pracma (version 1.4.5)

polylog: 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< code="">.

n

base of polylogarithm, integer greater or equal -4.

Value

• Returns the function value (not vectorized).

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|

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