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  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\C^n$ be a domain with $C^\\infty$ boundary.\nLet $\\rho$ be a [[smooth|0V33BNKZ9S]] defining function of $\\Omega$, as in the [[definition of the Levi form|5S30FAJNGM]].\nLet $D \\subseteq \\C$ be the unit disc.\nFor a function $g$ of class $C^\\infty$ near $0 \\in \\C$, let $\\nu(g)$ be its order of vanishing at $0$.\nThis is the smallest $k$ such that some partial derivative of $g$ of order $k$ is nonzero at $0$, and $\\nu(g) = \\infty$ if there is none.\nFor $p \\in \\partial\\Omega$, the *type* of $\\partial\\Omega$ at $p$ is\n$$\\Delta(p) = \\sup_{\\gamma} \\frac{\\nu(\\rho \\circ \\gamma)}{\\nu(\\gamma - p)},$$\nwhere $\\gamma \\colon D \\to \\C^n$ runs over the nonconstant [[holomorphic maps|3MQTWKBBBT]] with $\\gamma(0) = p$.\nHere $\\nu(\\gamma - p)$ is the least order of vanishing of the components of $\\gamma - p$.\nThe domain $\\Omega$ is *of finite type* if $\\Delta(p) \u003c \\infty$ for every $p \\in \\partial\\Omega$.\n\n## Remarks\n\nThe number $\\Delta(p)$ does not depend on the choice of $\\rho$, since two defining functions differ by a smooth positive factor.\nIt is also called the D'Angelo type, or the $1$-type.\nExample: at a strongly pseudoconvex point $\\Delta(p) = 2$, so a bounded [[strongly pseudoconvex domain|Z7RVXBA2KB]] is of finite type.\nFor $\\Omega = \\set{\\abs{z}^2 + \\abs{w}^{2m} \u003c 1}$ in $\\C^2$, with $m \\ge 1$ an integer, the disc $\\gamma(t) = (1, t)$ gives $\\rho \\circ \\gamma(t) = \\abs{t}^{2m}$, so $\\Delta((1, 0)) \\ge 2m$.\nNon-example: if $\\partial\\Omega$ contains a nonconstant holomorphic disc $\\gamma$ through $p$, then $\\rho \\circ \\gamma = 0$, so $\\Delta(p) = \\infty$.\nWhen $n = 2$, the type agrees with the type of Kohn, which is defined by iterated brackets of complex tangent vector fields.\n",
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