the local degree of a holomorphic map

Stand at a point and ask: how many times does a holomorphic function wrap a tiny neighborhood around the value it takes there? At most points it wraps exactly once — the map is locally one-to-one. But at special points it folds the neighborhood over itself several times, like winding a strip twice around a pole. That fold count is the local degree: the number of times the map covers values just next to f(z_0).

Precisely, if f is holomorphic near z_0 with f(z_0) = w_0, the local degree of f at z_0 is the order m of the zero of f(z) - w_0 at z_0 — equivalently the smallest m with f^(m)(z_0) not zero, equivalently the multiplicity with which z_0 solves f(z) = w_0. By the local mapping theorem this m has a vivid meaning: for every value w sufficiently close to w_0 but not equal to it, the equation f(z) = w has exactly m distinct solutions in a small disk around z_0. So the local degree literally counts how many nearby points map to a generic nearby value. You compute it by differentiating: m = 1 when f'(z_0) is not zero (a regular point), and m greater than 1 at a critical point where f' has a zero of order m - 1.

The local degree is the bridge between the analytic and topological pictures. Summed over all preimages of a value inside a contour, it reproduces the global count in the argument principle. It tells you exactly when local inversion is possible (local degree 1 means locally invertible and conformal), and it pinpoints the branch points of the map (local degree m greater than 1). It is the precise reason angles get multiplied by m at a critical point. The caveat worth stating: local degree is always a positive integer for a non-constant holomorphic map at any point — there is no fractional or negative local degree here, unlike signed topological degrees that can cancel; holomorphic maps are orientation-preserving, so all the local contributions add with the same sign.