## Polylogarithm

The function

 (1)

Also known as Jonquière's Function. (Note that the Notation is also used for the Logarithmic Integral.) The polylogarithm arises in Feynman Diagram integrals, and the special case is called the Dilogarithm. The polylogarithm of Negative Integer order arises in sums of the form
 (2)

where is an Eulerian Number.

The polylogarithm satisfies the fundamental identities

 (3)

 (4)

where is the Riemann Zeta Function. The derivative is therefore given by
 (5)

or in the special case , by

 (6)

This latter fact provides a remarkable proof of the Wallis Formula.

The polylogarithm identities lead to remarkable expressions. Ramanujan gave the polylogarithm identities

 (7)

 (8)

 (9)

 (10)

 (11)

(Berndt 1994), and Bailey et al. show that
 (12)

 (13)

 (14)

 (15)

 (16)

and

 (17)

No general Algorithm is know for the integration of polylogarithms of functions.

References

Bailey, D.; Borwein, P.; and Plouffe, S. On the Rapid Computation of Various Polylogarithmic Constants.'' http://www.cecm.sfu.ca/~pborwein/PAPERS/P123.ps.

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 323-326, 1994.

Lewin, L. Polylogarithms and Associated Functions. New York: North-Holland, 1981.

Lewin, L. (Ed.). Structural Properties of Polylogarithms. Providence, RI: Amer. Math. Soc., 1991.

Nielsen, N. Der Euler'sche Dilogarithms. Leipzig, Germany: Halle, 1909.