the local mapping theorem

Zoom in close enough on a holomorphic function near a point, and it always looks like one of two simple model maps. Either it behaves like a clean stretch-and-rotate that opens out fan-like (when the derivative is nonzero), or it behaves like the power map z mapsto z^m that folds a neighborhood over itself m times (when the derivative vanishes to order m - 1). The local mapping theorem says these are the only local pictures — every holomorphic map, magnified, is locally a power map.

The statement: suppose f is holomorphic near z_0 and f(z_0) = w_0, and let m be the order of the zero of f(z) - w_0 at z_0 (so m is the first index with f^(m)(z_0) not zero; m = 1 exactly when f'(z_0) is not zero). Then near z_0 there is a holomorphic change of coordinate phi with phi(z_0) = 0 and phi'(z_0) not zero so that f(z) = w_0 + phi(z)^m. In words: after a conformal relabeling of the input, f is exactly the m-th power map. The construction extracts an m-th root of (f(z) - w_0) / (leading coefficient), which is legitimate because that quotient is nonvanishing and equals 1 at z_0, so a holomorphic m-th root exists locally. Consequently, for every value w near w_0 (but not equal), the equation f(z) = w has exactly m distinct simple solutions near z_0 — the map is locally m-to-1.

This is the structural heart behind several theorems. When m = 1 the map is locally injective and conformal (preserving angles), which is the holomorphic inverse function theorem. When m is bigger than 1, z_0 is a critical point (f'(z_0) = 0) and angles get multiplied by m — a quarter turn at z_0 becomes a half turn in the image — so conformality fails exactly at critical points. The open mapping theorem is the immediate consequence (the power map is open), and the count m is the local degree. The honest point: 'locally' means in a small enough neighborhood; globally a holomorphic function can be wildly many-to-one, and the clean power-map picture only holds after zooming in.