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Kramers-kronig RelationsIn mathematics and physics, the Kramers-Kronig relations describe the relation between the real and imaginary part of a certain class of complex-valued functions. The requirements for a function to which they apply can be interpreted as that the function must represent the Fourier transform of a linear and causal physical process. If we write - ,
where and are real-valued "well-behaving" functions, then the Kramers-Kronig relations are -
f_1(\omega) = \frac{2}{\pi} \int_0^{\infty} \frac{\omega' f_2(\omega') d\omega'}{\omega^2 - \omega'^2} -
\frac{f_1(\omega') d\omega'}{\omega^2 - \omega'^2} . 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.
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