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  "body": "\n## Definition\n\nLet $a \\in \\C^n$.\nConsider pairs $(U, f)$ where $U \\subseteq \\C^n$ is an open [[neighbourhood|0ZCY0Z80B6]] of $a$ and $f \\colon U \\to \\C$ is a [[holomorphic function|507C34W0ZT]].\nTwo pairs $(U, f)$ and $(V, g)$ are equivalent if there is an open neighbourhood $W \\subseteq U \\cap V$ of $a$ with $f = g$ on $W$.\nA *germ of a holomorphic function* at $a$ is an equivalence class for this relation.\nThe germ of $(U, f)$ is written $f_a$, and the set of all germs at $a$ is written $\\mathcal{O}_{n,a}$, or $\\mathcal{O}_a$.\n\n## Remarks\n\nThis is the notion of [[germ|Y98MD3WVTG]] applied to holomorphic functions.\nSums and products of representatives on a common neighbourhood make $\\mathcal{O}_{n,a}$ a commutative ring, and evaluation $f_a \\mapsto f(a)$ is well defined.\nExample: the pairs $(\\C \\setminus \\set{1}, 1/(1 - z))$ and $(\\set{\\abs{z} \u003c 1}, \\sum_{k \\ge 0} z^k)$ define the same germ at $0 \\in \\C$, since the two functions agree on the unit disc.\nNon-example: the pairs $(\\C, z)$ and $(\\C, z + z^2)$ do not define the same germ at $0$, since $z^2 \\ne 0$ at every point $z \\ne 0$ near $0$.\nBy the identity theorem, two holomorphic functions on a connected open set containing $a$ have the same germ at $a$ only if they are equal.\nTaking Taylor series at $a$ gives an isomorphism of $\\mathcal{O}_{n,a}$ onto the ring of convergent power series in $z_1 - a_1, \\dots, z_n - a_n$.\nThe germs at all points of an open set $D$ are the stalks of the sheaf $\\mathcal{O}$ of holomorphic functions on $D$.\n",
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