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  "body": "\n## Definition\n\nLet $W_k$, for $k \\in \\Z$, be the branches of the [[Lambert W function|FW2JKAN14B]] over $\\C$.\nFor $z \\in \\C$, let $K(z) = \\ceil{\\frac{\\operatorname{Im} z - \\pi}{2\\pi}}$ be the unwinding number.\nThe *Wright omega function* is\n$$\\omega(z) = W_{K(z)}(e^z).$$\n\n## Remarks\n\nExample: for real $x$, $\\operatorname{Im} x = 0$ and $K(x) = \\ceil{-\\tfrac12} = 0$, so $\\omega(x) = W_0(e^x)$.\nIt is the unique real $y \u003e 0$ with $y + \\ln y = x$, since $y e^y = e^x$ is the same equation and $y \\mapsto y + \\ln y$ is a strictly increasing bijection from $(0, \\infty)$ onto $\\R$.\nFor instance $\\omega(1) = 1$, and $\\omega(0) = W_0(1) \\approx 0.5671$, the omega constant.\n\nNon-example: $W_0(e^z)$ is not $\\omega(z)$ in general.\nFor $z = 4\\pi i$, $K(z) = \\ceil{3/2} = 2$, so $\\omega(4\\pi i) = W_2(1)$, which is not real, while $W_0(e^{4\\pi i}) = W_0(1)$ is real.\n\nThe function $\\omega$ is single-valued.\nWith the principal branch of the logarithm, $\\omega(z) + \\ln \\omega(z) = z$ holds for every $z$ except possibly on the two half-lines $\\set{t \\pm i\\pi : t \\le -1}$.\n",
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