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  "body": "\n## Statement\n\nLet $U \\subseteq \\mathbb{C}$ be an open set and let $f : U \\to \\mathbb{C}$ be [[holomorphic|0DG0Z72CQD]] on $U$.\nLet $u = \\operatorname{Re} f$ and $v = \\operatorname{Im} f$, as real functions of $(x, y)$ on $U$, with $z = x + iy$.\nThen $u$ and $v$ are twice continuously differentiable, and\n$$\\frac{\\partial^2 u}{\\partial x^2} + \\frac{\\partial^2 u}{\\partial y^2} = 0, \\qquad \\frac{\\partial^2 v}{\\partial x^2} + \\frac{\\partial^2 v}{\\partial y^2} = 0$$\non $U$. Thus $u$ and $v$ are harmonic functions on $U$.\n\n## Remarks\n\nProof: $f$ is [[infinitely differentiable|0QEBN8DGVE]]. By the [[Cauchy–Riemann equations|0T71SSYZ39]], $u_{xx} = v_{yx} = v_{xy} = -u_{yy}$, and the same computation works for $v$.\nFor example, $\\operatorname{Re}(z^2) = x^2 - y^2$ is harmonic, and so is $\\operatorname{Re}(e^z) = e^x \\cos y$.\nThe function $v$ is a [[harmonic conjugate|0CC56MJ4FG]] of $u$.\n\n## Explanations\n\n### L1 human\n\nA harmonic function is a function of two real variables whose two pure second derivatives add up to $0$.\nEach holomorphic function gives two harmonic functions for free: its real part and its imaginary part.\n",
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