---
tag: Q8K5MWEXB9
kind: concept
name: Bloch–Wigner function
msc: ["33B30", "11G55"]
sources:
  - url: https://en.wikipedia.org/wiki/Polylogarithm
---

## Summary

The Bloch–Wigner function $D$ is a real-valued function on the complex plane built from the imaginary part of the [[dilogarithm|G0X0NWSDB7]] and a logarithm correction.
Unlike the dilogarithm, it is single-valued and continuous on all of $\C$.
It satisfies a clean five-term relation and gives the volume of an ideal tetrahedron in hyperbolic space.
This makes it central in hyperbolic geometry and in the theory of the Bloch group.

## Explanations

### L1 human

The dilogarithm is a multivalued function: going around the point $1$ changes its value.
The Bloch–Wigner function takes the imaginary part and adds just the right correction so that these jumps cancel.
The result is a smooth real function on the plane with the points $0$ and $1$ removed, and it extends continuously to them with value $0$.
It is zero on the real line and changes sign under complex conjugation.
Its largest value, about $1.0149$, is reached at $z = e^{i\pi/3}$, and this number is the volume of the regular ideal tetrahedron in hyperbolic space.