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  "body": "\n## Statement\n\nLet $U \\subseteq \\mathbb{C}$ be an open set, let $f : U \\to \\mathbb{C}$ be [[holomorphic|0DG0Z72CQD]], and let $z_0 \\in U$.\nIf $f'(z_0) \\neq 0$, then $f$ is [[conformal|0VHNKP1N3K]] at $z_0$.\nIn particular, if $f'(z) \\neq 0$ for each $z \\in U$, then $f$ is conformal on $U$.\n\n## Remarks\n\nWrite $f'(z_0) = r e^{i\\varphi}$ with $r \u003e 0$; then the real derivative of $f$ at $z_0$ is $h \\mapsto r e^{i\\varphi} h$, a rotation by $\\varphi$ followed by a scaling by $r$.\nThe converse also holds: a map that is conformal at each point of an open set $U$ is holomorphic on $U$ with a nonzero derivative.\nThe condition $f'(z_0) \\neq 0$ is necessary: $z \\mapsto z^2$ doubles the angles between curves through $0$.\nThe exponential function is conformal at each point but not injective, so a conformal map need not be injective.\n\n## Explanations\n\n### L1 human\n\nNear $z_0$, a [[complex differentiable|0D1C9Y1EXA]] function acts almost like $z \\mapsto f(z_0) + f'(z_0)(z - z_0)$.\nMultiplication by the complex number $f'(z_0)$ turns the plane and scales it.\nTurning and scaling keep all angles, so $f$ keeps the angles between curves through $z_0$.\n",
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    "name": "Holomorphic functions with nonzero derivative are conformal",
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        "locator": "In two dimensions",
        "url": "https://en.wikipedia.org/wiki/Conformal_map"
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  "msc": [
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  "name": "Holomorphic functions with nonzero derivative are conformal",
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