Polylogarithm

In quantum electrodynamics, polylogarithms of positive integer order arise in the calculation of processes represented by higher-order Feynman diagrams.

The special case s = 1 involves the ordinary natural logarithm, Li1(z) = −ln(1−z), while the special cases s = 2 and s = 3 are called the dilogarithm (also referred to as Spence's function) and trilogarithm respectively.

For nonpositive integer orders s, the polylogarithm is a rational function.

by the above series definition and taken to be continuous except on the positive real axis, where a cut is made from

Going across the cut, if ε is an infinitesimally small positive real number, then:

Both can be concluded from the series expansion (see below) of Lis(eμ) about μ = 0.

The derivatives of the polylogarithm follow from the defining power series:

The square relationship is seen from the series definition, and is related to the duplication formula (see also Clunie (1954), Schrödinger (1952)):

Kummer's function obeys a very similar duplication formula.

This is a special case of the multiplication formula, for any positive integer p:

which can be proved using the series definition of the polylogarithm and the orthogonality of the exponential terms (see e.g. discrete Fourier transform).

Any of the following integral representations furnishes the analytic continuation of the polylogarithm beyond the circle of convergence |z| = 1 of the defining power series.

For |z| ≫ 1, the polylogarithm can be expanded into asymptotic series in terms of ln(−z):

As usual, the summation should be terminated when the terms start growing in magnitude.

Wood (1992, § 11) describes a method for obtaining these series from the Bose–Einstein integral representation (his equation 11.2 for Lis(eμ) requires −2π < Im(μ) ≤ 0).

The following limits result from the various representations of the polylogarithm (Wood 1992, § 22):

Wood's first limit for Re(μ) → ∞ has been corrected in accordance with his equation 11.3.

The limit for Re(s) → −∞ follows from the general relation of the polylogarithm with the Hurwitz zeta function (see above).

An alternate integral expression of the dilogarithm for arbitrary complex argument z is (Abramowitz & Stegun 1972, § 27.7):

A source of confusion is that some computer algebra systems define the dilogarithm as dilog(z) = Li2(1−z).

In the case of real z ≥ 1 the first integral expression for the dilogarithm can be written as

This is immediately seen to hold for either x = 0 or y = 0, and for general arguments is then easily verified by differentiation ∂/∂x ∂/∂y.

Known closed-form evaluations of the dilogarithm at special arguments are collected in the table below.

The reflection formula was already published by Landen in 1760, prior to its appearance in a 1768 book by Euler (Maximon 2003, § 10); an equivalent to Abel's identity was already published by Spence in 1809, before Abel wrote his manuscript in 1826 (Zagier 1989, § 2).

The designation bilogarithmische Function was introduced by Carl Johan Danielsson Hill (professor in Lund, Sweden) in 1828 (Maximon 2003, § 10).

Don Zagier (1989) has remarked that the dilogarithm is the only mathematical function possessing a sense of humor.

Leonard Lewin discovered a remarkable and broad generalization of a number of classical relationships on the polylogarithm for special values.

Polylogarithm ladders occur naturally and deeply in K-theory and algebraic geometry.

Polylogarithm ladders provide the basis for the rapid computations of various mathematical constants by means of the BBP algorithm (Bailey, Borwein & Plouffe 1997).

The monodromy group for the polylogarithm consists of the homotopy classes of loops that wind around the two branch points.