# §25.12 Polylogarithms

## §25.12(i) Dilogarithms

The notation $\mathop{\mathrm{Li}_{2}\/}\nolimits\!\left(z\right)$ was introduced in Lewin (1981) for a function discussed in Euler (1768) and called the dilogarithm in Hill (1828):

 25.12.1 $\mathop{\mathrm{Li}_{2}\/}\nolimits\!\left(z\right)=\sum_{n=1}^{\infty}\frac{z% ^{n}}{n^{2}},$ $|z|\leq 1$. Defines: $\mathop{\mathrm{Li}_{2}\/}\nolimits\!\left(\NVar{z}\right)$: dilogarithm Symbols: $n$: nonnegative integer, $x$: real variable and $z$: complex variable A&S Ref: 27.7.2 (with $z=1-x$) Referenced by: §25.12(i), §25.12(i) Permalink: http://dlmf.nist.gov/25.12.E1 Encodings: TeX, pMML, png See also: Annotations for 25.12(i)
 25.12.2 $\mathop{\mathrm{Li}_{2}\/}\nolimits\!\left(z\right)=-\int_{0}^{z}t^{-1}\mathop% {\ln\/}\nolimits\!\left(1-t\right)\mathrm{d}t,$ $z\in\mathbb{C}\setminus(1,\infty)$.

Other notations and names for $\mathop{\mathrm{Li}_{2}\/}\nolimits\!\left(z\right)$ 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 $\mathop{\mathrm{Li}_{2}\/}\nolimits\!\left(z\right)$ has a branch point at $z=1$. The principal branch has a cut along the interval $[1,\infty)$ and agrees with (25.12.1) when $|z|\leq 1$; see also §4.2(i). The remainder of the equations in this subsection apply to principal branches.

 25.12.3 $\mathop{\mathrm{Li}_{2}\/}\nolimits\!\left(z\right)+\mathop{\mathrm{Li}_{2}\/}% \nolimits\!\left(\frac{z}{z-1}\right)=-\frac{1}{2}(\mathop{\ln\/}\nolimits\!% \left(1-z\right))^{2},$ $z\in\mathbb{C}\setminus[1,\infty)$. Symbols: $[\NVar{a},\NVar{b})$: half-closed interval, $\mathbb{C}$: complex plane, $\mathop{\mathrm{Li}_{2}\/}\nolimits\!\left(\NVar{z}\right)$: dilogarithm, $\in$: element of, $\mathop{\ln\/}\nolimits\NVar{z}$: principal branch of logarithm function, $\setminus$: set subtraction, $x$: real variable and $z$: complex variable A&S Ref: 27.7.5 (is a modified form with $z=1-x$) Permalink: http://dlmf.nist.gov/25.12.E3 Encodings: TeX, pMML, png See also: Annotations for 25.12(i)
 25.12.4 $\mathop{\mathrm{Li}_{2}\/}\nolimits\!\left(z\right)+\mathop{\mathrm{Li}_{2}\/}% \nolimits\!\left(\frac{1}{z}\right)=-\frac{1}{6}\pi^{2}-\frac{1}{2}(\mathop{% \ln\/}\nolimits\!\left(-z\right))^{2},$ $z\in\mathbb{C}\setminus[0,\infty)$.
 25.12.5 $\mathop{\mathrm{Li}_{2}\/}\nolimits\!\left(z^{m}\right)=m\sum_{k=0}^{m-1}% \mathop{\mathrm{Li}_{2}\/}\nolimits\!\left(ze^{2\pi ik/m}\right),$ $m=1,2,3,\dots$, $|z|<1$.
 25.12.6 $\mathop{\mathrm{Li}_{2}\/}\nolimits\!\left(x\right)+\mathop{\mathrm{Li}_{2}\/}% \nolimits\!\left(1-x\right)=\frac{1}{6}\pi^{2}-(\mathop{\ln\/}\nolimits x)% \mathop{\ln\/}\nolimits\!\left(1-x\right),$ $0.

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

 25.12.7 $\mathop{\mathrm{Li}_{2}\/}\nolimits\!\left(e^{i\theta}\right)=\sum_{n=1}^{% \infty}\frac{\mathop{\cos\/}\nolimits\!\left(n\theta\right)}{n^{2}}+i\sum_{n=1% }^{\infty}\frac{\mathop{\sin\/}\nolimits\!\left(n\theta\right)}{n^{2}}.$

The cosine series in (25.12.7) has the elementary sum

 25.12.8 $\sum_{n=1}^{\infty}\frac{\mathop{\cos\/}\nolimits\!\left(n\theta\right)}{n^{2}% }=\frac{\pi^{2}}{6}-\frac{\pi\theta}{2}+\frac{\theta^{2}}{4}.$ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function, $n$: nonnegative integer and $\theta$: phase A&S Ref: 27.8.6 (second series) Permalink: http://dlmf.nist.gov/25.12.E8 Encodings: TeX, pMML, png See also: Annotations for 25.12(i)

By (25.12.2)

 25.12.9 $\sum_{n=1}^{\infty}\frac{\mathop{\sin\/}\nolimits\!\left(n\theta\right)}{n^{2}% }=-\int_{0}^{\theta}\mathop{\ln\/}\nolimits\!\left(2\mathop{\sin\/}\nolimits\!% \left(\tfrac{1}{2}x\right)\right)\mathrm{d}x.$ Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\mathop{\ln\/}\nolimits\NVar{z}$: principal branch of logarithm function, $\mathop{\sin\/}\nolimits\NVar{z}$: sine function, $n$: nonnegative integer, $x$: real variable and $\theta$: phase A&S Ref: 27.8.6 (integration of first series) Referenced by: §25.19 Permalink: http://dlmf.nist.gov/25.12.E9 Encodings: TeX, pMML, png See also: Annotations for 25.12(i)

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).

