The dilogarithm
is a special case of the polylogarithm for . Note that the notation is unfortunately similar to that for the logarithmic
integral .
There are also two different commonly encountered normalizations for the function, both denoted , and one of which is known as the Rogers
L-function.

There are several remarkable identities involving the dilogarithm function. Ramanujan gave the identities

(25)

(26)

(27)

(28)

(29)

(Berndt 1994, Gordon and McIntosh 1997) in addition to the identity for , and Bailey et al. (1997) showed that

(30)

Lewin (1991) gives 67 dilogarithm identities (known as "ladders"), and Bailey and Broadhurst (1999, 2001) found the amazing additional dilogarithm identity

Abramowitz, M. and Stegun, I. A. (Eds.). "Dilogarithm." §27.7 in Handbook
of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.
New York: Dover, pp. 1004-1005, 1972.Andrews, G. E.; Askey,
R.; and Roy, R. Special
Functions. Cambridge, England: Cambridge University Press, 1999.Bailey,
D. H.; Borwein, P. B.; and Plouffe, S. "On the Rapid Computation of
Various Polylogarithmic Constants." Math. Comput.66, 903-913,
1997.Bailey, D. H. and Broadhurst, D. J. "A Seventeenth-Order
Polylogarithm Ladder." 20 Jun 1999. http://arxiv.org/abs/math.CA/9906134.Bailey,
D. H. and Broadhurst, D. J. "Parallel Integer Relation Detection:
Techniques and Applications." Math. Comput.70, 1719-1736, 2001.Berndt,
B. C. Ramanujan's
Notebooks, Part IV. New York: Springer-Verlag, pp. 323-326, 1994.Borwein,
J. M.; Bradley, D. M.; Broadhurst, D. J.; and Lisonek, P. "Special
Values of Multidimensional Polylogarithms." Trans. Amer. Math. Soc.353,
907-941, 2001.Bytsko, A. G. "Fermionic Representations for
Characters of ,
, and Minimal Models and Related Dilogarithm and Rogers-Ramanujan-Type
Identities." J. Phys. A: Math. Gen.32, 8045-8058, 1999.Campbell,
J. M. "Some Nontrivial Two-Term Dilogarithm Identities." Irish
Math. Soc. Bull., No. 88, 31-37, 2021.Erdélyi, A.; Magnus,
W.; Oberhettinger, F.; and Tricomi, F. G. "Euler's Dilogarithm." §1.11.1
in Higher
Transcendental Functions, Vol. 1. New York: Krieger, pp. 31-32,
1981.Gordon, B. and McIntosh, R. J. "Algebraic Dilogarithm
Identities." Ramanujan J.1, 431-448, 1997.Kirillov,
A. N. "Dilogarithm Identities." Progr. Theor. Phys. Suppl.118,
61-142, 1995.Lewin, L. Dilogarithms
and Associated Functions. London: Macdonald, 1958.Lewin, L.
Polylogarithms
and Associated Functions. New York: North-Holland, 1981.Lewin,
L. "The Dilogarithm in Algebraic Fields." J. Austral. Math. Soc. Ser.
A33, 302-330, 1982.Lewin, L. (Ed.). Structural
Properties of Polylogarithms. Providence, RI: Amer. Math. Soc., 1991.Lima,
F. M. S. "New Definite Integrals and a Two-Term Dilogarithm Identity."
Indag. Math.23, 1-9, 2012.Nielsen, N. "Der Eulersche
Dilogarithmus und seine Verallgemeinerungen." Nova Acta Leopoldina, Abh.
der Kaiserlich Leopoldinisch-Carolinischen Deutschen Akad. der Naturforsch.90,
121-212, 1909.Watson, G. N. "A Note on Spence's Logarithmic
Transcendent." Quart. J. Math. Oxford Ser.8, 39-42, 1937.