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  "body": "\n## Definition\n\nLet $\\operatorname{Li}_2$ be the [[dilogarithm|G0X0NWSDB7]], taken as its principal branch on $\\C \\setminus [1, \\infty)$.\nFor $z \\in \\C \\setminus [1, \\infty)$, let $\\arg(1 - z) \\in (-\\pi, \\pi)$ be the principal argument of $1 - z$, as for the [[complex logarithm|02QGGD5G87]].\nThe *Bloch–Wigner function* is\n$$D(z) = \\operatorname{Im}\\big(\\operatorname{Li}_2(z)\\big) + \\arg(1 - z) \\, \\log \\abs{z}, \\qquad z \\in \\C \\setminus \\big(\\set{0} \\cup [1, \\infty)\\big),$$\nextended to all of $\\C$ by continuity, with $D(0) = D(1) = 0$.\n\n## Remarks\n\nThe function is real-analytic on $\\C \\setminus \\set{0, 1}$ and continuous on $\\C$.\nAcross $(1, \\infty)$ 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.\n\nExample: 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|AVS6G54711]].\nIn particular $D(e^{i\\pi/3}) = \\operatorname{Cl}_2(\\pi/3) \\approx 1.01494$.\n\nNon-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.\n\nSymmetries: $D(\\bar z) = -D(z)$, and\n$$D(z) = D\\left(1 - \\tfrac1z\\right) = D\\left(\\tfrac{1}{1-z}\\right) = -D\\left(\\tfrac1z\\right) = -D(1 - z) = -D\\left(\\tfrac{z}{z-1}\\right).$$\nFor $\\operatorname{Im} z \u003e 0$, $D(z)$ is the hyperbolic volume of the ideal tetrahedron with vertices $0, 1, \\infty, z$.\n",
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