Polylogarithm

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

(1)

defined in the complex plane over the open unit disk. Its definition on the whole complex plane then follows uniquely via analytic continuation.

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 of the integrals of the Fermi-Dirac distribution

(3)

where is the gamma function, and the Bose-Einstein distribution

(4)

The special case reduces to

(5)

where is the Riemann zeta function. Note, however, that the meaning of for fixed complex is not completely well-defined, since it depends on how is approached in four-dimensional -space.

The polylogarithm of negative integer order arises in sums of the form

			(6)
			(7)

where is an Eulerian number. Polylogarithms also arise in sum 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 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 is itself a polylogarithm,

(18)

Bailey et al. showed that

(19)

A number of remarkable identities exist for polylogarithms, including the amazing identity satisfied by , where (OEIS A073011) is the smallest Salem constant, i.e., 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.