dilogarithm function

The dilogarithm function

${\text{Li}}_2(x)=:{\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{\displaystyle \frac{x^n}{n^2}},$ (1)

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

$${\text{Li}}_s(x)=:\sum _{n=1}^{\mathrm{\infty }}\frac{x^n}{n^s}.$$

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  $0\le x\le 1$,  the equation (1) is apparently equivalent to

${\text{Li}}_2(x)=:-{\displaystyle {\int }_0^x}{\displaystyle \frac{\ln (1-t)}{t}}dt,$ (2)

(cf. logarithm series of $\ln (1-x)$).  The analytic continuation of ${\text{Li}}_2$ for  $|z|\ge 1$  can be made by

${\text{Li}}_2(z)=:-{\displaystyle {\int }_0^z}{\displaystyle \frac{\log (1-t)}{t}}dt.$ (3)

Thus ${\text{Li}}_2(z)$ is a multivalued analytic function of $z$.  Its principal branch is single-valued and is got by taking the principal branch of the complex logarithm; then

For real values of $x$ we have

$\text{Im}({\text{Li}}_2(x))=\{\begin{array}{cc}\mathrm{\hspace{0.33em}0}\hspace{1em}\hspace{1em}\text{for}x\le 1,\hfill & \\ -\pi \ln x\text{for}x>1.\hfill & \end{array}$

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

$${\text{Li}}_2^{\prime }(x)=\frac{-\ln (1-x)}{x}.$$

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

(4)

Some functional equations and values

$${\text{Li}}_2(z)+{\text{Li}}_2(-z)=\frac{1}{2}{\text{Li}}_2(z^2),$$

$${\text{Li}}_2(z)+{\text{Li}}_2\left(\frac{1}{z}\right)=-\frac{1}{2}{(\log (-z))}^2-\frac{{\pi }^2}{6},$$

$${\text{Li}}_2(iz)-i{\text{Li}}_2(z)=\frac{1}{4}{\text{Li}}_2(-z^2),$$

$${\text{Li}}_2(1)=\frac{{\pi }^2}{6},{\text{Li}}_2(2)=\frac{{\pi }^2}{4}-i\pi \ln 2,{\text{Li}}_2(i)=-\frac{{\pi }^2}{48}-iG$$

Here, $G$ is Catalan’s constant.

Title	dilogarithm function
Canonical name	DilogarithmFunction
Date of creation	2013-03-22 19:34:58
Last modified on	2013-03-22 19:34:58
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	14
Author	pahio (2872)
Entry type	Definition
Classification	msc 30D30
Classification	msc 33B15
Synonym	Spence’s function
Related topic	ApplicationOfLogarithmSeries
Defines	polylogarithm function