Five-term relation for the dilogarithm

Remarks

For the real form, fix $y$ and differentiate both sides in $x$. Using $\frac{d}{dx}\operatorname{Li}_2(x) = -\frac{\log(1-x)}{x}$, the derivatives of the two sides agree, and both sides tend to $L(y)$ as $x \to 0^+$, since $L(x) \to 0$. The complex form follows in the same way from the differential of $D$, which is an explicit real-analytic $1$-form on $\C \setminus \set{0, 1}$.

Example: with $y = x$, the third argument is $\frac{x(1-x)}{1-x^2} = \frac{x}{1+x}$, so the relation gives $$2L(x) = L(x^2) + 2L\left(\frac{x}{1+x}\right), \qquad 0 < x < 1.$$

The five arguments in the complex form are cross-ratios of five points on the Riemann sphere taken four at a time. Geometrically, $D(z)$ is the hyperbolic volume of the ideal tetrahedron with vertices $0, 1, \infty, z$. The relation says that two ways of cutting an ideal polyhedron with five vertices into ideal tetrahedra give the same volume.

The correction term $\frac12 \log x \log(1-x)$ is needed: the plain dilogarithm $\operatorname{Li}_2$ satisfies a five-term relation only with extra products of logarithms.

Going deeper

For a student

The relation is usually stated for the Rogers dilogarithm on $(0, 1)$ or for the Bloch–Wigner function on $\C \setminus \set{0, 1}$, because these versions have no elementary correction terms. It is the defining relation of the Bloch group in algebraic K-theory, and it explains why sums of values $D(z_i)$ compute volumes of hyperbolic $3$-manifolds. Its quantum analogue is the pentagon identity of the quantum dilogarithm.