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  "body": "\n## Definition\n\nLet $U$ be an [[open set|02BH3MCBQB]] in $\\C$.\nLet $\\mu \\colon U \\to \\C$ be a measurable function with $\\norm{\\mu}_\\infty \\le k$ for some constant $k \u003c 1$.\nThe *Beltrami equation* with *Beltrami coefficient* $\\mu$ is\n$$f_{\\bar z} = \\mu f_z,$$\nwhere $f_z$ and $f_{\\bar z}$ are the [[Wirtinger derivatives|WMXDVF36ZT]] of $f$.\nA *solution* is a function $f \\in W^{1,2}_{\\mathrm{loc}}(U)$, that is, with locally square-integrable weak first partial derivatives, that satisfies the equation almost everywhere on $U$.\n\n## Remarks\n\nExample: for $\\mu = 0$ the equation is $f_{\\bar z} = 0$, which is the Cauchy–Riemann system, and its solutions are the [[holomorphic functions|0NAHXR1FRH]] on $U$.\nExample: for a constant $c$ with $\\abs{c} \u003c 1$, the function $f(z) = z + c \\bar z$ has $f_z = 1$ and $f_{\\bar z} = c$, so it solves the equation with $\\mu = c$.\nNon-example: the coefficient $\\mu = 1$ fails the condition $\\norm{\\mu}_\\infty \u003c 1$.\nThen the equation reads $f_{\\bar z} - f_z = i f_y = 0$, and every function of $x$ alone solves it, so the equation is degenerate and its solutions are never homeomorphisms.\nA homeomorphic solution is a [[quasiconformal mapping|SB8KG46NHP]] whose complex dilatation is $\\mu$.\n",
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