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  "body": "\n## Summary\n\nThe Cauchy–Riemann operator is the first-order differential operator $\\partial / \\partial \\bar z = \\tfrac12 (\\partial_x + i \\partial_y)$ on complex-valued functions of $z = x + iy$.\nA continuously differentiable function $f$ is [[holomorphic|0NAHXR1FRH]] exactly when $\\partial f / \\partial \\bar z = 0$, which is a compact form of the [[Cauchy–Riemann equations|0T71SSYZ39]].\nIt is the simplest elliptic operator, with fundamental solution $1/(\\pi z)$.\nIts several-variable version, the $\\bar\\partial$ operator, is a main tool of complex analysis and complex geometry.\n\n## Explanations\n\n### L1 human\n\nA function of $z = x + iy$ can depend on $x$ and $y$ in any way, for example through $\\bar z = x - iy$.\nThe Cauchy–Riemann operator measures how much a function depends on $\\bar z$.\nHolomorphic functions, such as polynomials in $z$, do not depend on $\\bar z$ at all, so the operator gives zero on them.\nFor example, it gives $0$ on $z^2$ and $1$ on $\\bar z$.\n",
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