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  "body": "\n## Definition\n\nLet $\\Omega \\subseteq \\C$ be an open set, and write $z = x + iy$ with $x, y$ real.\nFor a function $f \\colon \\Omega \\to \\C$ that has [[partial derivatives|09EGC6F8PT]] $\\partial_x f$ and $\\partial_y f$, the *Cauchy–Riemann operator* applied to $f$ is\n$$\\frac{\\partial f}{\\partial \\bar z} = \\frac12 \\left( \\frac{\\partial f}{\\partial x} + i \\frac{\\partial f}{\\partial y} \\right).$$\nIt is also written $\\bar\\partial f$.\n\n## Remarks\n\nWriting $f = u + iv$ with $u, v$ real, $\\partial f / \\partial \\bar z = 0$ is equivalent to $u_x = v_y$ and $u_y = -v_x$, the [[Cauchy–Riemann equations|0T71SSYZ39]].\nSo a continuously differentiable function $f$ on $\\Omega$ is [[holomorphic|0NAHXR1FRH]] if and only if $\\partial f / \\partial \\bar z = 0$ on $\\Omega$.\nExample: $\\partial z / \\partial \\bar z = \\tfrac12 (1 + i \\cdot i) = 0$, so $z$ is in the kernel.\nNon-example: $\\partial \\bar z / \\partial \\bar z = \\tfrac12 (1 + i \\cdot (-i)) = 1 \\ne 0$, so $\\bar z$ is not holomorphic.\nThe companion operator $\\partial / \\partial z = \\tfrac12 (\\partial_x - i \\partial_y)$ satisfies $4 \\, \\partial_z \\partial_{\\bar z} = \\Delta$.\nIn distributions on $\\C$, $\\partial_{\\bar z} \\big( 1/(\\pi z) \\big) = \\delta_0$, the Dirac measure at $0$.\n",
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