Talk:Kramers–Kronig relations

message: Since the Kramers-Kroning dispersion relations
relate to so many topics in signal theory and physics,
a more general discussion might be appropriate.
Why not start with a general causal transfer function
and use this to show the relations.
Specific examples for such a transfer function \chi
such as filter, susceptibilities, etc. could be listed at the end of the article.
ben
Personally, I would prefer stating the relations as
\chi(\omega)=\frac{1}{\pi} P \int_{-\infty}^{\infty} d\omega' \frac{\chi'(\omega')}{\omega-\omega'}
ben
May be
{\rm Im}[\chi(\omega)]=\frac{1}{\pi}~ {-}\!\!\!\!\!\!\int_{0}^{\infty} {\rm d}\Omega \frac{{\rm Re}[\chi(\Omega)]}{\omega-\Omega}
would look better?
Is the current version equivalent of this?
--dima 23:28, 8 August 2006 (UTC)
thread_title: Untitled

message: Why did somebody remove the reference to the important article by Toll? It didn't hurt, did it, to
have a pointer to causality and Kramers-Kronig? (It took Kronig to 1942 before he saw the connection clearly). --P.wormer 10:47, 23 February 2007 (UTC)
thread_title:  Toll 

message: This article was automatically assessed because at least one WikiProject had rated the article as start, and the rating on other projects was brought up to start class. BetacommandBot 09:57, 10 November 2007 (UTC)
thread_title:  WikiProject class rating

message: It seems there is an error with the sign. As a simple example the Fourier transform of the Heaviside function can be used:
\mathrm{H}(t)\to\frac{1}{2}\delta(f)-\frac{\imath}{2\pi f}
Then finding the imaginary part from the real part leads with the expressions proposed to
\frac{-2f}{\pi}\int_0^{+\infty}\frac{\delta(f)}{2}\times\frac{1}{\nu^2-f^2}\mathrm{d}\nu = \frac{-f}{2\pi}\int_{-\infty}^{+\infty}\frac{\delta(f)}{\nu^2-f^2}\mathrm{d}\nu = \frac{-f}{2\pi}\times\frac{1}{-f^2} = \frac{1}{2\pi f}
which is actually the opposite of what it is expected.
--193.54.84.254 (talk) 15:15, 21 November 2008 (UTC)
thread_title:  Problem of sign 

message: 'The segment at infinity vanishes since we assume χ(ω) vanishes as we take |\omega| \rightarrow \infty' --- I'm not sure if it's true in general...  —Preceding unsigned comment added by 93.105.182.2 (talk) 11:26, 25 December 2008 (UTC)
thread_title:  Over-simplification? 

message: The canonical URL for this article has three unicode characters in between Kramers and Kronig. http://en.wikipedia.org/wiki/Kramers%E2%80%93Kronig_relation
Any reason for this? Shouldn't the canonical URL be simply http://en.wikipedia.org/wiki/Kramers-Kronig_relation (i.e. an ASCII dash/minus '-' 0x2D in hex) Thanks! Woz2 (talk) 11:45, 14 May 2009 (UTC)
replies: text: I do not know anything about canonical URL, but there should be an en dash between Kramers and Kronig, per WP:DASH. Looking here that is indeed E2 80 93 in UTF-8 hex, so it seems to be correct. -- Crowsnest (talk) 14:43, 14 May 2009 (UTC)
replies: text:  Thanks! (BTW, by "canonical URL" I meant "non-redirected URL": i.e. the one and only one that doesn't have a "(Redirected from..." notice on it. By counter example:
Kramers-Kronig_relation
which contains:
[]
is a non-canonical or redirect URL) Woz2 (talk) 16:02, 14 May 2009 (UTC)
thread_title:  Canonical URL? 

message: The section on the time-domain proof states:
"This proof covers slightly different ground from the one above in that it connects the real and imaginary frequency domain parts of any function that is causal in the time domain, and bypasses the condition about the function being analytic in the upper half plane of the frequency domain."
I believe that any function that is causal in the time domain is analytic in the upper half plane in the frequency domain. Certainly the proofs in the time- and frequency-domains have to be equivalent. Can anyone confirm or refute this and does anyone know a good reference for it?
--DJIndica (talk) 16:13, 4 November 2010 (UTC)
replies: text: Yes. I'll fix it. Woz2 (talk) 01:24, 9 April 2014 (UTC)
thread_title:  Time-domain proof 

message: \chi(t) is used for the response function, \chi(\omega) for its Fourier transform. To me this seems to call for two different function names or a hat,  or is it customary to specify a function by its independent variable alone? But I don't know what choice to make for a suitable letter that fits with conventional choices. 84.227.225.36 (talk) 10:42, 7 April 2014 (UTC)
replies: text: I've seen proofs that use lower case for time domain and upper case for the equivalent frequency domain. Woz2 (talk) 23:36, 16 April 2014 (UTC)
replies: text:  The notation described is very common. Mathematicians wouldn't use it, but in many settings it is unambiguous enough to work. 178.39.122.125 (talk) 15:33, 8 February 2017 (UTC)
thread_title: Symbol for Fourier transform

message: From the article:
:Fortunately, in most systems, the positive frequency-response determines the negative-frequency response because χ(ω) is the Fourier transform of a real quantity χ(t),...
What would be typical examples, situations, or a characterization of when real-ness is violated? I thought χ(t) and the signals it operates on (gets convolved with) were ALWAYS real. 178.39.122.125 (talk) 15:31, 8 February 2017 (UTC)
thread_title:  What class of systems? 

message: added a section based on this paper by Horsley et al.: https://doi.org/10.1038%2Fnphoton.2015.106 . I am deleting it. I don't really have anything bad to say about the text, nor about the Horsley paper, which I think is a lovely paper. My complaint is that discussing it is undue emphasis. There are literally tens of thousands of papers in the literature that use or elaborate on Kramers-Kronig, there is an entire book written about Kramers-Kronig. Describing this one particular paper, one out of those tens of thousands, feels like very disproportionate emphasis, and sorta misleads readers who are trying to get a balanced introduction to Kramers-Kronig in general. I feel like this article could be 10 times longer than it is now, while only covering topics much more important to understanding Kramers-Kronig than the Horsley paper is, e.g. aspects of Kramers-Kronig that are widely-used in theory and practice for decades. --Steve (talk) 01:03, 22 June 2018 (UTC)
thread_title:  Deleting "Spatial Kramers-Kronig" section