Spence's function

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"Li2" redirects here. For the molecule with formula Li2, see dilithium.

In mathematics, Spence's function, or dilogarithm, denoted as Li2(z), is a particular case of the polylogarithm. Two related special functions are referred to as Spence's function:

  • the dilogarithm itself:
\operatorname{Li}_2(z) = -\int_0^z{\ln|1-\zeta| \over \zeta}\, \mathrm{d}\zeta = \sum_{k=1}^\infty {z^k \over k^2};
  • the dilogarithm with its argument multiplied by − 1:
F(z)=\operatorname{Li}_2(-z) = \int_0^z{\ln(1+\zeta) \over \zeta}\, \mathrm{d}\zeta = \sum_{k=1}^\infty {(-z)^k \over k^2}.

Here the series can only be used for | z | < 1, inside its radius of convergence.

A computer routine to compute the dilogarithm using approximation by truncated Chebyshev series is available, for example, as TMath::DiLog() in the open-source ROOT data analysis package.


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