holomorphic function

A holomorphic function is the central character of complex analysis: a function that is complex differentiable not just at one point but at every point of an open region. Think of it as a function so smoothly woven into the complex plane that, once you know it on a tiny patch, you know it almost everywhere — its local behavior dictates its global behavior with uncanny tightness. These are the functions for which the whole rich theory works.

Formally, f is holomorphic on an open set U if its complex derivative f'(z) exists at every z in U. A landmark theorem (a consequence of Cauchy's integral formula) is that this single hypothesis forces f to be infinitely complex differentiable and, moreover, to equal its own convergent power series in a neighborhood of every point. Thus 'holomorphic' and 'complex analytic' describe exactly the same class of functions — a coincidence with no parallel in real analysis, where smooth functions can fail badly to be analytic.

Holomorphicity is rigid in striking ways. Two holomorphic functions on a connected open set that agree on a tiny set with a limit point must agree everywhere (the identity theorem). A holomorphic function has no interior maximum of modulus unless it is constant. Its zeros are isolated. None of these statements is remotely true for general real differentiable functions, which can be bent and patched at will.