---
tag: 7PJ8KVZZCQ
kind: concept
name: Germ of a holomorphic function
msc: [32A10, 32B05]
specializes: [VZXF7RNFKS]
sources:
  - url: https://en.wikipedia.org/wiki/Germ_(mathematics)
---

## Summary

A germ of a holomorphic function at a point $a \in \C^n$ is a [[holomorphic function|507C34W0ZT]] defined near $a$.
Two such functions are identified when they agree on some neighbourhood of $a$.
It is the special case of a [[germ|VZXF7RNFKS]] for holomorphic functions.
The germs at $a$ form a ring $\mathcal{O}_{n,a}$, which is isomorphic to the [[ring of convergent power series|EVPZ7ZG1HB]] in $n$ variables.
Local analytic geometry studies this ring and its ideals.

## Explanations

### L1 human

A germ keeps only what a function does arbitrarily close to one point.
For holomorphic functions this is a lot of information.
By the identity theorem, the germ at $a$ determines the function on every connected open set containing $a$ where it is defined.
For 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 $0$.