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The question is physics related, but my issue stems from the math so I figured this was an appropriate place. I'm looking at the (electric) susceptibility $\chi$, in both the time $t$ and frequency $\omega$ domains. From physical reasoning (causality) one knows that in the time domain $\chi(t) = 0$ for all times $t < 0$, and also that it is a real function, $\chi(t) = \chi(t)^*$.

Okay, so lets go into the frequency domain! We define $\chi(\omega) = \int_{-\infty}^{\infty}{\chi(t)e^{-i\omega t}dt}$

Now, my textbook says that because we have that because the time domain susceptiblity is zero valued for all $t<0$, we must have a complex valued $\chi(\omega)$. I feel like this should be a trivial consequence of the above definitions (and the fourier transform), but I just don't see it. Can someone help me on the way?

user3183724
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  • Note: $\chi(t)=0$ for all $t$ gives a real-valued result. – Joel Jan 20 '16 at 15:26
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    $FT\chi$ is complex valued, its complex argument is not constant : because if it was then $e^{i\theta} FT\chi$ would be real for all $\omega$ so that $e^{i\theta}\chi(t)$ would be even : $(e^{i\theta} \chi(-t))^* = e^{i \theta} \chi(t)$ – reuns Jan 20 '16 at 15:38
  • @Joel You are completely right, I should say for non-trivial time domain susceptibility, my mistake. – user3183724 Jan 20 '16 at 15:39
  • Hm, I've found a possible solution myself, but it's not super complicated so perhaps it doesn't really warrant an answer. Should I close the question instead? I'm not entirely sure what the policy is. – user3183724 Jan 20 '16 at 15:40
  • the fourier (or inverse) transform of a real valued function is even : $F(-\omega) = F(\omega)^*$ – reuns Jan 20 '16 at 15:42
  • @user1952009 Yes, that is true. But if $F(\omega)$ was real in what you write, that would still be true, right? – user3183724 Jan 20 '16 at 15:46
  • I used the converse : if your fourier transform had a constant complex argument, then your $\chi(t)$ function would be "even" in the sense I wrote. that's all we can say about that fourier transform for now and that's what your textbook meant. – reuns Jan 20 '16 at 15:50
  • @user1952009 Right, so you mean that that is in conflict with causality. That makes sense. – user3183724 Jan 20 '16 at 15:59

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