PolyLog

The polylogarithm , the slice of the Lerch transcendent.

PolyLog(s, z)the polylogarithm at integer order .

PolyLog(s, z)extends the same call to non-integer and complex order , for .

Domain: Special functions

Details

in terms of the Lerch transcendent — see LerchPhi, which is how the non-integer orders are computed.
: the ordinary zeta lives on the edge. See Zeta.
Low orders are elementary: , , , and every negative integer order is a rational function.
The dilogarithm has the special values , , and .
Convergence: the series converges for , and on for . Outside the disk the function continues analytically, with a branch cut along — so is complex.
Integer orders (including the continuation past ) are compute-engine's own. `@enumeratio/analytic` adds non-integer and complex by summing the Lerch series, which reaches only; outside that disk a non-integer order is left unevaluated rather than guessed.
As with the other numeric heads, an exact argument stays symbolic under plain evaluation — pair with N(), or pass an inexact argument.

Examples