Bloch–Wigner function

Remarks

The function is real-analytic on $\C \setminus \set{0, 1}$ and continuous on $\C$. Across $(1, \infty)$ the term $\operatorname{Im} \operatorname{Li}_2$ jumps by $2\pi \log x$ at the point $x$, and the term $\arg(1 - z) \log \abs{z}$ jumps by $-2\pi \log x$, so the extension is well defined.

Example: for $z = e^{i\theta}$ one has $\log \abs{z} = 0$, so $D(e^{i\theta}) = \operatorname{Im} \operatorname{Li}_2(e^{i\theta}) = \operatorname{Cl}_2(\theta)$, the Clausen function. In particular $D(e^{i\pi/3}) = \operatorname{Cl}_2(\pi/3) \approx 1.01494$.

Non-example: the function $\operatorname{Im} \operatorname{Li}_2(z)$ alone does not extend continuously across $(1, \infty)$, because of the jump $2\pi \log x$ described above.

Symmetries: $D(\bar z) = -D(z)$, and $$D(z) = D\left(1 - \tfrac1z\right) = D\left(\tfrac{1}{1-z}\right) = -D\left(\tfrac1z\right) = -D(1 - z) = -D\left(\tfrac{z}{z-1}\right).$$ For $\operatorname{Im} z > 0$, $D(z)$ is the hyperbolic volume of the ideal tetrahedron with vertices $0, 1, \infty, z$.