﻿ polylog | Enterprise Architect User Guide

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# polylog

Polylogarithms.

SYNOPSIS:

double x, y, polylog();
int n;
y = polylog(n, x);

The polylogarithm of order n is defined by the series:

inf   k
-   x
Li (x)  =    >   ---  .
n          -     n
k=1   k

For x = 1,

inf
-    1
Li (1)  =    >   ---   =  Riemann zeta function (n).
n          -     n
k=1   k

When n = 2, the function is the dilogarithm, related to Spence's integral:

x                      1-x
-                        -
| |  -ln(1-t)            | |  ln t
Li (x)  =    |    -------- dt    =    |    ------ dt    =   spence(1-x) .
2        | |       t              | |    1 - t
-                        -
0                        1
References:

Lewin, L., Polylogarithms and Associated Functions,
North Holland, 1981.

Lewin, L., ed., Structural Properties of Polylogarithms,
American Mathematical Society, 1991.

ACCURACY:
Relative error:
arithmetic   domain   n   # trials      peak         rms
IEEE      0, 1     2     50000      6.2e-16     8.0e-17
IEEE      0, 1     3    100000      2.5e-16     6.6e-17
IEEE      0, 1     4     30000      1.7e-16     4.9e-17
IEEE      0, 1     5     30000      5.1e-16     7.8e-17