# Hilbert transform on the Bethe Lattice

In order to compute the local Green's function on the Bethe lattice, it is necessary to perform a Hilbert transform

$$G_{loc}(\omega) = \int \frac{DOS(\varepsilon)}{\omega-\varepsilon} \text{d}\varepsilon$$

Where, for $$|\varepsilon| < 2t$$ (0 otherwise)

$$DOS(\varepsilon) = \frac{\sqrt{4t^2 - \varepsilon^2}}{2\pi t^2}$$

Where $$t$$ is the hopping amplitude.

Often, in textbooks or in papers, it is said that the result of the integral takes a simple form (even if in some definitions, the sgn function is omitted) :

$$G_{loc}(\omega) = \frac{\omega - sgn(\Im \omega) \sqrt{\omega^2 - 4t^2}}{2t^2}$$

I've been trying to find the derivation to get to this formula for quite some time now and never found a way to get there (and never found it online). Even simply plugging the equation into Mathematica doesn't work. I've seen this question asked on other forums but never get an answer I hope here it can get one.

Does anyone know the steps to do this integral ?