Integral relations that, for a linear, time‑invariant, causal and stable system whose frequency‑domain response is analytic in the appropriate half of the complex frequency plane and decays sufficiently at infinity, uniquely link the real and imaginary parts of that response via Hilbert transforms (principal‑value integrals). They express the requirement that amplitude (dispersion/absorption) and phase (reactive/dissipative) parts are not independent but constrained by causality and analyticity.

Principle

Causality plus linearity and analyticity in the complex frequency domain imply that the real and imaginary parts of the frequency response are Hilbert transforms of one another; knowledge of one over all frequencies determines the other up to the conditions required for convergence (or up to subtraction constants if subtracted forms are used).

Demonstration

Illustrative scenario: A material’s measured imaginary permittivity (absorption) Im[ε(ω)] is available across frequencies. Applying the Kramers–Kronig principal‑value integral to Im[ε(ω)] yields the dispersion Re[ε(ω)] provided the response is causal and appropriate extrapolations are supplied for unmeasured bands; mismatched extrapolation or noncausal behavior breaks the computed Re[ε(ω)].

Misapplication

Treating the relations as applicable to non‑causal, nonlinear, time‑varying or active systems (e.g., systems with poles in the analytic half‑plane used) or applying them to truncated experimental data without addressing extrapolation and subtraction leads to incorrect phase/magnitude reconstruction; the semantic error is assuming locality of frequency data is sufficient without the global analyticity requirement.

Consequence

They provide a consistency test for measurements and a method to recover phase/dispersion from amplitude/absorption (or vice versa) when the required analytic and decay conditions hold; in practice they force model extrapolation or subtraction choices when data are finite, and can expose violations of causality or measurement error.

Reversal

If the frequency response is not analytic in the assumed half‑plane (for example an active system with upper‑half–plane poles), or does not decay so integrals converge, the standard Kramers–Kronig forms fail and must be replaced by modified (subtracted) relations or by formulations appropriate to the system’s analyticity domain.

Boundary

Applies to linear, time‑invariant, causal and stable single‑input single‑output responses whose Fourier/Laplace transforms are analytic in the designated half‑plane and satisfy decay conditions; excludes inherently nonlinear responses, time‑varying systems, and cases where analytic continuation or extrapolation is impossible or ill‑posed without additional assumptions.

Semantic Tension

Causality/analyticity ↔ Finite and noisy measurements: the mathematical relations require global frequency information (or justified extrapolation), but practical measurement provides only finite, noisy bands, forcing a tradeoff between theoretical consistency and empirical feasibility.

Synthesis

Kramers–Kronig relations convert the physical principle of causality into explicit integral constraints in the frequency domain: amplitude and phase are two aspects of the same analytic function and cannot be chosen independently without violating causality or analyticity assumptions.