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polylog -- the polylogarithm function

Introduction

polylog(n,x) represents the polylogarithm function Li(n, x) of index n at the point x.

Call(s)

polylog(n, x)

Parameters

n - an arithmetical expression representing an integer
x - an arithmetical expression

Returns

an arithmetical expression.

Overloadable:

x

Side Effects

When called with a floating point argument x, the function is sensitive to the environment variable DIGITS which determines the numerical working precision.

Related Functions

dilog, ln

Details

Example 1

Explicit results are returned for integer indices n<=1:

>> polylog(-5, x), polylog(-1, x), polylog(0, x), polylog(1, x)
                2       3       4    5
        x + 26 x  + 66 x  + 26 x  + x      x        x
        ------------------------------, --------, -----, -ln(1 - x)
                          6                    2  1 - x
                   (1 - x)              (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)
                                  2
          polylog(2, x), polylog(n  + 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)
                -0.2326930882, 104.9131863 + 11.44600047 I

An error occurs if n is a numerical value, but not an integer:

>> polylog(5/2, x)
      Error: first argument must be an integer [polylog]

Some special symbolic values are implemented:

>> polylog(4, 1), polylog(5, -1), polylog(2, I)
                      4                              2
                    PI     15 zeta(5)              PI
                    ---, - ----------, I CATALAN - ---
                    90         16                  48
>> assume(n <> 1): polylog(n, -1)
                                          1 - n
                          - zeta(n) (1 - 2     )
>> unassume(n): polylog(n, -1)
                              polylog(n, -1)

Example 2

For indices n>=1, the real interval [1,infinity) 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)
                              0.1044301529 I

Example 3

The functions diff, float, limit, and series handle expressions involving polylog:

>> diff(polylog(n, x), x), float(polylog(4, 3 + I))
              polylog(n - 1, x)
              -----------------, 3.177636803 + 1.859135861 I
                      x
>> series(polylog(4, sin(x)), x = 0)
                      2       3       4         5
                     x    25 x    13 x    1523 x       6
                 x + -- - ----- - ----- + ------- + O(x )
                     16    162     768    405000

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