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

if  and by analytic continuation otherwise.  The index a can be any complex number.  If , the point  is a singularity.

For all indices a, the point  is a branch point for all branches, and in Maple, the branch cut is taken to be the interval ().  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  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 () is crossed in the counterclockwise direction, subtract . Add this quantity if the branch cut is crossed in the clockwise direction.

Each time the branch cut () is crossed in the counterclockwise direction, add  to each  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 , goes clockwise around , then counterclockwise around , then clockwise around  again to return at , the formula for the branch of polylog thus obtained would be

where polylog(a, z) indicates the principal branch and  means the principal branch of the logarithm.

Maple only evaluates the principal branch.

Maple's dilog function is related to polylog by the relation .

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