Kramers-Kronig relations via Laplace formalism and $L^1$ integrability

TL;DR

This paper provides a rigorous, simplified proof of the Kramers-Kronig relations using Laplace formalism, requiring only $L^1$ integrability of the transfer function, making the mathematical understanding more accessible to physicists.

Contribution

It offers a new, simpler proof of the Kramers-Kronig relations based on Laplace formalism, requiring $L^1$ integrability instead of the traditional $L^2$ condition.

Findings

The proof is rigorous and relies on $L^1$ integrability.

Simplifies the mathematical understanding of Kramers-Kronig relations.

Makes the relations more accessible to physicists.

Abstract

Kramers-Kronig relations link the real and imaginary part of the Fourier transform of a well-behaved causal transfer function describing a linear, time-invariant system. From the physical point of view, according to the Kramers-Kronig relations, absorption and dispersion become two sides of the same coin. Due to the simplicity of the assumptions underlying them, the relations are a cornerstone of physics. The rigorous mathematical proof was carried out by Titchmarsh in 1937 and just requires the transfer function to be square-integrable ($L^2$), or equivalently that the impulse response of the system at hand has a finite energy. Titchmarsh's proof is definitely not easy, thus leading to crucial steps that are often overlooked by instructors and, occasionally, prompting some authors to attempt shaky shortcuts. Here we share a rigorous mathematical proof that relies on the Laplace formalism and requires a slightly stronger assumption on the transfer function, namely its being Lebesgue-integrable ($L^1$). While the result is not as general as Titchmarsh's proof, its enhanced simplicity makes a deeper knowledge of the mathematical aspects of the Kramers-Kronig relations more accessible to the audience of physicists.