The Kramers-Kronig relations connect the real part and imaginary part of the frequency response of a causal
linear system (Kronig 1926, Kramers 1927, Toll 1956). If 
extends to a suitably
decaying analytic function in the upper
half-plane of the complex frequency plane, then
 |  |  |
(1)
|
 |  |  |
(2)
|
where 
denotes the Cauchy principal value. Thus
and 
are Hilbert transforms of one another and cannot
be specified independently.
The analyticity condition follows from causality when 
is the Fourier transform
of a response that vanishes before the applied stimulus. In optical applications,
the relations connect dispersion and absorption.
