  The important consequence of the Kramers-Kronig relations (KKrs) is that
dissipative behavior in material media inevitably implies the existence of
dispersion, i.e., a frequency dependence in the constitutive equations.
Basically, the relations are the frequency-domain expression of causality and
correspond mathematically to pairs of Hilbert transforms. The relations have
many forms and can be obtained with diverse mathematical tools. Here, two
different demonstrations are given in the electromagnetic case, illustrating
the eclectic mathematical apparatus available for this purpose. Then, we apply
the acoustic (mechanical)-electromagnetic analogy to obtain the elastic
versions. Finally, we discuss the concepts of stability and passivity and
provide a novel algorithm to compute the relations numerically by using the
fast Fourier transform.