## §25.12(ii) Polylogarithms

For real or complex $s$ and $z$ the polylogarithm $\mathop{\mathrm{Li}_{s}\/}\nolimits\!\left(z\right)$ is defined by

 25.12.10 $\mathop{\mathrm{Li}_{s}\/}\nolimits\!\left(z\right)=\sum_{n=1}^{\infty}\frac{z% ^{n}}{n^{s}}.$ Defines: $\mathop{\mathrm{Li}_{\NVar{s}}\/}\nolimits\!\left(\NVar{z}\right)$: polylogarithm Symbols: $n$: nonnegative integer, $s$: complex variable and $z$: complex variable Permalink: http://dlmf.nist.gov/25.12.E10 Encodings: TeX, pMML, png See also: Annotations for 25.12(ii)

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

The notation $\mathop{\phi\/}\nolimits\!\left(z,s\right)$ was used for $\mathop{\mathrm{Li}_{s}\/}\nolimits\!\left(z\right)$ 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: $\mathop{\zeta\/}\nolimits\!\left(s\right)=\mathop{\mathrm{Li}_{s}\/}\nolimits% \!\left(1\right)$.

### Integral Representation

 25.12.11 $\mathop{\mathrm{Li}_{s}\/}\nolimits\!\left(z\right)=\frac{z}{\mathop{\Gamma\/}% \nolimits\!\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}}{e^{x}-z}\mathrm{d}x,$

valid when $\Re{s}>0$ and $\left|\mathop{\mathrm{ph}\/}\nolimits\!\left(1-z\right)\right|<\pi$, or $\Re{s}>1$ and $z=1$. (In the latter case (25.12.11) becomes (25.5.1)).

Further properties include

 25.12.12 $\mathop{\mathrm{Li}_{s}\/}\nolimits\!\left(z\right)=\mathop{\Gamma\/}\nolimits% \!\left(1-s\right)\left(\mathop{\ln\/}\nolimits\frac{1}{z}\right)^{s-1}+\sum_{% n=0}^{\infty}\mathop{\zeta\/}\nolimits\!\left(s-n\right)\frac{(\mathop{\ln\/}% \nolimits z)^{n}}{n!},$ $s\neq 1,2,3,\dots$, $|\mathop{\ln\/}\nolimits z|<2\pi$,

and

 25.12.13 $\mathop{\mathrm{Li}_{s}\/}\nolimits\!\left(e^{2\pi ia}\right)+e^{\pi is}% \mathop{\mathrm{Li}_{s}\/}\nolimits\!\left(e^{-2\pi ia}\right)=\frac{(2\pi)^{s% }e^{\pi is/2}}{\mathop{\Gamma\/}\nolimits\!\left(s\right)}\mathop{\zeta\/}% \nolimits\!\left(1-s,a\right),$

valid when $\Re{s}>0$, $\Im{a}>0$ or $\Re{s}>1$, $\Im{a}=0$. When $s=2$ and $e^{2\pi ia}=z$, (25.12.13) becomes (25.12.4).

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 $\displaystyle F_{s}(x)$ $\displaystyle=\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(s+1\right)}\int_{0}^{% \infty}\frac{t^{s}}{e^{t-x}+1}\mathrm{d}t,$ $s>-1$, Defines: $F_{s}(x)$: Fermi–Dirac integral (locally) Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $\mathrm{d}\NVar{x}$: differential of $x$, $\mathrm{e}$: base of exponential function, $\int$: integral, $x$: real variable and $s$: complex variable Referenced by: §25.12(iii), §25.19, §25.20 Permalink: http://dlmf.nist.gov/25.12.E14 Encodings: TeX, pMML, png See also: Annotations for 25.12(iii) 25.12.15 $\displaystyle G_{s}(x)$ $\displaystyle=\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(s+1\right)}\int_{0}^{% \infty}\frac{t^{s}}{e^{t-x}-1}\mathrm{d}t,$ $s>-1$, $x<0$; or $s>0$, $x\leq 0$, Defines: $G_{s}(x)$: Bose–Einstein integral (locally) Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $\mathrm{d}\NVar{x}$: differential of $x$, $\mathrm{e}$: base of exponential function, $\int$: integral, $x$: real variable and $s$: complex variable Referenced by: §25.12(iii) Permalink: http://dlmf.nist.gov/25.12.E15 Encodings: TeX, pMML, png See also: Annotations for 25.12(iii)

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

In terms of polylogarithms

 25.12.16 $\displaystyle F_{s}(x)$ $\displaystyle=-\mathop{\mathrm{Li}_{s+1}\/}\nolimits\!\left(-e^{x}\right),$ $\displaystyle G_{s}(x)$ $\displaystyle=\mathop{\mathrm{Li}_{s+1}\/}\nolimits\!\left(e^{x}\right).$

For a uniform asymptotic approximation for $F_{s}(x)$ see Temme and Olde Daalhuis (1990).