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.