Dilogarithm

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:

${\displaystyle \operatorname {Li} _{2}(z)=-\int _{0}^{z}{\ln |1-\zeta | \over \zeta },円\mathrm {d} \zeta =\sum _{k=1}^{\infty }{z^{k} \over k^{2}};}${\displaystyle \operatorname {Li} _{2}(z)=-\int _{0}^{z}{\ln |1-\zeta | \over \zeta },円\mathrm {d} \zeta =\sum _{k=1}^{\infty }{z^{k} \over k^{2}};}

• the dilogarithm with its argument multiplied by −1:

${\displaystyle F(z)=\operatorname {Li} _{2}(-z)=-\int _{0}^{z}{\ln(1+\zeta ) \over \zeta },円\mathrm {d} \zeta =\sum _{k=1}^{\infty }{(-z)^{k} \over k^{2}}.}${\displaystyle F(z)=\operatorname {Li} _{2}(-z)=-\int _{0}^{z}{\ln(1+\zeta ) \over \zeta },円\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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