The Schwarz Reflection Principle is a method for extending the domain of a complex analytic function beyond its original region of definition, a form of analytic continuation. It states that a function holomorphic on the upper half of the complex plane and continuous with real values along the real axis can be extended to the lower half plane by reflecting its values across that axis, with the extended function remaining analytic on the whole plane. Named for Hermann Schwarz, the principle also adapts to extend harmonic functions and, in a weakened form, functions permitted to carry certain singularities.
Facts
StatementIf an analytic function is defined on the upper half-plane and has well-defined (non-singular) real values on the real axis, then it can be extended to the conjugate function on the lower half-plane, by F(conj z) = conj F(z). 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.
Proved By
Source Schwarz reflection principle (Wikipedia)
Sources
1. Schwarz reflection principle (Wikipedia)
Introduction, sentence 2
if an analytic function is defined on the upper half-plane, and has well-defined (non-singular) real values on the real axis, then it can be extended to the conjugate function on the lower half-plane
Proved By: Hermann Schwarz, Lead paragraph
In mathematics, the Schwarz reflection principle is a way to extend the domain of definition of a complex analytic function, i.e., it is a form of analytic
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