# §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 $\mathrm{Li}_{2}\left(z\right)\equiv\sum_{n=1}^{\infty}\frac{z^{n}}{n^{2}},$ $|z|\leq 1$. ⓘ Defines: $\mathrm{Li}_{2}\left(\NVar{z}\right)$: dilogarithm Symbols: $\equiv$: equals by definition, $n$: nonnegative integer, $x$: real variable and $z$: complex variable Keywords: definition Source: Maximon (2003, (3.1), p. 2808) 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 Ch.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 Keywords: definite integral, integral representation Source: Maximon (2003, (2.1), p. 2807) 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 Ch.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 Keywords: connection formula Source: Maximon (2003, (3.4), p. 2808) 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 Ch.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. ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Li}_{2}\left(\NVar{z}\right)$: dilogarithm, $\ln\NVar{z}$: principal branch of logarithm function and $x$: real variable Keywords: connection formula Source: Maximon (2003, (3.3), p. 2808) Permalink: http://dlmf.nist.gov/25.12.E6 Encodings: TeX, pMML, png See also: Annotations for §25.12(i), §25.12 and Ch.25

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

 25.12.7 $\mathrm{Li}_{2}\left(e^{i\theta}\right)=\sum_{n=1}^{\infty}\frac{\cos\left(n% \theta\right)}{n^{2}}+i\sum_{n=1}^{\infty}\frac{\sin\left(n\theta\right)}{n^{2% }}.$ ⓘ Symbols: $\cos\NVar{z}$: cosine function, $\mathrm{Li}_{2}\left(\NVar{z}\right)$: dilogarithm, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin\NVar{z}$: sine function, $n$: nonnegative integer and $\theta$: phase Keywords: infinite series, series representation Source: Maximon (2003, (8.7), p. 2813) Referenced by: §25.12(i) Permalink: http://dlmf.nist.gov/25.12.E7 Encodings: TeX, pMML, png See also: Annotations for §25.12(i), §25.12 and Ch.25

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 Keywords: infinite series Source: Maximon (2003, (8.7), p. 2813) 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 Ch.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 Keywords: definite integral, infinite series Source: Maximon (2003, (8.7), (8.8), p. 2813) A&S Ref: 27.8.6 (integration of first series) Referenced by: 1st item Permalink: http://dlmf.nist.gov/25.12.E9 Encodings: TeX, pMML, png See also: Annotations for §25.12(i), §25.12 and Ch.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 Keywords: definition, infinite series, series representation Source: Lewin (1981, p. 236) Referenced by: (25.14.3) Permalink: http://dlmf.nist.gov/25.12.E10 Encodings: TeX, pMML, png See also: Annotations for §25.12(ii), §25.12 and Ch.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 $\mathrm{Li}_{s}\left(z\right)\equiv\frac{z}{\Gamma\left(s\right)}\int_{0}^{% \infty}\frac{x^{s-1}}{e^{x}-z}\mathrm{d}x,$

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 $\mathrm{Li}_{s}\left(z\right)=\Gamma\left(1-s\right)\left(\ln\frac{1}{z}\right% )^{s-1}+\sum_{n=0}^{\infty}\zeta\left(s-n\right)\frac{(\ln z)^{n}}{n!},$ $s\neq 1,2,3,\dots$, $|\ln z|<2\pi$, ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\ln\NVar{z}$: principal branch of logarithm function, $\mathrm{Li}_{\NVar{s}}\left(\NVar{z}\right)$: polylogarithm, $n$: nonnegative integer, $s$: complex variable and $z$: complex variable Keywords: infinite series, series representation Source: Erdélyi et al. (1953a, (1.11.8), p. 29); take $v=1$ and use (25.14.3), (25.11.2) Permalink: http://dlmf.nist.gov/25.12.E12 Encodings: TeX, pMML, png See also: Annotations for §25.12(ii), §25.12(ii), §25.12 and Ch.25

and

 25.12.13 $\mathrm{Li}_{s}\left(e^{2\pi ia}\right)+e^{\pi is}\mathrm{Li}_{s}\left(e^{-2% \pi ia}\right)=\frac{(2\pi)^{s}e^{\pi is/2}}{\Gamma\left(s\right)}\zeta\left(1% -s,a\right),$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\zeta\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $(\NVar{a},\NVar{b})$: open interval, $\mathrm{Li}_{\NVar{s}}\left(\NVar{z}\right)$: polylogarithm, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable Source: Erdélyi et al. (1953a, (1.11.16), p. 31); with $\mathrm{Li}_{s}\left(z\right)=F(z,s)$, $z={\mathrm{e}}^{2\pi\mathrm{i}a}$ Referenced by: §25.12(ii), §25.12(ii) Permalink: http://dlmf.nist.gov/25.12.E13 Encodings: TeX, pMML, png See also: Annotations for §25.12(ii), §25.12(ii), §25.12 and Ch.25

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 natural logarithm, $\int$: integral, $x$: real variable and $s$: complex variable Keywords: Fermi–Dirac integral, definition, improper integral, integral representation Source: Dingle (1957b, (1), p. 226) Referenced by: (25.12.16), §25.12(iii), 3rd item, 5th item Permalink: http://dlmf.nist.gov/25.12.E14 Encodings: TeX, pMML, png See also: Annotations for §25.12(iii), §25.12 and Ch.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 natural logarithm, $\int$: integral, $x$: real variable and $s$: complex variable Keywords: Bose–Einstein integral, definition, improper integral, integral representation Source: Dingle (1957a, p. 240) Referenced by: (25.12.16), §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 Ch.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 $\displaystyle F_{s}(x)$ $\displaystyle=-\mathrm{Li}_{s+1}\left(-e^{x}\right),$ $\displaystyle G_{s}(x)$ $\displaystyle=\mathrm{Li}_{s+1}\left(e^{x}\right).$ ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $\mathrm{Li}_{\NVar{s}}\left(\NVar{z}\right)$: polylogarithm, $x$: real variable, $s$: complex variable, $F_{s}(x)$: Fermi–Dirac integral and $G_{s}(x)$: Bose–Einstein integral Source: Derivable from (25.12.11), (25.12.14), (25.12.15). Referenced by: §25.12(iii) Permalink: http://dlmf.nist.gov/25.12.E16 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §25.12(iii), §25.12 and Ch.25

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