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  "body": "\n## Summary\n\nThe Wirtinger derivatives $\\partial / \\partial z$ and $\\partial / \\partial \\bar z$ are two first-order differential operators on functions of $z = x + iy$, see the [[definition|698R12MY4G]].\nThey let one treat $z$ and $\\bar z$ as if they were independent variables.\nA continuously differentiable function is [[holomorphic|0NAHXR1FRH]] exactly when its $\\partial / \\partial \\bar z$ derivative is zero, which is a compact form of the [[Cauchy–Riemann equations|0T71SSYZ39]].\n\n## Explanations\n\n### L1 human\n\nA function of a complex variable is also a function of two real variables $x$ and $y$.\nThe Wirtinger derivatives repackage the two partial derivatives in $x$ and $y$ into two complex ones.\nOn a polynomial in $z$ and $\\bar z$, the derivative in $z$ differentiates in $z$ while holding $\\bar z$ fixed.\nThe derivative in $\\bar z$ does the opposite.\nFor example, $\\abs{z}^2 = z \\bar z$ has derivative $\\bar z$ in $z$ and derivative $z$ in $\\bar z$, just as if $z$ and $\\bar z$ were separate letters.\n",
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