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  "body": "\n## Statement\n\nLet $\\Omega \\subseteq \\C$ be an open set that is symmetric with respect to the real axis: $\\bar z \\in \\Omega$ for each $z \\in \\Omega$.\nLet $\\Omega^+ = \\set{z \\in \\Omega : \\operatorname{Im} z \u003e 0}$ and $\\sigma = \\Omega \\cap \\R$.\nLet $f$ be continuous on $\\Omega^+ \\cup \\sigma$, [[holomorphic|0DG0Z72CQD]] on $\\Omega^+$, and real-valued on $\\sigma$.\nDefine $F(z) = f(z)$ for $z \\in \\Omega^+ \\cup \\sigma$ and $F(z) = \\overline{f(\\bar z)}$ for $z \\in \\Omega$ with $\\operatorname{Im} z \u003c 0$.\nThen $F$ is holomorphic on $\\Omega$.\n\n## Remarks\n\nProof idea: $F$ is holomorphic on $\\Omega \\setminus \\R$, since $z \\mapsto \\overline{f(\\bar z)}$ satisfies the Cauchy–Riemann equations when $f$ does.\n$F$ is continuous on $\\Omega$, because $f$ is real on $\\sigma$, so the two definitions agree there.\nThe integral of $F$ over each small triangle in $\\Omega$ is $0$: for triangles meeting $\\R$, approximate by triangles slightly off the axis.\n[[Morera's theorem|0VDD7W0EKZ]] then gives that $F$ is holomorphic.\nExample: $f(z) = e^{z}$ on the upper half-plane is real on $\\R$, and its reflection is $\\overline{e^{\\bar z}} = e^{z}$, as expected.\nThe hypothesis that $f$ is real on $\\sigma$ is needed: $f(z) = i$ is continuous up to $\\R$ but its reflection is $-i$ below the axis, which is not even continuous.\nBy a Möbius transformation the same principle holds for reflection across a circle, with $\\bar z$ replaced by the inverse point.\n\n## Explanations\n\n### L1 human\n\nIf a holomorphic function on the upper half-plane takes real values on the real axis, it extends to the lower half-plane by mirror symmetry.\nThe value at the mirror point $\\bar z$ is the complex conjugate of the value at $z$.\nThe two halves fit together into one holomorphic function.\n\n### L2 human\n\nThe reflection principle is a basic tool of [[analytic continuation|0M3YQJ81NB]].\nIt is used to extend conformal maps whose boundary goes to a line or a circle, as for the Schwarz–Christoffel formula.\nIt also shows that a conformal map between domains bounded by analytic arcs extends across those arcs.\nThere is a version for harmonic functions that vanish on a line, which is the main step in some proofs.\n",
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