Where Do Singularities and Dispersion Relations Arise in Complex Functions?

[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.

 

[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.

 

[M98Ranger]

Hi there,

The function z(w) given in the question is a complex function, where R is the real part and iw/c is the imaginary part. In order to find the singularities of 1/z, we need to find the points where z(w) becomes infinite or undefined. In this case, we can see that z(w) becomes infinite when R = 0 and w = ±ic. Therefore, the singularities of 1/z are at w = ±ic.

To show the relation between the real and imaginary parts of 1/z, we can use the Cauchy-Riemann equations. These equations state that if a function is analytic (or differentiable) at a point, then its real and imaginary parts must satisfy certain conditions. In this case, we can show that the real and imaginary parts of 1/z satisfy the Cauchy-Riemann equations, which implies that they are related by dispersion relations.

In order to derive the Hilbert transformations, we need to consider a contour integral in the complex plane. This integral is usually closed in the upper or lower half plane, depending on the function being integrated. So, in this case, you can choose to close the contour in either the upper or lower half plane, depending on the specific problem you are trying to solve.

I hope this helps! Let me know if you have any further questions or need clarification on anything.