nLab
dilogarithm

Euler’s dilogarithm is a complex valued function ${\mathrm{Li}}_2$ given by

$${\mathrm{Li}}_2(x)=\sum _{n=1}^{\mathrm{\infty }}\frac{x^n}{n^2}$$Li_2(x) = \sum_{n=1}^\infty \frac{x^n}{n^2}

The dilogarithm is a special case of the polylogarithm ${\mathrm{Li}}_n$. The Bloch–Wigner dilogarithm is defined by

$$D(z):=\mathrm{Im}({\mathrm{Li}}_2(z))+\arg (1-z)\log \mid z\mid$$D(z) := Im(Li_2(z)) + arg(1-z) log |z|

The dilogarithm has remarkable relations to many areas of mathematics and mathematical physics including scissors congruence, Reidemeister’s torsion, regulators in higher algebraic K-theory, the Bloch group, CFT, Liouville's gravity?, hyperbolic geometry? and cluster transformations.

See also the references at mathworld and P.P. Cook’s blog and the related entry quantum dilogarithm.

• Don Zagier, The dilogarithm function, in Frontiers in Number Theory, Physics, and Geometry II, pp. 3–35 (2007) MR2290758 doi:10.1007/978-3-540-30308-4; Polylogarithms, Dedekind zeta functions, and the algebraic K-theory of fields, in Arithmetic Algebraic Geometry, Progr. Math. 89, Birkhäuser, Boston, 1990, 391–430 MR1085270

• A. N. Kirillov, Dilogarithm identities, Progr. Theoret. Phys. Suppl. 118 (1995), 61–142 hep-th/9408113 MR1356515 doi; Identities for the Rogers dilogarithm function connected with simple Lie algebras, J. Soviet Math. 47 (1989), 2450–2458.

• B. Richmond, G. Szekeres, Some formulas related to dilogarithm, the zeta function and the Andrews-Gordon identities, J. Aust. Math. Soc. 31 (1981), 362–373 MR633444 doi

• S. Bloch, Applications of the dilogarithm function in algebraic K-theory and algebraic geometry, in: Proc. Int. Symp. on Alg. Geometry, Kinokuniya, Tokyo 1978.

• Tomoki Nakanishi, Dilogarithm identities for conformal field theories and cluster algebras: Simply laced case, Nagoya Math. J. 202 (2011), 23-43, MR2804544 doi

• W. Nahm, Conformal field theory and torsion elements of the Bloch group, in Frontiers in Number Theory, Physics, and Geometry, II, Springer, Berlin, 2007, 67–132 MR2290759 doi

category: analysis