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 Li_{2}(*z*) and Li_{3}(*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, Li_{s}(*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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Tags: Dirichlet Eta Function, Math, Mercator Series, Polylogarithm, Riemann Zeta Function

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