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  "body": "\n## Definition\n\nLet $D \\subseteq \\C^n$ be an [[open set|02BH3MCBQB]].\nThen $D$ is a *domain of holomorphy* if there are no nonempty open sets $U_1, U_2 \\subseteq \\C^n$ such that:\n\n1. $U_2$ is [[connected|0SPE8WNWFV]] and $U_2 \\not\\subseteq D$;\n2. $U_1 \\subseteq D \\cap U_2$;\n3. for every [[holomorphic function|507C34W0ZT]] $f$ on $D$ there is a holomorphic function $g$ on $U_2$ with $f = g$ on $U_1$.\n\n## Remarks\n\nExample: every convex open set $D \\subsetneq \\C^n$ is a domain of holomorphy.\nFor a boundary point $p$ there is a complex linear function $\\ell$ with $\\operatorname{Re} \\ell(z) \u003c \\operatorname{Re} \\ell(p)$ for all $z \\in D$, and $f_p(z) = 1/(\\ell(z) - \\ell(p))$ is holomorphic on $D$ and unbounded near $p$.\nSuppose $U_1, U_2$ satisfied the three clauses, and let $W$ be a component of $D \\cap U_2$ that meets $U_1$.\nSince $U_2$ is connected and not contained in $D$, it contains a boundary point $p$ of $W$, and $p$ lies on the boundary of $D$.\nThe extension $g$ of $f_p$ equals $f_p$ on $W$ by the identity theorem, and $g$ is continuous at $p$.\nSo $f_p$ is bounded near $p$ on $W$, which is a contradiction.\n\nNon-example: $D = \\set{z \\in \\C^2 : 0 \u003c \\abs{z} \u003c 1}$.\nTake $U_2 = \\set{\\abs{z} \u003c 1}$ and $U_1 = D$.\nClauses 1 and 2 hold, and clause 3 holds by [[Hartogs's extension theorem|FSGES670AZ]] with $K = \\set{0}$.\n\nEvery open subset of $\\C$ is a domain of holomorphy.\n",
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