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 Hans Kramers. In mathematics these relations are known under the names Sokhotski–Plemelj theorem and Hilbert transform. 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 Hans Kramers. In mathematics these relations are known under the names Sokhotski–Plemelj theorem and Hilbert transform. Let χ ( ω ) = χ 1 ( ω ) + i χ 2 ( ω ) {displaystyle chi (omega )=chi _{1}(omega )+ichi _{2}(omega )} be a complex function of the complex variable ω {displaystyle omega } , where χ 1 ( ω ) {displaystyle chi _{1}(omega )} and χ 2 ( ω ) {displaystyle chi _{2}(omega )} are real. Suppose this function is analytic in the closed upper half-plane of ω {displaystyle omega } and vanishes like 1 / | ω | {displaystyle 1/|omega |} or faster as | ω | → ∞ {displaystyle |omega | ightarrow infty } . Slightly weaker conditions are also possible. The Kramers–Kronig relations are given by