The notation was introduced in Lewin (1981) for a function discussed in Euler (1768) and called the dilogarithm in Hill (1828):
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
The right-hand side is called Clausen’s integral.
For real or complex and the polylogarithm is defined by
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: .
Further properties include
valid when , or , . When and , (25.12.13) becomes (25.12.4).
The Fermi–Dirac and Bose–Einstein integrals are defined by
|, ; or , ,|
respectively. Sometimes the factor is omitted. See Cloutman (1989) and Gautschi (1993).
In terms of polylogarithms
For a uniform asymptotic approximation for see Temme and Olde Daalhuis (1990).