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  "body": "\n## Statement\n\nLet $U, V \\subseteq \\C$ be open sets and let $\\varphi \\colon U \\to V$ be a [[holomorphic function|0NAHXR1FRH]].\nLet $h \\colon V \\to \\R$ be a [[harmonic function|0DEB0JMN0W]].\nThen $h \\circ \\varphi$ is harmonic on $U$.\nIn particular, let $\\varphi$ be a [[conformal map|0YYT36R4YD]] of $U$ onto $V$.\nThen $h$ is harmonic on $V$ if and only if $h \\circ \\varphi$ is harmonic on $U$.\n\n## Remarks\n\nProof idea: near a point $\\varphi(z_0)$, [[harmonic conjugates exist locally|CVDGP7DKZ3]], so $h = \\operatorname{Re} F$ for a holomorphic $F$ on a disc around $\\varphi(z_0)$.\nThen $h \\circ \\varphi = \\operatorname{Re}(F \\circ \\varphi)$ near $z_0$, and $F \\circ \\varphi$ is holomorphic.\nA direct computation gives $\\Delta (h \\circ \\varphi) = \\abs{\\varphi'}^2 \\, (\\Delta h) \\circ \\varphi$, where $\\Delta$ is the [[Laplacian|0E0B6RA8K0]].\nExample: $h(w) = \\log\\abs{w}$ is harmonic on $\\C \\setminus \\set{0}$, and with $\\varphi(z) = z^2$ one gets $h(\\varphi(z)) = 2 \\log\\abs{z}$, again harmonic.\nHolomorphy of $\\varphi$ is needed: for the smooth map $\\varphi(x, y) = (x, y^2)$ and $h(s, t) = s^2 - t^2$, the function $h \\circ \\varphi = x^2 - y^4$ has Laplacian $2 - 12 y^2 \\ne 0$.\nThe statement also holds for antiholomorphic $\\varphi$, because $h \\circ \\bar\\varphi$ equals $\\tilde h \\circ \\varphi$ with $\\tilde h(w) = h(\\bar w)$, which is harmonic.\n\n## Explanations\n\n### L1 human\n\nHarmonic functions in the plane survive a change of coordinates by a holomorphic map.\nSo one may move a problem about harmonic functions from a complicated region to a simple one, such as a disc, solve it there, and move the answer back.\n\n### L2 human\n\nThis is the basis of conformal methods in planar potential theory.\nWith the Riemann mapping theorem, the Dirichlet problem on a simply connected domain reduces to the disc, where the Poisson integral solves it.\nThe same invariance makes harmonic measure and Green's functions conformally invariant.\nIt also shows that harmonicity makes sense on a Riemann surface, since the transition maps between charts are holomorphic.\n",
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