Kramers–Kronig relations

The Kramers–Kronig relations are bidirectional mathematical relations, connecting the real and imaginary parts of any complex function that is analytic in the upper half-plane. These relations are often used to calculate the real part from the imaginary part (or vice versa) of response functions in physical systems, because for stable systems, causality implies the analyticity condition, and conversely, analyticity implies causality of the corresponding stable physical system. The relation is named in honor of Ralph Kronig and Hendrik Anthony Kramers. In mathematics these relations are known under the names Sokhotski–Plemelj theorem and Hilbert transform.

Let ${\displaystyle \chi (\omega )=\chi _{1}(\omega )+i\chi _{2}(\omega )}$ be a complex function of the complex variable ${\displaystyle \omega }$, where ${\displaystyle \chi _{1}(\omega )}$ and ${\displaystyle \chi _{2}(\omega )}$ are real. Suppose this function is analytic in the closed upper half-plane of ${\displaystyle \omega }$ and vanishes like ${\displaystyle 1/|\omega |}$ or faster as ${\displaystyle |\omega |\rightarrow \infty }$. Slightly weaker conditions are also possible. The Kramers–Kronig relations are given by

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