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where and are real-valued "well-behaving" functions, then the Kramers-Kronig relations are
The Kramers-Kronig relations are related to the Hilbert transform, and are most often applied on the permittivity of materials. However, it must be noticed that in this case,
where is the electric susceptibility of the material. The susceptibility can be interpreted as the Fourier transform of the time-dependent polarization in the material after an infinitely short pulsed electric field, in other words the impulse response of the polarization.
Complex analysis