Causality in time becomes analyticity in frequency. Absorption and dispersion are two sides of the same complex function — one cannot change without the other.
χ'(ω) = (2/π) P ∫₀ⁿ ω' χ''(ω') / (ω'² − ω²) dω' χ''(ω) = −(2ω/π) P ∫₀ⁿ χ'(ω') / (ω'² − ω²) dω'
Any causal, linear, time-invariant response — an electric susceptibility, a refractive index, a loudspeaker transfer function, a microwave S-parameter — is a complex-valued function of frequency. The Kramers–Kronig relations state that the real and imaginary parts of such a response cannot be chosen independently. Each is the Hilbert transform of the other.
The same theorem appears in electromagnetism, dielectric spectroscopy, network analysis, rheology, seismic attenuation, filter theory, room acoustics, and fibre-optic communications. Every discipline dresses it differently, but the mathematics is one.
If the system cannot respond before it is poked, then measuring how much it absorbs determines how much it disperses — and vice versa — up to an integral over all frequencies.
Write a causal impulse response as h(t) with h(t)=0 for t<0. Its Fourier transform (physics convention) is
The frequency variable ω is real on physical grounds — it is the angular frequency of a steady-state Fourier component — but the integral on the right is a perfectly good function of ω regarded as a complex variable. Promoting ω = ωr + iωi with ωr, ωi ∈ ℝ turns χ(ω) into a candidate analytic function on the complex plane, and the K–K argument lives or dies on where that candidate is well-defined.
For Im ω > 0 — i.e. ωi > 0, the upper half of the complex frequency plane — the factor eiωt = eiωrt e−ωit decays exponentially as t → ∞, so the integral converges absolutely (causality, h(t)=0 for t<0, is what lets us drop the lower half of the time axis where this exponential would blow up). A convergent power-series definition in an open region is holomorphic. Therefore χ is analytic in the upper half complex-frequency plane. Apply Cauchy's integral formula on a closed contour along the real axis (with a small indentation at ω=ω₀) plus a large arc at infinity, and the two Kramers–Kronig integrals fall out.
The K–K integrals themselves only ever evaluate χ on the real axis (real ω), which is what experiments measure. The complex extension is scaffolding: it exists so that Cauchy's theorem has somewhere to live, and the conclusions are then read off on its real boundary.
The same logic in the engineering sign convention e+jωt swaps the role of upper and lower half planes. The content is identical; only the sign of an i flips.
Three columns × two rows. Top row shows real parts; bottom row shows imaginary parts. Left: complex impulse response h(t) = e−γ(t−τ) eiω₀(t−τ) for t ≥ τ, zero otherwise. Centre: real and imaginary parts of its spectrum χ(ω) — the “truth”. Right: the K–K prediction (gold) of each part computed from the other one via the Hilbert transform — Re χ = H[Im χ] on top, Im χ = −H[Re χ] on the bottom. The right column should match the centre column when h is causal (τ ≥ 0) and visibly disagree as soon as h leaks into negative time.
h(t)The P is the Cauchy principal value — it handles the pole on the real axis. The subtracted form is essential for optical constants where n(∞)→1, not zero.
A single damped harmonic oscillator driven by an electric field gives a susceptibility with one pair of complex-conjugate poles:
Splitting χ = χ' + iχ'' gives
You can verify by direct integration that these satisfy K–K. Because the poles sit in the lower half plane (physics convention), χ(ω) is analytic in the upper half plane — just as causality demands.
Same complex function, three conventions. The electrical-engineer's Bode plot, the dielectric-spectroscopist's Cole–Cole semicircle, and the physicist's χ'/χ'' spectra all carry identical information.
The Kramers–Kronig derivation is a single contour integral. Walk along the real axis from −∞ to +∞, indent with a small semicircle above the pole at ω₀, then close the loop with a giant semicircle in the upper half plane. The enclosed region contains no singularities, so Cauchy says the total integral is zero.
Large arc at R→∞ vanishes by Jordan's lemma. Small indent at ω₀ contributes −iπ χ(ω₀) by the Sokhotski–Plemelj formula (half of a full residue). The rest is the principal-value integral along ℝ.
Rearranging:
Split into real and imaginary parts and you have the relations.
One of the reasons the Kramers–Kronig literature is so hard to read across fields is that every community chose its own notation. Here is the translation table.
| Discipline | Quantity | Decomposition | Typical plot |
|---|---|---|---|
| Optics / condensed matter | ε(ω), ñ(ω) = n + iκ | dispersion + absorption | n, κ vs ℏω |
| Dielectric spectroscopy | ε* = ε' − jε'' (eng) or ε' + iε'' (phys) | storage + loss | Cole–Cole semicircle |
| Microwave / RF | Sij(ω), Z(ω) | real & imag, or mag & phase | Smith chart, Bode |
| Controls / signals | H(jω) = |H|ejφ | gain & phase | Bode, Nyquist |
| Rheology | G*(ω) = G' + iG'' | elastic + loss moduli | log–log vs ω |
| Particle physics | f(ω) scattering amplitude | real + i·(σtot/4π) | forward amplitude vs energy |
| Audio / DSP | H(ejω) | log-magnitude & phase | dB / linear phase |
Physics writes ε = ε' + iε'' with ε'' > 0 for absorption. Some engineering texts write ε = ε' − jε'' with the same ε'' > 0. Both describe a lossy medium — the sign flip lives in e−iωt vs e+jωt. Always check the kernel before copying a formula across a disciplinary boundary.
|H(ejω)|, the K–K relations on log H recover phase uniquely — the discrete-time Bode theorem. Core of the complex cepstrum.R(ω); K–K on log r recovers the phase, and Fresnel inverts to n + iκ. Workhorse of solid-state optics.ε(ω) to Lorentzian / Debye / Drude poles — poles in the lower half plane are automatically K–K-consistent, and yield stable ADE or PLRC updates.ε'(ω) and ε''(ω) don't obey K–K, something in the calibration is wrong.Each chapter is a standalone Markdown page in the chapters/ directory. The five parts build from mathematical foundations, through the theorem itself, to the applications above.