Mathematical Addendum to Physics Friday 92

The polylogarithms are the functions
;
the sum converges in the complex plane over the open unit disk (and can be extended to the whole plane via analytic continuation).
The specific cases Li2(z) and Li3(z) are known respectively as the dilogarithm and trilogarithm. Note that from the Mercator series , we see that . Similarly, .

Taking the derivative, we see:
.
Combining this with the above, we see when the parameter s is a negative integer, the polylogarithm is a rational function:

,
and so on. Similarly,

,
and so on.

Examining the series, we see right away that and that . Further, for s>1

,
where is the Riemann zeta function and is the Dirichlet eta function.
Combining these with the derivative relation,

.

For s≥0, Lis(z) is monotonically increasing (with z<1 for convergence of the series).

With regards to Fermi-Dirac and Bose-Einstein statistics, I show here that
and
.
[More generally, we have a pair of integral definitions:
.]

Using all the above, we see that our functions and used in the Bose gas analysis have the following properties:

  • and for small ξ.
  • and .
  • and , which diverges (infinite slope).
  • , with equality only at ξ=0
  • Expanding to first order in ξ, .
  • and are both monotonically increasing functions for |ξ|≤1

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