Can One Prove the Lorentz Distribution Identity?

[Niles]

Hi guys

Can one prove the identity

<br /> <br /> \frac{\epsilon}{x^2 + \epsilon^2} \underset{\epsilon\to 0^+}{\to} \pi \delta(x)<br /> <br />

or is it just intuitively clear (by looking at a graph)?

 

[Tinyboss]

First show that \int_{-\infty}^\infty\frac{\epsilon}{x^2+\epsilon^2}dx=\pi for any \epsilon&gt;0. Then show that for any fixed x\ne0, \lim_{\epsilon\to0^+}\frac{\epsilon}{x^2+\epsilon^2}=0.

 

[Niles]

Ahh, I see. Is this called the Sokhatsky–Weierstrass theorem? http://en.wikipedia.org/wiki/Sokhatsky–Weierstrass_theorem

 

[Mute]

Alternatively, consider

\lim_{\epsilon \rightarrow 0^+} \int_{-\infty}^{\infty} dx~\frac{\epsilon}{x^2 + \epsilon^2} f(x)
and consider the change of variables y = x/\epsilon, and show that the result is \pi f(0). It is in this sense that

\frac{\epsilon}{x^2 + \epsilon^2} \underset{\epsilon\to 0^+}{\to} \pi \delta(x).