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  "body": "\n## Summary\n\nA germ of a holomorphic function at a point $a \\in \\C^n$ is a [[holomorphic function|507C34W0ZT]] defined near $a$.\nTwo such functions are identified when they agree on some neighbourhood of $a$.\nIt is the special case of a [[germ|VZXF7RNFKS]] for holomorphic functions.\nThe germs at $a$ form a ring $\\mathcal{O}_{n,a}$, which is isomorphic to the [[ring of convergent power series|EVPZ7ZG1HB]] in $n$ variables.\nLocal analytic geometry studies this ring and its ideals.\n\n## Explanations\n\n### L1 human\n\nA germ keeps only what a function does arbitrarily close to one point.\nFor holomorphic functions this is a lot of information.\nBy the identity theorem, the germ at $a$ determines the function on every connected open set containing $a$ where it is defined.\nFor example, $1/(1 - z)$ on $\\C \\setminus \\set{1}$ and the series $\\sum_{k \\ge 0} z^k$ on the unit disc have the same germ at $0$.\n",
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