Summation rules for the antenna input impedance

The Kramers-Kronig relations are probably the most well known dispersion relations [1]. These relations relate the real and imaginary parts of the constitutive relations. Summation rules are closely related to dispersion relations, and they ar important for consistency checks as well as deriving various physical bounds. In [2, 3, 4, 5], a summation rule that relates the absorption and scattering cross sections to the polarizability dyadics is analyzed, and e.g., used to derive physical bounds for antennas. Similar relations are of great importance in quantum mechanics [1], in circuit and system theory with early contributions by Fano [6], and planar radar absorbers in [7].