In linear response theory, the generalized susceptibility $\chi(\omega)$ is defined as $$\chi(\omega)=\int\limits_{0}^{\infty}\phi(t) e^{i\omega t} dt, ~~t\geq 0\tag{1}$$ where $\phi(t)$ is the response function$^1$. If it is assumed that $\chi(\omega)$ exists for all real, non-negative $\omega$, then its integral representation as given in (1) suggests that $\chi(\omega)$ also exists when ${\rm Im}\omega\geq 0$ i.e., in the complex upper half plane of $\omega$. This is because an additional damping factor increases the convergence of the integral (1). From (1), how can one argue that $\chi(\omega)$ is also analytic in the complex upper half plane of $\omega$? It's crucial in deriving the so-called Kramer's-Kronig relations in physics.

$^1$ On physical grounds, $\phi(t)$ is smooth (which @AFT enquired in his comment) and is also bounded as $t\to\infty$.

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    $\begingroup$ What is $\phi(t)$? What are its analytic properties? Is it smooth? $\endgroup$ – AccidentalFourierTransform Aug 2 '18 at 19:06
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    $\begingroup$ In addition to providing sufficient information about $\phi$ to answer the question, could you also explain how this is a better fit for physics than for Mathematics? $\endgroup$ – ACuriousMind Aug 2 '18 at 19:07
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    $\begingroup$ @ACuriousMind, this is a mathematical question, but it arises most naturally in the study of dispersion/absorption theory and response functions. So it would be useful to have good answers here on physics.se. $\endgroup$ – Ján Lalinský Aug 2 '18 at 20:53
  • $\begingroup$ Doesn't the analyticity property follow from causality, I.e. The fact that $\phi(t)$ is the response function of a physical system? $\endgroup$ – user197851 Aug 4 '18 at 7:45

I believe that this follows from causality, namely that $\phi(t)=0$ for $t<0$. Physically, the response of the system cannot precede the perturbation that causes it.

In your integral defining $\chi(\omega)$, the lower limit of integration may therefore be extended to $-\infty$, making $\phi(t)$ and $\chi(\omega)$ a Fourier transform pair. Then the Titchmarsh theorem applies: $\chi(\omega)$ is analytic in the upper half complex plane of $\omega$.

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  • $\begingroup$ I think this is correct. A useful discussion is in D. Forster, "Hydrodynamic Fluctuations, Broken Symmetry, And Correlation Functions". It is a very good book. $\endgroup$ – Fitzgerald Creen Jul 4 '19 at 15:57

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