Show that $$\lim_{y\to\infty}\frac{1}{\pi}\frac{y}{y^2+x^2} = \delta(x)$$

I do not know where the $\pi$ arise.

  • $\begingroup$ I don't know either shouldn't that limit always equal to 0? $\endgroup$ – gamma Oct 30 '15 at 2:58
  • $\begingroup$ @frank000: The limit is not to be taken pointwise, but in the usual topology on the space of distributions. $\endgroup$ – Nate Eldredge Oct 30 '15 at 3:03
  • 2
    $\begingroup$ Hint: what do you get if you integrate the function on the right side over $\mathbb{R}$ (essentially pairing it, as a distribution, with the function 1)? What do you get if you leave out the $\pi$? $\endgroup$ – Nate Eldredge Oct 30 '15 at 3:05

The constant $\pi$ is to normalize the area under the curve to unity.

$$\int_{-\infty}^{\infty}{\frac{y}{y^2+x^2}}dx = \pi$$


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