R: Polylogarithm Function

polylog {pracma} | R Documentation

## 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

∑_{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.

### 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

[Package pracma version 2.0.4 Index]