the germ of a function

A function element carries a whole disk, but most of that disk is redundant if all we really care about is the function's behaviour right at one point. A germ is the distilled essence: it is what is left of a holomorphic function near a point z_0 once you agree to ignore how big the disk is and forget any two functions that happen to coincide on some (possibly tiny) neighbourhood of z_0. Two function elements through z_0 define the SAME germ if they agree on some little disk around z_0, no matter how small. So a germ is an equivalence class of elements — a 'point-sized' piece of analytic behaviour.

Concretely, near a point z_0 a holomorphic function is completely determined by its Taylor coefficients there: a_0 = f(z_0), a_1 = f'(z_0), a_2 = f''(z_0)/2!, and so on. The germ is, in effect, exactly this infinite list of coefficients (with positive radius of convergence). Two functions have the same germ at z_0 precisely when all their derivatives agree at z_0. This is why germs and power series are almost interchangeable: the germ IS the convergent power series, stripped of any claim about how far out it stays valid.

Germs are the cleanest building blocks for the global theory. The collection of all germs of all continuations of one starting function, glued together by the relation 'these two germs are connected by a chain', forms a space — and that space, made geometric, is exactly the Riemann surface of the function. Talking in germs lets us say, without ambiguity, what it means for two routes of continuation to 'arrive at the same local function' (same germ) or at a 'different branch' (a different germ over the same base point).