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  "body": "\n## Definition\n\nLet $U \\subseteq \\C$ be an open set and write $z = x + iy$ with $x, y$ real.\nLet $f \\colon U \\to \\C$ be a function whose [[partial derivatives|05AXG3QRKR]] $\\partial f / \\partial x$ and $\\partial f / \\partial y$ exist at a point $z \\in U$.\nThe *Wirtinger derivatives* of $f$ at $z$ are\n$$\\frac{\\partial f}{\\partial z} = \\frac{1}{2} \\left( \\frac{\\partial f}{\\partial x} - i \\frac{\\partial f}{\\partial y} \\right), \\qquad \\frac{\\partial f}{\\partial \\bar z} = \\frac{1}{2} \\left( \\frac{\\partial f}{\\partial x} + i \\frac{\\partial f}{\\partial y} \\right).$$\n\n## Remarks\n\nExample: for $f(z) = \\abs{z}^2 = x^2 + y^2$, one gets $\\partial f / \\partial z = \\frac{1}{2}(2x - 2iy) = \\bar z$ and $\\partial f / \\partial \\bar z = \\frac{1}{2}(2x + 2iy) = z$.\nNon-example of holomorphy: since $\\partial f / \\partial \\bar z = z \\neq 0$ for $z \\neq 0$, the function $\\abs{z}^2$ is not [[holomorphic|0DG0Z72CQD]] on any open set.\nFor $f$ of class $C^1$ on $U$, the equation $\\partial f / \\partial \\bar z = 0$ on $U$ is equivalent to the [[Cauchy–Riemann equations|0T71SSYZ39]], so it holds exactly when $f$ is holomorphic on $U$; then $\\partial f / \\partial z = f'$.\nThe operators obey the rules $\\partial z / \\partial z = 1$, $\\partial \\bar z / \\partial z = 0$, $\\partial z / \\partial \\bar z = 0$ and $\\partial \\bar z / \\partial \\bar z = 1$, together with linearity and the product rule.\nThe Laplacian is $\\Delta f = 4 \\, \\dfrac{\\partial^2 f}{\\partial z \\, \\partial \\bar z}$ for $f$ of class $C^2$.\nThe notations $\\partial_z f$ and $\\partial_{\\bar z} f$ are also common.\n",
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