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  "body": "\n## Statement\n\nLet $U \\subseteq \\C$ be an open set and let $u \\colon U \\to \\R$ be a [[harmonic function|0DEB0JMN0W]].\nLet $D = \\set{z \\in \\C : \\abs{z - a} \u003c r}$ be an open disc with $D \\subseteq U$.\nThen there is a [[holomorphic function|0NAHXR1FRH]] $f \\colon D \\to \\C$ with $\\operatorname{Re} f = u$ on $D$.\nIts imaginary part $v = \\operatorname{Im} f$ is a [[harmonic conjugate|050F2FMTKF]] of $u$ on $D$, and it is unique up to an additive real constant.\n\n## Remarks\n\nProof idea: write $z = x + iy$ and set $g = u_x - i u_y$.\nThe equation $u_{xx} + u_{yy} = 0$ and the symmetry $u_{xy} = u_{yx}$ are exactly the [[Cauchy–Riemann equations|0T71SSYZ39]] for $g$, so $g$ is holomorphic on $U$.\nA holomorphic function on a disc [[has a primitive|02AC7JR6G6]] $F$, and $F' = g$ gives $\\nabla(\\operatorname{Re} F) = \\nabla u$.\nSo $\\operatorname{Re} F - u$ is a real constant $c$ on $D$, and $f = F - c$ works.\nThe same proof works on every [[simply connected domain|0QW64ARSHB]] in place of the disc.\nThe result is local: on $U = \\C \\setminus \\set{0}$ the function $u(z) = \\log\\abs{z}$ is harmonic.\nOn each disc that avoids $0$ it has the conjugate $v = \\arg z$ for a continuous branch of the argument.\nIt has no harmonic conjugate on all of $U$, because no continuous argument exists on a circle around $0$.\n\n## Explanations\n\n### L1 human\n\nA harmonic function in the plane looks like the real part of a holomorphic function.\nThis theorem says that this is always true, at least on small discs.\nSo questions about harmonic functions in two variables can often be turned into questions about holomorphic functions, where many tools are available.\n\n### L2 human\n\nThe theorem lets one transfer results from complex analysis to planar harmonic functions.\nFor example, the mean value property and the maximum principle for $u$ follow from those for a local holomorphic $f$ with $\\operatorname{Re} f = u$.\nGlobally, a harmonic conjugate exists exactly when the period $\\oint (-u_y \\, dx + u_x \\, dy)$ vanishes on each closed curve in $U$.\nThese periods are the obstruction on multiply connected domains, as the example $\\log\\abs{z}$ shows.\n",
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