The Identity Theorem of complex analysis states that if two functions holomorphic on the same connected open region agree on a set of points that has an accumulation point within that region, then the two functions must be identical everywhere on the region. It is a foundational rigidity result showing that a holomorphic function is far more constrained than a general smooth real function, since agreement on even a small but accumulating set of points forces global agreement, and it underlies the standard technique of analytic continuation.
Facts
StatementGiven functions f and g analytic on a domain D (open and connected), if f = g on some subset S of D that has an accumulation point in D, then f = g on D. 1 Classification
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Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Sources
1. Identity theorem (Wikipedia)
Theorem statement, sentence 1Quote, Theorem statement, sentence 1
where S has an accumulation point in D, then f = g on D
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