Polylogarithm

The polylogarithm , also known as Jonquière's function, is the function

(1)

defined for complex order and . For each fixed , this series is an analytic function of on the open unit disk. Analytic continuation requires a choice of branch. The formulas below use the principal branch, with the real interval excluded from its analytic domain in the complex plane (NIST).

Note that the similar notation is used for the logarithmic integral.

The polylogarithm is also denoted and equal to

(2)

where is the Lerch transcendent (Erdélyi et al. 1981, p. 30). The polylogarithm arises in Feynman diagram integrals (and, in particular, in the computation of quantum electrodynamics corrections to the electrons gyromagnetic ratio), and the special cases and are called the dilogarithm and trilogarithm, respectively. The polylogarithm is implemented in the Wolfram Language as PolyLog[n, z].

The polylogarithm also arises in the closed-form solution of the integrals of the Fermi-Dirac distribution

(3)

for real and , where is the gamma function and denotes the real part of . For the Bose-Einstein distribution,

(4)

for real and , or for and (NIST).

The special case reduces to

(5)

where is the Riemann zeta function and . Outside this range, a value obtained by continuing in the order need not equal a limit as . For example, diverges as , while .

The polylogarithm of negative integer order arises, for a positive integer and , in sums of the form

			(6)
			(7)

where is an Eulerian number. Polylogarithms also arise in sums of generalized harmonic numbers as

(8)

for .

Special forms of low-order polylogarithms include

			(9)
			(10)
			(11)
			(12)

At arguments and 1, the general polylogarithms become

			(13)
			(14)

where is the Dirichlet eta function and is the Riemann zeta function. The first identity holds for every complex , while the second requires . The polylogarithm for argument can also be evaluated analytically for small ,

			(15)
			(16)
			(17)

No similar formulas of this type are known for higher orders (Lewin 1991, p. 2). appears in the third-order correction term in the gyromagnetic ratio of the electron.

The derivative of a polylogarithm can be expressed using a polylogarithm,

(18)

for in the analytic domain, with a removable singularity at , where the derivative is 1. Bailey et al. (1997) showed that, for , 2, ..., 5,

(19)

where any term containing the factorial of a negative integer in its denominator is omitted.

Bailey and Broadhurst (1999) found an empirical integer relation at order 17 involving 125 constants, including polylogarithms and powers of and , and checked it to more than 59000 decimal digits. Here (OEIS A073011) is the smallest known Salem constant, namely the largest positive root of the polynomial in Lehmer's Mahler measure problem (Cohen et al. 1992; Bailey and Broadhurst 1999; Borwein and Bailey 2003, pp. 8-9).

No general algorithm is known for integration of polylogarithms of functions.