Kramer-Kronig Relations

Below there is a proof of the Kramer-Kronig relations. I remember, sitting in the middle of the small lecture room in U.C.C. when Dr. Padraig MhicCarthaigh introduced us this very useful mathematical equivalence to switch between the real and imaginary part of an analytic complex functions. They were typically used in response formulations and, in the context of the course (a physics course – advanced electromagnetism), it was used for flipping between the real and complex parts of the dielectric permittivity. I wanted to put together a clear, expandable proof of the Kramer-Kronig relation, cobbled together from a number of textbook examples.