# Dirac delta function mathematical expression proof

In a discussion of the second order transitions in graphene this mathematical expression is used. $$\left|\frac{1}{\varepsilon + i \Gamma/2}\right|^2 = \frac{2\pi}{\Gamma}\delta(\epsilon)$$ And I'm kind of confused right now. Can someone prove this equation?

• Which discussion? – Emilio Pisanty Apr 17 at 8:26
• It's about the mechanism of the 2D' peaks in graphene Raman spectrum. There the author split the second order Fermi golden rule with this equation into 4 parts. – Levon Apr 17 at 8:38
• No, that wasn't the question. You should always provide a full bibliographic reference to the documents you're talking about. – Emilio Pisanty Apr 17 at 9:32

OP's formula is derived from the Poisson kernel representation $$\delta(\varepsilon)~=~\lim_{\Gamma \to 0^+}\frac{1}{\pi}\frac{\Gamma/2}{|\varepsilon+i\Gamma/2|^2}$$ of the Dirac delta distribution.