Germ of a holomorphic function

Remarks

This is the notion of germ applied to holomorphic functions. Sums 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. Example: the pairs $(\C \setminus \set{1}, 1/(1 - z))$ and $(\set{\abs{z} < 1}, \sum_{k \ge 0} z^k)$ define the same germ at $0 \in \C$, since the two functions agree on the unit disc. Non-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 $0$. By the identity theorem, two holomorphic functions on a connected open set containing $a$ have the same germ at $a$ only if they are equal. Taking Taylor series at $a$ gives an isomorphism of $\mathcal{O}_{n,a}$ onto the ring of convergent power series in $z_1 - a_1, \dots, z_n - a_n$. The germs at all points of an open set $D$ are the stalks of the sheaf $\mathcal{O}$ of holomorphic functions on $D$.