Polylogarithm Function
Computes the n-based polylogarithm of z: Li_n(z).
polylog(z, n)
z |
real number or vector, all entries satisfying |
n |
base of polylogarithm, integer greater or equal -4. |
The Polylogarithm is also known as Jonquiere's function. It is defined as
∑_{k=1}^{∞}{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.
Returns the function value (not vectorized).
Based on some equations, see references. A Matlab implementation is available in the Matlab File Exchange.
V. Bhagat, et al. (2003). On the evaluation of generalized BoseEinstein and FermiDirac integrals. Computer Physics Communications, Vol. 155, p.7.
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
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