# dilogarithm function

 $\displaystyle\mbox{Li}_{2}(x)\;=:\;\sum_{n=1}^{\infty}\frac{x^{n}}{n^{2}},$ (1)

studied already by Leibniz, is a special case of the polylogarithm function

 $\mbox{Li}_{s}(x)\;=:\;\sum_{n=1}^{\infty}\frac{x^{n}}{n^{s}}.$

The radius of convergence of the series (1) is 1, whence the definition (1) is valid also in the unit disc of the complex plane.  For  $0\leq x\leq 1$,  the equation (1) is apparently equivalent to

 $\displaystyle\mbox{Li}_{2}(x)\;=:\;-\int_{0}^{x}\frac{\ln(1\!-\!t)}{t}\,dt,$ (2)

(cf. logarithm series of $\ln(1\!-\!x)$).  The analytic continuation of $\mbox{Li}_{2}$ for  $|z|\geq 1$  can be made by

 $\displaystyle\mbox{Li}_{2}(z)\;=:\;-\int_{0}^{z}\frac{\log(1\!-\!t)}{t}\,dt.$ (3)

Thus $\mbox{Li}_{2}(z)$ is a multivalued analytic function of $z$.  Its is single-valued and is got by taking the principal branch of the complex logarithm; then

 $z\;\in\;\mathbb{C}\!\smallsetminus\![1,\,\infty[,\quad 0<\arg(z\!-\!1)<2\pi.$

For real values of $x$ we have

 $\displaystyle\mbox{Im}(\mbox{Li}_{2}(x))\;=\;\begin{cases}\;0\qquad\mbox{for}% \;\;x\leq 1,\\ -\pi\ln{x}\;\;\mbox{for}\;\;x>1.\end{cases}$

According to (2), the derivative of the dilogarithm is

 $\mbox{Li}^{\prime}_{2}(x)\;=\;\frac{-\ln(1\!-\!x)}{x}.$

In terms of the Bernoulli numbers, the dilogarithm function has a series expansion more rapidly converging than (1):

 $\displaystyle\mbox{Li}_{2}(x)\;=\;\sum_{n=1}^{\infty}B_{n-1}\frac{(-\ln(1\!-\!% x))^{n}}{n!}\qquad(|\ln(1\!-\!x)|<2\pi)$ (4)

Some functional equations and values

 $\mbox{Li}_{2}(z)+\mbox{Li}_{2}(-z)\;=\;\frac{1}{2}\mbox{Li}_{2}(z^{2}),$
 $\mbox{Li}_{2}(z)+\mbox{Li}_{2}\left(\frac{1}{z}\right)\;=\;-\frac{1}{2}(\log(-% z))^{2}-\frac{\pi^{2}}{6},$
 $\mbox{Li}_{2}(iz)-i\mbox{Li}_{2}(z)\;=\;\frac{1}{4}\mbox{Li}_{2}(-z^{2}),$
 $\mbox{Li}_{2}(1)\;=\;\frac{\pi^{2}}{6},\quad\mbox{Li}_{2}(2)\;=\;\frac{\pi^{2}% }{4}-i\pi\ln{2},\quad\mbox{Li}_{2}(i)\;=\;-\frac{\pi^{2}}{48}-i\,G$

Here, $G$ is Catalan’s constant.

## References

• 1 Anatol N. Kirillov: Dilogarithm identities (1994). Available http://arxiv.org/pdf/hep-th/9408113v2.pdfhere.
• 2 Leonard C. Maximon: The dilogarithm function for complex argument.  – Proc. R. Soc. Lond. A 459 (2003) 2807–2819.
Title dilogarithm function DilogarithmFunction 2013-03-22 19:34:58 2013-03-22 19:34:58 pahio (2872) pahio (2872) 14 pahio (2872) Definition msc 30D30 msc 33B15 Spence’s function ApplicationOfLogarithmSeries polylogarithm function