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
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":
.
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).