## 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