Dispersion relations question

1. Nov 20, 2004

Ed Quanta

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. Nov 20, 2004

Tide

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 $R = R(\omega)$ and your "iw" term is really a product of $\omega$ with a damping rate $\nu$ or somesuch. If that is the case then R will have complex zeros.