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Dispersion relations question

  1. Nov 20, 2004 #1
    So here are my questions

    If z(w)= R + iw/c, then 1/z = 1/(R + iw/c)

    Where does 1/z have singularities? I mean, there doesn't appear to be a point where R= -iw/c since R is real and the other term is imaginary.

    And how do I show the Real and Imaginary parts of 1/z are related by dispersion relations? And do I have to close the contour in the upper or lower half plane for this derivation.

    It seems to me that what I am looking for is a derivation of the Hilbert transformations, but get at me if you have any suggestions as to what I should do.
  2. jcsd
  3. Nov 20, 2004 #2


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    Since the title of your post includes "dispersion relations" it would appear to me that you are misstating the problem. Generally I would think that [itex]R = R(\omega)[/itex] and your "iw" term is really a product of [itex]\omega[/itex] with a damping rate [itex]\nu[/itex] or somesuch. If that is the case then R will have complex zeros.
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