Integral of a delta function from -infinity to 0 or 0 to +infinity

[maximummman]

i think i have a missing point in the delta function

if
$$\delta_{}n (x) = (n/\Pi) . (1/(1+n^{}2 x^{}2 ))$$

so how can we show that
$$\int\delta_{}n (x) d(x) = 1$$

 

[CompuChip]

Assuming you tried to say
$$\delta_n(x) = \frac{n}{\pi} \frac{1}{1 + n^2 x^2}$$
you can easily work out that
$$\int_{-a}^{a} \delta_n(x) \, \mathrm{d}x = \frac{2}{\pi} \operatorname{arctan}(a n)$$.
This is no coincidence of course, it is the reason the ugly factor of 1/pi was added in the first place.
Of course, for $a \to \infty$ this converges to 1. So if we define
$$\int_{-\infty}^\infty \delta_n(x) \, \mathrm dx = \lim_{a \to \infty} \int_{-a}^{a} \delta_n(x) \, \mathrm{d}x = \frac{2}{\pi}$$
it follows.

One has to be careful in applying limits on integrals though, in particular
$$0 = \int_{-\infty}^{\infty} \lim_{n \to \infty} \delta_n(x) \, \mathrm dx \neq \lim_{n \to \infty}\left( \int_{-\infty}^\infty \delta_n(x) \, \mathrm dx \right) = 1$$