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### 7.11.1 Real Argument

Function: double gsl_sf_dilog (double x)
Function: int gsl_sf_dilog_e (double x, gsl_sf_result * result)

These routines compute the dilogarithm for a real argument. In Lewin's notation this is Li_2(x), the real part of the dilogarithm of a real x. It is defined by the integral representation Li_2(x) = - \Re \int_0^x ds \log(1-s) / s. Note that \Im(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)=Li_2(1-x) as the dilogarithm rather than Li_2(x).