# Kramers-Kronig relations

In 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 [itex]f(\omega)[itex] 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

[itex]f(\omega) = f_1(\omega) + i f_2(\omega)[itex],

where [itex]f_1[itex] and [itex]f_2[itex] are real-valued "well-behaving" functions, then the Kramers-Kronig relations are

[itex]

f_1(\omega) = \frac{2}{\pi} \int_0^{\infty} \frac{\omega' f_2(\omega') d\omega'}{\omega^2 - \omega'^2} [itex]

[itex]f_2(\omega) = -\frac{2 \omega}{\pi} \int_0^{\infty}

\frac{f_1(\omega') d\omega'}{\omega^2 - \omega'^2} [itex].

The Kramers-Kronig relations are related to the Hilbert transform, and are most often applied on the permittivity [itex]\epsilon(\omega)[itex] of materials. However, it must be noticed that in this case,

[itex] f(\omega) = \chi(\omega) = \epsilon(\omega)/\epsilon_0 - 1[itex],

where [itex]\chi(\omega)[itex] 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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