function) is a special function Lis(z) that is defined by the infinite sum, or power series: It is in general not an elementary function, unlike …   Wikipedia

• Incomplete Fermi–Dirac integral — In mathematics, the incomplete Fermi–Dirac integral for an index j is given by:F j(x,b) = frac{1}{Gamma(j+1)} int b^infty frac{t^j}{exp(t x) + 1},dt.This is an alternate definition of the incomplete polylogarithm. See also * Complete Fermi–Dirac… …   Wikipedia

• List of mathematical functions — In mathematics, several functions or groups of functions are important enough to deserve their own names. This is a listing of pointers to those articles which explain these functions in more detail. There is a large theory of special functions… …   Wikipedia

• Debye function — In mathematics, the family of Debye functions is defined by The functions are named in honor of Peter Debye, who came across this function (with n = 3) in 1912 when he analytically computed the heat capacity of a solid. His method is now called… …   Wikipedia

• List of mathematics articles (I) — NOTOC Ia IA automorphism ICER Icosagon Icosahedral 120 cell Icosahedral prism Icosahedral symmetry Icosahedron Icosian Calculus Icosian game Icosidodecadodecahedron Icosidodecahedron Icositetrachoric honeycomb Icositruncated dodecadodecahedron… …   Wikipedia

• Lerch zeta function — In mathematics, the Lerch zeta function, sometimes called the Hurwitz Lerch zeta function, is a special function that generalizes the Hurwitz zeta function and the polylogarithm. It is named after Mathias Lerch [http://www groups.dcs.st… …   Wikipedia

• Harmonic number — The term harmonic number has multiple meanings. For other meanings, see harmonic number (disambiguation) .In mathematics, the n th harmonic number is the sum of the reciprocals of the first n natural numbers::H n=… …   Wikipedia

• Complete Fermi–Dirac integral — In mathematics, the complete Fermi–Dirac integral, named after Enrico Fermi and Paul Dirac, for an index j  is given by This is an alternate definition of the polylogarithm function. The closed form of the function exists for j = 0 …   Wikipedia