dilogarithm function

The dilogarithm function

Li2(x)=:n=1xnn2, (1)

studied already by Leibniz, is a special case of the polylogarithm function


The radius of convergenceMathworldPlanetmath of the series (1) is 1, whence the definition (1) is valid also in the unit disc of the complex plane.  For  0x1,  the equation (1) is apparently equivalent to

Li2(x)=:-0xln(1-t)tdt, (2)

(cf. logarithm series of ln(1-x)).  The analytic continuation of Li2 for  |z|1  can be made by

Li2(z)=:-0zlog(1-t)tdt. (3)

Thus Li2(z) is a multivalued analytic functionMathworldPlanetmath of z.  Its principal branchMathworldPlanetmath is single-valued and is got by taking the principal branch of the complex logarithm; then


For real values of x we have

Im(Li2(x))={ 0  forx1,-πlnxforx>1.

According to (2), the derivative of the dilogarithm is


In terms of the Bernoulli numbersDlmfDlmfPlanetmath, the dilogarithm function has a series expansion more rapidly converging than (1):

Li2(x)=n=1Bn-1(-ln(1-x))nn!  (|ln(1-x)|<2π) (4)

Some functional equations and values


Here, G is Catalan’s constant.


  • 1 Anatol N. Kirillov: Dilogarithm identities (1994). Available http://arxiv.org/pdf/hep-th/9408113v2.pdfhere.
  • 2 Leonard C. Maximon: The dilogarithm function for complex argument.  – Proc. R. Soc. Lond. A 459 (2003) 2807–2819.
Title dilogarithm function
Canonical name DilogarithmFunction
Date of creation 2013-03-22 19:34:58
Last modified on 2013-03-22 19:34:58
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 14
Author pahio (2872)
Entry type Definition
Classification msc 30D30
Classification msc 33B15
Synonym Spence’s functionMathworldPlanetmath
Related topic ApplicationOfLogarithmSeries
Defines polylogarithm function