§25.12 Polylogarithms

§25.12(i) Dilogarithms

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):

 25.12.1 $\mathrm{Li}_{2}\left(z\right)=\sum_{n=1}^{\infty}\frac{z^{n}}{n^{2}},$ $|z|\leq 1$. ⓘ Defines: $\mathrm{Li}_{2}\left(\NVar{z}\right)$: 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 and 25
 25.12.2 $\mathrm{Li}_{2}\left(z\right)=-\int_{0}^{z}t^{-1}\ln\left(1-t\right)\mathrm{d}t,$ $z\in\mathbb{C}\setminus(1,\infty)$. ⓘ Symbols: $\mathbb{C}$: complex plane, $\mathrm{d}\NVar{x}$: differential of $x$, $\mathrm{Li}_{2}\left(\NVar{z}\right)$: dilogarithm, $\in$: element of, $\int$: integral, $\ln\NVar{z}$: principal branch of logarithm function, $(\NVar{a},\NVar{b})$: open interval, $\setminus$: set subtraction, $x$: real variable and $z$: complex variable A&S Ref: 27.7.1 (is a modified form with $z=1-x$) Referenced by: §25.12(i) Permalink: http://dlmf.nist.gov/25.12.E2 Encodings: TeX, pMML, png See also: Annotations for 25.12(i), 25.12 and 25

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 $[1,\infty)$ 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 $\mathrm{Li}_{2}\left(z\right)+\mathrm{Li}_{2}\left(\frac{z}{z-1}\right)=-\frac% {1}{2}(\ln\left(1-z\right))^{2},$ $z\in\mathbb{C}\setminus[1,\infty)$. ⓘ Symbols: $[\NVar{a},\NVar{b})$: half-closed interval, $\mathbb{C}$: complex plane, $\mathrm{Li}_{2}\left(\NVar{z}\right)$: dilogarithm, $\in$: element of, $\ln\NVar{z}$: 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 and 25
 25.12.4 $\mathrm{Li}_{2}\left(z\right)+\mathrm{Li}_{2}\left(\frac{1}{z}\right)=-\frac{1% }{6}\pi^{2}-\frac{1}{2}(\ln\left(-z\right))^{2},$ $z\in\mathbb{C}\setminus[0,\infty)$.
 25.12.5 $\mathrm{Li}_{2}\left(z^{m}\right)=m\sum_{k=0}^{m-1}\mathrm{Li}_{2}\left(ze^{2% \pi ik/m}\right),$ $m=1,2,3,\dots$, $|z|<1$.
 25.12.6 $\mathrm{Li}_{2}\left(x\right)+\mathrm{Li}_{2}\left(1-x\right)=\frac{1}{6}\pi^{% 2}-(\ln x)\ln\left(1-x\right),$ $0.

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

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

The cosine series in (25.12.7) has the elementary sum

 25.12.8 $\sum_{n=1}^{\infty}\frac{\cos\left(n\theta\right)}{n^{2}}=\frac{\pi^{2}}{6}-% \frac{\pi\theta}{2}+\frac{\theta^{2}}{4}.$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$: 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), 25.12 and 25

By (25.12.2)

 25.12.9 $\sum_{n=1}^{\infty}\frac{\sin\left(n\theta\right)}{n^{2}}=-\int_{0}^{\theta}% \ln\left(2\sin\left(\tfrac{1}{2}x\right)\right)\mathrm{d}x.$ ⓘ Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\ln\NVar{z}$: principal branch of logarithm function, $\sin\NVar{z}$: 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), 25.12 and 25

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 $\mathrm{Li}_{s}\left(z\right)$ is defined by

 25.12.10 $\mathrm{Li}_{s}\left(z\right)=\sum_{n=1}^{\infty}\frac{z^{n}}{n^{s}}.$ ⓘ Defines: $\mathrm{Li}_{\NVar{s}}\left(\NVar{z}\right)$: 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), 25.12 and 25

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 $z$, $\mathrm{Li}_{s}\left(z\right)$ is defined by analytic continuation.

The notation $\phi\left(z,s\right)$ 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: $\zeta\left(s\right)=\mathrm{Li}_{s}\left(1\right)$.

Integral Representation

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

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

Further properties include

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

and

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

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 $\displaystyle F_{s}(x)$ $\displaystyle=\frac{1}{\Gamma\left(s+1\right)}\int_{0}^{\infty}\frac{t^{s}}{e^% {t-x}+1}\mathrm{d}t,$ $s>-1$, ⓘ Defines: $F_{s}(x)$: Fermi–Dirac integral (locally) Symbols: $\Gamma\left(\NVar{z}\right)$: 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 and 25 25.12.15 $\displaystyle G_{s}(x)$ $\displaystyle=\frac{1}{\Gamma\left(s+1\right)}\int_{0}^{\infty}\frac{t^{s}}{e^% {t-x}-1}\mathrm{d}t,$ $s>-1$, $x<0$; or $s>0$, $x\leq 0$, ⓘ Defines: $G_{s}(x)$: Bose–Einstein integral (locally) Symbols: $\Gamma\left(\NVar{z}\right)$: 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), 25.12 and 25

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

In terms of polylogarithms

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

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