§25.12 Polylogarithms

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Notes:: See Maximon (2003), Lewin (1981), Dingle (1957a), and Dingle (1957b).
Permalink:: http://dlmf.nist.gov/25.12
See also:: Annotations for Ch.25

§25.12(i) Dilogarithms
§25.12(ii) Polylogarithms
§25.12(iii) Fermi–Dirac and Bose–Einstein Integrals

§25.12(i) Dilogarithms

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Keywords:: Clausen’s integral, analytic properties, definition, dilogarithms, graphics, principal branch (or value)
Notes:: See Maximon (2003, pp. 2807–2813). The graphics were constructed at NIST.
Referenced by:: 2nd item, 4th item, §25.21(vi)
Permalink:: http://dlmf.nist.gov/25.12.i
See also:: Annotations for §25.12 and Ch.25

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

25.12.1 $\operatorname{Li}_{2}\left(z\right)\equiv\sum_{n=1}^{\infty}\frac{z^{n}}{n^{2}},$ $|z|\leq 1$ .

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Defines:: $\operatorname{Li}_{2}\left(\NVar{z}\right)$ : dilogarithm
Symbols:: $\equiv$ : equals by definition, : nonnegative integer, : real variable and : complex variable
Keywords:: definition
Source:: Maximon (2003, (3.1), p. 2808)
A&S Ref:: 27.7.2 (with )
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 $\operatorname{Li}_{2}\left(z\right)=-\int_{0}^{z}t^{-1}\ln\left(1-t\right)\,% \mathrm{d}t,$ $z\in\mathbb{C}\setminus(1,\infty)$ .

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Symbols:: $\mathbb{C}$ : complex plane, $\,\mathrm{d}\NVar{x}$ : differential of , $\operatorname{Li}_{2}\left(\NVar{z}\right)$ : dilogarithm, $\in$ : element of, $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $(\NVar{a},\NVar{b})$ : open interval, $\setminus$ : set subtraction, : real variable and : 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 )
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 $\operatorname{Li}_{2}\left(z\right)$ 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 $\operatorname{Li}_{2}\left(z\right)$ has a branch point at . 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 $\operatorname{Li}_{2}\left(z\right)+\operatorname{Li}_{2}\left(\frac{z}{z-1}% \right)=-\frac{1}{2}(\ln\left(1-z\right))^{2},$ $z\in\mathbb{C}\setminus[1,\infty)$ .

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Symbols:: $[\NVar{a},\NVar{b})$ : half-closed interval, $\mathbb{C}$ : complex plane, $\operatorname{Li}_{2}\left(\NVar{z}\right)$ : dilogarithm, $\in$ : element of, $\ln\NVar{z}$ : principal branch of logarithm function, $\setminus$ : set subtraction, : real variable and : complex variable
Keywords:: connection formula
Source:: Maximon (2003, (3.4), p. 2808)
A&S Ref:: 27.7.5 (is a modified form with )
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 $\operatorname{Li}_{2}\left(z\right)+\operatorname{Li}_{2}\left(\frac{1}{z}% \right)=-\frac{1}{6}\pi^{2}-\frac{1}{2}(\ln\left(-z\right))^{2},$ $z\in\mathbb{C}\setminus[0,\infty)$ .

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $[\NVar{a},\NVar{b})$ : half-closed interval, $\mathbb{C}$ : complex plane, $\operatorname{Li}_{2}\left(\NVar{z}\right)$ : dilogarithm, $\in$ : element of, $\ln\NVar{z}$ : principal branch of logarithm function, $\setminus$ : set subtraction and : complex variable
Keywords:: connection formula
Source:: Maximon (2003, (3.2), p. 2808)
Referenced by:: §25.12(ii)
Permalink:: http://dlmf.nist.gov/25.12.E4
Encodings:: TeX, pMML, png
See also:: Annotations for §25.12(i), §25.12 and Ch.25

25.12.5 $\operatorname{Li}_{2}\left(z^{m}\right)=m\sum_{k=0}^{m-1}\operatorname{Li}_{2}% \left(ze^{2\pi ik/m}\right),$ $m=1,2,3,\dots$ ,

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{Li}_{2}\left(\NVar{z}\right)$ : dilogarithm, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, : nonnegative integer, : nonnegative integer and : complex variable
Source:: Maximon (2003, (3.12), p. 2809)
Permalink:: http://dlmf.nist.gov/25.12.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §25.12(i), §25.12 and Ch.25

25.12.6 $\operatorname{Li}_{2}\left(x\right)+\operatorname{Li}_{2}\left(1-x\right)=% \frac{1}{6}\pi^{2}-(\ln x)\ln\left(1-x\right),$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{Li}_{2}\left(\NVar{z}\right)$ : dilogarithm, $\ln\NVar{z}$ : principal branch of logarithm function and : 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 $\operatorname{Li}_{2}\left(e^{i\theta}\right)=\sum_{n=1}^{\infty}\frac{\cos% \left(n\theta\right)}{n^{2}}+i\sum_{n=1}^{\infty}\frac{\sin\left(n\theta\right% )}{n^{2}}.$

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Symbols:: $\cos\NVar{z}$ : cosine function, $\operatorname{Li}_{2}\left(\NVar{z}\right)$ : dilogarithm, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\sin\NVar{z}$ : sine function, : 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 $\sum_{n=1}^{\infty}\frac{\cos\left(n\theta\right)}{n^{2}}=\frac{\pi^{2}}{6}-% \frac{\pi\theta}{2}+\frac{\theta^{2}}{4}.$

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, : 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 $\sum_{n=1}^{\infty}\frac{\sin\left(n\theta\right)}{n^{2}}=-\int_{0}^{\theta}% \ln\left(2\sin\left(\tfrac{1}{2}x\right)\right)\,\mathrm{d}x.$

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $\sin\NVar{z}$ : sine function, : nonnegative integer, : 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).

