general polylogarithm function
The polylogarithm of index a at the point z is defined by
if z<1 and by analytic continuation otherwise. The index a can be any complex number. If ℜ⁡a≤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,∞). 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,∞) is crossed in the counterclockwise direction, subtract 2⁢I⁢π⁢ln⁡za−1Γ⁡a. Add this quantity if the branch cut is crossed in the clockwise direction.
Each time the branch cut (−∞,0) is crossed in the counterclockwise direction, add 2⁢I⁢π to each ln⁡z 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=12, goes clockwise around z=1, then counterclockwise around z=0, then clockwise around z=1 again to return at z=12, the formula for the branch of polylog thus obtained would be
where polylog(a, z) indicates the principal branch and ln⁡z means the principal branch of the logarithm.
Maple only evaluates the principal branch.
Maple's dilog function is related to polylog by the relation dilog⁡z=polylog⁡2,1−z.
Lewin, L. Polylogarithms and Associated Functions. Amsterdam: North Holland, 1981.
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