Deriving Kramers Kronig relation and Hilbert transform pair

Kramers-Kronig and Hilbert transform relations are fundamental mathematical relations that establish a fundamental connection between the real and imaginary parts of a complex response function, such as the refractive index or dielectric constant of a material as a function of frequency. We explain what is a linear, time invariant, causal system, and why the generalized response function is analytic over the upper or lower half of the complex frequency plane. With the Cauchy integral principle, we can derive the Hilbert transform relation, connecting the real and imaginary part of the response function. Following this, we can derive the well known Kramers Kronig relation. We end with showing a few concrete examples of Hilbert transform pairs.