polylog: Polylogarithm Function

In pracma: Practical Numerical Math Functions

Polylogarithm Function

Description

Computes the n-based polylogarithm of z: Li_n(z).

Usage

polylog(z, n)

Arguments

z	real number or vector, all entries satisfying abs(z)<1.
n	base of polylogarithm, integer greater or equal -4.

Details

The Polylogarithm is also known as Jonquiere's function. It is defined as

\sum_{k=1}^{\infty}{z^k / k^n} = z + z^2/2^n + ...

The polylogarithm function arises, e.g., in Feynman diagram integrals. It also arises in the closed form of the integral of the Fermi-Dirac and the Bose-Einstein distributions.

The special cases n=2 and n=3 are called the dilogarithm and trilogarithm, respectively.

Approximation should be correct up to at least 5 digits for |z| > 0.55 and on the order of 10 digits for |z| <= 0.55.

Value

Returns the function value (not vectorized).

Note

Based on some equations, see references. A Matlab implementation is available in the Matlab File Exchange.

References

V. Bhagat, et al. (2003). On the evaluation of generalized BoseEinstein and FermiDirac integrals. Computer Physics Communications, Vol. 155, p.7.

Examples

polylog(0.5,  1)    # polylog(z, 1) = -log(1-z)
polylog(0.5,  2)    # (p1^2 - 6*log(2)^2) / 12
polylog(0.5,  3)    # (4*log(2)^3 - 2*pi^2*log(2) + 21*zeta(3)) / 24
polylog(0.5,  0)    # polylog(z,  0) = z/(1-z)
polylog(0.5, -1)    # polylog(z, -1) = z/(1-z)^2

pracma documentation built on Nov. 10, 2023, 1:14 a.m.