What is the integral of a square of Dirac delta function?

[Just some guy]

Homework Statement

Hi there, I'm stuck at a problem where I have (sorry i don't know how to use mathtype so I'll try my best at making this clear) the integral of a dirac delta function squared:

int[delta(x*-x)^2] between minus infinity and infinity (x*=constant)

I know that the function equals x*(int[delta(x)^2]) but I'm at a bit of a loss as to why :/

The Attempt at a Solution

I'm pretty sure the function int[delta(x-x*)f(x)dx]=f(x*) has something to do with it but I can't quite figure it out.

If f(x)=delta(x-x*) then f(x*)=delta(x*-x*) :? can f(x) even be another delta function?

Thanks for any replies,
JSG

 

[mda]

I am not a mathematician, but as far as I know int[delta(x)^2] is infinite, hence your integral should also be infinite.

One way to deal with these sort of problems is to replace the delta with a finite approximant and take the limit as the function width goes to zero.

 

[pam]

$$\delta[f(x-x*)]=\delta(x-x*)/|df/dx|_{x=x*}$$.
This integral diverges. Perhaps the problem was for $$\delta(x^2-x*^2)$$.

 

[Mute]

$$\delta[f(x-x*)]=\delta(x-x*)/|df/dx|_{x=x*}$$.
This integral diverges. Perhaps the problem was for $$\delta(x^2-x*^2)$$.

The question asked is actually for the square of the delta function, not its argument.

The problem with the square of a delta function is that by the property

$$\int_{-\infty}^{\infty} dx f(x) \delta(x - x_0) = f(x_0)$$

one would expect the answer to be the delta function at x_0 if one just blindly substitutes the delta function for f(x) there, but since the delta function is not really defined outside of an integral, that won't work. So, slightly less naively than that I would say there is no well defined answer and that integral just doesn't make sense. I believe I once found a pdf somewhere on the internet that stated that.

However, I once asked my supervisor about this, and he gave some sort of shifty physicist explanation of it, which I think resulted a result like

$$\int_{-L}^{L} dx \delta^2(x) f(x) = 2 \pi L \int_{-L}^{L} dx \delta(x) f(x)$$

where I think he used the Fourier integral representation for one of the delta functions in the first integral. Now, I probably didn't recall this correctly and the RHS is probably wrong, but that was the idea, or something. I know I wasn't really convinced by it, and I think it had something to do with him using a Fourier representation of the delta function using finite limits of +/- L, which I didn't like, so presummably L would be tending to infinity in the above expression. But again, I can't remember what my prof did, and it came off as shifty physics logic, so this may not help you much.