How to Prove \(\lim_{y\rightarrow0}\frac{y}{x^2+y^2}=\pi\delta(x)\)?

[daudaudaudau]

What is the way to show that

$$\lim_{y\rightarrow0}\frac{y}{x^2+y^2}=\pi\delta(x)$$
?

 

[mathman]

Integrate y/(x² + y²) with respect to x over an interval containing 0. You will have a function of y. Let y ->0, and see if you get π. Further integrate over an arbitrary interval not including 0, then the limit should be 0.

 

[daudaudaudau]

But isn't that a bit hand-waving? What if you have something a little harder, like showing that
$$\frac{1}{x+i\eta}=P\frac{1}{x}-i\pi\delta(x)$$
? Don't you need a more systematic way of doing it?

 

[arildno]

No, it is no hand-waving involved.

It is precisely the limit behaviour mathman points to that you need to prove is present, in order to legitimize the introduction of the Dirac Delta-formalism.