# §25.12 Polylogarithms

## §25.12(i) Dilogarithms

The notation was introduced in Lewin (1981) for a function discussed in Euler (1768) and called the dilogarithm in Hill (1828):

25.12.1.

Other notations and names for include (Kölbig et al. (1970)), Spence function (’t Hooft and Veltman (1979)), and (Maximon (2003)).

In the complex plane has a branch point at . The principal branch has a cut along the interval and agrees with (25.12.1) when ; see also §4.2(i). The remainder of the equations in this subsection apply to principal branches.

When , , (25.12.1) becomes

The cosine series in (25.12.7) has the elementary sum

25.12.8

By (25.12.2)

25.12.9

The right-hand side is called Clausen’s integral.

For graphics see Figures 25.12.1 and 25.12.2, and for further properties see Maximon (2003), Kirillov (1995), Lewin (1981), Nielsen (1909), and Zagier (1989).

 Figure 25.12.1: Dilogarithm function , Symbols: : dilogarithm and : real variable Referenced by: §25.12(i) Permalink: http://dlmf.nist.gov/25.12.F1 Encodings: pdf, png Figure 25.12.2: Absolute value of the dilogarithm function , , . Principal value. There is a cut along the real axis from 1 to . Symbols: : dilogarithm and : real variable Referenced by: §25.12(i) Permalink: http://dlmf.nist.gov/25.12.F2 Encodings: VRML, X3D, pdf, png

## §25.12(ii) Polylogarithms

For real or complex and the polylogarithm is defined by

25.12.10

For each fixed complex the series defines an analytic function of for . The series also converges when , provided that . For other values of , is defined by analytic continuation.

The notation was used for in Truesdell (1945) for a series treated in Jonquière (1889), hence the alternative name Jonquière’s function. The special case is the Riemann zeta function: .

### ¶ Integral Representation

valid when and , or and . (In the latter case (25.12.11) becomes (25.5.1)).

See also Lewin (1981), Kölbig (1986), Maximon (2003), Prudnikov et al. (1990, §§1.2 and 2.5), Prudnikov et al. (1992a, §3.3), and Prudnikov et al. (1992b, §3.3).

## §25.12(iii) Fermi–Dirac and Bose–Einstein Integrals

The Fermi–Dirac and Bose–Einstein integrals are defined by

25.12.14,
25.12.15, ; or , ,

respectively. Sometimes the factor is omitted. See Cloutman (1989) and Gautschi (1993).

For a uniform asymptotic approximation for see Temme and Olde Daalhuis (1990).

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