polylog
Polylogarithm function
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polylog(n
, x
)
polylog(n,x)
represents the polylogarithm
function Li_{n}(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 Li_{n} with
the integral logarithm function Li
which is displayed using the same
symbol (without an index).
If n
is an integer and x
a
floatingpoint number, then a floatingpoint result is computed.
If n
is an integer less or equal to 1,
then an explicit expression is returned for any input parameter x
.
If n
is an integer larger than 1 or
if n
is a symbolic expression, then an unevaluated
call of polylog
is returned, unless x
is
a floatingpoint 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 Li_{n}(0)
= 0 and are
implemented for any n.
Furthermore, for
any n ≠ 1.
Li_{n}(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.
Mathematically, polylog(2,x)
coincides with dilog
(1x)
.
When called with a floatingpoint argument x
,
the function is sensitive to the environment variable DIGITS
which
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 floatingpoint 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 polylog
jump
when crossing this cut:
polylog(3, 1.2 + I/10^1000)  polylog(3, 1.2  I/10^1000)
The functions diff
, float
, limit
, and series
handle expressions
involving polylog
:
diff(polylog(n, x), x), float(polylog(4, 3 + I))
series(polylog(4, sin(x)), x = 0)

An arithmetical expression representing an integer 

Arithmetical expression.
x
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 Li_{n}(0) = 0 and Li_{1}(x) =  ln(1  x). Li_{n}(x) is a rational function in x for n ≤ 0.
Li_{n} 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”:
.