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polylog(n,x) represents the polylogarithm
function Lin(x) of
index n at
the point x.
For a complex number x of modulus |x| < 1, the polylogarithm function of index n is defined as
This function is extended to the whole complex plane by analytic
continuation. Do not confuse the polylogarithms Lin with
the integral logarithm function
Li which is displayed using the same
symbol (without an index).
n is an integer and
floating-point number, then a floating-point result is computed.
n is an integer less or equal to 1,
then an explicit expression is returned for any input parameter
n is an integer larger than 1 or
n is a symbolic expression, then an unevaluated
polylog is returned, unless
a floating-point number. If
n is a numerical value,
but not an integer, then an error occurs.
Some special values for n =
2 are implemented (cf.
dilog). The values Lin(0)
= 0 and are
implemented for any n.
any n ≠ 1.
Lin(x) has a singularity at the point x = 1 for indices n ≤ 1. For indices n ≥ 1, the point x = 1 is a branch point. The branch cut is the real interval . A jump occurs when crossing this cut. Cf. Example 2.
polylog(2,x) coincides with
When called with a floating-point argument
the function is sensitive to the environment variable
determines the numerical working precision.
Explicit results are returned for integer indices n ≤ 1:
polylog(-5, x), polylog(-1, x), polylog(0, x), polylog(1, x)
An unevaluated call is returned if the index is an integer n > 1 or a symbolic expression:
polylog(2, x), polylog(n^2 + 1, 2), polylog(n + 1, 2.0)
Floating point values are computed for integer indices n and floating-point arguments x:
polylog(-5, -1.2), polylog(10, 100.0 + 3.2*I)
Some special symbolic values are implemented:
polylog(4, 1), polylog(5, -1), polylog(2, I)
assume(n <> 1): polylog(n, -1)
unassume(n): polylog(n, -1)
For indices n ≥
1, the real interval is
a branch cut. The values returned by
when crossing this cut:
polylog(3, 1.2 + I/10^1000) - polylog(3, 1.2 - I/10^1000)
diff(polylog(n, x), x), float(polylog(4, 3 + I))
series(polylog(4, sin(x)), x = 0)
L. Lewin, “Polylogarithms and Related Functions”, North Holland (1981). L. Lewin (ed.), “Structural Properties of Polylogarithms”, Mathematical Surveys and Monographs Vol. 37, American Mathematical Society, Providence (1991).
The polylogarithms are characterized by in conjunction with Lin(0) = 0 and Li1(x) = - ln(1 - x). Lin(x) is a rational function in x for n ≤ 0.
Lin has a branch cut along the real interval for indices n ≥ 1. The value at a point x on the cut coincides with the limit “from below”: