Dilogarithm

Summary

The dilogarithm is $\operatorname{Li}_2(z) = \sum_{k \ge 1} \frac{z^k}{k^2}$, the polylogarithm of order $2$. It equals $-\int_0^z \frac{\ln(1 - u)}{u}\, du$ and continues analytically to $\C \setminus [1, \infty)$. It satisfies remarkable functional equations, such as Euler's reflection formula and the five-term relation. It appears in hyperbolic geometry, through the Bloch–Wigner function, and in algebraic K-theory.