rigidity and uniqueness from the maximum principle

Holomorphic functions are stiff: small constraints propagate into total control. 'Rigidity' is the umbrella word for this. The maximum-modulus principle is one of its sharpest expressions — because |f| cannot peak inside, a holomorphic function pinned down on a boundary is pinned down everywhere, and any leftover freedom collapses. Where flexible real functions can be nudged, holomorphic ones snap to a single shape.

The cleanest uniqueness argument runs like this. Suppose f and g are holomorphic on a bounded region, continuous up to its boundary, and equal on the boundary. Set h = f - g. Then h is holomorphic and zero on the boundary, so by the maximum-modulus principle the maximum of |h| (which is on the boundary) is 0; hence |h| is at most 0 everywhere, meaning h = 0 and f = g throughout. Equality on the edge forces equality inside — there is exactly one holomorphic function with given boundary values, not a family of them.

This rigidity reverberates across the subject. It powers the uniqueness half of the Riemann mapping theorem (a conformal map of a region onto the disk, once normalized, is unique), the Schwarz lemma's equality case (equality forces a rotation), and uniqueness for the Dirichlet problem. The reasonable warning: rigidity is a feature, not a bug, but it means holomorphic functions cannot be locally edited — you cannot change one on a small patch and keep it holomorphic, unlike a bump function in real analysis.