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 f(ω) 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

f(ω) = f₁(ω) + if₂(ω),

where f₁ and f₂ 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 ε(ω) of materials. However, it must be noticed that in this case,

f(ω) = χ(ω) = ε(ω) / ε₀ - 1,

where χ(ω) 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.