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 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
- .
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.