The notation ${\mathrm{Li}}_2\left(z\right)$ was introduced in Lewin (1981) for a function discussed in Euler (1768) and called the dilogarithm in Hill (1828):

Other notations and names for ${\mathrm{Li}}_2\left(z\right)$ include $S_2(z)$ (Kölbig et al. (1970)), Spence function $\mathrm{Sp}(z)$ (’t Hooft and Veltman (1979)), and ${\mathrm{L}}_2(z)$ (Maximon (2003)).

In the complex plane ${\mathrm{Li}}_2\left(z\right)$ has a branch point at $z=1$. The principal branch has a cut along the interval $[1,\mathrm{\infty })$ and agrees with (25.12.1) when $|z|\le 1$; see also §4.2(i). The remainder of the equations in this subsection apply to principal branches.

When $z={\mathrm{e}}^{\mathrm{i}\theta }$, $0\le \theta \le 2\pi$, (25.12.1) becomes

The cosine series in (25.12.7) has the elementary sum

By (25.12.2)

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).

For real or complex $s$ and $z$ the polylogarithm ${\mathrm{Li}}_s\left(z\right)$ is defined by

For each fixed complex $s$ the series defines an analytic function of $z$ for . The series also converges when $|z|=1$, provided that $\mathrm{\Re }s>1$. For other values of $z$, ${\mathrm{Li}}_s\left(z\right)$ is defined by analytic continuation.

The notation $\varphi (z,s)$ was used for ${\mathrm{Li}}_s\left(z\right)$ in Truesdell (1945) for a series treated in Jonquière (1889), hence the alternative name Jonquière’s function. The special case $z=1$ is the Riemann zeta function: $\zeta \left(s\right)={\mathrm{Li}}_s\left(1\right)$.

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

respectively. Sometimes the factor $1/\mathrm{\Gamma }\left(s+1\right)$ is omitted. See Cloutman (1989) and Gautschi (1993).

In terms of polylogarithms

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