Causal function Hilbert transform

The transformation defined by (11.56) is called Hilbert transform, and we have found that the real and imaginary parts of a function that is analytic in half of the complex plane and vanishes at infinity on that plane are Hilbert transforms of each other. Thus, causality, by which response functions have such analytical properties, also implies this relation. On the practical level this tells us that if we know the real (or imaginary) part of a response function we can find its imaginary (or real) part by using this transform. 

Mathematically, integral Kramers-Kronig relations have very general character. They represent the Hilbert transform of any complex function s(co) = s (co) + s"(co) satisfying the condition s (co) = s(—co)(here the star means complex conjugate). In our particular example, this condition is applied to function n(co) related to dielectric permittivity s(co). The latter is Fourier transform of the time dependent dielectric function s(f), which takes into account a time lag (and never advance) in the response of a substance to the external, e.g. optical, electric field. Therefore the Kramers-Kronig relations follow directly from the causality principle. 

If the transfer function H is in accordance with the causality rule, the components R and X are no longer independent of each other. Causality in the meaning of system theory forces couplings between the real and imaginary part, which are known as Kramers-Kronig relations (KKT) or Hilbert relations (HT), for details see Section 3.1.2.9 (The use of Kramers-Kronig Transforms). 

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See also in sourсe #XX -- [ ]

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