Relations of Kramers-Kronig

In mathematical and Physical, the Relations of Kramers-Kronig describe the relation which exists between the real part and the imaginary part of certain complex functions. The condition so that they apply to a function $f\; (\backslash \; Omega)$ is that this one must represent the Transformée of Fourier of a linear and causal physical process. If one writes

$f\; (\backslash \; Omega)\; =\; f\_1\; (\backslash \; Omega)\; +\; I\; f\_2\; (\backslash \; Omega)$,

with $f\_1$ and $f\_2$ of the real functions " sympathiques" , then the relations of Kramers-Kronig are

$$

f_1 (\ Omega) = \ frac {2} {\ pi} \ int_0^ {\ infty} \ frac {\ Omega f_2 (\ Omega)}{\ Omega^2 - \ omega^2} D \ Omega

$f\_2\; (\backslash \; Omega)\; =\; -\; \backslash \; frac\; \{2\; \backslash \; Omega\}\; \{\backslash \; pi\}\; \backslash \; int\_0^\; \{\backslash \; infty\}$

\ frac {f_1 (\ Omega)}{\ Omega^2 - \ omega^2} D \ Omega .

The relations of Kramers-Kronig are related to the Transformée of Hilbert, and are generally applied to the Permittivité $\backslash \; epsilon\; (\backslash \; Omega)$ of materials. However, in this case, it should be noted that

$F\; (\backslash \; Omega)\; =\; \backslash \; chi\; (\backslash \; Omega)\; =\; \backslash \; epsilon\; (\backslash \; Omega)/\backslash \; epsilon\_0\; -\; 1$,

with $\backslash \; chi\; (\backslash \; Omega)$ the electric Susceptibility of material. Susceptibility can be interpreted like the transform of Fourier of the temporal response of material to an infinitely short excitation, i.e. its Impulse response.

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