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 <math>f(\omega)<math> 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

<math>f(\omega) = f_1(\omega) + i f_2(\omega)<math>,

where <math>f_1<math> and <math>f_2<math> are real-valued "well-behaving" functions, then the Kramers-Kronig relations are

<math>

f_1(\omega) = \frac{2}{\pi} \int_0^{\infty} \frac{\omega' f_2(\omega') d\omega'}{\omega^2 - \omega'^2} <math>

<math>f_2(\omega) = -\frac{2 \omega}{\pi} \int_0^{\infty}

\frac{f_1(\omega') d\omega'}{\omega^2 - \omega'^2} <math>.

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

<math> f(\omega) = \chi(\omega) = \epsilon(\omega)/\epsilon_0 - 1<math>,

where <math>\chi(\omega)<math> 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.