Identity theorem

The identity theorem states that two holomorphic functions on a connected domain are equal everywhere if they agree on a set with an accumulation point inside that domain.

Zero of a holomorphic function: A point where a holomorphic function takes the value zero. The theorem reduces equality of two functions to zeros of their difference.
Accumulation point: A point every neighborhood of which contains a point of a set distinct from itself. The agreement set must accumulate at a point inside the domain.
Schwarz reflection principle: A method for extending certain holomorphic functions across a boundary using symmetry of their boundary values. Uniqueness from the identity theorem ensures competing reflected extensions coincide.
Identity theorem for real analytic functions: A uniqueness result stating that real-analytic functions agreeing on a set with an interior accumulation point agree on a connected interval. Its real-variable setting contrasts with the complex-domain theorem and its stronger zero-set rigidity.
Isolated zeros theorem: A nonzero holomorphic function has isolated zeros in its domain. An accumulation of zeros forces the difference function to vanish identically.
Domain (mathematics): A nonempty connected open subset of a mathematical space, especially the complex plane. The theorem applies on a connected open domain, not an arbitrary set.
Uniqueness theorem for holomorphic functions: A result that prescribed values on a suitable set determine a holomorphic function uniquely. The identity theorem is the central uniqueness result for values accumulating inside a domain.
Unique continuation: The principle that a solution to a differential equation is determined throughout a region by its values on a smaller set. It generalizes identity-type rigidity to solutions beyond holomorphic functions.
Taylor series: A power series that represents a function near a point through its derivatives there. A holomorphic function with accumulating zeros cannot have a first nonzero Taylor coefficient.
Open set: A set in a topological space in which every point has a neighborhood contained in the set. Holomorphic functions are defined on open sets, allowing local power-series arguments.

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