Kramers-Kronig relations with logarithmic kernel and application to the phase spectrum in the Drude model

Standard Kramers-Kronig relations are formulated on the premise that the response functions are well behaved asymptotically. In certain physical problems in which the functions are logarithmic, one may then need to reformulate these relations. This was recently pointed out very explicitly in an optical context by Nash, Bell, and Alexander [J. Mod. Opt. 42, 1837 (1995)]. Much earlier this issue was discussed more generally. We examine in some detail the mathematical problem by considering the phase spectrum in the Drude model. Comparison is made between the standard and the reformulated forms of Kramers-Kronig relations.