# Proving the scaling property of the Delta function

caesius

## Homework Statement

Prove that $$\delta(at)=\frac{1}{abs(a)}\delta(t)$$

Hint: Show that $$\int\phi(t)\delta(at)dt=\frac{1}{abs(a)}\phi(0)$$

(the limits of integration are from -inf to +inf btw, I couldn't find how to put them in..)

## The Attempt at a Solution

Ok. I understand that the integral is only defined for at = 0, i.e., t = 0. But if I follow this logical then I evaluate the integral to be phi(0), not phi(0)/a. Where does the a come from?

And I'm not entirely sure how proving that relationship helps with the proof of the scaling function...

Homework Helper
That trick only works when the argument to the delta is the bare integration variable.
So try substituting u = a t, so you get something like
$$\int_{-\infty}^{\infty} f(u) \delta(u) \, du = f(0)$$
(and click the formula to see how I did the boundaries)

caesius
Ok I've substituted x = at, now I get

$$\int_{-\infty}^{\infty}\phi(\frac{x}{a})\delta(x)d\frac{x}{a} = \frac{1}{a}\int_{-\infty}^{\infty}\phi(\frac{x}{a})\delta(x)dx = \frac{1}{a}\phi(\frac{0}{a}) = \frac{1}{a}\phi(0)$$

I don't quite see how I'm allowed to take the 1/a out of the d/dx operater and put it at the front though. And why should it be the abs(a) not a?

And I still can't see how this helps prove anything.

Cheers, Benjamin

$$\int \tfrac12 x^2 \, dx = \tfrac12 \int x^2 \, dx$$