Delta function of a function with multiple zeros

[maverick280857]

Hi everyone,

I was wondering how to deal with delta functions of functions that have double zeros.

For instance, how does one compute an integral of the form

$$\int_{-\infty}^{\infty}dx g(x)\delta(x^2)$$

where g(x) is a well behaved continuous everywhere function?

In general how does one find

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

where f(x) has a finite number of multiple zeros along with some simple zeros. I know that

$$\delta(f(x)) = \sum_{i=1}^{N}\frac{\delta(x-x_{i})}{|f'(x_{i})|}$$

where $x_{i}$'s (for i = 1 to N) are simple zeros of $f(x)$ and it is known that $f(x)$ has no zeros of multiplicitiy > 1.

but this is of course not valid here. Using this, however I could write

$$\delta(x^2-a^2) = \frac{1}{2|a|}\left(\delta(x-a) + \delta(x+a)\right)$$

But the limit of this as $a \rightarrow 0$ tends to infinity.

Any ideas?

Thanks in advance.
Cheers.

 

[haushofer]

Maybe this link helps you:

http://ocw.mit.edu/NR/rdonlyres/Physics/8-07Fall-2005/40686CBE-369E-4373-95AD-BAEA1F2B8A37/0/deltafun.pdf

 

[maverick280857]

Thank you haushofer.