Figure 25.12.1: Dilogarithm function $\operatorname{Li}_{2}\left(x\right)$ , $-20\leq x<1.$ Magnify 3D Help

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Symbols:: $\operatorname{Li}_{2}\left(\NVar{z}\right)$ : dilogarithm and : real variable
Referenced by:: §25.12(i)
Permalink:: http://dlmf.nist.gov/25.12.F1
Encodings:: pdf, png
See also:: Annotations for §25.12(i), §25.12 and Ch.25

Figure 25.12.2: Absolute value of the dilogarithm function $\left|\operatorname{Li}_{2}\left(x+iy\right)\right|$ , $-20\leq x\leq 20$ , $-20\leq y\leq 20$ . Principal value. There is a cut along the real axis from 1 to $\infty$ . Magnify 3D Help

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Symbols:: $\operatorname{Li}_{2}\left(\NVar{z}\right)$ : dilogarithm, $\mathrm{i}$ : imaginary unit and : real variable
Referenced by:: §25.12(i)
Permalink:: http://dlmf.nist.gov/25.12.F2
Encodings:: Vizualization, pdf, png
See also:: Annotations for §25.12(i), §25.12 and Ch.25

§25.12(ii) Polylogarithms

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Defines:: $\phi\left(\NVar{z},\NVar{s}\right)=\operatorname{Li}_{s}\left(z\right)$ : notation used by (Truesdell, 1945)
Keywords:: Riemann zeta function, analytic properties, definitions, polylogarithms, relations to other functions, series expansions
Notes:: See Lewin (1981, Section 7.12).
Referenced by:: §25.14(i)
Permalink:: http://dlmf.nist.gov/25.12.ii
See also:: Annotations for §25.12 and Ch.25

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

25.12.10 $\operatorname{Li}_{s}\left(z\right)=\sum_{n=1}^{\infty}\frac{z^{n}}{n^{s}}.$

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Defines:: $\operatorname{Li}_{\NVar{s}}\left(\NVar{z}\right)$ : polylogarithm
Symbols:: : nonnegative integer, : complex variable and : 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 the series defines an analytic function of for . The series also converges when , provided that $\Re s>1$ . For other values of , $\operatorname{Li}_{s}\left(z\right)$ is defined by analytic continuation.

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

Integral Representation

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Keywords:: Hurwitz zeta function, Riemann zeta function, integral representations, polylogarithms, relations to other functions
See also:: Annotations for §25.12(ii), §25.12 and Ch.25

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

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Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of , $\equiv$ : equals by definition, $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{Li}_{\NVar{s}}\left(\NVar{z}\right)$ : polylogarithm, : real variable, : complex variable and : complex variable
Keywords:: improper integral, integral representation
Source:: Lewin (1981, (7.188), p. 236)
Referenced by:: (25.12.16), §25.12(ii)
Permalink:: http://dlmf.nist.gov/25.12.E11
Encodings:: TeX, pMML, png
See also:: Annotations for §25.12(ii), §25.12(ii), §25.12 and Ch.25

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

Further properties include

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

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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 ), $\ln\NVar{z}$ : principal branch of logarithm function, $\operatorname{Li}_{\NVar{s}}\left(\NVar{z}\right)$ : polylogarithm, : nonnegative integer, : complex variable and : complex variable
Keywords:: infinite series, series representation
Source:: Erdélyi et al. (1953a, (1.11.8), p. 29); take 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 $\operatorname{Li}_{s}\left(z\right)+{\mathrm{e}}^{\pi\mathrm{i}s}\operatorname% {Li}_{s}\left(\frac{1}{z}\right)=\frac{(2\pi)^{s}{\mathrm{e}}^{\pi\mathrm{i}s/% 2}}{\Gamma\left(s\right)}\zeta\left(1-s,\frac{\ln z}{2\pi\mathrm{i}}\right),$ $z\in\mathbb{C}\setminus[0,\infty)$ .

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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, $[\NVar{a},\NVar{b})$ : half-closed interval, $\mathbb{C}$ : complex plane, $\in$ : element of, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function, $\operatorname{Li}_{\NVar{s}}\left(\NVar{z}\right)$ : polylogarithm, $\setminus$ : set subtraction, : real or complex parameter, : complex variable and : complex variable
Source:: Erdélyi et al. (1953a, (1.11.16), p. 31); with $\operatorname{Li}_{s}\left(z\right)=F(z,s)$
Referenced by:: §25.12(ii), Erratum (V1.2.3) for Equation (25.12.13)
Permalink:: http://dlmf.nist.gov/25.12.E13
Encodings:: TeX, pMML, png
Modification (effective with 1.2.3):: In this update we replaced ${\mathrm{e}}^{2\pi\mathrm{i}a}$ by . In this way we can remove the constraints on and .
See also:: Annotations for §25.12(ii), §25.12(ii), §25.12 and Ch.25

When (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).