I am interesting in learn easy facts about new issues of complex analysis. I've read in Wikipedia the statement of Kramers–Kronig relations.
Question. Please can you provide us a simple example for mathematicians of such theorem? I am asking thus about a $\chi_1(\omega)$ being the real part, and $\chi_2(\omega)$ the imaginary part respectively, of a complex function $\chi(\omega)$ satyisfying the hypothesis of the theorem, and how one gets from such example one of those integrals. Since I can read the identities from Wikipedia, only are required the calculations for a simple example inspired in a mathematical function. Thanks in advance.
In the article of spanish Wikipedia there are different expressions for such identities (you can use these or previous).
In the english version of Wikipedia are referenced the genuine works due to Kramers and Kroning.
Kramers Kronig relations are used in general to find the realtion between the real and imaginary parts of any complex numbers, e.g. if you know the real part then you can derive the imaginary part and vice versa. but there are some constraints to in these relations to be appicable for any complex function, so not all of the complex functions are KK transformable. you can take a look at this link, i think it is helpfull http://lampx.tugraz.at/~hadley/ss2/linearresponse/causality.php
also this http://www.iam.kit.edu/wet/plainhtml/Download/Derivation_Kramers-Kronig.pdf