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The function

\mathop{\rm Li}\nolimits _n(z)\equiv \sum_{k=1}^\infty {z^k\over k^n},
\end{displaymath} (1)

Also known as Jonquière's Function. (Note that the Notation $\mathop{\rm Li}\nolimits (z)$ is also used for the Logarithmic Integral.) The polylogarithm arises in Feynman Diagram integrals, and the special case $n=2$ is called the Dilogarithm. The polylogarithm of Negative Integer order arises in sums of the form
\sum_{k=1}^\infty k^n r^k=\mathop{\rm Li}\nolimits _{-n}(r) ...
...}} \sum_{i=1}^n \left\langle{n\atop i}\right\rangle{} r^{n-i},
\end{displaymath} (2)

where $\left\langle{n\atop i}\right\rangle{}$ is an Eulerian Number.

The polylogarithm satisfies the fundamental identities

-\ln(1-2^{-n})=\mathop{\rm Li}\nolimits _1(2^{-n})
\end{displaymath} (3)

\mathop{\rm Li}\nolimits _s(-1)=-(1-2^{1-s})\zeta(s),
\end{displaymath} (4)

where $\zeta(s)$ is the Riemann Zeta Function. The derivative is therefore given by
{d\over ds} \mathop{\rm Li}\nolimits _s(-1)=-2^{1-s}\zeta(s)\ln 2-(1-2^{1-s})\zeta'(s),
\end{displaymath} (5)

or in the special case $s=0$, by

\left[{{d\over ds} \mathop{\rm Li}\nolimits _s(-1)}\right]_{...
...tyle{1\over 2}}\ln(2\pi)=\ln\left({\sqrt{2\over\pi}\,}\right).
\end{displaymath} (6)

This latter fact provides a remarkable proof of the Wallis Formula.

The polylogarithm identities lead to remarkable expressions. Ramanujan gave the polylogarithm identities

\mathop{\rm Li}\nolimits _2({\textstyle{1\over 3}})-{\textst...
...)={\textstyle{1\over 18}}\pi^2-{\textstyle{1\over 6}}(\ln 3)^2
\end{displaymath} (7)

\mathop{\rm Li}\nolimits _2(-{\textstyle{1\over 2}})+{\texts...
...tstyle{1\over 2}}(\ln 2)^2-{\textstyle{1\over 3}}(\ln 3)^2\eno
\end{displaymath} (8)

\mathop{\rm Li}\nolimits _2({\textstyle{1\over 4}})+{\textst...
...}}\pi^2+2\ln 2\ln 3-2(\ln 2)^2-{\textstyle{2\over 3}}(\ln 3)^2
\end{displaymath} (9)

\mathop{\rm Li}\nolimits _2(-{\textstyle{1\over 3}})-{\texts...
...=-{\textstyle{1\over 18}}\pi^2+{\textstyle{1\over 6}}(\ln 3)^2
\end{displaymath} (10)

\mathop{\rm Li}\nolimits _2(-{\textstyle{1\over 8}})+\mathop...
...ver 9}})=-{\textstyle{1\over 2}}(\ln {\textstyle{9\over 8}})^2
\end{displaymath} (11)

(Berndt 1994), and Bailey et al. show that
\pi^2=36\mathop{\rm Li}\nolimits _2({\textstyle{1\over 2}})-...
...ver 8}})+6\mathop{\rm Li}\nolimits _2({\textstyle{1\over 64}})
\end{displaymath} (12)

12\mathop{\rm Li}\nolimits _2({\textstyle{1\over 2}})=\pi^2-6(\ln 2)^2
\end{displaymath} (13)

{\textstyle{35\over 2}}\zeta(3)-\pi^2\ln 2=36\mathop{\rm Li}...
...over 8}})+\mathop{\rm Li}\nolimits _3({\textstyle{1\over 64}})
\end{displaymath} (14)

2(\ln 2)^3-7\zeta(3)=-24\mathop{\rm Li}\nolimits _3({\textst...
...over 8}})-\mathop{\rm Li}\nolimits _3({\textstyle{1\over 64}})
\end{displaymath} (15)

10(\ln 2)^3-2\pi^2\ln 2=-48\mathop{\rm Li}\nolimits _3({\tex... 8}})-3\mathop{\rm Li}\nolimits _3({\textstyle{1\over 64}}),
\end{displaymath} (16)


${\mathop{\rm Li}\nolimits _m({\textstyle{1\over 64}})\over 6^{m-1}}-{\mathop{\r...
...thop{\rm Li}\nolimits _m({\textstyle{1\over 2}})\over 9}-{5(-\ln 2)^m\over 9m!}$
$ +{\pi^2(-\ln 2)^{m-2}\over 54(m-2)!}-{\pi^4(-\ln 2)^{m-4}\over 486(m-4)!}-{403\zeta(5)(-\ln 2)^{m-5}\over 1296(m-5)!} =0.\quad$ (17)

No general Algorithm is know for the integration of polylogarithms of functions.

See also Dilogarithm, Eulerian Number, Legendre's Chi-Function, Logarithmic Integral, Nielsen-Ramanujan Constants


Bailey, D.; Borwein, P.; and Plouffe, S. ``On the Rapid Computation of Various Polylogarithmic Constants.''

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 323-326, 1994.

Lewin, L. Polylogarithms and Associated Functions. New York: North-Holland, 1981.

Lewin, L. (Ed.). Structural Properties of Polylogarithms. Providence, RI: Amer. Math. Soc., 1991.

Nielsen, N. Der Euler'sche Dilogarithms. Leipzig, Germany: Halle, 1909.

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© 1996-9 Eric W. Weisstein