Dilogarithm¶

gsl_sf_dilog(x)

This routine computes the dilogarithm for a real argument. In Lewin’s notation this is $$\operatorname{Li}_2(x)$$, the real part of the dilogarithm of a real $$x$$. It is defined by the integral representation $$\operatorname{Li}_2(x) = -\operatorname{Re}\int_0^x (\log(1-s) / s) ds$$. Note that $$\operatorname{Im}(\operatorname{Li}_2(x)) = 0$$ for $$x <= 1$$, and $$-\pi\log(x)$$ for $$x > 1$$.

Note that Abramowitz & Stegun refer to the Spence integral $$S(x)=\operatorname{Li}_2(1-x)$$ as the dilogarithm rather than $$\operatorname{Li}_2(x)$$.