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.