What did Kramers and Kronig do and how did they do it?

Over time the account of how the Kramers–Kronig (dispersion) relations between the real and imaginary parts of response functions were derived in 1926 and 1927 has been transmogrified into anecdotes about what might have been done but was not. Although Kramers obtained both members of a pair of relations, Kronig obtained only one. Both authors appealed to specific models of an atomic gas rather than to the general arguments about linearity, causality and analyticity in modern model-independent derivations. Kramers merely speculated on whether the specific results he obtained might have a more general validity. Neither author showed that a signal cannot travel faster than cin any medium for which the dispersion relations are satisfied. Indeed, they did not mention, even obliquely, signal speeds and causality. Despite their magical aura, Kramers–Kronig relations are translations into somewhat cryptic frequency language of statements clearer in time language. Although advanced undergraduate students or beginning graduate students are likely to be taught about Kramers–Kronig relations in courses in electromagnetic theory, these relations usually are presented as mathematical theorems that, like Athena from the head of Zeus, emerged fully formed from the heads of their eponyms. These relations, however, were physically motivated, not the result of investigations of the properties of analytic functions. The purpose of this paper is to set the historical record straight and to make the Kramers– Kronig relations more palatable by showing that they are statements translated from a language that more clearly reveals their physical origins. Kramers–Kronig relations (or dispersion relations) have an aura of magic because they are not physically transparent although they do rest on physical foundations: linearity, causality and the inability of physical systems to respond to excitation at indefinitely high frequencies. Suppose that a physical quantity X at time t depends on another quantity Y at all other times t′ by way of the linear functional relationship