The Kramers-Kronig Relations connect the real and imaginary parts of any complex-valued function that is analytic in the upper half of the complex plane, expressing each part as a specific integral transform, a Hilbert transform, of the other. Named for Hendrik Kramers and Ralph Kronig, they apply to any physical response function that respects causality, such as a material's frequency-dependent optical or electrical response, and let one part of the response be recovered entirely from a full knowledge of the other.
Facts
StatementThe real and imaginary parts of a complex function analytic in the upper half-plane are related by Hilbert transforms of each other. 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.
Sources
1. Kramers-Kronig relations (Wikipedia)
FormulationQuote, Formulation
the real and imaginary parts of any complex function that is analytic in the upper half-plane
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