The polylogarithm of index a at the point z is defined by

if $\left|z\right|<1$ and by analytic continuation otherwise.  The index a can be any complex number.  If $\mathrm{\Re }\left(a\right)\le 1$, the point $z=1$ is a singularity.

For all indices a, the point $z=1$ is a branch point for all branches, and in Maple, the branch cut is taken to be the interval ($1,\mathrm{\infty }$).  For the branches other than the principal branch (which is given on the unit disk by the series above, and hence is analytic at 0), the point $z=0$ is also a branch point, and the branch cut is taken to be the negative real axis.  The formula for a particular branch can be determined with the following rules:

Each time the branch cut ($1,\mathrm{\infty }$) is crossed in the counterclockwise direction, subtract $\frac{2\mathrm{I}\mathrm{\pi }{\ln \left(z\right)}^{a-1}}{\mathrm{\Gamma }\left(a\right)}$. Add this quantity if the branch cut is crossed in the clockwise direction.

Each time the branch cut ($-\mathrm{\infty },0$) is crossed in the counterclockwise direction, add $2\mathrm{I}\mathrm{\pi }$ to each $\ln \left(z\right)$ term in the current formula.  Subtract this quantity if the branch cut is crossed in the clockwise direction.

For example, if one traverses a path which starts at $z=\frac{1}{2}$, goes clockwise around $z=1$, then counterclockwise around $z=0$, then clockwise around $z=1$ again to return at $z=\frac{1}{2}$, the formula for the branch of polylog thus obtained would be

where polylog(a, z) indicates the principal branch and $\ln \left(z\right)$ means the principal branch of the logarithm.

Maple only evaluates the principal branch.

Maple's dilog function is related to polylog by the relation $\mathrm{dilog}\left(z\right)=\mathrm{polylog}\left(2\&comma;1-z\right)$.

$\mathrm{polylog}\left(a\&comma;0\right)$

$0$

$\mathrm{polylog}\left(2\&comma;1\right)$

$\frac{{\mathrm{\pi }}^2}{6}$

$\mathrm{polylog}\left(3\&comma;1\right)$

$\mathrm{\zeta }\left(3\right)$

$\mathrm{polylog}\left(2\&comma;\mathrm{I}\right)$

$-\frac{{\mathrm{\pi }}^2}{48}+\mathrm{I}\mathrm{Catalan}$

$\mathrm{diff}\left(\mathrm{polylog}\left(a\&comma;x\right)\&comma;x\right)$

$\frac{\mathrm{polylog}\left(a-1\&comma;x\right)}{x}$

$\mathrm{combine}\left(\mathrm{polylog}\left(a\&comma;x\right)+\mathrm{polylog}\left(a\&comma;-x\right)\&comma;\mathrm{polylog}\right)$

$2^{1-a}\mathrm{polylog}\left(a\&comma;x^2\right)$

$\mathrm{polylog}\left(4\&comma;x\right)+\mathrm{polylog}\left(4\&comma;\frac{1}{x}\right)$

$\mathrm{polylog}\left(4\&comma;x\right)+\mathrm{polylog}\left(4\&comma;\frac{1}{x}\right)$

$\mathrm{combine}\left(\&comma;\mathrm{polylog}\right)\mathbf{assuming}1<x$

$-\frac{{\ln \left(-x\right)}^2{\mathrm{\pi }}^2}{12}-\frac{7{\mathrm{\pi }}^4}{360}-\frac{{\ln \left(-x\right)}^4}{24}$

$\mathrm{combine}\left(\&comma;\mathrm{polylog}\right)\mathbf{assuming}x::\mathrm{RealRange}\left(-1\&comma;1\right)$

$-\frac{{\ln \left(-\frac{1}{x}\right)}^2{\mathrm{\pi }}^2}{12}-\frac{7{\mathrm{\pi }}^4}{360}-\frac{{\ln \left(-\frac{1}{x}\right)}^4}{24}$

$\mathrm{polylog}\left(a\&comma;z^5\right)$

$\mathrm{polylog}\left(a\&comma;z^5\right)$

$\mathrm{expand}\left(\right)$

$\frac{5^a\mathrm{polylog}\left(a\&comma;{\left(\mathrm{-1}\right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}z\right)}{5}+\frac{5^a\mathrm{polylog}\left(a\&comma;{\left(\mathrm{-1}\right)}^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}z\right)}{5}+\frac{5^a\mathrm{polylog}\left(a\&comma;-{\left(\mathrm{-1}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}z\right)}{5}+\frac{5^a\mathrm{polylog}\left(a\&comma;-{\left(\mathrm{-1}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$5$}\right.}z\right)}{5}+\frac{5^a\mathrm{polylog}\left(a\&comma;z\right)}{5}$

$x≔'x'\&colon;$

$\mathrm{polylog}\left(1\&comma;x\right)$

$-\ln \left(1-x\right)$

$\mathrm{polylog}\left(2\&comma;\frac{1}{3}\right)$

$\mathrm{polylog}\left(2\&comma;\frac{1}{3}\right)$

$\mathrm{evalf}\left(\right)$

$0.3662132299$

$\mathrm{polylog}\left(-3.7+2.2\mathrm{I}\&comma;1.5+2.7\mathrm{I}\right)$

$\mathrm{-188.9091729}+104.0046999\mathrm{I}$

Lewin, L. Polylogarithms and Associated Functions. Amsterdam: North Holland, 1981.